{"id":"ba4f4e2a-8eec-464c-839f-9bf31ccaaa8b","arxiv_id":"2608.11320","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Boolean function on the Grassmann scheme over F2 that is close to a degree 1 function is close to a canonical point/hyperplane indicator sum, up to complement.","lead":"The paper proves a stability version of the classical Friedgut-Kalai-Naor theorem for Boolean functions on the binary Grassmann scheme. It shows that functions close to degree 1 must be close, relative to their variance, to indicator sums defined by points and hyperplanes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition of h_eta in Section 3.3 is missing a complement; as written Claim 3.11 is false and the proof of Theorem 1.3 stalls unless h_eta is corrected to f times the indicator of the complement of E_eta.","rationale":"I read the whole argument in good faith. The central claim is plausible and the proof strategy is coherent, but the most load-bearing weakness is not the one identified by the reader. The reader's weakest assumption, Lemma 3.7's uniformity of (A+L, A+L') as a random Grassmann edge, is actually true: the distribution is GL_n(F_2)-invariant and all edges are in the support, so it is uniform. The real problem is the definition of h_eta in Section 3.3. As written, h_eta=f 1_{E_eta} has mu_x(h)=mu_x(f)>eta for x in X_eta, so Claim 3.11 is false. The proof only works if h_eta is the complement indicator, f 1_{overline{E_eta}}, which is confirmed by the surrounding proof sketch and by Eq. (3), where f-f_1 equals the leftover outside E_eta. This is a typo-level but load-bearing error: without the correction, Claim 3.12 and hence Theorem 1.3 are unsupported. There is also an algebraic slip in Eq. (8), where the coefficient of E[f] should come from the exact probability (2^{n-ell}-1)/(2^n-1), giving -1/(2^n-1) rather than -1/2^{n-1}; this too appears repairable. Because both issues are concrete, localized, and fixable without changing the theorem's statement, I agree with the reader's conditional verdict, but not with their identification of the weakest assumption.","tokens_in":15040,"tokens_out":47696,"duration_ms":438906,"concrete_test":"Perform the following check on Section 3.3. Let f(L)=1_{x_0 in L}, fix eta<1, so X_eta(f) contains x_0 and E_eta = {L : x_0 in L}. With the literal definition h=f 1_{E_eta}, compute mu_{x_0}(h)=1>eta, contradicting Claim 3.11. Then repeat with h=f 1_{overline{E_eta}} and verify that Claim 3.11, the derivation in Claim 3.12, and Eq. (3) all become consistent. This settles whether the concern is a harmless typo or a substantive gap in the proof of Theorem 1.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing flaw is a sign error in the definition of h_eta in Section 3.3. The text defines h_eta(L)=f(L)1_{E_eta}(L), where E_eta is the event that L contains a point of X_eta(f) or is contained in a hyperplane of W_eta(f). Claim 3.11 then asserts that h_eta is (1,eta)-global, and its proof begins 'For x in X_eta(f) we have that mu_x(h)=0.' This is false under the literal definition: if x in X_eta(f), then every L containing x automatically satisfies E_eta, so h(L)=f(L) and mu_x(h)=mu_x(f)>eta. For example, take f(L)=1_{x_0 in L} with eta<1; then X_eta contains x_0, E_eta is the set of L containing x_0, and h=f, so mu_{x_0}(h)=1>eta. The intended definition must be h_eta(L)=f(L)1_{overline{E_eta}}(L): then Claim 3.11 is correct, and Claim 3.12 proves that the part of f outside E_eta has small norm, exactly what is needed for Eq. (3). With the literal definition, Claim 3.12 would prove the opposite bound, so the proof of Theorem 1.3 does not go through as written. Although this appears to be a typo-level error that can be repaired without changing the theorem, it is load-bearing and must be fixed before the central claim is established.