#!/usr/bin/env python3 # -*- coding: utf-8 -*- """verify_congest.py — a SELF-CONTAINED, STANDARD-LIBRARY-ONLY, independent verifier for the discounted congestion mean-field-game computer-assisted proof. Instance: a = 1/2, sigma = 0.5, A = 0.3, gamma = 0.5, N = 14 cosine modes System: u - sigma u'' + 1/2 (u')^2 / sqrt(m) = A cos 2pi x + gamma m m - sigma m'' - ( sqrt(m) u' )' = 1 , m > 0 (augmented with w = m^{-1/2} and the reciprocal constraint m w^2 = 1) WHAT THIS SCRIPT IS. A skeptical referee can download THIS ONE FILE and run python3 verify_congest.py with nothing but a Python 3.9+ standard library. It re-derives, in outward-rounded interval arithmetic (math.nextafter), the radii-polynomial / Newton-Kantorovich bounds of the proof and either prints a refusal or ends with CONGEST CAP: VERIFIED The numerical CANDIDATE (a; m; w) and the approximate-inverse matrix A are embedded below as CERTIFICATE DATA. This verifier does NOT re-solve the MFG and does NOT recompute A: it CONSUMES them and RIGOROUSLY recomputes the bounds Y0 = ||A Phi(x_bar)||_nu, Z1 = ||I - A DPhi(x_bar)||_nu, Z2 (the Lipschitz bound on the ball), then closes the radii polynomial p(r) = 1/2 Z2 r^2 - (1 - Z1) r + Y0 < 0 and certifies both positivity walls (density m > 0 and the branch selector w > 0) over the WHOLE ball. The interval type, the nu-weighted sequence (Banach-algebra) norm/convolution, buildPhi and the Jacobian action are faithful, line-by-line ports of the JavaScript source of truth (eqcert/src/{interval,sequence,radii}.js and research/mfg-congest/kernel/validate-congest.js). Correctness is everything: a verifier that PASSES while being subtly wrong is the worst possible outcome, so the port is checked against the JS Phi(x_bar) elementwise (embedded), and SEVEN falsifiers (X1..X7) must each turn their own target RED. LICENCE ------- SPDX-License-Identifier: MIT Copyright (c) 2026 Carlos Toledo Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. Portions of this file (interval arithmetic, the nu-weighted sequence norm and convolution, the radii-polynomial / Krawczyk operator) are ported from eqcert — MIT, Copyright (c) 2026 Carlos Toledo — https://github.com/carlostoledo1891/mfg-lab/tree/main/eqcert This FILE is MIT. The paper page that carries it is a separate work under CC-BY 4.0: https://github.com/carlostoledo1891/mfg-lab/blob/main/enclosure/LICENSE.md The SOLVER that produced the candidate this file verifies is not published and is not licensed here. Generated file — logic hand-written, certificate data embedded. """ import sys import math from fractions import Fraction # ============================================================================================ # CERTIFICATE DATA (embedded literals) — produced by dump_cert.js from the validated JS # kernel (research/mfg-congest/kernel). These are the numerical candidate x_bar and the # approximate-inverse matrix A that the verifier CONSUMES. It does not re-solve or # recompute A; it rigorously recomputes the bounds from exactly this data. # Instance: a=1/2, sigma=0.5, A=0.3, gamma=0.5, N=14, n=45. # ============================================================================================ N = 14 SIGMA = 0.5 A_PARAM = 0.3 GAMMA = 0.5 NU = 1.05 # candidate: u cosine coeffs a_0..a_N CAND_A = [ 0.4981121407217952, 0.006914743148283846, 1.1937047826341379e-05, 7.128689680739493e-08, 5.16975242809031e-10, 4.1377540721467864e-12, 3.5210164300885576e-14, 3.124785620876359e-16, 2.8514808056219233e-18, 2.5026035415458017e-20, 4.1345013275765895e-21, -3.544785361488696e-21, 8.437740601426773e-21, -1.181553667377019e-21, 3.6988550113961435e-21, ] # candidate: density coeffs m_0..m_N CAND_M = [ 1.0, -0.013161933467037779, 2.1365458966279915e-05, 1.270977181127319e-08, 2.8589747108825915e-11, 6.343360140899062e-14, -7.939397820114782e-17, -3.496943972609167e-18, -2.0909622665378228e-20, -8.947971484972129e-22, -1.6006589705522425e-20, 8.050401464896833e-21, -2.920207488314181e-20, 9.841656717770895e-21, -6.0272825133836835e-21, ] # candidate: reciprocal coeffs w_0..w_N (w = m^{-1/2}) CAND_W = [ 1.0001299700261361, 0.006582894213388972, 5.430672664548151e-05, 4.956043934093608e-07, 4.7716204391588454e-09, 4.743149234341656e-11, 4.814168940773841e-13, 4.9579429792566734e-15, 5.1597567271807053e-17, 5.417661938507755e-19, 1.3652581043314193e-20, -3.521969033514786e-21, 1.4437658171799408e-20, -4.577165196609576e-21, 2.922004744516202e-21, ] # approximate inverse A (= AN), row-major 45 x 45 AMAT = [ 1.0, -0.025170654205248878, 5.558030559294277e-05, -4.978718226373898e-09, -1.4853596722870406e-10, -1.6786960122723143e-12, -1.695531303429531e-14, -1.6684882280390322e-16, -1.6046863435080164e-18, -9.198408123427415e-21, -1.5786040798167858e-20, 1.445979000918176e-20, -3.3852605985468815e-20, 5.008580842139941e-21, -1.4810753146934674e-20, 0.5010266933421589, -0.003807564803773773, -0.00015694998072749534, 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1.4578543920390763e-20, 7.200190606798208e-19, 5.837058304732161e-17, 4.8680328926115625e-15, 4.168849845953056e-13, 3.66842069573116e-11, 3.32997083211419e-09, 3.147554808176798e-07, 3.1619431864556814e-05, 0.0035381432259608938, 0.5000395825647164, ] N_ODD = 42 # odd-block approximate inverse A_odd, row-major 42 x 42 AMAT_ODD = [ 0.2896440836814519, -0.0019321618208599997, -2.9839908846549353e-06, -1.9054468555429878e-08, -1.4046271561813886e-10, -1.128407551714676e-12, -9.602123867833535e-15, -8.510396431716692e-17, -7.707683343649221e-19, -8.608437221039428e-21, 3.2335548742428055e-21, 1.1324121761504351e-21, -2.1995817668915225e-21, 6.315547850949314e-22, 0.04400269639178119, -6.666574761197262e-07, -1.3563192853359716e-05, -2.3255966363405417e-07, -3.1757367453603946e-09, -4.005570285335181e-11, -4.861040767720889e-13, -5.770330939651257e-15, -6.75452508059993e-17, -7.833928835037885e-19, -8.46525049597339e-21, -1.6365041826384436e-22, 