{
 "record": "ember-defect",
 "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": "2026-09-02T14:40:18.724Z",
 "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": "‖∂νu‖_{L²(∂Ω)} ≤ defectUpper for the frozen exact-Helmholtz trial (order-2 interval Taylor jets along each edge, Bessel-ODE closure, certified midpoint-Taylor cell rule; per-edge pointwise sup|∂νu| certified from the same jets).",
 "trial": {
  "lambdaTilde": 12.021687243,
  "K": 10,
  "nbasis": 40,
  "coefficients": [
   -4.545775382608376,
   2.8330116650432635,
   -4.509915893922395,
   5.482102152814437,
   -1.4636900101070296,
   -305.91585186501334,
   -10056.802899419534,
   -47904.88140401989,
   -19422250.462553024,
   0,
   -2.563491429443502,
   -4.431859785272341,
   2.992436933644638,
   0.8339443625872769,
   10.969816536110812,
   -145.75298142893962,
   1662.8327044169346,
   -65644.6585718384,
   220429.63552093506,
   -97872974.99588013,
   -1.265117939989023,
   -1.9684508947825117,
   -4.416535224354552,
   5.371873297105727,
   -23.149279641834216,
   166.8255105398423,
   -866.141494514623,
   4681.439152488425,
   -61215.296606430114,
   221526.37881804164,
   0.02850593640194605,
   3.524420879499283,
   -0.9026485368823367,
   0.9059845094861736,
   22.890598021283623,
   55.94928954571824,
   160.84605250617355,
   -3186.7570923009284,
   -15051.251885127087,
   -190509.41980976844
  ],
  "floatDefect": 0.000009014713885671235,
  "note": "coefficients are frozen exact doubles from the deterministic float MPS proposer (trial.js); downstream stages regenerate and compare element-exact"
 },
 "cells": 700,
 "defectUpper": 0.000012368817442805293,
 "perEdge": [
  {
   "edge": 0,
   "fluxSq": 5.521468387773315e-11,
   "supFlux": 0.00001376039502151031
  },
  {
   "edge": 1,
   "fluxSq": 2.4729983539375695e-11,
   "supFlux": 0.000009134794826721813
  },
  {
   "edge": 2,
   "fluxSq": 2.6734472584880956e-11,
   "supFlux": 0.000012444767078947143
  },
  {
   "edge": 3,
   "fluxSq": 4.630850493145453e-11,
   "supFlux": 0.000013910038614144616
  }
 ],
 "supFluxPerEdge": [
  0.00001376039502151031,
  0.000009134794826721813,
  0.000012444767078947143,
  0.000013910038614144616
 ],
 "checks": [
  {
   "name": "bridge edge0 t=0.31",
   "ok": true,
   "detail": "float=5.526e-6 iv=[5.486e-6,5.565e-6]"
  },
  {
   "name": "bridge edge0 t=0.5",
   "ok": true,
   "detail": "float=5.275e-6 iv=[5.250e-6,5.301e-6]"
  },
  {
   "name": "bridge edge0 t=0.77",
   "ok": true,
   "detail": "float=3.823e-6 iv=[3.794e-6,3.852e-6]"
  },
  {
   "name": "bridge d/dτ edge0",
   "ok": true,
   "detail": "num=-6.244e-6 iv=[-6.497e-6,-5.995e-6]"
  },
  {
   "name": "bridge edge1 t=0.31",
   "ok": true,
   "detail": "float=1.208e-6 iv=[1.192e-6,1.224e-6]"
  },
  {
   "name": "bridge edge1 t=0.5",
   "ok": true,
   "detail": "float=-1.656e-6 iv=[-1.671e-6,-1.642e-6]"
  },
  {
   "name": "bridge edge1 t=0.77",
   "ok": true,
   "detail": "float=-5.598e-6 iv=[-5.614e-6,-5.582e-6]"
  },
  {
   "name": "bridge d/dτ edge1",
   "ok": true,
   "detail": "num=-1.567e-5 iv=[-1.578e-5,-1.555e-5]"
  },
  {
   "name": "bridge edge2 t=0.31",
   "ok": true,
   "detail": "float=-5.145e-6 iv=[-5.168e-6,-5.122e-6]"
  },
  {
   "name": "bridge edge2 t=0.5",
   "ok": true,
   "detail": "float=-7.380e-6 iv=[-7.399e-6,-7.360e-6]"
  },
  {
   "name": "bridge edge2 t=0.77",
   "ok": true,
   "detail": "float=-6.105e-6 iv=[-6.127e-6,-6.084e-6]"
  },
  {
   "name": "bridge d/dτ edge2",
   "ok": true,
   "detail": "num=1.158e-5 iv=[1.144e-5,1.173e-5]"
  },
  {
   "name": "bridge edge3 t=0.31",
   "ok": true,
   "detail": "float=-2.336e-6 iv=[-2.349e-6,-2.324e-6]"
  },
  {
   "name": "bridge edge3 t=0.5",
   "ok": true,
   "detail": "float=2.302e-6 iv=[2.287e-6,2.318e-6]"
  },
  {
   "name": "bridge edge3 t=0.77",
   "ok": true,
   "detail": "float=6.987e-6 iv=[6.948e-6,7.025e-6]"
  },
  {
   "name": "bridge d/dτ edge3",
   "ok": true,
   "detail": "num=1.362e-5 iv=[1.348e-5,1.376e-5]"
  },
  {
   "name": "certified defect within sane factor of float (sanity)",
   "ok": true,
   "detail": "certified 1.2369e-5 vs float 9.015e-6"
  }
 ],
 "secs": 4.3
}