Expositions · Premise notes · PRM-013
PRM-013 · DET-7 structural postulate
Register page /premises/PRM-013/ · kind POSTULATE · claims declared to rest on it today: LIB2-019, LIB2-020, LIB2-051, LIB2-066, LIB2-356
Drafted by the ChatGPT drafting lane (A1634), verified by the house, published under R76, R78 and R81; house insertions are marked. What a premise note is.
Premise
Register note (R78-d; R81: no change). Status words differ in print: "structural postulate" (App E) and "named structural input, not promoted to an axiom" (App X). Recorded for the PI; the PI ruled no change to the row.
Register kind: POSTULATE. Located suite statement:
DET-7 enters this appendix as a structural \emph{postulate}, not a derived necessity; the result is therefore a \emph{conditional} selector theorem.Appendix E · Status of (H2) — honest label · PDF p.164 · appendix_e_vacuum_selector.tex:98.
What kind of premise this is
POSTULATE is the register kind. Appendix E names H2 as the Gram equality and explicitly makes its selector conditional. H0 is unit norm; Route A adds inversion symmetry, while Route B adds a golden-field restriction. Neither route derives the necessity of seven. The first-principles origin remains open in Appendix X, whose separate structural-input wording should not be silently rewritten into a uniform axiom declaration. Source status words: structural \emph{postulate} (appendix_e_vacuum_selector.tex:98).
Where the suite invokes it
Located invocations in the read excerpts, not an exhaustive full-source-tree census. Pages refer to saved Rev32.7 page-delimited text, not PDF-image authentication.
| # | locator | quote (verbatim, ≤ 30 words) | role | argument |
|---|---|---|---|---|
| 1 | appendix_e_vacuum_selector.tex:44Appendix E · Setup and hypotheses · PDF p.163 | $\det\mathrm{Gram}(\vac,\vac^{\#})=7$ | STATED | DET-7 hypothesis H2 |
| 2 | appendix_e_vacuum_selector.tex:98Appendix E · Status of (H2) — honest label · PDF p.164 | DET-7 enters this appendix as a structural \emph{postulate}, not a derived necessity | STATED | Status of H2 |
| 3 | appendix_e_vacuum_selector.tex:65Appendix E · DET-7 invariant in elementary form · PDF p.164 | Setting this equal to $7$ gives \eqref{eq:det7-16}. | LOAD_BEARING | Gram-product normalization |
| 4 | appendix_e_vacuum_selector.tex:79Appendix E · Route A: symmetry + DET-7 · PDF p.164 | A real positive diagonal $\vac\in\J$ satisfying \emph{(H0)--(H2)} is unique up to permutation: | CONDITIONAL | Symmetry-form vacuum selector |
| 5 | appendix_e_vacuum_selector.tex:117Appendix E · Route B: golden-field units + DET-7 · PDF p.165 | Then \emph{(H2)} forces $\vac=\diag(\phi,1,\phi^{-1})$ up to permutation. | CONDITIONAL | Arithmetic vacuum selector |
| 6 | appendix_e_vacuum_selector.tex:121Appendix E · Route B: arithmetic proof · PDF p.165 | By Lemma~\ref{lem:e-det7}, (H2) reads | LOAD_BEARING | Route-B gap classification |
| 7 | p3_ckm_pmns_mixing.tex:582Paper 3 · Mixing summary table · PDF p.55 | AX2 eliminated: $\|J\|^2=4$ from DET-7+$N(J)=1$ | LOAD_BEARING | Internal mismatch normalization |
| 8 | appendix_x_zero_parameter_input_ledger.tex:65Appendix X · Fit-free-interior ledger — C1 coefficient · PDF p.319 | Exact $c=1/2$ from DET-7, Frobenius norm, and Peirce orthogonality. | LOAD_BEARING | C1 generator coefficient |
| 9 | appendix_x_zero_parameter_input_ledger.tex:73Appendix X · Fit-free-interior ledger — Quark 2–3 entry · PDF p.320 | DET-7 ``7 unification''; inherits the loaded atmospheric attachment | LOAD_BEARING | Retained quark 2–3 relation |
| 10 | appendix_e_vacuum_selector.tex:164Appendix E · Spectral corollary and the solar-angle theorem · PDF p.166 | From $\rho^2+\rho^{-2}=3$ at $\rho=\phi$ (Theorem~\ref{thm:routeA}). | CONDITIONAL | Selected-spectrum identity |
