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 The state is recomputed from the uses on every build and is withdrawn if they stop justifying it.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-5f8e7521e2913d1bb26be0e2 · master 7570ddad604d5b37 · S342.

LIB2-020 · Route B selects the positive golden-unit triple under DET-7

class CLAIMmath status (as printed) NOT_PRINTEDattachment (as printed) NOT_PRINTEDscorecard tier word (as printed) none printedhouse tier not yet assignedreview state PUBLISHEDentry version 9

Label sentence at the locator (PDF p.165 (bytes of record)):

Let Jvac = diag(a, b, c) with a, b, c > 0 algebraic integers in Q( 5) and abc = 1. Then (H2) forces Jvac = diag(ϕ, 1, ϕ−1 ) up to permutation.

Statement

For positive algebraic integers a,b,c in Q(√5) with abc = 1 and DET-7, Jvac = diag(a,b,c) has spectrum {ϕ,1,ϕ⁻¹} up to permutation.

Scope

carrier	Positive diagonal Jvac = diag(a,b,c)
domain	Appendix E Route B with the replaced admissible-gap proof.
hypotheses	a,b,c are positive algebraic integers in Q(√5).; abc = 1, so the entries are units of Z[ϕ].; H2: det Gram(Jvac,Jvac#) = 7.
quantifier	Every triple satisfying the printed arithmetic hypotheses.
real form	J3(Os)

What this does not show. The arithmetic hypothesis and DET-7 remain premises; this result is not a derivation of the ordered physical frame or of an action.

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 E, §4, Theorem 2: arithmetic selector — PDF p.171 (bytes of record)
review state: REVIEWED · role SUPPORTS · warrant: grok read THE_RECORD → SUPPORTS · gemini read SUITE_TEXT_EXTRACT → SUPPORTS · grok read THE_RECORD → SUPPORTS · gemini read SUITE_TEXT_EXTRACT → SUPPORTS

Let Jvac = diag(a, b, c) with a, b, c > 0 algebraic integers in Q( 5) and abc = 1. Then (H2) forces Jvac = diag(ϕ, 1, ϕ−1 ) up to permuta- tion.

Appendix E, §4, Remark 4: distinct residual input — PDF p.172 (bytes of record)
review state: REVIEWED · role CONTEXT · warrant: grok read THE_RECORD → CONTEXT · gemini read SUITE_TEXT_EXTRACT → CONTEXT

Route A’s residual input is the inversion symmetry (H1); Route B’s residual input is the golden-field hypothesis a, b, c ∈ Z[ϕ]. Each, together with (H0) and DET-7, removes all continuous freedom.

Appendix E, §9, Ordered-frame boundary — PDF p.179 (bytes of record)
review state: REVIEWED · role CONTEXT · warrant: grok read THE_RECORD → CONTEXT · gemini read SUITE_TEXT_EXTRACT → CONTEXT

The ordered physical Peirce frame remains loaded: a frame-free F4 -invariant selects only the spectral orbit (item (vi), s978/s981), so ordering requires a loading or symmetry-breaking tensor.

Evidence (availability is not outcome)

kind	reference	line / key	availability	note
analytic	Appendix E §4, Theorem 2, p.165	—	shipped	The printed gap-pair/Lucas argument is the source; the retired spread-monotonicity step is not used 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
`5`	PDF p.165 (bytes of record)	closed form: Theorem 2's field, determinant hypothesis and golden spectral conclusion
`1`	PDF p.165 (bytes of record)	closed form: Theorem 2's field, determinant hypothesis and golden spectral conclusion
`7`	PDF p.165 (bytes of record)	closed form: Theorem 2's field, determinant hypothesis and golden spectral conclusion
`−1`	PDF p.165 (bytes of record)	closed form: Theorem 2's field, determinant hypothesis and golden spectral conclusion

Premises this record rests on (qualified uses)

Each row is a house-declared use of a registered premise (R37). DECLARED is the house's editorial judgment only; a use becomes REVIEWED when two non-drafting lanes, reading different objects, agree the premise is LOAD_BEARING for this record's result. REJECTED rows were retired by ruling and are shown for provenance.

