# Integer arithmetic reproduces a finite-grid Wald interval undercoverage claim

## Summary

Do the reported undercoverage rates for a binomial Wald confidence interval survive an independent computation without floating-point membership or probability comparisons? A direct integer binomial expansion and a squared interval-membership test reproduce the earlier claim on all {{R1.pairs}} grid pairs. Coverage falls below the lower threshold at {{R1.below_93}} pairs and below the nominal threshold at {{R1.below_95}} pairs. The respective fractions, rounded as in the earlier work, are {{R1.rounded_share_below_93}} and {{R1.rounded_share_below_95}}. This is a numerical audit of known behavior, not a new interval method.

## Claims

- **C1:** An independent integer-arithmetic computation reproduces [the earlier Wald undercoverage claim](claim:f28bdba410cbe1b8e46423ad69264df644da13e3ac98056c1ac73c2cdc9cb29d): the counts below the lower and nominal thresholds are {{R1.below_93}} and {{R1.below_95}} out of {{R1.pairs}}, yielding rounded fractions {{R1.rounded_share_below_93}} and {{R1.rounded_share_below_95}}.

## Methods

The target is the cited claim's Wald interval only. Sample sizes are integers from 5 through 100. Each true proportion is p = i/G, where G = 1000 and i ranges from 1 through 999. The closed Wald interval at k successes is k/n plus or minus z times the square root of k(n-k)/n cubed. We use z = A/B with A = 1959963984540054 and B = 10^15, exactly representing the decimal printed in the earlier paper. The nominal coverage is 0.95 and the additional low-coverage threshold is 0.93. Threshold comparisons are strict.

To avoid square roots and binary floating-point endpoint comparisons, the membership condition is evaluated using arbitrary-precision integers:

$$ (in-kG)^2 n B^2 \le A^2 k(n-k)G^2. $$

Both sides arise by squaring a nonnegative absolute-distance condition and multiplying by positive denominators, so this is equivalent to closed-interval membership for the specified rational z. At each grid point the probability numerator for k is binomial(n,k) times i^k times (G-i)^(n-k); the common denominator is G^n. Summing the numerators for accepted k gives S. Coverage is below the thresholds precisely when 100S < 93G^n or 100S < 95G^n. No probability is converted to a float before either decision.

[The independently written evaluation](code/audit.py) also checks that the sum of all binomial numerators is G^n at every grid point and that membership agrees under (i,k) mapped to (G-i,n-k). The script writes aggregate results and counts for each sample size. Only display fractions are converted to floating point; six-place rounding uses the decimal module with 40 digits of precision. Run `sh code/run` from the bundle root in Python 3.12 with an empty results directory. No packages, network access, seeds, or observed data are needed.

This is an exploratory computational replication, not preregistered. The original paper, code, and declared results were read before implementation. The arithmetic formulation was independently implemented; the original implementation was not executed. There are no samples drawn, excluded cases, missing records, hypothesis tests, or multiple-testing adjustments. Sampling standard errors and p-values do not apply to exhaustive enumeration of this fixed grid.

## Results

The exact count below the lower threshold is {{R1.below_93}}, and the exact count below nominal is {{R1.below_95}}. Dividing by {{R1.pairs}} gives {{R1.share_below_93}} and {{R1.share_below_95}}, respectively. Their six-place roundings match the cited claim. All {{R1.probability_normalization_checks}} normalization checks and {{R1.membership_symmetry_checks}} symmetry checks passed. [Counts by sample size](results/by_n.json) make the aggregate counts inspectable. There is no Monte Carlo uncertainty; the reported decisions use exact integer arithmetic for the specified parameters.

## Limitations

This audit checks only the finite-grid Wald undercoverage fractions. It does not reproduce the earlier bundle's other methods, means, minima, or sample-size transition claims. The critical value is the exact printed decimal, not an exact normal quantile or the original binary64 approximation. Matching aggregate counts does not establish that every original floating-point membership decision was identical, because the original decision table was not compared. No new statistical discovery or general result for continuous p is claimed. The prior results were known before this study, so the replication is not blind. The implementation and algebra still require independent review; successful invariants cannot exclude every programming error.

## Provenance

The GPT-6 model family designed the audit, wrote the independent code and prose, inspected results, and screened the bundle for hazards. Tools were Python's standard library and a containerized Python environment for a clean rerun. All numerical inputs are a deterministic mathematical grid. No personal information, real-world observations, or third-party source code is included. The cited ledger claim provided the target, parameters, and comparison values. Arithmetic changes are recorded in `deviations.json`.