| 11 | appendix_b_independent_verification.tex:131Appendix B · B.5 theorem summary · PDF p.153 | Gram determinant $7$, fixing the mismatch geometry and the normalization of the $(1,3)$ sector. | MENTIONED | Evaluation on the selected vacuum |
| 12 | appendix_e_vacuum_selector.tex:34Appendix E · Abstract · PDF p.163 | Route~B re-derives the same vacuum arithmetically as the unique golden-field unit triple compatible with DET-7. | CONDITIONAL | Arithmetic selector summary |
| 13 | appendix_e_vacuum_selector.tex:94Appendix E · Route A residual-input remark · PDF p.164 | The only structural hypotheses are (H1) inversion symmetry and (H2) DET-7; (H0) is the normalization. | MENTIONED | Named Route-A assumptions |
| 14 | appendix_e_vacuum_selector.tex:106Appendix E · Status of H2 · PDF p.164 | A first-principles derivation of DET-7 itself remains open and is carried as such. | MENTIONED | Unresolved origin of the postulate |
| 15 | appendix_e_vacuum_selector.tex:111Appendix E · Origin of phi remark · PDF p.165 | The golden ratio enters solely through the selector theorem above (unit determinant $+$ inversion/palindromic symmetry $+$ DET-7) | MENTIONED | Selector rather than richer origin claims |
| 16 | appendix_e_vacuum_selector.tex:144Appendix E · Route-B residual-input remark · PDF p.165 | Each, together with (H0) and DET-7, removes all continuous freedom. | MENTIONED | Shared equality input of two routes |
| 17 | appendix_e_vacuum_selector.tex:509Appendix E · Status · PDF p.172 | conditional on the named hypotheses (unit norm; inversion symmetry, or the golden-field restriction; DET-7) | CONDITIONAL | Current selector status |
| 18 | appendix_e_vacuum_selector.tex:513Appendix E · Status · PDF p.172 | It shares DET-7 / unit-coupling as structural inputs and does \emph{not} upgrade the tier of this selector | MENTIONED | Separately scoped variational probe |
| 19 | appendix_x_zero_parameter_input_ledger.tex:63Appendix X · Fit-free-interior ledger · PDF p.319 | uniquely forced by unit determinant $+$ inversion symmetry $+$ DET-7 (field-free) | CONDITIONAL | Working-vacuum ledger |
| 20 | appendix_x_zero_parameter_input_ledger.tex:64Appendix X · Fit-free-interior ledger · PDF p.319 | carried as a named structural input, not promoted to an axiom. | MENTIONED | Ledger wording of DET-7 status |
| 21 | appendix_x_zero_parameter_input_ledger.tex:69Appendix X · Fit-free-interior ledger · PDF p.319 | DET-7 mismatch geometry exact; lower-octant registration selected against data | LOAD_BEARING | Internal geometry versus physical registration |
| 22 | appendix_x_zero_parameter_input_ledger.tex:96Appendix X · Fit-free-interior ledger · PDF p.324 | DET-7 Gram-ratio bridge. \emph{Independent route}: | MENTIONED | Gram route versus vacuum-free dimension route |
| 23 | p0_framework_foundations.tex:701Paper 0 · Mixing provenance ledger · PDF p.14 | $n_{23}=7$ from $\det G(\mathrm{diag}(\phi,1,\phi^{-1}))=7$ (DET-7 theorem; Gram matrix criterion); AX2 eliminated | MENTIONED | Selected-vacuum Gram evaluation |
| 24 | p2_mass_hierarchy_resolvent_quintics.tex:517Paper 2 · Seven-unification bridge · PDF p.39 | The DET-7 theorem then confirms $n_{23}=7$ for $J_{\rm vac}=\mathrm{diag}(\phi,1,\phi^{-1})$ | MENTIONED | Independent-route consistency check |
| 25 | p3_ckm_pmns_mixing.tex:576Paper 3 · Mixing summary table · PDF p.54 | $|V_{cb}|$ & $1/(9\sqrt7)$ via DET-7 `7~unification' | LOAD_BEARING | Repeated quark 2–3 provenance |
House additions — R78-c (S327i)
The house read the rest of the suite for this premise. The material sent to the drafting lane showed at most twelve passages per premise, so A1634 FINDINGS §7 lists 14 hit lines for PRM-013 that the lane never saw. Under R78-c the house read all 14 in the full suite TeX: 12 passages are added below, 2 are TeX source comments. Every quote was matched against the TeX at its locator. With the lane's table above, every hit of this premise's search keys is now accounted for.
