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arxiv: 2606.28473 · v1 · pith:3WWV74W3 · submitted 2026-06-26 · math.NT · cs.IT· math.CO· math.IT

Classification of Boolean Cubic Forms in Ten Variables

Kirill Khoruzhii , Patrick Gel\ss , Sebastian Pokutta This is my paper

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classification math.NT cs.ITmath.COmath.IT
keywords boolean cubic formsGL(10,2) equivalenceorbit classificationalternating trilinear formsfinite fieldsBurnside lemmaenumeration
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The pith

Boolean cubic forms in ten variables over GF(2) fall into exactly 3691560 nonzero orbits under GL(10,2).

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

The paper enumerates and lists all distinct nonzero Boolean cubic forms in ten variables over the two-element field, where two forms count as the same if one is obtained from the other by an invertible linear change of variables. For each of the 3691560 orbits the authors give a short representative polynomial, the size of its stabilizer inside GL(10,2), and its alternating rank together with an explicit decomposition into linear factors. The list is built by generating forms rank by rank and is checked for completeness and uniqueness with two separate counting arguments from group theory. The same data immediately supplies the first full classification of alternating trilinear forms in ten dimensions over GF(2) by polarization. A fast-to-compute invariant is also supplied that decides equivalence without searching the group.

Core claim

We classify Boolean cubic forms in ten variables up to GL(10,2)-equivalence. The catalog contains all 3691560 nonzero orbits. For every orbit we provide a representative with small monomial count, the stabilizer order, and the alternating rank together with an explicit decomposition. The classification is obtained by rank-stratified enumeration. We verify completeness by the Burnside orbit count and independently by the orbit-stabilizer identity. We also provide a fast, complete GL(10,2)-invariant. By polarization, this gives the first complete classification of alternating trilinear forms in dimension 10 over GF(2).

What carries the argument

Rank-stratified enumeration of orbits under the natural action of GL(10,2) on cubic polynomials over GF(2), cross-checked by Burnside orbit counting and the orbit-stabilizer theorem.

If this is right

  • Every nonzero cubic has a representative whose monomial support is minimal within its orbit.
  • Stabilizer orders are known for all 3691560 orbits.
  • Alternating rank and an explicit decomposition into linear factors are recorded for every orbit.
  • Equivalence of any two cubics can be decided by evaluating the supplied invariant.
  • Alternating trilinear forms over GF(2) in ten dimensions receive an explicit orbit classification for the first time.

Where Pith is reading between the lines

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

  • The explicit list makes it feasible to test any property of cubic forms that is invariant under linear change of variables by checking only the representatives.
  • The same enumeration strategy can be rerun for eleven variables once sufficient computing time becomes available.
  • The catalog supplies a concrete database for studying the zero sets or correlation properties of all inequivalent cubics.

Load-bearing premise

The rank-stratified generation procedure together with the Burnside count and the orbit-stabilizer identity together guarantee that every orbit appears exactly once.

What would settle it

Exhibiting any nonzero cubic polynomial in ten variables over GF(2) whose orbit is missing from the catalog, or exhibiting any two distinct representatives in the catalog that become identical after multiplication by a single matrix in GL(10,2), would show the list is incomplete or contains duplicates.

Figures

Figures reproduced from arXiv: 2606.28473 by Kirill Khoruzhii, Patrick Gel\ss, Sebastian Pokutta.

Figure 1
Figure 1. Figure 1: Boolean cubic forms in six variables. a) Examples of forms on F 6 2, drawn as hypergraphs on a labeled hexagon; each triangular face corresponds to one monomial xixjxk. b) Examples of the GL(6, 2)-action. The basis change is displayed as a binary matrix, with blue squares for 1 and empty squares for 0. c) Orthogonality graphs Gf for two orbits; equivalent forms determine isomorphic graphs. d) One represent… view at source ↗
read the original abstract

We classify Boolean cubic forms in ten variables up to GL(10,2)-equivalence. The catalog contains all 3691560 nonzero orbits. For every orbit we provide a representative with small monomial count, the stabilizer order, and the alternating rank together with an explicit decomposition. The classification is obtained by rank-stratified enumeration. We verify completeness by the Burnside orbit count and independently by the orbit--stabilizer identity. We also provide a fast, complete GL(10,2)-invariant. By polarization, this gives the first complete classification of alternating trilinear forms in dimension 10 over GF(2).

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 classifies all nonzero Boolean cubic forms in ten variables over GF(2) up to GL(10,2)-equivalence. It enumerates 3691560 orbits, supplying for each a representative of small monomial support, the stabilizer order, the alternating rank, and an explicit decomposition. The enumeration proceeds by rank stratification and is verified by an independent Burnside orbit count together with the orbit-stabilizer relation applied to the listed representatives; a fast, complete GL(10,2)-invariant is also given. Polarization yields the first complete classification of alternating trilinear forms in dimension 10 over GF(2).

Significance. If the enumeration and its two independent verifications hold, the result supplies the first exhaustive catalog in dimension 10, together with an explicit invariant and a corollary classification of alternating trilinear forms. The combination of rank-stratified enumeration, Burnside verification, and orbit-stabilizer cross-check constitutes a reproducible computational classification whose reliability is strengthened by the multiple consistency checks.

minor comments (2)
  1. [Abstract] The abstract states the orbit count but does not indicate the range of alternating ranks or the maximal stabilizer orders appearing in the catalog; a single sentence summarizing these statistics would improve immediate readability.
  2. [Introduction] Notation for the alternating rank and the explicit decomposition is introduced without a forward reference to the section where the definitions are formalized; a parenthetical pointer would aid navigation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation and recommendation to accept the manuscript. The report accurately summarizes the main results and verification methods.

Circularity Check

0 steps flagged

No significant circularity; enumeration independently verified

full rationale

The derivation consists of rank-stratified enumeration of cubic forms over GF(2), cross-checked by the standard Burnside orbit-count formula and the orbit-stabilizer theorem applied to listed representatives. Both verification methods are external group-theoretic identities whose computation does not depend on the enumerated list or any fitted parameters. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the explicit invariant and polarization consequence are direct consequences of the classification rather than circular inputs. The work is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification rests on standard facts about the action of GL(10,2) on the space of cubic forms and on the correctness of the Burnside lemma and orbit-stabilizer theorem; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math GL(10,2) acts on the vector space of cubic forms over GF(2) by linear substitution.
    Invoked implicitly by the statement of classification up to GL(10,2)-equivalence.
  • standard math Burnside's lemma and the orbit-stabilizer theorem correctly count orbits when applied to this action.
    Used to verify completeness of the enumeration.

pith-pipeline@v0.9.1-grok · 5633 in / 1351 out tokens · 21092 ms · 2026-06-30T01:25:22.452726+00:00 · methodology

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