Claims & Evidence Registrar › H02
Label sentence at the locator (PDF p.165 (bytes of record)):
Theorem 2 (Vacuum-selector, √ arithmetic form). 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.
For positive golden-field units a, b, c with abc = 1, the Route-B classification gives F = 16 only at the exponent multiset {1, 0, −1}; apart from the equal-spectrum case, every other admissible gap pair has F ≥ 41.
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.
| carrier | Positive diagonal Jvac = diag(a, b, c) |
|---|---|
| domain | Appendix E, Route B; F is the gap functional defined in the proof of Theorem 2. |
| hypotheses | a, b, c are positive units of Z[ϕ], with abc = 1.; Write a = ϕ^n1, b = ϕ^n2, c = ϕ^n3 with n1 + n2 + n3 = 0; order n1 ≥ n2 ≥ n3.; The nonnegative gaps p = n1 − n2 and q = n2 − n3 obey p + 2q ≡ 0 (mod 3). |
| quantifier | All admissible gap pairs in the printed arithmetic classification. |
| real form | J3(Os) |
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.
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: admissible-gap case count — PDF p.171 (bytes of record)
review state: REVIEWED · role SUPPORTS · warrant: grok read THE_RECORD → SUPPORTS · chatgpt read SUITE_TEXT_EXTRACT → SUPPORTS · gemini read SUITE_TEXT_EXTRACT → SUPPORTS · grok read THE_RECORD → SUPPORTS · grok read THE_RECORD → SUPPORTS · gemini read SUITE_TEXT_EXTRACT → SUPPORTS
(p, q) = (1, 1) — the triple {1, 0, −1} — gives F = 3 + 3 + 3 + 7 = 16; no other pair with p + q ≤ 2 is admissible (p + 2q ≡ 0 (mod 3) excludes (1, 0), (0, 1), (2, 0), (0, 2)); for p + q = 3 the admissible pairs are (3, 0) and (0, 3), each giving F = 3 + 18 + 2 + 18 = 41 (attained at {2, −1, −1}); and for p + q ≥ 4, Ap+q ≥ 47 gives F ≥ 3 + 47 + 2 + 2 = 54
Appendix E, §9, Status: ordered-frame boundary — PDF p.179 (bytes of record)
review state: REVIEWED · role CONTEXT · warrant: grok read THE_RECORD → CONTEXT · chatgpt read SUITE_TEXT_EXTRACT → CONTEXT · gemini read SUITE_TEXT_EXTRACT → CONTEXT · grok read THE_RECORD → CONTEXT · grok read OUR_QUOTE_AS_PRINTED → CONTEXT · gemini read OUR_QUOTE_AS_PRINTED → 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.
Appendix E, §9, Status: pointer to the stronger variational form — PDF p.179 (bytes of record)
review state: REVIEWED · role CONTEXT · warrant: gemini read SUITE_TEXT_EXTRACT → CONTEXT · grok read THE_RECORD → CONTEXT · grok read OUR_QUOTE_AS_PRINTED → CONTEXT · gemini read OUR_QUOTE_AS_PRINTED → CONTEXT
kernel s1085 verifies the replacement and proves the stronger variational form
| kind | reference | line / key | availability | note |
|---|---|---|---|---|
| analytic | Appendix E §4, Theorem 2, PDF p.165: printed admissible-gap case count | — | shipped | The proof text is supplied. This record restates its gap classification; no new calculation or independent proof audit is claimed. |
| computational | s1085, named in Appendix E §9, PDF p.172 | — | unknown | The suite cites a stronger variational form, but neither the kernel nor its output is attached to D1253. |
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 |
|---|---|---|
1 | PDF p.165 (bytes of record) | closed form: abc = 1 and exponent multiset {1, 0, −1}; printed in Appendix E §4, Theorem 2 |
16 | PDF p.165 (bytes of record) | closed form: F = 16 at the golden exponent multiset; printed in Appendix E §4, Theorem 2 |
0 | PDF p.165 (bytes of record) | closed form: exponent multiset {1, 0, −1}; printed in Appendix E §4, Theorem 2 |
−1 | PDF p.165 (bytes of record) | closed form: exponent multiset {1, 0, −1}; printed in Appendix E §4, Theorem 2 |
41 | PDF p.165 (bytes of record) | closed form: F ≥ 3 + 18 + 2 + 18 = 41 for the remaining admissible pairs; printed in Appendix E §4, Theorem 2 |
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-0022 | REVIEWED by lanes (R49c) | PRM-024 (PRM-024/r1) — Route B golden-unit restriction (positive algebraic integers in Q(√5), abc = 1) | hypothetical_assumption | Proposed by the drafting lane (A1634): The gap-pair classification uses exponent triples obtained from this positive golden-unit domain. Its zero-sum exponent condition comes from abc=1; the field restriction supplies the unit-power reduction. — House reading: House read App E:117–124: the classification is stated for positive golden-field units with abc = 1 (the premise's exact domain); the unit-power reduction a = φ^n uses it. The premise is a hypothesis of the result.
|
| type | target | licensed by |
|---|---|---|
related_to (never evidential) | LIB2-020 Route B selects the positive golden-unit triple under DET-7 | — |
cited_by_section | E-C04.1.claims-used (Exposition section page; use UNREVIEWED) | TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0046 |
cited_by_section | E-C04.1.derivation (Exposition section page; use UNREVIEWED) | TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0058 |
cited_by_section | E-C04.1.prior-art (Exposition section page; use UNREVIEWED) | TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0054 |
cited_by_section | E-C04.1.sources-receipt (Exposition section page; use UNREVIEWED) | TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0061 |
rests_on_premise | PRM-024 Route B golden-unit restriction (positive algebraic integers in Q(√5), abc = 1) | appendix_e_vacuum_selector.tex:121 — "positivity removes the sign, so $a=\phi^{n_1}$, $b=\phi^{n_2}$, $c=\phi^{n_3}$" |
| 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 |
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-F2A1632 | PROOF | CONFIRMED | The Route-B lower-bound step (A_{p+q} ≥ 18 ⇒ F ≥ 3+18+2+18) is invalid as written; the conclusion F ≥ 41 holds by a three-case count. | appendix_e:137–138; house enumeration of admissible gaps p,q ≤ 7: F = 9, 16, 41, 64, 176, 361. paper: REV32.9_CHANGELOG S327·A1632 item 2 (landed in Rev32.9) | no — the predicate is unaffected |
| 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 |
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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.
| March library (v1) antecedents | no legacy antecedent identified |
|---|---|
| Track-2 claim ids | A-VAC-02 — 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 | s1085 |
| candidate ids (frozen list S304) | C2-111 |
Corrected reissue of G-005 after the Part 1 ruling: scorecard_tier_word is null; the local theorem heading is not a scorecard tier. C2-111 remains narrowed to the explicit gap classification on p.165. The full stronger variational statement is not printed at the p.172 s1085 pointer (ADD1-07). No determinant-as-output or physical-selection claim is added. The navigation link now targets the separately drafted Route-B selector G-024.
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 cb263cc9d60e7eefdc392d9a5e109dcfa3ee16aa24105056057b1226784566ac
Reviewer verdict line (one per entry, as text): LIB2-001 | 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