d3168488f0f9fde6d768595c141db50f35014ec80ea7ec0db7ba5e0bc0d51039, and this page is generated from exactly those bytes — the build refuses to run if they do not match the served MANIFEST. Every page below is its own anchor, so a Registrar record can link to the page it cites instead of to the top of a 300 KB file. ★ Built because a cold review lost six of fourteen tendon verdicts to unpinnable page markers (D1264 → A1593), and declined to guess rather than mis-cite — which was the right call and the house's cost to bear.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
# SGTOE Rev33.1_S369 suite — PAGE-PRESERVING TEXT EXTRACTION — PART 1 (pp 1–100) (house, S369 2026-09-26). Source PDF: SGTOE_Rev33.1_S369_suite.pdf, 15800235 B, 433 pp, # sha256 b64ddcd9e16a6dcdc7ba41c97c5e1b8d087e72e82201ba581bc28db946779c34 (the sealed release PDF; the served copy is byte-identical). Tool: pdftotext -layout, one call per part split on form feeds. # Every page begins with a line '===== PDF PAGE n / 433 =====' — cite that n as the PDF page. Layout text is lossy (math symbols, tables); where a quote matters, # the tex file:line in the release bundle is authoritative. This extraction is a READING AID pinned to the PDF above; it is not a second edition of the suite.
Leibniz Quantum Beats Newton
Paper 0: SuperGrokTOE Rev33.1 Framework Foundations
Tom O’Sieg
August 2026
Abstract
Paper 0 is the entry point for the J3 (Os ) programme. The suite revision is Rev33.1 (the
authoritative suite label, declared here, on the Executive State Map below, and in each component’s
title; point releases roll all three together). Revision stamps inside bodies mark the revision at
which a result entered the programme; titles and this sentence roll at every release, including point
releases (Rev31 → Rev31.1 → Rev31.2 were point releases of the sealed Rev31); references to
earlier revisions (“Rev12”, “Rev19”, “Rev25”, . . . ) in the bodies below are retained as change-log
stamps naming when a given result first entered the programme, not as the current revision label
(Rev16 added Appendix I and the binding claim-language convention; Rev17 the consolidated 7/3
finite-layer result; Rev18 the in-house dynamics layer; Rev19 the engine re-stamp (ms derived, me
Koide value); Rev20 the structural integration layer (Appendices J/K/L); Rev21 the order-five
clock and selector fences (Appendices M/N); Rev22 the honest-negatives fold; Rev23 the rest-
mass program record (Appendix O); Rev24 the S188–S207 reconciliation fold; Rev25 adds the
nuclear cluster tier (Paper 9: seventeen sharp binding energies deuterium through calcium against
pre-frozen bars plus one typed obstruction (28 Si; recount Rev27, re-adjudicated Rev28 — see the
Gate 1 note in Paper 9, Typed features and open items), with a closed-form composition ladder),
the fifth-direction theorem tier (Appendix P), the bounded golden-vacuum spectrum window
(Appendix Q), the closure of the rest-mass program as ratios-complete with the framework’s
one dimensionful input registered by theorem (Appendices O/X), and the occupation selector’s
promotion to derived-conditional with a live public falsifier (Appendix M)).
“Occupation” is bound here at first use, and the binding holds throughout
this suite. It means occupation of golden mass-ladder rungs — the coordinate locations
O = {0, 3, 11, 17} on a logarithmic mass scale — and not occupation of atomic orbitals,
nor of states by particles. The shared noun invites a false parity with atomic subshell
filling; that parity is claimed nowhere in this suite, and reading it in was the proximate
cause of a type error in an earlier internal plan, caught independently by two review lanes
[A1484, A1485].
The leading mixing-sector result of the present revision is that the full mixing sector is zero-fit inside
the specified correspondence map: no independently fitted PMNS or CKM angle, phase, or mixing
normalization remains inside that map, with two entries (θ13 and δCKM ) readout-conditioned
rather than independently fit. The broader suite still uses explicit external inputs. Given two
measured scales (mτ and vEW ), one hadronic anchor (ΛG2 ) — inventories reconciled under the
Appendix X input-count convention: one independent dimensionful input class after registered
identifications, two operational calibrations, three named anchors — the QED input α, the external
empirical Koide rule K = 2/3, the independent AX6 orbit selector for the Weinberg-angle branch,
Route B selection, normal ordering, and a Dirac-neutrino assumption — with the working vacuum
Jvac = diag(ϕ, 1, ϕ−1 ) now fixed (algebraically, up to permutation) by the conditional selector
1
theorem of Appendix E rather than carried as an external assumption — the algebraic structure
J3 (Os ) generates the operative PMNS/CKM mixing ledger and the current charged-lepton /
heavy-quark mass proxies. [P-017, P-019, P-031, P-032] The headline formulas are
√ √
3 7 π 3−2 2
tan θ12
PMNS
= 2, sin2 θ23 = , sin2 θ13 = sin4 = ,
ϕ 16 8 8
2π
δCP = − √ = −160.997◦ ,
5
with the CKM entries constructed from the Route B Peirce map. This is a zero-fit claim inside the
specified algebraic map, not a claim that the full paper suite has no measured anchors, empirical
selection rules, or independent postulates. [LIB2-233] The specified correspondence map is
itself a chosen (loaded) embedding, not shown to be forced or unique by the algebra
— the selectors listed above (Jvac , Route B, AX6, ordering, Dirac-ν) are the loading. [P-012] So
“zero-fit” is a statement about the map’s internal parameter economy once that map is granted, not
a first-principles prediction that the algebra alone selects these angles; the derivation of the map
itself from a physical readout functor is the program’s named head wall (Appendix R). The same
honesty applies one level up, to the gauge content itself: the charge/gauge provenance program
has reached a settled terminal — possible-not-necessary, i.e. constructible but not forced —
triangulated across five independent attacks and recorded as final [A1262, A1265]. [LIB2-342] The
gauge group is therefore carried as an input throughout this suite (Paper 1; Appendix F), not as a
derived consequence. [LIB2-342] (Rev25/26 note on the dimensionful anchors: the framework carries √
exactly one registered dimensionful input, the depth-cylinder Bargmann class, with vEW / σ
reclassified from bare input to a conditional loaded-depth registration, φ 105/8
(registered at β = 1
from the two-element menu β ∈ {1, 7} of Appendix N’s electroweak-permanence proposition — no
intrinsic selector fixes β = 1); see Appendices N/O/X.)
The algebraic core remains the cubic-invariant Yukawa structure
Y(Ψ, ΦH ) = Tr Jvac ◦ (Ψ × ΦH ) , (1)
with Jvac = diag(ϕ, 1, ϕ−1 ). This projects onto the middle Peirce idempotent and fixes the
ultraviolet tree-level boundary value yt (MPl ) = 1, while standard one-loop running carries that
value to the low-energy effective theory. The tree-level eigenvalue and the running coupling remain
distinct objects.
Rev29 also compresses the charged-lepton mass sector into a mixed-provenance chain (algebraic
anchor + structural ratio + Koide closure; see below — tiers per the Appendix O re-tier), not
a derivation of all three masses from J3 (Os ) alone. The authoritative charged-lepton ledger is
Appendix O (“The authoritative charged-lepton ledger”); every shorter form in this suite defers√to
it. The charged-lepton ordering from the Jordan automorphism exp(t∗ Thalf ), with t∗ = 2π/ 5,
reverses the resolvent ordering R1 , R2 , R3 7→ R3 , R2 , R1 and yields the assignment d = 1 → τ ,
d = 2 → µ, d = 3 → e. The muon-to-tau ratio (Structural tier, Appendix O — the short form
is a loaded correspondence, not a zero-parameter proof) is
8/3 √
2
mµ ϕ
= √ = 0.059684,
mτ 5 10
where p = 8/3 is an exact dimension ratio read as a topological
√ winding ratio (target-first
provenance; the derivation is not closed — Paper 7 §4) and 2/10 is fixed by the J2 (Os ) two-
generation subalgebra. With mτ = 1776.93(9) MeV this gives mtree
µ = 106.05 MeV; one-loop QED
matching gives mµ,phys = 105.72 MeV, and the Koide closure then gives
me = 0.5076 MeV,
within −0.66% of the PDG value. The honest provenance of the charged-lepton chain is therefore
mixed: mτ is the external anchor, mµ is Structural (Appendix O re-tier) via the resolvent
2
short form, and me is Koide-closed ⋆ ⋆ ⋆⋆ under the external empirical rule K = 2/3, not derived
from J3 (Os ) alone.
The suite remains explicit about what is not yet closed. The Higgs vacuum expectation value
is measured rather than derived. The weak-angle bridge requires threshold-sector completion. The
heavy-quark 7/3 lepton–quark bridge is a proved algebraic invariant (dim Im Os /rank J3 = 7/3, and
equivalently det Gram(Jvac , Jvac
#
)/rank = 7/3), but its promotion to a physical S-matrix residue is
open and demonstrably hard: a dedicated literature-and-construction scout (three independent
model reviews plus four in-house computations, S116) found no theorem mapping a Jordan/cubic
Gram determinant to an LSZ residue, OPE coefficient, or crossing ratio, and an explicit canonical
Peirce-trilinear LSZ calculation returns mass-residue ratio 1 and inclusive colour ratio 3, not
7/3. The obstruction has a sharp structural cause: the “7” is the dimension of Im Os , not a
trace of the split-signature metric (whose signed value on Im Os is −1, signature (3, 4)); the
channel coefficients (1, 1, 1, −1, −1, −1, −1) give positive Hilbert count 7 but signed trace −1, so
a residue channel-sum weighted by the physical metric does not return +7. The sub-percent
heavy-quark residuals additionally carry a weak-isospin T3 signature that is structural, not derived
(both the carrier search and a boundary-condition revision closed negative, A715–A717). The
7/3 residual is now sharply named: it is the response of a cotangent Yukawa source B ∈ J12 ∨
entering the (still unwritten) gauged E7(7) /J3 (Os ) fermion action as the source term Sedge − ⟨B, D⟩
— a structural source term, not a derived amplitude (Appendix I). The suite is therefore best
read as a sharply structured algebraic correspondence scheme whose PMNS/CKM mixing sector
carries no fitted mixing coefficients inside the loaded Route-B correspondence — after the Rev29
row-by-row re-tiering (Rev29 re-tiering, Paper 3 § “Row-by-row re-tiering”; kernel s959), two of
the eight observables stand at Derived-conditional or above (θ12 PMNS
, δCP ; the physical θ23 row
is Loaded-correspondence since A1566, its internal 7/16 identity theorem-grade), the CKM
first row is Loaded because its route was selected against kaon data, δCKM is Coincidence-class,
θ13 is structural, |Vcb | is Loaded-correspondence (physical attachment) / Structural formula
(A1566), and the CKM β consistency is unresolved (Paper 3); no row may be cited as zero-
parameter — and whose charged-lepton sector is structurally pinned with mixed provenance,
while bosonic extraction, AX6, and the formal heavy-quark amplitude proof remain the principal
tasks.
The post-Rev14 work (Sessions 106–108) is recorded in the new Appendix H: a structural “time
layer” on the certified E7(7) substrate in which the past↔future moment is realised as a genuine
symplectic element T ∈ E7(7) with T 2 = −1, the gauge-side moment dial and the E6 × U (1)
grading circle are shown to be one E7(7) -conjugate circle, and the clean-lepton mass residual on the
golden ladder is identified (empirically) with the Koide relation. Consistent with the Definition-A
posture, these are classical, finite statements on the substrate; no public observable moves.
Rev16 added Appendix I, a speculative probe dynamics that addresses the “no dynamics” gap
identified by external review: a classical geometric bounce (the Koecher–Vinberg cone metric places
the singular Det = 0 boundary at infinite distance from the vacuum), an overdetermined golden √
vacuum (pinned independently by the invariant selector det G = 7, whose integer encodes 5
and hence ϕ, and by a grading-coupled potential at unit coupling), and an E7(7) clock connection
whose holonomy is the time-reversal element T with T 2 = −1. It is fenced as a diagonal-slice probe
with det G = 7 and unit coupling as structural inputs and the mass scale as an Input; no public
observable moves. Rev17 extends Appendix I with two fenced additions: a consolidated record of
the 27-run 7/3 amplitude-edge probe (A843–A863), which exhausts the tested finite endpoint/self-
adjoint/source layer — pinning the edge operator, its readout, its uniqueness, and its physical
object (a cotangent Yukawa source B ∈ J12 ∨
) while excluding the minimizer/integrate-out classes,
leaving an explicitly non-finite residual (the generation assignment plus the full E7(7) /J3 (Os )
action); and a moment–annulus synthesis (T 2 = −I on the 56; a Fibonacci charge↔mass bridge;
a moment-graded dual collapse whose future/past curvature asymmetry is shown not to be the
cosmological constant). Both remain fenced and move no public observable. Rev18 folded the
in-house dynamics layer (S120–S124) into Appendix I: a derived dynamical vacuum (golden =
tachyon-free Minkowski critical point of dyonic ω = π/4 SO∗ (8)), the derived σ-odd-defect selection
of ϕ modulo the AX1 unit (Coleman–Weinberg shown σ-even and barred from golden at every
3
layer), and the partial stabilization status (the σ-even residual lift reduced to one named field-
dependent tensor Ciab = 12 ∂a ∂b m2i plus a no-scale dilaton). Selection is derived modulo AX1; full
stabilization is not yet closed; no public observable moves.
Notation (σ). Three unrelated objects share this letter across the suite and all three appear here.
√
Where an involution is meant it is σframe , Appendix I’s Jordan frame involution ρ 7→ 1/ρ. σ is
the string tension. A numeral before the letter — 2.7σ, 0.36σ — is a standard deviation. Two
further involutions are named apart elsewhere and never appear bare here: Appendix H’s throat
involution σC and Appendix F’s time-mirror σ = Ω; the split-octonion σoct does not appear in the
typeset suite.
Rev33.1 Executive State Map (one page — read this first)
Reading rule (Rev32.1; carried at every mass/mixing table): tier words are the
mathematical-status axis and physical attachment is priced separately; no row may
be cited as zero-parameter; cross-scheme σ are comparator distances, not tensions
(A1531/T2).
How to read revision labels: the suite revision is Rev33.1 (point releases roll this sentence
and every title; the family label is Rev33). “Rev12”/“Rev19”/“Rev25” etc. inside paper bodies
are historical change-log stamps (when a result entered), not the current status; current status is
controlled by this page, the claim tiers, and Appendix X. Titles roll at every release, including point
releases; body stamps do not.
New at Rev31 / Rev31.2 (the selector arc, S274–S280).
• The selector arc ran end-to-end (Appendix√E, Appendix S): the parent-action closure at complete-
conditional tier with the 10+1 census; the 6 clock identity proved via the golden bridge Hφ ; the
26 ⊕ 10 split grammar-forced at zero price; the rank-character variational representation theorem
upgraded derived-in-class; and the selection obstruction win-obstructed: the four-relation
principle Φdiag is adopted as a declared structural principle, not derived. [P-007]
√
• The two tens are proved inequivalent (Casimirs 1/4 vs 1/6) and the intertwiner numeral 1/ 6 is
derived as the selector’s own Casimir.
• Point releases Rev31.1/Rev31.2 (S280): the explicit-destination link rewrite (the S274 link audit),
suite hygiene, and the frozen Rev31.2 seal that the S281 cold round reviewed. No public observable
moved anywhere in the Rev31 family.
New at Rev32 (the hostile-review + verification fold, S281–S288).
• The A-1 hostile review ran 3/3 lanes against the sealed Rev31.2 and every confirmed defect is
repaired in this revision: the revision-label sentence you are reading (point-release semantics),
the Appendix E Route-B monotonicity proof replaced with the verified gap proof (conclusion
unchanged), stale tier language re-tiered, and the referee dictionary restated at current status.
• Comparators refreshed to PDG 2026 / NuFIT 6.1 with the cross-scheme adjudication: percent
offsets are primary and scheme-labelled; bare cross-scheme σ are retyped as naive comparator
4
distances that set no status (Appendix X, Papers 2/3/5). The d = 1 reading of the ∆m2 ratio
remains the one genuine falsified row, now printed with its dataset; the suite’s own d = 2 rung
(R2 ) is printed beside it as a below-floor edge target, not a hit (Paper 5, Rev32.2).
• The clause-27 carrier grammar is printed: three tens named (physical = ∼ selector at Casimir 1/6
̸= boundary ad Kdyn at 1/4), the adjoint wall and quadratic-bridge census (Appendix S), and the
dynamics-seed reconciliation beside the no-energy-order fence (Appendix X).
• Selection frontier: the C0 six-coefficient independence theorem is banked at countermodel grade
(scoped); F-SEL(c) closes negative (scoped) — Casimir-naturality, the lead refined-family
candidate, excludes Φdiag rather than deriving it, so Φdiag remains a priced adoption and the live
selection frontier is the G7 motivation (Appendix E).
• One independent dimensionful input class — the three printed counts are reconciled under the
input-count convention (Appendix X): one input class, two operational calibrations, three named
anchors; the noun after the number changed, no count disappeared.
New at Rev28 (the twenty-round fold, D1126–D1146).
• The boundary question is closed at its tier (Appendix S): the ladder is the affine universal cover,
the unpointed interface is obstructed, the affine support theorem recovers the finite projector
from the continuum, and AX-MG is proved a genuine independent boundary axiom in
both clauses (coefficient-one and degree-one) by exact countermodel. Nothing derivation-shaped
remains there. The 0.0331562% row is dead twice over with located cause; 0.2744490308% stands
as a legal miss.
• The geometry spine is separated and closed (Appendix T): “W ” was two operators (Wsig preserves
the KK grading, Sshell reverses it); there is no third cascade and no A4 completion; one shared
radius; the transition cocycle is derived at zero continuous parameters; the two signature
sheets are not glued at the F4 tier (Sylvester obstruction) but are two real structures in one
complexified Albert fibre.
• A four-dimensional Cartan carrier closes inside the Albert signature sector (so(5), an S 4 Cartan
geometry), with metric and orientation trace-derived and triality acting as the imaginary unit; a
MacDowell–Mansouri parent action for that sector follows as a candidate, explicitly not the full
particle/gauge/mass action.
• A reflection-positive state exists at finite regulator, together with a positive nonlinear simplicity
measure — so gravity-measure existence is closed at that tier, while the pure MM/EH weight is
proved not to be a positive coercive covariance. [P-014] Einstein universality is the remaining
named object.
• The observation functor is built (Appendix T, OMOS ): modular score Krel = Dφ (the log leg a
theorem, not an axiom), unital complete positivity fixing the normalization, an exact POVM, a
unique minimal dilation, and Pin/OS evenization — unique under seven premises with verified
teeth, and carrying no continuous modulus. What the microscopic theory has not done is
select it. OMOS is the record-quotient functor only; the mass/generation observation map Ogen is
a distinct, unbuilt object (Appendix T).
• Actuality splits. The record type is derived (history ⇐⇒ character ⇐⇒ boundary record are
one object); the realized value is proved to be sample data, not a parameter. Appendix L’s
firewall stays for the specific realized outcome.
5
• Freedom is retyped. The bits-plus-dimensions headline is withdrawn pending typed rebuild
(its 17-dimensional decomposition is internally inconsistent as typed) and replaced by a typed
ledger with no total — because the observation-functor faithfulness wall has no computed kernel.
What can be said: exactly one continuous modulus (ρ) survives anywhere in the end-state ledger.
• A laboratory notebook is added (Appendix Y): all ninety adversarial rounds from S228 to S253 —
dispatch, session, headline result, and the component that carries it — plus a reverse index from
printed object to certifying round, the casualty register, and a disclosure. The disclosure: before
Rev28 this suite cited no assessment above A1357, so the Rev26 and Rev27 folds had entered
the tex without their citations. Nothing printed was unsupported; much of it was unfindable.
Appendix Y is the audit trail, and four rounds (A1414, A1416–A1418) that had no ledger row
anywhere are named in it.
• Unchanged by all of it: KILL-NODE 2 fired, now off shell and for the full microscopic loading; χ
value-open; the engine unchanged; α and colour fenced; VCKM = I; no public observable moved.
Promoted at Rev25.
• Nuclear cluster tier → Paper 9: seventeen sharp binding energies deuterium→calcium at pre-
frozen bars (worst untyped 0.77%, 3 H; 0.37% is the worst of the A ≥ 5 non-silicon continuation)
plus one typed obstruction (28 Si at 0.334%: pre-answer bar 1% (D1060) PASSED; entered through
the frozen disjunctive refinement gate (D1061 3(b), pre-send sha 85c69f1f. . . ) on its obstruction
branch (A1357) — neither a sharp nor a miss at a pre-answer bar), the Hoyle pass, eleven correct
sign calls, two glueball bands; closed-form composition ladder Tn = (n − 2) + trφ (Rn ) with two
blind vault matches (φ2 at 26 ppm; T9 at 0.10%). Known soft flank stated in the Paper 9 posture
box: charge magnitudes open.
• The scale input, registered: exactly one dimensionful input by theorem (dim H 2 = 1, unconditional
√
census) — the depth-cylinder Bargmann class, σ/me = 870.84 (±1.51%); sealed null guard
(Appendices O/X). Appendix O closed ratios-complete.
• Occupation selector → Derived-Conditional: three-layer derivation (mod-2/5/7), Appendix M. [LIB2-
029]
• Fifth-direction theorem tier (Appendix P): 5D-necessity two-sided; tick = fifth momentum mod 4
(exact); flicker = NS spin structure.
• Anomaly audit closed exactly (T3-P; Paper 1/Paper 7/Appendix F) and the E7 -closure theorem
(Paper 1).
Added, deliberately bounded. Appendix Q: the golden vacuum’s classical spectrum closes at
machine zero — conditional tier throughout (κphys = structural candidate; one recorded blind loss
kept on the books); the live dynamics lane stays out until Rev26.
Promoted at Rev26 (the dynamics-frontier fold, S228–S240).
• The interacting vertex → Appendix R: the head wall (“no parent action”) attacked directly and re-
counted as a package — one merged loaded import ([σCD ] holonomy welded to its induced Cartan
metric η+ ), three continuous data {g3 , Csub , rC }, and one loaded non-equivariant placement kernel
Π5←6 — carrying the program’s first derived, parameter-free (discrete, derived) relative
vertex sign, A(4, 5; 1)/A(5, 4; 1) = −1 (structurally a trilinear antisymmetry; amputated, F4-
fenced, not an amplitude). Conditional/candidate tier.
6
• The σ-bridge scale registration → Appendix O: the electroweak hierarchy is paid at conditional/loaded-
√
depth tier, vEW / σ = φ105/8 = 553.30 (framed-depth operator 13 + 18 ) (registered at β = 1 from
the two-element menu β ∈ {1, 7} of Appendix N’s electroweak-permanence proposition — no
intrinsic selector fixes β = 1), the suite’s first foregrounded dimensionful result — a framed regis-
tration relative to the marked compact-positive carrier, not a frame-free prediction (the frame-free
√
route is closed negative, Appendix F) — with mτ / σ = 4 its exact dimensionless companion.
• Theorem-tier upgrades: the frame-free Higgs-line no-go now derived in-house (Appendix F,
E7(7) ); the Marked Carrier Theorem (Paper 4, selection level); the 7/3 finite-carrier theorem
(Appendix K).
Re-priced / added at Rev27 (the priced-selector fold, S242–S245).
• The discrete selector wall, closed as a priced ledger → Appendix S: the D1116–D1121 arc types
the missing parent-action selector as [jA ] ∈ RP2 (plus two magnitudes, a relative-phase/lift datum,
and the standing orientation bit), with six independence theorems certifying that no banked
principle selects it. One conditional deduction is new — the V-side quadrature z− = ±i z+ , forced
by the candidate shape action (itself unpromoted). Conditional/candidate tier; no observable
moves.
• The exact φ-ladder mass completion is REFUTED at the ratio tier (Appendix O): no single-χ
additive closure reproduces the occupancy-support ratios (0/24 bijections); the ordered φ-depth
skeleton and the few-percent shape survive, the exact completion does not. The support {3, 11, 17}
is independently corroborated as the adjacent-integer minimax (Appendix M).
• Four in-house over-claims caught and corrected (casualties #4–#7, Appendices S/X): the
independent-channel doorway (retired for the paired-source theorem); “the action votes for
the root line” (now common-lift-conditional); the blanket ≤ 2-parameter mass no-go (withdrawn
— a fitted period-70 harmonic passes; watch item banked); and the depth-exclusion cut (retired).
Fenced / input (unchanged). α (five-legged fence, Appendix N); the compact colour real form;
the gauge group at group level (two proven walls + uniqueness-of-completion, Appendix F); the
generation count (input, with a forced-conditional skeleton); the external Koide rule K = 2/3. [P-011]
Still open (named). The 7/3 S-matrix/LSZ bridge (Paper 7); the charge-magnitude sector (Paper 9,
§8 — signs and counts derived, magnitudes open); the two-circulation sector (the typed 28 Si residual);
the continuum/Yang–Mills functor; the θ13 readout forcing and the δCKM 5φ−5 derivation; the
√
σ-bridge is paid at conditional/loaded-depth tier (vEW / σ = φ105/8 (registered at β = 1 from the
two-element menu β ∈ {1, 7} of Appendix N’s electroweak-permanence proposition — no intrinsic
selector fixes β = 1), promoted above, Appendix O) — what stays open upstream is the origin of the
loaded boundary Bphys , and the loaded placement kernel Π5←6 of Appendix R.
Live kill conditions. {31, 39} (hash-stamped occupation falsifier, Appendix M); a second underivable
dimensionful ratio (falsifies the one-input theorem chain, Appendix X); DUNE-era ≥ 3σ exclusion of
δCP or the 7/16 octant branch (Paper 7); any scored miss at a pre-answer frozen bar, or any untyped
miss at any frozen gate, or a broken sign call in the Paper 9 protocol; a confirmed non-hadronic aµ
anomaly (associator null, Paper 7).
7
Claim-language convention (binding across the suite)
Every load-bearing statement in this suite carries exactly one of five provenance tiers, and the language
used for it is constrained by that tier. This convention is binding suite-wide: it governs the title
pages, the abstracts, the section prose, and the status stars. The full per-quantity provenance ledger
is Appendix X; the table below fixes the words each tier may and may not use, so that a claim’s
wording cannot outrun its evidence.
Tier Meaning Allowed words Banned words
Proved Forced by the Jordan/Freudenthal proved, theorem, —
algebra with no fit to particle data; a forced, derived
uniqueness/selection theorem closes (from the algebra),
it. closed
Certified Machine-verified computation on the computed, cer- proved (analytically),
certified E7(7) /J3 substrate, repro- tified, “Theo- closed, “derived from
ducible from the bundled kernels; rem/Proposition nothing”
a computational result (often condi- (certified computa-
tional on a named substrate/axiom), tion)”, reproducible
not an analytic uniqueness theorem.
Structural Value correct and algebraically moti- structural, alge- proved, theorem,
vated, but permitted rather than se- braically motivated, derived, closed,
lected; no uniqueness theorem closes permitted, target, “100/100”, “zero
it. correspondence, free parameters”,
conditional (under a emergence
named axiom)
Input Genuinely external: a measured di- input, external, mea- derived, predicted,
mensionful scale, an empirical selec- sured, assumed, em- emergent, forced,
tion rule, or a discrete datum the pirical, declared “from nothing”
algebra does not fix.
Historical Retained legacy record, superseded historical, legacy, su- current, holds, estab-
by later work; kept for provenance perseded, retained lished, the present
only. for record state
Enforcement rule (S112). A claim may use Proved-tier language only when a stated
theorem closes it with no external input. A conditional theorem must name its load-bearing
axiom inline (e.g. “conditional theorem under the channel-expectation readout axiom”); it is a
Structural claim with a derivation path, not a Proved one. Phrasings such as “100/100,”
“all green,” “zero free parameters” (for the suite as a whole), “closed,” and “emergence” are
retired except where a Proved-tier theorem licenses them. [LIB2-233, LIB2-234] The dimensionful
mass scale is Input throughout (the Honest Boundary; Appendix X), and no relabelling of
Jvac from input to output is asserted until the full-J3 dynamical lift succeeds (Appendix I).
8
Two-axis provenance (Rev27): mathematical status × physical attachment
A single provenance axis conflates two independent questions: (i) is the algebraic or computational
statement established? and (ii) is its identification with a physical observable derived? A quantity
can be theorem-grade on (i) while its physical attachment on (ii) remains loaded. From Rev27
every load-bearing claim therefore carries both a mathematical status — the five tiers above
(Proved / Certified / Structural / Input / Historical) — and an orthogonal physical
attachment: Derived-attachment (the map to the observable is proved), Conditional-readout
(attaches only under a named readout/axiom), Loaded-correspondence (true inside the chosen
loaded map, whose boundary is a declared input), Fitted (matched to data), or External-input
(measured/empirical). Physical attachment never inherits the mathematical tier: an exact theorem
about the algebra is not, by itself, a physical prediction until its attachment is stated. [LIB2-322]
Internal statement Mathematical status Physical attachment
det G = 7 exact algebraic theorem Loaded-correspondence: does not by
itself give sin2 θ23 = 7/16 (kinematic nor-
malization, Paper 3)
7/3 = Tr(1⊕3⊕ 3̄)/Tr(3) exact trace/index theorem Conditional-readout as an index; not
an LSZ residue — the metric-weighted-
residue route is closed-negative (s902,
App. I)
8/3 = dim Os /rank J3 exact dimension ratio Loaded-correspondence: not, alone, a
mass exponent
√
Thalf frequency 5/2 exact computation Conditional-readout: identifying the
orbit parameter with the measured Dirac
phase needs the readout map
z− = ±iz+ exact conditional extremum eq. Conditional-readout: conditional on
Sshape , an unpromoted candidate (App. S)
Crosswalk to the scorecard tier column (Rev32.9, S304.5). The scorecard and several paper
tables carry a single tier column with a different vocabulary. It is a compressed reading of the two
axes above, not a third system; each row’s own two-axis statement governs where they differ.
9
Scorecard tier Mathematical status Physical attachment
Axiom Input (a stated premise) not applicable
Theorem Proved or Certified Derived-attachment only where the
map to the observable is proved; other-
wise the row also names its attachment
Derived-conditional Proved or Certified Conditional-readout
Structural Structural stated per row (Conditional-readout
or Loaded-correspondence)
Loaded any a discrete choice made against data (a
route, an octant, an ordering): Fitted
in the discrete sense
Loaded-correspondence any Loaded-correspondence
Reproduced Structural or Certified Loaded-correspondence: the construc-
tion reproduces data it was built toward
Coincidence-class Structural (candidate) none established: a numeral agreement
without a derivation
REFUTED Historical withdrawn
This axis is what makes the “zero-fit” statements precise: they are Loaded-correspondence claims
(fit-free interior, declared-input boundary; Appendix X), not Derived-attachment ones. The
per-quantity mathematical tier remains as tagged throughout the suite; the physical attachment is
stated in the Appendix X evidence column and, where load-bearing, inline.
Definition-A Boundary Ledger (Rev25)
So that no claim’s wording can outrun its evidence, the load-bearing claims are ledgered here by tier
before the reader enters the suite. Every use of “derived,” “selected,” “barred,” or “closed” in the
body points to a row below; a row marked open or conditional is never written as closed. The
dynamics rows (I.1–I.6) refer to Appendix I; their reproduction artifacts ship in machinery/ and are
checked by verify_rev19.sh.
10
Claim Tier Status / inputs / repro artifact
Derived (no external selector)
I.1 Golden = tachyon-free Certified (cond. reproducible: s120c + shipped e7_substrate/; Hes-
Minkowski critical point of dyonic on the E7 sub- sian (0− , 340 , 36+ ). Obs. moved: no
ω = π/4 SO∗ (8) strate)
I.2 Coleman–Weinberg cannot Certified no-go s123b, s124b; bein-independent (Aut(J3 )-
uniquely select golden in the equivariance forced). Obs.: no
tested σ-equivariant layers
I.6 Ng = 3 = χ(OP 2 ) as a Morse Certified count s124c; cell indices {0, 8, 16}. The
count (within the OP 2 generation→family / A1→Peirce assignment
carrier) is open. Obs.: no
Derived modulo the AX1 unit
I.3/I.4 σ-odd defect: F (ρ) = 1 ⇔ Certified modulo s123c, s124a; unit magnitude γ = 1 ≡ AX1 (named
AX1; + 12 ∥B∥2 completion; radial AX1 unit, not derived from nothing). Obs.: no
m2 = 7/4
Named structural / source term (amplitude open)
7/3 heavy-quark bridge = cotan- Structural / HeavyQuark_T3; proved algebraic invariant
gent Yukawa source B ∈ J12
∨
named source (dim Im Os /rank); amplitude promotion open
(unwritten E7(7) /J3 action). Obs.: no
Deferred build (open)
I.5 Lift of the 8 σ-even residual Open (reduced-to) undecidable from the point spectrum alone; needs
flats = Ciab = 21 ∂a ∂b m2i the certified complex 56-bein. The no-scale dilaton
is a separate scale/lift input. Obs.: no
Declared inputs (external / named)
AX1 ρ2 − ρ − 1 = 0 (the golden Input (named ax- assumed, not derived from nothing. Obs.: no
unit) iom)
Mass scale + anchors: mτ , vEW , Input declared in the INPUTS sheet / Appendix X (Honest
ΛG2 , α, Koide K = 2/3, Ng = 3- Boundary). Obs.: no
as-engine-input
Rev19 / S136 release boundary. Relative to Rev18 the engine advanced: ms is now derived
(ΛG2 φ 31/4 , S130) so mc = (7/3)ms reads 1291.9 MeV (+1.54% → +1.48%); the electron
Koide value reads me = 0.5076 MeV (S134); and the Outputs workbook gains five landscape
explainer tabs (S133). The SGTOE_Machine.xlsx hash at Rev19 was sha256 7aef7f20...
(prior 3d2c7a0a, 437ce51b, 1fc4757e). These are the only displayed-observable changes.
The Appendix I dynamics fold (introduced Rev18) is unchanged: the golden critical point is
computed on the dyonic SO∗ (8) substrate; σ-even Coleman–Weinberg cannot uniquely select
golden within the tested σ-equivariant point-spectrum layers; the σ-odd defect selection is
derived modulo the AX1 unit; and full stabilization is not closed — the 8 σ-even residual flats
require the field-dependent tensor Ciab , and the dilaton remains a separate scale/lift input.
AX1 is named, not hidden; Ciab stays in the open column until computed.
How to read this suite: four layers
This suite is not one document but four, bound together and now separated in the PDF’s bookmark
panel. Knowing which layer a page belongs to tells you what kind of claim you are reading and how
11
much weight it carries.
1. Authoritative papers (Papers 0–9). The claims. Anything the framework asserts about
physics is stated here, with its tier and its premises attached. If a statement is not in this layer,
it is not being claimed.
2. Theorem and derivation appendices (Appendices A–T). The work behind the claims:
constructions, proofs, no-go theorems and the fences that scope them. A paper cites this layer;
this layer does not enlarge on what the paper claims.
3. Input and price ledger (Appendix X). What the framework assumes. Every declared input,
every unpriced clause, and for each one what would end the argument. A reader checking whether
a result is genuinely zero-parameter should start here and not in the papers.
4. Historical notebook (Appendix Y). What was actually done, round by round, including the
rounds that failed — currently A1300–A1572 (its span is printed in its own title; through Rev32.4
it stopped at A1500, sixty-two rounds short, and said so only there; Rev32.6 adds A1563–A1572).
It is a locator and an audit trail, not a second statement of the physics. Its “Carried by” column
names the component that carries each round’s content — or records that nothing carries it yet.
Two things this separation deliberately does not yet do (Rev30). Paper 8 is the
junkyard — deleted content, dead ends and junked ideas — so by function it belongs to the
historical layer, but it is numbered and ordered as a paper and is left there rather than moved.
Likewise a “computational supplement” would gather Appendices B, D and Q, which sit inside
the theorem run. Grouping either would make the bookmark tree jump backwards through
the page order, which is worse navigation rather than better; completing the separation for
those two requires changing the canonical component order, and that is a papering decision,
not a formatting one. Both are recorded as open.
