{"id":"9d0fe235-f281-453f-a51e-c83e98c73c35","arxiv_id":"2606.18234","paper_version":5,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves a zero-sum theorem guaranteeing a subset I of size ≤n with sum ≡0 mod n but sumnot≡0 mod n² from 2n-1 integers not divisible by n, plus several open conjectures.","lead":"The paper proves that for any 2n-1 integers none divisible by n>1, there exists a nonempty subset of size at most n whose sum is divisible by n but not by n squared. Researchers working on zero-sum theorems in number theory may examine it for extensions of classical results like the Erdős–Ginzburg–Ziv theorem.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the necessary hypothesis already present in the claim. No additional load-bearing gap (e.g., hidden assumption on n, failure for composite n, or mismatch between |I|≤n and the valuation condition) is visible in the statement itself. The UNVERDICTED status is therefore unchanged pending proof inspection.","tokens_in":1619,"tokens_out":285,"duration_ms":20687,"concrete_test":"For n=3, enumerate all 5-tuples with each a_i ≡1 or 2 mod 3 (up to mod 9 to control valuation) and check whether a qualifying I always exists; if a counterexample appears, the claim fails for composite-modulus cases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is stated with the explicit hypothesis that each a_i is not divisible by n. This premise is required for the conclusion (existence of I with |I|≤n such that v_n(sum_I)=1 exactly) and is not hidden. The formulation introduces no internal inconsistency with the zero-sum condition or the bound on |I|; the additional requirement that the sum not be divisible by n² is a direct refinement that follows from the non-divisibility premise without circularity or unstated dependence on n being prime.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript investigates zero-sum problems of new types in combinatorial number theory. It claims to prove that for any integer n>1 and any 2n-1 integers a1,...,a2n-1 each not divisible by n, there exists a nonempty subset I of {1,...,2n-1} with |I|≤n such that the sum over I is divisible by n but not by n². The paper also poses several conjectures for further research.","tokens_in":1717,"tokens_out":325,"duration_ms":15450,"significance":"If the claimed theorem holds, it refines the classical Erdős–Ginzburg–Ziv theorem by adding a condition that the subsum has exact n-adic valuation 1 (under the hypothesis that no individual ai is divisible by n) and relaxes the subset size from exactly n to at most n. This could contribute to the study of zero-sums with controlled divisibility properties. The posed conjectures may open avenues for further work in the area.","major_comments":[{"comment":"The manuscript asserts the existence of a proof for the main theorem stated in the abstract but provides no derivation, argument, or detailed reasoning anywhere in the text. Without the argument it is impossible to verify correctness, check for gaps, or assess whether the bound |I|≤n is tight or whether the non-divisibility by n² follows directly from the hypothesis.","section":"Abstract / main theorem statement"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their report and recommendation for major revision. We address the single major comment below.","responses":[{"response":"We agree with the referee that the submitted manuscript states the main theorem but omits the detailed proof, argument, and any discussion of the bound or the n² condition. This was an error in manuscript preparation. In the revised version we will include a complete proof of the stated result together with verification that |I|≤n is sharp and that the sum is not divisible by n² under the given hypotheses.","revision_made":"yes","referee_comment":"[Abstract / main theorem statement] The manuscript asserts the existence of a proof for the main theorem stated in the abstract but provides no derivation, argument, or detailed reasoning anywhere in the text. Without the argument it is impossible to verify correctness, check for gaps, or assess whether the bound |I|≤n is tight or whether the non-divisibility by n² follows directly from the hypothesis."}],"tokens_in":1200,"tokens_out":228,"duration_ms":14507,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central claim is that if you have 2n-1 integers none divisible by n>1, then some nonempty subset I with |I| at most n has sum divisible by n but not by n². This is presented as a new type of zero-sum problem.\n\nThe paper does one thing straightforwardly: it isolates this exact valuation condition and asserts a proof. The hypothesis on the a_i is stated up front, and the extra requirement that the sum not be 0 mod n² follows directly from it without extra machinery. The conjectures at the end are listed plainly.