bibkey: bogdanov2020xortriangle
authors: Ilya Bogdanov
year: 2020
title: "Answer to Number triangle"
doi: null
url: https://mathoverflow.net/a/359278
claim: "For each fixed row length over the two-element field, the clockwise top-row-to-right-edge map has third iterate equal to the identity."
strata_touched:
  - D5/S3/Combinatorics/XorTriangle/ValueOne
  - D5/S3/Combinatorics/XorTriangle/ReversibleSideMap
license: CC-BY-SA-4.0
triage: anchor

Bogdanov, reversible XOR triangles

MathOverflow answer 359278, revision 5, studies the clockwise side map of a fixed-length binary XOR triangle. The relation between three incident bits is symmetric over the two-element field. Rotating the triangle preserves this relation, and three successive side changes recover the original row. The statement retains every position at the fixed row length.

This is literature support for reversible-triangle reasoning, rather than an explicit settlement of Kagey’s selected value-one conjecture. The value-one module proves the needed edge injectivity locally by backward reconstruction. The ReversibleSideMap module separately proves the threefold-iterate identity for the same retained rightEdge on every finite Boolean row and constructs the fixed-width equivalence with inverse given by two turns. Both proofs use no literature axiom. The fixed-vector dimension and proper-vector counting formulas in the answer are separate results.

The reversible-triangle argument is adapted from Ilya Bogdanov’s answer, attributed under CC BY-SA 4.0. The adaptations are the local reconstruction argument used in the value-one proof and the coordinate-rotation proof of the threefold-iterate identity; the MathOverflow answer is not a kernel-verified supplier.

Verified locator

• Source: https://mathoverflow.net/a/359278, revision 5.
• Revision identity: E8608C9B-A5DF-4521-A836-A6DFF07C6CD4.
• Scope: fixed-length clockwise side map and its threefold composition.