The picture in three layers (Rev32.6; A1567 T3, house-edited). The suite carries
two strong geometries that are not yet one object. The first is a real hyperbolic scaling
spine: the reciprocal Jordan vacuum generates Peirce spectra, resolvent ratios, logarithmic
mass depths, boost rungs, braid iterates and the shell-wide moiré law, connected by exact
identities and changes of basis — while the same spectral law on different carriers is, by
the suite’s own rule, not a carrier identification. The second is a compact phase spine: on
the live 56 the generator K gives a bounded E7√clock circle and the half-turn T 2 = −I;
on V3 , Thalf gives the leptonic rotation with Ω = 5/2 = ϕ − 12 , an equality specific to the
golden (1, 3) eigenvalue that does not turn a real rung multiplier into a complex one. The
third layer is missing: no theory-selected normal operator combines a real rung step with a
nonzero compact phase (Appendix N, Part N.E), and the physical readout/basepoint layer
remains loaded or model-only. The older wormhole/black-hole–moiré and “beating” language
survives only with the verbs models, carries and is compatible with (Appendices O, H, E);
the obstruction is primarily in the physics, and this paragraph exists so the prose no longer
hides the separation.
1 Introduction
Rev29 marked the first point in the programme where the full mixing sector could be presented as
closed rather than merely promising; Rev29 withdraws the word “closed”. Sessions 22–30 fixed
12
the CKM sector, the PMNS reactor angle, and the leptonic Dirac phase inside the specified algebraic
correspondence map, so the suite carries eight mixing observables with no fitted mixing coefficients
inside that map. After the Rev29 row-by-row re-tiering (Rev29 re-tiering, Paper 3 § “Row-by-row
re-tiering”; kernel s959) two of the eight stand at Derived-conditional or above (θ12 PMNS , δ
CP ; the
physical θ23 row and |Vcb | are Loaded-correspondence since A1566/D1244, Rev32.6, the internal
7/16 mismatch identity staying theorem-grade) and none may be cited as zero-parameter. The
purpose of Paper 0 is to state that algebraic core cleanly, keep the measured-versus-derived boundary
explicit, and give the reader one place where the current status of the entire bundle can be read
without navigating the specialist papers first.
The companion manuscripts remain internal working papers rather than external review articles.
Their job is to expose the algebra, the bookkeeping, the benchmark comparisons, and the still-open
bridges in a form that can be audited and revised. Rev29 therefore uses a deliberately explicit
vocabulary: some quantities are theorems, some are structural algebraic outputs, some are measured
anchors, and some remain open. The new Appendix X, “Input Ledger” (titled “Zero-Parameter Input
Ledger” through Rev32.8; renamed Rev32.9, R-13), lists the algebraic inputs of the zero-fit mixing
sector and separates them from measured anchors in one place.
2 Operational axioms and current reading
AX1: Algebraic vacuum. The working vacuum is
√
−1 1+ 5
Jvac = diag(ϕ, 1, ϕ ), ϕ= . (2)
2
It fixes the Peirce spectrum λ1 = ϕ, λ2 = 1, and λ3 = ϕ−1 .
AX2: Vacuum selector. The preferred vacuum Jvac = diag(ϕ, 1, ϕ−1 ) is fixed (algebraically,
up to permutation) by the conditional selector theorem of Appendix E: unit determinant,
inversion symmetry, and the DET-7 condition force ρ2 + ρ−2 = 3, hence ρ = ϕ (equivalently,
the golden-field units of Z[ϕ] together with DET-7 give {1, 0, −1} uniquely). The vacuum
ratio ϕ is therefore an algebraically forced structural feature given the Appendix E premises
(modulo the AX1 unit) — not a free parameter, but not premise-free either: Appendix E
types DET-7 as a structural postulate and its selector theorem as conditional (Rev32.9;
through Rev32.8 this said “not . . . an externally carried assumption”).
AX3: Minkowski balance. The operational scheme uses the current vacuum-balance condition
as part of the structural framework and not as a free fit parameter.
AX4: Non-associative CP source. The basic CP source is
[e1 , e2 , e3 ] ≡ i, (3)
so CP violation is treated as a structural consequence of the split-octonion sector rather
than an added phase convention.
Current reading (entered Rev29). AX1, AX3, and AX4 remain active operational inputs.
The vacuum selector is no longer carried as a free parameter in the working bundle. The
suite still does not present the electroweak scale, the weak-angle threshold bridge, or the
full rest-mass programme as solved. Those bridges remain named explicitly and are carried
forward as the next research tasks.
13
3 Peirce decomposition and the cubic invariant
Relative to the Jordan frame {e1 , e2 , e3 }, the algebra splits into three diagonal idempotent lines and
three off-diagonal Peirce blocks. The identity
ϕ2 + ϕ−2 = 3 (4)
remains a structural through-line. It gives the transparent form tan2 θ12 = 3/ϕ4 , it underlies several
resolvent-family ratios, and it is the simplest exact algebraic identity linking the outer Peirce weights.
The Yukawa sector has two layers. The same-block Peirce carrier invariant (the unit-eigenvalue
theorem of A674) is
Ycarrier (Ψ, ΦH ) = Tr Jvac ◦ (Ψ × ΦH ) , (5)
while the literal one-generation SM Yukawa is the cross-block trilinear {J12 (H), J13 (ΨL ), J23 (ΨcR )}
with the physical Higgs in the vector 10 = J12 ⊕ Rc1 ⊕ Rc2 (A686–A689). Placing both Ψ and ΦH in
the (1, 3) Peirce block (the carrier reading) forces
Ψ × ΦH = −2 Re(ψ h̄) e2 , (6)
so contraction with Jvac extracts the middle eigenvalue λ2 = 1 and therefore fixes
yt (MPl ) = 1 (7)
at tree level.
Peirce-to-physics dictionary (C3)
For a referee unfamiliar with the Jordan-algebra language, the following one-to-one map between
Peirce sectors and physical observables summarises the mixing-sector ledger. All eight carry no fitted
mixing coefficients inside the correspondence map.
Rev29 supersession — read before the table. As of Rev32 the status markings in
the third column are restated at the current two-axis status (the pre-Rev29 “THEOREM
⋆100” markings survive only in sealed revisions Rev31.2 and earlier); the authoritative source
remains the row-by-row re-tiering (Rev29 re-tiering, Paper 3 § “Row-by-row re-tiering”;
kernel s959). After propagation and the A1566 retype (Rev32.6), two of the eight sit at
Derived-conditional or above (θ12 PMNS , δ
CP , each with its premises printed; the internal
7/16 mismatch identity is theorem-grade and its physical θ23 attachment, with |Vcb |, is
Loaded-correspondence); the CKM first row is Loaded because its route was selected
against kaon data; δCKM is Coincidence-class; θ13 is structural and |Vcb | is Loaded-
correspondence (physical attachment) / Structural formula (A1566). No row below
may be cited as zero-parameter. No number in the table changes.
14
Algebraic object Physical observable Formula and status
√
(2, 3) Peirce block Solar mixing angle θ12 tan θ12 = 3/ϕ2 ; Derived-
(8 off-diagonal degrees conditional (Rev29 re-tier;
of freedom) premises in Paper 3); ∥Jvac ∥2 =
ϕ2 + 1 + ϕ−2 = 3
(1, 2) Peirce block Atmospheric angle θ23 sin2 θ23 = 7/16; internal mis-
DET-7 Gram metric, match identity theorem-grade;
det G = 7 physical octant attachment
Loaded-correspondence
(A1566/D1244, Rev32.6;
s501 a candidate map, not
a selector); n23 = 7 from
det G(diag(ϕ, 1, ϕ−1 )) = 7
(DET-7 theorem; Gram matrix
criterion); AX2 eliminated
(1, 3) Peirce block Reactor angle θ13 sin
√ θ13 = sin (π/8) = (3 −
2 4
D4 triality 8v ↔ 8s 2 2)/8; STRUCTURAL ⋆ ⋆ ⋆
(round-trip framing T 2 , exponent-
2; forcing pending [s502/A1275]);
Jvac at π/4 in 8v , triality maps
π/4 → π/8 Stamps
√
V3 rotation orbit Leptonic CP phase δCP δCP = −2π/ 5 = −160.997◦ ;
V3 ,
Thalf |√ frequency Derived-conditional (Rev29
Ω = 5/2 re-tier; structural support; full
eigenvalue–angular-velocity
equivalence lemma forthcoming,
App B.7); 0.36σ central-value,
NuFIT 6.1 (NH), err +26/−36◦
√
Peirce block nesting CKM magnitudes |Vus |√ = sin θ12 / 6; |Vcb | √=
W13 ⊂ W12 ⊕ W23 |Vus |, |Vcb |, |Vub | 1/(9 7); |Vub | = |Vus ||Vcb |/ 6;
pre-Rev29 marking ⋆100, super-
seded — the CKM first row is
Loaded (route selected against
kaon data) and |Vcb | is Loaded-
correspondence (physical at-
tachment) / Structural formula
(A1566)
Sub-leading Jvac de- CKM Dirac phase δCKM δCKM = arctan 3(ϕ + 5ϕ−5 ) =
p
formation 68.1297◦ ; Coincidence-class
Split-octonion (4, 4) (rating pending the G7 motiva-
coupling weights tion; number retained); 0.8σ
γ-fit / 1.6σ direct
in this table ( ⋆100, THEOREM, Proved) are the Rev29-era marking, superseded en bloc by the Rev29 re-tier (Paper 3
§“Row-by-row re-tiering”); the current tier of each row is the re-tier table’s. Footer added Rev32.5.
The three named operational anchors vEW , mτ , and ΛG2 fix the electroweak scale, the lepton-tower
15
anchor, and the hadronic scale respectively (anchor inventory per the Appendix X input-count
convention; αs is a measured constant, not an anchor); the remaining entries are J3 (Os )-algebraic
correspondences at the tiers stated in each cell (Rev29 two-axis status; no row here is citable as
zero-parameter).
4 Technical clarifications
Tree-level vs. running Yukawa
Rev8 derives a tree-level eigenvalue and nothing stronger than that. Below the ultraviolet matching
scale, the top Yukawa obeys the standard one-loop Standard Model equation
dyt yt
= βt (yt , g1 , g2 , g3 , . . .), (8)
d ln µ 16π 2
so that the operational reading is
yt (MPl ) = 1, yt (MZ ) ≈ 0.967. (9)
The tree-level eigenvalue is exact inside the algebra; the loop evolution is the ordinary Standard
Model bridge.
Split-octonion algebra and verification
All products are taken in the split-octonion algebra Os with the Cayley–Dickson construction
(p, q)(r, s) = (pr + s∗ q, sp + qr∗ ), the split inner product ⟨x, y⟩CD = k<4 xk yk − k≥4 xk yk , and the
P P
standard Jordan product x ◦ y = 12 (xy + yx). The Freudenthal cross product is
x × y = x ◦ y − 12 Tr(x) y − 21 Tr(y) x + 12 [Tr(x)Tr(y) − Tr(x ◦ y)] I. (10)
The suite uses these definitions consistently in the prose, in the appendices, and in the local verification
kernels. The generator Thalf of the V3 -rotation has been confirmed to lie in Der(J3 (Os )) = f4(4) with
derivation error 3.4 × 10−15 and rotation residual ∥ exp(t∗ Thalf ) · Jvac − Jvac
# ∥ = 2.1 × 10−11 (kernel v3.9
E121, Session 33 — the kernel of that era; the shipped kernel is named in Appendix X).
Algebra selection: metric signatures and Os (C new)
A natural question is why J3 (Os ) specifically, rather than some other Jordan algebra. The answer is
a short chain of physical requirements standing on one adopted premise at its head (Rev29 ). Earlier
revisions of this section described the chain as following “with no free choices”; that phrasing is
withdrawn. It is not true of Step 1, and Step 1 is where the load sits. Paper 1 (§“Why J3 (Os )?” and
abstract) already carries the corrected statement; this paper now matches it verbatim.
Step 1 ( adopted premise, not a derivation) — both Lorentz signatures are physically equivalent, and
we choose a convention-neutral fibre. [P-001] The two metric conventions (1, 3) and (3, 1) describe the
same physical spacetime; no experiment distinguishes them. We adopt a sign-convention-neutral fibre:
placing both conventions on equal footing inside a single composition algebra motivates an inner
product of signature (1, 3) + (3, 1) = (4, 4). Rev29 correction. The neutral (4, 4) signature is a model
16
choice that places both conventions on equal footing, not a consequence forced by the convention
equivalence alone; this is stated as an adopted premise, not a derivation. (Same wording, Paper 1
abstract and Paper 1 §“Why J3 (Os )?” item (2).) The equivalence of the two conventions does not by
itself require an algebra carrying both at once: either convention alone is a consistent choice, and a
theory built on one of them is not thereby refuted. What the premise buys is neutrality, not necessity.
Step 2 — the only real normed composition algebra with split signature (4, 4) is Os . Among the four
normed division algebras (R, C, H, O) and their split counterparts, the unique 8-dimensional algebra
whose norm form has neutral signature (4, 4) is the split octonion algebra Os . The compact octonion
algebra O has positive-definite norm (+ · · · +) and is excluded because it does not accommodate the
split signature required in Step 1.
Step 3 — three generations fix J3 (under the three-generation condition). The cohomology H ∗ (OP 2 ; Z)
of the octonionic projective plane is non-trivial in exactly three degrees (0, 8, 16), and the Atiyah–
Hirzebruch spectral sequence yields exactly three non-trivial generators, fixing the matrix size at
3 × 3. The unique 3 × 3 Hermitian matrix algebra over Os is J3 (Os ).
Kernel confirmation (Session 33). The requirement is not merely conceptual: using the Cayley–
Dickson split-octonion product with a Euclidean inner product (signature (8, 0)) in the Jordan
product gives dim Der(J) = 12, not the expected dim f4(4) = 52. Replacing the inner product with
the (4, 4)-signature split inner product ⟨x, y⟩CD = 3k=0 xk yk − 7k=4 xk yk restores dim Der(J) = 52
P P
exactly, with maximum residual |M · NS| < 2 × 10−15 . The algebra itself rejects the wrong signature.
Summary. Physical equivalence of (1, 3) and (3, 1) ⇝ adopted premise: a convention-neutral fibre of
signature (4, 4) ⇒ Os uniquely (under the added requirement of an eight-dimensional real normed
composition algebra) ⇒ J3 (Os ) uniquely (three-generation condition). Only the second and third
links are implications; the first is written ⇝ because it is the adopted premise, not an inference.
The argument is therefore premise-plus-uniqueness — a stated model choice followed by two genuine
uniqueness steps — rather than a selection principle derived from experiment. So read, it remains
independent of, and separate from, the numerical agreement of the mixing-sector predictions.
Rev29 re-tiering: the (1, 3) ∼ = (3, 1) premise. The claim that convention equivalence
forces the (4, 4) fibre, and the accompanying phrase “no free choices,” are withdrawn. The
corrected statement, held in common with Paper 1 and Paper 4: the neutral (4, 4) signature
is a model choice that places both conventions on equal footing, not a consequence forced
by the convention equivalence alone; this is stated as an adopted premise, not a derivation.
Steps 2 and 3 are unaffected — given the (4, 4) premise and an eight-dimensional real normed
composition algebra, Os is unique, and given the three-generation condition, J3 (Os ) is unique.
The kernel confirmation below is likewise unaffected: it tests the algebra’s response to the
signature, not the provenance of the signature. Nothing downstream of the fibre changes tier;
what changes is that the entry point no longer advertises the fibre as free of choices.
The (4, 4) signature is not merely a static convention choice; it has a dynamical reading. The two metric
conventions (1, 3) and (3, 1) each correspond to a distinct vacuum-collapse cycle; their simultaneous
presence creates an interference pattern — a moire of the two cycles — whose combinatorial structure
is precisely the (4, 4) split-octonionic algebra. The SO(8) triality automorphism τ : 8v → 8s → 8c
of so(4, 4) is the order-3 twist connecting the two sectors (it is not an automorphism of (Os , ×));
its stabilizer G∗2 = G2(2) ⊂ SO(4, 4) acts as the band-gap symmetry at the moire interface. Bosons
have zero winding in this moire (VCW bos ≡ 0, proved); fermions carry a Möbius half-twist (V ferm =
CW ̸ 0,
proved), with the vacuum minimum at r⋆ = φ2 selecting the golden-ratio hierarchy. The three Peirce
17
idempotents {f1 , f2 , f3 } are the three topological winding sectors; the three fermion generations are
conjectured to be three winding levels of the moire. This is physical intuition, not a theorem. In
particular, algebraic replication of three generations inside a single J3 (Os ) is excluded: three families
need 3 × 16 = 48 fermionic dimensions, which do not fit in the 27 (8+8+8 = 24 off-diagonal, and
triality maps 8+ → 8− → 8v , so the blocks are not three gauge-identical copies) [A691]. [LIB2-298] The
natural alternative — a continuous topological realization in which the three generations are Dirac
zero-modes over a moire background glued by the order-3 triality τ — is also excluded [A693]: the
τ -invariant zero-mode Ψ0 = 13 (x + τ x + τ 2 x) is forced to carry an equal 1/3 weight in the vector
block 8v = J12 (kernel-verified), mixing a boson into the fermionic mode and ruining its Standard
Model charges; moreover a one-cycle holonomy gives dim ker(1 − τ ) ≤ 16 and cannot multiply a
16-dimensional fiber into the 48 states three generations require. The only direction not excluded
is a discrete (noncommutative-geometry) reading in which {f1 , f2 , f3 } define a three-point internal
space and the Dirac operator is a finite matrix; there the number three is an input, not yet derived.
That discrete branch is now built (Rev32.3; lane A1549, house s1133/s1134): the replication map
ι : C ∗ (|M |) → C ∗ (Jvac ), the unique unital ∗-isomorphism with ι(Pλ ) = fλ , realises the discrete Kan
extension T ⊕ T ⊕ T with a common diagonal factor — aligned by construction, V = I — so the
moire story’s surviving branch is a construction, not a to-do; it picks no branch, scale or scheme, and
the number three remains an input. [LIB2-298] The open object is the noncentral defect (A1550/A1551):
a registered off-diagonal source inducing [Yu , Yd ] ̸= 0, located by three coordinates — a staged
intertwiner, the E6 /F4 basepoint (Appendix T, a declared placeholder), and one Appendix-T door.
The moire picture is therefore a span narrative whose discrete branch is built and whose continuous
branch is refuted; it does not establish three generations by any continuous-topology or algebraic
mechanism.
Remark (the shell-wide moire law; A1551 rider, Rev32.3). On the reciprocal 27 ⊕ 27′ shells,
Sd = exp(d ln φ DKK ) carries the eigenvalue pair (φ2d , φ−2d ) on every symplectically paired direction,
and the resolvent pair per direction is (Rd , −(1 + Rd )) with one conserved −1 per pair, Tr56 = −28.
The resolvent family, the S197 knot anchor and the moire two-cycle thus carry the same reciprocal
spectral law, lifted to all 28 pairs of the 56 — one law seen on three carriers, not an identification of
the carriers.
Lorentz-signature category (T4-H, partial resolution). The neutral (4, 4) signature is carried
by the split-octonion/Clifford fibre, not by an eight-dimensional physical spacetime. The ghost-safe
physical readout is a frame restriction:
′
V4,4 = W1,3 ⊕ W3,1 , Πphys : V4,4 → W1,3 , Π2phys = Πphys .
Only W1,3 is assigned physical Lorentz propagation; the complement W3,1 ′ is an algebraic dual-frame
sector, not a tower of propagating extra-time fields. The two-time ghost problem is thus avoided by
category: the extra directions are never quantized as physical coordinates. The subgroup preserving
the split is
Spin(4, 4) ⊃ Spin(1, 3)phys × Spin(3, 1)alg , so(4, 4) = so(1, 3) ⊕ so(3, 1) ⊕ (4phys ⊗ 4alg ),
with 28 = 6 + 6 + 16 (kernel-verified). The off-block 16 is a representation-theoretic complement
only (the vector–vector coset of [A714]); it is not identified with the 42 − 26 scalar-count gap
and is not a physical tower. This category was adjudicated by three independent assessments
(two cold) with unanimous convergence [A727–A727c]; the survivor is two-layered — the fibre-level
Spin(1, 3) × Spin(3, 1) frame group and, at the Jordan layer, the compact rank-four Standard-Model
18
input (which cannot embed in so(4, 4) itself: the maximal compact is SU (2)4 , all A1 factors). The
compact Standard Model gauge group thus remains the independent rank-four input used elsewhere
in the suite; B − L is constructible but not gauged at this stage [A710]. A dynamical reduction
(gauge-quotient) route would require a constraint algebra removing three extra timelike directions
and is not constructed; the holographic/interface reading remains physical intuition. The external
frame selector leaves the Peirce blocks, Jvac , and the proved Coleman–Weinberg dichotomy untouched
[A728], so the suite’s proved results are insulated from this category choice.
A decorative combinatorial illustration (the Boolean-hypercube / I Ching trigram indexing of the
Os basis, formerly Box C.1) has been moved to Paper 8 §Curios; it is an observation on shared
combinatorics [A626], not load-bearing structure.
Cubic invariant terminology
The phrase “cubic invariant” is used in two related senses in the literature. The global Jordan cubic
norm Det(Q) classifies orbits. The local trilinear form Tr[Jvac ◦ (Ψ × ΦH )] is the unique cubic object
used here to close the Yukawa sector while preserving the off-diagonal Higgs geometry. Rev8 separates
those two roles explicitly.
5 Charged-lepton and rest-mass mechanism: current Rev29 status
The electron baseline inherited from Rev8,
π
√
e = MPl exp −
m(0) ϕ4 2 = 0.5036 MeV, (11)
8α
now functions as a historical semiclassical cross-check rather than as the operative closure mechanism.
The current Rev29 charged-lepton chain is the QED-corrected Koide closure:
3α Λ2G2
δQED = ln = +0.3125%, (12)
4π m2µ
106.05
mµ,phys = = 105.72 MeV, (13)
1.003125
followed by
√ √ √ 3
( me + mµ + mτ )2 = (me + mµ + mτ ), (14)
2
which yields
me = 0.5076 MeV (−0.66% PDG). (15)
Thus mτ is the single charged-lepton anchor, mµ is fixed by the resolvent short form (Structural
tier, Appendix O) plus one-loop QED matching, and me is fixed by Koide using the QED-corrected
muon mass. No phenomenological electron phase dressing remains in the current charged-lepton
sector.
6 Operational Rev29 ledger
19
Table 1: Operational Rev29 ledger. Rev32.1 reading rule:
tiers name mathematical provenance; physical attach-
ments are stated separately; no row may be cited as
zero-parameter; displayed σ values are named com-
parator distances, not profile likelihoods.
Quantity Expression or source Status Comment
yt Tr[Jvac ◦ (Ψ × ΦH )] Algebraic Tree-level eigenvalue 1 at MPl ;
runs to yt (MZ ) ≈ 0.967.
Vus , Vud , Vub Peirce J12 block and hi- Algebraic First-row CKM relations car-
erarchy relation ried structurally inside the cur-
rent map.
δCP AX4 non-associator Algebraic Geometric phase target used
phase map in the PMNS sector.
sin2 θW tree-level exceptional ra- Algebraic RG bridge to the measured Z-
tio pole quantity remains a sepa-
rate QFT step.
SU (3)c × Peirce decomposition Algebraic Working gauge structure of the
SU (2)L ×U (1)Y and slot reduction suite.
me QED-corrected Koide Koide- me = 0.5076 MeV (−0.66%
chain consistent PDG); depends on external
empirical K = 2/3; no elec-
tron phase dressing remains.
v = 246 GeV electroweak scale Measured Measured, not derived, in
Rev8.
AX2 vacuum se- structural fixed-point Theorem Working vacuum
lector theorem for ϕ diag(ϕ, 1, ϕ−1 ) no longer
treated as a free parameter in
Rev29.
7 Roman surface model of generation mixing
The off-diagonal Peirce closure Pij ◦ Pjk ⊂ Pik in J3 (Os ) has a concrete geometric realisation as the
Roman surface (Steiner surface, 1840) [3], a degree-two map f : RP2 → R3 :
(x′ , y ′ , z ′ ) 7−→ (y ′ z ′ , z ′ x′ , x′ y ′ ). (16)
This replaces the earlier intuition picture with a mathematically precise object. The structural
correspondences are:
• Three double-point lines of the Roman surface ↔ three off-diagonal Peirce blocks (Z22 grading).
• Six pinch points (at (± 12 , 0, 0) and cyclic permutations) ↔ rank-drop points of the diagonal
Jordan adjoint; the determinant-zero locus maps to the Roman double-line skeleton [A680].
• Non-orientability ↔ Möbius boundary condition on (1, 3)-representation spinors (Theorem A.7.4).
20
Setting diagonal anchors x′ ∼ φ, y ′ ∼ 1, z ′ ∼ φ−1 for Generations 3, 2, 1 respectively, the cross-
product inverts the hierarchy in the quark sector: Gen 1 quarks map to the largest Peirce block P23
(λ23 = φ1/2 ), acquiring the largest confinement-driven constituent mass. Leptons follow the primary
anchor hierarchy; quarks follow the inverted cross-product hierarchy. This lepton/quark duality is a
rigid algebraic consequence of the embedding, not a phenomenological assumption. [A632]
8 Known gaps, deferrals, and tensions
Known gaps and deferrals.
(a) Vacuum theorem carried. The working bundle no longer treats ϕ as a free input,
and the derivational presentation is printed: Appendix E gives two explicit proofs —
Route A (printed and valid at the conditional-selector tier) and Route B (printed, with
the uniqueness step repaired at Rev32 via the verified order-gap argument; conclusion
unchanged).
(b) Higgs VEV. The electroweak value v = 246 GeV is measured rather than derived. The
algebra fixes dimensionless structure; the weak scale remains an external anchor.
(c) Weak-angle RG bridge. The tree-level weak-angle output requires renormalization-
group running, threshold corrections, and scheme choice before comparison to the measured
Z-pole quantity.
(d) Electron mass: Koide-closed under external K = 2/3. The electron mass is
Koide-closed ⋆ ⋆ ⋆⋆ via the QED-corrected Koide chain [A636, A637]: me = 0.5076 MeV
(−0.66% PDG). This result is conditional on the external empirical Koide rule K = 2/3,
not derived from J3 (Os ); an in-framework derivation
√ of the value K = 2/3 is not yet
available. The shape relation Q = 2/3 ⇔ r = 2 is derived (Appendix O, “The Koide
shape”): the shape is derived, the value is an external selector, and the two are not the
same claim. The Rev8 instanton-phase ansatz and associated bookkeeping are superseded
and retired.
(e) Charged-lepton chain. The charged-lepton sector is an advanced compressed chain
with mixed provenance: mτ is the algebraic anchor, mµ = 106.05 MeV is Structural
(Appendix O re-tier) via the resolvent short form, and me = 0.5076 MeV is Koide-
closed ⋆ ⋆ ⋆⋆ under external K = 2/3. It is not a derivation of all three charged-lepton
masses from J3 (Os ) alone.
9 Path to falsifiability
Rev29 turns the remaining rest-mass validation steps into an explicit protocol rather than a future
aspiration.
(1) Use the mµ /mτ structural formula (Appendix O tier) as the charged-lepton family anchor for
the next mass-operator kernel.
(2) The electron mass me = 0.5076 MeV (−0.66% PDG) is Koide-closed ⋆⋆⋆⋆ under external K = 2/3
via the QED-corrected Koide chain [A636, A637] — conditional on Koide, not a J3 (Os )-internal
21
derivation. The next mass-sector task is the formal S-matrix proof of the 7/3 amplitude bridge
for mb and mc .
(3) Construct the formal S-matrix proof that the Peirce-closure amplitude equals n23 /rank(J3 ) = 7/3
for the heavy-quark bridge.
(4) Treat failure of the 7/3 S-matrix amplitude derivation as a failure of the heavy-quark bridge
mechanism, not of the fit-free PMNS/CKM mixing sector or the algebraic mµ /mτ ratio.
10 Conclusion
Paper 0 states the Rev29 programme in its current honest form. The cubic invariant, the Peirce
decomposition, and the benchmark ledger remain the algebraic core. The weak-angle AX6 selector and
the heavy-quark S-matrix amplitude bridge remain the principal open items, while the PMNS/CKM
mixing sector carries no fitted mixing coefficients inside the loaded Route-B correspondence (six
quantities take exact algebraic formulae there, of which two — θ12 and δCP — survive the Rev29 row-
by-row re-tiering and the A1566 retype (Rev32.9: through Rev32.8 this said three) (Rev29 re-tiering,
Paper 3 § “Row-by-row re-tiering”; kernel s959) at Derived-conditional or above; the CKM first
row is Loaded, δCKM is Coincidence-class, θ13 is structural, |Vcb | is Loaded-correspondence
(physical attachment) / Structural formula (A1566), and no row may be cited as zero-parameter;
and the CKM β = 22.5◦ vs. reconstructed 23.44◦ consistency is unresolved, Paper 3) and the charged-
lepton mass sector is structurally pinned with mixed provenance (anchor mτ + algebraic mµ /mτ
+ Koide-closed me under external K = 2/3; the compact ratio’s relation to the two-gap operator
of Appendix O is a pending dual-route reconciliation, not a completed derivation). [LIB2-235] The
framework is presented as a sharply structured algebraic correspondence scheme. The two principal
open items—the weak-angle AX6 selector and the formal S-matrix promotion of the 7/3 lepton–quark
bridge—are now characterized rather than merely outstanding: the latter is established as an algebraic
invariant and shown to require genuinely new exceptional-field-theory machinery for its amplitude
form (Appendix I). Neither blocks publication of the fit-free PMNS/CKM mixing sector or the
structurally pinned charged-lepton sector; both are stated as falsifiable, well-posed problems rather
than completed results.
References
[1] C. Barton and A. Sudbery, “Magic squares and matrix models of Lie algebras,” Adv. Math. 180
(2003) 596–647.
[2] G. Bossard, Y. Michel, and B. Pioline, “Extremal black holes, nilpotent orbits and the true fake
superpotential,” JHEP 01 (2010) 038.
[3] R. Bryant and R. Kusner, “The Roman surface and its symmetries,” unpublished notes; see also
J. Steiner, “Über solche algebraischen Curven,. . . ,” J. reine angew. Math. 21 (1840) 33–66.
[4] S. Coleman and E. Weinberg, “Radiative corrections as the origin of spontaneous symmetry
breaking,” Phys. Rev. D 7 (1973) 1888–1910.
[5] E. Cremmer and B. Julia, “The SO(8) Supergravity,” Nucl. Phys. B 159 (1979) 141–212.
22
[6] DUNE Collaboration, “Deep Underground Neutrino Experiment (DUNE) Technical Design
Report,” 2020.
[7] K. McCrimmon, A Taste of Jordan Algebras, Springer, 2004.
[8] I. Esteban, M.C. González-García, M. Maltoni, I. Schwetz, and A. Zhou, “NuFit-6.0: updated
global analysis of three-flavor neutrino oscillations,” JHEP 12 (2024) 216, arXiv:2410.05380;
updated values from NuFIT 6.1 (Nov. 2025), www.nu-fit.org.
[9] Particle Data Group, “CKM quark-mixing matrix,” 2025 review pages.
[10] Particle Data Group, “Review of Particle Physics,” electroweak and Standard Model review
pages, 2025 update.
[11] I. Todorov and M. Drenska, “Octonions, exceptional Jordan algebra and the role of the group
F4 in particle physics,” Adv. Appl. Clifford Algebras 28 (2018) 82, arXiv:1805.06739.
[12] E. Witten, “Constraints on Supersymmetry Breaking,” Nucl. Phys. B 202 (1982) 253.
23
Leibniz Quantum Beats Newton
Paper 1 (Rev33.1): Peirce Decomposition, Algebra Selection, and
Gauge-Group Assignment
Tom O’Sieg
August 2026
Abstract
This paper isolates the Peirce decomposition of the split exceptional Jordan algebra and states
the gauge-group assignment — the input label identified (not produced) by the Peirce frame —
in its updated form. The central point is that the working algebra is not chosen ad hoc: the
physical equivalence of the (1, 3) and (3, 1) Lorentz-signature conventions motivates adopting
a sign-convention-neutral (4, 4) fibre, and under the added requirement of an eight-dimensional
real normed composition algebra this selects the split octonion algebra Os . (The neutral (4, 4)
signature is a model choice that places both conventions on equal footing, not a consequence
forced by the convention equivalence alone; this is stated as an adopted premise, not a derivation.)
The three-generation condition then fixes the 3 × 3 Hermitian Jordan algebra J3 (Os ). All local
split-octonion products are understood in Cayley–Dickson form with the SplitCD inner product;
the older Fano/sign-flip surrogate is not used. Once the vacuum Jvac = diag(ϕ, 1, ϕ−1 ) is fixed,
the three Peirce blocks carry inequivalent algebraic roles that organize colour, weak, and Yukawa
structure. Legacy Route A flavour identifications are retired; the live bundle uses the Route B
canonical CKM entry carried in Paper 3, which Paper 3 types Loaded (selected against kaon
data; Rev32.9 — through Rev32.8 this said “theorem-level”).
1 Introduction
Paper 1 states the gauge-sector claim as cleanly as possible. Within the working J3 (Os ) map used by
the suite, the Peirce decomposition of the chosen vacuum identifies the operative gauge assignment
SU (3)c × SU (2)L × U (1)Y (1)
without any additional gauge-group-selection axiom. The gauge group is thus identified by the algebra
as a group-level input — not produced by it: its emergence as a maximal-supergravity gauging is
obstructed by two representation-theoretic walls, and its low-energy completion is unique only within
a stated restricted class, up to U (1)B−L and global form (Appendix F). The orbit selector AX6pol
(Λ⋆ = 3P/Q(Y ) = 6ϕ − 1 on the exact-colour orbit; the degree-six Det2 object cos(6ψ) = 12 once
printed under the same name is a distinct, retired selector — Paper 4, Rev32.2) is an independent
postulate required to fix the Weinberg-angle branch within the identified group, and is discussed
separately in Paper 4. In Rev29 this claim is strengthened by an upstream algebra-selection argument:
J3 (Os ) is the unique 3 × 3 Hermitian Jordan algebra over the unique real normed composition algebra
compatible with a neutral (4, 4) norm. The same corrected Cayley–Dickson algebra now underlies the
whole bundle.
1
2 Why J3 (Os )?
The algebraic core is fixed by a short chain of physical requirements.
(1) The metric conventions (1, 3) and (3, 1) are physically equivalent.
(2) We adopt a sign-convention-neutral fibre; placing both conventions on equal footing in a single
composition algebra motivates a neutral (4, 4) norm (a model choice, not forced by the equivalence
alone).
(3) Under the added requirement of an eight-dimensional real normed composition algebra, the unique
signature-(4, 4) realization is Os .
(4) The three-generation condition fixes the 3 × 3 Hermitian Jordan algebra over Os .
Kernel verification sharpens the same statement: with the wrong (Euclidean) inner product the
derivation algebra has dimension 12, whereas the SplitCD inner product gives dim Der(J3 (Os )) =
52 = dim f4(4) .
3 Peirce decomposition at the golden-ratio vacuum
Let
Jvac = ϕe1 + 1 · e2 + ϕ−1 e3 (2)
in a Jordan frame {e1 , e2 , e3 }. Then
3
M
J3 (Os ) = Rei ⊕ J12 ⊕ J13 ⊕ J23 . (3)
i=1
Each off-diagonal block is eight-real-dimensional and carries split-octonion data; the diagonal lines
carry the spectral weights λ1 = ϕ, λ2 = 1, and λ3 = ϕ−1 .
The E6(6) → SO(5, 5) × SO(1, 1) branching 27 → 16 + 10 + 1 has three conjugate realizations,
one per primitive idempotent ck (A684): the suite uses the c3 -selected branch, 16 = J13 ⊕ J23 ,
10 = J12 ⊕ Rc1 ⊕ Rc2 , 1 = Rc3 , consistent with tR ∈ J13 (A674) and J12 as the CKM sector. The
three branches are related by the S3 outer automorphism (triality) of D4 . The Jordan adjoint map
−1 (Weyl reflection c ↔ c , A680) sends the active vacuum to the conjugate branch while
X # = Jvac 3 1
leaving the spinor sector J13 ⊕ J23 invariant: J12 is the vector (bosonic) block in both the c3 and c1
branches.
3.1 Roman surface and the diagonal Jordan adjoint [proved, A680]
Let D = {diag(x, y, z) : x, y, z ∈ R} ⊂ J3 (Os ) be the diagonal cubic Jordan subalgebra. For
X = diag(x, y, z), the quadratic Jordan adjoint is
X # = diag(yz, xz, xy). (4)
Thus the classical Roman-surface map
T : S 2 → R3 , T (x, y, z) = (yz, xz, xy),
2
is precisely the diagonal coordinate form of the Jordan adjoint X # , restricted to the unit sphere: the
homogeneous map X 7→ X # is quadratic, so on S 2 it acts on X/∥X∥ (domain stated Rev32.9, A1587).
Since T (−x, −y, −z) = T (x, y, z), the map descends through the antipodal quotient S 2 /{±1} = RP2 ,
and its image is the compact Roman (Steiner) surface — the bounded parametrized image. That
image is a proper subset of the real zero locus of the Steiner quartic y 2 z 2 + x2 z 2 + x2 y 2 − xyz = 0,
which also contains the unbounded coordinate axes; verifying the quartic therefore does not by itself
establish equality with the image (Rev32.9, A1587).
