Exact arithmetic testbed

### PDF page 13 Daniel J. Murray Revised September 2026 The earlier asymptotic-floor question remains open for actual gap-word statistics. Resolution co- variance says that equivalent full representations have the same values at the same scale. It does not determine the limit of those values as 𝑥 → ∞ , and small differences between two truncations cannot rule out a common limiting error. The present contribution replaces an unconstrained floor fit with explicit approximation components and tests; it does not prove that the floor is zero. 16. Conclusion Prime divisibility admits exact recursive sieve refinement. A Hardy–Littlewood tuple functional also admits exact transfer of a prime coordinate between explicit wheel state and an unresolved Euler factor, and this identity survives full finite inclusion–exclusion. A finite-universe positivity theorem states when the algebra supports a probability interpretation and therefore the conditional-mixture and predictive-risk identities. The archived pair model closely matches eight inherited reversal contrasts and records a prospective comparison in the next disjoint range. The numerical agreement is preserved, while its inferential scope is explicit. The remaining task is controlled prediction for actual primes: coherent finite com- putations, independent enumeration, dependence-aware residual analysis, and uniform arithmetic bounds that survive rare-event conditioning. None of these follows from representation covariance alone.

Recursive Predictive Closure in the Prime Sieve Resolution covariance, finite probability consistency, and gap-word prediction Daniel John Murray 7 September 2026 — revised research manuscript Independent Researcher, Melbourne, Australia ORCID: 0009-0005-1794-5945 to unconditional positivity and prove that, on any fixed finite offset universe and fixed finite family of wheels, sufficiently small candidate density produces a consistent probability law. Conditional mixture and predictive-risk identities then follow on that jurisdiction. Classical sieve recursion supplies a separate exact result: the next survivor after 𝑝𝑘 is 𝑝𝑘+1, and 𝑝2 𝑘+1 is the first composite surviving the 𝑝𝑘 primorial. A two-point truncation is compared with eight previously selected prime gap-word reversal observables. The archived prospective record for 1.024 × 1011 < 𝑝 < 2.048 × 1011 gives 𝑅2 = 0.998190 across those contrasts, with two discrepancies exceeding three reported naive Bernoulli standard errors. These are descriptive diagnostics; dependence-aware calibration and numerical error control are needed before a formal rejection or attribution to higher interactions. Exact representation covariance, finite probability consistency, arithmetic asymptotics, and empir- ical validation are separate claims. The unresolved problem is controlled conditional prediction for actual primes, including growing offset windows, interaction truncation, and conditioning on rare gap words. Keywords: prime sieve; singular series; predictive state; consecutive primes; inclusion–exclusion; Chinese remainder theorem; gap words; finite probability consistency. 1. Introduction Prime numbers are deterministic, while finite samples of consecutive-prime residue patterns show strong biases. Lemke Oliver and Soundararajan developed a conjectural explanation from Hardy– ulo 30 change when the observed state is lifted to finer primorial wheels or supplemented by ordered gap words. This motivates a more precise question than extrapolating a residual against wheel size: how does a prediction change when a divisibility coordinate moves from the unresolved interaction law into the explicit state?

as algebra; a separate finite-universe theorem establishes when the same expressions can be used as probabilities. The empirical component retains all eight inherited gap-word contrasts and the archived prospective record. Arithmetic recombination, predictor reproducibility, timestamp evidence, and independent enumeration are distinguished. A high correlation across selected contrasts is valuable evidence of descriptive agreement, but it does not establish a prime-tuple theorem, identify a unique omitted interaction, or eliminate an asymptotic residual floor. 2. Claim status and scope and verification scope Archived counts and frozen predictions Open target A proposed improvement or unresolved inference Higher-order accuracy; asymptotic floor and conditional-variance identity are established mathematics. The contribution is their explicit organization into a resolution-transfer construction, its probability-consistency boundary, and a testable prediction hierarchy. We do not claim a proof of the Hardy–Littlewood conjecture, the Riemann hypothesis, an efficient formula for the 𝑛th prime, or uniqueness of this representation among all possible predictive models. 3. Definitions Let 𝑝1 = 2, 𝑝2 = 3, … be the primes and 𝑄𝑘 = ∏ 𝑘 𝑖=1 𝑝𝑖 = 𝑝𝑘#. A wheel 𝑄 is a positive squarefree