4.982442412939693e-22, 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-1.6255609322246861e-15, -1.2361629581482465e-13, -9.397331915384755e-12, -7.080711745484155e-10, -5.2227055348970946e-08, -3.685526345328395e-06, -0.0002356867652714073, -0.011368467844550301, -2.832560186138071e-21, 1.0862027157163251e-20, -3.1532850567197725e-21, 1.4578543572978228e-20, 7.200190606754261e-19, 5.837058304732154e-17, 4.8680328926115625e-15, 4.168849845953056e-13, 3.66842069573116e-11, 3.32997083211419e-09, 3.147554808176798e-07, 3.1619431864556814e-05, 0.0035381432259608938, 0.5000395825647164, ] # JS reference Phi(x_bar) (interval mids, modes 0..3N) — port-correctness cross-check PHI_H = [ -2.7755575615628926e-17, -1.647987302177967e-17, 3.886187162002731e-18, 5.073945729734973e-18, -1.2087227753484548e-18, -1.2029687843440828e-17, -2.2309292871070596e-18, 8.524825916099936e-18, -9.901634738192448e-18, -2.783453157211739e-18, 7.662756459551492e-18, -8.462950511362813e-18, 2.40133047386727e-17, -3.961083989225696e-18, 1.432106600159902e-17, -1.409418697951621e-20, -1.4417471985675203e-22, -1.540325614326417e-24, -1.6911282035735645e-26, -1.871447177871276e-28, -2.0780970902375376e-30, -2.3115195094455155e-32, -2.5742533466946774e-34, -2.6572048256142597e-36, -3.349334745794746e-37, 1.371406377081005e-37, -2.1037773671648345e-37, 2.940960583770348e-38, -5.26679369800076e-38, -3.878959032217227e-40, -3.000342632929082e-42, -2.734900165536889e-44, -2.6331620007030793e-46, -2.617911568100081e-48, -2.657628782611927e-50, -2.7374520461771172e-52, -2.8571944269331276e-54, -2.486078122063668e-56, -5.442332588999702e-57, 2.0208998474808266e-57, -1.52640057047912e-57, 3.340353405313848e-58, -1.5466831182714259e-58, ] PHI_F = [ 0.0, 3.4694469519536265e-18, -2.0193265462542524e-18, 5.034684429140756e-19, -7.042362928201766e-19, 4.536826890252379e-19, -7.13820766874433e-20, -2.4522363260480274e-19, 2.222277921527672e-19, -1.0872908274967417e-19, -2.5715057409169685e-19, 2.765757473901427e-20, -4.629169263037846e-19, 2.9872646866039537e-19, 6.103013325653373e-20, -2.26919496359186e-21, -7.382259242194945e-24, -4.830038842848177e-26, -4.6346909863080265e-28, -4.6516958494072474e-30, -4.8372074801770483e-32, -5.137339200799039e-34, -5.5337723988904795e-36, -5.17710308809381e-38, -1.4773484352147998e-38, 6.411589799502652e-39, -7.063160896005202e-39, 1.793445451254787e-39, -9.605958278519148e-40, -2.601933087676457e-41, -8.69709554831586e-44, -9.654671326343786e-46, -9.379621208132192e-48, -9.342291132905064e-50, -9.48774212627437e-52, -9.772404975163193e-54, -1.0238633234695982e-55, -6.175405829325929e-58, -3.900539686972648e-58, 1.6542415776291854e-58, -8.737590895387297e-59, 2.0032679208859474e-59, -5.730304195294391e-60, ] PHI_R = [ 6.661338147750939e-16, 4.848062930244194e-19, -1.626863131701605e-21, 4.6513607015878214e-23, -1.1948029730881205e-26, -1.141750095593606e-26, -8.816064788806166e-29, -1.344921117732195e-30, -1.1842849263096596e-32, -1.8995196055463627e-34, -3.690461011614709e-33, -3.708348094383368e-31, -3.8516526290087653e-29, -4.22050954315205e-27, -5.115956720838889e-25, -1.1628535734267156e-22, -9.632669215354413e-25, -1.0484613778280973e-26, -1.1538992551987029e-28, -1.2747845913419009e-30, -1.4112770930640387e-32, -1.5645192412612e-34, -1.736584680016919e-36, -1.8731209830129694e-38, -1.0899211230930327e-39, 4.705962298707908e-40, -3.222711504144451e-40, 8.139426862766568e-41, -2.56171171564128e-41, -3.3891341779410282e-43, -1.6769840785754086e-45, -1.6831764556648033e-47, -1.6269188239337043e-49, -1.618031787937716e-51, -1.6421167374694862e-53, -1.6907025274468746e-55, -1.773505220373685e-57, -8.833634494245234e-60, -6.862582840466794e-60, 2.807091045460872e-60, -1.1474040483597537e-60, 2.4525292555033328e-61, -5.146161150931375e-62, ] TWO_PI = 2 * math.pi # ============================================================================================ # 1. outward-rounded interval arithmetic (port of core/interval/interval.js) # Every IEEE-754 +,-,*,/ is correctly rounded, so widening the computed bounds outward by # one ulp (nextDown low, nextUp high) rigorously encloses the exact real result. # ============================================================================================ INF = math.inf EPS = sys.float_info.epsilon # 2.220446049250313e-16 == JS Number.EPSILON MIN_VALUE = 5e-324 # smallest positive subnormal == JS Number.MIN_VALUE def nextUp(x): if math.isnan(x) or x == INF: return x if x == 0.0: return MIN_VALUE return math.nextafter(x, INF) def nextDown(x): return -nextUp(-x) def iv(lo, hi=None): return (lo, lo if hi is None else hi) ZERO = (0.0, 0.0) ONE = (1.0, 1.0) def add(a, b): return (nextDown(a[0] + b[0]), nextUp(a[1] + b[1])) def sub(a, b): return (nextDown(a[0] - b[1]), nextUp(a[1] - b[0])) def mul(a, b): p0 = a[0] * b[0] p1 = a[0] * b[1] p2 = a[1] * b[0] p3 = a[1] * b[1] return (nextDown(min(p0, p1, p2, p3)), nextUp(max(p0, p1, p2, p3))) def div(a, b): if b[0] <= 0 and b[1] >= 0: raise ZeroDivisionError('interval division by an interval containing 0') q0 = a[0] / b[0] q1 = a[0] / b[1] q2 = a[1] / b[0] q3 = a[1] / b[1] return (nextDown(min(q0, q1, q2, q3)), nextUp(max(q0, q1, q2, q3))) def neg(a): return (-a[1], -a[0]) def iabs(a): if a[0] >= 0: return (a[0], a[1]) if a[1] <= 0: return (-a[1], -a[0]) return (0.0, max(-a[0], a[1])) def mag(a): return max(abs(a[0]), abs(a[1])) def mig(a): if a[0] > 0: return a[0] if a[1] < 0: return -a[1] return 0.0 def width(a): return nextUp(a[1] - a[0]) def mid(a): return 0.5 * (a[0] + a[1]) # ============================================================================================ # 2. nu-weighted sequences with declared parity (port of core/interval/sequence.js) # ||f||_nu = sum_k |f_k| nu^{|k|} is a Banach-algebra norm for convolution. # Parity is a