The three Roman coordinates carry a direct Peirce-triality reading: under S3 permutation of (x, y, z),
the output coordinates (yz, xz, xy) are permuted equivariantly, mirroring the three pairwise Peirce
channels J23 , J13 , J12 . (Note: S3 is the equivariance group of the map, not the quotient group
producing RP2 ; the projective topology arises from the antipodal Z2 .)
The Jordan determinant N (X) = xyz vanishes on the zero-divisor locus. Under T , this locus maps
to the three self-intersection double-line axes of the Roman surface. [LIB2-333] The six pinch points (at
(± 12 , 0, 0) and cyclic permutations) arise from the rank-drop points of dT .
The vacuum Jvac = diag(ϕ, 1, ϕ−1 ) satisfies Jvac
# = diag(ϕ−1 , 1, ϕ) = J −1 (Jordan inverse), consistent
vac
with det Jvac = 1. The homogeneous adjoint thus sends the golden-ratio vacuum to its Weyl reflection;
the unit-sphere Roman map sends Jvac /2 (since ∥Jvac ∥2 = 4) to Jvac# /4 — the same direction at a
quarter of the scale (Rev32.9).
Since T is quadratic (degree 2), the antipodal invariance T (−u) = T (u) mirrors the fundamental
property of spin- 12 states: physical observables (mass terms, currents) are quadratic in the spinor
amplitude and hence invariant under ψ → −ψ, requiring a 4π rotation to return the spinor to itself.
Rev29 correction. The antipodal Z2 on the Peirce diagonal models the spinor double-cover parity. The
Z2 producing RP2 = S 2 /{±1} and the Z2 generating the spinor double cover Spin(3) → SO(3) are
abstractly isomorphic and carry the same sign-parity bookkeeping; that is an interface-type statement
— a correspondence between two Z2 actions on two different spaces — and not an identity of the
two Z2 ’s. Earlier revisions asserted that they are “the same Z2 ”; that assertion is withdrawn. The
antipodal involution here acts on the diagonal S 2 of Jordan eigenvalue directions, not on a spinor
space, and no spinor-bundle map realizing the correspondence is constructed (the explicit Möbius
spinor bundle for the Peirce blocks remains [OPEN] below). The half-spinor 8+ representation of
Spin(4, 4) carrying tR (with CX (8+ ) = 4 [proved, A670]) exhibits the 720◦ property independently;
the diagonal Jordan adjoint provides a geometric model of that Z2 structure at the level of the Peirce
diagonal, and nothing beyond a model is claimed.
Roman–Jordan adjoint: [PROVED]. S3 equivariance of T : [STRUCTURAL]. Antipodal
Z2 models (does not equal) the spinorial double cover parity: [STRUCTURAL], interface-
type (Rev29 ). Explicit Möbius spinor bundle for Peirce blocks: [OPEN]. LQG monad-cell
interpretation: [SPECULATIVE].
3
4 Gauge-group assignment (the input label and its two-sided bound-
ary)
Gauge-group reading (claim superseded; see boundary box below). The Peirce
decomposition identifies and constrains the working gauge group SU (3)c × SU (2)L × U (1)Y
through the reduction of the slot symmetry from S3 to a colour-preserving branch — a labeling
of derived structure, not an emergence or selection mechanism. An earlier formulation of this
box could be read as an emergence claim; that reading is superseded by the theorem-grade
gauge boundary below (and Appendix F): the gauge group is an input, identified by the
Peirce frame but not produced by it. No extra gauge-group-selection mechanism is used in
the Rev29 observable map.
Charge/gauge provenance: the terminal is possible-not-necessary, and it is final.
The gauge content is algebraically constructible and physically realizable, but not forced. Two
halves. (i) Algebra — closed: the full FTS conformal sl(2) acting on 56 = J3 ⊕ R ⊕ R ⊕ J3 is
the spin-3/2 charge clock (56 ↓= V3/2 ⊕ 26 V1/2 ), so the clock target is canonical once the full
FTS local system is admitted; the scalar-pair-only sl(2) is a distinct class carrying no V3/2
and cannot export the relevant structure [A1262, verified in-house]. (ii) Physical realization —
contingent: the bare Möbius throat carries only the trace-0 half-twist S, whereas the clock
is the trace-3 figure-eight M3 = T 3 S; these are disjoint SL(2, Z) conjugacy classes (elliptic
versus hyperbolic), so no change of basis bridges them. The missing datum is the T 3 trefoil
/ three-generation winding, which enters as framing-blind homotopy input. Whether that
input is forced was attacked five independent ways — static forcing principles, necessity audit,
worldline composition coherence, algebraic reachability, and framed-ribbon geometry — and
all five returned not forced [A1260, A1261, A1265, A1267, A1266]. This is a settled negative,
not an open question, and it is the formal reason the gauge group is carried as an input in
the box above rather than derived.
The logic is operational. The three slots begin with an S3 permutation symmetry; the chosen vacuum
breaks that symmetry spectrally because ϕ ̸= 1 ̸= ϕ−1 . Colour acts on the block triplet, the weak
doublet is read from the off-diagonal Higgs/Yukawa block, and hypercharge is the residual diagonal
weighting compatible with the same frame. No legacy Route A flavour mechanism is invoked here;
the current bundle treats Route A as dead and uses the Route B normalization of the mixing paper,
which that paper types Loaded (selected against kaon data).
4
Gauge-boundary update (S75–S80; full statements and proofs in Appendix F).
The status of the gauge group as an input is now bounded on both sides by theorem-grade
results obtained on the certified E7(7) embedding-tensor substrate. (i) Obstruction (two
walls): no gauging of maximal N =8, D=4 supergravity, in any symplectic frame, has a gauge
algebra containing a compactly embedded su(3) ⊕ su(2) — the branching-wall theorem —
and no gauging contains a unitary-class su(3) (the matter-colour embedding 8 = 3 ⊕ 5 · 1) —
the charge-wall theorem; only vector- and adjoint-class su(3)’s are ever gauging-realizable.
Gauge-group emergence as a gauging is closed negative; su(2)L is exactly the piece that can
never fit, and the right algebra, where it fits, provably never carries the right representation.
(ii) Uniqueness of completion: among compact low-energy completions gauging the full derived
colour + weak action on exactly the derived sixteen fermions with the derived hypercharges
and 10H -only scalar closure, GSM is unique up to U (1)B−L (extra Cartan computed unique,
3(B−L)) and up to global structure Γ ∈ {1, Z2 , Z3 , Z6 }. Uniqueness-of-completion is not
emergence; the input label stands. (iii) Shadow selection: the maximal derivable remnant,
an su(3) ⊕ u(1) ⊕ u(1) of shadow type, is selected dynamically — vacuum stability of the
missing-16 scalar sector fires uniquely at SO(6, 2) within the scanned family, whose dyonic
completion is a published Minkowski vacuum with residual SO(6) × SO(2); the SM charge
labels for the two u(1)’s failed their test at the origin point (S78, rep-level; pre-registered
demotion applied), were excluded for the entire 36-parameter vacuum family by counting
(S78b), and are excluded for every gauging by the charge wall (S80): the shadow is structural,
not the SM in disguise. None of this changes the observable map of this paper; it fixes the
epistemic status of its gauge input.
5 Structure-algebra viewpoint
The reduced structure algebra and derivation algebra are
Str0 (J3 (Os )) ∼
= e6(6) , Der(J3 (Os )) ∼
= f4(4) . (5)
At the physical-map level the sector-reduction chain is
e6(6) ⊃ f4(4) ⊃ g2(2) . (6)
The gauged colour subgroup SU (3)c is the compact real form inside the maximal compact K =
Sp(1) × Sp(3) of F4(4) (the stabilizer of the trace), following the A513 chain
K ⊃ 1 × Sp(3) ⊃ U (3) ⊃ SU (3). (7)
The colour, weak, and hypercharge labels are the low-energy names attached to that reduction.
6 SM field-content bookkeeping
The representation-level dictionary is now fixed for one generation by the explicit B-map (A688/A689);
it was only partial through Session 49. The low-energy field content is the Standard Model set
QL , uR , dR , LL , eR , (8)
with a Dirac-neutrino extension νR assumed in the neutrino paper. [P-032]
The one-generation Peirce-block assignments under the explicit B-map are (Session 50; see Paper 6
§4.4 and A685–A689 for derivations):
5
Peirce block D4 rep Standard Model content Status
J13 = 8+ half-spinor left-handed doublets QL = (tL , bL ), PROVED under B
LL = (νL , eL ) [A685/A689]
J23 = 8− half-spinor c c c
right-handed conjugates u , d , e , ν c PROVED under B
[A685/A689]
J12 ⊂ 10 vector Higgs doublets Hu , Hd (in 10 = J12 ⊕ PROVED under B
Rc1 ⊕ Rc2 ) [A686/A688]
Notes: (i) The left–right split 8+ = Q ⊕ L and 8− = uc ⊕ dc ⊕ ec ⊕ ν c is fixed by the SM-commuting
chirality ΓQL = Γ3 Γ4 Γ8 Γ9 [A685]. (ii) The Higgs lives in the vector 10, not in J13 ; the physical
one-generation Yukawas are the cross-block trilinear {J12 (H), J13 (ΨL ), J23 (ΨcR )} [A686–A689], with
J13 in the middle slot. The earlier “tR , bcR ∈ J13 ” placement (A674/A675) is the pre-B Peirce same-
block carrier convention, not the literal SM Yukawa container; it survives only as the unit-eigenvalue
carrier theorem. (iii) Three Peirce eigenvalues (ϕ, 1, ϕ−1 ) map to tree-level carrier strengths [A674
Peirce–Yukawa theorem].
Status under B: all one-generation quark and lepton placements and all sixteen hypercharges are
PROVED under the explicit B-map (A689). [LIB2-108] The remaining open items are: the canonical
selection of B from J3 (Os ) alone (D690); the explicit three-generation embedding; and the algebraic
anomaly check — the matter content fills one SO(10) 16 per generation, which is anomaly-free, but
the in-framework calculation is not yet written. (Superseded, S64 note: the canonical-B question
was closed by the A707/A708 audits — the continuous boost is excluded by compact gauge and the
exhaustive signed-permutation enumeration leaves one frame component after quotienting, so AB is
discharged relative to the compact SM gauge embedding — and the one-generation anomaly calculation
is now written and kernel-verified [A696]; see the later status updates in this paper.) [LIB2-108]
A678/A679 lepton finding (the obstruction the B-map later resolved): For all q ∈ { 16 , 13 , 12 },
the 26F4 branch via U Sp(6)×U Sp(2) contains no (1, 2)Y lepton doublet and no (1, 1)−1 charged-lepton
singlet (kernel-verified, A678). The structure group Str0 (J3 (Os )) ≃ E6(6) acts on the full 27 = J3 (Os );
the 27 ↓ F4(4) = 26 ⊕ 1 adds only a neutral singlet (1, 1)0 (A679). At the complexified E6 GUT level,
the 27 does contain lepton slots (inside the SO(10) spinor 16), but the compact SO(10) chain is not
a real subgroup chain of the split E6(6) . The next candidate is SO(5, 5) × SO(1, 1) ⊂ E6(6) , under
which 27 → 16 + 10 + 1 with a split-real spinor (A681). This route is the one that succeeded: under
the explicit B-map the leptons LL and ec , ν c are placed in J13 ⊕ J23 and all lepton Yukawas and
hypercharges are reproduced (A688/A689). The simple-26 obstruction above is therefore resolved;
the only residual freedom is the canonical selection of the B-map orientation (D690).
7 Cubic invariant and the Peirce carrier theorem
The same-block cubic
Lcarrier ∝ Tr Jvac ◦ (Ψ × ΦH ) , (9)
with Ψ, ΦH in the same off-diagonal Peirce block, is the Peirce unit-eigenvalue carrier theorem, not
the literal SM Yukawa: the Freudenthal cross product isolates the middle diagonal idempotent and
contraction with the vacuum extracts λ2 = 1, yielding the tree-level carrier eigenvalue yt (MPl ) = 1
[A674]. The literal one-generation SM Yukawa is instead the cross-block trilinear
{J12 (H), J13 (ΨL ), J23 (ΨcR )} = Re (x̄y)z̄ c2
(10)
6
with the Higgs Hu,d in the vector 10 = J12 ⊕ Rc1 ⊕ Rc2 , verified component-by-component for QHu uc ,
QHd dc and LHd ec under the explicit B-map (A687–A689). In this way the gauge and (carrier)
Yukawa readouts share one Jordan frame, while the physical 16 × 10 × 16 Yukawa is the cross-block
triple.
8 Anomaly-cancellation status
With the explicit B + SU (5) embedding, one generation of fermions is
QL = (3, 2)1/6 , uc = (3̄, 1)−2/3 , dc = (3̄, 1)1/3 , L = (1, 2)−1/2 , ec = (1, 1)1 , ν c = (1, 1)0 ,
i.e. the SO(10)-spinor branching 16 → 10 ⊕ 5̄ ⊕ 1 under SU (5), with the same index hypercharge rule
used by the B-map (Y0,1,2 = − 13 , Y3,4 = + 12 ). In the left-handed Weyl convention the local anomaly
coefficients all vanish (kernel-verified, cross-checked in house, A696):
SU (3)3 : 2 − 1 − 1 = 0, SU (3)2 U (1) : 1 1 1
6 − 3 + 6 = 0, SU (2)2 U (1) : 1 1
4 − 4 = 0,
U (1)3 : 1 8 1 1
36 − 9 + 9 − 4 + 1 = 0, U (1)-gravity : 1 − 2 + 1 − 1 + 1 = 0.
The Witten SU (2) global anomaly is absent: three colour quark doublets plus one lepton doublet give 4
left-handed SU (2) doublets per generation, an even number. [LIB2-109] The singlet ν c contributes nothing
to the Standard Model gauge anomalies; it completes the SO(10) 16 and supplies the right-handed-
neutrino slot. The split-real origin (Spin(4, 4), Spin(5, 5), E6(6) ) does not affect this calculation, which
depends only on the compact SU (3)c ×SU (2)L ×U (1)Y acting on the chiral content. The identified one-
generation matter content is therefore anomaly-free [PROVED under explicit B + SU (5)]. Anomaly
cancellation holds for any number of complete generations and does not by itself fix the generation
count, which remains an independent input. [LIB2-109]
Rev25 update: the complete exact audit (kernel s588, S214, upgrading the structural check s572)
verified that all SM gauge and gravitational anomaly sums cancel exactly from the framework’s own
derived hypercharges, formally closing the hostile-board item T3-P; see Paper 7 (Rev25 additions)
and Appendix F. [LIB2-109]
9 Rev25 structural additions (fold of S209/S223 results)
Novelty label: new specialization proved here.
Theorem 9.1 (E7 -closure of the congruence dressing; s679, S223; independently re-proved by the
hostile lane, A1343). Let p = span(S70 ) ⊂ Sym(56), k = [p, p], e7 = k ⊕ p (Cartan pair, machine-
verified). [LIB2-197] Then SPD ∩ E7(7) = exp(p) exactly, and for any two E7 -Grams A = exp(2SA ),
M = exp(2Φ) with SA , Φ ∈ p, the unique SPD solution B of BAB = M satisfies log B ∈ p —
i.e. B ∈ E7(7) identically. [LIB2-197] (Five-line proof, each assumption machine-verified; membership
residuals ≤ 3.6 × 10−14 at all eleven measured points; the control test has teeth: a random grade-
compatible symmetric matrix has p-residual 0.982.)
This upgrades the measured grade anatomy of the dictionary program from observed to derived:
the dressing lives on the E7 /SU (8) coset, always — a frame-free kinematic reduction from 56 × 56
functions to 70.
7
Conditional remark (labeled; s520/s521/s523, S209). An independent algebraic chain
reproduces the one-generation content from division-algebra data: one generation from the
Witt ideal of Cl(6); the anomaly-free 16; and C ⊗ H ⊗ O (the Dixon algebra) yielding the full
SM gauge group with the correct hypercharges. The algebra is robust and kernel-verified,
but its physical attachment to the suite is conditional on the κphys candidate mechanism
and the ιψ gluing — both open (Appendix Q posture). Recorded as a convergence signal, not
banked as a derivation of the field content.
10 Status and caveat
The gauge-group claim is algebraic inside the Rev29 map, but it does not yet amount to a universal
selector theorem excluding every rival subgroup assignment in every extension. What the present
paper claims is narrower and precise: (a) the Peirce decomposition fixes the working gauge assignment
used by the suite; (b) no extra gauge-group-selection axiom is added on top of the Jordan frame; (c)
the independent AX6 orbit selector fixes the Weinberg-angle branch within the already identified
gauge group, not the gauge group itself; and (d) the identification with the Standard Model remains
part of the programme’s falsifiable correspondence map.
11 Conclusion
Paper 1 is the shortest route from the abstract algebra to the familiar gauge labels of particle physics.
In Rev29 it also makes explicit why the underlying algebra is J3 (Os ) rather than some other Jordan
algebra: the same Lorentz-signature equivalence argument that selects Os also selects the non-compact
real form f4(4) needed by the rest of the suite. An independent published precedent for the F4 + J3 (O)
route to Standard Model quantum-number classification is provided by Todorov and Drenska [1].
References
[1] I. Todorov and M. Drenska, “Octonions, exceptional Jordan algebra and the role of the group
F4 in particle physics,” Adv. Appl. Clifford Algebras 28 (2018) 82, arXiv:1805.06739.
8
Leibniz Quantum Beats Newton
Paper 2 (Rev33.1): Mass-Matrix Hierarchy and Resolvent Quintics
Tom O’Sieg
August 2026
Abstract
This paper reorganises the fermion-mass sector around the Rev29 resolvent family, the cubic-
invariant Yukawa coupling, and the clarified renormalization-group story. The central statement is
that the hierarchy law
1
Rd = 2d (1)
ϕ −1
acts as the common algebraic kernel behind the second-to-third generation mass ratios, while the
local cubic invariant fixes the tree-level top-Yukawa eigenvalue at yt = 1. The d-index assignment
is selected, not derived (Paper 5; ruling R-23, Rev32.9): no printed theorem fixes which index goes
with which ratio, and the two assignments this paper prints are different objects, related in §3.1.
1 Introduction
Rev29 keeps the mass sector compact while adding the Sprint 3 blind-drop charged-lepton update.
The local cubic invariant fixes the top Yukawa. The resolvent family organises the hierarchy ratios.
The remaining issue is not whether the dimensionless structure is present, but how the measured
electroweak scale and low-energy running convert that structure into GeV numbers.
2 Cubic-invariant Yukawa closure
The same-block Peirce carrier coupling (unit-eigenvalue theorem, A674; the physical SM Higgs Hu,d
sits in the vector 10 and the literal Yukawa is the cross-block triple, A686–A689) is
Lcarrier ∝ Tr Jvac ◦ (Ψ × ΦH ) . (2)
With Ψ and ΦH placed in the (1, 3) Peirce block, the Freudenthal cross product projects onto e2 and
the contraction with Jvac = diag(ϕ, 1, ϕ−1 ) extracts λ2 = 1. Therefore
yt (MPl ) = 1. (3)
At low energy this boundary value runs under the ordinary one-loop Standard Model equations to
yt (MZ ) ≈ 0.967, so the tree/loop separation is now explicit rather than implicit.
1
3 Resolvent family
The common hierarchy kernel is
1
Rd = . (4)
ϕ2d − 1
The first four members are
1
R1 = = ϕ−1 , (5)
ϕ2 − 1
1 1
R2 = = , (6)
ϕ4 − 1
ϕ(ϕ + 2)
1 1
R3 = 6 = 3, (7)
ϕ −1 4ϕ
1
R4 = 8 . (8)
ϕ −1
Numerically these are R1 ≈ 0.6180, R2 ≈ 0.1708, R3 ≈ 0.0590, and R4 ≈ 0.0217. These are normalized
mass ratios inside the resolvent family, not the raw resolvent denominators.
3.1 RGE Preservation and Structural Stability (A3 closure)
The resolvent family is built on the Peirce eigenvalue spectrum {λ1 = ϕ, λ2 = 1, λ3 = ϕ−1 }. The
operational d-index assignment used in this paper is (Rev32.9, ruling R-23 : through Rev32.8 this
paragraph said the spectrum “structurally fixes” the assignment; it does not — the assignment is a
selected, post-hoc feature rule, as Paper 5 types d(ν) = 2):
d = 1 → CKM transfer sector,
d = 2 → mν2 /mν3 ,
(9)
d = 3 → mµ /mτ ,
d = 4 → ms /mb .
The A3 closure result is that these dimensionless ratios preserve to sub-percent accuracy under
standard one-loop running from the ultraviolet matching scale to MZ . The reason is representation-
theoretic: same-representation fermions carry identical leading Casimirs, so the anomalous-dimension
corrections cancel in the ratios. That protects the ratios under running; it does not select the
assignment. How the two printed assignments relate (Rev32.9, H202). The table above pairs d = 3
with the ratio mµ /mτ as a label: the scored value 0.059684 is the monomial of Eq. (17), not the
resolvent member R3 = 1/(ϕ6 − 1) = 0.059017. The reversal theorem of §3.2 pairs leptons with
resolvent indices (d = 1 → τ , d = 2 → µ, d = 3 → e). The two are different objects; neither selects
the other.
3.2 Sprint 3: Jordan-automorphism reversal theorem and charged-lepton blind
drop
Session 30 supplies the first nontrivial charged-lepton blind drop built from the full 27 × 27 Al-
bert/Freudenthal machinery. Let
DR = diag(R1 , R2 , R3 ) (10)
2
and let LDR denote Jordan multiplication by DR on J3 (Os ). The Sprint 3 kernel constructs the
Jordan automorphism
2π
UThalf = exp(t∗ Thalf ), t∗ = √ , (11)
5
and verifies the conjugated operator
−1
LMlepton = UThalf LDR UThalf . (12)
Two structural results follow.
First, LMlepton is isospectral to LDR , with the Albert-Freudenthal spectrum
{R1 , R2 , R3 , 12 (Ri + Rj ) × 8}, (13)
verified numerically to a maximum eigenvalue mismatch of 9.01 × 10−14 . Second, the action of UThalf
reverses the diagonal ordering
(R1 , R2 , R3 ) 7−→ (R3 , R2 , R1 ), (14)
so the charged-lepton assignment is forced algebraically to be
d = 1 → τ, d = 2 → µ, d = 3 → e. (15)
This is the UThalf reversal theorem: the heaviest lepton is paired with the smallest resolvent index
because the Jordan automorphism literally reverses the diagonal ordering set by Jvac .
With the single external anchor
mτ = 1776.93(9) MeV, (16)
the exact tree-level muon-to-tau ratio is
√
mµ ϕ 8/3 2
= √ × = 0.059684 PDG: 0.059461, miss + 0.37% , (17)
mτ 5 10
giving
mµ = 106.05 MeV (+0.37% PDG), (18)
where the exponent p = 8/3 = dim(Os )/rank(J3 ) is an exact dimension ratio [A616], read as a topo-
logical winding√ratio (8 octonionic dimensions per 3 generation winding levels), and the normalisation
constant C = 2/10 is read √ off the two-generation sub-algebra J2 (Os ) [A618]: dim(J2 ) = 10 and
off-diagonal trace norm = 2.
Rev29 correction. Both ingredients were previously typed Proved, and the formula was cited
as carrying zero free parameters. That typing is withdrawn. [LIB2-247] Paper 0’s tier/attachment
table already prints the honest row — the exact dimension ratio 8/3 = dim Os /rank J3 is Loaded-
correspondence: “not, alone, a mass exponent” — and the same holds of C. The rows of this
subsection therefore stand at Structural (mathematical tier), Loaded-correspondence (physical
attachment) and Reproduced (numerical status): the ratio is reproduced, not derived. [LIB2-122,
LIB2-247, LIB2-265, LIB2-266] Physical attachment never inherits the mathematical tier; an exact dimension
count is not by itself a mass exponent or a mass normalisation.
Provenance: target-first, and the derivation was never closed. The record runs backwards from the
datum, not forwards from the algebra. [D616:99–102] computes 0.0595/0.4220
√ = 0.141 from PDG and
asks whether the number is algebraic in φ. [A616:50–63] then offers 2/10 at 0.37% while stating
in the same breath that “the algebraic origin is not yet proved.” [A618:47–51] supplies dim J2 = 10
3
afterward, i.e. the dimension is matched to a coefficient already in hand. [A618:134–140] concedes
that the divide-by-dimension step “is argued by analogy.” [A618:147] lists the formal derivation as
BLOCKING — and that item is closed nowhere in the corpus. [LIB2-122]
The +0.37%
√ residual is the fit gap. The coefficient the data require
√ is Creq = 0.140894; the algebra
offers 2/10 = 0.141421. C enters the ratio multiplicatively, so 2/10 / Creq = 1.00374 (recomputed
Rev32.9 with mτ = 1776.93, mµ = 105.6583755 MeV; the pair 0.140900/1.00370 printed through
Rev32.8 reproduces the superseded mτ = 1776.86) — the +0.37% miss in Eq. (17) is precisely the
distance between the required coefficient and the offered one. It is not an independent residual sitting
on top of a closed derivation, and any downstream absorption of +0.37% (next paragraph) is an
absorption of the coefficient gap itself and must be read as such. [LIB2-123]
Rev29 re-tiering: mµ /mτ . Proved / zero-parameter is withdrawn. New typing:
Structural · Loaded-correspondence · Reproduced. Grounds: (i) √ provenance is target-
first (D616 asks the question from the PDG number; A616 supplies 2/10 while recording
that its algebraic origin “is not yet proved”; A618 supplies dim J2 = 10 afterward and concedes
the divide-by-dimension step “is argued by analogy”); (ii) A618:147 lists the formal derivation
as blocking and no component of the corpus closes it; (iii) the +0.37% √ residual is the
multiplicative gap between the required C = 0.140894 and the offered √ 2/10 = 0.141421,
i.e. the fit gap itself. No number is deleted — p = 8/3 and C = 2/10 stand exactly as
printed; what changes is what may be claimed for them. [LIB2-122] This row may not be cited
as zero-parameter.
[Retracted, Rev32:] an earlier sentence here claimed the +0.37% residual is “fully accounted for by
a one-loop QED correction [A637]”. That claim contradicts the authoritative ruling sixteen lines
above (and Appendix O): the +0.37% residual is the fit gap on C, and the QED correction below is a
separate, smaller matching effect (+0.3125%) applied to the tree value — it does not account for the
coefficient gap. [LIB2-123, LIB2-126] The ledger ruling wins. The complete five-step electron derivation is:
(1) mtree
µ = 106.05 MeV from the re-tiered ratio formula above (Structural / Loaded-correspondence
/ Reproduced; Rev29 ). [LIB2-125]
Λ2
(2) QED running: δQED = 3α
4π ln m2
G2
= +0.3125%.
µ,phys
(3) Physical muon mass: mµ,phys = mtreeµ /(1 + δQED ) = 105.7233 MeV (+0.061% PDG; Appendix X
carries the rounded +0.06%). [LIB2-126] Rev32.9: the four digits are needed — the Koide root from
the rounded 105.72 displays as 0.5078, while the chain’s own value gives the 0.5076 below.
(4) The Koide sum rule (K = 2/3) is applied to the physical muon mass at the ΛG2 boundary [A637];
the current attachment carries the Koide value and its branch as external (Appendix O). Rev32.9:
through Rev32.8 this item said the UV boundary condition “forces” the rule.
(5) me = 0.5076 MeV (−0.66% PDG; PDG: 0.5110 MeV).
Koide branch note. Two positive roots satisfy K = 2/3 for the {mτ , mµ , me } system. [LIB2-127]
The Rev29 mass chain selects the light root me = 0.5076 MeV, matching PDG at the −0.66%
level. [LIB2-127] The heavy root (≈ 43.7 GeV) is excluded as outside the charged-lepton mass
spectrum.
4
The Artin bridge (Appendix A §A.7.8) guarantees that the 1-loop QED self-energy is associative
(two-generator subalgebra) and hence exact at this order [A633]. The QED running direction is
unambiguous: mass grows UV in QED, so dividing mtree
µ by (1 + δ) gives the lower physical mass.
The status of the charged-lepton mass sector is now split honestly (authoritative ledger: Appendix O,
“The authoritative charged-lepton ledger”): mτ is the external anchor; mµ follows from the algebraic
ratio formula [A616, A618], re-typed Structural / Loaded-correspondence / Reproduced
above and not citable as zero-parameter (Rev29 ); and me is Koide-consistent ⋆ ⋆ ⋆⋆. [LIB2-265,
LIB2-266, LIB2-268, LIB2-269] The electron mass is determined by the external empirical relation K = 2/3
acting on that mµ chain and the measured mτ anchor [A636, A637]. [LIB2-127, LIB2-267]
3.3 Generation structure as topological winding sectors
The three Peirce idempotents {f1 , f2 , f3 } of J3 (Os ) have a topological interpretation consistent with
the mass ratio formula above. Each idempotent fi is a winding sector of the (4, 4) moire vacuum:
the three generations are three distinct topological winding levels in the split-octonionic fibre. The
√ p = 8/3 then reads as: 8 split-octonionic dimensions per 3 winding levels. The normalisation
exponent
√ transition amplitude: J2 (Os ) = Rf1 ⊕ Rf2 ⊕ P12 , of dimension
C = 2/10 is read as the 2-generation
10, with off-diagonal trace norm 2. Rest mass is therefore the topological energy of a winding
that cannot be removed by continuous deformation: fermions carry a Möbius half-twist in the (4, 4)
vacuum (bosons do not), and this half-twist generates VCW ferm ̸= 0 with minimum at r ⋆ = φ2 . This
picture is a physical narrative consistent with all proved results (see Appendix A §A.7.4); it is not
itself a theorem at the present programme state.
3.4 Light quark constituent masses and the confinement scale
The Quark Gauge-Protection Theorem (Appendix A, Theorem A.7.5) forces the light quarks (u, d, s)
to acquire mass exclusively through the G2 confinement potential ∆VG2 , not through Coleman–
Weinberg radiative corrections. The mechanism that produces this G2 confinement scale at weak
coupling — a spectral singularity of the cubic-norm cone rather than a strong-coupling limit — is
given in Appendix I (Confinement as the spectral geometry of the cubic norm). The constituent mass
formula derived from the chiral gap equation driven by ∆VG2 is:
mconst
u ≈ mconst
d ≈ ΛG2 φ1/2 ≈ 331 MeV, (19)
with ΛG2 ≈ 260 MeV the framework’s hadronic calibration — an external dimensional calibration in
the sense of the Appendix X input-count box, its value taken from hadronic phenomenology; not a
derived scale, and not a measured ΛQCD [A618] (Rev32.9, R84-h). [LIB2-128, P-029] The PDG constituent
mass targets are mu ≈ 336 MeV (−1.57%) and md ≈ 340 MeV (−2.73%; both at the full-precision
value 330.73 MeV, Rev32.9), consistent with the formula at leading order.
The approximate degeneracy md ≈ mu is structural: the (4, 4) sign flip η = −1 for down-type quarks
represents a phase alignment in the pseudo-Riemannian vacuum, not a magnitude suppression. The
physical isospin splitting md − mu ≈ 4 MeV (current masses) is of QED and electroweak loop origin,
outside the tree-level J3 (Os ) geometry.
For the strange quark, incorporating the G2 colour Casimir modifier KG2 /KSU (3) = 3 [A631] via
the G2 Langlands-self-dual geometric mean of the long-/short-root scales (the geometric mean
5
m2s = mshort mlong is exact under G2 root-system self-duality, s194) yields:
!1/4
KG2
mconst
s = ΛG2 φ × = 553.7 MeV, (20)
KSU (3)
placing the strange constituent mass at the threshold of the G2 versus SU (3) colour strength
transition. [A631, A632]
3.5 Top quark: fit-free construction from the Jordan idempotent, carrying a
recorded cross-scheme comparator distance (dcmp = +5.57, no status — Rev32.1,
A1531/T2)
Rev29 correction. This subsection was formerly headed “zero-parameter prediction.” It is a fit-free
construction: the boundary condition yt (MPl ) = 1 is fixed without a fitted parameter, but the resulting
tree value sits +0.87% above the PDG 2026 direct-measurement (MC) mass (172,600 ± 270 MeV; the
cross-section pole mass is a distinct object — Appendix X, Rev32.9), a naive cross-scheme comparator
distance dcmp = +5.57 — experimental-error units only, not a tension and not status-setting
(Rev32.1 retype, A1531/T2; the honesty-ledger BLACK typing of this row is retyped accordingly). A
construction whose scheme-matched significance is not yet computable is not a successful prediction
and is not reported as one here.
The top quark Yukawa coupling is fixed to yt (MPl ) = 1 by the cubic-invariant Yukawa theorem
(Appendix A); the Peirce idempotent derivation in the remark following Theorem A.6 provides a
complementary proof [A632]. What the theorem fixes is the carrier eigenvalue yt = 1; its reading as
the physical top Yukawa at the Planck scale is the declared attachment W01 (Appendix A, interface
marker W01) — an attachment, not a derivation (Rev32.5, A1564). [LIB2-139, P-027] This is a UV
boundary condition. Standard RG evolution carries the coupling to yt (MZ ) ≈ 0.967 in the low-energy
effective theory. [LIB2-139] The tree-level relation
vEW 246.22 GeV
mt = √ = √ ≈ 174,100 MeV, (21)
2 2
should therefore be read as the electroweak-scale tree estimate after inserting the measured Higgs
VEV, not as simultaneous equality of yt (MPl ) and yt (MZ ). [LIB2-121, LIB2-139, P-028] The resulting offset
from the PDG 2026 direct-measurement (MC) mass (172,600 ± 270 MeV; +0.87%; naive cross-scheme √
dcmp = +5.57, no status — Rev32.1 refresh, A1531/T2) is recorded in Appendix X. [LIB2-121] The 2
factor is structurally identical to the off-diagonal norm of J2 (Os ) [A618]: both arise from the Peirce
decomposition of the same algebra.
The suppression of yb ≈ 0.024 and yc ≈ 0.007 relative to yt = 1 is explained by the complement rule
fk ◦ x = 0 for x ∈ Pij (k ̸= i, j): the bare coupling of δj3 to P12 is exactly zero. Physical yb and yc are
generated by the Peirce closure mixing insertion Pij ◦ Pjk ⊂ Pik , which suppresses them by the mixing
amplitude. Their precise values require MS matching in the low-energy field theory. Artin’s theorem
ensures algebraic closure inside the subalgebra generated by any two octonionic elements; this is an
algebraic protection statement, not a theorem about loop diagrams. RG running and renormalisation
are addressed separately in Paper 6 and Appendix D.
3.6 Heavy-quark unification: the 7/3 bridge
The DET-7 invariant n23 = det(Gram(Jvac , Jvac# )) = 7 and the Jordan rank rank(J ) = 3 together
3
force the ratio n23 /rank = 7/3 [A635, A637]. Applied to the heavy-quark sector, this ratio acts as a
6
generation-independent bridge between the lepton and quark mass scales:
mb = 37 mτ = 73 × 1776.93 MeV = 4146 MeV (−0.95% vs PDG 2026; dcmp = −6.64, no status; full precision 4146.
(22)
mc = 73 mconst
s = 73 × 553.7 MeV = 1291.9 MeV (+1.49% vs PDG 2026; dcmp = +4.22, no status).
(23)
Rev29 correction, retyped Rev32.1 (A1531/T2) — report the comparator distance,
and type it honestly. The mb row must be quoted with its offset: mb = (7/3)mτ = 4146 MeV
against PDG 2026 mb (mb ) = 4186 ± 6 MeV is −0.95% and dcmp = −6.64 (cross-scheme, no
status; full precision, Rev32.9). [LIB2-130, LIB2-270, LIB2-271, LIB2-272] Rev29 typed this row
BLACK from the bare σ; the honesty ledger has since retyped those rows as comparator-
distance records (the Rev32 adjudication below), and Rev32.1 propagates that retype to
this box — the earlier text here cited the ledger for a BLACK status the ledger no longer
carries, and compared against the superseded PDG 4183 ± 7. Earlier revisions of this paper
printed only the percentage; the corrected house style is: percent primary, both schemes
stated, dcmp as non-status-setting audit metadata, at this and every other mb site in this
paper. Appendix X states the governing rule: percentage agreement and σ-level tension are
not equivalent when PDG measurements are sub-percent precise. At ±6 MeV on the PDG
2026 value 4186 MeV the bottom mass is a 0.14% measurement, so the −0.95% bridge offset
corresponds to a naive cross-scheme comparator distance dcmp = −6.64 — experimental-error
units only, cross-scheme, not a tension and not status-setting (Rev32 retype, A1531/T2).