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an FKN-type stability theorem for Boolean functions on the vertices of the binary Grassmann graph Gr(F_2^n, ℓ). Assuming a Boolean function f is close in squared L2 to its degree-≤1 projection, the theorem concludes that either f or 1−f is close to a function g(L) = Σ_{x∈X} 1_{x∈L} + Σ_{W∈W} 1_{L⊆W} for some set of points X and set of hyperplanes W, and that f has small variance. The proof has three stages: a fixed-dimension compactness argument converts the exact classification of Boolean degree-1 functions into a dimension-dependent stability bound; a random-restriction argument removes the dimension dependence and yields a coarse dichotomy that f is nearly constant; and a global hypercontractivity argument identifies the points and hyperplanes on which f has noticeable conditional expectation and shows that these conditional expectations are close to 1. The final section gives the rounding argument that produces the Boolean approximation g.","tokens_in":15431,"tokens_out":31472,"duration_ms":237474,"significance":"If the proof is repaired, this is the first robust classification of approximately degree-1 Boolean functions on the Grassmann scheme, extending the classical FKN theorem to a setting relevant to 2-to-1 games and short-code constructions. The high-level architecture—compactness, dimension reduction by restrictions, and global hypercontractivity—is coherent and likely to generalize, as the discussion suggests. The paper is honest about its black boxes: the exact classification [FI19b, Ihr24, Fil26] and the bilinear global hypercontractivity [KMS23] are prior results, and their use is not circular. The fixed-dimension stability lemma (Lemma 3.1) is elegant. However, several load-bearing technical points are currently incorrect or unjustified, so the contribution is conditional on repair.","major_comments":[{"comment":"The definition h_eta(L) = f(L)1_{E_eta}(L) makes Claim 3.11 false. For x in X_eta(f), every ℓ-subspace L containing x satisfies E_eta, so h(L)=f(L) on those L and mu_x(h)=mu_x(f)>eta, contrary to the proof's assertion that mu_x(h)=0. With the literal definition, Claim 3.12 bounds the wrong part of f, and Eq. (3) in Section 3.4 would not hold. The intended definition must be h_eta(L)=f(L)1_{overline{E_eta}}(L); with this correction, the proofs of Claim 3.11, Claim 3.12, and Eq. (3) are valid. This is a load-bearing error and must be fixed.","section":"Section 3.3, definition of h_eta and Claim 3.11"},{"comment":"Claim 2.3 states p=P[x in L] = 2^{ℓ-1}/(2^n-1) and q=P[L subseteq W] = 2^{n-ℓ-1}/(2^n-1); the correct values are (2^ℓ-1)/(2^n-1) and (2^{n-ℓ}-1)/(2^n-1). The incorrect q is used in Appendix A immediately before Eq. (8): substituting q=2^{n-ℓ-1}/(2^n-1) gives 2^ℓ q - 1 = -(2^{n-1}-1)/(2^n-1), so the simplification to -1/(2^n-1) E[f] in Eq. (8) is algebraically false. With the correct q one obtains exactly the claimed simplification. The same incorrect p and q appear in Section 3.4 in the computation of E[g(g-1)], where the equality P[x in L subseteq W] = (2^{n-1}/(2^{n-1}-1)) pq holds only for the correct p,q. Please correct Claim 2.3 and propagate the correction.","section":"Section 2 (Claim 2.3) and Appendix A (Eq. (8))"},{"comment":"The proof of Lemma 3.7 states without justification that sampling a random t-restriction and then adjacent L,L' subseteq C yields (A+L, A+L') as a uniformly random edge of Gr(F_2^n, ℓ). This step is load-bearing: it converts the restriction-based disagreement bound into the lower bound var(f) ≤ 2⟨f,(I-T)f⟩, which underlies Lemma 1.6. The claim is true—the sampling distribution is GL(n,2)-invariant and the action on adjacent pairs is transitive—but a rigorous proof or citation should be supplied, since a non-uniform edge distribution would break the variance bound.","section":"Section 3.2 (Lemma 3.7)"},{"comment":"In the proof of Theorem 1.3, ε is set to min(ε1, ε2, ε3, δ′), where ε3 is the L1-parameter from Lemma 1.6. The available bound is ∥f−P^{≤1}f∥_1 ≤ ∥f−P^{≤1}f∥_2 ≤ (ε var(f))^{1/2} ≤ √ε/2, so the hypothesis of Lemma 1.6 is satisfied only if ε ≤ 4ε3^2. As written, ε≤ε3 is insufficient; for instance, with ε3=10^{-6} and ε=10^{-3} the L1 norm may be about 0.016, far above ε3. This is a load-bearing parameter error, though it is repairable by taking ε = min(ε1, ε2, ε3^2/4, δ′).","section":"Section 3.4 (proof of Theorem 1.3)"}],"minor_comments":[{"comment":"The statement of Claim 3.12 should