{"id":"7167645a-f0de-4f22-a6d9-4d2088e39b05","arxiv_id":"2606.04677","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves an explicit threshold M < 1.78×10^32 such that every m ≥ M with gcd(m,9)=1 is the digit sum of a prime, with a lower bound on the count and an application to iterated digit-sum primes.","lead":"This paper derives an explicit bound M less than 1.78 times 10 to the 32 such that every larger integer m coprime to 9 is the digit sum of some prime. A smart generalist might read it to see how analytic number theory converts existence results into concrete numerical thresholds.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Accuracy of explicit constant-tracking in Type-II minor-arc estimates and prime exponential-sum replacement determines whether the numerical M holds.","rationale":"The reader's weakest_assumption directly identifies the same point of numerical explicitness that must hold for the headline claim. No internal inconsistency or missing logical step is visible from the abstract and stated method; the risk is purely whether the tracked constants are correct, which the proposed recomputation would settle without altering the overall verdict category.","tokens_in":1793,"tokens_out":312,"duration_ms":24899,"concrete_test":"Independently recompute the explicit major-arc and Type-II minor-arc bounds from the paper's stated lemmas using the same normalization and truncation parameters; if the resulting admissible error term exceeds the main term for any m near the claimed threshold, the numerical M fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the fully explicit versions of the DMR prime exponential-sum bound, the major-arc estimates, and the constant-tracked Type-II estimates produce error terms small enough to guarantee the surjectivity threshold at the stated numerical value of M. Any underestimation of the implicit constants or overestimation of the decay rates in those estimates would invalidate the specific bound M < 1.78 × 10^{32} even if the asymptotic framework remains valid. The paper asserts these explicit replacements suffice, but the load-bearing step is the numerical fidelity of the constant derivations rather than the existence of some (non-explicit) threshold.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper extends the circle-method framework of Drmota-Mauduit-Rivat on digital restrictions for primes by making all relevant constants explicit. It proves there exists an explicit integer M < 1.78 × 10^{32} such that every integer m ≥ M with gcd(m,9)=1 is realized as the decimal digit sum s(p) of at least one prime p. It also establishes an explicit lower bound A_m(10^{2m/9}) ≥ C_q(m) 10^{2m/9}/m^{3/2} with C_q(m) positive and bounded away from zero in admissible classes, and derives consequences including the infinitude of OEIS A070027.","tokens_in":1921,"tokens_out":579,"duration_ms":30464,"significance":"If the explicit constant derivations hold, the result supplies the first published numerical surjectivity threshold for digit sums of primes (previously known only to exist by Harman via the DMR asymptotics). The explicit lower bound and applications to additive primes and iterated digit-sum chains are direct consequences of the same estimates.","major_comments":[{"comment":"Abstract and § on Type-II minor-arc estimates: the numerical value M < 1.78 × 10^{32} is load-bearing on the fully explicit replacement for DMR's prime exponential-sum bound and the constant-tracked Type-II estimates; the manuscript must supply the complete error-term derivations and numerical verifications of those constants (including all implicit factors from the circle-method major/minor arc decompositions) so that the bound on M can be independently checked.","section":"Type-II minor-arc estimates"},{"comment":"The explicit major-arc estimates and the replacement for the DMR prime exponential-sum input are asserted to suffice for the stated error terms, but without tabulated intermediate constants or a verification appendix the specific numerical threshold cannot be confirmed; any underestimation of an implicit constant would invalidate the concrete M even if the asymptotic framework is correct.","section":"major-arc estimates and prime exponential-sum replacement"}],"minor_comments":[{"comment":"Notation for C_q(m) should be defined explicitly in the statement of the lower bound rather than only in the surrounding text.","section":"lower bound statement"},{"comment":"The application to OEIS A070027 would benefit from a short explicit statement of the iterated digit-sum condition in terms of the surjectivity result.","section":"applications"}],"recommendation":"major_revision","confidential_remarks":"The absence of any verification details or intermediate constant tables in the abstract raises a legitimate concern about numerical fidelity; the manuscript should be revised to include them before the specific value of M can be accepted at face value."