An FKN Theorem for the Binary Grassmann Scheme

Anqi Li; Dor Minzer; Yuval Filmus

REVIEW 4 major objections 4 minor 30 references

An FKN Theorem for the Binary Grassmann Scheme

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A stability theorem for Boolean functions on the binary Grassmann scheme: near degree one forces a point-or-hyperplane test.

desk verdict Fixable sign and probability errors hide a likely-true, genuinely new stability theorem for the binary Grassmann scheme. read the letter →

arxiv 2608.11320 v1 pith:DE5JE3FI submitted 2026-08-11 cs.CC math.CO

Yuval Filmus , Anqi Li , Dor Minzer This is my paper · ORCID

classification cs.CCmath.CO MSC 05E3006E3068Q17

keywords GrassmannschemeFKNtheoremBooleanfunctionsdegreeonerandomrestrictionsglobalhypercontractivityPoincaréinequalityFourieranalysisongraphs

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

Soft 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.

Verdict: 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[Section 3.3, definition of h_eta and Claim 3.11] 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 2 (Claim 2.3) and Appendix A (Eq. (8))] 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 3.2 (Lemma 3.7)] 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 3.4 (proof of Theorem 1.3)] 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′).

minor comments (4)

[Section 3.3, Claim 3.12] 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.
[Appendix A] 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 3.4] In the parameter-choice paragraph, 'Pick ... T4 from Claim 3.11' should read 'from Claim 3.12', since Claim 3.11 has no parameters.
[Lemma 3.16] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the robust FKN theorem is derived from external exact-classification and hypercontractivity results used as black boxes.

full rationale

The derivation chain is self-contained in the sense required by the circularity check. Lemma 3.1 converts the exact degree-1 classification (Theorem 1.2) into a finite-dimensional stability constant C(n,ell) by enumeration over a finite function space; that classification is cited from [FI19b, Ihr24, Fil26], where [FI19b] and especially the independent [Ihr24] establish the F2 case without assuming the robust statement of Theorem 1.3. Lemma 3.7's use of random t-restrictions depends on the combinatorial fact that (A+L, A+L') is a uniformly random edge of the Grassmann graph; this is an asserted fact about the sampling process, not an equation that identifies the output with an input. The global hypercontractivity input (Theorem 3.10) is proven in Appendix A by reduction to [KMS23, Lemma 2.8 and Theorem 2.13], parameter-free results about the bilinear scheme published independently of the present approximation theorem; using them does not presuppose closeness of f to X/W sums. Claims 3.12-3.15 and Lemma 3.16 apply these black boxes with Cauchy-Schwarz and size estimates; no fitted parameter is relabeled as a prediction. The only flagged issue I find is a definitional bug in Section 3.3: h_eta is written as f 1_{E_eta}, making Claim 3.11's 'mu_x(h)=0' for x in X_eta false, since then mu_x(h)=mu_x(f)>eta; the intended complement indicator is needed. This is a correctness error that must be repaired, but it is not a circular reduction of the theorem to its assumptions. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; eps, delta, T are existential quantifiers and C(n,l) is a finite maximum. The proof assumes three external results: the exact degree-1 classification, the global hypercontractivity theorem for the bilinear scheme, and the Grassmann second-eigenvalue bound. Two structural facts (uniform random edge distribution, v_W in J^{<=1}) are asserted without proof. No new entities are postulated.

assumptions (5)

domain assumption Exact classification of Boolean degree 1 functions on the Grassmann scheme (Theorem 1.2)
Imported from [FI19b, Ihr24, Fil26]; the base case for Lemma 3.1 and the entire compactness argument.
domain assumption Global hypercontractivity for basis-invariant functions on the bilinear scheme ([KMS23, Theorem 2.13])
Used to prove Theorem 3.10 in Appendix A; supports Claim 3.12.
standard math Second eigenvalue bound on the Grassmann graph (Poincare inequality, Fact 2.2)
Quoted from [BCN89, GM16]; used in Lemma 3.7.
domain assumption Uniformity of the induced edge distribution from random t-restrictions
Asserted in Lemma 3.7 without a counting proof; needed to connect restriction bounds to Grassmann edge probabilities.
standard math Hyperplane functions v_W(L) = 1_{L subseteq W} belong to the degree-1 space J^{<=1}
Used without proof in Appendix A; follows from Theorem 1.2 or standard Grassmann scheme duality.

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Pith. "Pith review of An FKN Theorem for the Binary Grassmann Scheme." pith.science (2026). https://pith.science/paper/DE5JE3FI

@misc{pith2026260811320,
  author       = {Pith},
  title        = {Pith review of: An FKN Theorem for the Binary Grassmann Scheme},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DE5JE3FI}},
  note         = {Machine review of arXiv:2608.11320}
}

abstract

A classical theorem due to Friedgut, Kalai and Naor asserts that if a function $f\colon \{0,1\}^n\to\{-1,1\}$ close to a degree $1$ function, then either $f$ or $-f$ is close to either the all $1$ function, or to $(-1)^{x_i}$ for some $i\in [n]$. We prove a version of their theorem for the Grassmann scheme over $\mathbb{F}_2$. More precisely, we prove if a function $f\colon \genfrac{[}{]}{0pt}{}{\mathbb{F}_2^n}{\ell}\to\{0,1\}$ is close to a degree $1$ function, then either $f$ or $1-f$ must be close to a function of the form $g(L) = \sum_{x\in\mathcal{X}}1_{x\in L}+\sum_{W\in\mathcal{W}}1_{L\subseteq W}$, where $\mathcal{X}\subseteq\mathbb{F}_2^n$ is a set of points and $\mathcal{W}$ is a set of hyperplanes in $\mathbb{F}_2^n$.

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Reviewed August 15, 2026 · model on record in the stance chip above.

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Bourgain, Jean , title =. Israel Journal of Mathematics , volume =. 2002 , doi =

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Filmus, Yuval , title =. Combinatorics, Probability and Computing , volume =. 2020 , doi =. 1809.03089 , archivePrefix =

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Discrete Analysis , volume =

Eldan, Ronen and Kindler, Guy and Lifshitz, Noam and Minzer, Dor , title =. Discrete Analysis , volume =. 2025 , doi =. 2204.06686 , archivePrefix =

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