specify that T depends on η as well as on ε, since the proof invokes Theorem 3.10, whose threshold depends on η.","section":"Section 3.3, Claim 3.12"},{"comment":"The expression 'c f⋆(u⊗v)' is ambiguous; it should be written as c \\hat f⋆(u⊗v) so that the Fourier coefficient is not confused with the function value.","section":"Appendix A"},{"comment":"In the parameter-choice paragraph, 'Pick ... T4 from Claim 3.11' should read 'from Claim 3.12', since Claim 3.11 has no parameters.","section":"Section 3.4"},{"comment":"The displayed bound '16E[f] 2^{n-1}/2^{ℓ-1}' should be written as 'O(E[f] (2^n-1)/(2^ℓ-1))'; the current expression drops a factor that is only absorbed later in the O(·) notation and is confusing as written.","section":"Lemma 3.16"}],"recommendation":"major_revision","confidential_remarks":"The two main technical errors (the h_eta complement issue and the incorrect p,q values) are local and repairable, but they currently invalidate the written proof of the central theorem. The manuscript also relies substantially on two prior results by subsets of the authors ([FI19b, Fil26] and [KMS23]); I do not regard this as circular, but it means the paper's contribution is the stability argument and the reduction rather than the classification or hypercontractivity themselves. I recommend inviting a major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before you read it. First: the paper proves a genuinely new stability statement for the binary Grassmann scheme—the first robust FKN-type theorem for that graph—and the overall strategy is coherent. Second: the proof as written does not go through, because of a load-bearing sign error in Section 3.3 plus some wrong probability factors in Claim 2.3. Both look repairable, but the current text is not a complete proof.\n\nThe new part is real. The exact classification for eps=0 was already in [FI19b, Ihr24, Fil26]; this paper adds the eps>0 version for large l and n-l, and the random t-restriction dimension reduction (Lemmas 3.6–3.7) is a nice idea. Lemma 1.6, the coarse-level result, is clean and well-motivated. The later coarse-to-fine argument using the KMS23 global hypercontractivity is plausible and, modulo the bugs, the high-level proof of Theorem 1.3 is in place.\n\nSoft spots, in order. (1) h_eta is defined as f*1_{E_eta}, but Claim 3.11 requires h_eta to vanish on points of X_eta. It doesn't as written: if x in X_eta, every L containing x satisfies E_eta, so mu_x(h)=mu_x(f)>eta. The intended definition is the complement, h_eta=f*1_{overline{E_eta}}; then the claim works. The stress-test note is correct, and Theorem 1.3's proof stalls until this is fixed. I read this as a typo, not a conceptual gap. (2) Claim 2.3 gives p=2^{l-1}/(2^n-1) and q=2^{n-l-1}/(2^n-1); the true values are (2^l-1)/(2^n-1) and (2^{n-l}-1)/(2^n-1). The main inequality of the claim survives. But the appendix uses q in the derivation of Eq. (8), and the cancellation to -E[f]/(2^n-1) only holds with the true q. So the appendix algebra is wrong as printed. (3) Lemma 3.7 states without proof that a random t-restriction followed by a random adjacent pair in C gives a uniform random edge. This is likely true, but it is load-bearing and needs a counting proof.\n\nVerdict: the central theorem is probably correct, and none of the identified problems look fatal. Would I accept? Yes—a serious referee should look at this, and the authors should be asked to fix the h_eta definition, the probability values, and the omitted counting proof before publication. It deserves a place in the literature once those are corrected.","headline":"Fixable sign and probability errors hide a likely-true, genuinely new stability theorem for the binary Grassmann scheme.","tokens_in":15957,"tokens_out":7216,"would_cite":true,"duration_ms":58500,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E30","06E30","68Q17"],"pacs":[],"model":"deepseek-v4-flash","headline":"A stability theorem for Boolean functions on the binary Grassmann scheme: near degree one forces a point-or-hyperplane test.","keywords":["Grassmann scheme","FKN theorem","Boolean functions","degree one functions","random restrictions","global