{
 "id": "LIB2-197",
 "entry_version": 8,
 "class": "CLAIM",
 "title": "The positive congruence dressing stays inside the registered E7 group",
 "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).",
 "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."
  ],
  "physical_target": null,
  "quantifier": "The registered construction and its stated premises.",
  "real_form": "E7(7)"
 },
 "source_labels_as_printed": {
  "math_status": "NOT_PRINTED",
  "attachment": "NOT_PRINTED",
  "scorecard_tier_word": null,
  "pdf_page": 29
 },
 "house_tier": null,
 "review_state": "PUBLISHED",
 "scientific_status": "UNDER_REVIEW (house status not yet adjudicated)",
 "evidence_status": [
  {
   "kind": "analytic",
   "availability": "shipped"
  },
  {
   "kind": "computational",
   "availability": "unknown"
  }
 ],
 "primary_source": {
  "component": "Paper 1",
  "section": "§9, Theorem 9.1: congruence closure",
  "pdf_page": 30,
  "quote": "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."
 },
 "caution": "The closure theorem does not apply to arbitrary Gram matrices or derive a physical mass or observation-map attachment.",
 "candidate_ids": [
  "C2-035"
 ],
 "basis": {
  "pdf_sha256_prefix": "b64ddcd9e16a6dcd",
  "release": "Rev33.1_S369"
 },
 "relationships": [
  {
   "type": "related_to",
   "target": "LIB2-006"
  },
  {
   "type": "cited_by_section",
   "target": "E-C03.1.claims-used"
  },
  {
   "type": "cited_by_section",
   "target": "E-C03.1.derivation"
  },
  {
   "type": "cited_by_section",
   "target": "E-C03.1.prior-art"
  },
  {
   "type": "cited_by_section",
   "target": "E-C03.1.sources-receipt"
  }
 ],
 "url": "/library-v2/LIB2-197/",
 "content_sha256": "111d6668d2b10e183adea93212c783f14cc037565b540ffbcb7984f7dcad8c22",
 "build_stamp": "BUILD_STAMP S371a · 2026-09-26T22:53Z · master 01d1a5873ede42f5 · cut 9dcc6ce9a8ee3e02"
}