The cocycle's joint spectral radius

• 2026-09-06 [Proved] The component cocycle of mixed Kronecker words is simultaneously triangularizable over Z and its joint spectral radius is the largest fill: in the frame (gamma, h, v, phi) = ((1,1,1,1), (1,1,2,2), (1,2,1,2), (1,2,2,4)) of components, horizontal runs, vertical runs and fill, every class matrix is nonnegative, integer and lower triangular with column sums (comp(A_c), r(c), s(c), fill_c) and diagonals (0,0,0,1), (0,1,0,2), (0,0,1,2), (0,0,0,2), (1,1,1,3), (1,2,2,4), so the cross-polytope P = conv{+/- gamma, +/- h, +/- v, +/- phi} satisfies M_c P subset fill_c P with exact integer residuals and its gauge is an extremal norm; hence JSR(F) = max fill and LSR(F) = min fill on all 2^15 - 1 subfamilies, the finiteness property holds with a one-letter spectrum maximizing product, and every word over {3, 6} has spectral radius exactly 2^level. Witness: connectivity.md THE JOINT SPECTRAL RADIUS OF THE COCYCLE, lab/py/jsr-schedules.
• 2026-09-06 [Refuted] That the joint spectral radius of a component-cocycle pair is a nontrivial invariant of the pair: the JSR depends on the alphabet only through its largest fill and the LSR only through its smallest, both attained by one-letter words, so on the 78 of 105 letter pairs whose fills differ the JSR rate log max_c k_c strictly exceeds chi at every interior frequency; an exhaustive scan of all 2^level words to level = 16 on {3, 6} and {3, 7} never improves on the one-letter rates 2 and 3, and the Blondel-Nesterov lifting equals (fill_1^k + fill_2^k)^(1/k) exactly, terminating at no finite k. Witness: connectivity.md THE JOINT SPECTRAL RADIUS OF THE COCYCLE, lab/py/jsr-schedules.