use	state	premise	role	why the house declared it · reads
`USE-0003`	REJECTED by PI (R78)	PRM-017 (PRM-017/r1) — Route B selector	convention	Route B is named in the title and statement; the record is the Route-B selection. DET-7 arrives transitively through uses_input LIB2-018.
`USE-0019`	REVIEWED by lanes (R49c)	PRM-013 (PRM-013/r1) — DET-7 structural postulate	hypothetical_assumption	DET-7 enters Route B as a hypothesis too. Both routes share (H0) unit determinant and DET-7 in the positive diagonal trace-form setting (App E Remark 4: "Each, together with (H0) and DET-7, removes all continuous freedom"); Route A adds inversion symmetry, Route B the golden-field restriction. [R129 D-5, S340: "the ONLY hypothesis the two routes share" corrected per A1704 F2 (house-checked, Rev32.8 part2:3239)] gemini · LOAD_BEARING · SUITE_TEXT_EXTRACT (D1350) grok · LOAD_BEARING · THE_RECORD (D1349)
`USE-0020`	REVIEWED by lanes (R49c)	PRM-024 (PRM-024/r1) — Route B golden-unit restriction (positive algebraic integers in Q(√5), abc = 1)	hypothetical_assumption	Route B instead restricts to positive golden units in Q(sqrt 5) with abc = 1 (H0 is bundled into PRM-024/r1). Route A uses no number-field restriction (the record says so in its own domain field). The two are separate sufficient hypothesis sets: conjoining them would narrow the domain needlessly and misstate which conditions each route needs, and no combination licenses dropping both distinguishing assumptions. [R129 D-5, S340: "unioning the two bundles would assert a theorem domain neither route proves" re-worded per A1704 F2b] gemini · LOAD_BEARING · SUITE_TEXT_EXTRACT (D1350) grok · LOAD_BEARING · THE_RECORD (D1349)
`USE-0021`	REJECTED by PI (R78)	PRM-017 (PRM-017/r1) — Route B selector	convention	Choosing Route B at all is a selector, not a hypothesis of the theorem — a different role again.

Typed links

type	target	licensed by
`uses_input`	LIB2-018 DET-7 is a structural postulate in the algebraic selector	Appendix E p.165, Theorem 2: H2 forces the stated golden triple under the arithmetic hypotheses.
`rests_on_premise`	PRM-013 DET-7 structural postulate	—
`rests_on_premise`	PRM-024 Route B golden-unit restriction (positive algebraic integers in Q(√5), abc = 1)	—
`cited_by_section`	E-C04.1.claims-used (Exposition section page; use UNREVIEWED)	TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0049
`cited_by_section`	E-C04.1.derivation (Exposition section page; use UNREVIEWED)	TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0059
`cited_by_section`	E-C04.1.prior-art (Exposition section page; use UNREVIEWED)	TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0055
`cited_by_section`	E-C04.1.sources-receipt (Exposition section page; use UNREVIEWED)	TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0064

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 (1)
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?
`A1632-F2b` A1632	PROOF	CONFIRMED	Inherits the Route-B proof step of A1632-F2; the conclusion holds.	as A1632-F2. paper: REV32.9_CHANGELOG S327·A1632 item 2 (landed in Rev32.9)	no — the predicate is unaffected
`A1632-F3b` A1632	QUOTE_RENDERING	CONFIRMED	The quote reads 'Q( 5)'; the source prints ℚ(√5) (ℚ(5) would be ℚ).	appendix_e:117.	no — the predicate is unaffected

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	A-VAC-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	A641
candidate ids (frozen list S304)	C2-102

Drafting-lane note

Arithmetic-route child of C2-102, separate from Route A and from C2-111's stronger variational formulation. G-005 records the explicit admissible-gap classification. The quote retains the extraction's missing adjacent radical rather than repairing it.

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 51dc78f16e2db545cbbde6fa79ec04f645ed89f162006264ab1bd6c608be7915

Review this entry

Reviewer verdict line (one per entry, as text): LIB2-020 | 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