{"id":"7274a5e2-c319-461e-af87-cfcefc58ce88","arxiv_id":"2601.02009","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"AMCCs are introduced as the correlation-space analogue of absolutely maximally entangled states, with PR boxes, GHZ correlations, and parity/CSP-built families as examples.","lead":"This paper defines a new class of multipartite nonlocal correlations called absolutely maximally contextual correlations (AMCCs), which are simultaneously maximally contextual and have maximally random marginals. It constructs examples using parity-check and constraint-satisfaction methods and sketches applications to secret sharing and randomness extraction.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Table II CF=1 claim is false: with seven interior parameters, only one parity constraint remains and a global section exists, so CF<1.","rationale":"The definition of AMCC and the PR-box/GHZ examples are not in question. The parity-check construction for (3,2,2) may still produce finitely many AMCCs. But the paper's explicit claim of an infinite family is central to its abstract and conclusion, and it is the least secure part: it is unsupported by proof or code and is demonstrably false as stated. The reader's conditional verdict is appropriate; no additional adjustment is needed from a stress-test perspective.","tokens_in":23676,"tokens_out":12535,"duration_ms":126280,"concrete_test":"Set p1=0.25, p2=...=p8=0.1 in Table II. (a) Verify that the global section with all six measurement values 0 lies in the support of every context. (b) Run the contextual-fraction linear program from [37] for this vector. The returned CF will be strictly less than 1 (the LP can be made to output the noncontextual fraction by calculating the max λ such that subtraction of the delta-model is nonnegative), contradicting the paper's 'CF remains 1' assertion. This single check settles whether the infinite-family claim holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that Table II provides an eight-parameter (hence infinite) family of AMCCs rests on Section III.A's assertion that if one parameter is 0.25, the other seven can take any values in [0,0.125] with CF=1. This is false. In Table II, a parameter strictly between 0 and 1/4 gives a context with full support, adding no possibilistic constraint. Take p1=0.25 and p2=...=p8=0.1. The only support restriction is the parity-even constraint for context (0,0,0). The global assignment (X1=X1'=X2=X2'=X3=X3'=0) satisfies this and is compatible with all full-support contexts, so Se is nonempty. By the paper's own equivalence (strong contextuality ⇔ CF=1, [37]), the model is not maximally contextual; moreover, the deterministic model concentrated on this global section can be subtracted with small weight, giving a positive noncontextual fraction and CF<1. Thus 'any values' is false. The continuous/infinite-family conclusion drawn from the eight free parameters is unsupported, even though specific parameter choices (e.g., other seven set to 0) may indeed give CF=1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a correlation-space analogue of absolutely maximally entangled (AME) states, called absolutely maximally contextual correlations (AMCC). An AMCC is defined as a no-signaling empirical model that is maximally contextual (CF=1) and has all marginals over fewer than n parties uniformly random (Defs. 7–8). The PR box is identified as the bipartite AMCC; GHZ correlations and the three-way nonlocal correlations of Ref. [61] are proposed as tripartite examples. The paper then presents a parity-check construction and a CSP-based construction for AMCCs and non-AMCCs, claims an eight-parameter family (hence infinitely many AMCCs) via Table II, and sketches applications to secret sharing and randomness extraction.","tokens_in":23924,"tokens_out":16118,"duration_ms":160671,"significance":"If the technical claims were correct, the paper would provide a useful conceptual bridge between entanglement theory and the resource theory of contextuality, with potential applications to device-independent randomness and secret sharing. The core definition is natural and builds directly on the established contextual-fraction framework; the parity-check construction gives a concrete finite family of strongly contextual, uniformly-marginal models, and Eq. (22) is an explicit valid AMCC. However, the claimed infinite family rests on a parameter-range assertion that is demonstrably false, and the GHZ example as written is not a valid probability distribution. These are load-bearing issues for the paper's central claims.","major_comments":[{"comment":"The statement that 'if one parameter is set to 0.25, the remaining seven can take any values in [0,0.125], and the CF remains 1' is false. Take p1=0.25 and p2=...=p8=0.1. In Table II each row with parameter p_i has support equal to an even-parity set if p_i=0.25, an odd-parity set if p_i=0, and all eight sections if 0<p_i<0.25. With the chosen values, only the (0,0,0) context imposes a parity constraint; contexts (0,0,1) through (1,1,1) have full support. The global assignment X1=X1'=X2=X2'=X3=X3'=0 satisfies the remaining constraint, so S_e is nonempty. By the paper's own equivalence strong contextuality ⇔ CF=1, CF<1. Hence the claimed eight-parameter/infinite AMCC family is unsupported. The reported CF values are also given without proof or reproducible code.","section":"Section III.A, Table II"},{"comment":"The GHZ correlation is not a valid probability distribution as written and does not demonstrate CF=1. If the second branch is read as assigning probability 1/8 uniformly to every outcome in the four listed contexts, those contexts impose no possibilistic constraint; the remaining four parity equations then admit a global solution, e.g. X1=1, X1'=1, X2=0, X2'=0, X3=0, X3'=1, so the support has a global section and CF<1. If the branch is instead read as adding 1/8 to the