| # | locator | quote (verbatim, TeX source) | role | argument |
|---|---|---|---|---|
| H1 | p0_framework_foundations.tex:621PDF p.12 | unit determinant, inversion symmetry, and the DET-7 condition force $\rho^2+\rho^{-2}=3$, hence $\rho=\phi$ | CONDITIONAL | Paper 0 framework: the vacuum selector of Appendix E |
| H2 | p2_mass_hierarchy_resolvent_quintics.tex:368-370PDF p.36 | The DET-7 invariant $n_{23}=\det(\mathrm{Gram}(J_{\rm vac},J_{\rm vac}^\#))=7$ and the Jordan rank $\mathrm{rank}(J_3)=3$ together force the ratio $n_{23}/\mathrm{rank}=7/3$ | LOAD_BEARING | The 7/3 heavy-quark bridge: n23 = 7 over rank 3 |
| H3 | p2_mass_hierarchy_resolvent_quintics.tex:413-414PDF p.37 | The DET-7 theorem gives the MS-bar mass directly at the self-consistent threshold $\mu=m_Q$ | LOAD_BEARING | The 7/3 bridge: the mass read at the self-consistent threshold |
| H4 | p2_mass_hierarchy_resolvent_quintics.tex:533-536PDF p.39 | This identity follows from the DET-7 Gram structure and Peirce orthogonality | LOAD_BEARING | Freudenthal complementarity R12 + R23 = 1 (E49) |
| H5 | p2_mass_hierarchy_resolvent_quintics.tex:669PDF p.41 | no status (Rev32.1 retype, A1531/T2; \S3.6, App.~X). DET-7 threshold. | MENTIONED | Mass ledger row: bottom, 7/3 unification, DET-7 threshold |
| H6 | p2_mass_hierarchy_resolvent_quintics.tex:670-674PDF p.41 | cross-scheme audit metadata, not status. DET-7 threshold. | MENTIONED | Mass ledger row: charm, 7/3 unification, DET-7 threshold |
| H7 | p2_mass_hierarchy_resolvent_quintics.tex:715-716PDF p.43 | Bottom and charm masses are unified via the 7/3 DET-7 bridge at STRUCTURAL $\star\star\star\star$ accuracy. | MENTIONED | Paper 2 summary: bottom and charm unified via the 7/3 bridge |
| H8 | p3_ckm_pmns_mixing.tex:322PDF p.50 | $\|J_{\rm vac}\|^2=4$ is now a theorem from DET-7+$N(J)=1$ (AX2 eliminated, D575), making $7/16$ a corollary of that theorem | LOAD_BEARING | The internal mismatch identity 7/16 as a corollary of |J_vac|^2 = 4 |
| H9 | p3_ckm_pmns_mixing.tex:362-375PDF p.50 | Proof uses only $\varphi^2+\varphi^{-2}=3$ and $\langle J_{\rm vac},J_{\rm vac}^\#\rangle=3$ (both forced by AX2). Consequently $n_{23}=7$ is derived | STATED | Theorem 3.2 (DET-7 / Moire derivation): det Gram = 7 = n23 |
| H10 | p3_ckm_pmns_mixing.tex:707-709PDF p.59 | $\|J_{\rm vac}\|^2=4$ is a theorem from DET-7 $+\,N(J)=1$, which eliminated AX2 (D575) and makes $7/16$ a corollary of that theorem | LOAD_BEARING | The internal mismatch identity 7/16 (retier section) |
| H11 | p5_neutrino_cp_violation.tex:267PDF p.84 | is proved exactly from the DET-7 ratio, Frobenius norm, and Peirce orthogonality | LOAD_BEARING | C1 theorem: the weight c = 1/2 in T_half |
| H12 | p5_neutrino_cp_violation.tex:283PDF p.84-85 | the internal F$_4$ mismatch theorem ($\|J_{\rm vac}\|^2=4$ from DET-7+$N(J)=1$, AX2 eliminated) is attacked only if that attachment is independently derived | MENTIONED | DUNE-era falsification table: the theta23 row |
House reading notes:
p0_framework_foundations.tex:621— WORDING CHECK (REV32.8 candidate): the same sentence ends "not a free parameter or an externally carried assumption", while Appendix E types DET-7 as a structural postulate (LIB2-018). Not assessed past the print.p2_mass_hierarchy_resolvent_quintics.tex:368-370— CONSIDERED, NOT PROPOSED: LIB2-130 (m_b = (7/3)m_tau) and LIB2-131 (m_c = (7/3)m_s^const) use this ratio, but appendix_x_zero_parameter_input_ledger.tex:96 prints an independent vacuum-free route to the same 7/3 (dim Im(O_s) / rank J3, A641/Q2A), so neither bridge holds ONLY given DET-7 (R78-a). The house first drafted this as a proposal and