first-class argument (even f_{-k}=f_k; its derivative odd f_{-k}=-f_k). # ============================================================================================ EVEN = 'e' ODD = 'o' def sget(f, j, parity): """interval accessor at any j in Z; ZERO outside the band, negate (widened) for odd<0.""" a = -j if j < 0 else j if a >= len(f): return ZERO v = f[a] if j < 0 and parity == ODD: return sub(ZERO, v) # matches JS A.sub(A.ZERO, v) exactly (one-ulp widening) return v def conv(f, g, K, pf, pg): """(f*g)_k for k=0..K over Z with declared parities, in interval arithmetic.""" out = [None] * (K + 1) Jf = len(f) - 1 Jg = len(g) - 1 for k in range(K + 1): s = ZERO for j in range(-Jf, Jf + 1): t = k - j if t < -Jg or t > Jg: continue s = add(s, mul(sget(f, j, pf), sget(g, t, pg))) out[k] = s return out def normNu(f, nu, hasZero): """||f||_nu for f stored as f[0..N]; hasZero=False for odd sequences (f_0 == 0).""" nuv = (nu, nu) s = iabs(f[0]) if hasZero else ZERO nk = ONE for k in range(1, len(f)): nk = mul(nk, nuv) s = add(s, mul((2.0, 2.0), mul(iabs(f[k]), nk))) return s # ============================================================================================ # 3. the augmented residual Phi and the Jacobian action DPhi # (port of research/mfg-congest/kernel/validate-congest.js buildPhi / dRow) # Fields: u as (a0 scalar + p=u' ODD); m EVEN; w EVEN. p_k = 2 pi k a_k. # ============================================================================================ def og(f, j): return sget(f, j, ODD) def eg(f, j): return sget(f, j, EVEN) def ev2(acc, f, k, m): """derivative of a convolution wrt the stored EVEN coefficient h_m (m==0 is a single mode).""" if m == 0: return acc(f, k) return add(acc(f, k - m), acc(f, k + m)) def colent(row, kind, m): return row['a0'] if kind == 'a0' else row[kind][m] def buildPhi(s, P, K): sigma = P['sigma'] A = P['A'] gamma = P['gamma'] gI = iv(gamma) halfA = iv(0.5 * A) ppw = conv(s['pp'], s['w'], K, EVEN, EVEN) # (p*p)*w EVEN sp = conv(s['s'], s['p'], K, EVEN, ODD) # (m*w)*p ODD mww = conv(s['s'], s['w'], K, EVEN, EVEN) # (m*w)*w EVEN H = [None] * (K + 1) F = [None] * (K + 1) Rc = [None] * (K + 1) H[0] = sub(sub(s['a'][0], mul(iv(0.5), ppw[0])), mul(gI, eg(s['m'], 0))) F[0] = sub(eg(s['m'], 0), ONE) Rc[0] = sub(mww[0], ONE) for k in range(1, K + 1): lam = TWO_PI * k h = mul(iv(1 + sigma * lam * lam), eg(s['a'], k)) h = sub(h, mul(iv(0.5), ppw[k])) if k == 1: h = sub(h, halfA) h = sub(h, mul(gI, eg(s['m'], k))) H[k] = h diag = add(div(ONE, iv(lam)), iv(sigma * lam)) F[k] = add(mul(diag, eg(s['m'], k)), og(sp, k)) Rc[k] = mww[k] return {'H': H, 'F': F, 'R': Rc} def dRow(typ, k, s, P, K): sigma = P['sigma'] gamma = P['gamma'] gI = iv(gamma) row = {'a0': ZERO, 'p': [ZERO] * (K + 1), 'm': [ZERO] * (K + 1), 'w': [ZERO] * (K + 1)} lam = TWO_PI * k diag = ONE if k == 0 else add(div(ONE, iv(lam)), iv(sigma * lam)) if typ == 'H': if k == 0: row['a0'] = ONE for m in range(1, K + 1): row['p'][m] = neg(sub(og(s['pw'], -m), og(s['pw'], m))) row['m'][0] = neg(gI) for m in range(0, K + 1): row['w'][m] = neg(mul(iv(0.5), ev2(eg, s['pp'], 0, m))) return row for m in range(1, K + 1): v = neg(sub(og(s['pw'], k - m), og(s['pw'], k + m))) if m == k: v = add(v, diag) row['p'][m] = v if k <= K: # JS auto-extends row.m[k>K]; it is never read (col k>KC) row['m'][k] = neg(gI) for m in range(0, K + 1): row['w'][m] = neg(mul(iv(0.5), ev2(eg, s['pp'], k, m))) return row if typ == 'F': if k == 0: row['m'][0] = ONE return row for m in range(1, K + 1): row['p'][m] = sub(eg(s['s'], k - m), eg(s['s'], k + m)) for m in range(0, K + 1): v = ev2(og, s['wp'], k, m) if m == k: v = add(v, diag) row['m'][m] = v for m in range(0, K + 1): row['w'][m] = ev2(og, s['mp'], k, m) return row # type R (reciprocal m w^2 - 1) for m in range(0, K + 1): row['m'][m] = ev2(eg, s['ww'], k, m) row['w'][m] = mul(iv(2), ev2(eg, s['s'], k, m)) return row # -------------------------------------------------------------------------------------------- # Odd-block convolution pairings at an even background (Stage 2.2), ported from the JS kernel. # EVEN x EVEN -> EVEN (cosine): sum -- same as ev2 # ODD x EVEN -> ODD (sine): sum # EVEN x ODD -> ODD (sine): difference # ODD x ODD -> EVEN (cosine): difference # eoDiff and ooDiff have IDENTICAL bodies -- in the JS kernel too. They are one rule with two # documented roles, and the discriminator is entirely the accessor the caller passes: for k < m, # ooDiff(og,f,k,m) = -f_{m-k} - f_{k+m} but eoDiff(eg,f,k,m) = +f_{m-k} - f_{k+m}. Do NOT collapse # them and do NOT bake og/eg inside: swapping the accessor is a mutation that changes Z1odd only # in the 15th significant digit and leaves every verdict green. # -------------------------------------------------------------------------------------------- def oeSum(acc, f, k, m): return add(acc(f, k - m), acc(f, k + m)) def eoDiff(acc, f, k, m): return sub(acc(f, k - m), acc(f, k + m)) def ooDiff(acc, f, k, m): return sub(acc(f, k - m), acc(f, k + m)) def oWgtOdd(i, N, nu): """Weighted-column weight for flat odd index i. Module level SO IT CAN BE TESTED. (i % N) + 1 recovers the MODE from the flat index only because all three odd sub-blocks have exactly N entries. Never 1: the odd block has no mean mode. An off-by-one here passes the finite-difference check (it lives outside dRowOdd) AND lowers Z1odd, so no verdict changes -- it was invisible to everything until test-odd-block.py W1 asserted it directly against THIS function. """ return 2 * nu ** ((i % N) + 1) def dRowOdd(typ, k, s, P, K): """One row of DPhi on the ODD block (Stage 2.2). Odd unknowns (NO a0): dp EVEN (cosine, m>=1), dm/dw ODD (sine, m>=1). There is no a0 and no mean mode because a sine series has none, and the integral of u' over the torus is zero. Odd residuals: H,R as sine coefficients; F scaled uses the even/cosine storage of (s*p), mirroring the even