{
 "id": "LIB2-172",
 "entry_version": 11,
 "class": "CLAIM",
 "title": "The positive completion cone has a strict coefficient domain",
 "statement": "The registered quadratic completion Bα,β = αBC + βBMM is positive definite exactly when α > 0 and |β| < α; its nonzero boundary α = |β| > 0 is chiral-null, and the apex α = β = 0 is the zero form.",
 "scope": {
  "carrier": "The two-parameter completion of the Cartan and MacDowell–Mansouri quadratic forms",
  "domain": "The displayed quadratic completion, not an established nonlinear gravitational continuum measure.",
  "group": null,
  "hypotheses": [
   "Use the forms BC and BMM on the registered generator space."
  ],
  "physical_target": null,
  "quantifier": "The registered construction and its stated premises.",
  "real_form": null
 },
 "source_labels_as_printed": {
  "math_status": "NOT_PRINTED",
  "attachment": "NOT_PRINTED",
  "scorecard_tier_word": null,
  "pdf_page": 305
 },
 "house_tier": null,
 "review_state": "PUBLISHED",
 "scientific_status": "UNDER_REVIEW (house status not yet adjudicated)",
 "evidence_status": [
  {
   "kind": "analytic",
   "availability": "shipped"
  }
 ],
 "primary_source": {
  "component": "Appendix T",
  "section": "§7, Proposition 7.4: positive completion cone",
  "pdf_page": 322,
  "quote": "Bα,β = αBC + βBMM > 0 ⇐⇒ α > 0, |β| < α,"
 },
 "caution": "Positive quadratic completion does not establish Einstein universality, a continuum limit or generic nonlinear reflection positivity.",
 "candidate_ids": [
  "C2-095"
 ],
 "basis": {
  "pdf_sha256_prefix": "b64ddcd9e16a6dcd",
  "release": "Rev33.1_S369"
 },
 "relationships": [
  {
   "type": "related_to",
   "target": "LIB2-171"
  },
  {
   "type": "related_to",
   "target": "LIB2-170"
  }
 ],
 "url": "/library-v2/LIB2-172/",
 "content_sha256": "91828e8945db9d78f87ad5e225f26c3868ea1fa19e51eef5088bc8129f2a9579",
 "build_stamp": "BUILD_STAMP S371a · 2026-09-26T22:53Z · master 01d1a5873ede42f5 · cut 9dcc6ce9a8ee3e02"
}