Irrationality of the Prime Binary Series

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This lead's results that are not drawn from a cited source are Plasma AI's, credited to Plasma AI.

Target and standing

Problem 251 asks whether $S=\sum_{n\ge1}p_n2^{-n}$ is irrational, $p_n$ being the $n$th prime. The problem is open; its page carries the status search, the current assessment and the two conditional claim pages. This lead records the project's route toward an unconditional proof. It contains no proof or disproof of the target. The counting criteria and model calculations are author-recorded. The matching-window alternative has the separately graded review described below; that review does not cover the whole route.

The editable proof sketch is the parity-prediction sketch. It states the whole argument from a rational value to a contradiction, with its held links, its single open step and a failure log. The other pages of the route support it:

• The zero-phase analysis builds the certificates the sketch needs from one congruence among consecutive primes and size bounds on the gaps after them, and measures the counts that known cluster theorems supply against the count it needs.
• The obstruction record states where fixed lengths, residue-pattern certificates, pointwise brackets, weighted means and combinations of tails stop, counts original gaps after filtering, and keeps the dependence among overlapping carry constraints explicit. It also compares the known consecutive-prime theorems.
• The matching-window note gives an alternative contradiction from repeated gap windows in different index classes, for odd denominator parts greater than one. Its unreviewed extension permits explicitly graded congruences of the halved gaps, and its unreviewed repeated-window lemma shows that equal windows with small later tails occur at all but $X^{o(1)}$ of at least $(3/4-o(1))N_X$ starts satisfying its fixed span bound. Its unreviewed preceding-gap congruence shows that, for every odd denominator part, the dyadic case included, such windows must also have preceding gaps in one class modulo $4$; the reversed form makes the tail constant on each class of graded congruences of the preceding gaps. Producing one late pair with equal windows and preceding gaps in different classes modulo $4$ is a missing input for every $d$; a pair in different index classes remains an alternative for $d>1$.
• The prediction-overlap comparison groups future windows by one weighted phase modulo two. Rationality bounds their minority preceding-gap parity count by $(1+\varepsilon)dX/2^{K+1}$. A lower bound exceeding every fixed multiple of $X/2^K$ would suffice, but is unproved; repetition and balanced marginals alone do not imply it.
• The doubled-phase comparison gives the sharp universal ceiling $3/2$ for finite normalized phase averages. The resonant references of P9 meet its hypotheses for every odd denominator. A strict saving requires a positive proportion of square energy, for which no actual-prime estimate is proved.
• The moment analysis separates aggregate residue switches from a prescribed pattern, and identifies the pair-count constant missing at modulus $4$. Its fixed-modulus consequence does not provide a growing-block estimate.
• The upper-count note shows that a rational value would repeat one tail value at a proportion of order $1/(d\log X)$ of the starts and would put most starts in a narrow carry window, and states the upper comparisons for consecutive primes that would contradict this. A comparison can lose $Cc^K$, $c<2$; the note proves phase bounds for independent and periodically restricted references and identifies a false termwise estimate for a signed majorant.
• The shorter-span note keeps a positive rationality lower count while enlarging the upper event to a monotone subset-existence test on actual sieve survivors. Its reference calculation is proved; its two arithmetic costs remain open.
• The survivor-moment note removes rough composites from bounded short-window events and the rooted second moment by a fixed-dimensional sieve. It identifies the unresolved centered estimate on the primes and proves that the stopped-sieve transport must cancel a nonzero base moment.
• The deletion-sign note uses the rational carry lattice to bound negative deletion mass at an enlarged span. The positive mass is then the final count up to $o(N_X)$. The finite-rank, small-factor upper test remains open; dropping integrality gives a distinct continuous-crossing count with too much mass. The fixed-wheel pointwise certificates specified in Proposition D24, with bounded potentials and nonnegative prime-pair costs, fail already at $d=1$. The sign bound supplies no estimate for the shorter-span event.
• The phase-transport note separates the original prime-root increments from the sum obtained by removing each root's phase. The corresponding increments agree under rationality; their unconditional endpoints differ. The original sum leads to a binary digit-change density test; the dephased sum retains a carry-dependent signed cofactor test. Both fixed-saving bounds are unknown. Its unconditional bound $Q\le2N_X+o(N_X)$ is too weak for the contradiction. Proposition P9 chooses a periodic reference and an odd multiple $b$ of each denominator $d$ so that the real reference phase is always $-1/2$. Rationality then forces $Q_{b,q}=(3/2+o(1))N_X$ for every odd $d$. The strict upper bound remains open; the sign issue at $d=7$ concerns only the more restricted reference family in P8.
• The earlier notes hold the truncation identities, the mean-tail bound and the single-phase comparison that the sketch reuses.

