{"id":"1cdb3507-a39f-4bcb-9668-8ceb2c78523d","arxiv_id":"2607.15510","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"The set of \"large\" zeros of integer linear recurrence sequences has null density, yielding a density-one Universal Skolem Set, and large zeros would be impossible under a strong Cramér conjecture.","lead":"This paper studies the Skolem Problem—whether a number sequence defined by a linear recurrence ever hits zero—and proposes that \"large\" zeros, if they exist, are extremely rare. It proves unconditionally that such special zeros are sparse enough to yield a density-one Universal Skolem Set, and conditionally (on a strengthened Cramér conjecture) that they do not exist at all, which would make the Skolem Problem decidable.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Smallness is not implied by large zeros: the 3/2 vs 3/4 arithmetic error in Thm 4.3 is not cosmetic, and at the boundary of the large-zero definition H exceeds the smallness threshold, so the P-bad argument in Thms 4.3 and 5.1 collapses.","rationale":"The reader's REJECT is justified. The paper advertises two main results: conditionally on Conjecture 4.2, decidability of the Skolem Problem, and unconditionally, a density-one Universal Skolem Set. Both rely on the smallness of any LRS that has a large zero: smallness is what links a non-zero norm expression to the definition of a bad prime. The reader's weakest_assumption identifies exactly this step. I agree, and add that the smallness claim is not merely unproved but false at the boundary of the large-zero definition. A minimal large zero has log log X ~ 10 H log H, making the smallness threshold X/2 approximately H * log H / (log H + log log H + log 10), which is less than H. Thus the arithmetic error (3/2 instead of 3/4) is symptomatic of a deeper problem, not a typo. Thm 5.1 simply asserts the same smallness via 'same reasoning', so its unconditional result is unsupported. In good faith, the framework of good primes and the counting via Amoroso-Viada may be salvageable with a different notion of smallness or a different way to handle non-small LRS, but as written the proofs do not establish the advertised claims. Therefore I do not change the reader's verdict.","tokens_in":11310,"tokens_out":22119,"duration_ms":187551,"concrete_test":"Analytic/numerical check of the smallness implication. Fix H=100, define n = ceil(exp(exp(10 H log H))), and choose the power of two X with X/2 <= n < X. Compute T = log log (X/2) / (10 log log log (X/2)). If T < H (as it will be), then a minimal large zero violates the smallness inequality (7), contradicting the proof's claim in Thm 4.3. Independently, redo the derivation in the proof of Thm 4.3 replacing the erroneous 3/2 by 3/4: show that exp exp(10 H log H) >= exp((log Y)^{3/4}) < Y for large Y, so the asserted contradiction with n < X disappears. If either computation confirms, the load-bearing smallness step is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central conditional theorem (Thm 4.3) hinges on proving that any large zero forces u to be 'small' at level Y=X/2, i.e. H < log log Y / (10 log log log Y). This is the step that lets the proof say a non-zero expression (8) would make P bad, and hence must vanish. The proof attempts this by contradiction, but the algebra is wrong: the negation of (7) yields 10 H log H >= (3/4) log log Y, not (3/2) log log Y. With the correct factor, exp exp(10 H log H) >= exp((log Y)^{3/4}), which for large Y is far smaller than Y (log Y)^{1/2}; no contradiction with n < X follows. This is not a harmless constant slip. At the boundary of the definition, the claimed implication is false. If n is only slightly above the large-zero bound, say n ~ exp exp(10 H log H), then X ~ n, so log log X ~ 10 H log H and log log log X ~ log(10 H log H). Then the smallness threshold at level X/2 is approximately H * log H / (log H + log log H + log 10), which is strictly smaller than H. Thus a minimal large zero is not small; H exceeds the threshold. Consequently, in Thm 4.3 the step 'expression (8) cannot be non-zero, otherwise P would be a bad prime' is invalid: bad primes are defined only for LRS that are small at the relevant level. The contradiction with Conjecture 4.2 therefore does not go through. Thm 5.1 invokes 'the same reasoning and notation as in the proof of Thm 4.3' without supplying a proof of smallness; for the same reason its reduction to bad primes fails, and the null-density conclusion is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a new approach to the Skolem Problem for integer linear recurrence sequences. It introduces the notion of a 'large zero' (a zero n with n ≥ exp exp(10H log