Prime binary series

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single_phase_comparison: Derives a signed tuple-count functional sufficient to exclude one rational denominator in the prime binary series. Its finite Bonferroni expansion retains lower-count obligations; no estimate for the functional is proved.

truncation_certificates: Certificate arguments for Problem 251: exact lattice and mean-tail tests, a congruence block supplied by a consecutive-prime theorem, the remaining tail and counting gaps, and the limits of the replacements examined.

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What the folder holds

Problem 251 asks whether $S=\sum_{n\ge1}p_n/2^n$ is irrational. Both conditional arguments filed on its claim pages use the weighted gap tails $G_n=\sum_{j\ge0}(p_{n+j+1}-p_{n+j})2^{-j-1}$. Rationality with denominator $b$ makes every $bG_n$ an integer. A prime estimate placing one of these tails strictly between adjacent lattice points would give a contradiction.

Truncation certificates holds the earlier certificate argument. It derives the lattice and averaged-tail tests, constructs phase-controlled consecutive blocks using congruences, and identifies the tail bound still missing from that construction. It also separates necessary cluster bounds from sufficient tail control. A single phase test derives the exact signed tuple-count functional for excluding one denominator and locates the lower estimate missing from a direct upper-sieve substitution. The source results keep their qualifications and canonical library homes. These deductions are author-recorded and not independently reviewed; they supply no new prime-distribution estimate and leave the problem open.

The parity-prediction sketch is the canonical editable sketch. It uses the same mean-tail estimate with $H_n=G_n/2$ for $n\ge2$ to isolate counting tests at $K\asymp\log\log X$, including an upper comparison that may lose $Cc^K$ for any $1\le c<2$. Its required prime estimates remain unproved. The accompanying comparison results preserve subcritical residue information while allowing a rational series value, even with density-one agreement with the actual prime positions. The changed positions need not be prime. The logarithmic-modulus and sparse support bounds prevent interval interpolation in their stated regimes; they exclude no individual rational target. All these deductions remain author-recorded.