A global girth obstruction for Garg--Mineyev taiko product structures

Abstract

Mineyev's taiko construction, in Garg--Mineyev's finite support-size formulation, gives a concrete route from finite support data to zero divisors and units in group rings of torsion-free CAT(0) groups over $\mathbb{F}_2$. We prove that this triple-girth product-structure route is globally closed: no product structure, even or odd, with support sizes $m,n\ge2$ admits a coherent orientation for which the no-fold and triple-girth conditions both hold. Consequently the Garg--Mineyev triple-girth product-structure assembly route produces neither zero-divisor nor unit counterexamples over $\mathbb{F}_2$ for any such support-size pair. The obstruction is structural, not a bounded-search artifact. High middle-link girth forces signed colors into a balanced near-disjoint rectangle decomposition of the board, with the single odd defect omitted. The product identity, pressure inequalities, Fisher inequalities, and a dual Fisher bound force the middle link to have girth $4$ or $6$; in the girth-six case, the minimum of the two horizontal-link girths is at most $5$. This dichotomy rules out every triple-girth branch. A weighted dual Fisher inequality and an exact finite certificate sharpen the frontier: if the middle link has girth $6$, the horizontal girth is at most $4$, and characteristic-two affine-plane constructions attain equality. Thus the Garg--Mineyev finite failures reflect a structural barrier in the taiko geometry itself. The finite certificate is used only for this sharper frontier, not for the no-$T_4$ obstruction.