{
 "record": "ember-theorem",
 "campaign": "ember — certified hot spots for a trapezoid outside every proven class",
 "specimen": {
  "vertices": [
   "(0, 0)",
   "(1, 0)",
   "(17/20, 9/10)",
   "(1/4, 9/10)"
  ],
  "note": "vertices are EXACT RATIONALS; convex; side slopes 6 and 18/5; no symmetry axis; not a lip domain"
 },
 "generated": "2026-09-02T16:28:04.629Z",
 "trustBase": [
  "instruments/interval rigor model (outward-rounded IEEE doubles; exact BigInt rationals)",
  "instruments/ivspecial (interval Γ Spouge + Bessel J_ν; battery: closed-form falsifiers + bigfloat/exact-rational cross-gates)",
  "Liu framework Thm 2.4 (Appl. Math. Comput. 2015), as quoted in You–Xie–Liu arXiv:1808.08148 §2 — LITERATURE INPUT, assumed not certified; pinned at corpus/sources/liu2018_arxiv-1808-08148.pdf (sha256 fb867aa5…), transcription beside the pin",
  "Liu Lemma 3.2 CR interpolation constant 0.1893·h_K, arXiv:1808.08148 p.8 (Lemma 3.2 quoting Liu 2015) — LITERATURE INPUT, assumed not certified; same pin",
  "classical facts: Neumann separation in a sector + H¹ regularity (excludes the singular Bessel family), Green identity, spectral theorem, Courant"
 ],
 "source": {
  "bench": "/Users/carlostoledo/Documents/frontier-apps/experiments/ember/",
  "note": "frontier-apps has NO git and is NOT a lift source; the bench scripts are pinned byte-for-byte in instruments/hotspots/frontier-ref/ and every proof-bearing quantity is recomputed HERE on instruments/interval + instruments/ivspecial"
 },
 "verdict": "VERIFIED",
 "statement": "THEOREM. Let Ω be the trapezoid with vertices A=(0,0), B=(1,0), C=(17/20,9/10), D=(1/4,9/10) — convex, side slopes 6 and 18/5, no axis of symmetry, not a lip domain; to our knowledge outside every class for which the hot spots conjecture was previously proven. The second Neumann eigenvalue μ1 is SIMPLE, with μ1 ∈ [12.020976137861565, 12.022398348138436], and the second Neumann eigenfunction attains its maximum and its minimum on ∂Ω ONLY. COROLLARY: the maximum is attained at vertex A and only there.",
 "corollary": {
  "statement": "The maximum of φ̂ over the closure is attained AT VERTEX A AND ONLY THERE, with φ̂(A) ∈ [2.1260290907622075, 2.1931576134707655] (= b₀(A) exactly, intersected with the witness chain). For triangles, extrema only at vertices is Judge–Mondal's refinement; for this quadrilateral the maximum-side analogue is now certified.",
  "maxAtVertex": "A",
  "phiAtA": [
   2.1260290907622075,
   2.1931576134707655
  ],
  "minimumLocation": "OPEN — φ̂(C) = b₀(C) ∈ [-2.020353458498882, -1.978946351838731] overlaps the observed boundary minimum (float −1.9998 at distance 0.0195 from C along the top edge); whether the cold spot is at vertex C or strictly inside the edge is undecided at current enclosure widths — a tighter corner extraction would decide it, and an off-vertex answer would contrast with the triangle behaviour."
 },
 "honestFraming": {
  "claim": "to our knowledge the first certified hot-spots domain outside every analytically proven class — ONE domain, ONE theorem; the quadrilateral census is future work and is not counted",
  "fences": [
   "Judge–Mondal: all triangles (Annals of Mathematics, 2020; and the 2022 erratum); earlier partial acute-triangle results (Siudeja, arXiv:1308.3005)",
   "lip domains (Atar–Burdzy)",
   "certain non-convex polygons: L-tiled domains (Hatcher, arXiv:2405.19508)",
   "symmetric quadrangle subcases (Deng–Gui–Jiang–Yang–Yao, arXiv:2604.19003, Apr 2026)",
   "THE OTHER DIRECTION: in sufficiently high dimension the conjecture is FALSE for convex sets (de Dios Pont, arXiv:2412.06344, \"Convex sets can have interior hot spots\") — the planar convex case is exactly where the conjecture remains expected, and this domain sits there",
   "computational antecedent, cited not fenced: the Polymath7 project developed a validated-numerics route to acute triangles (numerics by Nigam) before the analytic triangle proof"
  ],
  "raceWatch": "arXiv weekly for quadrilateral hot-spots claims",
  "status": "machine-derived, not peer-reviewed, not independently rerun"
 },
 "mu1": [
  12.020976137861565,
  12.022398348138436
 ],
 "simple": true,
 "chain": {
  "spectrum": {
   "mu1CR": [
