Parity prediction for the prime binary series

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consecutive_block_counts: For Problem 251, a rational sum would make one weighted gap-tail value repeat about N_X/(d log X) times among starts X<p_n<=2X, and put most starts in a narrow carry window for about log_2 log X consecutive gaps. An upper comparison tolerates a loss c^K with c<2 at longer blocks. Exact model bounds, a signed majorant, a survivor count at the square-root sieve level and a signed deletion sum isolate the missing prime estimate.

critical_length_obstructions: Limits of specified phase-counting mechanisms for Problem 251, a rational interpolation theorem preserving residues below the critical length, and a comparison with known consecutive-prime theorems.

evidence/: Finite observations of the parity-prediction rule on the primes below 2 * 10^7: the rule's two forms agree on every start, mispredicted fractions stay near one half, and later residues add almost nothing to predicting the parity of a gap; and a separately graded mathematical review of the conditional matching-window argument.

harmonic_phase_ceiling: Finite normalized mixtures of doubled phase deficits have optimal universal ceiling 3/2, attained by weights 2/3 and 1/3. For a resonant prime-tail reference, the deficit below this ceiling is an exact nonnegative energy. A positive proportion of that energy remains an arithmetic hypothesis, and integer carries make every doubled site weight identical.

matching_gap_windows: Exact repeated gap windows with small enough tails would force a congruence between their starting indices and, for every odd denominator including the dyadic case, between their preceding gaps modulo 4. Such windows occur in abundance unconditionally; producing them in different index classes or preceding classes remains open.

phase_transport: Composite deletion yields two signed sums with different unconditional endpoints. Integer carries make their corresponding root increments agree. Removing each root's phase retains the actual carry coefficient. A periodic reference and an odd multiple of every denominator give the common rationality endpoint 3N/2. The signed cofactor estimate needed to improve that endpoint remains unproved.

prediction_overlap: Rationality forces the preceding half-gap parity to be determined by the weighted future window modulo two, except at O(dX/2^K) starts. A scalar minority count gives a sufficient occurrence target using less window information. Repetition, PNT growth and balanced parity marginals alone do not supply the required overlap.

proof_sketch: The canonical sketch for Problem 251, from a rational sum through an integer carry lattice to a contradiction conditional on a consecutive prime-block estimate. An upper comparison may lose c^K for any c<2. Parity prediction, zero-phase alternatives, model computations and an exact failure log distinguish the held reductions from the missing arithmetic estimate.

rational_deletion_signs: A rational prime binary series forces only o(N_X) downward residue transitions when composites are deleted from a periodic set, so the positive mass is the final count up to o(N_X). Gains are decided at the prime endpoints; high ranks cost arbitrarily little and rough sites at most log 2. The small-factor estimate is open. Dropping integrality gives too much mass, and certificates with nonnegative prime-pair costs fail on stationary integer transitions.

residue_switch_moments: A moment criterion turns a bound for same-class prime pairs into a positive proportion of residue switches. The inspected tuple sieve gives the criterion when phi(q)>8; reaching modulo 4 needs a smaller pair constant. This fixed-length information does not supply the growing prime-block estimate for Problem 251.

short_span_sieve: A rational prime binary series forces at least N_X/24 qualifying blocks in a shorter span. Their monotone enlargement is rare in the reference model and can be posed for actual sieve survivors. The survivor estimate is unproved. Its centered-moment cost is equivalent to a prime-rooted short-interval estimate. A damped signed form isolates the remaining truncated phase cost, whose required bound is also unproved. The band-loss alternative has an exact divisor expansion but no arithmetic estimate.

survivor_moments: A fixed-dimensional sieve bounds rough-composite contamination of the rooted second moment by O(N_X) and of bounded local counts by O(N_X/K). The centered-moment input is therefore equivalent to its prime-rooted version. At the lower sieve cutoff, a positive base moment must be canceled by the signed transport. Neither missing estimate is proved.

zero_phase_certificates: For Problem 251, one congruence among K+1 consecutive primes would force a large following tail under rationality; a two-cutoff count turns this into a criterion, the counts that known cluster theorems supply fall short of it, and under the Riemann hypothesis a frequency bound for the congruence alone would suffice.

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What the folder holds

Problem 251 asks whether $S=\sum_{n\ge1}p_n/2^n$ is irrational. This folder continues the earlier notes on the prime binary series, reusing their identities and mean-tail bound without repeating the proofs.

