Checks · Approximate Antiunitary Symmetry as a Matching Problem

Recorded output of the numerical verification, September 28, 2026

The numerical verification program wrote this file on September 28, 2026; its stamp completed_utc is in universal time, which puts it on September 29. It lists the library versions it ran under and its own SHA-256 (verify_sha256), which matches the published program, then one block per check: matching_vs_exhaustive (1,000 cases, no failures), haar_unitaries (4,800 random unitary matrices, none below the theorem's minimum beyond roundoff), independent_skew_contractions (240 matrices, no failures), scalar_dynamic_program (1,500 cases, no failures), continuous_optimization (120 runs of a local optimizer: 96 reached the exact minimum, none went below it, the worst ended 48 above it, every run listed under details), covariance_and_perturbation (70 commuting triples) and exact_controls (the two examples).

Output of: The numerical verification program (checks/verify.py), shown beside it on that page

Written by: verify.py, a program by GPT-6 Astra (OpenAI)
Size: 31,201 bytes
SHA-256: a75f278a30394d381d0fe31be3dc393d6d96257d34f96b0cab4f8ed2c5c50bbd

The paper's page Every file published with it This file on GitHub

Tables

seed	20260928
scope	Finite falsification and implementation checks; not a formal proof
completed_utc	2026-09-29T03:23:07.317390+00:00
verify_sha256	3cf140d5aff5ce5f25cc925571119e8dc41615ac36f21a797af6734e4280c016

versions

python	3.12.3
numpy	2.5.3
scipy	1.18.1
networkx	3.7

matching_vs_exhaustive

cells	1000
dimensions	[1, 10]
failures	0
max_witness_error	3.552713678800501e-15

haar_unitaries

samples	4800
dimensions	[1, 12]
min_energy_minus_exact_minimum	-1.4210854715202004e-14
max_decomposition_error	1.7053025658242404e-13
max_odd_operator_norm_error	1.5543122344752192e-15
max_odd_set_constraint_excess	0.0
failures	0

independent_skew_contractions

samples	240
failures	0

scalar_dynamic_program

cells	1500
dimensions	[1, 30]
failures	0

continuous_optimization

trials	120
gradient_check_error	1.4437805510484203e-06
reached_global_within_1e-7	96
optimizer_success_count	80
min_gap	-2.1316282072803006e-14
max_gap	48.000000000000014

