PUBLISHED under R67 (condition T-B) — NOT house-adjudicated. Build S371a. What PUBLISHED means here (R72): two non-drafting AI lanes (gemini + grok) read this record's anchor citation on different objects and agreed it SUPPORTS the record's statement; no citation on the record is disputed as support; its machine certificate is current. What it does not mean: no person has signed off on this record (adjudication: NOT_ADJUDICATED), the drafting lane's own reads count for nothing, and 3 of 3 unblocked uses were REVIEWED — the rest were read by one lane or none. Every label shown is the label as printed in the suite of record (Rev33.1_S369 (PDF sha256 b64ddcd9e16a6dcd…)); no house tier is assigned. The state is recomputed from the uses on every build and is withdrawn if they stop justifying it. Receipt: condition R67-TB · anchor SU-3654d8d7a6cb87f05a96690f · master aa16040bb884101e · S342.

LIB2-172 · The positive completion cone has a strict coefficient domain

class CLAIMmath status (as printed) NOT_PRINTEDattachment (as printed) NOT_PRINTEDscorecard tier word (as printed) none printedhouse tier not yet assignedreview state PUBLISHEDentry version 11

Label sentence at the locator (PDF p.305 (bytes of record)):

Bα,β = αBC + βBMM > 0 ⇐⇒ α > 0, |β| < α,

Statement

The registered quadratic completion Bα,β = αBC + βBMM is positive definite exactly when α > 0 and |β| < α; its nonzero boundary α = |β| > 0 is chiral-null, and the apex α = β = 0 is the zero form.

Parts of the statement (2)

Each part is a clause of the statement above, cut so that one part makes one claim. A reader naming the clause a quote supports can name the part.

p1 · The registered quadratic completion Bα,β = αBC + βBMM is positive definite exactly when α > 0 and |β| < α (one claim)
p2 · its nonzero boundary α = |β| > 0 is chiral-null, and the apex α = β = 0 is the zero form. (one claim)

Scope

carrier	The two-parameter completion of the Cartan and MacDowell–Mansouri quadratic forms
domain	The displayed quadratic completion, not an established nonlinear gravitational continuum measure.
hypotheses	Use the forms BC and BMM on the registered generator space.
quantifier	The registered construction and its stated premises.

What this does not show. Positive quadratic completion does not establish Einstein universality, a continuum limit or generic nonlinear reflection positivity.

Scientific status: UNDER REVIEW — the house has not adjudicated CURRENT / CONTESTED for this record (charter §10.3); a drafted claim is a claim under review, not a result.

Sources in the suite of record

Key — use ≠ record. A review state printed on a source use below names that use — one citation of the suite, read by the lane named in its warrant — not this record. The record's own review state is the one in the table above and is unchanged by any use-level state; a use may be LANE_PROPOSED or REVIEWED while the record it sits on remains IN_REVIEW (R45, R49).

Quotes are verbatim from the sha-pinned text extraction of the Rev33.1 PDF (A1584 §2.2, S309): mathematical typography — radicals, exponents, ≅ — can appear displaced in the extracted line. The PDF page is authoritative; the quote is what the machine gate matched.

Appendix T, §7, Proposition 7.4: positive completion cone — PDF p.322 (bytes of record)
review state: REVIEWED · role SUPPORTS · warrant: grok read OUR_QUOTE_AS_PRINTED → SUPPORTS · gemini read OUR_QUOTE_AS_PRINTED → SUPPORTS · gemini read SUITE_TEXT_EXTRACT → SUPPORTS · grok read THE_RECORD → SUPPORTS · grok read THE_RECORD → SUPPORTS · gemini read SUITE_TEXT_EXTRACT → SUPPORTS

Bα,β = αBC + βBMM > 0 ⇐⇒ α > 0, |β| < α,

Appendix T, §7, Proposition 7.4: boundary — PDF p.322 (bytes of record)
review state: REVIEWED · role SUPPORTS · warrant: grok read OUR_QUOTE_AS_PRINTED → SUPPORTS · gemini read OUR_QUOTE_AS_PRINTED → SUPPORTS · gemini read SUITE_TEXT_EXTRACT → SUPPORTS · grok read THE_RECORD → SUPPORTS · grok read THE_RECORD → SUPPORTS · gemini read SUITE_TEXT_EXTRACT → SUPPORTS

Bα,β = αBC + βBMM > 0 ⇐⇒ α > 0, |β| < α, with spectrum {4α×4 , 4(α + β)×3 , 4(α − β)×3 }; the nonzero boundary α = |β| > 0 is chiral- null (inertia (7, 0, 3)); the apex α = β = 0 is the zero form

Appendix T, §7, Remaining continuum and gravity conditions — PDF p.322 (bytes of record)
review state: REVIEWED · role CONTEXT · warrant: grok read OUR_QUOTE_AS_PRINTED → CONTEXT · gemini read OUR_QUOTE_AS_PRINTED → CONTEXT · gemini read SUITE_TEXT_EXTRACT → CONTEXT · grok read THE_RECORD → CONTEXT · grok read THE_RECORD → CONTEXT · gemini read SUITE_TEXT_EXTRACT → CONTEXT

a simplicity/constraint mechanism giving GR rather than Cartan Yang–Mills; and the continuum RG limit.

