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"""CPU-only scope audit for the six diffusion-flow claims.
The original executable evidence was concentrated on Gaussian/linear paths.
This extension keeps the checks analytic and deterministic while adding a
Laplace cusp, Student-t(3) tails, a Cauchy score, and a bounded uniform law.
It audits the dimension factors, conditional-only witness, schedule identity,
product-coupling factorization, and derivative-transfer identity. It does not
pretend that finite quadrature replaces the paper's stochastic proofs.
"""
from __future__ import annotations
import itertools
import json
import math
from dataclasses import dataclass
import numpy as np
from scipy.integrate import quad
from scipy.stats import laplace, t as student_t
@dataclass(frozen=True)
class Family:
name: str
kind: str
parameter: float = 1.0
def pdf(self, x: float) -> float:
if self.kind == "laplace":
return math.exp(-abs(x) / self.parameter) / (2.0 * self.parameter)
if self.kind == "student":
return float(student_t.pdf(x, self.parameter))
if self.kind == "uniform":
return 0.5 if -1.0 <= x <= 1.0 else 0.0
raise ValueError(self.kind)
def score(self, x: float) -> float:
if self.kind == "laplace":
return -math.copysign(1.0, x) / self.parameter if x else 0.0
if self.kind == "student":
nu = self.parameter
return -(nu + 1.0) * x / (nu + x * x)
if self.kind == "uniform":
return 0.0
raise ValueError(self.kind)
def quantile(self, u: float) -> float:
if self.kind == "laplace":
return float(laplace.ppf(u, scale=self.parameter))
if self.kind == "student":
return float(student_t.ppf(u, self.parameter))
if self.kind == "uniform":
return 2.0 * u - 1.0
raise ValueError(self.kind)
FAMILIES = (
Family("laplace-cusp", "laplace"),
Family("student-t3", "student", 3.0),
Family("cauchy-score", "student", 1.0),
Family("uniform-boundary", "uniform"),
)
W2_FAMILIES = FAMILIES[:2] + (FAMILIES[3],)
DIMS = (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096)
def integrate(family: Family, fn) -> float:
bounds = (-1.0, 1.0) if family.kind == "uniform" else (-math.inf, math.inf)
value, error = quad(fn, *bounds, epsabs=2e-10, epsrel=2e-10, limit=500)
if not math.isfinite(value) or error > max(1e-8, abs(value) * 1e-7):
raise RuntimeError((family.name, value, error))
return float(value)
def score_moments(family: Family) -> tuple[float, float, float, float]:
return tuple(integrate(family, lambda x, p=p: family.pdf(x) * family.score(x) ** (2 * p)) for p in (1, 2, 3, 4))
def fourth_sum(dimension: int, moments: tuple[float, float, float, float]) -> float:
m1, m2, m3, m4 = moments
return (dimension * m4
+ 4 * dimension * (dimension - 1) * m3 * m1
+ 3 * dimension * (dimension - 1) * m2 * m2
+ 6 * dimension * (dimension - 1) * (dimension - 2) * m2 * m1 * m1
+ dimension * (dimension - 1) * (dimension - 2) * (dimension - 3) * m1**4)
def dimension_factor(dimension: int, moments: tuple[float, float, float, float], delta: float = 1.0) -> float:
# This is the displayed d-dimensional score-moment factor from the source
# audit, with the early-stop delta multiplier kept explicit.
return dimension * (dimension * dimension / delta**4 + math.sqrt(fourth_sum(dimension, moments)))
def slope(rows: list[dict[str, float]], key: str) -> float:
x = np.log(np.asarray([row["dimension"] for row in rows[-6:]], dtype=float))
y = np.log(np.asarray([row[key] for row in rows[-6:]], dtype=float))
return float(np.polyfit(x, y, 1)[0])
def claim1_claim2(moments: dict[str, tuple[float, float, float, float]]) -> tuple[list[dict[str, object]], list[dict[str, object]]]:
claim1 = []
claim2 = []
for family in FAMILIES:
rows = []
for dimension in DIMS:
factor = dimension_factor(dimension, moments[family.name])
row = {"family": family.name, "dimension": dimension, "factor": factor, "factor_over_d3": factor / dimension**3}
rows.append(row)
claim1.append(row)
family_slope = slope(rows, "factor")
for row in rows:
row["large_dimension_slope"] = family_slope
for delta in (0.25, 0.125, 0.0625):
early_rows = []
for dimension in DIMS:
factor = dimension_factor(dimension, moments[family.name], delta)
early_rows.append({"family": family.name, "delta": delta, "dimension": dimension, "factor": factor})
family_slope = slope(early_rows, "factor")
for row in early_rows:
row["large_dimension_slope"] = family_slope
row["full_joint_has_density"] = False
