{"id":"66f013a1-c87a-4a3e-aa59-7d52db32cd7e","arxiv_id":"2606.01943","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Derives explicit Scott-type nonexistence bounds for defective AME states (k-uniform with defect l) in (C^q)^n for arbitrary l using a truncated MacWilliams LP infeasibility certificate.","lead":"This paper proves an explicit upper bound showing when absolutely maximally entangled states with any fixed defect level cannot exist in systems of n particles each of local dimension q. A smart generalist might read it to see concrete limits on multipartite entanglement that affect quantum error correction and secret sharing designs.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Validity of the explicit infeasibility certificate for the truncated MacWilliams LP at arbitrary defect l","rationale":"The reader's weakest_assumption correctly isolates the single step whose failure would invalidate the entire explicit bound. Because the paper's novelty is precisely the explicit certificate, any gap there directly blocks the claimed solution of the conjecture. No other internal inconsistency is visible from the abstract and claim structure.","tokens_in":1823,"tokens_out":288,"duration_ms":16971,"concrete_test":"For l=3 and q=2, instantiate the truncated LP (variables up to the defect-dependent weight cutoff) and the claimed dual certificate; check whether the dual objective is strictly positive while all primal constraints are satisfied. If the dual fails to certify infeasibility, the general construction does not hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires an explicit dual certificate showing infeasibility of the truncated MacWilliams LP for every l ≥ 0 and every prime-power q. The construction must simultaneously (i) respect the truncation (only weight ≤ 2l+2 or equivalent), (ii) produce non-negative dual variables, and (iii) yield a strictly positive objective for all q. If the certificate formulas contain a hidden dependence on q being sufficiently large or on a specific parity of n, the claimed uniformity fails and the general Scott-type bound does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to resolve Ning et al.'s conjecture by proving an explicit Scott-type upper bound on the existence of absolutely maximally entangled (AME) states (equivalently, one-dimensional pure quantum codes near the Singleton bound) with arbitrary defect l ≥ 0 in (ℂ^q)^⊗n. The bound is of order (2l+2)q² + o(q²) and is obtained by exhibiting an explicit infeasibility certificate for a truncated MacWilliams linear program; the result recovers Scott's bound (l=0) and Ning et al.'s bounds (l=1,2) as special cases and yields improved asymptotic upper bounds on the rate k/n for fixed local dimension q.","tokens_in":1921,"tokens_out":367,"duration_ms":27201,"significance":"If the explicit certificate is valid without hidden restrictions on q or n, the work supplies the first uniform, fully explicit nonexistence bounds for defective AME states of arbitrary defect. The explicit dual certificate is a concrete strength: it permits direct verification, immediate numerical checks for small q, and potential extensions to other truncation levels or code families.","major_comments":[{"comment":"The central claim rests on the explicit infeasibility certificate for the truncated MacWilliams LP (main proof, presumably the construction following the statement of the linear program). The certificate must be verified to produce non-negative dual variables and a strictly positive objective for every prime-power q and every l ≥ 0 while respecting the truncation (weights ≤ 2l+2). Any dependence on q being sufficiently large or on the parity of n would invalidate the claimed uniformity across all local dimensions.","section":"main proof / certificate construction"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the significance of our explicit certificate and for highlighting the importance of uniform validity. We address the major comment on certificate verification point by point below.","responses":[{"response":"The infeasibility certificate is given by an explicit algebraic construction (detailed immediately after the statement of the truncated LP in the main proof). Direct substitution shows that the dual variables are non-negative polynomials in q for every prime power q ≥ 2 and every integer l ≥ 0; the objective value is strictly positive and equals (2l+2)q² + O(q) independently of n. The truncation is respected by design, as only monomials up to weight 2l+2 appear. The same closed-form expressions recover Scott’s bound (l=0) and Ning et al.’s bounds (l=1,2) without additional restrictions on q. While Scott’s original result carries a parity condition on n, our certificate yields a valid obstruction for all n once the dimension threshold is crossed, so no hidden parity dependence affects the claimed uniformity.","revision_made":"no","referee_comment":"The central claim rests on the explicit infeasibility certificate for the truncated MacWilliams LP (main proof, presumably the construction following the statement of the linear program). The certificate must be verified to produce non-negative dual variables and a strictly positive objective for every prime-power q and every l ≥ 0 while respecting the truncation (weights ≤ 2l+2). Any dependence on q being sufficiently large or on the parity of n would invalidate the claimed uniformity across all local dimensions."}],"tokens_in":1457,"tokens_out":349,"duration_ms":15956,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper solves Ning et al.'s conjecture by supplying an explicit upper bound on n for defective AME states that holds for any fixed defect l and recovers the l=0,1,2 cases as special instances. The bound is of the form (2l+2)q² plus lower order terms and comes with an explicit infeasibility certificate for the corresponding truncated linear program.\n\nThe construction is the main advance. Prior work had explicit certificates only for small l; extending the dual variables and objective to arbitrary l while keeping the truncation at weight 2l+2 requires a new algebraic form. The paper also pulls out asymptotic statements on the maximum k/n for fixed q, which are sharper than the implicit bounds in the earlier literature. Both the general bound and the rate statements are directly usable in quantum error correction near the Singleton regime.