{"id":"5065398c-4ed5-4fb7-80c6-541e24eafc6a","arxiv_id":"2511.06624","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Bell inequalities have a canonical correlator form that is invariant under the L2 projection of weakly signalling data onto the no-signalling affine hull, and this projection can be computed by a simple closed formula.","lead":"This paper shows that every Bell inequality for n parties with m binary-outcome settings can be rewritten in a canonical correlator form whose value is unchanged by the standard L2 projection that removes weak signalling from experimental data. It provides a closed-form formula for that projection and a simple three-step computation for general (n,m,2) scenarios, plus a weighted generalization.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Invariance holds only for the unconstrained affine-hull projection: when Π_A(f) has negative entries (§2.2), enforcing non-negativity breaks it, so the abstract's 'onto the polytope' claim and the invariant-violation measure are regime-dependent.","rationale":"Reading the paper in good faith, the mathematical core is sound: Proposition 1's kernel-membership argument checks out, Corollary 2 is a direct consequence, and the T3T2T1 decomposition is a valid closed-form projector. My concern is not with the algebra but with the bridge from the theorem to the advertised application. The claimed 'projection-invariant measure of nonlocal violation' is invariant only for the unconstrained affine projection; the moment the projection is required to be a genuine probability behavior (the QP of §2.2), invariance is lost. The manuscript's own limitation statement (§2.2) makes this concrete, so the abstract's 'onto the no-signalling polytope' is an overstatement: the polytope-constrained projection does not preserve canonical Bell values in general. This is the same load-bearing assumption the reader identified, and I agree it is the weakest link; it limits the method's scope to regimes where Π_A(f) is non-negative, which the paper asserts but does not quantify. The I_3322 canonical form in Eq. (36) is a separate, concrete error that corroborates the recipe's delicacy: it averages one-party marginals over only two of the three settings, so it is not the UMC form and not projection-invariant, though §4.3's LOSR form (38) does implement the correct averaging. Neither concern falsifies the central theorem; both are addressable. The verdict should therefore remain CONDITIONAL (UNCHANGED): accept once the abstract is corrected to 'affine hull,' Eq. (36) is replaced by the genuine UMC form, and the non-negativity regime is either bounded or empirically validated.","tokens_in":23671,"tokens_out":30379,"duration_ms":237892,"concrete_test":"Compute Π_A(f) via the closed form T3T2T1 (or direct projector) for the Table 1 dataset and for subsampled/bootstrap versions at N ≈ 10^2–10^6 trials per setting. Record (i) the minimum component of Π_A(f) and (ii) the gap between B_can(f) and the value at the QP projection onto A ∩ R^d_+ for tilted CHSH (§4.1) and I_3322 (§4.2). If negative entries and nonzero gaps occur at experimentally attainable N, the applied claim fails at that scale; if Π_A(f) stays in the simplex down to the relevant N, the concern is mitigated. As a companion algebraic control, re-derive Eq. (36) from (35) by replacing C^1_x with (1/3)Σ_{y=0}^2 C^1_x(y) and C^2_y with (1/3)Σ_{x=0}^2 C^2_y(x); term-by-term comparison with the printed u,v form settles whether (36) is the claimed UMC canonical form.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's theorem concerns Π_A, the L2 projection onto the affine hull A = aff(P_NS), not onto the polytope P_NS = A ∩ R^d_+. Corollary 2 and the §4 canonical-form recipe are exact for Π_A. The stated application, however, requires a valid probability behavior: the projected estimate must be non-negative. Section 2.2 (paragraph after Eq. (7)) concedes the affine projection 'can contain negative probabilities' and proposes, when it does, replacing (7) with a convex QP over A ∩ R^d_+. That QP is a different map, and B_can is not invariant under it. Hence in precisely the regime the paper acknowledges, the 'projection-invariant' canonical Bell value need not equal B(p) for any no-signalling p, so it does not 'isolate the violation attributable to strictly no-signalling nonlocality' as claimed in §1. The abstract states the projection is 'onto the no-signalling polytope' and that the violation is unchanged — literally false when the QP is required. Nothing in Proposition 1 or the closed formula (20) bounds the distance from f to the simplex; non-negativity of Π_A(f) is asserted for high-count experiments, not proved. Corroborating that the recipe is delicate, §4.2's Eq. (36) is not the UMC form of I_3322: under the stated 'like-for-like substitution,' the marginals of (35) become (1/3)Σ_y C^1_x(y) and (1/3)Σ_x C^2_y(x) (averaging over all three settings), whereas (36), via u^T p_xy = C^{12}_xy + C^1_x(y)/2 − C^2_y(x)/2, averages over only settings 0 and 1 — so (36) is not projection-invariant.