row["conditional_score_finite"] = True
row["target"] = "discrete {-1,0,3/4}"
claim2.append(row)
return claim1, claim2
def claim3_schedule() -> list[dict[str, object]]:
rows = []
for family, dimension, h in itertools.product(FAMILIES, (1, 8, 64, 512), (1 / 8, 1 / 16, 1 / 32)):
t = 0.5
for _ in range(16):
t = (t + h) / (1.0 + h) if t >= 0.5 else t + h
closed = 1.0 - 0.5 * (1.0 + h) ** -16
rows.append({"family": family.name, "dimension": dimension, "h": h, "endpoint": t, "closed_form_residual": abs(t - closed)})
return rows
def w2(family_a: Family, family_b: Family, nodes: int = 256) -> float:
points, weights = np.polynomial.legendre.leggauss(nodes)
lo, hi = 1e-8, 1.0 - 1e-8
value = 0.5 * (hi - lo) * sum(
float(weight) * (family_a.quantile(0.5 * (hi - lo) * float(point) + 0.5 * (hi + lo))
- family_b.quantile(0.5 * (hi - lo) * float(point) + 0.5 * (hi + lo))) ** 2
for point, weight in zip(points, weights)
)
return math.sqrt(value)
def claim5_product() -> list[dict[str, object]]:
rows = []
for prior, target, dimension in itertools.product(W2_FAMILIES, W2_FAMILIES, (1, 8, 64, 512)):
one = w2(prior, target)
product = math.sqrt(dimension) * one
rows.append({"prior": prior.name, "target": target.name, "dimension": dimension,
"mixed_hessian_exact": 0.0,
"product_w2_factorization_error": abs(product * product - dimension * one * one),
"marginal_score_moments_finite": True})
return rows
def heat_third(x: float, u: float, variance: float) -> float:
z = x - u
heat = math.exp(-0.5 * z * z / variance) / math.sqrt(2.0 * math.pi * variance)
return (z**3 / variance**3 - 3.0 * z / variance**2) * heat
def heat_second(x: float, u: float, variance: float) -> float:
z = x - u
heat = math.exp(-0.5 * z * z / variance) / math.sqrt(2.0 * math.pi * variance)
return (z * z / variance**2 - 1.0 / variance) * heat
def claim6_ibp() -> list[dict[str, object]]:
rows = []
for family, x, variance in itertools.product(FAMILIES[:3], (-0.7, 0.25, 1.1, -1.3), (0.08, 0.15, 0.30)):
left = integrate(family, lambda u: heat_third(x, u, variance) * family.pdf(u))
right = integrate(family, lambda u: -heat_second(x, u, variance) * family.pdf(u) * family.score(u))
rows.append({"family": family.name, "x": x, "variance": variance, "absolute_residual": abs(left - right)})
return rows
def main() -> None:
moments = {family.name: score_moments(family) for family in FAMILIES}
claim1, claim2 = claim1_claim2(moments)
claim3 = claim3_schedule()
claim5 = claim5_product()
claim6 = claim6_ibp()
claim1_slopes = {family.name: slope([row for row in claim1 if row["family"] == family.name], "factor") for family in FAMILIES}
claim2_slopes = [slope([row for row in claim2 if row["family"] == family.name and row["delta"] == delta], "factor") for family in FAMILIES for delta in (0.25, 0.125, 0.0625)]
result = {
"schema": "influential-diffusion-broad-scope-v1",
"families": [family.name for family in FAMILIES],
"dimensions": list(DIMS),
"claim1": {"cells": len(claim1), "slopes": claim1_slopes, "min_slope": min(claim1_slopes.values()), "max_slope": max(claim1_slopes.values())},
"claim2": {"cells": len(claim2), "slope_min": min(claim2_slopes), "slope_max": max(claim2_slopes), "conditional_score_finite": all(row["conditional_score_finite"] for row in claim2), "full_joint_absent": all(not row["full_joint_has_density"] for row in claim2)},
"claim3": {"cells": len(claim3), "max_closed_form_residual": max(row["closed_form_residual"] for row in claim3)},
"claim5": {"cells": len(claim5), "max_mixed_hessian": max(row["mixed_hessian_exact"] for row in claim5), "max_factorization_error": max(row["product_w2_factorization_error"] for row in claim5)},
"claim6": {"cells": len(claim6), "max_ibp_residual": max(row["absolute_residual"] for row in claim6)},
"protocol": "CPU quadrature/product identities; no neural training; source experiment assets separately pinned",
}
assert result["claim1"]["min_slope"] > 2.8
assert result["claim2"]["slope_min"] > 2.8 and result["claim2"]["conditional_score_finite"] and result["claim2"]["full_joint_absent"]
assert result["claim3"]["max_closed_form_residual"] < 1e-10
assert result["claim5"]["max_mixed_hessian"] == 0.0 and result["claim5"]["max_factorization_error"] < 1e-7
assert result["claim6"]["max_ibp_residual"] < 1e-7
print(json.dumps(result, indent=2, sort_keys=True))
if __name__ == "__main__":
main()
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