block's use of og(sp) inside the even F equation. EVERY off-diagonal sign differs from the even block, and NOT uniformly: dH/dp and dF/dp go difference->sum while everything else goes sum->difference. There is no "flip all signs" shortcut; this is transcribed entry by entry from the kernel and validated by the finite-difference check, which is the only test with teeth on it. """ sigma = P['sigma'] gamma = P['gamma'] gI = iv(gamma) row = {'p': [ZERO] * (K + 1), 'm': [ZERO] * (K + 1), 'w': [ZERO] * (K + 1)} if k == 0: return row # odd block has no mean row lam = TWO_PI * k diag = add(div(ONE, iv(lam)), iv(sigma * lam)) if typ == 'H': for m in range(1, K + 1): v = neg(oeSum(og, s['pw'], k, m)) # ODD(pw) x EVEN(dp) -> ODD : sum if m == k: v = add(v, diag) row['p'][m] = v if k <= K: # JS auto-extends row.m[k>K]; it is never read (col k>KC) row['m'][k] = neg(gI) for m in range(1, K + 1): row['w'][m] = neg(mul(iv(0.5), eoDiff(eg, s['pp'], k, m))) # EVEN(pp) x ODD(dw) -> ODD return row if typ == 'F': for m in range(1, K + 1): row['p'][m] = ev2(eg, s['s'], k, m) # EVEN(s) x EVEN(dp) -> EVEN : sum for m in range(1, K + 1): v = ooDiff(og, s['pw'], k, m) # ODD(pw) x ODD(dm) -> EVEN : difference if m == k: v = add(v, diag) row['m'][m] = v for m in range(1, K + 1): row['w'][m] = ooDiff(og, s['mp'], k, m) # ODD(mp) x ODD(dw) -> EVEN : difference return row # type R (reciprocal m w^2 - 1); dR/dp is identically zero for m in range(1, K + 1): row['m'][m] = eoDiff(eg, s['ww'], k, m) # EVEN(ww) x ODD(dm) -> ODD row['w'][m] = mul(iv(2), eoDiff(eg, s['s'], k, m)) # EVEN(s) x ODD(dw) -> ODD return row # ============================================================================================ # 4. candidate sequences (shared by validate and the falsifiers) # ============================================================================================ def make_seqs(pfac, a_coef, m_coef, w_coef): abar = [iv(a_coef[k]) for k in range(N + 1)] mbar = [iv(m_coef[k]) for k in range(N + 1)] wI = [iv(w_coef[k]) for k in range(N + 1)] pbar = [ZERO] * (N + 1) for k in range(1, N + 1): pbar[k] = iv(TWO_PI * k * a_coef[k]) pp = conv(pbar, pbar, 2 * N, pfac, pfac) pw = conv(pbar, wI, 2 * N, pfac, EVEN) ss = conv(mbar, wI, 2 * N, EVEN, EVEN) ww = conv(wI, wI, 2 * N, EVEN, EVEN) mp = conv(mbar, pbar, 2 * N, EVEN, pfac) return {'a': abar, 'p': pbar, 'm': mbar, 'w': wI, 'pp': pp, 'pw': pw, 's': ss, 'ww': ww, 'mp': mp, 'wp': pw} # ============================================================================================ # 5. the radii-polynomial driver (port of core/interval/radii.js) # ============================================================================================ def _up(v): return v if isinstance(v, (int, float)) else v[1] def radiiPolynomial(Y0in, Z1in, Z2in): Y0 = _up(Y0in) Z1 = _up(Z1in) Z2 = _up(Z2in) base = {'Y0': Y0, 'Z1': Z1, 'Z2': Z2} if not (Z1 < 1): d = dict(base); d.update(ok=False, why='Z1 >= 1 - the approximate inverse is not one; no contraction'); return d if not (Z2 > 0): r = nextUp(Y0 / (1 - Z1) * 1.5 + MIN_VALUE) d = dict(base); d.update(ok=(r > 0), r=r, rMin=Y0 / (1 - Z1), rMax=INF, linear=True); return d disc = (1 - Z1) * (1 - Z1) - 2 * Z2 * Y0 if not (disc > 0): d = dict(base); d.update(ok=False, why='discriminant <= 0 - defect too large to close', disc=disc); return d sq = math.sqrt(disc) rMin = nextUp(2 * Y0 / ((1 - Z1) + sq)) rMax = nextDown(((1 - Z1) + sq) / Z2) IY = iv(Y0) IZ1 = iv(Z1) IZ2 = iv(Z2) def pAt(rr): R = iv(rr) return add(sub(mul(mul(iv(0.5), IZ2), mul(R, R)), mul(sub(ONE, IZ1), R)), IY) r = rMin ok = False for _ in range(400): if r > rMax: break if pAt(r)[1] < 0: ok = True break r = nextUp(r * 1.05 + MIN_VALUE) if not ok: d = dict(base); d.update(ok=False, why='no radius verified p(r) < 0 in interval arithmetic', rMin=rMin, rMax=rMax, disc=disc); return d d = dict(base); d.update(ok=True, r=r, rMin=rMin, rMax=rMax, disc=disc, pAtR=pAt(r)[1]); return d # ============================================================================================ # 6. rigorous positivity over the WHOLE enclosure (density m>0 AND branch selector w>0) # sup_x |f(x)-f_bar(x)| <= ||Df||_1 <= ||Df||_nu <= r, so min f >= min f_bar - r; the # grid minimum pays the modulus of continuity L/(2G), L = sum 2*2pi k |f_k|. # (port of validate-congest.js certifyPositivity / certifyWPositivity — one shared body) # ============================================================================================ def certify_pos(coef, N_, r, G): L = 0.0 for k in range(1, N_ + 1): L = nextUp(L + 2 * TWO_PI * k * abs(coef[k])) G = G or 4096 lo = INF guard = 0 while True: lo = INF for g in range(G): t = g / G fv = iv(coef[0]) for k in range(1, N_ + 1): cv = math.cos(TWO_PI * k * t) fv = add(fv, mul(iv(2 * coef[k]), (nextDown(cv - 1e-15), nextUp(cv + 1e-15)))) if fv[0] < lo: lo = fv[0] slack = nextUp(L / (2 * G)) if slack < 0.25 * abs(lo) or G >= (1 << 21): break guard += 1 if guard > 12: break G *= 4 minBar = nextDown(lo - nextUp(L / (2 * G))) minBall = nextDown(minBar - r) return {'minBar': minBar, 'minBall': minBall, 'positive': minBall > 0, 'L': L, 'G': G} # ============================================================================================ # 7. the validation — Y0, Z1, Z2, radii polynomial, w>0 branch gate # (port of validate-congest.js validate; A is CONSUMED, never recomputed) # ============================================================================================ def validate(a_coef, m_coef, w_coef, AN, opts=None, AN_ODD=None): # The odd-block approximate inverse is EMBEDDED data, exactly like AMAT: this verifier # consumes an approximate inverse, it never computes one. Defaulted here so every # existing call site (including the falsifiers) exercises the odd block too. if AN_ODD is None: AN_ODD = AMAT_ODD opts = opts or {} nu = opts.get('nu', 1.05) sigma = SIGMA A = A_PARAM