parity-satisfying outcomes in those contexts, the contexts do not normalize (total 3/2). The standard GHZ correlation, with parity constraints in all eight contexts, should be written explicitly to support the claimed AMCC status.","section":"Eq. (20), Section III.A"}],"minor_comments":[{"comment":"The explanatory argument that combining the first and second parity equations 'yields X3 = X3′' is incorrect: adding the two equations gives X3 ⊕ X3′ = 1, i.e. X3 ≠ X3′, which is not a contradiction. The contradiction only appears after summing all eight equations, as in Eq. (35).","section":"Section IV.B, paragraph before Eq. (35)"},{"comment":"The numerical claims — 'exactly 240' AMCCs among 256 parity assignments and '2,401' CSP Boolean formulas — are stated without proof, code, or a reproducible enumeration procedure. These counts should be backed by a derivation or an explicit computational artifact.","section":"Section IV.C"},{"comment":"The randomness-extraction section contains undefined notation (Hmin(xy|XY)) and Eq. (45) does not follow from Eq. (44). The correct statement is about k-party output min-entropy given inputs; the 'global min-entropy' bound is not established.","section":"Section V.B"},{"comment":"The secret-sharing protocol is only a sketch. No adversary model, correctness condition, or security proof is given; the statement that the secret is recoverable only when all players cooperate is not derived from the AMCC properties.","section":"Section V.A"},{"comment":"There are several typos and presentation issues, including 'Secrete Sharing', 'Popescu-Rohrich', inconsistent notation for measurement settings, and the ambiguous sentence in Section III.A: 'if two parameters are fixed to any values (including zero) within the allowed range, the CF is always 1.' This needs rewording and proof.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The core definition and the parity-check construction may be salvageable, but the paper currently makes two load-bearing claims that are demonstrably problematic: the eight-parameter CF=1 family in Table II and the GHZ distribution in Eq. (20). I would ask the authors to provide explicit formulas or machine-checked code for all CF=1 parameter ranges, to correct the GHZ example, and to reassess the 'infinite family' claim in the abstract and conclusion. The numerical counts in Section IV.C also need to be reproducible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the AMCC definition—maximally contextual plus uniform marginals—is a clean way to name an intersection that the literature has not explicitly labelled. The parity-check and CSP constructions are real, and the paper connects them properly to the existing FNS/AVN line of work. PR boxes as bipartite AMCCs is correct, and the CSP enumeration of asymmetric non-AMCC strongly contextual tables is a useful addition.\n\nThe main problem is Table II. The claim that with one parameter set to 0.25 the other seven can be any values in [0,0.125] with CF=1 is false. Take p1=0.25, p2=...=p8=0.1. The first context then enforces even parity, but all other contexts have full support, so the all-zero global assignment satisfies every context; Se is nonempty and CF<1. The stress-test note is right. The paper provides no code or proof for its CF ranges, so the 'infinite family' conclusion is unsupported. The parity-check construction is on much firmer ground: if the parity equations are unsatisfiable, strong contextuality follows, and the symmetric zero pattern gives uniform marginals. That part alone generates genuine AMCCs, including the 240 tripartite cases mentioned.\n\nAlso worth flagging: the GHZ correlation in Eq. (20) has 1/4 and 1/8 entries and a parity function with quadratic terms. For the stated state |000>+|111> with X/Y measurements, the standard correlation gives 1/2 on two outcomes per context, so the formula looks wrong. The applications in Section V are sketches: they are direct consequences of maximal marginals and lack any formal security statement or adversary model.\n\nThe definition and the parity/CSP constructions deserve to be in the literature, but the Table II parameter claims and the GHZ example need to be corrected first. I would send this to peer review rather than desk-reject, but with the expectation of a substantial revision.","headline":"AMCC is a reasonable definition and the parity/CSP constructions are genuinely useful, but the claimed infinite family in Table II is wrong as stated and the GHZ example is garbled.","tokens_in":24456,"tokens_out":3794,"would_cite":false,"duration_ms":42448,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper defines absolutely maximally contextual correlations (AMCCs)—correlations that are maximally contextual and have uniformly random reduced marginals—and constructs infinite families of them, with PR boxes and GHZ correlations as t","keywords":["AMCC","contextual fraction","maximal marginals","PR box","GHZ correlations","parity-check construction","constraint satisfaction problem","randomness extraction"],"falsifier":"Evaluate the contextual fraction for the eight-parameter tripartite table at p1=0.25 and p2=p3=...