actual primes. Those are separate questions. Finite offset windows used by a computation therefore need their own probability or signed-functional status declared. 8. Conditional mixtures and predictive heterogeneity Suppose the full cylinder functionals on a common finite universe are nonnegative probability laws at 𝑄ℓ and its lifts. Their average defines the coarse law by (8). Let 𝐵 be any event in that universe with positive coarse probability, and let 𝐴 be another event. Cylinder formulas extend to arbitrary events by summing disjoint atoms.

a probability distribution. Zero-mass lifts also have zero mass for 𝐴 ∩ 𝐵 and are omitted; their conditional ratios are never evaluated. □ Theorem 8 (conditional predictive value). For 𝑌 = 1𝐴, 𝑝 = ∑𝑗 𝑤𝑗𝑝𝑗, and squared-error loss, the reduction in Bayes risk obtained by revealing the lifted phase is 𝑝(1 − 𝑝) − ∑ 𝑗 𝑤𝑗𝑝𝑗(1 − 𝑝𝑗) = ∑ Proof. Expand the squares and use ∑𝑗 𝑤𝑗 = 1 and ∑𝑗 𝑤𝑗𝑝𝑗 = 𝑝. □ For nested finite coordinate revelations, conditional expectation gives a telescoping variance decom- position under one coherent joint law. Each increment measures additional predictive information for the specified event and sampling law. Heterogeneity among fine phases means the coarse state loses predictive information relative to those phases. It does not alone prove temporal memory. If histories ℎ, ℎ′ change fine-phase weights to 𝑤𝑗(ℎ), 𝑤𝑗(ℎ′)while the fine future probabilities remain 𝑝𝑗, observable history dependence requires ∑ 𝑗 [𝑤𝑗(ℎ) − 𝑤𝑗(ℎ′)]𝑝𝑗 ≠ 0. (15) Different weights can cancel in prediction; a latent coordinate can also be independent of every observed history. Markov lumpability concerns a specified transition kernel and observation map [9]. No finite-wheel Markov kernel for actual primes is proved here. Equation (15) states exactly the extra condition required for the projection-induced-memory interpretation. 9. Two directions of refinement and the conditional arithmetic bridge

Daniel J. Murray Revised September 2026 There is no theorem exchanging one extra wheel prime for precisely one fewer interaction order. Near equality of two pair-truncated predictions is a useful diagnostic but does not bound their common error relative to the all-order functional or actual primes. 9.1 What a number-theoretic approximation must control For a sampling interval 𝐽 and phase 𝑐, let 𝑀𝐽,𝑄,𝑐 (𝑈 )be the exact empirical tuple frequency for a uniformly sampled eligible integer congruent to 𝑐; endpoints must be kept in the stated observation 𝜀𝐹 ∪𝑇. (18) The number-theoretic task is to prove suitable bounds on the right. A named uniform Hardy– Littlewood hypothesis must specify which offsets, wheels, tuple sizes and ranges it covers. Fixed finite patterns differ from windows growing on the scale of log 𝑥, as occur in consecutive-prime predictions [2,4,11]. Conditioning needs a further denominator bound. If an actual event numerator and denominator satisfy 0 ≤ 𝑁 ≤ 𝐷 , and | ̂𝑁 − 𝑁 | ≤ 𝜂𝑁 𝐷, | ̂𝐷 − 𝐷| ≤ 𝜂𝐷𝐷, 0 ≤ 𝜂 𝐷 < 1, then This follows by subtracting the ratios and using ̂𝐷 ≥ (1 − 𝜂𝐷)𝐷. Absolute errors tending to zero are not enough when the conditioning event itself becomes rare. For fixed required prime offsets, raw frequencies often tend to zero anyway. A meaningful closure hypothesis therefore controls errors relative to conditioning-event frequency, together with window-tail and interaction-truncation errors. No such complete bound is established here for the empirical observables below. 10. Gap-word reversal observables For consecutive primes 𝑝𝑛, put 𝑔𝑛 = 𝑝 𝑛+1 − 𝑝𝑛. In a declared interval, retain starting primes for