withdrew it on reading App X:96; A1634 FINDINGS reached the same point.p2_mass_hierarchy_resolvent_quintics.tex:413-414— See the row at line 368: no edge proposed (vacuum-free 7/3 route, App X:96).p2_mass_hierarchy_resolvent_quintics.tex:533-536— No Registrar record carries this identity (searched by statement and title).p3_ckm_pmns_mixing.tex:362-375— WORDING CHECK (REV32.8 candidate): Theorem 3.2 prints n23 = 7 as "derived" from phi^2 + phi^-2 = 3 "(both forced by AX2)", while Appendix E types DET-7 as a structural postulate and this paper prints AX2 as eliminated by DET-7 (lines 322, 708). Its input phi^2 + phi^-2 = 3 is the DET-7 equation at rho = phi, so the house does not read this as a discharge.p5_neutrino_cp_violation.tex:267— Edge declared S327i (R76, from A1634).
Hit lines that are not this premise:
p3_ckm_pmns_mixing.tex:437— A % source comment about theorem numbering; it is not printed.p3_ckm_pmns_mixing.tex:439— A % source comment about theorem numbering; it is not printed.
Arguments that rest on it
LIB2-018 — DISPUTE. This record states how DET-7 is classified and used; it does not assume the equality to prove a mathematical conclusion. A record ABOUT the premise should retain that association rather than automatically receiving a rests_on_premise proof edge. Step locator: appendix_e_vacuum_selector.tex:98 (STATED).
LIB2-019 — CONFIRM. H2 is explicitly among the positive-diagonal symmetry-form theorem hypotheses. This is conditional spectrum selection, not a derivation of the necessity of DET-7. Step locator: appendix_e_vacuum_selector.tex:79 (CONDITIONAL).
LIB2-020 — CONFIRM. The arithmetic theorem explicitly invokes H2 after its golden-field/unit assumptions. Confirm this route-specific equality input, not the CKM Route-B convention. Step locator: appendix_e_vacuum_selector.tex:117 (CONDITIONAL).
LIB2-356 — CONFIRM. The source derives the normalization from DET-7 plus N(J)=1 for the internal mismatch. It separately leaves physical atmospheric registration loaded. Step locator: p3_ckm_pmns_mixing.tex:582 (LOAD_BEARING).
LIB2-001 — ADD. The printed gap classification is performed with F=16 supplied by H2 through the Gram lemma. The record is that arithmetic classification, not an independent CKM construction. Step locator: appendix_e_vacuum_selector.tex:121 (LOAD_BEARING).
LIB2-051 — ADD. The coefficient ledger explicitly gives DET-7, Frobenius norm and Peirce orthogonality as sources of the exact C1 weight carried by this record. Step locator: appendix_x_zero_parameter_input_ledger.tex:65 (LOAD_BEARING).
LIB2-066 — ADD. The retained formula’s seven is explicitly attributed to DET-7. Its atmospheric readout and integer nine remain separately loaded; this edge does not derive either. Step locator: appendix_x_zero_parameter_input_ledger.tex:73 (LOAD_BEARING).
LIB2-046 — ADD. The selected-spectrum identity is proved here by the Route-A theorem. This proposal records that printed selector-route provenance; it is not a claim that every proof of a golden-number identity must assume DET-7. Step locator: appendix_e_vacuum_selector.tex:164 (CONDITIONAL).
What breaks without it
LIB2-019, LIB2-020 and LIB2-001: the displayed proof sets the Gram determinant to seven to obtain the product normalization sixteen (appendix_e_vacuum_selector.tex:65,121–123). Removing H2 removes that specified step; no replacement selector is printed.
LIB2-356 and LIB2-051: their displayed normalization/coefficient provenance explicitly includes DET-7 (p3_ckm_pmns_mixing.tex:582; appendix_x_zero_parameter_input_ledger.tex:65). NOT ASSESSED — replacement values without H2 are not supplied.