The forced structure

Write $g_n=p_{n+1}-p_n$ and $G_n=\sum_{j\ge0}g_{n+j}2^{-j-1}$. For $n\ge2$ the $n$th binary digit of $S$ is the parity of $\lfloor G_{n+1}\rfloor$, so $S$ is rational exactly when these parities are eventually periodic. If $S=a/(2^sd)$ with $d$ odd, every tail $G_n$ with $n\ge\max(s+2,3)$ lies in the lattice $(2/d)\mathbb Z$; replacing the gaps by $p_{n+t}-p_n$, with $t$ the order of $2$ modulo $d$ and $t=1$ when $d=1$, puts the corresponding tails in $2\mathbb Z$. Then one congruence among the positions of $K+1$ consecutive primes, a zero phase, forces the halved tail $G_{n+K}/2$ after them to be at least $2^K/d$, about $2^{K+1}/(d\log X)$ times its average near $X$. The prime number theorem bounds these tails only on average, so a rational value would force at most $O(dX/2^K)$ exceptions to an explicit parity rule among the starts with $X<p_n\le2X$, at every length $K$.

Current gap

The main proposed contradiction is an upper count. Put $I_X=\{n:X<p_n\le2X\}$, $N_X=|I_X|$, $K=\lceil A\log_2\log X\rceil$, $R=\lfloor4d\log X\rfloor$ and $D=\lfloor16K\log X\rfloor$. Let $B_d(X;K)$ count starts in $I_X$ with span at most $D$ and with the least positive residue of $-d\sum_{j<K}2^{K-1-j}g_{n+j}/2$ modulo $2^K$ at most $R$. Rationality and the prime number theorem force $B_d(X;K)\ge(13/16-o(1))N_X$. An upper bound $B_d(X;K)/N_X\le Cc^K R/2^K+o(1)$, for $1\le c<2$ and $A>1/\log_2(2/c)$, tends to zero and would contradict this. Theorem U8 proves the implication, including $d=1$ and every preperiod. The upper estimate for actual consecutive primes is unknown. Constants may depend on $d$; uniformity in $d$ is unnecessary. The same count for the integers with no prime factor up to $X^{1/2-1/K^2}$ differs from $B_d(X;K)$ by $O(N_X/K)$, so the estimate may be posed for those survivors. The lower sieve cutoffs examined leave an unknown base count and signed sums over deleted composites whose terms depend on about $K$ nearby survivors (Lemmas U10 and U11 and Proposition U12 of the upper-count note). Proposition U15 gives an exact alternative with no unknown base count: start with the odd integers avoiding one fixed prime $q$ with $2^{q-1}-1>d$, whose small-carry count is zero, and delete composites in increasing order. The resulting identity is $B_d(X;K)=\sum\Phi(m)$, with each weight determined by the preceding $K$ actual primes and a fixed periodic continuation. The signed upper bound (PD*) remains open. For $K=\lceil A\log_2\log X\rceil$ with fixed $A>2$ and sufficiently large $X$, every weight is at most a constant depending on $d$ and $q$, so the composites with two large prime factors contribute at most $(1/4+o(1))N_X$ (Lemma U16 and Corollary U17). The remaining sum equals $B_d(X;K)$ minus that part, so it is not a weaker target than the upper estimate itself.