𝐹 = {0, 𝑎, 𝐻, ℎ} ∪ 𝑈 . There are 𝑘 + 1 required primes including the starting prime. Sum 𝑊2 over these subsets, phases, and endpoints ℎ ≤ 𝐻 max. Divide by the prefix denominator to obtain a truncated prediction for

window checks without Monte Carlo. 11.2 Recorded choices and remaining numerical controls The archived prospective prediction uses 𝑄 = 30030, 𝜑(𝑄) = 5760and 𝐿 = 25.734082860811313, giving 𝜌 = 0.202592868565214. Recompiling the unchanged source and setting 𝐻max = 900 repro- duces all eight frozen predictions with byte-identical printed output. The archive does not specify a derivation rule for its effective 𝐿. It is therefore recorded explicitly as a numerical input, rather than reconstructed by assuming a geometric or arithmetic midpoint. There are no recurrence-contrast coefficients fitted in the archived formula. Nevertheless 𝑄, 𝐿, 𝐻max, the retained interaction order and the selected observables are choices. The appropriate description is an arithmetic predictor with no fitted recurrence-contrast parameters. For a future freeze, the scale rule, interval averaging, cutoff convergence and positivity or signed-functional status must also be fixed in advance. 12. Numerical record and its interpretation 12.1 Retrospective high range denominators reproduce the observed contrasts below. Model values retain the original manuscript precision. Observable Observed Pair prediction Observed minus predicted 5 ∶ (2, 10) +0.0046323 +0.0046218 +0.0000104 5 ∶ (2, 6) −0.0005932 −0.0005918 −0.0000014 5 ∶ (10, 20) −0.0061474 −0.0062681 +0.0001207 8 ∶ (10, 30) +0.0103134 +0.0100630 +0.0002504 Rerunning the archived source with 𝐿 = 25.04093568025137and 𝐻max = 900 gives Pearson 𝑟 = 0.9997133466and 𝑅2 = 0.9991174650, reproducing the original rounded statistics. This range informed the decision to freeze the next prediction, so it is a retrospective model-development check. 12.2 Archived prospective range The record identifies 1.024 × 1011 < 𝑝 start < 2.048 × 1011 as a prospective test. It contains four count summaries, the combined integer record, frozen predictions, and source/prediction hashes. Matching hashes establish content identity. The archive’s statement that the predictions preceded the scan is a provenance claim; an independent timestamped publication of the freeze is not supplied by the hashes alone.

Observable 𝑛𝑎𝑏 𝑛𝑏𝑎 Observed Frozen pair prediction Error / naive SE 5 ∶ (2, 10) 21,973,823 21,973,601 +0.0041602471 +0.0041159334 +0.465 5 ∶ (2, 6) 16,198,984 16,194,299 −0.0011196849 −0.0009380950 −1.582 ∑𝑖(𝑦𝑖 − ̄ 𝑦)2 = 0.9981900063, 𝑟 = 0.9991117462, RMSE = 2.0696863 × 10−4. (23) The headline absolute error is 4.4313751 × 10−5. These are descriptive comparisons over eight chosen contrasts, not estimates of predictive accuracy across all prime patterns. 𝑅2 is defined by the displayed error formula and is not simply Pearson 𝑟2. Figure 1: Archived prospective agreement and residual scale. Numbers identify rows in the same order as the table. The right panel shows naive standardized discrepancies; it does not display calibrated significance tests. 12.3 Uncertainty and the two largest residuals The archived standard error is