The parity-prediction sketch is the editable proof sketch. A rational $S$ with odd denominator part $d$ would make an explicit rule predict the parity of a halved prime gap from the next $K-1$ gap residues, with at most $O(dX/2^K)$ exceptions among starts $X<p_n\le2X$. A count exceeding this bound at $K=\log_2\log X+O_d(1)$ would exclude $d$; such a count for every odd $d$ would prove irrationality. It remains unproved. A single hypothesis free of $d$, that these residue laws retain a fixed amount of overlap with a one-coordinate translate, would also suffice. The sketch opens with the digit identity, which makes the binary digits of $S$ from the second on the parities of the integer parts of the gap tails, and with a step lemma, which turns every rational value into a dyadic case for the gaps $p_{n+t}-p_n$. The zero-phase analysis builds mispredicted starts from one congruence among $K+1$ consecutive primes and size bounds on the gaps after them. It states the count this needs, shows that the counts proved by the known cluster theorems fall short of it, and shows that under the Riemann hypothesis a frequency bound for the congruence alone would suffice. The lead page summarizes the route for Problem 251. The upper-count note turns the lattice into two lower counts, a repeated tail value and a narrow carry window for most blocks. A comparison for the latter may lose $Cc^K$ for any $c<2$ at a suitable length $K=A\log_2\log X+O(1)$; the repeated-tail criterion is free of $d$. Independent and periodically restricted references have the needed phase bound. No comparison with actual consecutive primes is proved. The note also gives an exact obstruction to bounding a signed majorant by its adverse termwise errors. The [[research/prime_binary_new_attack/rational_deletion_signs|conditional sign analysis]] retains the actual integer carries in an increasing composite-deletion process. It bounds downward residue transitions by $o(N_X)$ at $K=\lceil A\log_2\log X\rceil$, $A>2$, and counts span losses separately. With span $K(\log X)^2$, the negative deletion weight is negligible under rationality, so the positive mass equals the final count up to $o(N_X)$. Its residue gains have an endpoint criterion; the full gains also retain their span cutoff. High ranks have arbitrarily small relative cost, bad gaps and multiple gains have negligible total cost, and sites with least prime factor above $X^{1/3}$ cost at most $(\log2+o(1))N_X$. The small-factor, low-rank bound remains open. The endpoint identity does not rule out estimating it. Proposition D19 rules out a specific relaxation instead: dropping integrality gives an ordinary real-tail crossing count with an unconditional lower bound above the proposed upper target. The phase-transport note distinguishes those original prime-root increments from the dephased sum $Q$. Integer carries make corresponding root increments agree; the two unconditional endpoints differ. The original sum gives a positive-density digit-change criterion; the dephased sum gives a signed cofactor estimate with a carry-dependent coefficient. Both fixed-saving targets are unknown, and the bound $Q\le2N_X+o(N_X)$ is insufficient. The obstruction record states, with their exact scope, why fixed block lengths, residue-pattern certificates, pointwise brackets, weighted means, combinations of tails and upper-bound counts do not supply that statement, and compares the known theorems on consecutive primes. For block lengths with $2^K/\log X\to0$, it also constructs comparison sequences with the same pointwise residue patterns, the same cumulative growth and a dyadic rational sum. Residue-window laws on the corresponding value ranges $X<p_n\le2X$ differ by at most $o(1)$ in total variation; this does not preserve relative counts of rare events.

L228 gives a stronger comparison with a rational sum: an interval of series values can be attained while retaining the actual nth prime outside one density-zero set, preserving nested moduli $o(\log n)$, and changing every gap by $o(\log n)$. Its proof uses paired positive and negative gap corrections and the classical prime-pair upper bound. The altered terms need not be prime.

L229 gives the complementary boundary at logarithmic moduli: when $q_n\ge c\log n$ eventually, all values attainable with density-one agreement and the same nested congruences form a set of Hausdorff dimension zero. This prevents interval interpolation in that regime, but does not exclude any specified rational value.

L230 limits further sparsification: even without congruences, changing only $o(N/\log\log N)$ of the first $N$ positions gives a set of values of dimension zero. It rules out that support bound for an interval construction, while leaving individual rational targets open. L228 achieves the corresponding big-oh bound for nested moduli $O((\log n)^\alpha)$ with fixed $\alpha<1$.

The parity-prediction and interpolation deductions above are author-recorded and not independently reviewed. The separate [[research/prime_binary_new_attack/matching_gap_windows|matching-window conditional]] has an independently accepted review at its stated conditional scope, with a localized wording correction disclosed there. None supplies a new estimate for the unaltered prime sequence, and the problem remains open. The remaining obligation is an estimate for the actual consecutive primes, or a different argument using information the comparisons discard.

The totient binary-series comparisons use the same test of which coefficient information permits rational sums, but preserve different hypotheses: prime-power values and fixed-prime multiplication laws, rather than increasing positions and prime congruences. No implication between the two irrationality questions follows. In each case the open route must use information the corresponding comparisons do not retain.