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

- class: CLAIM · review_state: PUBLISHED · house_tier: not yet assigned
- labels as printed: math_status NOT_PRINTED · attachment NOT_PRINTED · scorecard tier word none (PDF p.29)
- basis: {"pdf_sha256_prefix": "b64ddcd9e16a6dcd", "release": "Rev33.1_S369"}

## 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).

## Caution

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

## Primary source

Paper 1, §9, Theorem 9.1: congruence closure, PDF p.30: "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."

## Machine

- url: /library-v2/LIB2-197/
- content_sha256: 111d6668d2b10e183adea93212c783f14cc037565b540ffbcb7984f7dcad8c22
- candidate_ids: C2-035

## Freshness

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