sion, is 54c19eff1803febf24bf70777335342f0760b7c425fc12aea9baf6856f7971db. The predictor pair_hl_general.cpp has SHA-256 48784950226e41411bf113cfc1af9dd516c1d738c4f920fbf1f669dce402e2e7. The frozen prediction file has SHA-256 fb39fe0ecb988bf4bbab6e37af14f09f9f58749bbd8757fa89f1dbf7fc3bfdd4. All three match the original manuscript. The archive contains an experimental triple implementa- tion, the derivation note, four prospective count summaries, combined tables, and previous-bin inte- ger counts. It does not contain the segmented-sieve enumeration source or the full prime sequence. Arithmetic recombination of archived summaries is consequently distinct from independently enu- merating the original intervals or checking every boundary-handling decision. The revision’s verification files provide exact rational checks of sieve jurisdiction, tuple and cylinder

Daniel J. Murray Revised September 2026 all eight prospective and all eight retrospective observed contrasts from integer counts, reproduce both sets of model predictions from unchanged source, and perform the three targeted endpoint checks above. Finite-product checks verify implementation identities; the accompanying mathemat- ical proofs establish the infinite-product identities. Neither constitutes a computational proof of Hardy–Littlewood asymptotics. For a complete independent replication, the deposit should additionally provide the enumeration source, compilation and execution commands, exact subrange boundaries and cross-boundary carry policy, per-block integer counts, effective-scale rule, cutoff convergence record, and independently timestamped prediction freeze. Counts alone cannot reconstruct the full overlap covariance. 14. Decisive next tests 1. Close the numerical specification. Freeze the effective-scale or interval-integration rule, 𝑄, interaction order, 𝐻max, endpoint convention and arithmetic precision. Check the returned conditional masses and compare cutoff changes with each claimed error tolerance. window sums with brute-force subset enumeration, and establish numerical convergence before using another prime range. A change in the desired direction alone is insufficient. 4. Generalize observables prospectively . Predeclare a wider family including small effects, zeros, different lags and observer moduli, with multiplicity handling if inferential tests are used. 5. Audit transfer quantitatively . Compare refined-wheel predictions after the same phase marginalization at fixed interaction order. Attribute discrepancies to bounded truncation or implementation errors before assessing an arithmetic hypothesis. 6. Prove a conditional approximation theorem. Supply a named uniform tuple-count assumption and explicit bounds through (18), conditioning through (19), and the growing- window tail. State all limits and their order. These tests have different outcomes. A code failure challenges an implementation; failure of the truncated predictor challenges that approximation; failure of a precisely specified uniform arithmetic hypothesis would challenge the proposed statistical bridge. An algebraic identity is assessed by proof or counterexample, not by a p-value. 15. Relation to previous work and broader interpretation Lemke Oliver and Soundararajan already use wheel-conditioned singular series, modified interac- tions and inclusion–exclusion to study consecutive residues [4]. The present transfer proof makes boundary results are compatible with that framework. Passing from candidate populations over an entire primorial cycle to actual primes in a specified interval remains a separate arithmetic inference. The probability-consistency theorem sharpens the broader predictive-state programme: a family of formal predictions must first belong to a common probability law before conditional independence, posterior weights or Bayes risk can be interpreted. On a coherent finite jurisdiction, the conditional mixture has an exact information-loss meaning. In the empirical prime setting, whether observed histories reveal the omitted coordinate must still be checked by (15).

factor, and this identity survives full finite inclusion–exclusion. A finite-universe positivity theorem states when the algebra supports a probability interpretation and therefore the conditional-mixture and predictive-risk identities. The archived pair model closely matches eight inherited reversal contrasts and records a prospective comparison in the next disjoint range. The numerical agreement is preserved, while its inferential scope is explicit. The remaining task is controlled prediction for actual primes: coherent finite com- putations, independent enumeration, dependence-aware residual analysis, and uniform arithmetic bounds that survive rare-event conditioning. None of these follows from representation covariance alone. Acknowledgments and disclosures