The charm bridge value mc = 1291.9 MeV lies +1.49% above the PDG 2026 MS comparator
mc (mc ) = 1272.9 ± 4.5 MeV. [LIB2-131, LIB2-273, LIB2-274, LIB2-275, LIB2-276] Dividing by the
comparator’s experimental error gives a naive cross-scheme comparator distance of +4.22, but
the bridge is constituent-anchored and no constituent-to-MS conversion or scheme uncertainty
is supplied; the number is therefore not a physics tension and does not carry a BLACK
classification. [LIB2-131, LIB2-273, LIB2-274, LIB2-276, LIB2-314] A scheme-matched significance awaits
an explicit conversion with uncertainty.
[LIB2-314]
Both values land in the < 3% band without additional parameters. Under the Rev32 cross-scheme
adjudication the percent offsets are the primary statement; the dcmp figures above are secondary audit
numbers that set no status, and the earlier “5.3σ BLACK” framing of the bottom row is retyped
accordingly (a scheme-matched Zmatched is not yet computable for either row — Appendix X, scheme
table). [LIB2-313] The DET-7 construction supplies the 7/3 bridge as a declared boundary value at the
self-consistent threshold µ = mQ ; reading that value as an MS mass at µ = mQ needs the unresolved
matching/scheme step; no QCD running from ΛG2 is applied (naive running from ΛG2 to mQ gives
∼ 1057 MeV, a factor ∼ 4 below the PDG value [A635]).
Generation-independence of the Peirce closure amplitude is proved ⋆ ⋆ ⋆ ⋆ ⋆ via Peirce idempotent
permutation symmetry: the three idempotents {f1 , f2 , f3 } are algebraically permutation-equivalent,
so no generation-dependent factor can appear in the bare coupling amplitude [A637/Q1B].
Status of the 7/3 bridge: Structural ⋆ ⋆ ⋆⋆. The formal S-matrix proof connecting the Gram-
determinant ratio to the physical heavy-quark coupling amplitude is the one open gap in the mass
sector; it does not block publication and is stated explicitly as an open programme item.
Reconciliation with Paper 7 (framing). The numerical mb , mc values above are the boundary-
value (threshold-matched) realization of the 7/3 ratio. Paper 7 treats the same ratio as an exact
7
high-scale matching boundary (KB = I7 from the cubic Jordan norm; I8 → I7 under SM covariance)
whose low-energy survival is a falsifiable 4321/twin-Pati–Salam prediction rather than a closed
low-energy mass theorem. The two statements are the same open object viewed at the two ends of
the running: 7/3 is exact at the matching scale, and its persistence to µ = mQ as a clean MS-bar
relation is precisely the S-matrix/threshold question named above. The mb , mc entries are therefore
reported as boundary-value predictions, not as a derived low-energy theorem; the Structural ⋆ ⋆ ⋆⋆
tier reflects exactly this. [LIB2-130, LIB2-270, LIB2-271, LIB2-272, LIB2-275, LIB2-313]
(0) √ (0)
Heavy-quark RG/matching audit. The heavy-quark boundary values mt = vEW / 2, mb =
7 (0) 7 const
3 mτ and mc = 3 ms were tested against standard SM/QCD running and matching with no
tuned matching scale. [LIB2-132] The result is negative. [LIB2-132] For the top, treating yt (vEW ) = 1 as an
MS Yukawa and applying the standard pole/MS conversion moves √ the prediction up (the pole mass
exceeds the running mass), the wrong direction; treating vEW / 2 as a pole-like estimate leaves the
+0.9% residual. For the bottom, the self-scale reading leaves the −0.9% residual while running the
boundary value from µ = mτ to mb worsens it; for charm, running from the strange constituent scale
overcorrects downward. [LIB2-132, LIB2-314] Each tension lies on a running curve only for a tuned scale not
fixed by the algebra. [LIB2-132, LIB2-312] Hence the mt , mb , mc tensions are genuine threshold problems,
not ordinary running artifacts (A713), with the required corrections in the top-down/bottom-up
sense. (The originally proposed 16-Goldstone source was subsequently excluded as a dimensional
coincidence, A714; see Paper 4 §4.2.)
Weak-isospin signature and the structural threshold term (S61–S62, A715–A717). The
residual pattern itself is structured. The needed corrections (mt −0.864%, mb +0.961%, mc −1.476%;
Rev32.11, PDG 2026 mt = 172.60, mb = 4186 ± 6 MeV) satisfy |δmt /mt | = |δmb /mb | to 0.67σ of the
PDG uncertainty on mb (|κt − κb | = 0.194 pp against σκb = 0.289 pp, ratio 0.67; at the superseded
comparators mt = 172.56 and PDG 2025 mb = 4183 ± 7 the two agreed to three significant figures,
0.003 pp — that agreement and the 0.148 pp printed through Rev32.10 both rest on that superseded
mt , s1190) with opposite sign, which is reproduced by a single weak-isospin coupling
δmf
= − κ T3 (f ), κ ≃ 1.824%, (24)
mf
the odd combination κb − κt carrying a 0.212 pp spread from the PDG errors (s1182), and by no
other single standard-model charge: electric charge predicts a 2:1 top/bottom ratio (data: 1:1) and
any universal (isospin-blind) coupling gives no sign flip; both are excluded (A715, confirmed by
two independent assessments). Charm, being up-type (T3 = + 12 ), requires an additional generation
factor χc ≃ 1.761 on top of the T3 weighting (not golden-ratio related: 1 + φ−1 = 1.618 ̸= 1.761).
This signature is structural, not derived: the only gauge-invariant operator carrying it, the weak-
adjoint Higgs spurion (H † τ a H) OYukawa
a built from the verified cross-block Yukawa trilinear, is
writable and charge-correct, but J3 (Os ) does not force its relative up/down sign — the conjugation
Hd = H̄u , the vacuum grading Jvac = diag(φ, 1, φ−1 ), and the Peirce structure constants each fail
to supply the required τ 3 weighting, which must be inserted by hand (A716, verified in the exact
A688/A689 √ component map). A boundary-condition revision fares no better (A717): the top boundary
mt = vEW / 2 (yt = 1) is rigid and its residual is not ordinary pole/MS matching, while the soft
down-type bridge revision 73 mτ (1 + κ/2) — numerically excellent, landing on mb (mb ) to 0.002%
with the same κ fixed by the top — reduces no parameter, since no existing algebraic quantity fixes
1 + κ/2. The honest ledger is therefore: bare algebraic boundary conditions plus a phenomenological
structural T3 threshold term, with κ and χc open. The equal-and-opposite t/b pattern is falsifiable:
8
improved PDG precision that breaks |δmt /mt | = |δmb /mb | would remove the T3 reading (Paper 7).
A separate environmental scan (five independent analyses, one with explicitly computed one-loop
self-energies; A719/A719b, S63) finds that known electroweak pole–MS/Yukawa matching effects are
percent-level and scheme-dependent in precisely this sector, with defensible scheme choices spanning
T3 -odd shifts of [−0.8, +3.8] pp around the observed κ ≃ 1.824%; the −κT3 reading is therefore
scheme-conditional — neither firm evidence of structure beyond the boundary conditions nor cleanly
absorbed by standard matching — and the operative falsification target remains the breaking of the
t/b equality at improved precision.
The custodial-odd one-loop matching shift, priced with its own scale ambiguity. The
scheme dependence just bounded can be computed rather than bounded. In the gaugeless limit
the custodial-odd combination ∆odd (µ) := δt − δb of the one-loop Yukawa-sector self-energies is
the T3 -odd shift this sector supplies at one loop, and the tadpole and Higgs wave-function pieces
cancel identically within it (s1186). Evaluated at the registered electroweak anchor µ = v it
gives ∆odd
√ (v)√= −1.4820 % ± 0.6467 %, the quoted error being the one-loop scale ambiguity across
µ ∈ [v/ 2, v 2] at d∆odd /d ln µ2 = −0.933 % per unit. Against the measured odd residual —
κodd ≃ 1.824 %, carrying the 0.212 pp spread from the PDG errors printed
√ above (s1182) — the two
agree at +0.50σ, the σ dividing by the single combined denominator 0.21212 + 0.64672 = 0.6806 %
that carries both errors (s1193). No σ for this comparison is printed on the experimental error alone.
The theory error is 3.05× the experimental one, so this is a consistency statement and not a test;
it becomes a test when the matching is carried to two loops. The scale is declared, not derived:
−κodd is reached exactly at µ⋆ = 295.8 GeV, a 20 % shift off v, and µ = v is not the best-fitting
choice — µ = 2mt agrees comparably, at −0.42σ on that same denominator, so the preference rests
on the distance to exactness and not on a σ gap. Charm is not covered by this combination, χc above
remaining open, and the reading leaves the weak-isospin signature typed a mechanism candidate
rather than a derivation.1
Independent algebraic derivation (S44, A641): An alternative route to the factor 7 bypasses
the Gram calculation entirely. Each off-diagonal Peirce block Pij ∼ = Os has dimension 8. The real
part (dim = 1) decouples into diagonal shifts; the 7 imaginary units of Os (the Fano-plane generators
of the off-diagonal interactions) carry the independent degrees of freedom. Therefore:
7 dim(Im(Os )) 7
= = . (25)
3 rank(J3 ) 3
This derivation requires no reference to Jvac or the Gram matrix. The DET-7 theorem then confirms
n23 = 7 for Jvac = diag(ϕ, 1, ϕ−1 ) specifically [A641/Q2A], making the two routes mutually consistent.
Uniqueness of the golden vacuum (A641/Q5A): For general diag(a, b, c) with abc = 1 the
Gram determinant evaluates to det(G) = (a2 + b2 + c2 )(a−2 + b−2 + c−2 ) − 9. Setting det(G) = 7
forces (a2 + b2 + c2 )(a−2 −2 −2
√ + b + c ) = 16. Among diagonal elements of J3 (Os ) with algebraic-
integer entries over Q( 5) and abc = 1, this condition is satisfied uniquely (up to permutation) by
{a, b, c} = {ϕ, 1, ϕ−1 }. Equivalently, the condition selects√uniquely the diagonal whose characteristic
polynomial is palindromic, i.e. satisfies T1 = T2 = 1 + 5 (self-adjugate symmetry of the golden
vacuum). No other algebraic-integer norm-1 diagonal element of J3 (Os ) satisfies both conditions
simultaneously.
1
s1186 evaluates
√ the matching at a constant yt , where a complete matching calculation wants yt (µ); both the pole
reading yt = 2 mt /v and the suite’s row-15 value yt = 0.967 are carried and the two differ in the third digit of
∆odd . The gaugeless limit drops the hypercharge odd running and the transverse-gauge finite pieces that carry no m2t
enhancement.
9
Freudenthal complementarity (E49): R12 + R23 = 1 exactly, with R12 = ϕ−4 and R23 = 1 − ϕ−4 . This
identity follows from the DET-7 Gram structure and Peirce orthogonality; it supplies the geometric
anchor for the 7/3 lepton–quark bridge.
Cross-sector envelope (scope note). The 7/3 bridge and the resolvent family above are within-
sector relations: they organize ratios inside the charged-lepton and inside the quark sectors. The much
larger cross-sector gap (e.g. mt /me ∼ e12.7 ) is a separate question. The octonion-index decomposition
Os → 1(e0 ) ⊕ 1(e7 ) ⊕ 3 ⊕ 3̄ under SU (3) ⊂ G2 supplies a rigorous lepton/quark band label (leptons and
the Higgs on the colour singlets, quarks on the colour triplets), and a Casimir-weighted localization
SU (3)
cf = c0 + α C2 (f ) would compress the cross-sector envelope to one coefficient. An explicit SU (3)-
structure Dirac-operator analysis shows this form is geometrically natural only under a homogeneous
adjoint-isotropy condition (generic curvature splits weights and the band law fails), while the coefficient
and sign remain free moduli. The cross-sector envelope is therefore reduced to one geometric
modulus under that assumption, not derived; the within-sector mechanisms of this section are
unaffected. See Appendix X (Color/singlet localization envelope) for the full status.
3.7 The clean-lepton residual: the golden-ladder skeleton and the Koide fine-
structure
Write each charged lepton’s golden beat-depth above the electron anchor as
√
φ = 1+2 5 ,
nf = logφ mf /me , (26)
so the electron is depth 0 (the pure-charge anchor) and mf = me φnf . Empirically the three charged
leptons sit close to the integer golden ladder {0, 11, 17}: nµ = 11.08, nτ = 16.94. But the integer
ladder alone is good only to a few percent, and the residual is structured — it changes sign between
generations: the pure integers predict mµ = me φ11 some 3.75% low and mτ = me φ17 some 2.70%
high (opposite signs), so the deviation is not a single uniform warp of the ladder.
The structured residual is the Koide relation. With
me + mµ + mτ
Q= √ √ √ 2 , (27)
me + mµ + mτ
the PDG pole masses give Q = 0.666661, i.e. Q − 23 = −6.2 × 10−6 : the cleanest known charged-lepton
mass relation. Fed the measured electron and muon masses, the constraint Q = 32 predicts the tau,
mpred
τ = 1776.97 MeV (vs. PDG 2026 1776.93 ± 0.09 MeV, +0.002%), (28)
which in golden-depth units is nτ = 16.9448 versus the observed 16.9447 — the Koide constraint
reproduces the tau residual to ∼ 10−4 in depth, parameter-free, where the pure-integer and half-integer
ladders miss by ∼ 0.055. So the gross hierarchy is the integer golden ladder (the charge/winding
skeleton of §3.3); the fine residual on top of it is Koide.
√ √
Geometrically Koide is a statement about the half-depth ladder. Writing mf = me φnf /2 , the
√ √ √
vector ( me , mµ , mτ ) makes a fixed angle with the diagonal (1, 1, 1):
P√
( mf )2 1 1
2
cos θ = P = = ⇐⇒ θ = 45◦ (measured 44.9997◦ ), (29)
3 mf 3Q 2
10
which holds exactly when Q = 23 . The relevant ladder is therefore the half-depth n/2 — the same
spinor “half” singled out by the live time-bridge T 2 = −1 (Appendix H, Theorem H.1): the lepton
mass fine-structure lives at the half-turn level.
Significance. The prediction is unambiguous: Koide(me , mµ ) = 23 has two tau roots, the heavy
one 1776.97 MeV (matches PDG) and a light one 3.32 MeV (excluded). It is not generic: among
random triples whose µ, τ depths lie in the same near-integer band as the data (±0.15 of {11, 17}),
only ∼ 0.05% — about 1 in 2,000 — reach |Q − 23 | < 6 × 10−6 , so the observed agreement is a thin
coincidence on top of the golden ladder, not a by-product of being near integers. Input count: given
me , the two free lepton masses carry two integer-ladder residuals; the ladder is a 0-parameter skeleton
and Koide is one equation that removes the tau residual exactly — 1 of 2 explained, no overfitting.
Honest limits. This is an empirical confrontation: the PDG pole masses and the Koide
relation are external; Koide is not (yet) derived from the substrate. The agreement is recorded
as a structural coincidence on the golden half-depth ladder, not a theorem. Koide removes one
of the two lepton residuals exactly (the tau, given e and µ); the remaining muon integer-offset
nµ − 11 = +0.0795 is the one number Koide does not fix and is left open (µoffset /κ ≈ 13 is
suggestive only, not claimed — three data points are not overfit). No public observable moves
to theorem grade; this strengthens the lepton rest-mass row of the scorecard (electron: Koide-
consistent; muon: Structural/Loaded-correspondence/Reproduced after the Rev29
re-tiering, §3.2, not Proved; tau: anchor) without changing the declared input status of the di-
mensionful scale. Evidence: s108a_lepton_residual.py + _S108a_lepton_residual.json
(keys a2_koide_Q, a3_depth_pred_minus_obs, a4_angle_deg); significance from s108e;
_S108_manifest.
Tiering note (S245). A separate in-house test closes off the cheapest hoped-for completion of this
ladder: an exact single-χ closure of the mass ratios on the occupancy depth support {0, 3, 11, 17} is
refuted at the ratio tier (no admissible bijection; Appendix O, the additive-χ closed-negative, and the
closed-selector ledger of Appendix S). The ordered golden-depth skeleton and the few-percent shape
on it are unaffected — the integer order is structural, the exact completion is not.
4 Resolvent quintics
For publication purposes the suite packages each sectoral mass as the positive real root of a monic
quintic
m
Qd (µ) = µ5 − Rd = 0, µ= , (30)
m⋆
where m⋆ is the relevant sectoral anchor. The quintic language is a bookkeeping device: it keeps the
common algebraic coefficient explicit without pretending that a unique ultraviolet dynamical quintic
has already been constructed.
5 Mass-sector assignments
11
Table 1: Operational mass assignments in Rev29. Rev32.1
reading rule: tiers name mathematical provenance;
physical attachments are stated separately; no row
may be cited as zero-parameter; displayed σ values are
named comparator distances, not profile likelihoods.
Sector Representative Algebraic factor Comment
Top mt yt = √ 1; mt = Fit-free construction, not a
vEW / 2 prediction (Rev29 ); +0.87%
vs PDG 2026, dcmp = +5.57
cross-scheme, no status
(Rev32.1 retype, A1531/T2);
pole/MS running moves
the wrong way — an open
threshold problem (§3.5, §3.6,
App. X), not a clean pass.
This row reads yt = 1 at the
electroweak scale; Appendix A
and Paper 0 read yt (MPl ) = 1,
whose own running endpoint
yt (MZ ) ≈ √
0.967 gives mt =
0.967 vEW / 2 = 168.36 GeV
(−2.46%) — two different
statements, not one (Rev32.9,
A1642 O2).
Strange (con- mconst
s ΛG2 φ(KG2 /KSU (3) )1/4 ≈ 554 MeV; G2 /SU (3) thresh-
stituent) old.
Up/Down (con- mconst
u,d ΛG2 φ1/2 ≈ 331 MeV; −1.57%/−2.73%
stituent) PDG (full precision 330.73).
7
Bottom mb 3 mτ = 4146 MeV 7/3 unification; −0.95% vs
PDG 2026 (mb (mb ) = 4186 ±
6 MeV), dcmp = −6.64 cross-
scheme, no status (Rev32.1 re-
type, A1531/T2; §3.6, App. X).
DET-7 threshold. STRUC-
TURAL ⋆ ⋆ ⋆⋆
7 const
Charm mc 3 ms = 7/3 unification; +1.49% vs.
1291.9 MeV PDG 2026 mc (mc ) = 1272.9 ±
4.5 MeV; dcmp = +4.22 is
cross-scheme audit metadata,
not status. DET-7 threshold.
STRUCTURAL ⋆ ⋆ ⋆⋆
12
Sector Representative Algebraic factor Comment
Neutrino hier- mν2 /mν3 R2 Solar-to-atmospheric ratio
archy channel: R22 = 0.02918 vs.
0.03002 (NuFIT 6.1 IC24
NO, ∆m221 /∆m23ℓ ); R2 sits
1.44–1.79σ below the m1 = 0
NO floor — a below-floor
EDGE target, not a hit
(Paper 5; s1162 certificate,
Rev32.6; s1104/A1546 for the
one-family result). d(ν) = 2
selected, not derived.
STRUCTURAL (ladder tier).
√ √
Charged lep- mµ /mτ (ϕ/ 5)8/3 2/10 Structural / Loaded-
tons correspondence / Repro-
duced (Rev29 ; Proved/zero-
parameter withdrawn —
provenance target-first, formal
derivation blocking at
A618:147, and the +0.37%
miss is the multiplicative
gap
√ Creq = 0.140894 vs.
2/10 = 0.141421, §3.2).
Gives mµ = 106.05 MeV from
mτ = 1776.93(9) MeV.
d-index assign- selected, not Rd = 1/(ϕ2d − 1) ratios RGE-preserved to < 1σ;
ment derived (R-23, the assignment itself is a post-
Rev32.9) hoc feature rule (Paper 5), and
the d = 3 entry for mµ /mτ is
a label, not R3 (§3.1).
Mass schemes and comparison scales
For all mass comparisons the scheme is part of the prediction. Rev29 therefore uses the following
comparison convention.
13
Mass Predicted value Scheme Reference scale PDG com-
parison
me 0.5076 MeV pole / physical on shell −0.66%
charged-lepton mass
mµ 105.7233 MeV pole / physical on shell +0.061%
physical after charged-lepton mass
QED matching
mτ 1776.93(9) MeV measured anchor; pole on shell input
/ physical charged-
lepton mass
mt 174.10 GeV tree-level electroweak scale +0.87% vs.
Yukawa/EW- PDG 2026;
boundary value dcmp =+5.57,
(not pole, not MS; no status
App. X scheme table)
mb 4.146 GeV tau-pole-anchored µ = mb −0.95% vs.
bridge (compared to PDG 2026;
MS, cross-scheme, dcmp =−6.64,
declared) no status
mc 1.292 GeV constituent-anchored µ = mc +1.49% vs.
bridge (compared to PDG 2026;
MS, cross-scheme, dcmp =+4.22,
declared) no status
mconst
s 554 MeV constituent mass ΛG2 hadronic scale structural
proxy proxy
6 Conclusion
The Rev29 mass sector structurally pins the charged-lepton sector with mixed provenance (not a
derivation of all three masses from J3 (Os ) alone) and establishes the 7/3 lepton–quark bridge. The
charged-lepton status is: mτ is the external anchor, mµ follows from the algebraic ratio formula
at Structural / Loaded-correspondence / Reproduced (Rev29 ; not zero-parameter — the
+0.37% miss is the coefficient fit gap, §3.2), and me = 0.5076 MeV is Koide-consistent ⋆ ⋆ ⋆⋆ from
the QED-corrected external Koide chain (−0.66% PDG). The top mass is fixed by the parameter-free
idempotent construction √ (an algebraic identity of the carrier, not a zero-parameter claim about the
observable) mt = vEW / 2, which sits +0.87% above the PDG 2026 value (dcmp = +5.57, cross-scheme,
no status — Rev32.1 retype, A1531/T2) — not a successful prediction (Rev29 ; §3.5, App. X). Bottom
and charm masses are unified via the 7/3 DET-7 bridge at STRUCTURAL ⋆ ⋆ ⋆⋆ accuracy. The
resolvent family is a structural Peirce consequence; the light-quark constituent sector is tied to the G2
confinement scale. The heavy-quark sub-percent residuals are genuine (A713), carry a weak-isospin
T3 signature, and are carried honestly as a phenomenological structural threshold term — neither a
derived carrier (A715/A716) nor a principled boundary-condition revision (A717) is available, and
both searches are recorded as closed negatives. The single remaining formal gap is the S-matrix proof
connecting the Gram-determinant ratio to the physical heavy-quark Peirce closure amplitude; this
14
does not block publication and is stated explicitly as an open programme item. The surviving open
work is concentrated in the bosonic extraction bridge, the AX6 axiom, and the not-derived {κ, χc }
threshold numbers of the heavy-quark sector.
15
Leibniz Quantum Beats Newton
Paper 3 (Rev33.1): CKM and PMNS Mixing in the J3 (Os ) Framework
Tom O’Sieg
August 2026
Abstract
Read this paper on two layers — a Rev29 fold notice. This paper contains two
bodies of work that are easy to mistake for one, and the Rev29 fold prints the distinction
rather than leaving a reader to infer it.
The Rev29 mixing ledger (Sections 2–5) is a√readout-conditioned
√ √ construction. Its
angle relations, its first-row Route B weight 1/ 6 = 1/ 2 · 1/ 3, its β = π/8 Shilov
conjecture (printed as a theorem through Rev32.8 and retyped at Rev32.9: the printed
proof computes γ, not β; §2.2) and its completion map all live downstream of the loaded
Route B correspondence. Where a row in that ledger is marked theorem it is a theorem
given that correspondence and its named readout rule — not a derivation from structure
with no measured input anywhere in the chain. The full 3 × 3 matrix is a completion
map: after the re-tiering of §5 and the A1566 retype, two of eight parameters stand at
PMNS
derived-conditional (θ12 , δCP ), each with its premise printed; the physical θ23 row
is loaded-correspondence (its internal 7/16 identity theorem-grade); θ13 is structural,
|Vcb | is Loaded-correspondence (physical attachment) / Structural formula (A1566),
the CKM first row is loaded, and the construction completes the rest. It is not an
eight-parameter prediction and was never claimed as one.
The structural layer (Section 8, folded at Rev29 from A1471) is upstream of all of that.
There, module naturality forbids generation mixing outright, the framework’s structural
value is VCKM = I, and any observed departure is typed as a controlled defect. This layer
makes no numerical mixing claim at all.
These do not contradict each other, but they are not the same claim, and the
suite has not previously said so in print. A structural layer that gives I and a loaded
readout layer that reproduces measured angles are compatible exactly to the extent that
the loading is honestly declared — which is what the A654 supersession in Section 8 exists
to do, and what the priced-clause register in Appendix X now enumerates.
What the re-tiering settled. That row-by-row audit has now been performed (§5, kernel
s959; A1566 retype, Rev32.6). Ten of eleven ledger rows carry a marking different from their
Rev29 column (the Rev29 hand count was six, treating the CKM first row as one row and
Algebraic as a non-tier). The CKM first row is loaded, because this paper selected Route B
over Route A by comparing both to kaon data; δCKM is coincidence-class while the
motivation for G7 is missing; the physical θ23 row and |Vcb | are loaded-correspondence
(A1566: the internal 7/16 mismatch identity stays theorem-grade; its octant registration is
unselected); and two rows survive at derived-conditional with their premises printed
PMNS
(θ12 , δCP ). No row in Sections 2–5 may be cited as zero-parameter, and the
agreement figures quoted there (1.3%, 1.8%, 0.8σ, 0.62σ) are agreement of a partly loaded
construction with data — not evidence for the structure.
1
This paper collects the Rev29 mixing-sector ledger in a single place. The quark matrix is
organised by Peirce-sector transition amplitudes; the quark Dirac phase is attached to the non-
associative sector; and the leptonic matrix is organised by spectral, Casimir, and Schur-type
projections on the same exceptional-Jordan background. The purpose of the paper is not to blur
the difference between exact algebraic structure and measured benchmark values. It is to state
both clearly.
The Rev29 update has three √ parts. First, the CKM first row is written in its canonical Route√B
PMNS
form with |Vus | =
√ sin θ √
12 / 6√and |Vud | fixed by first-row
√ unitarity. The Route-B weight
√ 1/ 6
factorises as 1/ 2 · 1/ 3 = 1/ 3! (s499): the 1/ 2 trace-form block times the 1/ 3 three-
generation normalisation — a derived product, independent of any kaon fit. [LIB2-050] Second, the
full 3×3 CKM matrix is carried in standard form and all six unitarity relations are
√ recorded explicitly.
PMNS
Third, the PMNS matrix is written with the exact angle set tan θ12 = 3/ϕ2 (PMNS solar
angle; proved as theorem), sin2 θ23 = 7/16 (F4 mismatch geometry; internal sin2 θmismatch = 7/16
theorem-grade; physical octant Loaded-correspondence
√ (A1566); the pre-Rev29 stamp was
THEOREM ⋆100, D575), sin2 θ13 = (3−2 2)/8 (double β=π/8; D572), and the authoritative
Rev29 V3 -rotation phase theorem for δCP . The Jordan-adjoint mixing carrier — called the neutrino
mass operator through Rev28, renamed at Rev29 because the literal mass reading is disproved
in Paper 5 — is defined as Xν = Jvac × Jvac = Jvac #
= diag(ϕ−1 , 1, ϕ) (Freudenthal cross-product,
Rev9 Session 11).
A closing reconciliation (§6, kernels s270–s282) records a hostile in-house audit of this sector
and its in-house adjudication: the parity-graded occupation clockwork and the dynamical cross-
generation spurion are both fenced, but the kinematic geometric derivation of the angles below is
verified to survive that audit; the theorem-grade internal constructions are unchanged (physical
tiers per §5).
1 Introduction
The Rev29 mixing story is a clean separation of source maps. The CKM matrix is the quark-sector
shadow of Peirce-sector transition amplitudes. The PMNS matrix is the lepton-sector shadow of
spectral, Casimir, and Schur-type projections on the same vacuum. The algebraic inputs are common;
the physical maps are not. The use of F4 and J3 (O) for Standard Model classification has independent
published precedent in Todorov and Drenska [1].
2 CKM sector
The CKM first row is canonically fixed by the Peirce J12 block through Route B:
PMNS
sin θ12
|Vus | = √ ≈ 0.22526. (1)
6
With
1 |Vus | |Vcb |
Vcb = √ ≈ 0.041996, Vub = √ ≈ 0.003862 (−0.18σ against the PDG 2026 direct comparator; first-r
9 7 6
(2)
Comparator, named (Rev32.2 ). Against the PDG 2026 direct value |Vub | = 0.00389 ± 0.00016, the
displayed 0.003862 gives −0.18σ under the stated central-value Gaussianization (full precision −0.1752;
the displayed 0.003862 sits on the −0.175 tie; Rev32.9). [LIB2-067, LIB2-215] The historical 0.003855
variant is retained only where explicitly labelled as that alternate rounding; it gives −0.22σ. The
2
former 0.52σ stamp is retired because its comparator was never recoverable from print. first-row
unitarity gives q
|Vud | = 1 − |Vus |2 − |Vub |2 ≈ 0.97429. (3)
Comparator (supplied Rev32 — this row previously had none anywhere in the suite): PDG 2026
|Vud | = 0.97367 ± 0.00032, central-value Gaussianized, places the prediction at +1.94σ — the second-
largest mixing-sector comparator distance in the suite. [LIB2-065] It is printed here so the row is
compared rather than left uncompared; the first-row Loaded typing (§5) applies. The first row is
Loaded: Route B was selected after comparison with kaon data. [LIB2-063, LIB2-237, LIB2-239, LIB2-293,
LIB2-294, LIB2-295, P-017] With the framework solar angle, the canonical current value is |Vus | = 0.225256
√
(the exact framework angle arctan( 3/ϕ2 ) = 33.4880◦ , printed from the kernel line s1162; the rounded
display angle 33.49◦ gives 0.225268, the figure printed through Rev32.4; the Rev32.5 “correction” to
0.225247/33.4877◦ was itself a hand calculation that matched no angle in the chain — the digit we
got wrong twice, corrected Rev32.6 from s1162, A1572), compared with PDG 2026 0.22431 ± 0.00085,
giving +1.11σ under the stated central-value Gaussianization. [LIB2-063, LIB2-237, LIB2-238, LIB2-239, LIB2-293,
LIB2-294, LIB2-295] Feeding the measured NuFIT solar angle through the same formula gives the separate
diagnostic value 0.22687. Against the |Vus | comparator error alone that is +3.01σ; propagating
the NuFIT θ12 error as well gives about +1.0σ (0.97–1.00, house recompute A1641), and only the
propagated figure√ is a distance (Rev32.9). [LIB2-064] These are different readout objects; neither changes
the derived 1/ 6 block weight, and neither upgrades the first-row provenance. [LIB2-064]
2.1 CKM matrix assembly and unitarity check (one angle under-specified)
Read first (tiering). “Assembly” is deliberate: this section inserts the framework’s four
independent entries into the standard unitary parameterisation and checks consistency — so
unitarity holds by construction, not as an independent success. [LIB2-071] The matrix is not fully
closed: the (2, 3) Peirce block has no explicit |Vtd | formula, and the β = π/8 conjecture (§2.2;
a theorem through Rev32.8, retyped at Rev32.9) disagrees with the angle βrecon = 23.44◦
reconstructed from these entries. [LIB2-072] The framework is under-specified here; see §2.2.
√
Using the standard parameterisation with the moduli-preserving √ insertion s13 =
√ |Vub | = |Vus ||Vcb |/ 6,
PMNS / 6 and |V | = 1/(9 7) (Rev32.9: through
s12 = |Vus |/c13 , s23 = |Vcb |/c13 , where |Vus | = sin θ12 cb
Rev32.8 the sines were set equal to the moduli, which drops c13 = 0.9999925 and would display
|Vus | = 0.22525; p the matrix below is ◦the moduli-preserving one; unitarity is unaffected), and
δCKM = arctan 3(ϕ + 5ϕ−5 ) = 68.1297 (Structural Result 2.1, ⋆ ⋆ ⋆; PDG 2026: +0.64σ vs. the
direct UT-angle γ and +1.40σ vs. the global-fit Dirac phase in its published radian representation
(R-29) — two distinct comparator objects, both named in §2.1), the magnitude matrix is
0.97429 0.22526 0.00386
|VCKM | ≈ 0.22512 0.97343 0.04200
. (4)
0.00878 0.04125 0.99911
3
One line, reconciling these magnitudes with VCKM = I (Rev29 ; the reconciliation
requested at S215 and open since). The framework’s structural value is VCKM = I — recovery-
fixed module naturality forbids generation mixing outright (§8) — while the matrix printed
immediately above is a readout-layer object delivered by the loaded Route B correspondence,
and the two are compatible exactly to the extent that the loading is declared: what is
tabulated above is the measured size of the controlled defect from I, read out through a
correspondence this paper types loaded, and it is not the framework structurally predicting
a mixing angle.
The σ figures in this suite are under reconciliation (Rev29 ). They disagree across
components and, for |Vub |, inside this paper. The state is recorded here rather than resolved
by silently picking one, because the failure mode this box guards against is silent divergence.
Comparators, named. Every σ in Sections 2–5 is central-value Gaussianized against a single-
number comparator — historically (Rev29 print) PDG 2025 review direct determinations
for the CKM magnitudes, NuFIT 6.1 (NH, w/SK) for the PMNS rows — and none is a
profile-likelihood confidence level. [LIB2-069, LIB2-070] Where the comparator uncertainty is
asymmetric, the error on the side facing the framework value is used (Rev32.9: through
Rev32.8 this sentence said “the lower error”, which is not what the printed numbers use —
δCKM against γ uses the upper +2.7◦ ). [LIB2-070]
Rev32.2 comparator reconciliation. The current scorecard uses one named PDG 2026
comparator for each CKM object and one rounding convention for each framework value:
|Vus | = 0.225256 (+1.11σ), |Vcb | = 0.041996 (+1.00σ), |Vub | = 0.003862 (−0.18σ).
The measured-angle |Vus | value and the 0.003855 |Vub | rounding remain explicitly labelled
controls. The historical 0.52/0.38/1.3σ stamps are retained only in audit prose and are not
current comparator results.
Rev32 comparator refresh (PDG 2026). The named CKM comparators move: |Vcb |
0.0408±0.0014 → 0.0407±0.0013, taking the prediction 0.04200 from +0.85σ to +1.00σ; |Vub |
0.00382 ± 0.00024 → 0.00389 ± 0.00016, and the deviation changes sign: 0.003862 moves from
+0.17σ to −0.18σ (the 0.003855 variant chain sits at −0.22σ). [LIB2-216, LIB2-230] For δCKM the
suite now names both PDG 2026 comparator objects explicitly rather than pairing a 2024-era
γ-fit with an unnamed global fit: the global-fit Dirac phase δ = 1.154 ± 0.025 rad gives +1.40σ
in that published representation, which is primary (R-29; the degree cells 66.12◦ ± 1.43◦
give +1.41σ), and the direct UT-angle γ = 66.4+2.7−2.8 gives +0.64σ (a numeral comparison: it
scores the inserted δ = 68.1297◦ , while the assembled matrix’s own angle is γ(V ) = 68.0924◦ ,
+0.63σ; Rev32.9) — two distinct objects, each labelled where quoted. [LIB2-069, LIB2-070, LIB2-222]
|Vud | gains its first comparator (0.97367 ± 0.00032, +1.94σ; see the first-row block). The
internal 0.52σ/0.38σ/1.3σ reconciliations above remain OPEN as recorded — this refresh
renames comparators, not the adjudications.
Because the matrix is built from the standard unitary parameterisation, all six unitarity relations
hold identically. [LIB2-071] In particular,
|Vud |2 + |Vus |2 + |Vub |2 = 1, (5)
2 2 2
|Vcd | + |Vcs | + |Vcb | = 1, (6)
2 2 2
|Vtd | + |Vts | + |Vtb | = 1, (7)
4
and likewise for the three column sums. Rev29 therefore carries CKM unitarity as a structural
internal consistency result rather than as a post-fit numerical accident.