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed report and for highlighting the need for full verifiability of the explicit constants. We agree that the numerical threshold M requires complete, checkable derivations of all error terms. Below we respond point-by-point and commit to adding a dedicated verification appendix in the revised manuscript.","responses":[{"response":"We agree that independent verification of M requires the full set of explicit error-term derivations. The manuscript already replaces DMR's implicit prime exponential-sum bound with an explicit version and tracks constants through the Type-II estimates, but the intermediate numerical factors from the circle-method decompositions are not collected in one place. We will add a new appendix that derives every error term from first principles, lists all implicit constants with their numerical values, and recomputes the final bound on M step-by-step.","revision_made":"yes","referee_comment":"[Type-II minor-arc estimates] Abstract and § on Type-II minor-arc estimates: the numerical value M < 1.78 × 10^{32} is load-bearing on the fully explicit replacement for DMR's prime exponential-sum bound and the constant-tracked Type-II estimates; the manuscript must supply the complete error-term derivations and numerical verifications of those constants (including all implicit factors from the circle-method major/minor arc decompositions) so that the bound on M can be independently checked."},{"response":"The major-arc estimates and the explicit replacement for the DMR prime exponential sum are stated with explicit forms in the text. We acknowledge, however, that without a tabulated list of all intermediate constants the numerical threshold cannot be independently confirmed. The revision will therefore include, in the same new verification appendix, a table of every constant appearing in the major-arc analysis together with its derivation and numerical value.","revision_made":"yes","referee_comment":"[major-arc estimates and prime exponential-sum replacement] The explicit major-arc estimates and the replacement for the DMR prime exponential-sum input are asserted to suffice for the stated error terms, but without tabulated intermediate constants or a verification appendix the specific numerical threshold cannot be confirmed; any underestimation of an implicit constant would invalidate the concrete M even if the asymptotic framework is correct."}],"tokens_in":1519,"tokens_out":476,"duration_ms":24456,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key point is that this work supplies the first published explicit numerical bound: every sufficiently large m coprime to 9 appears as the decimal digit sum of at least one prime, with the threshold pinned at M below 1.78 times 10 to the 32. They achieve this by replacing the implicit inputs in the Drmota-Mauduit-Rivat setup with fully explicit versions, including the prime exponential-sum bound, major-arc estimates, and constant-tracked Type-II minor-arc estimates.\n\nWhat the paper does well is carry the bookkeeping all the way through to a concrete number and then apply it to prove infinitude for the primes in OEIS A070027, where the iterated digit-sum chain remains prime until it hits a single-digit prime. The explicit lower bound on the counting function A_m is also stated with a positive constant C_q(m) that stays away from zero in admissible classes. This is the sort of effective consequence that the original asymptotic theory points toward but rarely delivers with numbers.\n\nThe soft spot is the size of M itself. Even if every constant is tracked correctly, a threshold at 10^32 has almost no computational reach, so the result stays largely theoretical. The load-bearing step is the accuracy of those explicit replacements and error-term calculations; a modest underestimate in one decay rate could shift the final M by orders of magnitude, though the underlying asymptotic framework would remain intact. The paper does not appear to introduce circularity or fitted parameters.