Conditioning on rationality also gives the common-carry sign bound (D12) of the deletion-sign note. At $D=\lfloor K(\log X)^2\rfloor$ with $A>2$, negative deletion mass is $o(N_X)$, whereas rationality forces the final count to be at least $(7/8-o(1))N_X$. A positive-mass upper bound $(7/8-\varepsilon_d)N_X$ would finish this variant; it is unproved. The negative bound makes the positive mass equal to the final count up to $o(N_X)$, so this target is the direct count bound in another form. The endpoint criterion concerns the residue gain; the full gain also keeps its span cutoff. High ranks have arbitrarily small relative cost, multiple gains at a site have negligible total cost, and rough sites cost at most $(\log2+o(1))N_X$. The remaining sufficient test is $P_{\rm SF}^{<T_0}\le N_X/6$ for a sufficiently large fixed $T_0$, still unknown. The identities do not rule out an arithmetic estimate for that sum. Proposition D19 instead rules out dropping integrality: the corresponding ordinary real-tail crossings have an unconditional lower count exceeding $N_X/6$. The actual residue event retains the origin $Mc_n$ of (D20). The exact locator (D21) expresses the remaining site through an increasing integer carry orbit, and (D22b) keeps all of its prime-gap restrictions. A residue predicate on that orbit is not an unweighted prime progression count. The artificial integer-tail ensemble (D23), with uniform endpoint classes modulo $30$ and $5/16$ of its rank-two sites divisible by $5$, rejects that marginal shortcut only; it is not a sequence of actual primes. The changed span and rationality hypothesis are part of this interface; neither the sign bound nor the telescoping identity estimates the shorter-span count below. A certificate for D24's enlarged crossing event, using a bounded potential in the carry and a fixed wheel and paying nonnegative costs by the prime-pair upper bound, cannot reach the budget: stationary integer transitions give every such certificate cost at least $(1-o(1))N_X$ (Proposition D24). That rejects only the stated class.

The shorter-span alternative takes $K=\lceil12\log_2\log X\rceil$ and $D_*=\lfloor(6/5)K\log X\rfloor$, keeping the same positive-residue cutoff $R$. Rationality forces at least $N_X/24-o(N_X)$ consecutive blocks. Counting roots with any compatible $K$-subset gives a monotone upper event, so an upper bound $N_X/48$ for that event on the actual survivors at $z=X^{1/2-1/K^2}$ would suffice. This is (MS) of the shorter-span note, unproved for every odd $d$. The weaker sufficient target (BC), also unproved, asks only for a fixed saving below $N_X/24$ for the bounded event on actual primes. Proposition S22 gives $N_X/32$ as a concrete target, with margin $N_X/96$, and an alternative single weighted band-error budget $N_X/64$. These bounds would contradict the forced lower count directly; neither is claimed to give the particular numerical bound in (MS). The aggregate band identity does not estimate its arithmetic error. The reference event is rare, while the untruncated sum of its certificates still has divergent reference expectation. Its two costs are located as follows. Bounding the roots with more than $5K/4$ later survivors through the centered survivor moment (VM) would imply a law of large numbers for sums of about $(6/5)K$ consecutive gaps after almost every prime, for which no proof is known here, even under the Riemann hypothesis. The survivor-moment note shows that (VM) differs from its prime-rooted version by only $O(N_X)$, and that composites change the bounded event by $O(N_X/K)$. Its lower-cutoff base moment has order $K^2N_X$, so the exact signed transport must cancel a nonzero contribution. Neither comparison estimates the remaining prime-rooted moment. At the other roots the specified nonnegative polynomial cutoff bounds are at least the untruncated subset count; the damped signed alternative leaves a growing-order correlation bound unproved. Its weights are too large for the affected-root estimate to transfer it at the needed precision. Both arithmetic costs remain open. Lowering the sieve level and proving the band-loss inequalities (PB) would address the bounded event directly and imply (MS). The exact losses have the semiprime expansion (S20). The attempted progression estimate leaves signed coefficients of growing degree and moduli outside the inspected Bombieri--Vinogradov ranges. Its weighted remainder and the required lower comparison remain unknown. The one-site calculation (S21) rules out simply replacing the full divisor sum by periodic density with an $o(N_X)$ error; it does not refute (PB). Prime-supported roots survive every band, consistently with (PB) implying the upper bound that would contradict rationality.

An alternative is more exceptions to the parity rule than the rational bound allows, along arbitrarily large $X$. The sketch names two sufficient counts, each open for every odd $d$, including $d=1$: a positive proportion of exceptions at length $\log_2(d\log X)+O(1)$, and a count of zero-phase starts with tame continuations, proposed at total length about $2\log_2\log X$. Both seek lower bounds for joint statistics of about $\log\log X$ consecutive gaps. The known theorems on consecutive primes, including bounded gaps, dense clusters, congruent strings and long gaps, stop at stated inequalities on the sketch's pages: they control fixed lengths, constant residue patterns or rare clusters whose proved counts fall short of the required count. Under the Riemann hypothesis, a lower bound for the frequency of zero phases alone at length $A\log_2\log X$, $A>4$, would suffice. Proposition Z8′ instead uses an explicit large-gap count $N(x,H)\ll x(\log x)^{B'}/H^2$ and proves the implication for $A>B'+1$, with its far tail controlled. That conditional refinement does not supply the zero-phase count. The conditional claims on the problem page assume uniform prime-tuple hypotheses and are not premises of this route.