3 PMNS sector
The operative leptonic angle relations are (Rev29; θ23 : internal sin2 θmismatch = 7/16 theorem-grade;
physical octant Loaded-correspondence (A1566) — pre-Rev29 stamp THEOREM ⋆100, Session 14,
D575; θ13 STRUCTURAL ⋆ ⋆ ⋆, round-trip framing T 2 , exponent-2; forcing pending [s502/A1275],
A608/E113)
√
3
PMNS
tan θ12 = 2 ⇒ θ12 PMNS
≈ 33.49◦ , (solar angle — THEOREM) (8)
ϕ
7
sin2 θ23 = ⇒ θ23 ≈ 41.41◦ , (F4 mismatch geometry — internal identity theorem-grade; physic
16
(9)
√
3−2 2
sin2 θ13 = ⇒ θ13 ≈ 8.42◦ . (double β=π/8; STRUCTURAL ⋆ ⋆ ⋆,
8
round-trip T 2 exponent-2 [s502/A1275]; A608/E113) (10)
The identity
ϕ2 + ϕ−2 = 3 (11)
shows immediately that
3
tan2 θ12 = = ϕ−2 + ϕ−6 , (12)
ϕ4
so the solar-angle numerator is structural rather than fitted. The derivational status of these
cross-generation angle relations under a recent hostile parity-grading audit is reconciled in §6.
#
3.1 The Jordan-adjoint mixing carrier Xν = Jvac (Rev9, D572; renamed at Rev29 )
Name change, not content change (Rev29 ). Through Rev28 this object was called the “neutrino
mass operator”. Paper 5 shows that the literal mass reading fails by ∼ 9× on ∆m221 /∆m231 (A712,
verified in house), so the physical-mass predicate named an interface this suite does not have. The
definition below is untouched and the object remains valid where it is constructed; Xν is carried as a
mixing/structural carrier only.
Novelty label: new specialization proved here.
Theorem 3.1 (Xν definition and self-duality). The Jordan-adjoint mixing carrier is, at tree level,
#
Xν = Jvac × Jvac = Jvac = diag(ϕ−1 , 1, ϕ),
# ) is self-dual: (J # )# = J
the Freudenthal adjoint of Jvac . The pair (Jvac , Jvac vac vac . Both operators sit
on the Freudenthal cubic N = 1 surface.
Proof sketch (Grok D572, ⋆100). Using the cross-product formula X × Y = X ◦ Y − 12 tr(X)Y −
1 1 2
2 tr(Y )X + 2 [tr(X) tr(Y ) − tr(X ◦ Y )]I with X = Y = Jvac : tr(Jvac ) = 2ϕ and tr(Jvac ) = 4 (using
ϕ2 = ϕ + 1, ϕ−2 = 2 − ϕ). Diagonal entries: (2ϕ − ϕ2 , 1, ϕ−2 + 2ϕ − 2) = (ϕ−1 , 1, ϕ). Off-diagonal
# )# = J
entries vanish because Jvac is purely diagonal. The double-adjoint (Jvac vac follows by the same
calculation with ϕ ↔ ϕ swapped. Both have N (Jvac ) = N (Jvac ) = ϕ · 1 · ϕ−1 = 1.
−1 #
5
Key geometric invariants.
#
⟨Jvac , Jvac #
⟩ = tr(Jvac ◦ Jvac ) = ϕ · ϕ−1 + 1 · 1 + ϕ−1 · ϕ = 3, (13)
∥Jvac ∥2 = ∥Jvac
# 2
∥ = 4, (14)
3 7
cos θmismatch = , sin2 θmismatch = , (15)
4 16
#
Jvac − Jvac = diag(1, 0, −1). (16)
Which premises the angle uses (Rev32.9; A1680, A1704). The equal norms above are those
of the selected vacuum. The mismatch angle needs only their product, sin2 θmismatch = 1 −
⟨J, J # ⟩2 /(∥J∥2 ∥J # ∥2 ), and DET-7 with N (J) = 1 fixes ⟨J, J # ⟩ = 3 and ∥J∥2 ∥J # ∥2 = 16, which gives
7/16 without equal norms. Equal norms ∥J∥2 = ∥J # ∥2 = 4 need in addition (H1), inversion √ symmetry
√ √
of the spectrum (Appendix E): the diagonal J = diag( x, x, 1/x) with x3 = (11 + 105)/4 has
N (J) = 1 and Gram determinant 7, but ∥J∥2 ≈ 3.818 and ∥J # ∥2 ≈ 4.191. The mismatch vector
diag(1, 0, −1) lies entirely in the (1, 3) Peirce sector (middle entry zero), so it directly seeds θ13 .
The internal identity sin2 θ23 = 7/16 (physical octant Loaded-correspondence, A1566) is the
(1, 3)-sector mismatch invariant (tree level, NH-like; NuFIT 6.1 NH (w/SK) global best fit is itself
lower-octant, sin2 θ23 = 0.470+0.017
−0.014 , 2.3σ from the 7/16 branch; the upper-octant local comparator
≈ 0.561 lies ≈ 9.5σ away); ∥Jvac ∥2 = 4 is now a theorem from DET-7+N (J) = 1+(H1) inversion
symmetry (AX2 eliminated, D575; the (H1) premise was not printed here through Rev32.8), and 7/16
follows from DET-7+N (J) = 1 alone through the norm product (remark above) (pre-Rev29 marking
THEOREM ⋆100, superseded — per §5 no row in Sections 2–5 may be cited as zero-parameter;
the row stands at Derived-conditional). The internal mismatch identity sin2 θmismatch = 7/16
is theorem-grade. Its attachment to the physical atmospheric angle retains an octant-registration
ambiguity: the chirality and octant Z2 torsors admit two equally equivariant identifications, and s501
selects negative writhe 7→ lower by convention rather than by a registered action or readout map
(A1566/D1244; s1156). Because the current census records the retained lower branch as selected
against data, the physical θ23 row is Loaded-correspondence under the override of §5. The value
7/16 is unchanged.
sin2 θ13 from the doubled D4 half-angle. The (1, 3) block hosts the D4 outer automorphism R
of Theorem A.2 (the half-angle π/8; its reading as the CKM angle β is the open conjecture of §2.2,
retyped at Rev32.9, and nothing below uses that reading). Applying the half-angle rotation twice —
# (the Freudenthal adjoint carrier) — gives the
once for Jvac (vacuum selector) and once for Xν = Jvac
quartic suppression
√ !2 √
2 4 π 2 − 2 3−2 2
sin θ13 = sin = = ≈ 0.02145. (17)
8 4 8
This prediction is theorem-level conditional on the named readout rule; the suite tier remains
Structural (the unit-endpoint-ray readout and the round-trip T 2 squaring (exponent-2) are not
yet derived from first principles): the D4 triality map 8v ↔ 8s sends the π/4 Cartan angle of Jvac to
the half-angle π/8, and the Jordan-manifold norm squares the resulting suppression to
√
2 4π 3−2 2
sin θ13 = sin = .
8 8
NuFIT 6.1 (NH w/SK, sin2 θ13 = 0.02248+0.00055
−0.00059 ) places the result at 1.8σ.
The leptonic Dirac phase is derived in Rev29 from the V3 rotation theorem:
2π
δCP = − √ ≈ −160.997◦ (JCP ≈ −0.011). (18)
5
6
The free parameter βδ = 11/(6π) is eliminated; three of the four PMNS parameters (θ12 , θ23 , δCP )
carry theorem-grade internal constructions (physically: θ12 and δCP Derived-conditional, the θ23
octant attachment Loaded-correspondence — §5, A1566), with θ13 STRUCTURAL ⋆ ⋆ ⋆ (round-
trip framing T 2 , exponent-2; forcing pending [s502/A1275]) (see Appendix B, §B.5 and Appendix C,
§C.5 for the proofs). NuFIT 6.1 (NH): 0.36σ (central-value σ from the NuFIT 6.1 NH global best fit
212+26 ◦ 2
−36 ; the full ∆χ profile is non-Gaussian; see Paper 7 for falsification discussion).
Theorem 3.2 (DET-7 on the selected vacuum; the Rev29 stamp ⋆100 is
#
⟨Jvac ,Jvac ⟩ ⟨Jvac ,Jvac ⟩
superseded by §5). Let G = 43
# # # = 34 be the Gram matrix of
⟨Jvac ,Jvac ⟩ ⟨Jvac ,Jvac ⟩
# ). Then det G = 4 · 4 − 9 = 7 = n . Proof uses only φ2 + φ−2 = 3 and
(Jvac , Jvac 23
# ⟩ = 3 on the selected vacuum. (Rev32.9: through Rev32.8 this read “both
⟨Jvac , Jvac
forced by AX2” and “n23 = 7 is derived”. AX2 is eliminated by DET-7 (D575), and
Appendix E types DET-7 as a structural postulate whose integer is 7, so det G = 7
is that postulate evaluated on the selected√vacuum, not a derivation of n23 .) The ‘7
√ share the integer n23 = 7. Note: The
structure’: sin2 θ23 = 7/16 and |Vcb | = 1/(9 7)
Rev29 authoritative CP phase is δCP = −2π/ 5 (V3 rotation); 2π/7 was a deprecated
PSL(2, 7) artifact and is not in the current derivation.
Candidate Construction 2.1 (G7 = ϕ + 5ϕ−5 and the retained CKM-phase numeral;
Sessions 22–23, A578+A579; ⋆ ⋆ ⋆). Status first (Rev32.1, A1538 F10): Coincidence-class /
candidate-class — step (B) below is house-proven circular as a derivation; the numeral and its
σ-comparisons stand. [LIB2-068] The CKM Dirac phase is constructed from the sub-leading eigenvalue
correction of J3 (Os ):
G7 = ϕ + 5ϕ−5 ,
q
δCKM = arctan 3(ϕ + 5ϕ−5 ) = 68.1297◦
◦
(+0.64σ from the PDG 2026 direct γ = 66.4+2.7
−2.8 ; +1.40σ from the PDG 2026 global-fit δ = 1.154 ± 0.025 rad)
Census note (R187-b; Appendix X, “N-1 blind formula census”, s1206 v3): in a fixed expression
language of complexity ≤ 6 this form is reached only at the ceiling (complexity 6), where every
in-band value is at least as simple; the bare-ϕ control 65.59◦ (−0.37σ, inside) is reached at
complexity 3 with rank 396. The correction moves δCKM away from the global fit and costs three
units of complexity. The correction 5ϕ−5 is obtained in four algebraic steps (step (B) circular as a
derivation — see the classification note below):
(C) The minimum closed Peirce-graph loop P13 → P13 requires n = 5 steps through Thalf ; Fibonacci
coefficient F5 = 5.
(B) The split-octonion (4, 4) signature of Os forces coupling weights wP13 ,k = 58 wtotal,k : compact
e1 , e2 , e3 cancel; scalar + non-compact e4 –e7 contribute (5 of 8 directions). Weighted resolvent
sums equalize: δMP13 = δMP22 = 5ϕ−4 .
(A) Equal shift ε = 5ϕ−4 on Peirce masses ϕ2 and 1 gives G7 = ϕ + ε/ϕ = ϕ + 5ϕ−5 .
(D) This deformation is the unique O(ϕ−4 ) perturbation of Jvac preserving the Albert-algebra
characteristic-polynomial form (no free parameter inside the deformation; the (c, k) = (5, 5)
choice itself is the menu selection recorded in the control below).
7
CKM magnitudes under the PDG 2026 scorecard: |Vus | = 0.225256 (+1.11σ, first row Loaded),
|Vcb | = 0.041996 (+1.00σ, Loaded-correspondence attachment, Structural formula — A1566),
and |Vub | = 0.003862 (−0.18σ, inherited first-row loading). The pre-Rev29 THEOREM/⋆100
stamps are historical and are superseded by §5; no row in Sections 2–5 may be cited as zero-
parameter. The CKM phase remains Coincidence-class: +0.64σ against the direct UT angle
and +1.40σ against the global-fit phase, with the first-principles origin of G7 open. Zero-correction
control, printed beside it (Rev32.2; √ house s1106, lane A1546 C4, PI ruling S294). The uncorrected
argument X = ϕ gives δ = arctan 3ϕ = 65.587◦ : −0.37σ against the global-fit 1.154 ± 0.025 rad
and −0.29σ (the lower error, the side facing the value) against the direct γ = 66.4+2.7 ◦
−2.8 — closer
than G7 on both comparators. [LIB2-075] The 5ϕ−5 correction (+2.54◦ ) is not supported by current
data, and 28 of its 64 siblings ϕ + c ϕ−k (c, k ∈ {1, . . . , 8}) land within 1σ of the global fit, so the
numeral’s agreement carries no significance beyond the √ √ of the menu; no further (c, k) scans
density
are made. √ The object type of δCKM is an angle (tan δ = 3 G7 has the PMNS-angle functor shape
tan θ = 3 Pk ); the phase-type invariant is the Jarlskog area, and the suite’s Jread = 3.30 × 10−5
is assembled post-insertion of the loaded CKM magnitudes — not an independent prediction
(PDG 2026: J = 3.16+0.13 −5
−0.11 × 10 ). Against that comparator (source-ledger row SL-10) the read
value sits at dcmp = +1.08 (+4.4%; ruling R147). The row is diagnostic: J is a function of
|Vus |, |Vcb |, |Vub | and δCKM , all four already scored, so it adds no evidence and is counted in
no tally of independent successes. Tier stays √ coincidence-class; reopen only by a registered
operator whose eigenvalue or invariant is G7 . [LIB2-316] Rev14 classification note: independent
formal scrutiny (D772/D773, artifact-backed) classifies the 5ϕ−5 correction as numerology-class
pending derivation: step (B)’s equal-shift property was shown to be an algebraic identity of the
construction (house-proven circular as a derivation), and an attempted kill of the integer 5 via
the α(0)/mτ error budget does not land (required coefficient 4.9635 ± 0.087, containing 5 at 1σ;
scheme-dependent). [LIB2-316] The numerical statement and its σ-comparisons are unchanged; the
derivational status is candidate-class — neither derived nor dead.
2.2 The CKM off-diagonal phase: β = π/8 (Rev9, E22–E26; retyped Rev32.9 as an
open conjecture)
Conjecture 3.3 (formerly Theorem 3.3; retyped Rev32.9, ruling R-31). The CKM
unitarity-triangle angle satisfies β = π/8 at leading order, from the D4 (1, 3)-block half-angle
identity. Why this is no longer a theorem. The argument printed below does not establish
β (A1641 F02; house kernel s1169, A1648): (1) Step 3 projects onto z = −Vud Vub ∗ /(V V ∗ ),
cd cb
whose argument is the angle γ, not β: arg z = 68.0924◦ on the matrix this paper prints,
under either insertion; (2) that matrix has β = arg[1/(1 − z)] = 23.444◦ , and none of the
rephasing-invariant angles checked (eighteen, under five transforms) equals π/8; β = π/8
would need δ = 51.62◦ or 83.53◦ at the registered moduli; (3) R(E4 ) = e1 + E4 is null in the
printed plane with coefficient angle π/4, so the halving to π/8 is asserted, not computed.
The numeral π/8 = 22.5◦ is kept as a conjecture with this proof gap. The reactor-angle
construction of §3.1 uses the D4 half-angle as a readout rule (Appendix B) and does not rest
on this conjecture.
The argument as printed through Rev32.8 (its gap is stated in the box above). Step 1: Peirce block
∼ 8s under the D4 triality
identification. The (1, 3) off-diagonal block of J3 (Os ) decomposes as J13 =
action carried by the Spin(4, 4) subalgebra of F4(4) (corrected Rev29 : the compact Spin(8), dim 28,
does not embed in the maximal compact sp(3) ⊕ su(2) of F4(4) , dim 24; the split real form so(4, 4) is
8
the one that acts here), where 8s is the positive-chirality spinor representation. The canonical unit
element is E4 , the highest-weight vector of 8s stabilised by the Jvac vacuum fixed-point condition; E4
is a split imaginary unit with norm N (E4 ) = −1.
Step 2: The outer automorphism R and charge conjugation. The D4 triality outer automor-
phism R exchanges 8s ↔ 8c while fixing 8v . It is the order-two outer automorphism of so(8) — the
diagram involution swapping the two spinor nodes of the D4 Dynkin diagram — and as such lies in
the disconnected component of Aut(so(8)) ∼ = so(8) ⋊ S3 . It therefore cannot be written as exp of a
Lie-algebra element (an exponential lands in the identity component); R is realised concretely by
the discrete triality (diagram) automorphism of Appendix A restricted to the spinor transposition
8s ↔ 8c . In Peirce-frame language R is the charge-conjugation operator C: it swaps particle and
antiparticle quantum numbers within the J13 ∼ = 8s block. The blocks J12 ∼ = 8v and J23 ∼= 8c remain
real under R (triality covariance), so CP violation is sourced exclusively by the J13 ∼
= 8s block.
Step 3: Im/Re ratio and the (4, 4) signature. Acting on E4 , the automorphism gives R(E4 ) =
e1 + E4 , a sum of one compact direction (N (e1 ) = +1) and the original split direction. Projecting
onto the CKM cross-ratio plane z = −Vud Vub∗ /(V V ∗ ),
cd cb
Im(R(E4 )) √
= 2 − 1 = tan π8 . (19)
Re(R(E4 ))
√
The 2 factor has a precise origin: the (4, 4) signature of J3 (Os ) forces R to mix exactly one compact
(N = +1) and one split (N = −1) direction in the highest weight of 8s . This is algebraically
identical
√ to the D4 half-angle formula: tan(θ/2) at θ = π/4 (the natural root system angle) gives
2 − 1 = tan(π/8).
Step 4: All-orders result. First-order CKM: Vub ∝ ⟨e1 |E4 |e3 ⟩ · ∆ and Vcb ∝ ⟨e1 |R(E4 )|e3 ⟩ · ∆,
where ∆ is the eigenvalue difference factor from the Peirce spectrum (ϕ2 , 1, ϕ−2 ). The ratio carries
arg(R(E4 )) = π/8. Higher-order corrections vanish because Jvac is an exact Freudenthal triple fixed
# ,J
point: {Jvac , Jvac 2 −2 cancel
vac } = Jvac (proved in Rev9, Session 8). The diagonal eigenvalues ϕ , 1, ϕ
in the ratio at every order. The printed conclusion was β = π/8 to all orders in the Jordan perturbation
expansion; by item (1) of the box above, the quantity this argument controls is arg z = γ.
[Rev29: the figures in this paragraph are agreement of a loaded construction with data — see the
two-layer notice at the head of this paper and Section 8.]
Shilov gaps. The conjectured tree-level value sin(π/8) = 0.3827 compares to sin βPDG ≈ 0.377 (gap
≈ 1.3%). Separately, |z| = ϕ−2 = 0.3820 vs PDG |z| ≈ 0.375 (gap ≈ 1.8%). Both gaps are confirmed
to share the same E6(6) Shilov-boundary correction origin (Session 8, Gemini+Grok): the framework
predicts the CKM triangle at the Shilov boundary; the PDG values are interior regularisation. The
two gaps are not independent.
Internal consistency note (S45/D642/A642). The D4 derivation above is algebraically indepen-
dent of the other Rev29 CKM elements (|Vus |, |Vcb |, |Vub |, δCKM ). A CKM matrix constructed from
those four elements under the standard parametrisation gives βrecon = 23.44◦ , not 22.5◦ . [LIB2-072]
The discrepancy (0.94◦ ) indicates the framework is currently under-specified: the (2, 3) Peirce block
(eigenvalue ϕ/2) has not yet been mapped to an explicit |Vtd | formula. [LIB2-072] Empirically, the
conjectured π/8 = 22.5◦ is 0.8σ from the PDG direct measurement (B → J/ψK: 22.1◦ ± 0.5◦ ), while
βrecon = 23.44◦ — the angle of the matrix this paper actually prints — is 2.7σ from it. Because
the conjecture is not derived (box above), the 0.8σ scores a numeral, not a theorem; the matrix the
suite prints is the object in 2.7σ tension (Rev32.9). Full-matrix unification via a complete D4 +Peirce
sweep is underway (fullboat kernel Part N, S45).
9
3.2 PMNS unitarity and oscillation consistency (C3 closure)
The PMNS matrix is written in the standard exact form
c12 c13 s12 c13 s13 e−iδCP
UPMNS = −s12 c23 − c12 s23 s13 eiδCP
c12 c23 − s12 s23 s13 eiδCP s23 c13 , (20)
s12 s23 − c12 c23 s13 eiδCP −c12 s23 − s12 c23 s13 eiδCP c23 c13
with cij = cos θij and sij = sin θij . Unitarity is exact by construction: U † U = I. [LIB2-074] Three of
the four PMNS parameters were at theorem level pre-Rev29: θ12 (THEOREM ⋆100), θ23 (THEO-
REM ⋆100), and δCP (THEOREM ⋆100, 0.36σ; A603/E110); after the re-tiering of §5 and the A1566
retype, θ12 and δCP are derived-conditional and the physical θ23 row is loaded-correspondence
(internal sin2 θmismatch = 7/16 theorem-grade; physical octant Loaded-correspondence (A1566)),
with their premises printed. θ13 is STRUCTURAL ⋆ ⋆ ⋆ (round-trip framing T 2 , exponent-2; forcing
pending [s502/A1275]; A608/E113). No fitted coefficients enter the PMNS sector; no row may be
cited as zero-parameter.
4 Operational table
10
Table 1: Rev29 mixing-sector ledger. The Status column
below — and every tier marking in the Note col-
umn, including the GREEN ⋆⋆⋆⋆⋆ stamps and
the phrase “zero fitted parameters” — is the Rev29
marking and is superseded en bloc by the re-tiering
of §5, which propagates each row’s declared derivation chain:
10 of these 11 rows carry a re-tiered marking different from
their printed Rev29 column (Rev32.6 count, generated from
the Result table of §5 by comparing the two columns; the
Rev29–Rev32.5 text said six, counting the CKM first row as
one row and Algebraic as a non-tier), the CKM first row to
loaded because its route was selected against data, δCKM to
coincidence-class, and the physical θ23 row with |Vcb | to
loaded-correspondence (A1566). Two of the eight mix-
ing observables survive at derived-conditional or above;
the internal 7/16 mismatch identity is theorem-grade and
recorded separately. No row in this table may be cited
as zero-parameter. No number in this table changes.
Quantity Algebraic source Status Note
Vus Peirce J12 block Loaded Framework-angle value
(Route √B: |Vus | = 0.225256 (s1162) against
sin θ12 / 6) PDG 2026 0.22431±0.00085:
+1.11σ. The measured-angle
readout 0.22687 is a sepa-
rate diagnostic (≈ +1.0σ
with the NuFIT θ12 error
propagated; +3.01σ against
the comparator error alone).
Route selection used kaon
data.
Vud first-row unitarity after Algebraic |V
pud | =
fixing |Vus | and |Vub | 1 − |Vus |2 − |Vub |2 .
|Vub | hierarchy √ relation Loaded 0.003862 against PDG 2026
|Vus ||Vcb |/ 6 0.00389 ± 0.00016: −0.18σ.
The 0.003855 variant is an
explicitly labelled rounding
control; the former 0.52σ
stamp is retired.
√
|Vcb | 1/(9 7) via DET-7 Loaded (physi- 0.041996 against PDG 2026
‘7 unification’ cal attachment, 0.0407 ± 0.0013: +1.00σ un-
A1566; Struc- der central-value Gaussian-
tural formula) ization.
11
Quantity Algebraic source Status Note
68.1297◦ ; PDG 2026 com-
p
δCKM arctan 3(ϕ + 5ϕ−5 ); Structural
G7 = ϕ + 5ϕ−5 from (⋆ ⋆ ⋆) parators, both named
J3 (Os ) sub-leading (Rev32): +1.40σ vs.
eigenvalue + Os (4, 4) the global-fit Dirac
coupling weights phase 1.154 ± 0.025 rad
(66.12◦ ± 1.43◦ ; R-29)
and +0.64σ vs. the direct
UT-angle γ = 66.4+2.7−2.8 ; the
first-principles algebraic
motivation for G7 remains
incomplete (Coincidence-
class pending G7 , §5).
Bare-ϕ control printed
beside it (Rev32.2): 65.59◦ ,
−0.37σ global / −0.30σ
direct; Jread post-insertion.
CKM first-row |Vud |2 + |Vus |2 + |Vub |2 Algebraic Framework predicts exact;
unitarity the current PDG global fit
shows ∼ 2.3σ tension from
unity (the first-row/Cabibbo
anomaly) — monitoring re-
quired.
PMNS PMNS =
√
θ12 tan θ12 3/ϕ2 Theorem PMNS solar angle (proved
as theorem, Rev9); nor-
mal ordering, conditional
on the Rd ladder (Paper 2
§3, Structural) and on its
identification with the prop-
agation eigenstates (inter-
face undeclared; Rev32.2,
s1108/A1546 C6).
12
Quantity Algebraic source Status Note
θ23 F4 mismatch geometry Loaded (phys- sin2 θ23 = 7/16 ≈ 0.44 (in-
#
Jvac vs Jvac ical; internal ternal sin2 θmismatch = 7/16
identity Theo- theorem-grade; phys-
rem) ical octant Loaded-
correspondence (A1566);
pre-Rev29 stamp THEO-
REM ⋆100, D575); AX2
eliminated: ∥J∥2 = 4 from
DET-7+N (J) = 1+(H1);
the 7/16 identity needs only
DET-7+N (J) = 1 (norm
product 16); NH tree-level;
NuFIT 6.1 NH global best
fit lower-octant (0.470),
2.3σ away; UO comparator
≈ 0.561 at ≈ 9.5σ.
θ13 D4 triality 8v ↔ 8s : Structural sin2 θ√
13 = sin4 (π/8) =
θ → π/8; (1, 3) Peirce (⋆ ⋆ ⋆) (3−2 2)/8 ≈ 0.0214
sector (STRUCTURAL ⋆ ⋆ ⋆,
round-trip framing T 2 ,
exponent-2; forcing pending
[s502/A1275]; A608/E113);
1.8σ NuFIT 6.1; no fit-
ted coefficients, but not
zero-parameter — see the
supersession note in the
caption.
13
Quantity Algebraic source Status Note
√
δCP V3 rotation: Thalf |V√3 3D Theorem δCP = −2π/ 5 =
rotation,√ Ω = 5/2, (⋆100) −160.997◦ (JCP ≈ −0.011);
t∗ = 2π/ 5 0.36σ NuFIT 6.1 (NH);
βδ eliminated; C1 Theo-
rem: c = 12 exact (A558);
Uniqueness Theorem:
Thalf unique up to scale
among G2 -invariant gen-
erators in the (1, 3) sec-
tor (A603/E110); 2/8
PMNS+CKM parameters
at derived-conditional
or above after the §5 re-
tiering, with their premises
printed (θ12PMNS , δ
CP ; the
physical θ23 row is Loaded-
correspondence since
A1566, its internal 7/16
identity theorem-grade); the
pre-Rev29 tally of 6/8 at
⋆100 is superseded.
PMNS unitarity exact by construction Algebraic U † U = I; not an indepen-
dent falsification target.
Stamps in this table ( ⋆100, THEOREM, Proved) are the Rev29-era marking, superseded en bloc by the Rev29 re-tier
(Paper 3 §“Row-by-row re-tiering”); the current tier of each row is the re-tier table’s. Footer added Rev32.5.
5 Re-tiering the Rev29 ledger against the structural layer
The table above was written before this paper carried a structural layer. It now does (§8), and the
two must be reconciled row by row rather than left to a reader to reconcile for themselves. This
section performs that audit. No result below is withdrawn and no number changes. What
changes is what several rows are entitled to be called.
5.1 The rule, fixed before any row was scored
A claim can be no stronger than the weakest link in its own derivation chain:
tier(row) = min tierown , min tier(d) ,
d ∈ deps
with one override that dominates everything else:
14
The override. If any selection in the chain — a route, a branch, an octant, an ordering
— was made by comparing candidates to measurement, the row is capped at loaded and
tagged route-selected-by-data, however exact the surviving formula is.
This is the entire point of the audit. A formula can be exact, its weight can be derived from
first principles, and the claim can still fail to be zero-parameter, because the decision to use
that formula rather than a sibling was made by looking at the answer. In this programme
“zero free parameters” is a claim about the whole derivation chain, not about the last step of
it.
The lattice is proved > derived-conditional > structural > loaded > coincidence-class.
Algebraic is not a tier: a row that is an exact consequence of its inputs adds no independent support
and inherits their tier exactly. The propagation is computed by kernel s959, which also verifies that
no row was re-tiered above its own declared tier — propagation only ever weakens.
5.2 Result
15
Row Printed Re-tiered Binding reason Fixed by (Rev32.2, s1107)
(Rev29)
PMNS
θ12 Theorem derived- normal ordering, assumed: mass ordering
cond. ladder- and interface- (NO | IO), ladder- and
conditional interface-conditional
θ23 , sin2 θ23 = 7/16 Theorem loaded (two exact internal mis- current lower branch selected
⋆100 axes: inter- match theorem; physi- against data; s501 is a non-
nal identity cal octant registration unique candidate map, not a
theorem- unselected selector (A1566/D1244)
grade;
physical
attachment
Loaded-
correspondence)
√
δCP = −2π/ 5 Theorem derived- (1, 3) framing, order- chosen: framing sector (1, 3);
⋆100 cond. ing assumed: Dirac
θ13 Structural structural unchanged; forcing chosen: exponent n =
⋆⋆⋆ pending 4 in sinn (π/8), by iden-
tity/heuristic
√
|Vcb | = 1/(9 7) Algebraic loaded Structural formula; in- by data: the integer 9 (small-
(physical herits the loaded θ23 integer menu; no numeric
attachment, attachment and the in- range was pre-declared —
A1566) teger 9 remains un- s1107 v2 states 1..9 as a lower
forced bound)
|Vus | Algebraic loaded route selected by by data: first-row route (A | B),
data kaon override
|Vub | Algebraic loaded inherits |Vus | inherits the route bit
|Vud | Algebraic loaded inherits |Vus | inherits the route bit
first-row unitarity Algebraic loaded inherits the first row inherits the route bit
δCKM Structural coincidence G7 motivation absent by fit: (c, k) = (5, 5) in ϕ +
⋆⋆⋆ cϕ−k , 64-menu, 28 within 1σ
PMNS unitarity Algebraic proved unchanged; exact by derived: no choice (singleton
construction hom-set)
Stamps in this table ( ⋆100, THEOREM, Proved) are the Rev29-era marking, superseded en bloc by the Rev29 re-tier
(Paper 3 §“Row-by-row re-tiering”); the current tier of each row is the re-tier table’s. Footer added Rev32.5.
10 of 11 rows carry a re-tiered marking different from their printed column (count generated from
the table above, Rev32.6; the earlier hand count of six treated the CKM first row as one row and
Algebraic as a non-tier). 1 row is unchanged (θ13 ), and PMNS unitarity — relabelled from the non-tier
Algebraic to proved — is if anything strengthened, because it was always exactly what it said it was.
The fifth column (Rev32.2; house s1105/s1107, lane A1546 C5, PI ruling S294). For each non-
theorem row it names the discrete choice(s) that cap the row — fixed by data, assumed, chosen, or
derived — with the menu stated in words. Every listed discrete choice affects at least one row. Rows
may inherit more than one choice; in particular |Vcb | inherits both the atmospheric-octant registration
and the independently unforced integer 9, and δCP inherits both the mass ordering and the framing
16
sector (s1107 v2 reverse census, A1566). [LIB2-066, LIB2-285, LIB2-286, LIB2-288] The menus are therefore
dependent, and no additive bit total is asserted. None is a continuous fit: the loaded content of the
sector is a handful of framing and orientation data (a route, an octant registration, an ordering, a
sector label, an exponent, an integer, a (c, k) pair) evaluated through theorem-grade maps. No bit
total is printed here: the nominal count over declared menus is dependent, not additive information,
and lives in the tracker as an √ attack list. One structural fact belongs in print: the√ naive S3 -orbit-
average functor F (θ) = sin θ/ 6 fails on |Vcb | by 6.6× (s1105: F (θ23 ) = sin θ23 / 6 = 0.270031 at
sin2 θ23 =√7/16, which √ is 6.635 times the PDG comparator 0.0407 and 6.430 times the framework’s
own 1/(9 7)), so 1/ 6 is a Peirce-block-dependent weight, not a functor; the object whose rule would
free |Vus |, |Vud |, |Vub | and |Vcb | together is that block-weight rule, and it is not registered. [LIB2-073]
5.3 The two findings that matter
1 · The CKM first row is route-selected. This paper states it in its own words:
“Route B is the canonical first-row prediction because it reduces the tension with direct kaon
data to 0.62σ.” Route A gave 0.22749, a 3.7σ tension, and was deprecated.
√ √
The 1/√ 6 weight is separately derived
√ and that derivation stands (s499: 1/ 2 trace-form
× 1/ 3 three-generation = 1/ 3!). But a derived weight sitting inside a route that was
chosen by comparing both routes to the measurement is not a zero-parameter prediction. [LIB2-
050] The free parameter is not in the formula; it is in the choice of formula.
|Vus |, and everything the first row builds on it — |Vub |, |Vud |, first-row unitarity — is
therefore loaded. [LIB2-287] This is the same defect the A654 supersession names in §8, found
independently and one layer further down.
2 · δCKM cannot hold a structural rating. The table’s own note reads: “the first-
principles algebraic motivation for G7 remains incomplete.” An expression in powers of φ
— G7 = φ + 5φ−5 — agreeing with measurement at 1.62σ, whose motivation is admitted to
be missing, is precisely what this programme’s standing rule against numerological support
exists to catch. It drops to coincidence-class and stays there until G7 is derived. The
number is not withdrawn; the rating is.
[LIB2-253, LIB2-316]
5.4 What survives, and it is not nothing
Two physical rows sit at derived-conditional, each with its premise printed beside it rather than
PMNS and δ
buried: θ12 CP . [LIB2-060]
√
δCP = −2π/ 5 is the strongest physically attached claim in this paper and the audit leaves it there.
It has the C1 theorem c = 12 exact (A558) and a genuine uniqueness theorem — Thalf is, up to scale,
the unique G2 -invariant√generator in the (1, 3) sector (A603/E110); the scale is the one that gives it
the V3 frequency Ω = 5/2 (Rev32.9). [LIB2-052] A uniqueness theorem is exactly what separates a
derivation from a coincidence, and no other row in the ledger has one.
The separate internal identity sin2 θmismatch = 7/16 remains a Jordan-geometry theorem: DET-7
+ N (J) = 1 fix ⟨J, J # ⟩ = 3 and the norm product ∥J∥2 ∥J # ∥2 = 16, which eliminated AX2 (D575)
and gives 7/16 as a corollary (the equal norms ∥Jvac ∥2 = 4 need (H1) inversion symmetry in addition,
Appendix E; Rev32.9) (no new free parameter enters at this step; the row itself is not citable as
17
zero-parameter, §5). [LIB2-356] A1566/D1244 finds that s501 does not uniquely register that mismatch
to the physical lower atmospheric octant: the chirality torsor and the octant torsor admit exactly
two equivariant bijections and s501 chooses one by convention. [LIB2-002, LIB2-356] The current physical
θ23 row is therefore Loaded-correspondence; the number and the internal theorem do not
move. [LIB2-002, LIB2-004, LIB2-048, LIB2-235, LIB2-240, LIB2-241, LIB2-243, LIB2-283, LIB2-284]
One tension that must print beside its tier, not beneath it. sin2 θ23 = 7/16 = 0.4375
sits 2.3σ from the NuFIT 6.1 NH global best fit of 0.470. That is real falsification pressure
on the framework’s sharp loaded lower-branch target, and a reader meeting a ⋆100 label
should meet the tension in the same breath. [LIB2-002] It is recorded here as a live falsifier,
not a footnote. (The upper-octant comparator at ≈ 0.561 lies ≈ 9.5σ away, so the selected
correspondence is at least on the correct side of the octant question.) Exclusion attacks the
current physical correspondence; it does not by itself refute the internal mismatch identity
unless the readout attachment is first derived (A1566).
5.5 The conclusion sentence does not survive
This paper’s conclusion states that “six of the eight mixing parameters are theorem-level”. After
propagation, two of eight sit at derived-conditional or above: θ12 PMNS and δ
CP (Rev32.5 printed
three, with θ23 ; A1566/D1244 retyped the physical θ23 row, Rev32.6). The theorem-grade internal
mismatch identity 7/16 is recorded separately; its physical atmospheric-octant attachment is Loaded-
correspondence. The sentence is corrected accordingly, and the corrected count is printed rather
than the claim quietly dropped.
What this audit did and did not do. It did not re-derive anything, refute anything, or
move any observable. Every formula, every number and every agreement figure in the Rev29
ledger stands exactly as computed. What the audit changes is the vocabulary: which rows
this programme is entitled to call theorems, which are conditional and on what, which are
loaded, and which are for now coincidences with the algebra still owed. The agreement figures
quoted in §§2–5 — 0.62σ, 1.62σ, 1.8σ, 0.36σ — are agreement of a partly loaded construction
with data. They are not, and should never have been read as, independent evidence for the
structure.