=p8=0.125. In that case every context except the first has full support, and the global assignment with all six observables equal to 0 satisfies the parity constraint of the first context, so the support admits a global section; a linear-programming calculation would then give CF<1, contradicting the paper's claim that CF remains 1 throughout this region.","tokens_in":23507,"feed_emoji":"🎲","tokens_out":9618,"duration_ms":88913,"temperature":0.7,"pith_summary":"The paper introduces absolutely maximally contextual correlations (AMCCs) as the correlation-side analogue of absolutely maximally entangled states: an (n,m,o) correlation is an AMCC if it is maximally contextual (its contextual fraction is 1) and every marginal over fewer than n parties is uniform. The bipartite PR box and suitably measured tripartite GHZ correlations are shown to be AMCCs, and the paper constructs infinite families using parity-check equations and constraint-satisfaction problems. It also constructs maximally contextual correlations that are not AMCCs, showing that maximal contextuality and maximal marginality are logically independent properties. If these constructions hold, AMCCs give a resource-theoretic counterpart to AME states, with direct applications to device-independent secret sharing and randomness extraction, since every k-party marginal is perfectly uniform.","feed_headline":"Meet the first AMCCs: PR box and GHZ correlations","feed_subtitle":"Maximally nonclassical and maximally random correlations, with infinite families from parity checks and CSPs.","key_machinery":"The central machinery is the contextual fraction CF, obtained by decomposing any empirical model as a convex mixture of a noncontextual part and a maximally contextual part; CF=1 means the model is strongly contextual, i.e. no global assignment is compatible with the support. Coupled with this is the maximal-marginal condition that every reduced distribution over fewer than n parties be uniform. For constructions, the parity-check method converts a set of linear equations over GF(2) into Boolean formulas whose joint unsatisfiability certifies CF=1, while the CSP method encodes supports directly as Boolean constraints; both yield probability tables by imposing zero constraints on the no-signa","core_discovery":"On the paper's own terms, the discovery is a definition plus a set of construction theorems: an (n,m,o) correlation is an AMCC precisely when its contextual fraction is 1 (so it is strongly contextual, with no global assignment compatible with its support) and all k-party marginals for k<n are uniform. The paper shows that in the (2,2,2) scenario exactly the eight PR boxes are AMCCs; that GHZ correlations measured in the X/Y bases are (3,2,2) AMCCs; that a parity-check method over GF(2) generates an infinite family of symmetric AMCCs; and that a CSP/SAT method generates both symmetric and asymmetric AMCCs as well as maximally contextual non-AMCCs. It then applies the uniform-marginals proper","pith_inferences":["If the stated CF=1 region for the eight-parameter table is verified, AMCCs form a continuous family on a face of the no-signaling polytope, which would allow noise-robust interpolation between PR-like correlations; this extends beyond what the paper proves.","The non-AMCC maximally contextual correlations may serve as a resource for randomness expansion where the adversary has partial information about outputs; studying their noise tolerance is a natural testable extension.","A consequence the paper leaves implicit is that the parity-check AMCC family shadows stabilizer states from quantum error correction, so AMCCs might act as the contextual analogue of quantum error-correcting codes.","Since four-party AMCCs exist even though 4-qubit AME states do not, the correlation picture of maximality is richer than the state picture; exploring five and more parties could reveal AMCCs that certify more than any quantum state can."],"forward_implications":["In any (n,2,2) scenario, a parity-check AMCC gives an all-versus-nothing proof of contextuality while every k-party marginal, k<n, is perfectly uniform, combining logical nonclassicality with maximum local randomness.","GHZ correlations with X/Y measurements are AMCCs, so the same correlations that power all-versus-nothing arguments also provide maximal marginal entropy in the tripartite setting.","Because every reduced marginal of an AMCC is uniform, the min-entropy of any k<n outputs is k bits, giving a device-independent bound on extractable randomness.","The parity-check construction yields AMCCs for any number of parties, including cases like four parties where absolutely maximally entangled quantum states are known not to exist.","Maximally contextual non-AMCCs demonstrate that CF=1 does not force uniform marginals, so the two properties are inequivalent resources."],"fun_headline_variants":["AMCC: the new max for nonlocal correlations","PR box and GHZ are first AMCCs, then infinite more","Parity-check and CSP methods build infinite AMCCs","Sheaf theory defines maximally contextual correlations","AMCCs for secret sharing and randomness extraction"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The key assumption is that certain probability tables with several non-zero entries remain maximally nonclassical—the paper states this for a whole range of parameters but supplies no proof, and a single counterexample in that range would collapse the claimed infinite family.","fun_headline_variants_meta":{"raw":{"variants":["AMCC: the new max for nonlocal correlations","PR box and GHZ are first AMCCs, then infinite more","Parity-check and CSP methods build infinite AMCCs","Sheaf theory defines maximally contextual correlations","AMCCs for secret sharing and randomness extraction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000436,"raw_usage":{"total_tokens":2109,"prompt_tokens":851,"completion_tokens":1258,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":1180}},"tokens_in":595,"tokens_out":1258,"duration_ms":14056,"temperature":1.0,"reasoning_tokens":1180,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T12:40:37.141063+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the contextual fraction for the eight-parameter tripartite table at p1=0.25 and p2=p3=...=p8=0.125. In that case every context except the first has full support, and the global assignment with all six observables equal to 0 satisfies the parity constraint of the first context, so the support admits a global section; a linear-programming calculation would then give CF<1, contradicting the paper's claim that CF remains 1 throughout this region.","supporting_citations":[],"review_version":1}