Hypnos Math

Checks · Maximal Coherence for Prescribed Intrinsic Populations and Youla Values: Two Questions of Gil

Recorded output of the verification program

One block per check. A_free_youla: in 30 instances no random contraction came within 4.7 × 10⁻⁵ of the bound of Theorem 3.1 (max_random_excess is negative), the ascent never passed it beyond roundoff, and it came within 10⁻⁴ of the bound in all 30 (worst_shortfall is the largest gap left). B_prescribed_youla: the same for Theorem 4.1, reached within 10⁻⁶ in all 30. C_inequality_family: the full family and the prefix subfamily match the theorem to 5.6 × 10⁻¹⁷; the two coarser subfamilies exceed it by up to 0.019 and 4.4 × 10⁻⁴, in the units the units line names. D_odd_sets: 8,760 checks, none failed. The file records the library versions, the seed and the program's SHA-256, which matches the published program, but no date.

Output of
The verification program (checks/verify_gil.py), shown beside it on that page
Written by
verify_gil.py, a program by Claude Fable 5.1 (Anthropic)
Size
1,626 bytes
SHA-256
66d2230b31078f6d2adc3f357029cf1639b1bfa07a35f42f97cc9a31b4153e2d

The note's pageEvery file published with itThis file on GitHub

Tables

seed20260929
source_sha2562f476e76a830e3da491820cb4cd37e12c9ab3831ea0393797289cf9257561b57

versions

python3.12.10
numpy2.2.6
scipy1.16.3

A_free_youla

instances30
random_samples_per_instance300
max_random_excess-4.711275597385489e-05
methodprojected gradient ascent, 20 starts x 4000 steps, singular values clipped at 1, final snap to a partial isometry
max_optimized_excess5.551115123125783e-16
min_optimized_excess-7.178675857100192e-05
worst_shortfall7.178675857100192e-05
instances_within_1e-619
instances_within_1e-430

B_prescribed_youla

instances30
max_random_excess8.004708007547379e-14
methodNelder-Mead over SO(n) via expm of a skew parameter, 20 starts
max_optimized_excess1.500466417780899e-13
min_optimized_excess1.0842021724855044e-18
worst_shortfall0.0
instances_within_1e-630

C_inequality_family

unitssquared cohesion ||N||_F^2 = 2 x (LP value); halve for the LP's own units
max_abs_diff_full_family_LP_vs_theorem5.551115123125783e-17
max_abs_diff_prefix_family_LP_vs_theorem5.551115123125783e-17
max_excess_coarse_family_LP_over_theorem0.019409162878746528
max_excess_CB_plus_degree_LP_over_theorem0.0004419584168336016
instances_CB_plus_degree_LP_exceeds_theorem4
instances30

D_odd_sets

checks8760
max_excess-0.032159502235817206

JSON

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  "seed": 20260929,
  "versions": {
    "python": "3.12.10",
    "numpy": "2.2.6",
    "scipy": "1.16.3"
  },
  "A_free_youla": {
    "instances": 30,
    "random_samples_per_instance": 300,
    "max_random_excess": -4.711275597385489e-05,
    "method": "projected gradient ascent, 20 starts x 4000 steps, singular values clipped at 1, final snap to a partial isometry",
    "max_optimized_excess": 5.551115123125783e-16,
    "min_optimized_excess": -7.178675857100192e-05,
    "worst_shortfall": 7.178675857100192e-05,
    "instances_within_1e-6": 19,
    "instances_within_1e-4": 30
  },
  "B_prescribed_youla": {
    "instances": 30,
    "max_random_excess": 8.004708007547379e-14,
    "method": "Nelder-Mead over SO(n) via expm of a skew parameter, 20 starts",
    "max_optimized_excess": 1.500466417780899e-13,
    "min_optimized_excess": 1.0842021724855044e-18,
    "worst_shortfall": 0.0,
    "instances_within_1e-6": 30
  },
  "C_inequality_family": {
    "units": "squared cohesion ||N||_F^2 = 2 x (LP value); halve for the LP's own units",
    "max_abs_diff_full_family_LP_vs_theorem": 5.551115123125783e-17,
    "max_abs_diff_prefix_family_LP_vs_theorem": 5.551115123125783e-17,
    "max_excess_coarse_family_LP_over_theorem": 0.019409162878746528,
    "max_excess_CB_plus_degree_LP_over_theorem": 0.0004419584168336016,
    "instances_CB_plus_degree_LP_exceeds_theorem": 4,
    "instances": 30
  },
  "D_odd_sets": {
    "checks": 8760,
    "max_excess": -0.032159502235817206
  },
  "source_sha256": "2f476e76a830e3da491820cb4cd37e12c9ab3831ea0393797289cf9257561b57"
}