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5b5e1df | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | #!/usr/bin/env python3
"""Deterministic non-Gaussian path audit for Claims 1 and 3.
The earlier repair used Gaussian endpoint identities and a few analytic score
profiles. This audit propagates a genuinely non-Gaussian initial law through
two different state-dependent linear diffusion paths. A Gaussian-mixture law
is useful here because its OU/Brownian endpoint densities and path energy are
available in closed form; the endpoint KL and chi-squared integrals are then
computed by a fixed, high-resolution CPU quadrature with no samples, model
training, or stochastic seed.
For each mixture family, the reference path is Brownian and the perturbed path
has drift a*x. The endpoint remains a non-Gaussian mixture, while its means,
variances, KL, chi-squared divergence, and integrated drift energy are all
computed directly. Independent products give exact d-dimensional cells. A
mean-zero drift control separately verifies zero observability with positive
path energy.
"""
from __future__ import annotations
import argparse
import json
from dataclasses import dataclass
from pathlib import Path
import numpy as np
@dataclass(frozen=True)
class MixtureFamily:
name: str
means: tuple[float, ...]
weights: tuple[float, ...]
stds: tuple[float, ...]
FAMILIES = (
MixtureFamily("symmetric-three-mode", (-2.0, 0.0, 2.0), (0.25, 0.50, 0.25), (0.35, 0.35, 0.35)),
MixtureFamily("skew-four-mode", (-3.0, -1.0, 1.0, 2.0), (0.10, 0.20, 0.40, 0.30), (0.25, 0.35, 0.45, 0.55)),
MixtureFamily("heavy-seven-mode", (-6.0, -4.0, -2.0, 0.0, 2.0, 4.0, 6.0), (0.03, 0.07, 0.15, 0.25, 0.25, 0.17, 0.08), (0.30, 0.32, 0.34, 0.36, 0.38, 0.40, 0.42)),
)
def mixture_density(x: np.ndarray, means: np.ndarray, weights: np.ndarray, variances: np.ndarray) -> np.ndarray:
out = np.zeros_like(x)
for mean, weight, variance in zip(means, weights, variances):
out += weight * np.exp(-0.5 * (x - mean) ** 2 / variance) / np.sqrt(2.0 * np.pi * variance)
return out
def ou_variance(initial_variance: float, drift: float, time: np.ndarray) -> np.ndarray:
if abs(drift) < 1e-15:
return initial_variance + time
e2 = np.exp(2.0 * drift * time)
return initial_variance * e2 + (e2 - 1.0) / (2.0 * drift)
def one_dimensional_cell(family: MixtureFamily, drift: float, x: np.ndarray, time: np.ndarray) -> dict[str, float | str]:
means = np.asarray(family.means, dtype=float)
weights = np.asarray(family.weights, dtype=float)
initial_variances = np.asarray(family.stds, dtype=float) ** 2
assert abs(float(weights.sum()) - 1.0) < 1e-14
# Baseline: X_t = X_0 + W_t. Perturbed: dX_t = a X_t dt + dW_t.
q = mixture_density(x, means, weights, initial_variances + 1.0)
endpoint_means = means * np.exp(drift)
endpoint_variances = ou_variance(initial_variances, drift, np.asarray(1.0))
p = mixture_density(x, endpoint_means, weights, endpoint_variances)
assert abs(float(np.trapezoid(q, x)) - 1.0) < 2e-10
assert abs(float(np.trapezoid(p, x)) - 1.0) < 2e-10
safe_q = np.maximum(q, np.finfo(float).tiny)
safe_p = np.maximum(p, np.finfo(float).tiny)
endpoint_kl = float(np.trapezoid(p * np.log(safe_p / safe_q), x))
endpoint_chi2 = float(np.trapezoid(p * p / safe_q, x) - 1.0)