continuous_optimization › details

n	case	tau	restart	exact	value	gap	optimizer_success	iterations
2	0	0.5	0	4.0	4.000000000000001	8.881784197001252e-16	true	8
2	0	0.5	1	4.0	3.999999999999999	-8.881784197001252e-16	true	6
2	0	0.5	2	4.0	4.000000000000001	8.881784197001252e-16	true	5
2	0	0.5	3	4.0	4.000000000000001	8.881784197001252e-16	true	7
2	0	2	0	16.0	16.000000000000004	3.552713678800501e-15	true	6
2	0	2	1	16.0	15.999999999999996	-3.552713678800501e-15	true	6
2	0	2	2	16.0	16.000000000000004	3.552713678800501e-15	true	4
2	0	2	3	16.0	15.999999999999996	-3.552713678800501e-15	true	5
2	0	8	0	16.0	16.000000000000004	3.552713678800501e-15	true	11
2	0	8	1	16.0	16.0	0.0	false	7
2	0	8	2	16.0	16.0	0.0	true	11
2	0	8	3	16.0	64.00000000000001	48.000000000000014	true	6
2	1	0.5	0	4.0	4.0	0.0	true	3
2	1	0.5	1	4.0	4.0	0.0	true	4
2	1	0.5	2	4.0	3.9999999999999987	-1.3322676295501878e-15	true	5
2	1	0.5	3	4.0	3.999999999999999	-8.881784197001252e-16	true	2
2	1	2	0	4.0	4.000000000000003	2.6645352591003757e-15	true	13
2	1	2	1	4.0	3.999999999999999	-8.881784197001252e-16	true	6
2	1	2	2	4.0	4.0	0.0	true	12
2	1	2	3	4.0	3.9999999999999996	-4.440892098500626e-16	true	12
2	1	8	0	4.0	4.0	0.0	true	11
2	1	8	1	4.0	4.000000000000001	8.881784197001252e-16	true	11
2	1	8	2	4.0	3.9999999999999973	-2.6645352591003757e-15	true	10
2	1	8	3	4.0	3.999999999999997	-3.1086244689504383e-15	false	12
3	0	0.5	0	6.0	5.999999999999997	-2.6645352591003757e-15	true	17
3	0	0.5	1	6.0	5.9999999999999964	-3.552713678800501e-15	true	15
3	0	0.5	2	6.0	5.999999999999997	-2.6645352591003757e-15	false	13
3	0	0.5	3	6.0	5.999999999999999	-8.881784197001252e-16	false	16
3	0	2	0	14.0	14.0	0.0	true	19
3	0	2	1	14.0	23.999999999999993	9.999999999999993	true	16
3	0	2	2	14.0	13.99999999999999	-1.0658141036401503e-14	true	18
3	0	2	3	14.0	23.999999999999996	9.999999999999996	true	12
3	0	8	0	38.0	38.0	0.0	true	22
3	0	8	1	38.0	38.0	0.0	true	18
3	0	8	2	38.0	37.999999999999986	-1.4210854715202004e-14	false	24
3	0	8	3	38.0	38.0	0.0	false	21
3	1	0.5	0	6.0	5.999999999999999	-8.881784197001252e-16	true	12
3	1	0.5	1	6.0	5.999999999999999	-8.881784197001252e-16	true	13
3	1	0.5	2	6.0	5.999999999999999	-8.881784197001252e-16	true	13
3	1	0.5	3	6.0	5.999999999999999	-8.881784197001252e-16	true	14
3	1	2	0	18.0	22.000000000000004	4.0000000000000036	true	16
3	1	2	1	18.0	21.999999999999996	3.9999999999999964	true	19
3	1	2	2	18.0	17.99999999999998	-2.1316282072803006e-14	false	36
3	1	2	3	18.0	18.0	0.0	true	24
3	1	8	0	42.0	41.99999999999998	-2.1316282072803006e-14	false	29
3	1	8	1	42.0	41.999999999999986	-1.4210854715202004e-14	true	22
3	1	8	2	42.0	41.999999999999986	-1.4210854715202004e-14	true	25
3	1	8	3	42.0	45.999999999999986	3.999999999999986	false	29
4	0	0.5	0	0.0	1.755365520504177e-18	1.755365520504177e-18	true	25
4	0	0.5	1	0.0	2.236000671753202e-19	2.236000671753202e-19	true	40
4	0	0.5	2	0.0	3.619163721399659e-18	3.619163721399659e-18	true	26
4	0	0.5	3	0.0	1.3073085525756968e-18	1.3073085525756968e-18	true	31
4	0	2	0	0.0	1.7180510548725808e-18	1.7180510548725808e-18	true	26
4	0	2	1	0.0	3.0687076957854155e-19	3.0687076957854155e-19	true	28
4	0	2	2	0.0	2.1206238976719378e-20	2.1206238976719378e-20	true	30
4	0	2	3	0.0	1.1697003684713522e-19	1.1697003684713522e-19	true	28
4	0	8	0	0.0	2.1548687270358594e-20	2.1548687270358594e-20	true	35
4	0	8	1	0.0	3.1422621202641936e-20	3.1422621202641936e-20	true	31
4	0	8	2	0.0	2.41173801043116e-20	2.41173801043116e-20	true	26
4	0	8	3	0.0	6.296709911629133e-19	6.296709911629133e-19	true	32
4	1	0.5	0	8.0	8.0	0.0	false	36
4	1	0.5	1	8.0	8.0	0.0	true	29
4	1	0.5	2	8.0	8.0	0.0	false	26
4	1	0.5	3	8.0	7.999999999999998	-1.7763568394002505e-15	true	14
4	1	2	0	28.0	28.0	0.0	true	26
4	1	2	1	28.0	28.000000000000004	3.552713678800501e-15	true	28
4	1	2	2	28.0	27.999999999999993	-7.105427357601002e-15	false	29
4	1	2	3	28.0	30.0	2.0	true	42
4	1	8	0	30.0	29.99999999999999	-1.0658141036401503e-14	false	81
4	1	8	1	30.0	29.999999999999996	-3.552713678800501e-15	false	63
4	1	8	2	30.0	29.999999999999993	-7.105427357601002e-15	true	80
4	1	8	3	30.0	29.999999999999993	-7.105427357601002e-15	false	60
5	0	0.5	0	6.0	5.999999999999999	-8.881784197001252e-16	true	38
5	0	0.5	1	6.0	6.000000000000001	8.881784197001252e-16	true	42
5	0	0.5	2	6.0	5.999999999999997	-2.6645352591003757e-15	true	32
5	0	0.5	3	6.0	5.999999999999997	-2.6645352591003757e-15	true	33
5	0	2	0	14.0	13.999999999999998	-1.7763568394002505e-15	true	30
5	0	2	1	14.0	16.000000000000007	2.000000000000007	false	36
5	0	2	2	14.0	15.999999999999996	1.9999999999999964	true	41
5	0	2	3	14.0	13.999999999999996	-3.552713678800501e-15	true	42
5	0	8	0	38.0	39.99999999999998	1.9999999999999787	false	57
5	0	8	1	38.0	37.99999999999998	-2.1316282072803006e-14	true	50
5	0	8	2	38.0	40.0	2.0	false	70
5	0	8	3	38.0	40.0	2.0	false	62
5	1	0.5	0	6.0	10.000000000000002	4.000000000000002	true	26
5	1	0.5	1	6.0	6.000000000000001	8.881784197001252e-16	true	36
5	1	0.5	2	6.0	5.999999999999999	-8.881784197001252e-16	false	28
5	1	0.5	3	6.0	5.999999999999999	-8.881784197001252e-16	false	29
5	1	2	0	16.0	16.000000000000007	7.105427357601002e-15	false	35
5	1	2	1	16.0	21.999999999999993	5.999999999999993	false	30
5	1	2	2	16.0	15.999999999999984	-1.5987211554602254e-14	true	44
5	1	2	3	16.0	21.999999999999993	5.999999999999993	true	32
5	1	8	0	40.0	39.999999999999986	-1.4210854715202004e-14	true	40
5	1	8	1	40.0	40.0	0.0	false	52
5	1	8	2	40.0	45.99999999999999	5.999999999999993	false	46
5	1	8	3	40.0	40.0	0.0	false	58
6	0	0.5	0	8.0	11.999999999999998	3.9999999999999982	true	26
6	0	0.5	1	8.0	8.0	0.0	false	37
6	0	0.5	2	8.0	7.999999999999998	-1.7763568394002505e-15	false	59
6	0	0.5	3	8.0	11.999999999999998	3.9999999999999982	true	29
6	0	2	0	20.0	19.999999999999993	-7.105427357601002e-15	true	60
6	0	2	1	20.0	25.999999999999993	5.999999999999993	true	49
6	0	2	2	20.0	19.999999999999993	-7.105427357601002e-15	true	51
6	0	2	3	20.0	19.99999999999999	-1.0658141036401503e-14	false	47
6	0	8	0	20.0	20.0	0.0	false	84
6	0	8	1	20.0	25.99999999999998	5.999999999999979	false	72
6	0	8	2	20.0	19.999999999999993	-7.105427357601002e-15	false	61
6	0	8	3	20.0	19.99999999999999	-1.0658141036401503e-14	false	88
6	1	0.5	0	12.0	11.999999999999998	-1.7763568394002505e-15	true	37
6	1	0.5	1	12.0	11.999999999999998	-1.7763568394002505e-15	false	28
6	1	0.5	2	12.0	11.999999999999998	-1.7763568394002505e-15	true	31
6	1	0.5	3	12.0	11.999999999999993	-7.105427357601002e-15	true	37
6	1	2	0	30.0	44.0	14.0	true	35
6	1	2	1	30.0	38.0	8.0	false	30
6	1	2	2	30.0	33.999999999999986	3.999999999999986	true	47
6	1	2	3	30.0	33.999999999999986	3.999999999999986	false	47
6	1	8	0	36.0	35.99999999999998	-2.1316282072803006e-14	false	107
6	1	8	1	36.0	35.999999999999986	-1.4210854715202004e-14	false	61
6	1	8	2	36.0	35.99999999999999	-7.105427357601002e-15	false	73
6	1	8	3	36.0	35.999999999999986	-1.4210854715202004e-14	false	117

