bibkey: ricardorr2026mo512655
authors: ricardorr (MathOverflow user 49534; real-name attribution unverified)
year: 2026
title: "Does the support of a cyclic trace of 2x2 sign matrices always span a hyperplane in Z_2^n?"
doi: null
url: https://mathoverflow.net/questions/512655
claim: "For every positive odd n and every nonzero b in F_2^n, the affine hull over F_2 of the nonzero support of the specified cyclic integer sign-matrix trace has dimension at least n-1."
strata_touched:
  - D5/S3/Combinatorics/CyclicBoolean/CyclicSignTrace
  - D5/S3/Combinatorics/CyclicBoolean/CyclicSignTrace/Prelude
  - D5/S3/Combinatorics/CyclicBoolean/CyclicSignTrace/Trace
  - D5/S3/Combinatorics/CyclicBoolean/CyclicSignTrace/BooleanCoefficients
  - D5/S3/Combinatorics/CyclicBoolean/CyclicSignTrace/RankAssembly
license: CC-BY-SA-4.0
triage: anchor

The cyclic sign-matrix trace support question

Verified locator

Source: https://mathoverflow.net/questions/512655

MathOverflow question 512655, revision 1, revision GUID E06A6743-2C15-41CD-9390-04D1862ABA02, by ricardorr (user 49534). The versioned question is attributed under CC BY-SA 4.0. The title is only a locator; the retained target uses the complete lower-bound assertion in the question body.

Mathematical scope

For every positive odd length and every nonzero binary mask, the source defines the cyclic exponents from the word being tested and takes the integer trace of the ordered product of the resulting 2 by 2 sign matrices. The question asks whether the affine hull over the binary field of the nonzero trace support has dimension at least one less than the length. Coincident cyclic indices at length one, every word, and the literal increasing-representative product order are part of the source convention.

Source boundary

The source reports a weaker unconditional bound, finite checks and a five-bit example. Those observations are context, not the formal result. The repository theorem supplies the complete universal lower bound and does not claim that all supports are hyperplanes or that the source’s surrounding inverse-chi discussion is itself a proved literature theorem.