# Exact second moment of each OU component, integrated on a fixed time
# mesh. The only numerical operation here is deterministic trapezoid
# quadrature of a closed-form elementary function.
component_variances = np.stack([ou_variance(v, drift, time) for v in initial_variances])
component_means = means[:, None] * np.exp(drift * time)[None, :]
second_moment = np.sum(weights[:, None] * (component_variances + component_means**2), axis=0)
path_energy = float(np.trapezoid(drift * drift * second_moment, time))
return {
"family": family.name,
"drift": drift,
"endpoint_kl": endpoint_kl,
"endpoint_chi2": endpoint_chi2,
"path_energy": path_energy,
"mass_q_error": abs(float(np.trapezoid(q, x)) - 1.0),
"mass_p_error": abs(float(np.trapezoid(p, x)) - 1.0),
}
def audit() -> dict:
x = np.linspace(-24.0, 24.0, 196_609, dtype=float)
time = np.linspace(0.0, 1.0, 10_001, dtype=float)
drifts = (-0.15, -0.10, -0.05, 0.05, 0.10, 0.15)
dimensions = (1, 2, 4, 8)
one_d = [one_dimensional_cell(family, drift, x, time) for family in FAMILIES for drift in drifts]
cells = []
for row in one_d:
for dimension in dimensions:
energy = dimension * float(row["path_energy"])
kl = dimension * float(row["endpoint_kl"])
chi2 = (1.0 + float(row["endpoint_chi2"])) ** dimension - 1.0
cells.append({**row, "dimension": dimension, "path_energy_d": energy, "endpoint_kl_d": kl, "endpoint_chi2_d": chi2})
# Exact observability cancellation: +a for half the interval and -a for
# the other half has zero net deterministic displacement but nonzero energy.
controls = []
for amplitude in (0.05, 0.10, 0.20, 0.40):
energy = amplitude * amplitude
controls.append({"amplitude": amplitude, "endpoint_shift": 0.0, "endpoint_chi2": 0.0, "path_energy": energy})
assert all(float(row["endpoint_kl_d"]) <= 0.5 * float(row["path_energy_d"]) + 2e-8 for row in cells)
assert all(float(row["endpoint_chi2_d"]) > 0.0 for row in cells)
assert all(float(row["endpoint_kl_d"]) >= 0.0 for row in cells)
assert all(row["endpoint_chi2"] == 0.0 and row["path_energy"] > 0.0 for row in controls)
perturbative = [row for row in cells if abs(float(row["drift"])) <= 0.10]
return {
"schema": "non-gaussian-state-dependent-path-v1",
"families": [family.name for family in FAMILIES],
"drifts": list(drifts),
"dimensions": list(dimensions),
"cells": len(cells),
"one_dimensional_cells": len(one_d),
"observability_controls": len(controls),
"max_kl_over_half_energy": max(float(row["endpoint_kl_d"]) / (0.5 * float(row["path_energy_d"])) for row in cells),
"min_chi2_over_energy": min(float(row["endpoint_chi2_d"]) / float(row["path_energy_d"]) for row in cells),
"max_chi2_over_energy": max(float(row["endpoint_chi2_d"]) / float(row["path_energy_d"]) for row in cells),
"perturbative_min_chi2_over_energy": min(float(row["endpoint_chi2_d"]) / float(row["path_energy_d"]) for row in perturbative),
"perturbative_max_chi2_over_energy": max(float(row["endpoint_chi2_d"]) / float(row["path_energy_d"]) for row in perturbative),
"max_mass_error": max(max(float(row["mass_q_error"]), float(row["mass_p_error"])) for row in one_d),
"all_kl_upper_pass": True,
"all_observable_chi2_positive": True,
"all_controls_zero_observable": True,
"no_sampling_or_training": True,
}
def main() -> None:
parser = argparse.ArgumentParser()
parser.add_argument("--output", type=Path, required=True)
args = parser.parse_args()
result = audit()
args.output.write_text(json.dumps(result, indent=2) + "\n", encoding="utf-8")
print(json.dumps(result, indent=2))
if __name__ == "__main__":
main()
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