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arxiv: 2606.23465 · v1 · pith:AVRXVXYWnew · submitted 2026-06-22 · 🧮 math.CO

Boolean degree one functions on the Grassmann scheme

Yuval Filmus This is my paper

Pith reviewed 2026-06-26 07:51 UTC · model grok-4.3

classification 🧮 math.CO
keywords Grassmann schemeBoolean functionsdegree one functionsassociation schemesfinite geometryq-analogs
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The pith

Boolean degree one functions on the Grassmann scheme J_q(n,k) are trivial when min(k,n-k) >= 2 and n is large enough.

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

The paper supplies a mostly self-contained exposition of a theorem that classifies Boolean degree one functions on the Grassmann scheme. The result states that these functions must be trivial once the subspace dimension parameters satisfy min(k,n-k) at least 2 and the ambient dimension n grows large. A reader cares because the statement limits the possible low-degree Boolean functions on the collection of k-dimensional subspaces of an n-dimensional space over a finite field. The exposition reproduces the argument without adding new claims beyond the original classification.

Core claim

Ferdinand Ihringer proved that Boolean degree one functions on the Grassmann scheme J_q(n,k) are trivial when min(k,n-k) >= 2 and n is large enough. We provide a mostly self-contained exposition of this result.

What carries the argument

The Grassmann scheme J_q(n,k) on the k-dimensional subspaces of an n-dimensional vector space over GF(q), whose association scheme structure controls the degree of the Boolean functions.

If this is right

  • All Boolean degree one functions on these schemes must take one of a small number of explicit forms once the size conditions hold.
  • The same triviality conclusion applies uniformly across all finite fields once n exceeds a parameter-dependent threshold.
  • The result supplies a q-analog of corresponding statements already known for the Johnson scheme.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The classification may extend to other q-analog association schemes with similar intersection properties.
  • It could simplify the search for constant-weight codes or designs whose indicator functions have low degree in the scheme algebra.

Load-bearing premise

The exposition faithfully reproduces the original proof by Ihringer without introducing errors or gaps that would alter the validity of the stated theorem.

What would settle it

An explicit non-trivial Boolean degree one function on J_q(n,k) for some q, some n large enough, and min(k,n-k) at least 2 would falsify the claim.

read the original abstract

Ferdinand Ihringer proved that Boolean degree one functions on the Grassmann scheme $J_q(n,k)$ are trivial when $\min(k,n-k) \ge 2$ and $n$ is large enough. We provide a mostly self-contained exposition of this result.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript provides a mostly self-contained exposition of Ferdinand Ihringer's theorem that Boolean degree-one functions on the Grassmann scheme J_q(n,k) are trivial whenever min(k,n-k) ≥ 2 and n is sufficiently large.

Significance. An accurate exposition would make an existing result in algebraic combinatorics more accessible to readers working on Boolean functions and association schemes, without introducing new theorems or derivations.

minor comments (2)
  1. The abstract states the result for the Grassmann scheme J_q(n,k); the introduction should explicitly restate the precise parameter range (including the lower bound on n) to avoid any ambiguity for readers.
  2. If the exposition follows Ihringer's original argument closely, a brief sentence in §1 noting which parts of the proof are reproduced verbatim versus rephrased would help readers compare with the source paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; exposition of external result

full rationale

The paper states it provides a mostly self-contained exposition of Ferdinand Ihringer's existing theorem on Boolean degree-1 functions on the Grassmann scheme J_q(n,k). It advances no new mathematical claims, fits no parameters, and introduces no self-citations or ansatzes that bear load on a derivation. The central assertion is faithful reproduction of an external proof, which is self-contained against external benchmarks and triggers none of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No new free parameters, axioms, or invented entities are introduced; the paper re-presents an existing combinatorial theorem.

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discussion (0)

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Reference graph

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