Materials-bound arithmetic fixture

Local modeled evidence only

This recipe does not establish a materials organ, a PDD registry, a prediction route, a receipt chain, or a signature. Current a11oy execution is UNAVAILABLE. The only executable claim below is standard-library arithmetic over explicitly modeled inputs.

The migrated source combined two distinct ideas:

• a Pointwise Distance Distribution (PDD) as a crystal-structure descriptor; and
• a PAC-Bayes upper-bound calculation for a modeled predictor.

Those ideas must be verified independently. A descriptor match is not proof that two crystals are identical, and arithmetic evaluation of a published bound is not evidence that a model, dataset, or runtime satisfies its assumptions.

Executable McAllester-style arithmetic fixture

For empirical risk (\hat R), divergence term (\mathrm{KL}(Q|P)), sample count (n), and confidence parameter (\delta), this page evaluates the displayed expression:

[ R(Q) \leq \hat R(Q) + \sqrt{\frac{\mathrm{KL}(Q|P) + \ln(2\sqrt{n}/\delta)}{2n}}. ]

The inputs are illustrative and labeled MODELED_FIXTURE; they are not measurements.

import json
import math


def modeled_bound(empirical_risk: float, kl: float, n: int, delta: float) -> float:
    if not 0.0 <= empirical_risk <= 1.0:
        raise ValueError("empirical_risk must be in [0, 1]")
    if kl < 0.0 or n <= 0 or not 0.0 < delta < 1.0:
        raise ValueError("require kl >= 0, n > 0, and delta in (0, 1)")
    penalty = math.sqrt((kl + math.log(2.0 * math.sqrt(n) / delta)) / (2.0 * n))
    return min(1.0, empirical_risk + penalty)


fixture = {
    "empirical_risk": 0.02,
    "kl": 1.5,
    "n": 1000,
    "delta": 0.05,
    "input_state": "MODELED_FIXTURE",
}
result = modeled_bound(
    fixture["empirical_risk"], fixture["kl"], fixture["n"], fixture["delta"]
)
report = {
    "bound": result,
    "input_state": fixture["input_state"],
    "runtime_state": "UNAVAILABLE",
    "signature_state": "UNAVAILABLE",
}
assert 0.0 <= report["bound"] <= 1.0
print(json.dumps(report, indent=2, sort_keys=True))

This code proves only that Python evaluated the displayed expression for the displayed inputs. It does not validate independence assumptions, data provenance, posterior construction, predictive performance, or a Lean theorem.

PDD boundary

The audited docs snapshot does not provide a pinned executable PDD implementation, registry snapshot, tolerance policy, or current route witness. Therefore no novelty verdict is produced here. Any future novel result must mean only "no match under this exact descriptor, registry, and tolerance" unless injectivity and all operational assumptions are separately established.

Evidence state

Surface	Current status
Displayed bound arithmetic	MODELED_LOCAL_FIXTURE
PAC-Bayes literature	CITED_PRIOR_ART
Exact model and evaluation dataset	UNAVAILABLE
PDD implementation and registry	SOURCE_ARTIFACT_UNAVAILABLE
Materials route, receipt, and signature	UNAVAILABLE
Lean closure for this bound/application	NOT_ESTABLISHED_HERE

References: D. McAllester, "PAC-Bayesian model averaging," COLT (1999); D. Widdowson and V. Kurlin, "Resolving the data ambiguity for periodic crystals," NeurIPS (2022).

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A local arithmetic result is not a certified model, materials verdict, receipt, or deployment.