6 Derivational status under Definition A: a hostile-audit reconcilia-
tion (s270–s282)
An in-house adversarial cycle (kernels s270–s283; the occupation side is folded in Appendix M)
tested whether the Standard-Model generation slots, the gauge groups, and the cross-generation
mixing of this paper can be derived from the pure bulk algebra with no phenomenological input,
and concluded they cannot, recommending that the PMNS/CKM angles be reclassified as empirical
boundary inputs and that the geometric V3 derivation be downgraded to coincidence-class. We have
verified that recommendation against the banked framework and partially accept it: the two genuinely
new honest-negatives are banked, but the proposed downgrade of the geometric angle theorems does
not go through, for the reasons recorded below. No public observable moves and no theorem rating in
this paper changes.
18
Banked (the valid honest-negatives). (i) Occupation is not bulk-forced (s270–s272): the slot
set {0, 3, 11, 17} is not fusion-closed (condensation fails), is no gapped-boundary subgroup of the
achiral double D(Z60 ) (none of the 12 Lagrangian algebras match), and is not reachable by any
bulk Jvac symmetry-breaking potential (no per-rung tuning). (ii) The weak sector is not native to
the diagonal frame (s273): the Eilenberg–Moore category of the Reynolds monad Pcode isolates the
Weyl-singlet diagonal (+, +) frame but annihilates the off-diagonal (−, −) Peirce spaces where the
non-commuting SU (2)L generators live, so SU (2)L is not extracted from the invariant frame. (iii) The
dynamical cross-parity spurion is fatal (s275–s282): using the odd defect Oodd = 12 diag(1, 0, −1)
as a propagating background VEV to bridge generations clears the kinematic hurdles but cannot
source the θ13 ≪ θ12 hierarchy (the golden weights on the two odd rungs differ by only ≈ 1.17),
and is killed outright by charge conservation in that mechanism. These sharpen the input ledger
(Appendices M, N, X): the occupation slots and the colour real-form remain named inputs.
Verified to stand (why the geometric downgrade is declined). The audit’s central move —
that an exact parity grading forces a block-diagonal mass matrix and hence θ12 = θ13 = 0 — does not
bind the derivation of §3, on three independent grounds.
(i) Layer separation. The parity-Z2 grading is a property of the discrete Z60 /Z2 occupation clock
and governs which rungs are stable (Appendix M, Theorem on the exact universal grading, s267).
The angles of §3 are fixed by the continuous Jordan/Freudenthal geometry (Jvac , Jvac # , the D
4
triality R, and the V3 rotation exp(t⋆ Thalf )Jvac = Jvac
# ). This paper invokes the parity grading
nowhere; importing a discrete occupation-stability rule as a selection rule on the continuous
mass geometry is a category error.
(ii) Already adjudicated in-house. The framework has separately established and recorded (s269,
Appendix M) that parity is falsified as a mixing rule — about two-thirds of the empirical CKM
weight, including |Vcs | ≈ 0.973 and |Vtb | ≈ 0.999, sits in parity-forbidden entries. “Parity forbids
cross-generation mixing” is therefore a position the project already rejected.
(iii) The hierarchy is the wrong target object. PMNS mixing is the misalignment of the neutrino
# with the charged-lepton basis, not charged-lepton↔charged-
flavour basis carried by Xν = Jvac
lepton mass mixing; parity protection of the charged-lepton mass eigenstates does not constrain
that misalignment. The mismatch diag(1, 0, −1) enters §3 as a kinematic angle between two
diagonal vacuum frames, not as an EM-charged propagating VEV, so the hypercharge/photon-
mass obstruction of s282 — which applies to the dynamical insertion, and uses a hypercharge
operator not defined in this suite — does not reach it. (The solar angle is moreover large and
well-measured, θ12 ≈ 33.4◦ ; a literal θ12 = 0 prediction would contradict both data and the
framework’s own result.)
Definition-A ledger (mixing sector). Derived/structural (unchanged): the geometric
PMNS/CKM angle relations of §3 (kinematic, continuous-geometry; survive the s270–s282
audit) — with the two-axis split of A1566: internal geometry, theorem-grade sin2 θmismatch =
7/16; physical PMNS attachment, Loaded-correspondence pending a canonical octant reg-
istration — and the parity-Z2 generation grading (occupation-stability bedrock; Appendix M).
Fenced inputs: the occupation slots {0, 3, 11, 17} (Appendix M), the colour real-form / gauge-
group assignment (Appendix N), and any dynamical cross-parity mass-mixing mechanism
(the Oodd spurion route is closed, s282). The verification adjudication is recorded for recheck
in the session notes; no public observable moves.
19
7 Conclusion
Rev29 presents the mixing sector as a precise set of source maps rather than as a bundle of numerically
isolated coincidences. The full 3 × 3 CKM matrix is best read as a completion map: two of the eight
mixing parameters sit at derived-conditional or above after the §5 re-tiering (θ12 PMNS , δ
CP , with
their premises printed; the theorem-grade internal mismatch identity 7/16 is recorded separately
and its physical θ23 attachment is Loaded-correspondence, A1566), θ13 is structural, |Vcb | is
Loaded-correspondence (physical attachment) with a Structural formula, the CKM first row is
loaded because its route was selected against data, δCKM is coincidence-class pending G7 , and
the standard-form construction completes the matrix from those inputs — it is not an independent
eight-parameter prediction. The CKM matrix is structurally unitary once the Peirce-derived angle set
is specified. The PMNS matrix is exactly unitary by construction. The open question is not whether
these matrices can be written down, but whether upcoming oscillation and flavour data continue to
track the particular algebraic map selected by the exceptional- Jordan framework.
References
[1] I. Todorov and M. Drenska, “Octonions, exceptional Jordan algebra and the role of the group
F4 in particle physics,” Adv. Appl. Clifford Algebras 28 (2018) 82, arXiv:1805.06739.
8 The structural home for VCKM = I, and a supersession
This paper has carried VCKM = I as a posture for several revisions: a statement of what the framework
does not derive, held in place because the alternative was to assert a mixing structure the derivation
did not support. At the Rev29 fold that posture acquires a structural reason, and one older claim is
formally superseded.
8.1 The controlled-defect frame
The module-Dirac census of A1471 (house-verified at s954) establishes the following under one named
premise.
Premise (recovery-fixed module naturality). The physical finite Dirac operator is a
fixed point of the recovery channel in the Heisenberg picture, and is natural for the exact
Fibonacci parent module action.
Theorem (A1471, forced under the premise). The recovery channel’s Heisenberg
fixed points form a 9-dimensional space; exact Fibonacci-module naturality cuts this
to exactly 3, spanned by D̂(A, B, C); and [DF , Agen ] = 0 — the finite Dirac operator
commutes with the generation algebra. The visible compression commutes with the census.
The immediate consequence is the one that matters for this paper. Strict module naturality forbids
generation mixing altogether. The mixing space is 3C /3R with Hilbert–Schmidt gap weights φ−2 , φ−4 , 1,
and every nonzero mixing is a departure from the naturality that made the three-family structure
forced in the first place.
20
Mixing is therefore not an output of this framework and not an omission from it. It is a controlled
defect: a measured deviation from an exact structural constraint, whose cheapest instance is the
(1 ↔ φ−1 ) link with penalty 12 φ−4 ε2 ∥K∥2 (Appendix X).
What this is not. This is not a derivation of the CKM matrix, and nothing here predicts a
mixing angle. The framework’s printed value remains VCKM = I, and the defect statement
is naturality-conditional — it inherits every condition of the premise above. The correct
reading is: the framework now says why it gives I, and what kind of object the observed
departure from I would have to be. It does not say how large that departure is. The φ
powers in the gap weights come from the module structure the theorem is stated in; they
are not numerological support for anything, and the programme’s standing rule against such
support applies here in full.
8.2 Supersession: the A654 zero-parameter CKM claim
At S46 an assessment (A654) recorded a complete, zero-free-parameter CKM determination
via the J12 ◦ J13 → J23 route, quoting |Vtd | and |Vts | at 0.25σ and 0.05σ. That claim is superseded.
It is superseded rather than retired because the underlying computation was real and is still on file.
What was wrong was its type. A zero-free-parameter claim in this programme means a quantity fixed
by structure with no measured input anywhere in its derivation chain. The route A654 used inherits
the loaded Route-B correspondence, whose six formulae are readout-conditioned; θ13 and δCKM sit
behind that conditioning, and CKM-β was never resolved. A result that agrees with experiment
to a quarter of a standard deviation while carrying an unaudited loaded correspondence is not a
zero-parameter derivation — it is a good fit whose parameters are somewhere else.
The arrow, printed once. A654 (S46, zero-free-parameter CKM, J12 ◦ J13 → J23 ) −→
the Rev29 posture: VCKM = I with mixing typed as a controlled defect under recovery-fixed
module naturality. The superseded claim may not be cited as live, and no revision of this suite
prints a zero-parameter CKM number. This is a statement about what “zero free parameters”
is allowed to mean here: it is a claim about the whole derivation chain, not about the last
step of one.
21
Leibniz Quantum Beats Newton
Paper 4 (Rev33.1): Higgs Sector and Electroweak Symmetry Breaking
Tom O’Sieg
August 2026
Abstract
The Higgs sector of the current J3 (Os ) programme is no longer presented as an isolated
Yukawa note. In the current reading it sits inside a larger algebraic chain whose first step is
already physically restrictive: the empirical equivalence of the (1, 3) and (3, 1) Lorentz-signature
conventions motivates a neutral (4, 4) inner product — an adopted premise (a stated model choice,
not a forcing argument; Paper 1, Rev29), and the unique real normed composition algebra with
that norm is the split-octonion algebra Os . Imposing a three-sector (rank-three) structure then
selects the 3 × 3 Hermitian Jordan algebra J3 (Os ) over Os — the generation count 3 is an external
input (algebraic replication into 3 × 16 is refuted; A691/A693/A694), and the rank-three Jordan
structure realizes, rather than derives, it. The Higgs sector is therefore not built on a merely
convenient algebraic choice; it is attached to the same adopted signature-neutral premise (plus
two genuine uniqueness steps) that underwrites the rest of the suite.
At the local level the same-block carrier coupling (Peirce unit-eigenvalue theorem, A674; the
physical SM Higgs Hu,d sits in the vector 10, and the literal Yukawa is the cross-block triple of
§Paper 6 §4.4) is the cubic invariant
Lcarrier ∝ Tr Jvac ◦ (Ψ × ΦH ) . (1)
With Jvac = diag(ϕ, 1, ϕ−1 ) this projects onto the middle Peirce idempotent and fixes the tree-level
boundary value yt (MPl ) = 1. The same Rev29 upgrade also integrates the corrected Cayley–
Dickson split-octonion multiplication and SplitCD inner product, the theorem-level vacuum
selector Jvac = diag(ϕ, 1, ϕ−1 ), the D=5 to D=4 architecture E6(6) /F → E7(7) /SU (8), and the
CKM+PMNS mixing ledger at its Rev29 re-tiered statuses (2/8 Derived-conditional or
above since the A1566 retype, Rev32.6; CKM first row Loaded; physical θ23 and |Vcb | Loaded-
correspondence; δCKM Coincidence-class).
What remains open is stated plainly. The D=5 scalar manifold E6(6) /F4(4) is pseudo-
Riemannian, while the D=4 E7(7) /SU (8) target is Riemannian and ghost-free. The F4 -invariant
tree-level potential has an exactly flat eigenvalue-ratio direction, so the golden-ratio hierarchy is
not selected at tree level and must come from a beyond-tree mechanism such as Coleman–Weinberg
lifting. The diagonal-vacuum weak-angle route is dead; the surviving electroweak route is the
non-diagonal exact-color orbit with algebraic target sin2 θW = ϕ−3 and one remaining selector
problem. Rev29 is therefore the honest EWSB statement of the suite: the Higgs geometry and the
cubic Yukawa are sharp, the mixing sector carries no fitted coefficients (θ13 and δCKM readout-
conditioned, the CKM β consistency open; Paper 3), and the weak-angle selector plus the full
bosonic extraction remain the principal open tasks.
1
1 Introduction
Paper 4 has a narrower purpose than Papers 0, 3, and 5. It does not attempt to re-derive the entire
architecture of the programme. Its job is to state the Higgs/EWSB layer in language a referee can
audit quickly: what the algebra fixes exactly, what the D=5 and D=4 scalar targets are, where the
ghost question is solved, what happened to the weak-angle story, and which pieces are still open.
Two changes since the old Rev7 paper are decisive. First, the upstream algebra has been corrected
and sharpened: all split-octonion products are now taken in Cayley–Dickson form, the SplitCD inner
product is the one compatible with the non-compact real form, and Thalf ∈ f4(4) is kernel-verified.
Second, the flavour sector is no longer an uncertain backdrop. Two of the eight CKM and PMNS
mixing parameters survive the Rev29 row-by-row re-tiering (Rev29 re-tiering, Paper 3 § “Row-by-row
re-tiering”; kernel s959; A1566 retype, Rev32.6) at Derived-conditional or above (θ12 PMNS , δ
CP ),
with no fitted coefficients inside the correspondence map; the CKM first row is Loaded because
its route was selected against kaon data, δCKM is Coincidence-class, θ13 is structural, and the
physical θ23 row and |Vcb | are Loaded-correspondence (the internal 7/16 mismatch identity stays
theorem-grade). The Higgs paper therefore sits on a firmer algebraic floor than any earlier revision.
2 Algebra selection and the vacuum theorem
The algebraic core is selected by a short chain of physical requirements.
(1) The metric conventions (1, 3) and (3, 1) describe the same Lorentzian physics.
(2) We therefore adopt, as a stated model choice, a convention-neutral algebraic framework accom-
modating both simultaneously — a neutral (4, 4) norm form. This is an adopted premise, not a
forcing argument: nothing proved here excludes a convention-fixing framework (Paper 1, corrected
form).
(3) Given that adoption, the composition-algebra demand then selects Os : it is the unique real
normed composition algebra with signature (4, 4).
(4) Imposing a rank-three (three-sector) structure then selects the 3 × 3 Hermitian Jordan algebra
J3 (Os ); the generation count 3 is an external input (replication refuted, A691/A693/A694),
realized but not derived by the rank-three Jordan structure.
No Higgs-sector statement in the present paper is prior to this selection principle.
The working vacuum is now carried at theorem level in the programme state rather than treated as a
freely fitted ansatz. √
−1 1+ 5
Jvac = diag(ϕ, 1, ϕ ), ϕ= . (2)
2
It is carried as the unique fixed point of the modular element M1 = T 3 S ∈ P SL(2, Z) ⊂ F4(4) acting
on the Peirce diagonal, with the companion Freudenthal fixed-point relation
#
{Jvac , Jvac , Jvac } = Jvac . (3)
The diagonal vacuum diag(ϕ, 1, ϕ−1 ) is proved at theorem level in session-level derivations [A641/Q5A:
uniqueness for n23 = 7]; the proof is not reproduced in the present suite and is the subject of a
forthcoming appendix. In the Higgs context this matters because the vacuum is no longer an
unconstrained ansatz whose role is merely to make the Yukawa extraction work.
2
3 Scalar manifold, Higgs placement, and local cubic coupling
The correct D=5 scalar manifold is
E6(6) /F4(4) , (4)
not F4(4) /K. The dimension count is 78 − 52 = 26, matching the split-magic-supergravity scalar
sector built from the N (h) = 1 constraint on the 27 of J3 (Os ). F4(4) is the isotropy group, not the
kinetic numerator.
Two readings must be kept distinct. In the carrier reading (the same-block construction of A674), a
Higgs-like field ΦH and a fermion doublet Ψ are placed in the same (1, 3) Peirce block; this is the
unit-eigenvalue carrier theorem, not the literal SM Higgs placement. The physical SM Higgs doublets
Hu,d instead live in the vector 10 = J12 ⊕ Rc1 ⊕ Rc2 , and the literal one-generation Yukawas are the
cross-block trilinear {J12 (H), J13 (ΨL ), J23 (ΨcR )} (A686–A689; Paper 6 §4.4). In the carrier reading
the full (1, 3) block is eight-real-dimensional, so the carrier doublet is not identified with the whole
block. Under
Spin(1, 3) × Spin(3, 1) ⊂ Spin(4, 4), (5)
the 8+ spinor decomposes into the same-chirality pairing
8+ −→ σ1 = ( 12 , 0)1 ⊗ ( 12 , 0)2 ⊕ σ2 = (0, 12 )1 ⊗ (0, 12 )2 . (6)
The carrier doublet is identified with one four-real-dimensional block; the other four real dimensions
are the orthogonal completion in the split-signature pair, not a second doublet. The dimensional
mismatch is therefore resolved by the (1, 3) + (3, 1) signature split, not ignored. The same-block
carrier coupling (the Peirce unit-eigenvalue theorem of A674, not the literal SM Yukawa) is
Lcarrier ∝ Tr Jvac ◦ (Ψ × ΦH ) , (7)
where the Freudenthal cross product is defined using the corrected Cayley–Dickson multiplication
and the SplitCD inner product. (The literal 16 × 10 × 16 SM Yukawa is the cross-block triple above,
with Hu,d ∈ 10.) For traceless (1, 3) fields one has
Ψ × ΦH = −2 Re(ψ h̄) e2 , (8)
so contraction with Jvac extracts the middle eigenvalue λ2 = 1 and yields the exact tree-level result
yt (MPl ) = 1. (9)
This remains a tree-level eigenvalue statement. Standard one-loop running carries it to the low-energy
effective theory,
dyt 1
= βt (yt , g1 , g2 , g3 , . . .), yt (MZ ) ≈ 0.967. (10)
d ln µ 16π 2
The same Peirce geometry also fixes the colour-safe subgroup placement used elsewhere in the suite:
SU (3)c ⊂ Sp(3) ⊂ K = Sp(1) × Sp(3) /{±1}. (11)
This becomes relevant again in the weak-angle discussion, where the exact-color slice is the only
surviving bosonic route. The anomaly-cancellation check for the identified Standard Model matter
content is not repeated here; it is closed in Paper 1 (the s588 audit, formally closing hostile-board
item T3-P) and in Appendix F under the explicit B + SU(5) map.
3
4 Ghost-freedom and the D=5 to D=4 architecture
The ghost question must be stated on two levels.
4.1 D=5 intermediate level
At the D=5 stage the scalar target is E6(6) /F4(4) with tangent-signature
(14, 12) (12)
on the traceless 26-dimensional sector. Equivalently, on the full 27-dimensional Jordan space the
quadratic form has signature (15, 12). This is the honest intermediate statement: the D=5 level is
pseudo-Riemannian and contains ghost-like directions if one mistakes the intermediate tangent metric
for the final physical scalar metric.
4.2 D=4 final level
After reduction on S 1 , the split-magic scalar count follows the chain
26 (split-magic D=5 scalars) + 27 (dualized vectors) + 1 (KK dilaton) = 54. (13)
The group-theoretically exact D=4 target of maximal N = 8 supergravity is
E7(7) /SU (8), dim = 133 − 63 = 70, (14)
where the standard formula uses 42 D=5 scalars from the maximal D=5 manifold E6(6) /U Sp(8)
(dim = 78 − 36 = 42), not the split-magic manifold E6(6) /F4(4) (dim = 78 − 52 = 26). The discrepancy
is
70 − 54 = 16 = (42 − 26) scalars outside the Jordan-algebra sector. (15)
These 16 additional degrees of freedom are present in maximal supergravity but are not directly
produced by the J3 (Os ) reduction. Their identification is an open architecture task (T1-N). SU (8)
is the maximal compact subgroup of E7(7) ; all 70 coset generators are non-compact and the coset
metric is positive-definite, so the final D=4 maximal target is Riemannian and ghost-free.
(Superseded, S73/S77 note: the identification is no longer open. On the certified E7(7) substrate the
s7→s·I
missing 16 is pinned — it is the identity-anchored Jordan complement (b) = 42 ⊖ 26 = ker p42 −−−−→
27 , orthogonal to the E6 /F4 orbit-tangent 26, and selected over the SO(5, 5)-spinor look-alike by the
kernel-identity lock (the two candidate 16-planes lie in different U Sp(8) orbits, stabilizers 12 vs 20).
The S77 potential layer then computes the dynamics of exactly this sector; see Appendix F. The T1-N
identification task is retired to the historical record.)
Status of the two-16 identification (S60, A714). A natural conjecture identified this 16-scalar
gap with the 16 broken generators of the vacuum-stabilizer breaking Spin(4, 4) → Spin(1, 3)×Spin(3, 1)
(28 − 12 = 16), via a putative branching 42 ↓ F4(4) = 26 ⊕ 16. This identification is now excluded
at the group-theory level (verified independently in house by two methods). First, F4 has no 16-
dimensional irreducible representation—its dimension spectrum begins 1, 26, 52, 273, . . . —and the 42
is the U Sp(8)-noncompact tangent of E6(6) /U Sp(8), not an F4(4) -module, so the proposed branching
is ill-posed. Under the commuting involutions defined by U Sp(8) and F4(4) (common subgroup
4
U Sp(6) × U Sp(2)), the 42 instead grades as 42 = 28 ⊕ 14 and the 26 as 26 = 12 ⊕ 14; no canonical
16-dimensional normal complement arises. Second, the Goldstone 16 is the well-defined vector ⊗ vector
representation (2, 2; 2, 2) of Spin(1, 3) × Spin(3, 1), whereas the natural spinorial candidate 8s ⊕ 8c
decomposes into spinor ⊗ spinor blocks (2, 1; 2, 1) ⊕ (1, 2; 1, 2) ⊕ (2, 1; 1, 2) ⊕ (1, 2; 2, 1); the two carry
different quantum numbers and are not equivariantly isomorphic. The equality 42 − 26 = 28 − 12 = 16
is therefore a dimensional coincidence, not an identification. Consequently the heavy-quark threshold
corrections of the rest-mass sector cannot be attributed to “eaten Goldstone scalars” of this gap;
that mechanism must be derived, if at all, from an explicit interaction sector rather than from the
scalar-count equality.
Status of the heavy-quark threshold carrier (S61–S62, A715–A717). The explicit-interaction-
sector search was subsequently carried to completion and closed negative at the derivation level. The
genuine (A713) residual pattern selects a weak-isospin coupling δmf /mf = −κ T3 (f ), κ ≃ 1.824%,
uniquely among single standard-model charges (A715, two independent assessments). The only
gauge-invariant carrier in the framework is the weak-adjoint Higgs spurion (H † τ a H) OYukawa a built
on the verified cross-block Yukawa trilinear {J12 (Hu,d ), J13 (Q), J23 (u /d )}; it is writable and charge-
c c
correct, but the sign-forcing test (A716, computed in the exact A688/A689 component map that
reproduces the standard-model Yukawas) shows that J3 (Os ) does not force its relative up/down
sign: the conjugation Hd = H̄u yields the same sector sign, the vacuum grading assigns λ2 = +1
to both channels, and the structure constants carry only component-level orientation signs. The
τ 3 weighting must therefore be inserted by hand, and the operator is Structural, not derived.
A boundary-condition revision is likewise excluded as a principled alternative (A717): it reduces
no parameter. The honest Rev29 statement is that the heavy-quark sector carries bare algebraic
boundary values plus a phenomenological structural T3 threshold term with κ and the charm factor
χc ≃ 1.761 not derived (see Paper 2 §3.6 for the full audit).
5
Ghost-free reading (Rev12 honest form, updated S77). The D=5 intermediate
manifold E6(6) /F4(4) is pseudo-Riemannian with signature (14, 12) on the traceless sector. The
split-magic KK reduction yields 26+27+1 = 54 D=4 scalars. The full maximal-SUGRA target
E7(7) /SU (8) with 70 scalars requires 16 additional fields beyond the Jordan-algebra sector;
these are now identified (S73): the missing 16 is the identity-anchored Jordan complement
42 ⊖ 26 inside the maximal-SUGRA coset, pinned on the certified E7(7) substrate. [LIB2-007]
The D=4 scalar content is ghost-free, and the full 70-scalar maximal completion is reached
operationally: the scalar potential of the gauged theory is computed on the full 70 at the
published normalization and certified against published vacua (Appendix F). The consistent-
truncation question (T4-C) is now settled, negatively and sharply (S78b, Appendix F): the
Jordan-54 is not a Lie-triple system in E7(7) /SU (8) (order-one leakage, computed), so no
consistent scalar truncation retains exactly the Jordan-visible 54 — while the missing-16
carries the corresponding local statement, re-typed at Rev29 : the missing-16 tangent plane is
a Lie triple system in e7(7) ⊖ su(8) at the basepoint, so by Cartan’s theorem it exponentiates
to a totally geodesic submanifold of E7(7) /SU (8) in a neighbourhood of that basepoint, locally
isometric to the symmetric space SO(4, 4)/(SO(4) × SO(4)), the split-octonion norm-form
Grassmannian. [LIB2-009] The local computation stands as printed and no number in
it is withdrawn; what is withdrawn is the scope. The global step — geodesic completeness
of the exponentiated leaf, closedness (as against a dense immersed image) of its embedding
in E7(7) /SU (8), and the resulting determination of the global isometry type, i.e. which real
form and which quotient by which discrete subgroup the leaf actually is — is not supplied
here or anywhere in the suite. The equality sign in “= SO(4, 4)/(SO(4) × SO(4))” is
therefore to be read as a local isometry, not as a global identification of manifolds. [LIB2-010]
The 54-scalar model is definitively a counting/structural correspondence; the full 70 is the
arena. (The missing-16 is thus identified; whether it couples dynamically to the golden vacuum
— the linear/quadratic portal tensors Ba , Hab at Jvac — is a separate, open computation,
pre-registered as the G3–G7 drill in Appendix I.)
The same architecture should also be read with one standard supergravity caveat: the full E7(7)
symmetry is on-shell, i.e. a symmetry of equations of motion plus Bianchi identities, not of the
off-shell local action in a naive sense.
The F4(4) -invariant potential on the 26-dimensional scalar manifold E6(6) /F4(4) contains the SM
Higgs doublet within a larger scalar spectrum. The additional scalars in the split-magic reduction
(26 + 27 + 1 = 54) and the 16-scalar gap to the maximal E7(7) /SU (8) target are not hidden Higgs
degrees of freedom already matched to the SM. Nor, as it turns out, are they projected out: the
S77 potential-layer computation finds no gauging in the scanned electric family that consistently
truncates to the Jordan 54 (no rank-zero leakage row), and at the dynamically selected SO(6, 2) point
the 16 extra scalars are instead stabilized — strictly positive missing-16 Hessian at the electric origin,
non-negative (quartic-lifted) at the dyonic Minkowski vacuum (Appendix F).
5 The Marked Carrier Theorem (selection level)
Once the compact-positive Standard-Model boundary Bphys is loaded, a sharper statement holds: the
boundary carries no continuous physical selection freedom. In the carrier 27 = 16 ⊕ 10H ⊕ 1 (with
the Higgs the SO(10) vector 10H → 5H ⊕ 5̄H , 5 → (3, 1)−1/3 ⊕ (1, 2)+1/2 ):
Novelty label: physical interpretation not established here.
6
Theorem 5.1 (Marked Carrier Theorem, selection level; S232). The loaded compact-positive boundary
selects each of the three marked loadings with zero continuous physical moduli:
• EW–σ: the requirement colour-singlet ∧ weak-doublet ∧ Y = + 12 picks the rank-2 Higgs doublet
uniquely; the raw vev-line CP1 is exactly one SU (2)L × U (1)Y gauge orbit, so the physical
direction moduli are 0.
• Heavy quark: SU (3)c is irreducible on the colour 3 (real commutant C · I), so by Schur the
invariant line metric is unique up to scale and Zcan = I — no form-factor family; Pb has rank
3, Pτ rank 1.
• Colour: 8 = 3 ⊕ 5 · 1; embeddings of the unitary 3 differ by the compact U (3) frame = gauge,
so 0 physical moduli (the gauging obstruction stands — colour is loaded, not gauged).
All three residuals are discrete markings and gauge orbits.
The theorem is selection level only: it does not derive the boundary Bphys itself (frame-free selection
is closed negative by the frame-free E7(7) no-go of Appendix F), does not gauge colour, and touches
no running, α, CKM, or scale. It is contingent on the discrete single-Higgs β = 1 (Z2 ) marking:
retaining both refinement lifts would leave an unaligned two-doublet model with a continuous tan β
residual, so the result leans on β = 1 being a genuine loaded Z2 selection, not a hidden tune. No
public observable moves.
6 Flat Direction Theorem and Coleman–Weinberg outlook
The tree-level potential is tightly constrained by F4(4) invariance. On J3 (Os ) the invariant ring is
generated by
1
T = Tr(X), Q = Tr(X 2 ), Det(X) = N (X), (16)
2
so the most general renormalizable F4 -invariant potential on the traceless subspace is built from Q
and Det alone. The Session 24 Lagrangian analysis establishes the key result.
Novelty label: new specialization proved here.
Theorem 6.1 (Flat Direction Theorem). On the traceless eigenvalue space, the F4 -invariant tree-level
potential has a rank-one Hessian in the (d1 , d2 ) block. Equivalently, the eigenvalue-ratio direction is
exactly flat: the ratio d1 /d2 cannot be fixed by any polynomial F4 -invariant tree-level potential.
At a representative point (a, b, 0) with a + b + 0 = 0 suppressed only after projection, the relevant
Hessian entries are
H11 = 8λ4 a2 , H12 = 8λ4 ab, H22 = 8λ4 b2 , (17)
with the upper 2 × 2 block proportional to the outer product [a, b]T [a, b]. Its eigenvalues are therefore
{8λ4 (a2 + b2 ), 0}, (18)
so the tangent direction to the constant-Q0 curve is exactly unlifted.
The physical consequence is clean. The golden-ratio hierarchy is not a tree-level output of the F4
potential. If the programme is right, d1 /d2 = ϕ must come from a beyond-tree mechanism, with
Coleman–Weinberg lifting from the 24 off-diagonal modes the most natural next step. The flat
7
direction is therefore not an embarrassment to be hidden; it is the theorem telling us precisely what
kind of mechanism the vacuum selector needs.
A second tree-level result is just as important for flavour: the same stationarity conditions force the
off-diagonal Peirce masses to vanish at quadratic order,
Hu = Hd = 0 at tree level, (19)
so any CKM or weak-angle route needing nontrivial Peirce mass splitting must live beyond tree level.
5.1 Coleman–Weinberg lifting: current status
The one-loop Coleman–Weinberg analysis of the 24 off-diagonal Peirce modes now has a substantial
chain of results [D332–D340]. The key theorems are as follows.
Bosonic sector: a graded index, not a determinant cancellation (Rev29 ). VCW bos ≡ 0
holds, and the number is not withdrawn — but it holds as an algebraic Γ-graded (Witten-type)
target-space index, not as a physical one-loop cancellation. Each Peirce block carries four positive
and four negative Hessian eigenvalues from the (4, 4) split signature of Spin(4, 4), the bosonic
modes sit in 8v = (4, 1) ⊕ (1, 4) under SO(4) × SO(4), and with the Z2 parity grading operator
Γ = diag(+1, +1, +1, +1, −1, −1, −1, −1) the graded trace Tr8v Γ M log M vanishes mode-by-
4 2
mode, (+4 − 4) m4 log m2 = 0 [D332]. That is a statement about the constructed graded object and
P
about nothing else. The physical one-loop bosonic effective potential sums all bosonic modes with a
single statistics sign; a ghost-free bosonic sector supplies no relative minus sign of its own. Promoting
the index to a determinant cancellation therefore requires one attachment, named here explicitly
and not supplied anywhere in this suite: an independent path-integral measure/BRST
derivation showing that the functional measure weights the 8v modes by the SO(4, 4)
grading operator Γ. Absent that derivation the claim is index-level only. Accordingly the earlier
reading printed here — that “the determinant contributions cancel exactly and the bosonic one-loop
correction vanishes to all orders,” promoting D=4 ghost-freedom to Green — is withdrawn to the
index tier; the dispatch numbers [D332, D333] and the value 0 stand unchanged, only their type
is corrected. This restores agreement with Appendix A §A.7.4 (“Algebraic index and topological
triviality of the bosonic CW sector,” Rev26 tier caveat), which already carried the correct tier; the
disagreement was an intra-suite contradiction, not a stale leftover.
Fermionic lifting and natural scale. VCW ferm ̸= 0: fermion loops break the ±η degeneracy and
create a local minimum. The renormalization-group stationarity (PMS) condition
∂κ∗
=0 (20)
∂log µ r∗ =ϕ2
is a pure function of the vacuum ratio r∗ = ϕ2 alone; the overall y 4 prefactor cancels from the
critical-point equation. Hence the unique RG-stationary scale at the golden-ratio vacuum is
h∨ (F4 ) 9
κPMS = log = log , (21)
Q0 4
determined by the Jordan vacuum geometry and the dual Coxeter number h∨ (F4 ) = 9 together with
Q0 = 4 (the exact Fibonacci identity ϕ2 + 1 + ϕ−2 = 4), independently of the Yukawa strength y 2 .
[Proved, D340.]
8
Natural Yukawa. The Yukawa coupling y 2 = 2/h∨ (F4 ) = 2/9 is derived from AX1 via the
canonical ’t Hooft coupling gF2 4 = 1/h∨ , the Dynkin index ℓ(8v , Spin(4, 4)) = 1, and the two-fold
Peirce eigenvalue degeneracy. No additional axiom is required. [Proved, D337.]
Loop coefficient structure. The perturbative corrections carry fully rational coefficients:
57 4 3249 1
δκ = y + y6 + y8 + · · · (22)
800 1,280,000 1260
evaluated at y 2 = 2/9, these give a three-loop residual of 1.6 × 10−6 relative to κPMS = log(9/4), con-
firming convergence. The full theorem statement and Peirce-sector derivation appear in Appendix A,
§A.7.3.
Current status. The AX2 CW gate is at Plausible ⋆ ⋆ ⋆ ⋆ ⋆. Steps 1, 3 and 4 of the proof
(flat direction, PMS scale, natural Yukawa) are Proved. Step 2 is re-typed at Rev29 : the bosonic
vanishing is Proved as a graded index and Unattached as a determinant cancellation,
pending the BRST/measure derivation of the Γ weight named above. Step 5 (exact perturbative
convergence to log(9/4)) carries a quantified 1.6 × 10−6 residual at three-loop and is Plausible;
four-loop confirmation or Borel resummation of the rational-coefficient series is the remaining task.
Path A retraction (A342). Assessment A342 retracted the Coleman–Weinberg/PMS convergence
argument as the primary derivation of κ = log(9/4) (Path A). The surviving and authoritative
source of the κPMS = log(9/4) value is Path B: the unique fixed point of the modular element
M1 = T 3 S ∈ P SL(2, Z) ⊂ F4(4) acting on the Peirce diagonal (see §3 above). The CW derivation
above is retained as corroborating evidence for the scale, not as the primary proof chain.
7 Weak angle: algebraic correspondence and precision status
Two earlier weak-angle routes are closed. A pure diagonal vacuum in the F4 derivation algebra gives
sin2 θW = 1/2 identically, permanently closing that route. The older 3/13 tree-level F4 trace-ratio
estimate is retired separately: it arises from a distinct algebraic path (trace-ratio counting of the 26
representation), not from the diagonal-vacuum route, and is no longer the operative Rev29 weak-angle
statement.
The surviving electroweak statement is the non-diagonal exact-colour orbit on the 56-dimensional
E7(7) side. The algebraic boundary value is the constant
√
sin2 θW = ϕ−3 = 5 − 2 ≈ 0.23607. (23)
In the corrected reading (A697, kernel-verified) ϕ−3 = 0.23607 is a ∼2% structural value lying on
the standard running curve of the effective weak mixing angle. That angle rises below MZ from
sin2 θ̂W (MS, MZ ) = 0.23122 ± 0.00006 (PDG 2026 EW review; the on-shell value is 0.22348 ± 0.00010
and the effective leptonic value 0.23161 ± 0.00004 — three distinct scheme objects). [LIB2-225] The
∼2% structural fence below is binding: this row is not converted to a σ distance. The effective
angle rises toward ≈ 0.2386 at q → 0, sub-GeV/GeV-order scale. A proper multi-threshold treatment
(S124; Appendix I §B4) makes this a scheme- and threshold-dominated crossing pushed sub-GeV
(µcross ∼ 0.15 GeV, band [0.09, 0.55]; the once-quoted 1.8 GeV lies outside it), so no fixed matching
scale is pinned. Running upward from this low-scale crossing to MZ still decreases the value to
0.23122 — the correct direction. The earlier very-large-significance “failure” framing was an artifact
of two retired errors: comparing ϕ−3 directly to the ultra-precise MZ anchor (the difference is
ordinary running), and assigning ϕ−3 to a high/VEV scale and running downward. At 246 GeV
9
the running value is ≈ 0.231, confirming that ϕ−3 is a low-energy, not VEV-scale, quantity. The
honest status is therefore a +2.10% structural offset from the PDG 2026 MS(MZ ) comparator, not
a precision prediction at that scale. that the matching scale µ⋆ is not yet derived from J3 (Os ) and
is scheme-dependent; until a structural reason fixes it, ϕ−3 is a structural target for the low-energy
weak angle rather than an absolute prediction. It is consistent with the atomic-parity-violation value
(0.4σ) but in mild (2–3σ) tension with the more precise Qweak/E158 determinations and is not the
q → 0 Thomson limit.