covariance_and_perturbation

cases	70
max_witness_covariance_error	7.105427357601002e-14
failures	0

exact_controls › triangle

n	3
c	0
tau	1
dual_odd_set_weight	8
matching_profit	8
minimum	4
degree_only_relaxation	0

exact_controls › rewiring

spectrum	[0, 2, 3, 5]
cardinality_costs	[0, 2, 16]
breakpoints	["1/4", "7/4"]

JSON

{
  "seed": 20260928,
  "scope": "Finite falsification and implementation checks; not a formal proof",
  "versions": {
    "python": "3.12.3",
    "numpy": "2.5.3",
    "scipy": "1.18.1",
    "networkx": "3.7"
  },
  "matching_vs_exhaustive": {
    "cells": 1000,
    "dimensions": [
      1,
      10
    ],
    "failures": 0,
    "max_witness_error": 3.552713678800501e-15
  },
  "haar_unitaries": {
    "samples": 4800,
    "dimensions": [
      1,
      12
    ],
    "min_energy_minus_exact_minimum": -1.4210854715202004e-14,
    "max_decomposition_error": 1.7053025658242404e-13,
    "max_odd_operator_norm_error": 1.5543122344752192e-15,
    "max_odd_set_constraint_excess": 0.0,
    "failures": 0
  },
  "independent_skew_contractions": {
    "samples": 240,
    "failures": 0
  },
  "scalar_dynamic_program": {
    "cells": 1500,
    "dimensions": [
      1,
      30
    ],
    "failures": 0
  },
  "continuous_optimization": {
    "trials": 120,
    "gradient_check_error": 1.4437805510484203e-06,
    "reached_global_within_1e-7": 96,
    "optimizer_success_count": 80,
    "min_gap": -2.1316282072803006e-14,
    "max_gap": 48.000000000000014,
    "details": [
      {
        "n": 2,
        "case": 0,
        "tau": 0.5,
        "restart": 0,
        "exact": 4.0,
        "value": 4.000000000000001,
        "gap": 8.881784197001252e-16,
        "optimizer_success": true,
        "iterations": 8
      },
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