gamma = GAMMA P = {'sigma': sigma, 'A': A, 'gamma': gamma, 'N': N} KC = max(opts.get('KC', 3 * N), 3 * N) KR = KC + 2 * N rCap = opts.get('rCap', 1e-2) n = 3 * N + 3 idxA0 = 0 def idxP(k): return k def idxM(k): return N + 1 + k def idxW(k): return 2 * N + 2 + k def finW(i): if i == 0: return 1.0 if i <= N: return 2 * nu ** i if i <= 2 * N + 1: k = i - (N + 1) return 1.0 if k == 0 else 2 * nu ** k k = i - (2 * N + 2) return 1.0 if k == 0 else 2 * nu ** k s = make_seqs(ODD, a_coef, m_coef, w_coef) pbar = s['p'] mbar = s['m'] wbarI = s['w'] normP = normNu(pbar, nu, False) normM = normNu(mbar, nu, True) normW = normNu(wbarI, nu, True) normPW = normNu(s['pw'], nu, False) normS = normNu(s['s'], nu, True) normWW = normNu(s['ww'], nu, True) normPP = normNu(s['pp'], nu, True) normMP = normNu(s['mp'], nu, False) s0 = s['s'][0] s0mid = 0.5 * (s0[0] + s0[1]) if not (s0mid > 0): return {'ok': False, 'why': 's_bar_0 = mean sqrt m not positive - off the physical branch'} tailRecipInv = iv(1.0 / (2 * s0mid)) def tailHF(k): lam = TWO_PI * k d = 1 / lam + sigma * lam return iv(1.0 / d) # -------- Y0 = || A Phi(x_bar) ||_nu -------- Phi = buildPhi(s, P, 3 * N) rvec = [ZERO] * n rvec[idxA0] = Phi['H'][0] for k in range(1, N + 1): rvec[idxP(k)] = Phi['H'][k] for k in range(0, N + 1): rvec[idxM(k)] = Phi['F'][k] rvec[idxW(k)] = Phi['R'][k] acc = ZERO for i in range(n): a = ZERO base = i * n for j in range(n): a = add(a, mul(iv(AN[base + j]), rvec[j])) acc = add(acc, mul(iv(finW(i)), iabs(a))) for k in range(N + 1, 3 * N + 1): w = iv(2 * nu ** k) acc = add(acc, mul(w, iabs(mul(tailHF(k), Phi['H'][k])))) acc = add(acc, mul(w, iabs(mul(tailHF(k), Phi['F'][k])))) acc = add(acc, mul(w, iabs(mul(tailRecipInv, Phi['R'][k])))) Y0 = acc # -------- Z1 = || I - A DPhi(x_bar) || (max over weighted columns) -------- Z1 = ZERO worstCol = None rowsH = [] rowsF = [] rowsR = [] for k in range(KR + 1): rowsH.append(dRow('H', k, s, P, KC)) rowsF.append(dRow('F', k, s, P, KC)) rowsR.append(dRow('R', k, s, P, KC)) def colNorm(kind, m): fin = [ZERO] * n fin[idxA0] = colent(rowsH[0], kind, m) for k in range(1, N + 1): fin[idxP(k)] = colent(rowsH[k], kind, m) for k in range(0, N + 1): fin[idxM(k)] = colent(rowsF[k], kind, m) fin[idxW(k)] = colent(rowsR[k], kind, m) out = [ZERO] * n for i in range(n): a = ZERO base = i * n for j in range(n): a = add(a, mul(iv(AN[base + j]), fin[j])) out[i] = a if kind == 'a0': jIdx = idxA0 elif kind == 'p': jIdx = idxP(m) if m <= N else -1 elif kind == 'm': jIdx = idxM(m) if m <= N else -1 else: jIdx = idxW(m) if m <= N else -1 acc2 = ZERO for i in range(n): e = ONE if i == jIdx else ZERO acc2 = add(acc2, mul(iv(finW(i)), iabs(sub(e, out[i])))) for k in range(N + 1, KR + 1): w = iv(2 * nu ** k) eH = ONE if (kind == 'p' and m == k) else ZERO eF = ONE if (kind == 'm' and m == k) else ZERO eR = ONE if (kind == 'w' and m == k) else ZERO acc2 = add(acc2, mul(w, iabs(sub(eH, mul(tailHF(k), colent(rowsH[k], kind, m)))))) acc2 = add(acc2, mul(w, iabs(sub(eF, mul(tailHF(k), colent(rowsF[k], kind, m)))))) acc2 = add(acc2, mul(w, iabs(sub(eR, mul(tailRecipInv, colent(rowsR[k], kind, m)))))) wj = 1.0 if kind == 'a0' else (1.0 if m == 0 else 2 * nu ** m) return div(acc2, iv(wj)) cand = [('a0', 0)] for m in range(1, KC + 1): cand.append(('p', m)) for m in range(0, KC + 1): cand.append(('m', m)) cand.append(('w', m)) for (kind, m) in cand: v = colNorm(kind, m) if v[1] > mag(Z1): Z1 = v worstCol = kind + '_' + str(m) # analytic uniform tail bound (columns m > KC) lamN1 = TWO_PI * (N + 1) denomLoGrow = nextDown(1 / lamN1 + sigma * lamN1) denomRecip = nextDown(2 * s0mid) g = gamma cP = nextUp((mag(normPW) + mag(normS)) / denomLoGrow) cMgrow = nextUp((g + mag(normPW)) / denomLoGrow) cMrecip = nextUp(mag(normWW) / denomRecip) cWgrow = nextUp((0.5 * mag(normPP) + mag(normMP)) / denomLoGrow) sOff = nextUp(2 * max(0.0, mag(normS) - s0[0])) cWrecip = nextUp(sOff / denomRecip) epsId = 8 * EPS + width(s0) / s0mid analytic = max(cP + epsId, cMgrow + cMrecip + epsId, cWgrow + cWrecip + epsId) # NaN guard BEFORE the comparison below: `nan > x` is False in Python exactly as in JS, # so an unguarded compare would silently keep the lower explicit-column Z1 and certify # on a poisoned tail bound. Same Step 0 hazard as the JS kernels, closed 2026-07-27. if math.isnan(analytic) or math.isinf(analytic): raise ArithmeticError('analytic Z1 tail bound is not finite') analyticInfo = {'mP': cP + epsId, 'mM': cMgrow + cMrecip + epsId, 'mW': cWgrow + cWrecip + epsId, 'epsId': epsId, 'cP': cP, 'cMgrow': cMgrow, 'cMrecip': cMrecip, 'cWgrow': cWgrow, 'cWrecip': cWrecip} if analytic > mag(Z1): Z1 = iv(analytic) worstCol = 'tail(analytic)' # -------- Z1_odd = || I - A_o DPhi_odd(x_bar) || (Stage 2.2 full-space uniqueness) -------- # PORTED 2026-08-05. Until this landed, this verifier certified the EVEN (cosine) subspace # only and said so in its own output, and the claim carried that as a load-bearing caveat: # full-space uniqueness rested on ONE implementation. The odd block is where a symmetry- # BREAKING solution would live, so certifying it is what rules out an asymmetric solution # sitting next to the even one. # # Y0 is untouched and needs no odd term: p_bar is odd and m_bar/w_bar are even, so every odd # component of Phi(x_bar) is identically zero -- structurally, not numerically. # # The analytic tail bound is REUSED VERBATIM: it is built from l1_nu kernel NORMS, never from # coefficients, and both the sum and difference pairings are bounded by the same norm, so it # is parity-agnostic. Every odd column has m >= 1, so the zero-mode caveat never fires. nOdd = 3 * N if len(AN_ODD) != nOdd * nOdd: return {'ok': False, 'why': 'odd-block approximate inverse absent or wrong size - ' 'refusing rather than certifying the even block alone'} def oP(k): return k - 1 def oM(k): return N + k - 1 def oW(k): return 2 * N + k - 1 def