Scheme identification (A698) sharpens this. Being a scale-free Peirce eigenvalue ratio, ϕ−3 is most
naturally the tree-level Lagrangian mixing parameter s20 = g ′2 /(g 2 + g ′2 ) at the algebraic boundary
— not the MS(MZ ), effective-leptonic, or on-shell value. [LIB2-199] Two routes to promote it to a
parameter-free prediction both fail: (i) as a running value the matching scale µ⋆ is scheme-dependent
and undetermined; (ii) as a tree value bridged by standard electroweak radiative corrections, the
required shift is not uniquely fixed — the ordinary on-shell→effective correction is +3.9% (the wrong
sign for the needed −2%), and forcing s2W = ϕ−3 on shell would require ∆r ≈ 0.073, about twice
the normal size. [LIB2-289, LIB2-290, LIB2-291, LIB2-292, LIB2-295, LIB2-310, LIB2-343] We therefore carry ϕ−3 as
a ∼ 2% structural tree-level target on the electroweak curve, not as a completed Z-pole precision
prediction. [LIB2-199, LIB2-289, LIB2-290, LIB2-291, LIB2-292, LIB2-294, LIB2-295, LIB2-310]
AX6 and the weak-angle boundary — two objects, one name (repaired Rev32.2).
The S47 algebraic result separates two statements. First, ϕ−3 = 1/(2(Ω12 + Ω23 )) is a constant
identity in the Peirce eigenvalues; it is not a function of any orbit angle. Second, the orbit
selector — and here two distinct objects have carried the one label “AX6” and the one
symbol ψ⋆ in print (house s1135, lane-cross-verified A1549; PI ruling S295). AX6pol is the
polarisation postulate Λ⋆ = 3P/Q(Y ) = 6ϕ−1 on the exact-colour I4 = ϕ3 orbit (Appendix A:
tan2 ψpol = (6ϕ − 1)/4, ψpol ≈ 55.87◦ ), whose registered readout sin2 θW = 3/(4 + Λ⋆ ) =
4 cos ψpol gives ϕ
−3 exactly (because 3 + 6ϕ = 3ϕ3 ; cos2 ψ
pol = 4/(3ϕ ) is quadratic over Q);
3 2 3
the three CW no-gos [A617,A620,A621] were evaluated at this angle. [LIB2-200, P-006] The Det2
object is the degree-six selector cos(6ψ) = 12 with its 50◦ branch; it is not a solution of AX6pol
(tan2 50◦ = 1.4203 ̸= 2.1771), gives 0.3099 under the readout, and is the sole object of A667’s
degree-three Galois argument. [P-026] The Det2 object is therefore retired as a selector of
AX6pol ; A667 remains correct for the 50◦ object only and does not constrain AX6pol . [LIB2-200,
LIB2-249, P-026] The value identity establishes the readout once AX6pol is assumed; it does not
derive or select AX6pol , which remains an independent postulate at tier.1 The remaining task
is to derive the (scheme- and threshold-dominated, sub-GeV/GeV-order) matching scale from
the algebra — the once-quoted 1.8 GeV is not pinned (Appendix I §B4); until then ϕ−3 is a
structural target for the low-energy weak angle, not an absolute prediction. [LIB2-200, LIB2-249,
LIB2-289, LIB2-292, LIB2-310] The earlier high-scale, downward-running test is superseded (A697).
8 The fit-free mixing sector and the common Thalf engine
The Higgs paper is no longer insulated from the flavour sector. The same algebra that supports the
cubic Yukawa carries the mixing ledger below.
1
Binding vocabulary
√ from Rev32.2: “AX6pol ” and “the Det2 object”; never a bare ψ⋆ . Readout identity: 3/(4 +
(6ϕ − 1)) = 1/ϕ = 5 − 2. Any earlier statement of “AX6” as cos(6ψ⋆ ) = 21 with ψ⋆ = 50◦ refers to the Det2 object.
3
10
Rev29 supersession — read before the table. The Status column is the pre-Rev29
marking and is superseded en bloc by the row-by-row re-tiering (Rev29 re-tiering, Paper 3
§ “Row-by-row re-tiering”; kernel s959; A1566 retype, Rev32.6): two of the eight stand
at Derived-conditional or above (θ12 PMNS , δ
CP ), the CKM first row is Loaded, δCKM
is Coincidence-class, θ13 is structural, and the physical θ23 row and |Vcb | are Loaded-
correspondence. No row may be cited as zero-parameter.
Observable Exact expression Status
√
PMNS
θ12 tan θ12 = 3/ϕ2 THEOREM ⋆100
PMNS
θ23 sin2 θ23 = 7/16 physical oc-
tant Loaded-
correspondence;
internal identity
theorem-grade
(A1566; pre-Rev29
stamp THEO-
REM ⋆100)
√
PMNS
θ13 sin2 θ13 = sin4 (π/8) = (3 − 2 2)/8 STRUCTURAL
⋆ ⋆ ⋆ (round-
trip framing T 2 ,
exponent-2 identi-
fied [s502/A1275]; Stamps in
forcing pending)
√
δCP δCP = −2π/ 5 = −160.997◦ THEOREM ⋆100
√
|Vus | |Vus | = sin θ12
PMNS
/ 6 THEOREM ⋆100
√
|Vcb | |Vcb | = 1/(9 7) Loaded-
correspondence
(physical attach-
ment) / Structural
formula (A1566)
(pre-Rev29 stamp
THEOREM ⋆100)
√
|Vub | |Vub | = |Vus ||Vcb |/ 6 THEOREM ⋆100
δCKM = arctan 3(ϕ + 5ϕ−5 ) STRUCTURAL
p
δCKM
⋆ ⋆ ⋆ (phase origin
open)
this table ( ⋆100, THEOREM, Proved) are the Rev29-era marking, superseded en bloc by the Rev29 re-tier (Paper 3
§“Row-by-row re-tiering”); the current tier of each row is the re-tier table’s. Footer added Rev32.5.
The Higgs-sector reason to record this here is not numerical decoration. The same generator that
drives the leptonic CP rotation is now proved to lie in the derivation algebra of J3 (Os ):
Thalf ∈ Der(J3 (Os )) = f4(4) , derivation error 3.39 × 10−15 . (24)
So CP violation is generated internally by an automorphism of the same algebraic structure that
fixes the Higgs placement and the cubic Yukawa. The CP source is not an externally attached phase
coefficient.
11
9 Mass pyramid and the Sprint 3 outlook
The suite now has a clear tier structure. Tier 1 is the zero-fit mixing sector (no fitted PMNS/CKM
angle, phase, or normalization inside the correspondence map; two entries readout-conditioned).
Tier 2 is the mass-sector construction now forming above it. The current mass operator is carried in
the Jordan-automorphism form
Mlepton = UThalf DR UT−1
half
, DR = diag(R1 , R2 , R3 ), (25)
with
1 2π
Rd = , t∗ = √ , UThalf = exp(t∗ Thalf ). (26)
ϕ2d − 1 5
The same generator that gives δCP therefore also drives the charged-lepton mass ordering.
The charged-lepton mass sector supplies one compact algebraic ratio. It was printed in earlier revisions
as a zero-free-parameter result; Rev29 re-tiers it to Structural / Loaded-correspondence /
Reproduced (see Paper 7 § “Lepton-family validation” for the provenance chain and the fit-gap
accounting). With the single external anchor
mτ = 1776.93 MeV, (27)
the tree-level ratio is 8/3 √
mµ ϕ 2
= √ = 0.059684, (28)
mτ 5 10
which gives
mµ = 106.05 MeV (+0.37% PDG). (29)
√
The exponent p = 8/3 is an exact dimension ratio read as a topological winding ratio, and C = 2/10
divides by dim J2 (Os ) = 10. Rev29: neither step is closed as a derivation — an exact dimension
ratio is not by itself a mass exponent, and A618 records the divide-by-dimension step as “argued by
analogy” and lists the formal derivation as Blocking, closed nowhere. The +0.37% above is the
fit gap,√not a loop correction: reproducing the measured ratio requires C = 0.140894 against the
offered 2/10 = 0.141421, and C enters multiplicatively. The same Jordan automorphism reverses
the resolvent ordering,
(R1 , R2 , R3 ) 7−→ (R3 , R2 , R1 ), (30)
so the generation ordering is forced algebraically rather than chosen by hand. The electron mass
is Koide-consistent ⋆ ⋆ ⋆⋆ under external K = 2/3: me = 0.5076 MeV (−0.66% PDG) from the
QED-corrected Koide chain [A636, A637]. The charged-lepton chain is structurally pinned with mixed
provenance (mτ anchor; mµ Structural / Loaded-correspondence / Reproduced at Rev29,
not zero-parameter; me Koide-consistent under the external rule); the individual masses are not jointly
derived, and the compact ratio’s relation to the Appendix O two-gap operator remains a pending
dual-route reconciliation. Authoritative ledger: Appendix O, The authoritative charged-lepton ledger.
10 Falsification criteria
The Higgs/EWSB layer now has both experimental and theoretical failure modes.
12
Criterion Present target Failure mode (Rev32.1 read-
ing rule: tiers name math-
ematical provenance; phys-
ical attachments are stated
separately; no row may
be cited as zero-parameter;
displayed σ values are
named comparator dis-
tances, not profile likeli-
hoods.)
Metric-signature foundation adopted neutral (4, 4) model If the adoption is rejected —
choice; the composition- i.e. if a convention-fixing frame-
algebra demand then selects work reproduces the same
Os structure — the physical moti-
vation for J3 (Os ) weakens up-
stream of numerics. The falsi-
fication target is the adoption,
not a forcing claim
Weak-angle route Exact-color orbit, algebraic ∼2% STRUCTURAL.
correspondence ϕ−3 ≈ 0.23607 ϕ−3 lies on the effective
sin2 θW (µ) running curve at a
sub-GeV, scheme-dominated
crossing (A697; corrected
S124/Appendix I §B4); the
PDG 2026 MS(MZ ) compara-
tor is 0.23122 ± 0.00006 and
the on-shell comparator is
0.22348 ± 0.00010. The match-
ing scale is not pinned, so this
is a structural target, not an
absolute Z-pole prediction; no
σ distance is assigned.
Leptonic CP δCP = −160.997◦ Stable disagreement at multi-
σ in long-baseline data
Atmospheric angle sin2 θ23 = 7/16 (phys- Stable exclusion of the 7/16
ical octant Loaded- branch in global fits — falsifies
correspondence, A1566) the loaded lower-octant regis-
tration first
√ √
Mass bridge mµ /mτ = (ϕ/ 5)8/3 2/10; Failure of the muon ratio at
me = 0.5076 MeV via QED- its Rev29 tier (Structural
Koide chain [A636,A637] / Loaded-correspondence;
not zero-parameter — the
+0.37% is the fit gap), or of
the QED-Koide chain giving
me outside 2% PDG once the
chain is fully specified
13
The Higgs quartic relation λH = 8/63 remains a low-priority conjectural item rather than a main
falsification channel.
11 Conclusion
Rev29 turns Paper 4 into the correct interface between the algebraic Higgs story and the rest of the
suite. The algebra follows from an adopted signature-neutral premise plus the composition-algebra
demand, the vacuum is theorem-level, the local cubic Yukawa fixes yt (MPl ) = 1, the D=4 scalar
target is ghost-free, the tree-level F4 potential has an exactly flat eigenvalue-ratio direction, the
diagonal weak-angle route is dead, the surviving bosonic route is the non-diagonal exact-color orbit,
and the CKM+PMNS sector stands at its Rev29 re-tiered statuses above the Higgs layer (2/8
Derived-conditional or above since A1566, Rev32.6; no “closed” claim is licensed). The result is
not a finished electroweak theory. It is the sharpest honest form of the present EWSB programme:
algebraically rigid where the suite is ready, explicit about the selector and loop-level tasks where it is
not.
12 Doublet–triplet splitting from the MRSS Higgs parity
The doublet–triplet splitting problem is the classic failure mode of Kaluza–Klein and grand-unified
constructions: the Higgs doublet that must stay light arrives in a multiplet with a coloured triplet
that must not, and no symmetry of the construction separates them.
At A1471 the splitting is derived in this framework, from the parity of the unique minimal chiral lift
rather than from an imposed mechanism.
Result (A1471, s954). The mouth-Real matrix-twist orbifold LMRSS is the unique
minimal admissible chiral lift (A1463, zero continuous parity moduli, house-proved). Its
Higgs-10 parity assignment is unique, and that unique parity retains the weak doublet
while projecting out the coloured triplet. The historic failure mode closes on the mouth
twist.
The boundary price associated with the branch is 6 microscopic gauge coefficients and 3 visible
combinations; the earlier endpoint shorthand from the A1463 era is retired.
The condition this rests on, stated in the same breath. At A1472 the uniqueness was
corrected: it is conditional on the loaded low-energy scalar spectrum — one weak
doublet, no coloured triplet. [LIB2-198] Read plainly, the parity is unique given the spectrum it
is being asked to produce. This is a real result and a real circularity risk in the same sentence,
and the register in Appendix X carries it as a loaded clause with its discharge condition
named: derive the low-energy scalar content and the Higgs parity from the parent boundary
dynamics, rather than selecting the parity by the desired spectrum. [LIB2-198] Until that is
done, this section states a consistency, not a derivation of the spectrum.
[LIB2-198]
14
Leibniz Quantum Beats Newton
Paper 5 (Rev33.1): Neutrino Physics and CP Violation
Tom O’Sieg
August 2026
Abstract
This paper isolates the neutrino and leptonic CP sector of the framework. It assigns exact
closed-form expressions to four PMNS parameters within a loaded correspondence: three carry
theorem-grade constructions on the algebraic structure of J3 (Os ) with F4(4) symmetry (marked
THEOREM ⋆100 pre-Rev29; Derived-conditional after the Rev29 re-tiering (Rev29 re-tiering,
Paper 3 § “Row-by-row re-tiering”; kernel s959)), the phase identification is conditional on the
stated readout map, and the reactor-angle attachment is STRUCTURAL ⋆ ⋆ ⋆ (round-trip framing
T 2 , exponent-2; forcing pending [s502/A1275]). No fitted parameters remain in the PMNS sector;
“no fit” is a statement about coefficients inside the map, not a claim that the correspondence map
itself is derived, and no row may be cited as zero-parameter. The three mixing angles and
the Dirac CP phase are:
√
3
tan θ12 = 2 =⇒ θ12 = 33.49◦ (THEOREM internally; DERIVED-CONDITIONAL physically — Paper 3 re-tier),
ϕ
7
sin2 θ23 = =⇒ θ23 = 41.41◦ (THEOREM internally; LOADED-CORRESPONDENCE as the physical octant — A
16
√
3−2 2
2 4
sin θ13 = sin (π/8) = =⇒ θ13 = 8.42◦ (STRUCTURAL ⋆ ⋆⋆),
8
2π
δCP = − √ = −160.997◦ (THEOREM internally, A603/E110; DERIVED-CONDITIONAL physically).
5
The θ13 attachment carries its qualifier in full: round-trip framing T 2 , exponent-2, with forcing
pending [s502/A1275]. Rev30 notation correction: these four lines previously chained a dimen-
sionless function value directly to an angle — e.g. “sin2 θ23 = 7/16 = 41.41◦ ”, which as written
asserts 7/16 = 41.41◦ . The implication arrows above separate the two statements; no value has
changed. The leptonic CP √ phase is derived from the V3 rotation: √Thalf |V3 is a 3D rotation with
angular frequency Ω = 5/2, giving half-period t∗ = π/Ω = 2π/ 5 (A603/E110, Appendix B,
§B.5). The free parameter βδ = 11/(6π) is eliminated. NuFIT 6.1 (NH, w/SK) comparison:
δCP = 212+26 ◦ ◦ ◦
−36 (−160.997 ≡ 199.003 ), deviation 0.36σ. Three of the four PMNS parameters were
marked THEOREM ⋆100 pre-Rev29 (θ12 , θ23 , δCP ). Two physical PMNS rows, θ12 and δCP , stand
at Derived-conditional after the Rev29 re-tiering (Rev29 re-tiering, Paper 3 § “Row-by-row
re-tiering”; kernel s959), each with its premises printed. The exact 7/16 mismatch identity remains
theorem-grade internally, while its physical lower-octant attachment is Loaded-correspondence
(A1566/D1244, Rev32.6); θ13 is STRUCTURAL ⋆ ⋆ ⋆ (round-trip framing T 2 , exponent-2; forcing
pending [s502/A1275]). No fitted coefficients enter the map; no row may be cited as zero-
parameter. Normal ordering is assumed; the operative benchmark assumes Dirac neutrinos —
the existing singlet permits an algebraic Majorana term absent an additional B − L principle —
and the representation content is exhaustive: the fermionic 16 = J13 ⊕ J23 is exactly saturated by
the SM states plus ν c , so the framework forbids light sterile neutrinos — a confirmed sterile would
falsify the verified assignment (A726; Paper 7).
1
1 Introduction
The neutrino sector is where the framework is most exposed to direct falsification. The paper therefore
keeps the structure simple: the angle set, the phase formula, the current comparison to NuFIT 6.1,
and the DUNE-era test scenario. [LIB2-345]
2 Three neutrino mixing angles
The operative angle relations are (normal ordering — conditional on the Rd ladder, Paper 2 §3,
and on its identification with the propagation eigenstates, an interface that is undeclared (Rev32.2);
pre-Rev29 markings: three of four PMNS parameters at THEOREM ⋆100, θ13 STRUCTURAL ⋆ ⋆ ⋆;
current: θ12 , δCP Derived-conditional, the physical θ23 octant Loaded-correspondence —
A1566):
√
PMNS 3
tan θ12 = 2, (PMNS solar angle — THEOREM) (1)
ϕ
7
sin2 θ23 = , (F4 mismatch geometry; THEOREM internally, physical octant LOADED-CO
16
(2)
√
3−2 2
sin2 θ13 = sin4 (π/8) = . (STRUCTURAL ⋆ ⋆ ⋆) (3)
8
The θ23 row is the sharp loaded lower-octant target of the selected correspondence (A1566); the θ13
row carries round-trip framing T 2 , exponent-2, with forcing pending [s502/A1275]. These give
PMNS
θ12 ≈ 33.49◦ , (4)
◦
θ23 ≈ 41.41 , (5)
◦
θ13 ≈ 8.42 . (6)
Precision and structural status of the angle predictions (A678). The reactor angle sin2 θ13 =
sin4 (π/8) is a structural ⋆ ⋆ ⋆ prediction: the formula reproduces the observed scale at < 1.8σ —
comparator NuFIT 6.1 (NH, w/SK), sin2 θ13 = 0.02248+0.00055−0.00059 , central-value Gaussianized on the
lower error and not a profile (Rev29 : comparator named) — and three compact closed forms (1/45,
sin4 (π/8), ϕ−8 ) are numerically close to the measured value, but not all within the comparator this
suite uses: against NuFIT 6.1 (Scorecard row 6) 1/45 sits at d = −0.44, the retained sin4 (π/8) at −1.75
and ϕ−8 at −2.02 (Rev32.9; through Rev32.8 this sentence said all three “lie within PDG uncertainty”);
no current Freudenthal theorem selects one. [LIB2-061] The atmospheric angle sin2 θ23 = 7/16 is the
sharp loaded lower-octant target of the selected correspondence (A1566): the correction required to
reach the NuFIT 6.1 NH upper-octant local comparator (≈ 0.561) is ≈ 0.124, far larger than any
natural Jordan subleading perturbation; the NuFIT 6.1 global best fit itself (0.470, lower octant) lies
only 0.0325 above the prediction. Watch DUNE/Hyper-K for octant resolution.
Scope of the Jordan-adjoint mixing carrier Xν (renamed at Rev29 ). The algebraic
# = diag(ϕ−1 , 1, ϕ) is the framework’s Jordan-adjoint mixing
invariant Xν = Jvac × Jvac = Jvac
carrier. Through Rev28 this suite called it the “neutrino mass operator”. That name asserted an
interface — from the constructed Jordan layer to the physical neutrino mass spectrum — which the
2
framework does not have and which the calculation in the next paragraph disproves in this very paper;
the object was carried across an absent interface under a physical name. The definition is unchanged
and the object remains valid in the layer where it is constructed. Only the predicate changes: from
mass operator to mixing carrier. As a mixing carrier it organizes the PMNS mixing-sector geometry:
the mismatch vector Jvac − Jvac# = diag(1, 0, −1) seeds θ #
13 and the (Jvac , Jvac ) Gram data seed the
atmospheric angle. Its eigenvalues do not furnish the physical neutrino mass spectrum — and this
is the ground of the rename, not a caveat appended to it: the literal reading mi ∝ λi (Xν ) predicts
normal ordering but gives the scale-free ratio
∆m221 1 − ϕ−2 ϕ−1
= = √ = 0.27639, (7)
∆m231 ϕ2 − ϕ−2 5
against the measured ratio 0.03002 (NuFIT 6.1 IC24 NO: ∆m221 = 7.537 × 10−5 eV2 , ∆m23ℓ =
2.511 × 10−3 eV2 — dataset named, Rev32) — too large by a factor 9.21 (A712, verified independently
in house). [LIB2-134]
Two readings, one family (Rev32.2; house s1104, lane A1546 C2, PI ruling S294; the exact-asymmetric
numbers below are certified by s1162, Rev32.6 — s1104’s −1.04σ was a symmetric calibration-grade
estimate and is superseded). [LIB2-140] The literal reading above is the d = 1 rung of the resolvent family,
m2 /m3 = ϕ−1 = R1 ; the suite’s own d-assignment (Paper 2 §3, Appendix X) places the neutrino
sector at d = 2, mν2 /mν3 = R2 = 1/(ϕ4 − 1) = 0.17082, so the ratio channel reads R22 = 0.02918
against the measured 0.03002 (−2.8%). [LIB2-140] Propagating the directional (asymmetric) NuFIT 6.1
marginal errors to first order (s1162; this is not an exact or q profile-likelihood distance — Rev32.9),
R2 lies 1.44–1.79σ below the m1 = 0 normal-ordering floor ∆m221 /∆m23ℓ = 0.17325 (without / with
SK-ATM); equality is impossible for any m1 ≥ 0, because the ratio is minimised at m1 = 0. This
row was typed, before Rev33.0, as a fixed below-floor edge target — a 1.4–1.8σ tension with
the m1 = 0 NO floor, printed as a tension. That typing is superseded. The suite now types the
row as a ∆m221 /∆m231 ratio prediction, asserting m1 = 0, under which m1 drops out of the ratio
and R22 = 0.029180 becomes a direct, JUNO-decidable statement rather than an edge target that no
measurement can meet. m1 = 0 is an assertion of this framework, not a derivation, and it is
registered as one. Against the comparators built from the printed rows the implied values stand at
−1.44σ (JUNO, 59.1 d) and +2.67σ (NuFIT 6.1): the two disagree in sign, and both are named —
no unqualified σ is printed for this row (s1184). The 1σ band of this ratio contains ten members of
the declared M1 monomial menu, a density of 186 per unit relative width, and that density is printed
with the row because a hit inside a band that dense is reported with its density or it is not reported.
What the typing costs, stated plainly: at JUNO’s projected design reach — a declared external
input taken from the collaboration’s own projection, and to be re-verified before it is relied on —
a −2.8% miss on this ratio becomes −7.73σ (s1184). The row is accordingly a falsifier, and it is
printed as one. It stands only with normal ordering and m1 ≲ 1 meV (m1 at −2σ: 0.86/0.51 meV
without / with SK-ATM, s1162). [LIB2-140] The d = 1 reading remains the one genuinely falsified row.
The assignment d(ν) = 2 is selected, not derived: it is reproduced by the post-hoc feature rule
d = 2 + [YR ̸= 0] + [coloured] and is the χ2 minimum of an injective assignment (∆χ2 = 17.8 to the
runner-up). [LIB2-141] Tier: structural (the ladder’s own tier); local-band look-elsewhere 10.9–13.9%
(s1162); no blind credit. [LIB2-141]
3
Falsifiable corollaries — of the m1 = 0 boundary, not of R2 . If the edge is taken at
the boundary m1 = 0 with normal ordering, the boundary itself fixes Σmν = 58.79 meV
and mβ = 8.74 meV. [LIB2-142] These are properties of the m1 = 0 NO boundary, which R2
forces only as an edge; they are not consequences of R2 . [LIB2-142] mββ = 0 is Dirac-loaded
(Appendix X), not predicted. Cosmology comparator, source-lined: DESI DR2+CMB baseline
Σmν < 64.2 meV (arXiv:2503.14744; margin 5.41 meV); the 53 meV frequentist-constrained
bound is model tension, not exclusion. [LIB2-142, LIB2-207, LIB2-208] A measured inverted ordering,
or m1 established above ∼ 1 meV, removes the edge.
Xν is therefore carried only as the Jordan-adjoint mixing carrier — a structural object of the
constructed layer, which predicates nothing whatever about masses — and no statement in this suite
may read it as a mass operator or as a mass ladder (Rev29 ; the matching rename is folded in Paper 3
§3.1). [LIB2-134] The absolute neutrino mass scale, the ∆m2ij splittings, and the Dirac/Majorana or
see-saw scale choice are not derived in this suite. The PMNS mixing predictions are independent
of those absolute-scale choices, with normal ordering — conditional on the Rd ladder and on the
undeclared ladder-to-eigenstate interface, never a ladder theorem (s1108; A1546 C6) — in the
benchmark tables.
The B-map dictionary (A689) does contain a right-handed-neutrino slot, ν c = e7 ∈ J23 , and the
cross-block Peirce trilinear yields a nonzero Dirac Yukawa,
τ (Hu , L, ν c ) = ϵαβ Huβ Lα ν c , (8)
the lepton analogue of the up-type coupling (A709, verified independently in house). The Dirac
partner is therefore present structurally. However, ν c is an SU (3)c × SU (2)L × U (1)Y singlet (Y = 0),
so a bare Majorana bilinear M ν c ν c is gauge invariant and algebraically nonzero. The renormalizable
Yukawa sector conserves an accidental global B − L (the Higgs is B − L-neutral, so the electroweak
vacuum does not break it), but no gauged or fundamental symmetry forbidding M ν c ν c is derived.
Neutrinos are therefore treated as Dirac by framework assumption; the seesaw/Majorana sector is
structurally available but is not included in the operative PMNS benchmark.
The relevant symmetry is constructible explicitly as the fifth SO(10) Cartan. On the spinor S+ =
Λeven R5 = 1 ⊕ 10 ⊕ 5, the grading X(I) = 5 − 2|I| together with the A689 hypercharge rule
P
Y (I) = i∈I Yi gives
B −L = 15 (X + 4Y ), (9)
which reproduces the standard one-generation charges B−L(Q) = + 13 , B−L(uc , dc ) = − 31 , B−L(L) =
−1, B −L(ec , ν c ) = +1, B −L(Hu,d ) = 0 (16/16, verified independently in house, A710), commutes
with SU (3)c × SU (2)L , and is linearly independent of Y . It conserves the Dirac Yukawa and assigns
the Majorana bilinear B −L(ν c ν c ) = +2, so an exact B −L would forbid it. However B −L is not
part of the gauged compact-SM group (rank 4; B −L is the extra SO(10) rank-5 Cartan), and no
B −L-charged VEV is present to break it; a gauged B −L would be anomaly-consistent with the
existing 16 but is neither forced nor realized here. Gauging B −L — extending the rank-4 SM to
the rank-5 SO(10)-type group — was the open gauge-emergence question (T4-B). Its modal status
was settled by the S65 dispatch ladder (D727–D729, three independent assessments, unanimous
[A729–A729c]): in the established frame-restriction category the gauge fields are input, not emergent,
and the B −L generator lives at the same structure-algebra/envelope layer as the successful one-
generation charge table (the simple F4 derivation route hosts a quark-sector candidate but no lepton
slots [A678]), so no categorical asymmetry forces the non-gauging. The absence of a gauged B −L is
therefore inherited from the compact rank-four input choice — a choice that is phenomenologically
4
′
required rather than algebraically forced: gauged-and-unbroken gives an excluded massless ZB−L ,
while gauged-and-broken needs a B−L-charged VEV not present here. Neutrinos accordingly remain
Dirac by framework benchmark assumption: a confirmed Majorana mass (0νββ) would falsify this
benchmark assumption, not the PMNS mixing algebra. The residual T4-B question is the selection
theorem for the input vector content — adj(GSM ) versus adj(GSM × U (1)B−L ) — currently answered
only by the Z ′ exclusion.
The PMNS solar angle may also be written as
3
tan2 θ12
PMNS
= = ϕ−2 + ϕ−6 , (10)
ϕ4
which makes the role of the exact identity ϕ2 + ϕ−2 = 3 explicit.
2.1 Framework vs. NuFIT comparison (C3 status)
Quantity Framework Comparator: NuFIT 6.1 Tension
(NH, w/SK) (central-value
Gaussianized)
θ12 THEOREM 33.49◦ 33.76+0.42
−0.41
◦ (NuFIT 6.1 0.66σ (the
IC24 NO — corrected earlier < 0.1σ
Rev32: the 33.41◦ ± 0.81◦ claim is struck)
previously printed here
was a pre-6.0-era value car-
rying a 6.1 label)
θ23 sin θ23 = 7/16: theorem-grade in- 43.3◦
2 (sin2 θ23 = 2.3σ (octant
+0.017
ternal identity; physical θ23 Loaded- 0.470−0.014 , lower oc- SK-ambiguous;
correspondence — the sharp, falsifi- tant, w/SK) UO comparator
able loaded lower-branch target of the se- 0.561 at ∼ 9.5σ)
lected correspondence (A1566; pre-Rev29
stamp THEOREM ⋆100) 41.41◦
θ13 STRUCTURAL
√ sin2 θ13 = sin4 (π/8) = 8.60◦ ± 0.13◦ (rounded 1.4σ in this vari-
(3 − 2 2)/8 (⋆ ⋆ ⋆; round-trip fram- symmetric degree quote) able; 1.8σ in
ing T 2 , exponent-2; forcing pending sin2 θ13 — see
[s502/A1275]; A608/E113), 8.42◦ the box below
(Rev29 )
√
δCP THEOREM ⋆100, −2π/ 5 = −160.997◦ 212+26
−36
◦ 0.36σ
(A603/E110)
Stamps in this table ( ⋆100, THEOREM, Proved) are the Rev29-era marking, superseded en bloc by the Rev29 re-tier
(Paper 3 §“Row-by-row re-tiering”); the current tier of each row is the re-tier table’s. Footer added Rev32.5.
5
θ13 : the two tensions this paper prints are the same fit (Rev29 ). The table above reads
1.4σ and §3 below reads 1.8σ; both are arithmetically correct, and they are one NuFIT 6.1
(NH, w/SK) result read in two variables. In sin2 θ13 the comparator is 0.02248+0.00055
−0.00059 against
the framework’s 0.021447, giving 1.75σ → 1.8σ on the lower error. In degrees, the same
comparator pushes forward to θ13 = 8.623+0.106 ◦ ◦
−0.115 , against which the framework’s 8.421 sits
at 1.76σ — also 1.8σ. [LIB2-228] The 1.4σ arises only from the rounded, symmetrized degree
quote 8.60◦ ± 0.13◦ , which is not the push-forward of the NuFIT 6.1 interval. The variable-
consistent figure is 1.8σ; 1.4σ is retained above and now carries the comparator that
produces it. All σ figures in this paper are central-value Gaussianized against a single-number
comparator, not profile likelihoods, and where the comparator is asymmetric the error on the
side facing the framework value is used (Rev32.9; through Rev32.8 this said “the lower error”,
which is not what every printed figure uses). Cross-suite: the CKM σ figures are separately
under reconciliation — see the reconciliation box in Paper 3, §2.1.
3 CP violation from PSL(2, 7) ⊂ G2 ⊂ F4(4)
Rev29 result: the leptonic Dirac phase is derived from the V3 rotation theorem (A603/E110). The
free parameter βδ = 11/(6π) is eliminated. The Uniqueness Theorem (Session 16, A558) establishes
that Thalf is the unique G2 -invariant generator in the (1, 3) Peirce sector of f4 .
2π
δCP = − √ ≈ −160.997◦ (JCP ≈ −0.011, rephasing-invariant). (11)
5
# , T
Mechanism (V3 rotation). The three-dimensional invariant subspace V3 = span{Jvac , Jvac half ·
Jvac } is closed under the adjoint
√ action of Thalf . Within V 3 , the operator Thalf
√ acts as a 3D rotation
with angular frequency Ω = 5/2. The half-period satisfies t∗ = π/Ω = 2π/ 5, giving
exp(t∗ Thalf ) · Jvac = Jvac
#
.
√
The CP phase is the accumulated rotation angle: δCP = −t∗ = −2π/ 5. The rotation angle is purely
algebraic — no free parameter enters t∗ — but its identification with the measured PMNS phase is a
conditional readout (frame, ordering, Dirac benchmark; no row may be cited as zero-parameter). (The
older PSL(2, 7) formula, − arccos(−1/3) − 2π/7 = −160.900◦ , remains as an independent cross-check;
the V3 formula is authoritative.)
exp
Experimental comparison (NuFIT 6.1, NH w/SK): δCP = 212+26 ◦ ◦
−36 ; with −160.997 ≡ 199.003 ,
◦
deviation = (212◦ − 199.003◦ )/36◦ = 0.36σ using the central-value Gaussianized (lower) error only.
The full NuFIT ∆χ2 profile is non-Gaussian; see Paper 7 for the falsification interpretation.
6
Status (Rev29; A1566 retype Rev32.6). 2/8 PMNS+CKM PARAMETERS
AT Derived-conditional OR ABOVE (θ12 PMNS , δ
CP ); THE CKM FIRST ROW IS
Loaded; THE PHYSICAL θ23 ROW AND |Vcb | ARE Loaded-correspondence (THE
INTERNAL 7/16 IDENTITY THEOREM-GRADE); δCKM IS Coincidence-class;
θ13 IS STRUCTURAL — NO FITTED MIXING COEFFICIENTS INSIDE THE
MAP, AND NO ROW MAY BE CITED AS ZERO-PARAMETER. The pre-Rev29
tally of 6/8 at ⋆100 is superseded (Rev29 re-tiering, Paper 3 § “Row-by-row re-tiering”; kernel
s959). √
δCP = −2π/ 5 = −160.997◦ (JCP ≈ −0.011) is Derived-conditional (Rev29 re-tiering;
the pre-Rev29 stamp was THEOREM ⋆100) from the V3 rotation (A603/E110); all M-
questions closed (A556, D576). NuFIT 6.1 (NH): 0.36σ — still the best-fitting parameter
in the suite. DUNE Phase II will test this target. The rephasing invariant itself is not yet
a constraint: the NuFIT 6.1 comparator, JPMNS = −0.0140 ± 0.0149, has a 16–84% band
[−0.0286, +0.0022] that includes zero (the comparator is 0.94σ from CP conservation). The
theory value JCP = −0.0107 is therefore consistent with the data and equally with no leptonic
CP violation. The row is diagnostic (a function of four PMNS rows already scored), is
counted in no tally of independent successes, and becomes a constraint only when the band
excludes zero (kernel s1185; Monte Carlo comparator, correlations ignored; ruling R147).
Normal ordering is assumed; the row is additionally conditional on the Structural Rd ladder
and on the undeclared ladder-to-propagation-eigenstate interface (§2); Majorana phases are
absent because the operative benchmark assumes Dirac neutrinos — an explicit framework
assumption, not a consequence of a derived symmetry forbidding M ν c ν c (A709). [LIB2-062,
LIB2-242, LIB2-243, LIB2-244, LIB2-277, LIB2-278, LIB2-279, LIB2-280, P-032, P-033]
√
θ13 : STRUCTURAL ⋆ ⋆ ⋆ (A608/E113): sin2 θ13 = sin4 (π/8) = (3 − 2 2)/8 ≈ 0.02145; D4
triality 8v ↔ 8s maps Jvac at π/4 to π/8; numerical agreement at 1.8σ (comparator NuFIT 6.1
NH w/SK, sin2 θ13 = 0.02248+0.00055
−0.00059 , central-value Gaussianized on the lower error — the
variable-consistent figure; cf. the 1.4σ degree-variable entry in the §2.1 table and the box
beneath it, Rev29 ); round-trip T 2 squaring identified (exponent-2, s502/A1275); forcing
pending.
