{"id":"1e8aeed7-018f-4dad-acf8-a28f3b69e562","arxiv_id":"2607.10654","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Pooled over all positions and all primes below 10^n, each decimal digit occurs with frequency 1/10 + O((log n)/n).","lead":"The paper proves that when you pool every decimal digit of every prime below 10^n, each digit 0–9 appears with frequency 1/10 plus an error that shrinks like (log n)/n. It is an averaged statement only: fixed positions, normality, and digit correlations remain open.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Lemma 3.2's exponential-sum bound is not rigorously justified for the short shell and unweighted sum, so the interior-digit equidistribution (and thus Theorem 2.2) rests on an incomplete estimate.","rationale":"The reader correctly isolates the single load-bearing gap: Lemma 3.2 is not a theorem as written. The remainder of the architecture (Erdős–Turán truncation, dilution of an O(log m) edge band, geometric shell summation via PNT) is standard and would go through once a correct short-interval exponential-sum estimate of comparable strength is supplied. Because that estimate is classical in spirit and almost certainly true after routine partial-summation and differencing arguments, the paper’s strategy remains viable; the present write-up simply does not contain it. Hence the verdict stays CONDITIONAL rather than REJECT. No stronger objection (e.g., an outright contradiction or a missing main-term cancellation) appears. The mis-cited references [18,19] are cosmetic and do not affect the logical chain.","tokens_in":10887,"tokens_out":736,"duration_ms":6804,"concrete_test":"Derive (or cite a standard reference for) a short-interval Vinogradov bound for the unweighted sum ∑_{Y/10 < p ≤ Y} e(hp/q) that is of the same strength as the claimed majorant, for all r ≥ (log Y)^{2A+8} and q = 10^{k+1} in the interior range. If no such bound is available at the stated strength, or if the resulting error after Erdős–Turán is only O(π_m / polyloglog Y) rather than O(π_m/(log Y)^A), the interior lemma fails and Theorem 2.2 must be weakened or withdrawn.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim (Theorem 2.2) is assembled from the single-shell estimate (Lemma 4.1), which in turn rests entirely on the interior-digit equidistribution of Lemma 3.3. That lemma is obtained by feeding the exponential-sum bound of Lemma 3.2 into the Erdős–Turán inequality (Lemma 3.1). Lemma 3.2 asserts\nS_m(h,q) ≪ (Y/√r + Y^{4/5} + √(Y r))(log Y)^4\nfor the unweighted sum over the short interval T_m = [10^{m-1},10^m). The only justification offered is a classical Vinogradov-type bound for the weighted cumulative sum ∑_{p≤Y} log p · e(αp), followed by the literal comparison “S_m(h,q) < ∑ log p e(·)”. This step is invalid: (i) the classical bound is for the cumulative sum up to Y, not the shell difference; (ii) the unweighted sum is not dominated by the weighted sum without partial summation or a lower bound on log p; (iii) the passage from the full sum to the short interval of length ~Y is never written. If the claimed majorant fails to hold uniformly for q=10^{k+1} in the interior range C_1 log m ≤ k ≤ m-C_2 log m, then the O(π_m/(log Y)^A) error of Lemma 3.3 collapses and the main theorem does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper claims an unconditional averaged equidistribution theorem for the decimal digits of primes: if one pools every digit position of every prime less than 10^n, the frequency of each digit d equals 1/10 + O((log n)/n) uniformly in d (Theorem 2.2). The argument proceeds by shells of m-digit primes, applies the Erdős–Turán inequality to the fractional parts {p/10^{k+1}} at each position k, bounds the resulting exponential sums over the shell via a classical Vinogradov-type majorant, obtains a power-saving error for all but O(log m) exceptional positions near the ends, dilutes those positions by a trivial estimate, and finally sums across shells using the geometric growth of shell mass under the Prime Number Theorem.","tokens_in":11190,"tokens_out":1117,"duration_ms":24092,"significance":"If the estimates are made rigorous, the result supplies a clean, elementary proof of the natural averaged form of digit equidistribution for primes, with an explicit power-saving error and a careful distinction from the still-open pointwise, normality, and correlation questions. The architecture (interior discrepancy + edge dilution + geometric domination) is transparent and uses only classical tools, so the paper would be a useful reference clarifying the precise scope of what is currently known. It does not supersede the deeper Mauduit–Rivat theory for sum-of-digits, but it fills a logically distinct and previously unrecorded gap.","major_comments":[{"comment":"Lemma 