hypercontractivity","Poincaré inequality","Fourier analysis on Grassmann graphs"],"falsifier":"One concrete check: pick small n and ℓ, fix any two adjacent ℓ-dimensional subspaces, enumerate all random slices (A,B,C) and all adjacent t-dimensional pairs L,L′ inside C that produce the chosen pair, and see whether the counts are equal across all edges; equality is exactly what Lemma 3.7 needs, and a computer search for small parameters would settle the uniformity quickly.","tokens_in":14845,"feed_emoji":"📐","tokens_out":7593,"duration_ms":65077,"temperature":0.7,"pith_summary":"The paper proves a stable analogue of the classical FKN theorem for Boolean functions on the binary Grassmann scheme, where the domain is the set of ℓ-dimensional subspaces of 𝔽₂ⁿ and 'degree one' means lying in the span of indicators of containing a fixed point and indicators of being contained in a fixed hyperplane. The main theorem says that when both ℓ and n−ℓ are large, any Boolean function whose squared distance from the degree-one projection is at most ε times its variance must, up to a small multiple of its variance, be close either to a sum of such point-containment and hyperplane-containment indicators or to the complement of such a sum. This turns the previously known exact classification of degree-one Grassmann functions into a stability statement, which is the form needed for applications in testing, coding, and hardness arguments. A sympathetic reader would care because stability theorems of this kind are the standard bridge from Fourier structure to concrete geometric structure.","feed_headline":"Near-degree-one Boolean functions are point-hyperplane tests","feed_subtitle":"On subspaces, Boolean functions close to degree one must nearly test a point or a hyperplane.","key_machinery":"The argument is carried by three mechanisms. The first is the degree decomposition on the Grassmann scheme: the space $J^{{≤1}}$ is spanned by indicators 1_{I⊆L} for subspaces I of dimension 0 or 1, and $P^{{≤1}}$ is the orthogonal projection onto it. The second is a dimension-reduction step based on random t-restrictions (A,B,C), where A is an (ℓ−t)-dimensional subspace, C⊆B is a 2t-dimensional subspace, and the restricted function is f_{A,B,C}(L)=f(A+L); the key Lemma 3.7 converts the near-constancy of random restrictions into the Poincaré lower bound Var(f)≤2⟨f,(I−T)f⟩, where T is the normalized adjacency operator of the Grassmann graph. The third is a global hypercontractivity bound for (1,η)-global functions, imported from the bilinear scheme, which controls how much L² mass a Boolean function can place on the first two levels when no point or hyperplane has a large conditional expectation. Together these pieces first show that f is close to constant at a coarse scale, then locate the points and hyperplanes on which the function's measure is noticeable, and finally show that the measure on almost all of those is close to one.","core_discovery":"On its own terms, the discovery is Theorem 1.3: for every δ>0 there exist ε>0 and T such that if ℓ,n−ℓ≥T and a Boolean function f on the ℓ-subspaces of 𝔽₂ⁿ satisfies ‖f−$P^{{≤1}}$f‖₂² ≤ ε·Var(f), then either f or 1−f is within δ·Var(f) in squared L² distance of a function g(L)=Σ_{x∈X}1_{x∈L}+Σ_{W∈W}1_{L⊆W}, where X is a set of points and W a set of hyperplanes. The theorem also forces Var(f)≤δ and that the Boolean rounding of g is O(Var(f)²) away from g, so the approximator is essentially Boolean. This extends the exact classification of degree-one Boolean functions on the Grassmann scheme to the robust regime, preserving the same list of possible shapes: constants, point tests, hyperplane tests, and sums of a point test with a hyperplane test for a point outside that hyperplane.","pith_inferences":["Beyond the paper, the same restriction-to-Poincaré mechanism could plausibly yield FKN-style stability for Grassmann schemes over larger finite fields, provided the edge-uniformity property behind Lemma 3.7 is verified by a direct counting argument.","A natural next target is the dependence of ε on δ: the authors did not optimize it and suspect δ=O(ε) may hold, which would