{
 "id": "LIB2-001",
 "entry_version": 11,
 "class": "CLAIM",
 "title": "Route-B admissible-gap classification",
 "statement": "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.",
 "scope": {
  "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.",
  "group": null,
  "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)."
  ],
  "physical_target": null,
  "quantifier": "All admissible gap pairs in the printed arithmetic classification.",
  "real_form": "J3(Os)"
 },
 "source_labels_as_printed": {
  "math_status": "NOT_PRINTED",
  "attachment": "NOT_PRINTED",
  "scorecard_tier_word": null,
  "pdf_page": 165
 },
 "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": "Appendix E",
  "section": "§4, Theorem 2: admissible-gap case count",
  "pdf_page": 171,
  "quote": "(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"
 },
 "caution": "This gap classification does not select the golden-field hypothesis or the ordered physical Peirce frame, and it does not supply a physical vacuum-selection law.",
 "candidate_ids": [
  "C2-111"
 ],
 "basis": {
  "pdf_sha256_prefix": "b64ddcd9e16a6dcd",
  "release": "Rev33.1_S369"
 },
 "relationships": [
  {
   "type": "related_to",
   "target": "LIB2-020"
  },
  {
   "type": "cited_by_section",
   "target": "E-C04.1.claims-used"
  },
  {
   "type": "cited_by_section",
   "target": "E-C04.1.derivation"
  },
  {
   "type": "cited_by_section",
   "target": "E-C04.1.prior-art"
  },
  {
   "type": "cited_by_section",
   "target": "E-C04.1.sources-receipt"
  },
  {
   "type": "rests_on_premise",
   "target": "PRM-024"
  }
 ],
 "url": "/library-v2/LIB2-001/",
 "content_sha256": "cb263cc9d60e7eefdc392d9a5e109dcfa3ee16aa24105056057b1226784566ac",
 "build_stamp": "BUILD_STAMP S371a · 2026-09-26T22:53Z · master 01d1a5873ede42f5 · cut 9dcc6ce9a8ee3e02"
}