\n\nThis is specialized work aimed at people who care about effective versions of analytic number theory results on digital restrictions of primes. A serious referee should see it because the explicitness is the actual contribution, and the calculations are checkable in principle even if the bound is large. I would send it to peer review.","headline":"The paper turns the DMR circle-method framework into the first explicit numerical threshold M < 1.78e32 for digit-sum surjectivity on primes, with a clean application to an OEIS sequence.","tokens_in":2380,"tokens_out":449,"would_cite":false,"duration_ms":20469,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11N05","11A63"],"pacs":[],"model":"grok-4.3","headline":"Every integer m at least M < 1.78 × 10^32 and coprime to 9 is the digit sum of some prime.","keywords":["digit sum","primes","surjectivity","circle method","explicit constants","number theory","additive problems"],"falsifier":"An integer m larger than or equal to M with gcd(m,9)=1 for which no prime p has digit sum equal to m would falsify the claim, or a verification that the error terms in the asymptotic formula exceed the main term for some range.","tokens_in":2680,"feed_emoji":"🔢","tokens_out":727,"duration_ms":34380,"temperature":0.7,"pith_summary":"The paper makes explicit the constants in the Drmota-Mauduit-Rivat circle method to prove that digit sums of primes are surjective onto all sufficiently large integers coprime to 9. It produces a concrete bound M below 1.78 times 10 to the 32 after which every such m appears as the sum of decimal digits of at least one prime. This turns a known existence result into an effective statement with a numerical threshold. The work also gives an explicit lower bound on the number of primes with given digit sum and applies it to show there are infinitely many primes whose digit sum chain stays prime.","feed_headline":"Primes achieve every large digit sum coprime to 9 above explicit M","feed_subtitle":"M below 1.78e32 is shown to work so that all m >= M with gcd(m,9)=1 equal s(p) for some prime p","key_machinery":"The circle method framework with explicit versions of major-arc estimates and Type-II minor-arc estimates, replacing DMR's implicit prime exponential sums.","core_discovery":"We exhibit an explicit integer M < 1.78 × 10^{32} such that every integer m ≥ M with gcd(m,9)=1 occurs as s(p) for at least one prime p. The proof uses explicit major-arc estimates, a fully explicit replacement for the implicit prime exponential-sum input, and constant-tracked Type-II minor-arc estimates.","pith_inferences":["This approach of tracking constants explicitly could be applied to other problems in additive number theory involving primes and digital conditions.","Improving the bound on M would require sharper explicit estimates in the minor arcs or better exponential sum bounds.","The existence of such an M implies that the set of digit sums of primes has positive density in the admissible classes for large values."],"forward_implications":["Infinitude of the sequence of primes whose iterated decimal digit sums remain prime down to a single digit prime.","Explicit lower bound A_m(10^{2m/9}) ≥ C_q(m) 10^{2m/9} / m^{3/2} with C_q(m) positive above the threshold.","Effective statements for additive primes and digit sum decompositions.","Every admissible residue class modulo 9 is eventually hit by digit sums of primes above the threshold."],"fun_headline_variants":["Explicit M under 1.78e32 for all large coprime digit sums from primes","Surjectivity of prime digit sums on coprimes to 9 above explicit M bound","Every large m coprime to 9 equals s(p) for some prime p after explicit M","M explicit below 1.78e32 guarantees prime digit sums cover all eligible m"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The explicit major-arc estimates, the replacement for the prime exponential-sum input, and the Type-II minor-arc estimates are accurate enough to give the stated error terms and the numerical value of M.","fun_headline_variants_meta":{"raw":{"variants":["Explicit M under 1.78e32 for all large coprime digit sums from primes","Surjectivity of prime digit sums on coprimes to 9 above explicit M bound","Every large m coprime to 9 equals s(p) for some prime p after explicit M","M explicit below 1.78e32 guarantees prime digit sums cover all eligible m"]},"model":"grok-4.3","cost_usd":0.008422,"raw_usage":{"total_tokens":3767,"prompt_tokens":743,"num_sources_used":0,"completion_tokens":93,"cost_in_usd_ticks":84215500,"prompt_tokens_details":{"text_tokens":743,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2931,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":743,"tokens_out":93,"duration_ms":25263,"temperature":1.0,"reasoning_tokens":2931,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T04:45:57.145348+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An integer m larger than or equal to M with gcd(m,9)=1 for which no prime p has digit sum equal to m would falsify the claim, or a verification that the error terms in the asymptotic formula exceed the main term for some range.","supporting_citations":[],"review_version":1}