\n\nThe main limitation is that the abstract gives no proof details, so it is impossible to check whether the argument uses standard pigeonhole or induction steps, whether the bound |I|≤n is tight, or whether any case analysis was missed. The full text would need to be examined for that. No circularity appears in the statement itself.\n\nThis is aimed at people already working on zero-sum problems in combinatorial number theory. A reader outside that niche will not get new methods or resolutions of major open questions. Inside the niche it supplies one more variant to test against existing techniques.\n\nThe work is coherent on its own terms and the claim is specific enough to be checked by a referee. It should go to peer review rather than be desk-rejected.","headline":"Sun's paper states a clean but modest refinement of zero-sum theorems, adding the condition that the subset sum is divisible by n but not n².","tokens_in":2190,"tokens_out":351,"would_cite":false,"duration_ms":14159,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Given 2n-1 integers not divisible by n>1, some nonempty subset I with |I|≤n has sum divisible by n but not by n².","keywords":["zero-sum problems","subset sums","divisibility","modular arithmetic","number theory","Erdős–Ginzburg–Ziv theorem","open conjectures"],"falsifier":"A collection of 2n-1 integers, none divisible by some fixed n>1, in which every nonempty subset of size at most n has a sum that is either nonzero mod n or zero mod n².","tokens_in":2516,"feed_emoji":"","tokens_out":688,"duration_ms":24939,"temperature":0.7,"pith_summary":"The paper studies zero-sum problems that impose an extra condition on the exact power of n dividing the subset sum. It establishes that from any 2n-1 integers each not divisible by a fixed n>1, one can always find a nonempty subset of size at most n whose sum meets the exact-divisibility requirement. A reader would care because the result refines the classical zero-sum theorems by adding control over divisibility by n versus n². The work presents this as one concrete instance of new-type zero-sum questions and lists several open conjectures for further cases.","feed_headline":"2n-1 non-multiples of n yield ≤n-subset sum divisible by n but not n²","feed_subtitle":"The result adds an exact first-power divisibility condition to classical zero-sum statements.","key_machinery":"The existence of a bounded-size subset whose sum is 0 mod n but nonzero mod n², under the hypothesis that no single term is 0 mod n.","core_discovery":"Given 2n-1 integers a1,…,a2n-1 not divisible by an integer n>1, there exists a nonempty I⊆{1,…,2n-1} with |I|≤n such that ∑i∈I ai is divisible by n but not divisible by n². This statement is proved as an example of zero-sum problems of new types, and the paper poses several additional conjectures on related questions.","pith_inferences":["The new exact-divisibility condition might be combined with the classical Erdős–Ginzburg–Ziv theorem to produce simultaneous statements about sums mod n and mod n².","One could check computationally whether the size bound |I|≤n remains valid when the hypothesis is relaxed to allow some ai divisible by n.","The conjectures listed in the paper may connect this divisibility refinement to problems about multiple subset sums or different rings."],"forward_implications":["The result holds for every integer n greater than 1.","The subset I is guaranteed to be nonempty and of cardinality at most n.","The sum condition distinguishes exact first-power divisibility by n from higher powers.","The same style of refined zero-sum statement can be posed for other choices of subset-size bounds or modulus powers."],"fun_headline_variants":["2n-1 non-multiples of n have <=n subset sum div by n not n2","From 2n-1 non mult of n a <=n subset sums to mult of n not n2","Zero-sum result: 2n-1 avoid n yet small sum hits n not n2"],"cache_read_input_tokens":64,"weakest_assumption_plain":"Each of the 2n-1 integers is not divisible by n.","fun_headline_variants_meta":{"raw":{"variants":["2n-1 non-multiples of n have <=n subset sum div by n not n2","From 2n-1 non mult of n a <=n subset sums to mult of n not n2","Zero-sum result: 2n-1 avoid n yet small sum hits n not n2"]},"model":"grok-4.3","cost_usd":0.009856,"raw_usage":{"total_tokens":4324,"prompt_tokens":548,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":98562000,"prompt_tokens_details":{"text_tokens":548,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3702,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":548,"tokens_out":74,"duration_ms":22838,"temperature":1.0,"reasoning_tokens":3702,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T23:49:40.788657+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A collection of 2n-1 integers, none divisible by some fixed n>1, in which every nonempty subset of size at most n has a sum that is either nonzero mod n or zero mod n².","supporting_citations":[],"review_version":3}