C1 Theorem (Session 16, A558): the weight c = 12 in Thalf = Tscalar + 12 Toct is proved
exactly from the DET-7 ratio, Frobenius norm, and Peirce orthogonality (3-step algebraic
chain; zero free parameters). [LIB2-051] Uniqueness Theorem (Session 16): Thalf is the unique
G2 -invariant generator in the (1, 3) Peirce sector of f4 . CKM sector (Sessions 22–23):
the three CKM magnitudes (|Vus |, |Vcb |, |Vub |) were marked Proved-tier (⋆100) within the
Route-B map pre-Rev29 — the first row is now Loaded (route selected against kaon data)
and |Vcb | is Loaded-correspondence (physical attachment) with a Structural formula
(A1566); the CKM phase δCKM is Coincidence-class / candidate construction — numerical
value retained, G7 selection absent; the Rev29 wording “conditional theorem under the
readout axiom F-δ, STRUCTURAL” is retired here (Rev32.5, A1563; Paper 3 Candidate
Construction 2.1; App B, §B.6). All 8 PMNS+CKM parameters from J3 (Os ) alone,
no fitted mixing coefficients inside the map: after the Rev29 re-tiering and the
A1566 retype 2/8 stand at Derived-conditional or above, the CKM first row is
Loaded, δCKM is Coincidence-class, θ13 is structural, the physical θ23 row and |Vcb |
are Loaded-correspondence (readout-axiom claims, named inline).
4 DUNE-era falsification scenario
7
Table 1: Rev29 neutrino-sector falsification targets. Rev32.1
reading rule: tiers name mathematical provenance;
physical attachments are stated separately; no row
may be cited as zero-parameter; displayed σ values are
named comparator distances, not profile likelihoods.
Quantity Framework target Adopted precision scenario Current reading
√
δCP −2π/ 5 = −160.997◦ DUNE-era O(10◦ ) at 1σ sharpest target; dis-
(JCP ≈ −0.011) (2030s) agreement beyond the
[THEOREM ⋆100; adopted band falsifies
A603/E110; βδ elimi-
nated; 0.36σ]
θ23 41.41◦ (sin2 θ23 = DUNE-era few-degree pre- exclusion falsifies the
7/16; Loaded- cision (2030s) loaded lower-octant reg-
correspondence istration first; the inter-
physical target, nal F4 mismatch theo-
internal identity rem (∥Jvac ∥2 = 4 from
theorem-grade — DET-7+N (J) = 1, AX2
A1566; pre-Rev29 eliminated) is attacked
stamp THEOREM only if that attachment
⋆100) is independently derived
θ12 33.49◦ solar/reactor combined consistency cross-check
precision
Stamps in this table ( ⋆100, THEOREM, Proved) are the Rev29-era marking, superseded en bloc by the Rev29 re-tier
(Paper 3 §“Row-by-row re-tiering”); the current tier of each row is the re-tier table’s. Footer added Rev32.5.
5 Conclusion
The Rev29 neutrino paper is intentionally direct. Three of the four PMNS parameters — θ12 , θ23 ,
δCP (V3 rotation; A603/E110) — were marked THEOREM ⋆100 pre-Rev29. Two physical PMNS
rows, θ12 and δCP , stand at Derived-conditional after the Rev29 re-tiering, with their premises
printed; the exact 7/16 mismatch identity remains theorem-grade internally, while its physical lower-
octant attachment is Loaded-correspondence (A1566/D1244, Rev32.6). The reactor angle θ13
is STRUCTURAL ⋆ ⋆ ⋆: the formula sin2 θ13 = sin4 (π/8) is in numerical agreement at 1.8σ against
NuFIT 6.1 (NH, w/SK), the release adopted throughout this suite — the 1.3–1.4σ quoted here
previously was against the superseded NuFIT 6.0 and is retained only as that earlier reading (Rev29 )
— but the round-trip T 2 squaring is identified (exponent-2, s502/A1275) but not yet forced from first
principles (D678). [LIB2-061, LIB2-281, LIB2-282, LIB2-315] The CKM magnitudes were marked Proved-tier
within Route B pre-Rev29; after the re-tiering the first row is Loaded and |Vcb | is a Structural
formula with Loaded-correspondence attachment (A1566; Paper 3; Appendix B, §B.6); the
CKM CP phase δCKM is Coincidence-class (candidate construction; value retained, G7 selection
absent) and the full matrix is a readout-conditioned completion map (first-principles gap; correct
numerically), and the 5ϕ−5 chain is candidate-class pending derivation. If long-baseline experiments
disagree decisively with these targets, the current exceptional-Jordan correspondence fails in the place
8
where it is easiest to test.
6 The neutrino posture, made explicit: Dirac-loaded
For five rounds the framework carried an unanswered question about its own neutrino typing —
whether the parent-layer structures it uses commit it to Dirac or Majorana neutrinos. The question
was raised as a standing rider and went unanswered through four dispatches. It was answered at
A1472 and the answer is recorded here in the paper that depends on it.
6.1 The posture
The framework is Dirac-loaded: the Dirac branch is carried explicitly as a loaded choice, not derived.
The charge controls are exact and house-verified:
X(L) + X(Hu ) + X(ν c ) = 0, X(ν c ν c ) = 10.
The first vanishing permits the Dirac Yukawa; the second is the obstruction that a Majorana mass
term would have to overcome.
6.2 A disambiguation that mattered
An earlier round (SD16, folded at A1452) derived the mouth-chiral grading {S, H} = 0 from what it
called a “branch-Real Majorana pair”. That phrase invited a reading it did not support. At A1472 it
is disambiguated: the pair is a parent-layer Real condition, C = τ1 K — an anti-linear structure
on the parent layer — and not a Standard-Model Majorana mass term. The two live at different
layers, and the earlier grading derivation is untouched by the neutrino typing.
The Dirac/Majorana collision watch opened during the S256 external reviews is resolved at this tier.
What the Majorana branch would cost. It is not closed off, and it is not free. Taking it
requires a ∆(B−L) = 2 source or seesaw block, and — because the loaded rows were audited
as a dependency ledger at A1472, separating Dirac-essential rows from ν-independent ones —
a complete re-audit of every row the Dirac loading touches. The branch is census-enlarging:
it does not merely substitute one mass term for another, it widens the space the framework’s
uniqueness arguments run over. Appendix X carries this as a loaded clause with that
discharge condition named.
9
Leibniz Quantum Beats Newton
Paper 6 (Rev33.1): Renormalization-Group Running, Threshold Corrections,
and the Electron-Mass Residual
Tom O’Sieg
August 2026
Abstract
Paper 6 records the renormalization-group bridge used by the Rev8 suite for the weak angle
and related threshold questions, and it adds the current electron-mass residual bookkeeping. The
main algebraic input remains a tree-level exceptional-Jordan ratio; the main physical statement is
that tree-level algebraic numbers and measured low-energy observables are connected only after
ordinary field-theory running, threshold effects, and scale identification.
Rev8 also makes a second bridge explicit. The baseline electron formula
h π i √
mbase
e = MPl exp − ϕ4 2 = 0.5036 MeV (1)
8α
undershoots the physical electron mass by 1.454%. The bundle currently tracks five physically
motivated correction phases, but their central values sum only to 1.304%, leaving a 0.15% remainder
unexplained. Rev12 S43 update: The electron-mass residual sections of this paper are superseded.
As of Session 43, me = 0.5076 MeV is Koide-consistent ⋆ ⋆ ⋆⋆ under external K = 2/3 via the
QED-corrected Koide chain [A636, A637]; the instanton-phase ansatz and all associated residual
bookkeeping are retired. Paper 6 is retained as a historical record of the Rev8 electron-mass bridge
programme.
Rev29 note (Sessions 16–18). The β-function and RGE content of Paper 6 are unchanged
from Rev9. The U (1)Y coefficient is carried in the GUT-normalised convention b1 = 41/10
throughout (KB9 fix, confirmed in both this paper and Appendix D). The Rev29 leptonic CP
results (δCP at Derived-conditional after the Rev29 re-tier) do not alter the RGE sector;
they are reported in Paper 5. The muon-to-tau mass ratio is Structural (re-tiered Rev29;
Appendix O is authoritative), and this does not alter the RGE equations or threshold bookkeeping
recorded here.
Rev32 restatement (superseding the Rev29 provenance overclaim that stood here,
per the authoritative ruling in this paper’s status section). The CKM Dirac phase δCKM re-
tains its number (68.1297◦ from G7 = ϕ+5ϕ−5 under the named readout axiom F-δ, Appendix B.6)
but its rating is Coincidence-class, and it stays there until G7 is derived: the number is not
withdrawn; the rating is. What this paper adds is RGE-irrelevance — renormalization-group
running and top-quark threshold corrections are confirmed irrelevant for δCKM (D594, A576) —
and RGE-irrelevance is a statement about running, not algebraic provenance. The CKM first
row is Loaded; |Vcb | is Loaded-correspondence (physical attachment) / Structural formula
(A1566); δCKM is Coincidence-class (canonical string, Rev32.1, amended Rev32.7 — the earlier
“three CKM magnitudes are Structural” overclaimed against the first-row route cap, A1538 F7) —
not a 100/100 closure of the sector.
1
1 Introduction
Historical note (Rev12 S43): The electron-mass closure problem described in this paper is
now resolved. me = 0.5076 MeV is Koide-consistent ⋆ ⋆ ⋆⋆ under external K = 2/3 via the
QED-corrected Koide chain (A636, A637). The content below is retained as the Rev8 historical
record; it does not represent the current programme state.
The weak-angle bridge is a standard field-theory problem: the exceptional algebra supplies a tree-level
ratio, and quantum field theory supplies running, scheme choice, and threshold corrections. Rev8
states that bridge openly. It also extends the same honesty to the electron-mass closure problem.
The current baseline formula is structurally sharp, but the residual percent-level phase system is not
yet derived from the Lagrangian. This paper exists to keep those two bridges conceptually separate.
2 Standard one-loop running
At one loop the gauge couplings satisfy
dgi bi 3 41 19
µ = g , (b1 , b2 , b3 ) = , − , −7 (2)
dµ 16π 2 i 10 6
— the Standard-Model one-loop coefficients, imported physics (tier Input; Rev32.9) — in the
GUT-normalised convention (SU(5)-embedding normalisation of U (1)Y , used throughout the D=5
framework; the alternative b1 = 41/6 is the SM-normalised value and is not used here — Appendix D
confirms b1 = 41/10). Equivalently,
dαi−1 bi
=− . (3)
d ln µ 2π
2.1 RGE preservation of Peirce structure
The resolvent family
1
Rd = (4)
ϕ2d − 1
remains stable under Standard Model running from the ultraviolet matching scale to MZ . The leading
anomalous dimensions are flavour-blind within a fixed representation, so the dominant corrections
cancel in the ratios. Rev8 therefore continues to carry the ratio sector as structurally stable under
ordinary one-loop running.
Artin bridge and four-point exposure [structural, A720]. By Artin’s theorem, every two-
generator subalgebra of Os is associative. Hence ordinary 2-point and 3-point QFT corrections —
propagators, self-energies, Yukawa vertices, gauge vertices, and one-loop beta functions — are com-
puted by the standard associative MS rules. The first place where split-octonion non-associativity can
(4)
enter is a four-point/box topology. A bounded power-counting audit (S63) gives δNA ∼ (g 2 /16π 2 )2 L
with an O(1) associator coefficient. For the QED-corrected muon chain this is 6 × 10−5 %, negligible
against the quoted mµ and me precisions; heavy-quark matching and the weak-angle structural row
are likewise protected. Long-log running from MPl to MZ gives ∼ 0.35%, so sub-percent RGE-stability
statements are not sharpened below that level without an explicit two-loop/box audit. The one
2
external observable with discovery or exclusion power is the muon anomalous moment: a light-by-
light-class associator insertion of O(1) coefficient would contribute ∼ 125 × 10−11 , already in tension
with the 2025 experiment–theory window of 38(63) × 10−11 . The normalization is FIXED to this LbL
class (A725): the colored lines are confined at ΛG2 , so the distortion runs at the hadronic loop scale
with no high-scale decoupling — an independent dipole-class mµ /Λ2UV suppression is precluded by the
framework’s own architecture. The insertion scales as C (α/π)3 (mµ /ΛG2 )2 Iloop ≈ C × 125 × 10−11
(chirality-preserving; the flip is supplied by mµ exactly as in standard hadronic light-by-light). No
headline claim of this suite is shifted at its quoted precision by four-point non-associativity.
Associator g − 2 null test [proved, A721, S63]. The {µ, γ} subsystem lies in the two-generator
singlet subalgebra ⟨e0 , e7 ⟩ of the verified field assignment (A688/A689), which is associative by Artin’s
theorem. Pure lepton–photon contributions to aµ are therefore free of split-octonion associator
insertions to all orders: the framework predicts aµ = aSM µ in this sector. Non-associative exposure can
enter only through colored internal lines, as a distortion of the hadronic light-by-light class. With the
2025 Theory Initiative values aexp SM −11 and aHLbL = 115.5(9.9) × 10−11 , current
µ − aµ = 38(63) × 10 µ
HLbL
data imply |Cassoc | ≲ 0.55 conservatively, or ≲ 0.086 within the present HLbL theory budget; with
the LbL normalization fixed (A725), a factor-2 window improvement tightens this to |C| ≲ 0.25, and
a stabilized null ∆aµ = 0 ± 15 × 10−11 would give |C| ≲ 0.12. A future confirmed non-hadronic aµ
anomaly would falsify the associative lepton–photon embedding (Paper 7).
3 Weak-angle bridge
Historical Rev8 content.
√ The weak-angle value in this section (3/13) is superseded by the Rev29 opera-
tive value ϕ−3 = 5 − 2 ≈ 0.23607. The current (S55, A697; corrected S124) reading is that
ϕ−3 is a ∼2% structural low-energy target lying on the effective sin2 θW (µ) running curve
at a sub-GeV/GeV-order, scheme- and threshold-dominated crossing (Appendix I §B4;
the once-quoted µ⋆ ≈ 1.8 GeV lies outside the band) — not a Higgs-VEV-scale boundary
condition and not a Z-pole prediction. Paper 4 §6 is authoritative; the Higgs-scale /
246 GeV language retained below is superseded. See Paper 4 §6.
The tree-level weak-angle output carried by the suite is
3
sin2 θW = = 0.2308. (5)
13
This is a tree-level algebraic ratio. The measured Z-pole weak angle is a renormalized quantity
obtained only after running, threshold corrections, and scheme choice.
Weak-angle bridge. The algebraic weak-angle ratio is a tree-level output. The measured
Z-pole quantity requires renormalization, threshold corrections, and scheme matching. Rev8
treats this as a standard QFT bridge problem rather than as a falsification of the tree-level
algebraic relation.
4 Threshold sector
The working threshold picture places heavy Peirce-sector modes near
Mth ∼ 5 TeV (6)
3
with order-of-magnitude effects
δα1 ∼ 10−3 , δ sin2 θW ∼ O(10−4 ). (7)
These are placeholder estimates — threshold calculation not performed. The sign, scale, and qualitative
mechanism are identified, but the detailed spectrum and loop integral are deferred.
sin2 θW√precision status (S55, A697 — corrected reading). The algebraic value sin2 θW =
ϕ−3 = 5 − 2 ≈ 0.23607 is a ∼2% structural target lying on the standard running curve of the effective
weak mixing angle, reached at a sub-GeV/GeV-order, scheme- and threshold-dominated crossing
(corrected S124/Appendix I §B4 — pushed sub-GeV, µcross ∼ 0.15 GeV; the once-quoted 1.8 GeV lies
outside the band); the difference to the Z-pole value 0.23122 ± 0.00006 (PDG 2026 EW review, MS
at MZ ; the ∼2% structural fence stays binding — no σ conversion) is then ordinary running in the
right direction. The matching scale is not derived from the algebra and the scheme is not fixed, so
ϕ−3 is a structural target, not a completed Z-pole prediction and not a theorem at MZ . (The earlier
high/VEV-scale downward-running test and the associated precision-failure/theorem language are
superseded; Paper 4 §6 is authoritative.) AX6 is an independent orbit selector, decoupled from the
value ϕ−3 .1
4.1 Route C: heavy X-boson threshold mechanism
SUPERSEDED (S59–S60, A713/A714) — retained for the record, not operative.
(Rev32.1 note: the σ figures quoted below are S59-era bare cross-scheme numbers, themselves
retyped as comparator distances by A1531/T2 — see Appendix X.) This subsection presented
the 16 eaten Goldstone bosons of Spin(4, 4) → Spin(1, 3)×Spin(3, 1) as a leading-log threshold
mechanism that closes the heavy-quark mass tensions (∆t = +ε, ∆b = −ε). That operative
reading is now retired: (i) A713 (S59) established that the heavy-quark comparator offsets
(mt : +0.87%, dcmp = +5.57; mb : −0.95%, dcmp = −6.64 (full precision, Rev32.9); mc :
+1.49%, dcmp = +4.22; PDG 2026, all cross-scheme and non-status-setting) are genuine
unresolved offsets and are not produced by standard RG/threshold matching — the “these
look like ordinary 1-loop matching corrections” framing is falsified (top pole/MS moves the
wrong way; mb /mc only fit on a tuned scale). (ii) A714 (S60) proved that the SUGRA
scalar-count gap 16 (42 − 26) and this Spin(4, 4) Goldstone 16 (28 − 12) are not the same
object (the equality is a dimensional coincidence; see Paper 4 §4.2), removing the proposed
carrier. The group-theory facts below (28 − 12 = 16, the triality Casimir CX = 4) remain
correct as representation theory, but they may not be invoked as the operative source of the
heavy-quark threshold. Any genuine threshold carrier must come from an explicit interaction
sector, not from this Goldstone count. The arithmetic is preserved here for traceability only.
(iii) A715–A717 (S61–S62) subsequently closed that explicit-sector search as well: the
residual pattern carries a unique weak-isospin T3 signature (δmf /mf = −κT3 (f ), κ ≃ 1.824%),
but no J3 (Os )-intrinsic operator forces it — the weak-adjoint Higgs spurion is writable but
its up/down sign is not forced (A716) — and a boundary-condition revision reduces no
parameter (A717). The heavy-quark sector is carried as bare algebraic boundary values plus
a phenomenological structural T3 threshold term (κ, χc ≃ 1.761 not derived); see Paper 2
§3.6 and Paper 4 §4.2.
1
Tracker-tier note (Rev25 fold; s539, S210; flagged crude): a one-loop exercise finds the MS running ŝ2W (µ) crossing
−3
ϕ near µ ≈ 237 GeV at the 0.03% level. It is a one-loop, scheme-naive computation and is recorded as a curiosity
only — it does not alter the corrected sub-GeV effective-angle reading above, and no scale is derived.
4
Rev12 update (Sessions 48–49) [superseded by A713/A714, S59–S60]. The threshold
sector now has a complete leading-log EFT mechanism. The Goldstone counting from the coset
Spin(4, 4)/[Spin(4) × Spin(4)] yields exactly 28 − 12 = 16 eaten Goldstone bosons, giving 16 heavy
X-vectors with a universal mass matrix MX 2 = g2 v2 1
X X 16 [proved, A669].
The Spin(4, 4) Casimir in the spinor representation is [proved, A670 kernel]:
CX (8v ) = CX (8+ ) = CX (8− ) = 4, (8)
a consequence of D4 triality. With the quark doublets assigned to J13 = 8+ (see Section 4.3), the
Dynkin index is
λX X
t = λb = CX (tL ) + CX (tR ) = 4 + 4 = 8. (9)
The leading-log threshold corrections to the top and bottom pole masses are:
3λXf MX
∆X
f =− 2
2
gX ln ⋆ (10)
16π µ
ay yt2 My
∆yt = + ln ⋆ , yt = 1, ay = 92 . (11)
16π 2 µ
The sign of ∆X f is negative by the MS matching-counterterm convention [convention, A673]; it is
not derived from the (2 − d) propagator numerator. The sign of ∆yt is positive (Yukawa mass upshift).
With cX = 3λX = 24:
Route C result [proved arithmetic, A673].
2 MX My
gX ln = 0.05836226069, ln = 0.6225307807 (ay = 92 ).
µ⋆ µ⋆
∆X X
t = ∆b = −ε, ∆yt = +2ε, ε ≈ 0.00887.
∆t = +ε, ∆b = −ε.
The sign split arises from the large top-Yukawa threshold (yt = 1); the bottom analogue is
suppressed by yb2 /yt2 ≈ 6 × 10−4 . The 16X gauge threshold is universal for top and bottom;
it does not arise from different Peirce block assignments.
Status table:
Claim Value Status
CX (8± ) = CX (8v ) 4 proved
λX
t = λb
X 8 proved
ay 9/2 (SM RGE) convention/import
Threshold signs MS counterterm convention
∆t = +ε, ∆b = −ε kernel-exact proved arithmetic
2 ℓ algebraic origin
gX open open
X
ay · ℓy = 2.801 algebraic origin open; lead: ℓy ≈ ϕ−1 (−0.72%, A678) open
5
4.2 Peirce–Yukawa selection theorem
Rev29 update (Session 49). The following theorem is proved for generic split-octonion elements
ψ, h ∈ Os (all 8 components; kernel-verified, A674).
Let
Jvac = λ1 c1 + λ2 c2 + λ3 c3 = ϕ c1 + c2 + ϕ−1 c3 .
For X, Y ∈ Jij , write their split-octonion entries as ψ, h ∈ Os , and let k be the remaining Peirce
index. The Freudenthal cross product satisfies
X × Y = −2 Re(ψ h̄) ck . (12)
Consequently
Tr[Jvac ◦ (X × Y )] = −2λk Re(ψ h̄). (13)
For distinct Peirce blocks Jij ̸= Jkl , the cross product lands in an off-diagonal Peirce block, hence
Tr[Jvac ◦ (Jij × Jkl )] = 0. (14)
The Peirce eigenvalues of Jvac give the tree-level Yukawa strengths:
J13 × J13 7→ −2λ2 = −2, J23 × J23 7→ −2λ1 = −2ϕ, J12 × J12 7→ −2λ3 = −2ϕ−1 . (15)
All nine trace contractions verified symbolically (differences= 0); see kernel d674_peirce_yukawa_
generic_S49.py.
Peirce–Yukawa correspondence [proved, A674]. Each same-block Peirce product
Jij × Jij extracts the vacuum eigenvalue λk of the missing idempotent ck . The three
Peirce eigenvalues (ϕ, 1, ϕ−1 ) are the three allowed tree-level Yukawa couplings in the direct
Freudenthal channel.
4.3 Fermion placement in J13 = 8+
Rev12 update (Sessions 48–49). The selection rule (14) constrains right-handed fermion place-
ment.
Top-quark placement (carrier channel). In the same-block carrier reading (A674; the literal
SM top Yukawa is the cross-block triple {J12 (Hu ), J13 (Q), J23 (uc )} with Hu ∈ 10, A686–A688), one
has QL = (tL , bL ) ∈ J13 and a carrier field ΦH ∈ J13 , and the carrier coupling
Lt ∝ Tr[Jvac ◦ (Ψt × ΦH )] = −2 Re(ψt h̄)
is nonzero if and only if tR ∈ J13 = 8+ . A cross-block assignment tR ∈ J23 gives Tr[Jvac ◦(J23 ×J13 )] = 0
(identically, for all ψ, h ∈ Os ).
tR ∈ J13 = 8+ [proved, A674] (16)
6
Bottom-quark placement. Since the conjugate Higgs Φ e H = iσ2 Φ∗ is an anti-linear weak-doublet
H
operation within the J13 subspace (octonionic conjugation preserves the Hermitian block structure;
e H ∈ J13 [structural, A675]. The same selection
the SU(2)L operation acts inside J13 ), we have Φ
rule then requires
bR ∈ J13 = 8+ [proved conditional, A675]. (17)
This placement is conditional on the direct Freudenthal bottom Yukawa; non-direct insertion mecha-
nisms are not ruled out. With tR , bR ∈ J13 , the 16X Casimir is universal (CX (tR ) = CX (bR ) = 4,
λX X
t = λb = 8) and the Route C sign split is produced by the Yukawa asymmetry yt = 1 ≫ yb ≈ 0.024,
not by different block assignments.
The bottom Yukawa coupling in the J13 channel reads
h i
Lb ∝ Tr Jvac ◦ (Ψb × Φ
eH) = −2ηb Re(ψb h),
e ηb = −1.
ηb
The phase ηb = −1 is currently a phase-convention/alignment datum [open/assumption, A675]; it is
not derivable from octonionic conjugation, Jordan conjugation, or any Z2 automorphism of J13 alone.
Bottom convention [convention, A678]. The J13,C colour decomposition is
J13,C → 1 ⊕ 1 ⊕ 3 ⊕ 3̄.
We use the convention that the 3 component represents the algebraic top singlet tR , while the
conjugate colour component represents the left-handed conjugate bottom field bcR . The physical
right-handed bottom field is bR = (bcR )c , appearing through the Hermitian-conjugate term in the
four-dimensional Lagrangian. This convention resolves the 8-real-dimensional counting problem and
is consistent with CX (8+ ) = 4 and λX
b = 8 [proved, A670/A677, A678]: the Casimir is a Spin(4, 4)
representation-theory quantity, unchanged by field redefinition (kernel-verified, A678). The down-type
phase ηb = −1 is not derived by this convention; it remains [open/assumption, A675].
Hypercharge embedding. With both tR and bR in J13 = 8+ , the hypercharge distinction
Y (tR ) = +2/3 vs Y (bR ) = −1/3 requires U (1)Y to act as an internal generator or projector within the
physical subspace of 8+ , rather than as a scalar on the whole block. The derivation of this projector
is addressed in Section 4.4.
4.4 Hypercharge inside the J13 = 8+ block
The direct Freudenthal Yukawa channel places both right-handed quarks in the same Peirce block,
tR , bR ∈ J13 = 8+ . This does not imply that hypercharge acts as a scalar on J13 . If it did, then
tR and bR would have identical hypercharge. Hence any viable embedding must realize U (1)Y as a
non-scalar internal generator or projector acting within the physical subspace extracted from 8+ .
For the complexified D4 weight lattice, the 8+ weights are
1
8+ : 2 (±e1 ± e2 ± e3 ± e4 ), #{minus signs} ≡ 0 (mod 2). (18)
(Kernel-verified, d676_hypercharge_projector_S49.py.) The weight set is centrally symmetric:
w 7→ −w maps 8+ to itself. Therefore every Cartan charge spectrum on 8+ is symmetric under
q 7→ −q. The SM-like charge multiset
(3, 1, + 23 ) ⊕ (3, 1, − 31 ) ⊕ (1, 2, − 12 )
7
is not q 7→ −q symmetric, and therefore cannot arise as a single 8+ Cartan branching [proved
obstruction, A676].
Furthermore, the physical compact group SU (3)c × SU (2)L × U (1)Y is not a compact subgroup of
Spin(4, 4): the maximal compact subgroup of Spin(4, 4) is Spin(4) × Spin(4) ≃ SU (2)4 , and no simple
su(3) subalgebra (dim 8) can embed in su(2)4 (each factor dim 3) [proved, A676]. The diagonal
Peirce operators Jvac , LJvac , and Π13 act as scalars within J13 and are incapable of separating tR
from bR by hypercharge [proved/structural, A676].
A chosen Cartan functional can distinguish individual 8+ weight-level charges. For example, HY =
( 12 , 21 , 16 , 16 ) assigns Y (w0 ) = +2/3 and Y (w6 ) = −1/3 (kernel-verified). However this direction is
chosen, not derived, and the full SM charge multiplicities cannot be realized by any single Cartan
direction (centrally symmetric obstruction above).
We therefore introduce the SM gauge embedding as an explicit structural axiom.
Hypercharge embedding datum (earlier A676/A677 form — superseded). This paragraph
records the pre-B structural datum (formerly “Axiom HY1”); it is superseded by the B + SU (5)
hypercharge derivation later in this section, where all sixteen hypercharges are derived rather than
assumed. It is retained only to motivate the SM embedding chain. The Standard Model hypercharge
generator was taken as a chosen compact generator Y ∈ f4(4) whose action on the physical subspace
of J13 = 8+ has eigenspaces
Y ΠtR = + 23 ΠtR , Y ΠbR = − 13 ΠbR . (19)
The projectors ΠtR and ΠbR are internal projectors inside the physical 8+ extraction, consistent with
the SM gauge embedding
ιSM : SU (3)c × SU (2)L × U (1)Y ,→ F4(4) . (20)
This datum is required because the diagonal Peirce operators act as scalars on J13 (Section 4.3).
Octonionic colour decomposition (A677). The Peirce entry x ∈ J13 = ∼ Os has 8 real components.
Choosing a distinguished compact imaginary unit e7 , right multiplication Re7 defines a compact
complex structure on the six-real complement W6 = spanR {e1 , . . . , e6 } (kernel-verified for compact O;
Re27 = −1). The +i eigenbasis
u1 = e1 − ie6 , u2 = e2 + ie5 , u3 = e3 + ie4
spans the colour triplet, giving the complexified decomposition
J13,C → 1 ⊕ 1 ⊕ 3 ⊕ 3̄ under SU (3)c .
The 3 ⊕ 3̄ is the complexification of one six-real colour module, not two independent colour triplets.
This resolves the dimensional puzzle: a single 8-real J13 block cannot contain both a full tR and
a full bR as independent colour triplets; instead tR ∼ 3 and bcR ∼ 3̄ share the same colour sector.
For the split algebra Os , compact SU (3)c is not a subgroup of split G2(2) (five of eight compact
su(3) generators fail as Os -derivations, kernel A677). Physical compact colour is therefore placed in
U Sp(6) ⊂ KF4 = U Sp(6) × U Sp(2).
8
Explicit ιSM chain (A677, kernel-verified). The embedding chain
SU (3)c × SU (2)L × U (1)q ,→ U (3) × U Sp(2) ,→ U Sp(6) × U Sp(2) ⊂ F4(4) (21)
is realised at the Lie-algebra level by
!
A 0
A ∈ su(3) 7→ ∈ usp(6),
0 −AT
which satisfies X † + X = 0 and X T Ω + ΩX = 0 for all eight su(3) generators (kernel-verified, max
error = 0). This gives 6U Sp(6) → 3q ⊕ 3̄−q and the 26-dimensional branching
26F4 → (8, 1)0 ⊕ (3, 1)−2q ⊕ (3̄, 1)+2q ⊕ (3, 2)+q ⊕ (3̄, 2)−q .
The branching contains SM-like colour-isospin structure, but the charge normalization q and the
physical projectors ΠtR , ΠbR are not fixed by the Jordan data. The embedding ιSM is therefore
[structural/assumption]—the chain is natural and explicit but not canonically derived. Note:
Todorov–Drenska (arXiv:1805.06739) obtain the SM gauge group as Spin(9) ∩ (SU (3) × SU (3)/Z3 ) ⊂
F4 using the compact F4 ; that intersection route is distinct from the U Sp(6) × U Sp(2) maximal-
compact route of F4(4) used here.
Hypercharge status (updated A689). The earlier datum (Eq. (19)) had status [STRUCTURAL/
ASSUMPTION] as of A677, but is now superseded: all sixteen Standard Model hypercharges are
DERIVED from the B + SU (5) exterior-index assignment (A688/A689; see below). The operative
statement is therefore YSM [derived under explicit B], with the only residual freedom the canonical
selection of the B-map orientation [OPEN]. “Axiom HY1” is accordingly retired as an independent
axiom. The explicit embedding chain (21) has been kernel-verified, and the Route C Casimir
CX (8+ ) = 4 (Eq. (8)) is independent of colour sublabelling and remains [PROVED] (A670, A677,
A678).
Lepton embedding status [open, A678/A679]. For all values of the charge unit q (q = 16 , 13 , 12 ),
the 26F4 branch above contains no color-singlet weak doublet (1, 2)Y and no charged-lepton singlet
(1, 1)−1 (kernel-verified, A678). The SM lepton projectors are therefore not fixed by the present
ιSM chain and remain an open structural datum. This is independent of the proved charged-lepton
mass-ratio formulas (Paper 2, A616/A618/A636/A637).
Status of the E6(6) extension [structural, A679]. The reduced structure group of J3 (Os )
is Str0 (J3 (Os )) ≃ E6(6) , and J3 (Os ) is its real 27-dimensional minimal representation. At the
complexified E6 level, the standard GUT branching 27 → 16 ⊕ 10 ⊕ 1 contains one full SM generation,
Q, uc , dc , L, ec , ν c , inside the SO(10) spinor 16. However, the compact SO(10) × U (1) chain does
not transfer as a real subgroup chain of E6(6) ; the split-form analogue is SO(5, 5) × SO(1, 1) ⊂ E6(6)
(kernel-verified, A679). Under the current F4(4) -nested branch 27 ↓ F4(4) = 26 ⊕ 1, the extra singlet
carries only (1, 1)0 quantum numbers. The E6(6) route is [STRUCTURAL]: it supplies the right
ambient representation, but an explicit intertwiner mapping the complex E6 lepton slots to the
suite’s Peirce quark embedding is not yet derived. The SO(5, 5) × SO(1, 1) sub-chain, under which
27 → 16 + 10 + 1 with the 16 a split-real spinor, is the next candidate; see below.
9
The SO(5, 5) spinor branch [structural, A681]. The split reduced-structure group E6(6) =
Str0 (J3 (Os )) contains the split subgroup SO(5, 5) × SO(1, 1), under which the real 27 branches as
(Hohm–Samtleben exceptional field theory)
27R → 16R ⊕ 10R ⊕ 1R .
The 16R is a real Majorana–Weyl spinor of Spin(5, 5) (Clifford-algebra kernel-verified, A681). After
complexification and a chosen SU(5) → SU (3)c × SU (2)L × U (1)Y embedding, one obtains
16C → (3, 2)1/6 ⊕ (3̄, 1)−2/3 ⊕ (1, 1)+1 ⊕ (3̄, 1)+1/3 ⊕ (1, 2)−1/2 ⊕ (1, 1)0 .
Thus the missing lepton doublet L = (1, 2)−1/2 and charged-lepton singlet ec = (1, 1)+1 are present in
the complexified SO(5, 5) spinor branch. However, compact SU (3)c is not contained in the maximal
compact Spin(5) × Spin(5) of Spin(5, 5) [proved obstruction, A681]; the result is [STRUCTURAL]
until the required complex structure and the Peirce-to-SM intertwiner are derived.
Peirce-sector compatibility via Spin(4, 5) [structural, A681]. The compatibility with the
F4(4) quark sector is controlled by the overlap
F4(4) ∩ SO(5, 5) ≃ Spin(4, 5), dim = 36.
Under this overlap the 26F4 branches as 16 ⊕ 9 ⊕ 1, and restricting to the previously used Spin(4, 4) ⊂
Spin(4, 5) gives
16 → 8+ ⊕ 8− , 9 → 8v ⊕ 1,
hence
26F4 → J13 ⊕ J23 ⊕ J12 ⊕ 1 ⊕ 1 = 8+ ⊕ 8− ⊕ 8v ⊕ 1 ⊕ 1,
matching the Peirce blocks plus the two traceless diagonal directions. The proved placement
tR ∈ J13 = 8+ is therefore compatible with the SO(5, 5) spinor route. The outstanding gap is
the explicit intertwiner
SU (5)
J13 ⊕ J23 = 8+ ⊕ 8− ←→ 16C = Q ⊕ uc ⊕ dc ⊕ L ⊕ ec ⊕ ν c .
Constructing this map was the target of D682; the result is below.
Peirce–SM intertwiner status [open, A682]. Using the exterior-algebra model S+ = Λeven R5 ,
the Spin(4, 4) chirality operator Γ4,4 = Γ0 Γ1 Γ2 Γ3 Γ5 Γ6 Γ7 Γ8 splits S+ = 8+ ⊕ 8− [proved, A682]. The
computed eigenspaces are
8+ : ν c ⊕ uc ⊕ Qweak,3 ⊕ Lweak,3 , 8− : Qweak,4 ⊕ ec ⊕ Lweak,4 ⊕ dc .
However, the SU(2)L generators (E34 , E43 ) map 8+ ↔ 8− : the quark and lepton doublets Q and L are
each split across the two chiralities [proved obstruction, A682]. Thus the naive Γ4,4 split does not
preserve complete SU(2)L multiplets and cannot yet be identified with J13 ⊕ J23 . The Peirce–Yukawa
theorem and the proved placement tR ∈ J13 = 8+ are unaffected by this obstruction. Three resolution
paths were explored in D683A–C.
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