make the theorem quantitatively match the classical FKN behavior.","The degree-2 counterexample in the discussion suggests that any Kindler–Safra-type structure theorem for constant-degree Grassmann functions must allow mixed point-hyperplane pairings rather than only decision-tree shapes; if the conjectured structure fails in degree 2, the general classification likely needs a richer list of approximators."],"forward_implications":["If f is close to degree one, its approximator g is a union test: it accepts a subspace L exactly when L contains a marked point or L is contained in a marked hyperplane.","The variance bound Var(f)≤δ means that a near-degree-one Boolean function that is not essentially constant has its nonzero mass concentrated on few geometric directions, matching the coarse-versus-refined dichotomy described in the paper's comparison with the p-biased cube.","The statement holds uniformly once ℓ and n−ℓ exceed the constant T, so it applies in the large-dimension regime relevant to short-code graphs and 2-to-1 games, where the non-robust exact classification is too rigid to use directly.","A degree-d analogue would follow along the same lines if an exact classification of degree-d Boolean functions on the Grassmann scheme were available; the paper identifies that classification as the missing ingredient."],"supporting_citations":[{"why":"The classical theorem over the Boolean cube that this paper generalizes; it sets the target shape of a robust degree-one classification.","marker":"[FKN02]"},{"why":"Supplies the exact classification of Boolean degree-one functions on the Grassmann scheme, used as the base for the dimension-dependent stability lemma.","marker":"[FI19b]"},{"why":"Extends the exact classification to arbitrary finite fields and is cited as part of the prior work behind Theorem 1.2.","marker":"[Ihr24]"},{"why":"The most recent exact classification result on the Grassmann scheme, cited alongside [FI19b, Ihr24] for the non-robust base theorem.","marker":"[Fil26]"},{"why":"Provides the global hypercontractivity bound and the bilinear-scheme level decomposition used to prove Theorem 3.10.","marker":"[KMS23]"},{"why":"Gives the second eigenvalue bound for the Grassmann graph behind the Poincaré inequality in Fact 2.2.","marker":"[BCN89]"},{"why":"Supplies the eigenvalue formulas for the Grassmann graph used for the same Poincaré inequality.","marker":"[GM16]"}],"fun_headline_variants":["FKN theorem for binary Grassmann scheme","Near-degree-one Grassmann functions are point-hyperplane tests","Robust FKN for Grassmann over F2","Grassmann FKN: near-linear is point or hyperplane","Point or hyperplane: robust classification on subspaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that when you slice the space at random into a fixed subspace plus a low-dimensional core, then pick two nearly intersecting subspaces inside that core, every pair of nearly intersecting ℓ-dimensional subspaces is produced equally often; this fact is stated without proof and converts the sliced picture into the variance bound.","fun_headline_variants_meta":{"raw":{"variants":["FKN theorem for binary Grassmann scheme","Near-degree-one Grassmann functions are point-hyperplane tests","Robust FKN for Grassmann over F2","Grassmann FKN: near-linear is point or hyperplane","Point or hyperplane: robust classification on subspaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1746,"prompt_tokens":974,"completion_tokens":772,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":693}},"tokens_in":590,"tokens_out":772,"duration_ms":6426,"temperature":1.0,"reasoning_tokens":693,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:15:37.353832+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete check: pick small n and ℓ, fix any two adjacent ℓ-dimensional subspaces, enumerate all random slices (A,B,C) and all adjacent t-dimensional pairs L,L′ inside C that produce the chosen pair, and see whether the counts are equal across all edges; equality is exactly what Lemma 3.7 needs, and a computer search for small parameters would settle the uniformity quickly.","supporting_citations":[],"review_version":1}