H), where H is the height of the LRS) and defines 'good' and 'bad' primes in terms of divisibility properties of algebraic norms attached to LRS that are 'small' at a given dyadic level. The authors prove that bad primes have null density among all primes. Assuming a Cramér-Granville-type conjecture for good primes (Conjecture 4.2), Theorem 4.3 claims that large zeros do not exist for sufficiently high LRS, which would imply decidability of the Skolem Problem. Theorem 5.1 claims unconditionally that the set of all possible large zeros has null density, yielding a Universal Skolem Set of density one.","tokens_in":11810,"tokens_out":21034,"duration_ms":187130,"significance":"If correct, the paper would be a major contribution: a conditional resolution of the Skolem Problem under a plausible strengthening of the Cramér conjecture, and the first unconditional Universal Skolem Set of density one. The counting framework, the use of Amoroso-Viada bounds and Jia's theorem, and the general strategy are coherent and potentially reusable. However, both main theorems currently rest on a smallness assertion whose proof is invalid. The central claims are therefore not established in the submitted form.","major_comments":[{"comment":"The algebraic manipulation is incorrect. From the negation of (7) one has H ≥ log log Y/(10 log log log Y) and log H ≥ 3/4 log log log Y, so 10H log H ≥ 3/4 log log Y, not 3/2 log log Y as printed. Consequently the lower bound obtained is exp exp(10H log H) ≥ exp((log Y)^{3/4}), which is far smaller than Y(log Y)^{1/2} and does not contradict n < 2Y. At the boundary H = log log Y/(10 log log log Y), a large zero with n ≈ Y is compatible with (6). Thus the proof does not establish that a large zero is small at level Y = X/2. Since this smallness is what makes expression (8) fall under the definition of a bad prime, the subsequent contradiction with Conjecture 4.2 collapses.","section":"Section 4, proof of Theorem 4.3, inequality (7)"},{"comment":"The step 'using the same notation and reasoning as in Thm. 4.3' is insufficient for the conclusion 'If the above expression is non-zero, then P is a bad prime.' Definition 3.1 requires the LRS u to be small at the relevant level X. No proof is given that a large zero n (with n ≥ exp exp(10H log H)) implies this smallness; as in Theorem 4.3, H can sit at the smallness threshold while satisfying the large-zero bound. Hence expression (9) non-zero does not imply P ∈ P_bad, and the bound #P_bad(X) = O(X^{2/3}) cannot be applied. The null-density claim and Corollary 5.2 are therefore unsupported. The gap is repairable in principle by strengthening the constant in (2) (any c > 10 suffices), but the current text does not contain the needed argument.","section":"Section 5, proof of Theorem 5.1"}],"minor_comments":[{"comment":"The constant 10 is explicitly chosen ad hoc; given the arithmetic error in Thm 4.3, it is too small for the claimed implication. Please either prove smallness for c = 10 or change the definition and all dependent statements accordingly.","section":"Footnote 9 and Eq. (2)"},{"comment":"The chain leading to n < exp exp(9H log H) relies on the inequality n < (n^{1/19}/(log n)^2)^{20} for sufficiently large n. This should be stated and justified explicitly, since it is not immediate.","section":"Section 5, proof of Theorem 5.1"},{"comment":"The claim that L is recursive should be justified: for a given n, the condition n ≥ exp exp(10H log H) bounds H, so only finitely many LRS need be checked.","section":"Corollary 5.2"},{"comment":"Typo: 'succeded' should be 'succeeded'.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The central ideas are promising and the counting framework is largely reusable. If the authors revise the constant in (2) and supply the missing smallness argument, the paper may become publishable. As it stands, both main theorems have invalid proofs, so I cannot recommend acceptance in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper deserves a serious referee, but not because the proofs are correct. The notion of \"large zeros\" and the good-prime/bad-prime machinery are genuinely new, and the unconditional density-one Universal Skolem Set would be an important result if it held. The problem is that the key estimates that make the machinery move are not actually derived.\n\nThe proof of Theorem 4.3 tries to show that a large zero forces the LRS to be \"small\" at level X/2. The algebra there is wrong. From the negation of (7) you get 10 H log H ≥ (3/4) log log Y, not (3/2) log log Y. With the correct factor the conclusion is exp exp(10 H log H) ≥ exp((log Y)^{3/4}), which is nowhere near Y (log Y)^{1/2}. The claimed contradiction with n < X evaporates. This is not a constant slip: at the boundary of the large-zero definition, H is actually larger than the smallness threshold, so the bad-prime argument doesn't have the smallness it needs. The stress-test note has the boundary computation right.