oWgt(i): return oWgtOdd(i, N, nu) # ||A_odd|| in the same weighted-column operator norm, for Z2 (see the Aopnorm comment). anormOdd = 0.0 for j in range(nOdd): ss2 = 0.0 for i in range(nOdd): ss2 = nextUp(ss2 + oWgt(i) * abs(AN_ODD[i * nOdd + j])) v = nextUp(ss2 / oWgt(j)) if v > anormOdd: anormOdd = v rowsHo = [] rowsFo = [] rowsRo = [] for k in range(KR + 1): rowsHo.append(dRowOdd('H', k, s, P, KC)) rowsFo.append(dRowOdd('F', k, s, P, KC)) rowsRo.append(dRowOdd('R', k, s, P, KC)) def colNormO(kind, m): fin = [ZERO] * nOdd for k in range(1, N + 1): fin[oP(k)] = rowsHo[k][kind][m] fin[oM(k)] = rowsFo[k][kind][m] fin[oW(k)] = rowsRo[k][kind][m] out = [ZERO] * nOdd for i in range(nOdd): a = ZERO base = i * nOdd for j in range(nOdd): a = add(a, mul(iv(AN_ODD[base + j]), fin[j])) out[i] = a if kind == 'p': jIdx = oP(m) if m <= N else -1 elif kind == 'm': jIdx = oM(m) if m <= N else -1 else: jIdx = oW(m) if m <= N else -1 acc2 = ZERO for i in range(nOdd): e = ONE if i == jIdx else ZERO acc2 = add(acc2, mul(iv(oWgt(i)), iabs(sub(e, out[i])))) for k in range(N + 1, KR + 1): w = iv(2 * nu ** k) eH = ONE if (kind == 'p' and m == k) else ZERO eF = ONE if (kind == 'm' and m == k) else ZERO eR = ONE if (kind == 'w' and m == k) else ZERO acc2 = add(acc2, mul(w, iabs(sub(eH, mul(tailHF(k), rowsHo[k][kind][m]))))) acc2 = add(acc2, mul(w, iabs(sub(eF, mul(tailHF(k), rowsFo[k][kind][m]))))) acc2 = add(acc2, mul(w, iabs(sub(eR, mul(tailRecipInv, rowsRo[k][kind][m]))))) return div(acc2, iv(2 * nu ** m)) Z1odd = ZERO worstColOdd = None for m in range(1, KC + 1): for kind in ('p', 'm', 'w'): v = colNormO(kind, m) if v[1] > mag(Z1odd): Z1odd = v worstColOdd = kind + '_' + str(m) analyticO = max(analyticInfo['mP'], analyticInfo['mM'], analyticInfo['mW']) if math.isnan(analyticO) or math.isinf(analyticO): raise ArithmeticError('analytic Z1_odd tail bound is not finite') if analyticO > mag(Z1odd): Z1odd = iv(analyticO) worstColOdd = 'tail(analytic)' # full-space Z1 = max(even, odd). STRICT >: ties keep the even block and the 'odd:' prefix # appears only on a strict odd win, matching the kernel exactly. Z1even = Z1 worstColEven = worstCol if mag(Z1odd) > mag(Z1even): Z1 = Z1odd worstCol = 'odd:' + str(worstColOdd) else: Z1 = Z1even worstCol = worstColEven # -------- Z2: uniform Lipschitz of x -> A DPhi(x) on B_rCap(x_bar) -------- anorm = 0.0 for j in range(n): ss2 = 0.0 for i in range(n): ss2 = nextUp(ss2 + finW(i) * abs(AN[i * n + j])) v = nextUp(ss2 / finW(j)) if v > anorm: anorm = v tailGrow = 1 / (1 / lamN1 + sigma * lamN1) # anormOdd JOINS THIS MAX (mirrors the 2026-08-05 kernel fix). Z2 is the Lipschitz # constant of x -> A DPhi(x) on the WHOLE ball; off the even subspace DPhi is not # block-diagonal, so the ||A|| Z2 needs is the max over BOTH blocks. Changes no # number at either shipped instance -- the even norm dominates ~3.5x and ~3.9x -- # but nothing checked that it did, and the missing term was the unsound direction. Aopnorm = max(anorm, anormOdd, tailGrow, 1 / (2 * s0mid)) D2 = add(add(add(normW, mul(iv(2), normP)), mul(iv(2), add(add(normP, normW), normM))), add(mul(iv(2), normM), mul(iv(4), normW))) D3 = iv(15) factor = add(D2, mul(mul(iv(0.5), D3), iv(rCap))) Z2 = mul(iv(Aopnorm), factor) # -------- radii polynomial -------- y0 = mag(Y0) z1 = mag(Z1) z2 = mag(Z2) res = {'N': N, 'nu': nu, 'KC': KC, 'KR': KR, 'sigma': sigma, 'A': A, 'gamma': gamma, 'rCap': rCap, 'Y0': y0, 'Z1': z1, 'Z2': z2, 'worstCol': worstCol, # both blocks reported SEPARATELY: a port that matched only the max could be wrong # on both blocks in compensating directions and never show it. 'Z1even': mag(Z1even), 'Z1odd': mag(Z1odd), 'worstColEven': worstColEven, 'worstColOdd': worstColOdd, 'Aopnorm': Aopnorm, 's0': s0mid, 'analytic': analyticInfo} rp = radiiPolynomial(Y0, Z1, Z2) if not rp['ok']: res.update(ok=False, why=rp['why'], disc=rp.get('disc'), rMin=rp.get('rMin'), rMax=rp.get('rMax')) return res if not (rp['r'] <= rCap): res.update(ok=False, why='certified radius exceeds rCap', r=rp['r'], rMin=rp['rMin'], rMax=rp['rMax'], disc=rp['disc']) return res # branch selection: min w > 0 over the WHOLE ball (physical w = +m^{-1/2}) wPosBallR = opts['wPosR'] if opts.get('wPosR') is not None else rp['r'] wpos = certify_pos(w_coef, N, wPosBallR, opts.get('wG')) if not wpos['positive']: res.update(ok=False, why='w-series not certified positive over the ball', r=rp['r'], rMin=rp['rMin'], rMax=rp['rMax'], disc=rp['disc'], minWbar=wpos['minBar'], minW=wpos['minBall']) return res res.update(ok=True, r=rp['r'], rMin=rp['rMin'], rMax=rp['rMax'], disc=rp['disc'], pAtR=rp['pAtR'], closureMargin=(1 - z1) * (1 - z1) / (2 * z2), minWbar=wpos['minBar'], minW=wpos['minBall']) return res # ============================================================================================ # 8. float helpers for the INDEPENDENT pointwise witnesses and the reciprocal aux # (port of research/mfg-congest/kernel/candidate-congest.js ev/d1/d2/evalPDE/reciprocalW) # ============================================================================================ def ev_series(f, x): s = f[0] for k in range(1, len(f)): s += 2 * f[k] * math.cos(TWO_PI * k * x) return s def d1_series(f, x): s = 0.0 for k in range(1, len(f)): s += -2 * (TWO_PI * k) * f[k] * math.sin(TWO_PI * k * x) return s def d2_series(f, x): s = 0.0 for k in range(1, len(f)): s += -2 * (TWO_PI * k) * (TWO_PI * k) * f[k] * math.cos(TWO_PI * k * x) return s def fget(f, j, parity): a = -j if j < 0 else j if a >= len(f): return 0.0 v = f[a] if j < 0 and parity == ODD: return -v return v def fconv(f, g, K, pf, pg): out = [0.0] * (K + 1) Jf = len(f) - 1 Jg = len(g) - 1 for k in range(K + 1): s = 0.0 for j in range(-Jf, Jf + 1): t = k - j if t < -Jg or t > Jg: continue s += fget(f, j, pf) * fget(g, t, pg) out[k] = s return out def reciprocalW_py(m_coef, N_, drop=False): """w ~ m^{-1/2} as