LIB2-066: the printed seven-source is DET-7, but atmospheric registration and integer nine remain loaded (appendix_x_zero_parameter_input_ledger.tex:73). NOT ASSESSED — no alternate quark readout is evaluated.
LIB2-046: the printed selected-spectrum proof cites Route A (appendix_e_vacuum_selector.tex:164); no universal necessity claim about all proofs is made.
Registrar sync
Registrar state after R76, R78 and R82 (S327j, master0e1418ad6501767d). Declared today: LIB2-019, LIB2-020, LIB2-051, LIB2-066, LIB2-356. Declared at S327i from A1634 after the house read each passage (R76): LIB2-051, LIB2-066. Retired, each recorded in full in the record's version history: LIB2-018 (R78-a) — The record states the status of DET-7 (it is ABOUT the premise); its result does not assume it. Proposed ADDs not declared: LIB2-001 — declined under R78-a — the classification finds where the DET-7 equation holds; it is about the premise, and its result does not assume it; LIB2-046 — refused by the house — φ² + φ⁻² = 3 is unconditional arithmetic about φ; only its use as a selector is premised. The R78-c reading found no further record whose result assumes this premise. The lane's lines below describe the master as it was supplied to the lane (d0aa26db154ff43f), before these changes.
Declared today in the supplied material, Registrar master d0aa26db154ff43f: LIB2-018, LIB2-019, LIB2-020, LIB2-356. Supplied transitive-only closure: LIB2-002, LIB2-065, LIB2-066, LIB2-067. It is not itself a new direct-edge declaration. Draft dispositions — ADD: LIB2-001, LIB2-051, LIB2-066, LIB2-046; CONFIRM: LIB2-019, LIB2-020, LIB2-356; DISPUTE: LIB2-018. LIB2-018 describes premise status; the disputed edge is not a disputed quotation. LIB2-336 and LIB2-337 have no record predicate and cannot carry a proposed argument. LIB2-002 and LIB2-065/067 concern physical registrations or inherited readouts. They are not automatically direct DET-7 edges. The bare 7/3 dimension route is explicitly vacuum-free at appendix_x_zero_parameter_input_ledger.tex:96; no unavoidable DET-7 dependence is inferred for that alternative.
Tier and what this does not show
A premise is not a result, and invocation count is not importance. An ADD remains a proposal until house verification under R76; a DISPUTE goes to the PI. No record or premise is upgraded, downgraded, or edited. Unlocated dependencies remain unknown.
Sources
TeX: appendix_b_independent_verification.tex:131; appendix_e_vacuum_selector.tex:106; appendix_e_vacuum_selector.tex:111; appendix_e_vacuum_selector.tex:117; appendix_e_vacuum_selector.tex:121; appendix_e_vacuum_selector.tex:144; appendix_e_vacuum_selector.tex:164; appendix_e_vacuum_selector.tex:34; appendix_e_vacuum_selector.tex:44; appendix_e_vacuum_selector.tex:509; appendix_e_vacuum_selector.tex:513; appendix_e_vacuum_selector.tex:65; appendix_e_vacuum_selector.tex:79; appendix_e_vacuum_selector.tex:94; appendix_e_vacuum_selector.tex:98; appendix_x_zero_parameter_input_ledger.tex:63; appendix_x_zero_parameter_input_ledger.tex:64; appendix_x_zero_parameter_input_ledger.tex:65; appendix_x_zero_parameter_input_ledger.tex:69; appendix_x_zero_parameter_input_ledger.tex:73; appendix_x_zero_parameter_input_ledger.tex:96; p0_framework_foundations.tex:701; p2_mass_hierarchy_resolvent_quintics.tex:517; p3_ckm_pmns_mixing.tex:576; p3_ckm_pmns_mixing.tex:582. Registrar: LIB2-001, LIB2-002, LIB2-018, LIB2-019, LIB2-020, LIB2-046, LIB2-051, LIB2-065, LIB2-066, LIB2-067, LIB2-336, LIB2-337, LIB2-356. Premise: PRM-013. Suite-cited kernels: s501.
Built by scripts/premise_notes_build.py from Coalition/library/textbook/premise_notes (MANIFEST verified) · Registrar master sha256 e64bfa7ef06444bf9ebd1abb9bf2c9b7562264ad903d1f881f65c38a19ce07be
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