\n\nTheorem 5.1 inherits the same problem. It says \"same reasoning as in Thm 4.3\" and then asserts that a P in the interval makes (9) a bad-prime norm, but bad primes are defined only for LRS that are small at the relevant level. The paper never proves that a large zero is small; in fact the large-zero bound alone gives H slightly above the threshold, not below. So the reduction to the bad-prime count is unsupported, and the null-density conclusion doesn't follow.\n\nWhat's good here is the shape of the idea. The bad-prime counting lemma (Prop. 3.2) is clean and likely usable. The use of Amoroso-Viada to count potential zeros is appropriate. And the paper is honest about the constants being chosen to make the argument work (footnote 9). If one strengthens the large-zero threshold to something like exp exp(20 H log H), the smallness gap might close, and the conditional and unconditional results might both survive. But that repair isn't in the manuscript.\n\nAs written, I would not rely on any of the two theorems. It's still the sort of work that should go to a knowledgeable referee rather than be desk-rejected, because the framework is original and the flaw is repairable. If the authors fix the smallness issue, the density-one result would settle an open question. So: send it to referees, but tell them to check the estimates from equation (7) onward carefully.","headline":"Attractive new framework for the Skolem Problem, but the central conditional theorem has a clear arithmetic mistake and the unconditional theorem rests on an unproved smallness claim; the advertised results don't follow as written.","tokens_in":12311,"tokens_out":4322,"would_cite":false,"duration_ms":42952,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B37","11N05","03D40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, under a strengthening of the classical prime-gap conjecture, the Skolem Problem is decidable, and unconditionally that the 'large' zeros of linear recurrence sequences are a density-zero set.","keywords":["Skolem Problem","linear recurrence sequences","decidability","prime gaps","good primes","Universal Skolem Set","large zeros","algebraic norms"],"falsifier":"Exhibit one non-degenerate integer LRS of height H with a zero at index n >= exp exp(10 H log H); the unconditional theorem would fall. A more targeted check: recompute the claimed contradiction in the proof of Theorem 4.3 with the corrected factor 3/4 — if the lower bound 10 H log H >= (3/2) log log Y no longer follows, the conditional proof has a gap that a single numerical counterexample to the derived inequality would expose.","tokens_in":11182,"feed_emoji":"🧮","tokens_out":7367,"duration_ms":68649,"temperature":0.7,"pith_summary":"The paper tackles the Skolem Problem — whether a given integer linear recurrence sequence has a zero term — a decidability question open for decades. It introduces the notion of a 'large' zero, one occurring at an index beyond a double exponential in the height of the recurrence, and claims two things. Conditionally on a strengthening of the classical prime-gap conjecture to a dense subset of primes called 'good' primes, large zeros cannot exist; this would give an effective bound on all zeros and hence make the problem decidable. Unconditionally, the paper proves that the set of indices that can be large zeros for some recurrence has density zero, which yields a Universal Skolem Set of density one — answering an open question. A sympathetic reader comes away with a concrete plan: verify the prime-gap hypothesis, or at least accept that the hard zeros are a negligible set.","feed_headline":"A prime-gap conjecture would make the Skolem Problem decidable","feed_subtitle":"The same argument shows, with no conjecture, that troubling zeros are a density-zero set and almost every index is decidable.","key_machinery":"The load-bearing mechanism is the good-prime/bad-prime dichotomy. A prime P is bad at scale X if, for some LRS that is 'small' at level X (height H below roughly log log X divided by 10 log log log X), some non-zero exponential-polynomial expression v_{m,σ} has norm divisible by P. Good primes are the complement; they have density one among primes. The proof of Theorem 4.3 reduces a large zero n = P+m modulo a prime ideal above a good prime P, forcing P to divide a norm; if the norm is non-zero, P is bad, and if zero, a known upper bound on the number of zeros of algebraic