# 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.