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers the (n,m,2) Bell scenario and studies the L2 projection onto A = aff(P_NS), the affine hull of the no-signalling polytope. Its main result (Proposition 1 and Corollary 2) is that the uniformly-averaged marginal correlator (UMC) coefficient vectors lie in the kernel of the equality-constraint matrix A_eq, so all full correlators and UMC terms are invariant under the affine projection. From this it derives a closed-form projection formula via three linear maps, and proposes a 'canonical form' of any Bell expression whose value is unchanged by the affine projection. Examples include CHSH, tilted CHSH, I_3322, and a LOSR-GTNL witness. The paper also sketches an extension to weighted L2 projections. The main derivation is elementary and essentially correct, but the paper contains an inaccurate abstract-level claim about projecting onto the polytope rather than its affine hull, and a concrete error in the I_3322 canonical form.","tokens_in":24143,"tokens_out":31794,"duration_ms":260032,"significance":"If repaired, the paper would make a useful contribution: it gives a clean, elementary proof of the kernel membership of UMC terms, a closed-form and computationally sparse projector, and a clear recipe for making Bell expressions insensitive to the affine projection step. The derivation is parameter-free and no circularity is present. However, the advertised invariance is only for the unconstrained affine-hull projection; the abstract's 'onto the no-signalling polytope' wording is false when non-negativity is imposed. In addition, the I_3322 example in §4.2 is not actually the UMC canonical form. These issues do not invalidate the central theorem but do affect the paper's main applications, so the manuscript needs substantial correction before it can be accepted.","major_comments":[{"comment":"The abstract states that the L2 projection is 'onto the no-signalling polytope' and that the violation of the canonical Bell inequality is unchanged. This is not what is proved: the projection is onto the affine hull A = aff(P_NS), and P_NS = A ∩ R^d_+. Equation (7) minimises over A, not over P_NS. As the paper itself notes after Eq. (7), the affine projection can contain negative probabilities; when non-negativity is enforced by the convex QP over A ∩ R^d_+, the canonical Bell value is no longer invariant. The headline claim should therefore be restricted to the affine projection, and the practical relevance should be qualified by the closeness assumption that the empirical vector is near P_NS.","section":"Abstract and §2.2 (Eq. (7), discussion after it)"},{"comment":"Equation (36) is not the UMC canonical form of the I_3322 inequality. The 'like-for-like substitution' described in §4.2 requires replacing C^1_x and C^2_y in (35) by (1/3)Σ_y C^1_{xy} and (1/3)Σ_x C^2_{xy}. The coefficient vector of (36) does not satisfy the no-signalling condition p(a_2|x=0,y=0)=p(a_2|x=2,y=0): for a_2=0, the sum of the coefficients over a_1 is -1 for the settings block p_{00} and 0 for p_{20}. Hence (36) is not invariant under Π_A and cannot be the canonical form claimed. This is a concrete error in a worked application, and it indicates that the examples in Section 4 need to be re-derived with the explicit UMC denominators.","section":"§4.2, Eq. (36)"},{"comment":"The weighted projection section claims that the D-invariant quantity ⟨c^I_{u_I}, p⟩_D is a non-uniform average of correlators that still reduces to the standard correlator on no-signalling behaviours. For a general positive diagonal D with entries reflecting settings probabilities, this is not correct: ⟨c^I_{u_I}, p⟩_D equals C^I_{u_I} times (1/m^{n-|I|}) Σ_{x_{\\bar I}} D_{(u_I,x_{\\bar I})} (assuming D is settings-dependent), which differs from C^I_{u_I} unless a special normalisation condition is imposed. The remark after Eq. (44) assumes such a normalisation without stating it, and the claimed generalisation of the Section 4 canonical-form construction to weighted projections is therefore unsupported as written.","section":"§5, Eqs. (39)-(44)"}],"minor_comments":[{"comment":"The phrase 'onto the no-signalling polytope' appears in the abstract and in Section 1; the precise object is the affine hull A. Please use 'affine hull' consistently and state the non-negativity caveat prominently.","section":"Throughout"},{"comment":"In Appendix A, the proof of Proposition 1 writes 'for all i∈[m]' in the recap of the no-signalling rows; this should be i∈[n]. There are also typos such as 'scenarios scenarios' in Section 1 and 'a I' in the definition of x_I.","section":"§2.3 and Appendix A"},{"comment":"The argument that (20) gives the orthogonal projection is terse. It would be clearer to state explicitly that the UMC/full-correlator moments separate points on A (which follows from the inversion identity (16)), so matching all these moments with a vector in A characterises the projection.","section":"§3, proof accompanying Eq. (20)"},{"comment":"The phrase 'termwise invariant under the projection' could be misread as applying to arbitrary Bell expressions. Please clarify that termwise invariance holds for the canonical UMC/full-correlator terms after the rewriting, not for the original coefficients in (24).","section":"§4, first paragraph"}],"recommendation":"major_revision","confidential_remarks":"The central mathematical