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 2 of 2 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 The state is recomputed from the uses on every build and is withdrawn if they stop justifying it.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-b2056c90a3d7b759ac72e772 · master a44d2c366ff3bbab · S339.

LIB2-197 · The positive congruence dressing stays inside the registered E7 group

class CLAIMmath status (as printed) NOT_PRINTEDattachment (as printed) NOT_PRINTEDscorecard tier word (as printed) none printedhouse tier not yet assignedreview state PUBLISHEDentry version 8

Label sentence at the locator (PDF p.29 (bytes of record)):

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.

Statement

For the registered Cartan pair and E7-Grams A = exp(2SA), M = exp(2Φ) with SA,Φ ∈ p, the unique positive-definite symmetric solution of BAB = M has log B ∈ p and hence B ∈ E7(7).

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 unique positive-definite symmetric solution of BAB = M has log B ∈ p (one claim)
p2 · hence B ∈ E7(7). (one claim)

Scope

carrier	The positive-definite symmetric congruence dressing of two registered E7-Grams
domain	The stated congruence construction under its E7-Gram hypotheses.
group	E7(7)
hypotheses	Use p = span(S70) ⊂ Sym(56), k = [p,p] and e7 = k ⊕ p as the source's Cartan pair.; Both A and M are E7-Grams of the stated exponential form; generic positive-definite matrices are not substituted.
quantifier	The registered construction and its stated premises.
real form	E7(7)

What this does not show. The closure theorem does not apply to arbitrary Gram matrices or derive a physical mass or observation-map attachment.

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.

Paper 1, §9, Theorem 9.1: congruence closure — PDF p.30 (bytes of record)
review state: REVIEWED · role SUPPORTS · warrant: grok read THE_RECORD → SUPPORTS · gemini read SUITE_TEXT_EXTRACT → SUPPORTS

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.

Paper 1, §9, Theorem 9.1: Cartan-pair hypothesis — PDF p.30 (bytes of record)
review state: REVIEWED · role CONTEXT · warrant: grok read THE_RECORD → CONTEXT · gemini read SUITE_TEXT_EXTRACT → SUPPORTS · the lanes disagreed; resolved by PI ruling R118 as CONTEXT — a ruling whose gloss the house owes; a use resolved by ruling never anchors PUBLISHED

Let p = span(S70 ) ⊂ Sym(56), k = [p, p], e7 = k ⊕ p (Cartan pair, machine-verified).

Evidence (availability is not outcome)

kind	reference	line / key	availability	note
analytic	Paper 1 §9, Theorem 9.1, p.29; A1343	—	shipped	The supplied theorem states the closure and its hypotheses; this drafting round does not independently re-prove it.
computational	s679, S223; Paper 1 §9, p.29	—	unknown	The source reports Cartan-pair and membership controls; their matrices and original outputs are not supplied.

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
`7`	PDF p.29 (bytes of record)	closed form: the two E7-Grams and congruence-closure conclusion in Paper 1, Theorem 9.1
`2`	PDF p.29 (bytes of record)	closed form: the two E7-Grams and congruence-closure conclusion in Paper 1, Theorem 9.1

Typed links

type	target	licensed by
`related_to` (never evidential)	LIB2-006 The registered symplectic exceptional substrate	—
`cited_by_section`	E-C03.1.claims-used (Exposition section page; use UNREVIEWED)	TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0013
`cited_by_section`	E-C03.1.derivation (Exposition section page; use UNREVIEWED)	TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0023
`cited_by_section`	E-C03.1.prior-art (Exposition section page; use UNREVIEWED)	TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0018
`cited_by_section`	E-C03.1.sources-receipt (Exposition section page; use UNREVIEWED)	TEXTBOOK_CITATIONS.jsonl (S312) — CIT-0028

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

states a predicate	yes
uses on the record · unblocked	2 · 2
uses any lane has read	2 of 2
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

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
serials	A1343 s679
candidate ids (frozen list S304)	C2-035

Drafting-lane note

Source-backed missing proposal T4-21 under C2-035. This late carrier entry retains both E7-Gram premises. The stated theorem is not converted to a scorecard tier or credited as a new calculation.

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 111d6668d2b10e183adea93212c783f14cc037565b540ffbcb7984f7dcad8c22

Review this entry

Reviewer verdict line (one per entry, as text): LIB2-197 | 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