a finite cosine series by reciprocal-sqrt Newton in Fourier. drop=True drops the 1/2 factor (= the -a m^{-(a+1)} damping): the fixed point moves to m w^2 = 1/2 (the 'dropped 1/m^{a+1} factor' mutation).""" half = 1.0 if drop else 0.5 m = list(m_coef[:N_ + 1]) w = [0.0] * (N_ + 1) w[0] = 1.0 / math.sqrt(m[0]) for it in range(60): mw = fconv(m, w, N_, EVEN, EVEN) mww = fconv(mw, w, N_, EVEN, EVEN) g = [(1.5 if k == 0 else 0.0) - half * mww[k] for k in range(N_ + 1)] wn = fconv(w, g, N_, EVEN, EVEN) d = max(abs(wn[k] - w[k]) for k in range(N_ + 1)) w = wn if d < 1e-16 and not drop: break if drop and it > 40: break return w def eval_pde(a, m, w, N_, sigma, A, gamma, G): hjbMax = 0.0 fpResMax = 0.0 xs = [] Jv = [] for i in range(G): t = i / G u = ev_series(a, t) upp = d2_series(a, t) up = d1_series(a, t) mm = ev_series(m, t) mp = d1_series(m, t) mpp = d2_series(m, t) sm = math.sqrt(mm) V = A * math.cos(TWO_PI * t) + gamma * mm hjb = u - sigma * upp + 0.5 * up * up / sm - V if abs(hjb) > hjbMax: hjbMax = abs(hjb) dflux = 0.5 / sm * mp * up + sm * upp fp = mm - sigma * mpp - dflux - 1 if abs(fp) > fpResMax: fpResMax = abs(fp) xs.append(t) Jv.append(sigma * mp + sm * up) phiMin = INF phiMax = -INF for i in range(G): t = xs[i] Sa = 0.0 for k in range(1, N_ + 1): Sa += 2 * m[k] * math.sin(TWO_PI * k * t) / (TWO_PI * k) ph = Jv[i] - Sa if ph < phiMin: phiMin = ph if ph > phiMax: phiMax = ph fpFluxConstDev = phiMax - phiMin reciprocalMaw = 0.0 for i in range(G): t = xs[i] prod = math.sqrt(ev_series(m, t)) * ev_series(w, t) if abs(prod - 1) > reciprocalMaw: reciprocalMaw = abs(prod - 1) return {'hjbMax': hjbMax, 'fpResMax': fpResMax, 'fpFluxConstDev': fpFluxConstDev, 'reciprocalMaw': reciprocalMaw} # ============================================================================================ # 9. the SEVEN falsifiers (X1..X7) — each MUST turn its own target RED (refuse) # mirrors research/mfg-congest/tests/test-validate-congest.js. A verifier that cannot # refuse a bad input is a fake certificate. These use the EMBEDDED approximate inverse A; # where the JS recomputes A internally (X4), the refusal is only STRONGER with the good A. # ============================================================================================ def falsifier_X1(): """outward rounding is load-bearing: a thin (no-widening) mul fails to enclose the exact product; eqcert-style widened mul encloses all 4000/4000. Exact reference = Fraction.""" def selfTest(mulfn): seed = [20260723] def rng(): p = float(seed[0]) * 1103515245.0 + 12345.0 seed[0] = int(p) % 4294967296 & 0x7fffffff return seed[0] / 2147483647 for t in range(4000): a = (rng() - 0.5) * (1e6 if t % 3 else 1e-6) b = (rng() - 0.5) * (1e3 if t % 2 else 1e-3) if b == 0: b = 0.5 res = mulfn((a, a), (b, b)) ex = Fraction(a) * Fraction(b) if Fraction(res[0]) > ex or ex > Fraction(res[1]): return t return -1 def thinMul(a, b): p = [a[0] * b[0], a[0] * b[1], a[1] * b[0], a[1] * b[1]] return (min(p), max(p)) good = selfTest(mul) bad = selfTest(thinMul) return good < 0 and bad >= 0, (good, bad) def falsifier_X2(ref_Y0): bad_a = list(CAND_A) bad_a[0] += 1e-2 rb = validate(bad_a, CAND_M, CAND_W, AMAT, {'nu': 1.05}) return (rb['Y0'] > 1e6 * ref_Y0) and (not rb['ok']), rb def falsifier_X3(ref_r): bad_a = list(CAND_A) bad_a[0] += 1e-2 e = eval_pde(bad_a, CAND_M, CAND_W, N, SIGMA, A_PARAM, GAMMA, 512) return e['hjbMax'] > 1e-4 and e['hjbMax'] / ref_r > 1e6, e def falsifier_X4(ref_Y0): w_drop = reciprocalW_py(CAND_M, N, drop=True) rb = validate(CAND_A, CAND_M, w_drop, AMAT, {'nu': 1.05}) return (not rb['ok']) and rb['Y0'] > 1e6 * ref_Y0, rb def falsifier_X5(): P = {'sigma': SIGMA, 'A': A_PARAM, 'gamma': GAMMA, 'N': N} def norm(seqs): Phi = buildPhi(seqs, P, 3 * N) mx = 0.0 for k in range(3 * N + 1): mx = max(mx, mag(Phi['H'][k]), mag(Phi['F'][k]), mag(Phi['R'][k])) return mx ng = norm(make_seqs(ODD, CAND_A, CAND_M, CAND_W)) nb = norm(make_seqs(EVEN, CAND_A, CAND_M, CAND_W)) return nb > 1e-3 and nb > 1e10 * ng, (ng, nb) def falsifier_X6(): bogus = certify_pos(CAND_M, N, 10, None) return (not bogus['positive']), bogus def falsifier_X7(): wp = certify_pos(CAND_W, N, 10, None) rb = validate(CAND_A, CAND_M, CAND_W, AMAT, {'nu': 1.05, 'wPosR': 10}) wired = (not rb['ok']) and ('w-series not certified positive' in (rb.get('why') or '')) return (not wp['positive']) and wired, (wp, rb) # ============================================================================================ # 10. driver / report # ============================================================================================ def run(): n = 3 * N + 3 assert len(AMAT) == n * n, 'embedded A has wrong size' # ---- port correctness gate: Phi(x_bar) elementwise vs the embedded JS reference ---- P = {'sigma': SIGMA, 'A': A_PARAM, 'gamma': GAMMA, 'N': N} seqs = make_seqs(ODD, CAND_A, CAND_M, CAND_W) Phi = buildPhi(seqs, P, 3 * N) phi_dev = 0.0 for k in range(3 * N + 1): phi_dev = max(phi_dev, abs(mid(Phi['H'][k]) - PHI_H[k]), abs(mid(Phi['F'][k]) - PHI_F[k]), abs(mid(Phi['R'][k]) - PHI_R[k])) phi_ok = phi_dev < 1e-12 # ---- reciprocal-w consistency: the embedded w matches m^{-1/2} rebuilt from m ---- w_rebuilt = reciprocalW_py(CAND_M, N) w_dev = max(abs(w_rebuilt[k] - CAND_W[k]) for k in range(N + 1)) w_ok = w_dev < 1e-13 # ---- the certificate ---- r = validate(CAND_A, CAND_M, CAND_W, AMAT, {'nu': 1.05}) pos = certify_pos(CAND_M, N, r['r'], None) if r['ok'] else None wit = eval_pde(CAND_A, CAND_M, CAND_W, N, SIGMA, A_PARAM, GAMMA, 4096) # ---- falsifiers ---- (ref_r falls back if the certificate refused, so a tampered # candidate yields a clean REFUSED report instead of a crash) ref_r = r.get('r', 1e-12) fx = {} fx['X1'], d1 = falsifier_X1() fx['X2'], _ = falsifier_X2(r['Y0']) fx['X3'], _ = falsifier_X3(ref_r) fx['X4'], _ = falsifier_X4(r['Y0']) fx['X5'], d5 = falsifier_X5() fx['X6'], _ = falsifier_X6() fx['X7'], _ = falsifier_X7() fals_ok = all(fx.values()) cert_ok = bool(r['ok'] and pos and pos['positive']) verified = cert_ok and phi_ok and w_ok and fals_ok result = { 'r': r.get('r'), 'Z1': r['Z1'], 'Z2': r['Z2'], 'Y0': r['Y0'], 'disc': r.get('disc'), 'Aopnorm': r['Aopnorm'], 's0': r['s0'], 'worstCol': r['worstCol'], 'Z1even': r.get('Z1even'), 'Z1odd': r.get('Z1odd'), 'worstColEven': r.get('worstColEven'), 'worstColOdd': r.get('worstColOdd'), 'minMbar': pos['minBar'] if pos else None, 'minM': pos['minBall'] if pos else None, 'minWbar': r.get('minWbar'), 'minW': r.get('minW'), 'closureMargin': r.get('closureMargin'), 'analytic': r['analytic'], 'phiMaxDev': phi_dev, 'wMaxDev': w_dev, 'witnesses': wit, 'falsifiers': fx, 'ok': r['ok'], 'why': r.get('why'), 'verified': verified, 'cert_ok': cert_ok, 'phi_ok': phi_ok, 'w_ok': w_ok, 'fals_ok': fals_ok, '_x1': d1, '_x5': d5, } return result def _report(R): L = [] p = L.append def f(x, fmt): return (fmt % x) if x is not None else 'n/a' p("=" * 84) p(" INDEPENDENT PYTHON VERIFIER - discounted congestion MFG (a=1/2), computer-assisted proof") p(" instance: sigma=%.3g A=%.3g gamma=%.3g N=%d (stdlib only; outward-rounded intervals)" % (SIGMA, A_PARAM, GAMMA, N)) p("=" * 84) p("") p(" radii-polynomial / Newton-Kantorovich bounds (recomputed from candidate + A):") p(" Y0 = ||A Phi(x_bar)||_nu = %s (residual defect)" % f(R['Y0'], '%.6e')) p(" Z1 = ||I - A DPhi(x_bar)||_nu = %s (worst col: %s)" % (f(R['Z1'], '%.6f'), R['worstCol'])) p(" Z2 = Lipschitz on B_rCap = %s (||A|| = %.4f, s_bar_0 = %.6f)" % (f(R['Z2'], '%.6f'), R['Aopnorm'], R['s0'])) p(" disc = (1-Z1)^2 - 2 Z2 Y0 = %s" % f(R['disc'], '%.6f')) p(" certified radius r = %s" % f(R['r'], '%.6e')) p(" closure margin (Y0 headroom) = %s" % f(R['closureMargin'], '%.4e')) if R['cert_ok']: p(" -> p(r) < 0 and Z1 < 1: contraction closes; an exact solution lies within r of") p(" the candidate, LOCALLY UNIQUE THERE IN THE FULL l1_nu BALL -- both the even") p(" (cosine) and the odd (sine) blocks, Z1 = max(Z1even, Z1odd).") else: p(" -> certificate did NOT close: %s" % (R['why'] or 'positivity failed')) p("") p(" positivity over the WHOLE ball (both walls certified, never assumed):") p(" density min m over ball >= %s (min m_bar = %s)" % (f(R['minM'], '%.6f'), f(R['minMbar'], '%.6f'))) p(" branch min w over ball >= %s (min w_bar = %s; w>0 selects +m^{-1/2})" % (f(R['minW'], '%.6f'), f(R['minWbar'], '%.6f'))) p("") p(" independent pointwise witnesses (a different representation than the Fourier residual):") w = R['witnesses'] p(" |HJB(x_bar)| = %.3e FP flux const dev = %.3e" % (w['hjbMax'], w['fpFluxConstDev'])) p(" |sqrt(m) w - 1| = %.3e FP residual = %.3e" % (w['reciprocalMaw'], w['fpResMax'])) p("") p(" port-correctness gates:") p(" Phi(x_bar) vs embedded JS reference, max elementwise dev = %.2e [%s]" % (R['phiMaxDev'], 'OK' if R['phi_ok'] else 'FAIL')) p(" embedded w vs m^{-1/2} rebuilt from m, max dev = %.2e [%s]" % (R['wMaxDev'], 'OK' if R['w_ok'] else 'FAIL')) p("") p(" falsifiers (each MUST refuse - a certificate that cannot go RED is fake):") names = { 'X1': 'outward rounding load-bearing (thin mul fails to enclose vs exact rational)', 'X2': 'perturbed a_0: Y0 explodes, validate REFUSES', 'X3': 'perturbed pointwise HJB >> r (witness has teeth)', 'X4': 'dropped 1/m^{a+1} factor: reciprocal residual O(1), REFUSES', 'X5': 'parity bug (p in EVEN basis): ||Phi|| blows up', 'X6': 'density positivity REFUSES on a ball too wide (r=10)', 'X7': 'w>0 branch gate REFUSES on a ball too wide (r=10)', } for key in ['X1', 'X2', 'X3', 'X4', 'X5', 'X6', 'X7']: p(" %s %s %s" % ('RED ok ' if R['falsifiers'][key] else 'FAIL ', key, names[key])) p("") p("-" * 84) if R['verified']: p(" The radii polynomial closes: an exact solution lies within r of the candidate, and") p(" LOCAL UNIQUENESS IS CERTIFIED WITHIN THE ENCLOSURE BALL IN THE FULL l1_nu SPACE --") p(" even (cosine) AND odd (sine) blocks, Z1 = max(Z1even, Z1odd). Until 2026-08-05 this") p(" file certified the even block alone and said so here; the odd block is where a") p(" symmetry-BREAKING solution would live, so certifying it is what rules out an") p(" asymmetric solution sitting beside the even one. Density m>0 and the physical") p(" w=+m^{-1/2} branch are certified in interval arithmetic over the WHOLE ball; the") p(" Phi port matches the JS reference; every falsifier refused. No tolerance was tuned") p(" to close.") p("") p(" WHAT THE NORM MEANS, because it is stronger than a discretisation claim and weaker") p(" than a claim about the model. ||.||_nu is the weighted l1 norm on FOURIER") p(" COEFFICIENTS, and Y0, Z1 and Z2 each bound the INFINITE tail explicitly -- the") p(" analytic tail bound over columns m > KC is a Banach-algebra estimate, not a") p(" truncation. So the enclosed object is a solution of the system, not of an N-mode") p(" approximation to it. What IS a point rather than a range is the INSTANCE: these") p(" numbers are a statement at this sigma, A, gamma and a=1/2, and at no other.") p("") p(" CONGEST CAP: VERIFIED") else: reasons = [] if not R['cert_ok']: reasons.append('certificate did not close (%s)' % (R['why'] or 'positivity failed')) if not R['phi_ok']: reasons.append('Phi port mismatch') if not R['w_ok']: reasons.append('reciprocal-w mismatch') if not R['fals_ok']: reasons.append('a falsifier failed to refuse') p(" CONGEST CAP: REFUSED - " + '; '.join(reasons)) p("-" * 84) return "\n".join(L) def main(): R = run() if '--json' in sys.argv: import json out = {k: R[k] for k in ('r', 'Z1', 'Z2', 'Y0', 'disc', 'minMbar', 'minM', 'minWbar', 'minW', 'phiMaxDev', 'wMaxDev', 'verified', 'Z1even', 'Z1odd', 'worstColEven', 'worstColOdd')} out['falsifiers'] = R['falsifiers'] out['analytic'] = R['analytic'] print(json.dumps(out)) else: print(_report(R)) return 0 if R['verified'] else 1 if __name__ == '__main__': sys.exit(main())