result (Proposition 1, Corollary 2, and the closed-form projector) is sound and should be publishable after revision. The main concerns are presentation-level but load-bearing: the polytope-versus-affine-hull overclaim in the abstract, the incorrect I_3322 canonical form in Eq. (36), and the unsubstantiated weighted-projection generalisation in Section 5. These are fixable without changing the core derivation, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main thing to know: the core result—that UMC coefficient vectors lie in the kernel of the equality constraint matrix, so full correlators and uniformly averaged marginals survive L2 projection onto the affine hull of the no-signalling set—is correct, and the closed-form three-map projection is a genuinely useful simplification. But the paper overstates its own scope and one worked example is wrong.\n\nThe abstract says projection onto the no-signalling polytope; the theorem is for the affine hull A, not the polytope A ∩ R_+^d. When the projection goes negative, the fix they propose is a different convex QP, and the invariance no longer holds. They acknowledge this in Section 2.2, but the abstract and introduction do not carry the caveat.\n\nThe I_3322 canonical form in Eq. (36) is not in UMC form. Replacing C^1_x(y) by its UMC means averaging over all three of y's settings; Eq. (36) averages over y=0,1 only. Same for C^2_y. That makes the expression not projection-invariant. Via u^T p_xy = C^{12}_xy + C^1_x(y)/2 − C^2_y(x)/2, summing y=0,1 leaves out y=2 in the marginals. So the headline example of the recipe is wrong. It is fixable by replacing the sums with averages over all three settings, but as written it is a genuine error, not a typo in signs.\n\nWhat is fresh: the elementary kernel proof for general (n,m,2), the explicit T3 T2 T1 decomposition, and the weighted-L2 generalization. The derivation is checkable and has no fitted parameters. They also cite the prior work (Bancal, Lin, Rosset) honestly; the novelty is the explicit general form, not the concept. The (2,2,2) invariance was known, and the general framework exists in Rosset et al., so the novelty is real but modest.\n\nWhere the paper is soft: the polytope/affine hull conflation in the abstract; the negative-probability regime; Eq. (36). None of these kills the main proposition. If the authors correct Eq. (36) and soften the abstract, the paper is a solid methods contribution for the device-independent point-estimation pipeline. The central argument holds; it is not circular, and the math is largely reproducible.\n\nThis is for experimentalists and DI protocol people who want a cheap closed-form denoiser before evaluating Bell values. It deserves a serious referee—send it out. The referee should verify Eq. (36) carefully.","headline":"Main theorem is right and the projection formula is useful, but the abstract overstates the result and the I_3322 canonical form in Eq. (36) is wrong as written.","tokens_in":24601,"tokens_out":3942,"would_cite":true,"duration_ms":34723,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Every Bell inequality in the (n,m,2) scenario has a canonical correlator form left invariant by the no-signalling L2 projection.","keywords":["Bell inequalities","no-signalling","L2 projection","correlators","finite statistics","weak signalling","canonical form","device-independent"],"falsifier":"Take a real or simulated Bell dataset with large spurious signalling (e.g., 40% differences between marginals) and compare the canonical Bell value after the affine L2 projection against the value after a constrained projection onto the no-signalling polytope: if they differ, the practical invariance does not extend to the constrained case. More directly, for any random empirical distribution f, compute the canonical Bell expression before and after the affine projection using the paper's closed formula; a discrepancy beyond numerical precision would disprove the central invariance claim.","tokens_in":23608,"feed_emoji":"⚛️","tokens_out":4625,"duration_ms":42123,"temperature":0.7,"pith_summary":"The paper aims to establish that the standard preprocessing step of projecting experimental Bell data onto the no-signalling set does not have to distort the Bell violation, provided the inequality is written in a canonical correlator form. The central claim is that in any n-party, m-setting, binary-outcome Bell experiment, the least-squares projection onto the no-signalling affine hull preserves all full correlators and all uniformly-averaged marginal correlators. Consequently every Bell inequality can be rewritten so that its value is exactly the same before and after the projection. This makes experimental Bell values a cleaner, signalling-robust measure of nonlocal violation, and it yields a simple three-map closed formula for the projection that avoids expensive dense linear algebra. A weighted extension adapts the result to experiments with non-uniform measurement settings.","feed_headline":"No-signalling projection preserves the key Bell violation","feed_subtitle":"A closed-form map cleans Bell data without changing a canonical inequality's value.","key_machinery":"The central object is the uniformly-averaged marginal