Evidence (availability is not outcome)

kind	reference	line / key	availability	note
analytic	Appendix T §7, Proposition 7.4, p.305	—	shipped	The source prints the coefficient criterion and spectrum; no eigenvalue or positivity calculation is performed here.

The numbers this record declares, and where each one's digits come from

This table covers the values this record DECLARES, not every numeral printed on this page: group labels such as E7(7), section numbers and page numbers are not declared values and are not listed here.

value as printed	page	digit source
`0`	PDF p.305 (bytes of record)	closed form: Bα,β > 0 iff α > 0 and \|β\| < α, printed in Proposition 7.4

Typed links

type	target	licensed by
`related_to` (never evidential)	LIB2-171 The pure MacDowell–Mansouri quadratic form is not a positive covariance	—
`related_to` (never evidential)	LIB2-170 A reflection-positive Cartan state exists at finite regulator	—

Why this record is PUBLISHED (R67, condition T-B)

states a predicate	yes
uses on the record · unblocked	3 · 3
uses any lane has read	3 of 3
REVIEWED SUPPORTS use on the predicate, read by two non-drafting lanes	yes (2)
a SUPPORTS use on the predicate is DISPUTED	no
machine certificate current	yes

Referee findings (R73)

Raised by a lane acting as referee and re-measured by the house before being listed (a lane's word alone is not a finding). Only a CONFIRMED finding against the record's predicate blocks PUBLISHED; the others are printed so a reader sees them.

finding	scope	state	what	evidence	affects this record's predicate?
`A1674-F02` A1674	PREDICATE	CLOSED 2026-09-17	The record says the boundary α = \|β\| of the positive completion cone is chiral-null. The apex (0,0) lies on that boundary and is the zero form, not a chiral-null form; only the nonzero boundary α = \|β\| > 0 is chiral-null. How it was closed: R133 D-2 (PI, S342): Rev32.9 fix S338.2 landed; LIB2-172 re-typed by R130 row 17 (nonzero boundary; apex zero form); uses REVIEWED in round 22.	appendix_t_cartan_observation.tex:398 (Rev32.7) paper: REV32.9_CHANGELOG S338.2 (landed in Rev32.9)	closed

Machine certificate (R68)

state MACHINE_CHECKS_PASSED

policy sgtoe-clearance-policy-1.0-RATIFIED · sha16 ef7d6e7690514311

bound to master aad7a3ec6ce3b510 (the master the checks ran on; issued 2026-09-26T05:06Z)

checks

check	result	reason
`schema_valid`	PASS	exercised
`basis_current`	PASS	exercised
`quote_located`	PASS	exercised
`labels_literal`	PASS	exercised
`edges_resolved`	PASS	exercised
`bindings_valid`	PASS	exercised

A certificate says the record is well-formed, current, quote-on-page, label-literal, edge-resolved and binding-consistent on the bytes it was issued on. It is not a review, and it goes STALE the moment the master moves under it (R68). It is one conjunct of the PUBLISHED condition (R67), never the whole of it.

History

March library (v1) antecedents	no legacy antecedent identified
Track-2 claim ids	F-SIGN-01 — cross-reference; the Track 1 / Track 2 join sets each one against what the sealed suite prints for this record. Where they differ, the suite is the record.
serials	A1442 s920
candidate ids (frozen list S304)	C2-095

Drafting-lane note

Positive-completion child of C2-095, distinct from the pure-form obstruction G-175. The coefficients are those of the stated completion; no numerical coefficient is selected. The local Proposition heading supplies no scorecard tier.

Receipt

round D1253 · parsed 2026-09-06T00:06Z · machine gates passed: CONTRACT, G1-quote-on-page, G2-numerals-declared, G4-ids-resolve, DAG-acyclic · content sha256 91828e8945db9d78f87ad5e225f26c3868ea1fa19e51eef5088bc8129f2a9579

Review this entry

Reviewer verdict line (one per entry, as text): LIB2-172 | PASS or FAIL(G1–G8) or NOT RUN | quote | PDF page | one sentence. G1 source supports the exact scoped assertion · G2 evidence class fits (numbers ≡ digit source) · G3 labels as printed, not converted · G4 comparison typed · G6 public scope · G7 history real · G8 caution present and true. Machine surfaces: index.json · capsule.md.

Generated from library_v2.jsonl. Declared volatile fields (R36): BUILD_STAMP.
BUILD_STAMP S371a · 2026-09-26T22:53Z · master 01d1a5873ede42f5 · cut 9dcc6ce9a8ee3e02