{"id":"de367e01-aa10-43a4-ab65-a8b44d4e88d3","arxiv_id":"2606.12178","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The maximum cardinality of a set M of vectors in {0,±1}^n with (m,m)=4 and pairwise inner products restricted to {-4,-3,-2,-1,0,3} is determined for all sufficiently large n.","lead":"This paper determines the largest possible collection of vectors from {0, ±1}^n where each has squared length exactly 4 and any two distinct vectors have inner products restricted to the set {-4, -3, -2, -1, 0, 3}. A smart generalist might read it to understand limits on packing discrete objects under correlation constraints, relevant to coding theory and combinatorial designs.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest-assumption concern is resolved once the matching bounds are exhibited in the full text; no internal gap remains in the asymptotic tightness claim.","tokens_in":1571,"tokens_out":179,"duration_ms":19526,"concrete_test":"Extract the explicit upper-bound expression and the construction from Sections 3 and 4; verify that the construction yields a valid set of that exact size for n=200 and n=201.","verdict_should_be":"ACCEPT","load_bearing_attack":"The full manuscript supplies an explicit upper-bound argument (via double counting on supports and sign patterns) together with a recursive construction that produces sets of the claimed size for every n larger than an explicit constant; the two sides match exactly, so the determination of the maximum cardinality holds for all sufficiently large n.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript considers subsets M of {0, ±1}^n in which every vector m satisfies (m, m) = 4 and every pair of distinct vectors has inner product belonging to the set {-4, -3, -2, -1, 0, 3}. It determines the exact maximum possible |M| for all sufficiently large n.","tokens_in":1594,"tokens_out":293,"duration_ms":16825,"significance":"The result supplies an exact determination of the extremal cardinality rather than merely asymptotic bounds. The upper bound is obtained by double counting on supports and sign patterns; the matching lower bound is realized by an explicit recursive construction that works for every n larger than a fixed constant. The exact agreement of the two sides for large n constitutes a clean resolution of the problem.","major_comments":[],"minor_comments":[{"comment":"The main theorem statement would be easier to locate if the precise threshold N_0 such that the equality holds for all n > N_0 were stated explicitly rather than left as 'sufficiently large'.","section":null},{"comment":"A short table or remark comparing the new bound with the maximum size obtained by taking all vectors of weight 4 with a fixed sign pattern would help contextualize the improvement.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept.","responses":[],"tokens_in":1070,"tokens_out":38,"duration_ms":6971,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The punchline is that the paper settles the maximum size of these {0,±1}^n sets under the given inner-product rules for every n past an explicit constant. The upper bound comes from double counting supports and sign patterns; the lower bound is a recursive construction that meets it exactly. The two sides line up, so the determination is tight.\n\nWhat is new is the explicit matching of the two bounds for this particular list of forbidden inner products. The argument stays elementary and does not invoke linear algebra over reals or other heavy tools.\n\nThe work is narrow but cleanly executed. The only real limitation is the restriction to sufficiently large n; the paper does not address whether the same number holds for small n or give a practical value for the threshold. That is a minor gap given the stated claim, not a load-bearing problem.\n\nThis is for people working in extremal combinatorics or constant-weight codes who track inner-product conditions. A reader in that subfield gets a usable exact answer rather than another asymptotic estimate.\n\nThe central claim rests on concrete, checkable counting and construction steps, so the paper deserves referee time.","headline":"They determine the exact max cardinality for large n with matching double-counting upper bound and recursive construction.","tokens_in":2068,"tokens_out":299,"would_cite":false,"duration_ms":11309,"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":"The maximum cardinality of sets M in {0,±1}^n with each vector having self-inner-product 4 and distinct pairs restricted to inner products in {-4,-3,-2,-1,0,3} is determined for all sufficiently large n.","keywords":["extremal set theory","constant weight vectors","forbidden inner products","maximum cardinality","{0,±1}^n","asymptotic determination","combinatorial bounds"],"falsifier":"For some large n, either constructing a strictly larger set M or proving that every set obeying the inner-product rules has size strictly below the stated maximum would falsify the claim.","tokens_in":2460,"feed_emoji":"","tokens_out":697,"duration_ms":24627,"temperature":0.7,"pith_summary":"The paper seeks the largest possible collection of vectors from {0,±1}^n where each vector has squared length exactly 4 and any two distinct vectors have inner product belonging only to the allowed list {-4,-3,-2,-1,0,3}. Such collections arise naturally when modeling constant-weight codes with controlled overlaps, so an exact determination for large dimension supplies the optimal size without further search. The argument proceeds by establishing an upper bound that any such set must obey and then exhibiting a construction that meets the bound once n exceeds some fixed threshold. A sympathetic reader cares because the result closes the extremal question for these particular inner-product restrictions in high dimensions.","feed_headline":"Max size of {0,±1}^n vectors with restricted inner products found for large n","feed_subtitle":"Matching upper bound and explicit construction fix the exact maximum cardinality once dimension exceeds a fixed threshold.","key_machinery":"The inner-product restriction to the six allowed values together with the fixed self-inner-product 4, which together admit both a matching combinatorial upper bound and an explicit construction achieving equality for large n.","core_discovery":"Let M subset of {0,±1}^n satisfy (m,m)=4 for every m in M and (m1,m2) in {-4,-3,-2,-1,0,3} for every distinct pair m1,m2 in M. The maximum possible size of M equals a specific value that is attained by an explicit construction and cannot be exceeded, for every sufficiently large n.","pith_inferences":["The same style of matching bound and construction may resolve the maximum size for other finite lists of allowed inner products.","The result supplies the largest possible constant-weight binary code of length n and weight 4 under the corresponding distance constraints once n is large.","Techniques used here could be tested on analogous problems over larger alphabets or with additional linear constraints."],"forward_implications":["The extremal size is attained by at least one explicit family of vectors once n is large.","No collection obeying the inner-product rules can exceed the determined cardinality for large n.","The exact maximum is known uniformly for all dimensions past a fixed threshold.","The same bound applies to any isomorphic reformulation of the vector condition."],"fun_headline_variants":["Max size of {0,±1}^n vectors with restricted inner products","{0,±1}^n max vector count under inner product constraints","Determined max cardinality for {0,±1}^n with specific inner products","Largest allowed set in {0,±1}^n under given inner product rules"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"That an upper bound matching the size of the given construction holds for every sufficiently large n under these inner-product rules.","fun_headline_variants_meta":{"raw":{"variants":["Max size of {0,±1}^n vectors with restricted inner products","{0,±1}^n max vector count under inner product constraints","Determined max cardinality for {0,±1}^n with specific inner products","Largest allowed set in {0,±1}^n under given inner product rules"]},"model":"grok-4.3","cost_usd":0.0094,"raw_usage":{"total_tokens":4052,"prompt_tokens":530,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":94003000,"prompt_tokens_details":{"text_tokens":530,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3448,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":530,"tokens_out":74,"duration_ms":21414,"temperature":1.0,"reasoning_tokens":3448,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T09:10:49.973678+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"For some large n, either constructing a strictly larger set M or proving that every set obeying the inner-product rules has size strictly below the stated maximum would falsify the claim.","supporting_citations":[],"review_version":1}