Evidence

../

verify/: Independent reports and separate grades for precisely delimited arguments concerning the prime binary series.

The entry point computes, for the primes $p_n$ with $X<p_n\le2X$ and $X\in\{10^6,10^7\}$, the exact phases of the parity-prediction sketch for $d\in\{1,3,5,7,9,11,15\}$ and $K=\lceil\log_2(d\log X)\rceil+C$ with $0\le C\le6$. It uses numpy, the root Python environment and the shared Checker, reads no files and writes no output, and finishes in a few seconds. Run it from the repository root:

uv run --no-sync python wiki/research/prime_binary_new_attack/evidence/main.py

--quick runs $X=10^6$ only and omits the third check. The checks are:

1. rule (P) of the sketch, computed once from the phase and once from the parity of $h_n$ against the rule $\beta_{n,K}$, agrees on every start in every cell;
2. every mispredicted fraction lies in $[0.49,0.51]$;
3. at $X=10^7$, the total variation $\delta_0$ of Theorem F for $K=2,3,4$ exceeds its value at $K=1$, the bias $|\Pr(h_n\text{ odd})-\Pr(h_n\text{ even})|$, by less than $0.01$.

The program prints the observed values. The fractions in check 2 lie between $0.4968$ and $0.5046$ at $X=10^6$ and between $0.4986$ and $0.5020$ at $X=10^7$, while a rational sum with odd denominator part $d$ would force them, asymptotically, below $2dX/(2^KN_X)$, which at $X=10^7$ lies between $0.016$ and $0.031$ for $C=6$. At $X=10^7$ the bias at $K=1$ is $0.1174$, and $\delta_0$ is $0.1174$, $0.1174$ and $0.1183$ for $K=2,3,4$. The program stops at $K=4$: at $K=5$ there are $2^{14}$ residue keys for about $6\times10^5$ starts, too few starts per key for the empirical total variation to be reliable, since sampling noise inflates it.

These are finite observations and exclude no denominator. This check supplies neither an explicit threshold $X_0$ for Theorem C nor a bound on the preperiod of a hypothetical rational value. It is separate from finite tail-certificate methods that can exclude specified denominators. It shows how far the primes in this range are from the concentration that a rational sum would force at large scales.

The independent matching-window review and grade retain the exact subject and mathematical derivations for that conditional argument. They do not examine or raise the standing of these finite numerical observations. Conversely, the numerical entry point supplies no evidence for the matching-window review's verdict.