3.2 asserts the unweighted shell sum S_m(h,q) ≪ (Y/√r + Y^{4/5} + √(Y r))(log Y)^4. The only justification offered is the classical weighted cumulative bound for ∑_{p≤Y} log p · e(αp), followed by the literal comparison “S_m(h,q) < ∑ log p e”. This step is invalid on three counts: (i) the classical bound is for the cumulative sum up to Y, not the short shell T_m = [10^{m-1},10^m); (ii) an unweighted sum is not dominated by a weighted sum without partial summation (or a uniform lower bound on log p together with control of the error); (iii) the passage from the full sum to the shell difference is never written. Because Lemma 3.3 feeds this majorant directly into Erdős–Turán, and Lemma 4.1 and Theorem 2.2 rest on Lemma 3.3, the gap is load-bearing. A correct write-up via partial summation plus differencing of two cumulative sums (or a short-interval form of the Vinogradov estimate) i","section":"§3.2, Lemma 3.2"},{"comment":"Even after the weighted-to-unweighted transfer is repaired, the range of the Dirichlet denominator r = q/gcd(h,q) ≥ (log Y)^{2A+8} is used to absorb the three terms of the majorant into Y/(log Y)^{A+4}. The argument assumes the same majorant constants remain valid uniformly for every q = 10^{k+1} with C_1 log m ≤ k ≤ m − C_2 log m. This uniformity should be stated explicitly (or a reference to a short-interval version of the bound supplied), since the subsequent choice of H and the admissible range of k depend on it.","section":"§3.3 and Lemma 3.3"}],"minor_comments":[{"comment":"The comparison symbol “<” in the display of Lemma 3.2 is notationally incorrect even if the intended majorization held; replace by the proper ≪ after partial summation.","section":"§3.2"},{"comment":"References [18] and [19] are the authors’ own cryptanalysis papers and have no bearing on digit equidistribution of primes; they should be removed or replaced by relevant surveys.","section":"References"},{"comment":"The title’s phrase “The Prime Digit Distribution Conjecture: A Formal Proof /*……*/” is slightly misleading; the body carefully distinguishes the averaged statement from the open pointwise conjecture. A milder title would better match the actual theorem.","section":"Title"},{"comment":"In the proof of Lemma 5.1 the constant 1/5 is arbitrary; any fixed ratio <1 works, but the text should note that the implied constant in N_n ≍ M_n depends on the choice.","section":"§5"}],"recommendation":"major_revision","confidential_remarks":"The central analytic gap in Lemma 3.2 is real but standard and repairable; once fixed the paper is a solid, modest contribution suitable for a general number-theory journal. The presence of two unrelated self-citations on RSA cryptanalysis is a minor red flag for citation hygiene but does not affect the mathematics."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know: this is a genuine averaged equidistribution theorem for individual decimal digits of primes (pooled over all places and all p < 10^n) with an explicit O((log n)/n) rate. That statement is not in Mauduit–Rivat, Maynard, or the other digit-sum literature they cite, and the authors correctly flag that it is weaker than pointwise normality or joint digit laws.\n\nWhat they do well is the global plan. Interior places are handled by Erdős–Turán plus classical exponential sums; an O(log m) edge band is diluted by trivial bounds; geometric growth of shell mass lets the top shell dominate. The conclusion section is unusually honest about scope. Circularity is zero—everything rests on PNT, Erdős–Turán, and Vinogradov-type estimates from Davenport. The rate is elementary but clean.\n\nThe soft spot is real and load-bearing, though fixable. Lemma 3.2 claims the unweighted shell sum S_m(h,q) inherits the classical majorant (Y/√r + Y^{4/5} + √(Y r))(log Y)^4. The write-up compares it to the weighted cumulative sum with a literal “<” and never writes the short-interval difference or partial summation. That gap sits under the interior-digit lemma and therefore under the main theorem. Standard tools (partial summation + Vaughan identity on [Y/10,Y]) almost certainly close it, but as printed the estimate is incomplete. Two self-citations ([18,19]) are also mis-described as surveys of the digit problem; that is sloppy but peripheral.