linear recurrences says there cannot be enough such m. The density-one unconditional result uses the same reduction toge","core_discovery":"The paper's central claim is that the Skolem Problem reduces to a question about prime gaps. It defines good primes as those that are not 'bad', where badness means that a prime divides the algebraic norm of a certain exponential-polynomial expression associated to a small LRS around a potential large zero. For any large zero of an LRS, the paper shows there must be an interval around the zero containing far fewer good primes than the prime-gap conjecture would predict; if the conjecture holds for good primes, this contradiction rules out large zeros. The unconditional theorem shows that across all LRS, the set of large-zero indices is so sparse that its complement still allows an algorithm","pith_inferences":["The arithmetic slip in the smallness derivation suggests a natural repair: if the intended factor 3/4 still yields a contradiction under a slightly larger constant in the prime-gap conjecture, the conditional result would survive the correction.","A singly exponential bound for the largest zero would be a much stronger statement; the paper itself notes that no family of LRS with singly exponential zeros is known, so the double-exponential threshold in the definition is likely a proof artifact.","The technique of filtering primes by divisibility properties of norms could be reused to obtain explicit bounds for other ineffective results in Diophantine approximation or automata theory.","If one only needs a practical termination prover, the density-one Universal Skolem Set already gives a sound-but-incomplete procedure that is correct for almost all time steps — an engineering payoff even before decidability is settled."],"forward_implications":["If the strengthened prime-gap conjecture is ever proved, the Skolem Problem is decidable: every zero of a non-degenerate LRS lies before a double-exponential bound in its height, and finitely many small LRS can be handled by an oracle.","Unconditionally, the complement of the set of large zeros is a recursive Universal Skolem Set of density one, so for every LRS and every index outside that set, whether the LRS vanishes there can be computed.","The set of large zeros has strong quantitative sparsity: its counting function is O(X/(log X)^B) for any fixed B, making large zeros negligible for any asymptotic purpose.","The connection between recurrence zeros and prime gaps suggests that decision problems in computer science can hinge on fine statistical properties of primes.","The results imply a finite, if astronomically large, search bound for the zeros of any sufficiently high LRS, conditional on the prime-gap conjecture."],"fun_headline_variants":["Prime gaps could crack the Skolem Problem","Skolem Problem: a density-zero escape hatch","Conjecture limits Skolem zeros to a null set","Skolem decidability via a prime-gap assumption","Almost every index is Skolem-decidable"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"Both theorems assume that any LRS with a large zero at n is 'small' at scale n, meaning its height H satisfies H < log log n divided by 10 log log log n; Theorem 4.3 tries to derive this from the large-zero bound but the derivation contains an arithmetic error (a factor 3/2 that should be 3/4), and Theorem 5.1 simply asserts smallness. If smallness does not hold, the bad-prime/norm argument that powers both proofs collapses.","fun_headline_variants_meta":{"raw":{"variants":["Prime gaps could crack the Skolem Problem","Skolem Problem: a density-zero escape hatch","Conjecture limits Skolem zeros to a null set","Skolem decidability via a prime-gap assumption","Almost every index is Skolem-decidable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000199,"raw_usage":{"total_tokens":1239,"prompt_tokens":808,"completion_tokens":431,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":552,"completion_tokens_details":{"reasoning_tokens":359}},"tokens_in":552,"tokens_out":431,"duration_ms":4477,"temperature":1.0,"reasoning_tokens":359,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T23:09:21.525317+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit one non-degenerate integer LRS of height H with a zero at index n >= exp exp(10 H log H); the unconditional theorem would fall. A more targeted check: recompute the claimed contradiction in the proof of Theorem 4.3 with the corrected factor 3/4 — if the lower bound 10 H log H >= (3/2) log log Y no longer follows, the conditional proof has a gap that a single numerical counterexample to the derived inequality would expose.","supporting_citations":[],"review_version":1}