    11.892662878023904,
    12.041819147219154
   ],
   "mu2lo": 13.95593603770136
  },
  "defect": {
   "defectUpper": 0.000012368817442805293
  },
  "eigenpair": {
   "mu1": [
    12.020976137861565,
    12.022398348138436
   ],
   "E_L2": 0.00006535174499233859,
   "Ctr": 2.642618944992971
  },
  "pointwise": {
   "witnesses": {
    "max": 2.1260290907622075,
    "minDeep": 1.9938107725305556
   },
   "coreSup": [
    2.086146764692134,
    1.9715110520750174
   ]
  },
  "collar": {
   "cells": 2272,
   "survivors": 0
  },
  "corner": {
   "valueRanges": {
    "A": [
     2.015278905106481,
     2.193226297094631
    ],
    "B": [
     0.7565549904171839,
     0.8965066841782028
    ],
    "C": [
     -2.0261839645411444,
     -1.8465050263108518
    ],
    "D": [
     -1.6642997952515894,
     -1.3514632695752773
    ]
   }
  },
  "cross": {
   "CtrBigfloat": [
    2.6426189449929693,
    2.64261894499297
   ],
   "p1Upper": 12.06415383201281
  }
 },
 "partition": {
  "grid": "1/100",
  "tipCells": 498,
  "coreCells": 4690,
  "collarCells": 2272,
  "exterior": 1540,
  "rules": "R1 entirely-in-tip (exact); core = corner-min depth ≥ 3/40 (exact, concavity); collar = the rest meeting Ω (exact SAT)"
 },
 "records": {
  "ember-spectrum.json": "9a6bd2aebb1119ce79ad2b10cf178a002d9a4d7739b1d7bf313615a588512afe",
  "ember-defect.json": "4947c112ff57d56a754bca9f1b8c5ce38864ec3e03c93ed93573a23170f1297a",
  "ember-eigenpair.json": "3448fc920e03252443954ed1c49c54a263d36c7e6dafb40817fbdd96534d6260",
  "ember-pointwise.json": "0600b4d3aa11367d97ec28ff3a6e1327d1f01d4bf1b9181c97c2740b6092fcf8",
  "ember-collar.json": "d0b0d5d769079cc9da6474d17bdfee61f74ceed6323e2da18e677b8e3794c3a1",
  "ember-corner.json": "78d3c62219518c343c58ee65ef47410226a9a68515ab5b01a3a1c617349852ac",
  "ember-cross.json": "9cc8a492d5a1c11f5d477e79e2d9f04b40165c158e73ed61e7e865da99793e0d"
 },
 "checks": [
  {
   "name": "eigenpair inputs = spectrum + defect outputs",
   "ok": true
  },
  {
   "name": "pointwise inputs = eigenpair outputs",
   "ok": true
  },
  {
   "name": "collar inputs = eigenpair + defect + pointwise outputs",
   "ok": true
  },
  {
   "name": "corner inputs = eigenpair + pointwise outputs",
   "ok": true
  },
  {
   "name": "μ1 (eigenpair) ⊆ μ1 (spectrum CR)",
   "ok": true
  },
  {
   "name": "spectral gap: μ2 lower > μ1 upper ⇒ μ1 SIMPLE",
   "ok": true
  },
  {
   "name": "PARTITION (rational re-decision): core cell count = pointwise record",
   "ok": true,
   "detail": "re-decided 4690 vs record 4690"
  },
  {
   "name": "PARTITION (rational re-decision): collar cell count = collar record",
   "ok": true,
   "detail": "re-decided 2272 vs record 2272"
  },
  {
   "name": "PARTITION: every grid cell classified exactly once (tip/core/collar/exterior)",
   "ok": true,
   "detail": "tip 498 + core 4690 + collar 2272 + exterior 1540 = 9000"
  },
  {
   "name": "Ω convex (exact rational cross products)",
   "ok": true
  },
  {
   "name": "core sweep: zero survivors",
   "ok": true
  },
  {
   "name": "collar sweep: zero survivors",
   "ok": true
  },
  {
   "name": "core margins positive (both sides)",
   "ok": true,
   "detail": "max 3.93e-2, min 5.61e-4"
  },
  {
   "name": "collar kills strict (both sides)",
   "ok": true
  },
  {
   "name": "tip A: max side by ∂rφ̂ < 0; min side by value",
   "ok": true
  },
  {
   "name": "tip B: value kill both sides",
   "ok": true
  },
  {
   "name": "tip D: value kill both sides",
   "ok": true
  },
  {
   "name": "tip C: max side by value; min side by min-form + wedge",
   "ok": true
  },
  {
   "name": "COROLLARY (i): w₊ lies inside tip A's sector (exact rational)",
   "ok": true
  },
  {
   "name": "COROLLARY (ii): φ̂ strictly radially decreasing on all of tip A",
   "ok": true,
   "detail": "cells worst -4.99e-2, inner disk -10.04"
  },
  {
   "name": "COROLLARY (iii): everything outside tip A is certified below φ̂(w₊)",
   "ok": true,
   "detail": "2.123592 < 2.126029"
  },
  {
   "name": "COROLLARY: φ̂(A) enclosure coherent (witness chain meets b₀(A))",
   "ok": true,
   "detail": "[2.126029, 2.193158]"
  },
  {
   "name": "cross-derivations record VERIFIED (I₀, C_tr bigfloat, P1 upper, two-annulus)",
   "ok": true
  }
 ],
 "secs": 0
}