\n\nThis is for analytic number theorists who care about digital statistics of primes. It is not a breakthrough, but it is a coherent unconditional corollary that the literature does not already contain. A serious editor should send it to referees; the repairs are routine and the claim is worth having on the record once the exponential-sum step is written properly. I would not cite it until that is fixed, but I would bring the cleaned version to reading group.","headline":"Averaged digit equidistribution for primes is a clean, modest new statement; the architecture works, but Lemma 3.2 needs a proper short-interval unweighted bound before the proof is complete.","tokens_in":11897,"tokens_out":529,"would_cite":false,"duration_ms":5287,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A63","11N05","11N13","11K38","11L07"],"pacs":[],"model":"grok-4.5","headline":"Pooled digits of primes below 10^n are equidistributed: each digit occurs with frequency 1/10 plus an O((log n)/n) error.","keywords":["sum of digits","prime digit distribution","equidistribution","exponential sums","Erdős–Turán inequality","Vaughan estimates","discrepancy theory","decimal digits of primes"],"falsifier":"Compute the empirical frequencies of digits 0–9 among all primes below 10^n for successive large n (say up to 10^12 or higher) and check whether the maximal deviation from 1/10 decays at least as fast as (log n)/n; a persistent larger deviation would refute the claimed error term.","tokens_in":11714,"feed_emoji":"π","tokens_out":778,"duration_ms":6735,"temperature":0.7,"pith_summary":"The paper claims that if you collect every decimal digit from every prime smaller than 10^n and count how often each digit 0 through 9 appears, the frequency of each digit is 1/10 plus an error that shrinks like (log n)/n. The argument works by first proving that almost every digit position inside an m-digit prime shell is equidistributed, then showing that the O(log m) positions near the leading and trailing ends contribute a vanishing share of the total digit mass, and finally summing the shells using the geometric growth of prime counts. A sympathetic reader cares because this is an unconditional, averaged form of the long-standing intuition that primes look random in their digits, obtained from classical tools without any unproved hypotheses. The authors carefully separate this pooled result from the still-open questions of normality in a fixed position and of joint digit statistics.","feed_headline":"Prime digits average to 1/10 with O((log n)/n) error","feed_subtitle":"Pooling every digit of every prime below 10^n yields asymptotic equidistribution, unconditionally.","key_machinery":"The Interior Digit Lemma: for positions k that lie between C1 log m and m - C2 log m inside an m-digit prime shell, the count of primes with k-th digit equal to d equals π_m/10 plus an error O(π_m/(log Y)^A). It is proved by feeding classical Vaughan–Vinogradov bounds on exponential sums over primes into the Erdős–Turán discrepancy inequality, then diluting the O(log m) exceptional end positions by averaging.","core_discovery":"For every digit d from 0 to 9, the total number of occurrences of d among all digits of all primes less than 10^n, divided by the total number of such digits, equals 1/10 plus an error of size O((log n)/n) as n tends to infinity, uniformly in d. The same limit therefore holds in total variation for the empirical digit measure.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Prime digits average to 1/10 with O((log n)/n) error","Pooled digits of primes <10^n equidistribute as 1/10 + O((log n)/n)","Average digit frequencies among primes hit 1/10 at rate O((log n)/n)","Unconditional average equidistribution of all prime digits","Every digit appears with density 1/10 in primes below 10^n"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The argument stands or falls on a classical bound for exponential sums over the primes in each m-digit shell; if that bound fails for the moduli that encode interior digit positions, the interior equidistribution step collapses.","fun_headline_variants_meta":{"raw":{"variants":["Prime digits average to 1/10 with O((log n)/n) error","Pooled digits of primes <10^n equidistribute as 1/10 + O((log n)/n)","Average digit frequencies among primes hit 1/10 at rate O((log n)/n)","Unconditional average equidistribution of all prime digits","Every digit appears with density 1/10 in primes below 10^n"]},"model":"grok-4.5","effort":"low","cost_usd":0.010734,"raw_usage":{"total_tokens":2401,"prompt_tokens":870,"num_sources_used":0,"completion_tokens":113,"cost_in_usd_ticks":107340000,"prompt_tokens_details":{"text_tokens":870,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1418,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":870,"tokens_out":113,"duration_ms":17669,"temperature":1.0,"reasoning_tokens":1418,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T10:11:35.396216+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the empirical frequencies of digits 0–9 among all primes below 10^n for successive large n (say up to 10^12 or higher) and check whether the maximal deviation from 1/10 decays at least as fast as (log n)/n; a persistent larger deviation would refute the claimed error term.","supporting_citations":[],"review_version":1}