The resonant phase-transport alternative writes $d=3^\nu d_0$ with $3\nmid d_0$ and takes $q=2\cdot3^\nu+1$ and $b=d_0(2^{q-1}-1)/3$, an odd multiple of $d$. Proposition P9 proves that the periodic reference has real phase $-1/2$ at every site, also when $q$ is composite. Under rationality the dephased sum is therefore $(3/2+o(1))N_X$. A strict upper bound below $3N_X/2$ on unbounded scales for each $d$ would finish the contradiction beyond every finite preperiod. This is (RFPT), unproved even for $d=1$. The actual carry coefficient, least-factor conditions and rank remainder of (P6) remain in the target. An upper proof may use integrality; an unconditional endpoint-correlation bound would also suffice. Rank by rank the complete site sums of (P6) are differences of prime-endpoint sums. This identity gives no bound for the parts selected by least prime factor and does not prohibit estimating them under rationality. The reference construction and the signed boundary exchange (P9h) supply no strict saving. The multiple-crossing payment (P9i) works for shifts $o(\sqrt{\log X})$, but the examined decoupling theorem's stated interval bound is too large at that range on the $N_X$ scale. A stronger estimate remains unproved. Frequency doubling leaves the same site weight, and the direct shift comparison needed by the examined decoupling method costs $O(uN_X)$. The original transport still has its different unconditional endpoint and the separate, unproved digit-change density test.

The finite-window attempts for the counting and phase targets reach roughly $\log_2\log X$ consecutive primes, with power-of-two moduli of order a power of $\log X$. The inspected fixed-length results and cluster bounds have not supplied the required estimates at that scale. This is a limitation of the attempts recorded here, not a necessary-input theorem for every proof of those targets. No proof of irrationality or required arithmetic saving is held.

There is also an upper criterion independent of $d$. A rational value would make one value of the tails $G_n$ recur at a proportion of order $1/(d\log X)$ of the starts with $X<p_n\le2X$. So a bound, within any fixed factor of an independent-gap model, on how often the weighted sums of the next $\log_2\log X+O(1)$ gaps fall in a short interval would exclude every rational value at once. The same kind of comparison motivates the carry bound above. The examined tuple sieve estimate has a factorial loss and does not keep the primes consecutive; this does not rule out a sharper direct count. Fixed nonnegative combinations of tuple indicators which majorize the event on every finite set, at the stated span, have reference mass nearly one (Lemma U13 and Proposition U14). A bound in that class needs an aggregate deficit of prime tuples relative to the reference; the argument does not exclude every arithmetic or signed majorant. An exact signed majorant retains consecutiveness, but its termwise adverse-error bound around independent Bernoulli sites is false: tuples excluded modulo $3$ already give too much error. Lemma U9 incorporates small odd-prime exclusions into a reference without losing its relative phase bound. Transferring that bound to the actual primes remains the missing theorem; the model calculation does not supply it. No equivalence between the signed estimate and a uniform prime-tuples conjecture is proved.

An unspecified selected subtuple cannot be made sufficient simply by designing a certificate for every possible selection: Lemmas Z6 and Z6b exhibit failing selections, including when the root is chosen inside the selected block afterward. Lemma Z6c tests grouped selection as well: sufficiently large prescribed groups admit representatives with all half-gaps too large for any internal score certificate. This concerns a universal guarantee, not which representatives actually occur as primes. The alternative remains an arithmetic estimate for favorable selections and their continuations. A single witness would suffice, but one built from a single residue class is either a chain of about $\log_2\log p_n$ gaps, each at least a constant times $\log p_n$, or is followed at once by a cluster denser than the mean (Proposition Z9); the stated chain and cluster theorems supply neither. Clearing the odd denominator does not reduce the required number of original gaps in the mean-based test (O9), and imposing carry integrality at every cut is one endpoint congruence rather than independent restrictions (O10).

Review status

No independent review covers the whole route. The separately graded review accepts only the conditional matching-window argument at the exact scope stated on its owning page, without a prime-pair existence theorem. The other deductions here remain author-recorded. The comparison results L228, L229 and L230, which show which local or sparse information about the primes cannot decide the question, are tier 0.