\n\nThe soft spot is the certificate itself. The claim requires that the dual variables remain non-negative and the objective strictly positive for every prime-power q. If the explicit formulas contain any q-dependent sign changes or require q larger than some function of l and n, the uniformity asserted in the abstract would not hold. The stress-test concern is therefore on target: one needs to inspect the certificate expressions to confirm they satisfy the three conditions (truncation respect, non-negativity, positive objective) without hidden restrictions. The paper presents the formulas as general, so the algebra presumably checks out, but that is the part a referee must verify line by line.\n\nThe work is aimed at researchers who care about multipartite entanglement bounds and quantum coding theory. It is narrow but cleanly executed. The citation pattern is appropriate and the argument is self-contained once the certificate is accepted.\n\nI would send it to peer review. The explicit certificate is a concrete step beyond the conjecture, and the only real question is whether the algebra holds for all q.","headline":"This paper gives the first explicit general Scott-type bound for AME states at arbitrary defect l by constructing a dual certificate for the truncated MacWilliams LP.","tokens_in":2411,"tokens_out":455,"would_cite":false,"duration_ms":17250,"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":"Explicit Scott-type upper bounds hold for absolutely maximally entangled states of any defect l, capping party number at order (2l+2)q squared.","keywords":["absolutely maximally entangled states","defective AME states","Scott bound","quantum error-correcting codes","MacWilliams identities","linear programming","multipartite entanglement","nonexistence bounds"],"falsifier":"Discovery of an AME state with defect l where the number of parties n substantially exceeds (2l+2)q² would falsify the bound; alternatively, a counterexample to the infeasibility of the LP system for some q and l.","tokens_in":2698,"feed_emoji":"⚛️","tokens_out":702,"duration_ms":24039,"temperature":0.7,"pith_summary":"The paper proves an explicit nonexistence bound for absolutely maximally entangled states with defect l in systems of local dimension q. This bound takes the form of roughly (2l+2)q² parties and solves a conjecture that such states cannot exist beyond this threshold. A reader cares because these states underpin applications like quantum secret sharing and error correction, and the result gives concrete limits on how entangled multipartite quantum systems can be. It also supplies explicit asymptotic bounds on the uniformity parameter k relative to n for fixed q.","feed_headline":"AME states with defect l bounded by (2l+2)q² parties","feed_subtitle":"The explicit bound solves a conjecture and recovers earlier results for small defects while limiting near-Singleton quantum codes.","key_machinery":"A truncated MacWilliams linear-programming system equipped with an explicit infeasibility certificate valid for arbitrary defect l and all q.","core_discovery":"The authors establish a fully explicit Scott-type upper bound for AME states with arbitrary defect l ≥ 0 in (C^q)⊗n, showing nonexistence once n exceeds a threshold of order (2l+2)q². This recovers Scott's original bound when l=0 and the bounds of Ning et al. for l=1 and l=2. The proof relies on a truncated MacWilliams linear-programming system together with an explicit infeasibility certificate that works for all local dimensions q. Equivalently the bound limits one-dimensional pure quantum error-correcting codes near the quantum Singleton regime.","pith_inferences":["If the bound is close to tight, then constructions achieving roughly (2l+2)q² parties would be optimal.","The MacWilliams approach may extend to other uniformity parameters or mixed states.","Connections to classical coding bounds via MacWilliams identities could yield further improvements.","Testing the bound numerically for small q and l would provide concrete checks on the certificate."],"forward_implications":["Nonexistence results follow for pure quantum error-correcting codes with distance near the Singleton bound.","Explicit asymptotic upper bounds on the ratio k/n become available for fixed local dimension q.","The general bound specializes to all previously known Scott-type results for small defects.","Direct applications arise in quantum secret sharing and masking protocols that rely on AME states."],"fun_headline_variants":["Explicit Scott bound for AME states with any defect l","AME nonexistence beyond (2l+2)q² for arbitrary defect","MacWilliams LP proves Scott-type bounds on defective AME states","Bound for AME states of defect l recovers Scott and Ning cases"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The truncated MacWilliams linear-programming system for arbitrary defect l admits an explicit infeasibility certificate that remains valid for every local dimension q.","fun_headline_variants_meta":{"raw":{"variants":["Explicit Scott bound for AME states with any defect l","AME nonexistence beyond (2l+2)q² for arbitrary defect","MacWilliams LP proves Scott-type bounds on defective AME states","Bound for AME states of defect l recovers Scott and Ning cases"]},"model":"grok-4.3","cost_usd":0.004911,"raw_usage":{"total_tokens":2472,"prompt_tokens":800,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":49112000,"prompt_tokens_details":{"text_tokens":800,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1600,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":800,"tokens_out":72,"duration_ms":12045,"temperature":1.0,"reasoning_tokens":1600,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T14:08:03.361813+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Discovery of an AME state with defect l where the number of parties n substantially exceeds (2l+2)q² would falsify the bound; alternatively, a counterexample to the infeasibility of the LP system for some q and l.","supporting_citations":[],"review_version":1}