correlator (UMC): for a subset of k parties, the average over all settings choices of the remaining n−k parties of the k-party parity expectation. Its coefficient vector is shown to be orthogonal to every equality constraint defining the no-signalling affine hull, and that orthogonality is the engine of the projection invariance. The closed formula for the projection is the composition of three sparse linear maps: conditional probabilities → parity correlators → uniformly averaged correlators → probabilities.","core_discovery":"The paper proves that in the (n,m,2) Bell scenario, the coefficient vectors of all full correlators and all uniformly-averaged marginal correlators lie in the kernel of the matrix encoding the no-signalling and normalisation constraints. Therefore the L2 projection onto the affine hull of the no-signalling polytope leaves every such correlator's value unchanged. Since these correlators span the space of linear functionals, any Bell expression admits a canonical correlator form that is termwise invariant under the projection, so the projected estimate attains exactly the same Bell value as the raw empirical distribution. The paper also gives a closed-form, three-step linear map for computing","pith_inferences":["Editorial inference: Because the projection is linear and closed-form, the same three-map pipeline could be embedded in real-time device-independent randomness generation to denoise data before probability estimation, though the statistical effect on the downstream randomness certification is not analysed here.","Editorial inference: The uniform averaging over remote settings is a design choice; other averages over the remote parties' settings would produce different projection-invariant functionals, potentially tailored to specific drift patterns in a particular experiment.","Editorial inference: The kernel-membership result suggests a broader principle: any linear functional whose coefficient vector is orthogonal to the equality constraints is projection-invariant. The correlators are a spanning set, but not the only one, so other invariant quantities may be constructed.","Editorial inference: For the constrained projection onto the no-signalling polytope (where non-negativity is enforced), the invariance does not hold in general; the method is therefore best suited to high-count experiments where signalling is small enough that the affine projection already yields valid probabilities."],"forward_implications":["Any Bell expression for the (n,m,2) scenario has a canonical form whose value is unchanged when the empirical data is L2-projected onto the no-signalling affine hull.","The projection can be computed via a sparse three-map pipeline instead of forming a dense Gram matrix and its inverse, scaling more easily to higher party and setting counts.","Experimental Bell violations reported in the canonical form become insensitive to weak finite-sample signalling, providing a standardised metric across experiments.","For experiments with non-uniform sampling of measurement settings, a weighted L2 projection leaves a correspondingly weighted correlator form invariant.","Full-correlator inequalities such as CHSH and Mermin are already in canonical form and thus automatically projection-invariant."],"fun_headline_variants":["Bell violation unchanged by no-signalling projection","Projection keeps Bell inequality value intact","No-signalling filter preserves Bell violations","Closed-form cleanup keeps Bell data honest","Bell tests pass through projection unscathed"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The method depends on the empirical data lying close enough to the no-signalling set that the affine projection produces a valid, non-negative probability distribution; otherwise enforcing non-negativity breaks the claimed projection invariance.","fun_headline_variants_meta":{"raw":{"variants":["Bell violation unchanged by no-signalling projection","Projection keeps Bell inequality value intact","No-signalling filter preserves Bell violations","Closed-form cleanup keeps Bell data honest","Bell tests pass through projection unscathed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000327,"raw_usage":{"total_tokens":1609,"prompt_tokens":633,"completion_tokens":976,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":377,"completion_tokens_details":{"reasoning_tokens":914}},"tokens_in":377,"tokens_out":976,"duration_ms":6964,"temperature":1.0,"reasoning_tokens":914,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T23:16:07.381458+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a real or simulated Bell dataset with large spurious signalling (e.g., 40% differences between marginals) and compare the canonical Bell value after the affine L2 projection against the value after a constrained projection onto the no-signalling polytope: if they differ, the practical invariance does not extend to the constrained case. More directly, for any random empirical distribution f, compute the canonical Bell expression before and after the affine projection using the paper's closed formula; a discrepancy beyond numerical precision would disprove the central invariance claim.","supporting_citations":[],"review_version":1}