# Kronecker word order

- 2026-08-28 [Verified] Boundary is order-blind at word length 2 and order-sensitive from length 3: on the `4 x 4` grid only the four central cells can be interior and their requirements pair up under the factor swap, `(S_1, S_4)` against `(S_4, S_1)` and `(S_2, S_3)` against `(S_3, S_2)`, so interior and hence boundary is symmetric in the two factors, exact on all 256 code pairs with the interior formula matching direct computation on all 256; at length 3 it fails on 36 of 210 multisets over the ten-code library of every code of fill 2 or 3. Witness: lab/rs/magic-words.
- 2026-08-28 [Verified] Connected components of a mixed Kronecker word are order-sensitive, with minimal witness the multiset `{3, 6}`: `comp(A_3 (x) A_6) = 4` against `comp(A_6 (x) A_3) = 2`, both factors of fill 2, because the inner tile's contacts decide whether adjacent outer copies merge (`A_6` is two isolated cells with no boundary contact, so four cells stay apart, while `A_3` is a connected vertical pair sitting in two non-adjacent outer copies), enumerated by two independently written renderers, the study's substitution pass and the crate's Kronecker factory, agreeing cell for cell. Witness: lab/rs/magic-words.
- 2026-08-28 [Proved] The `fill = 2` designs at base 2, `dim = 2` split into adjacent (codes 3, 5, 10, 12) and diagonal (codes 6, 9), and commutativity of the component count follows the split: adjacent times adjacent commutes, diagonal times diagonal gives 4 in both orders, and adjacent times diagonal never commutes, always 4 against 2, by contact geometry, the diagonal pair having neither a face-adjacent cell nor a contact in either direction, and on all 15 pairs among the six codes with zero violations. Witness: connectivity.md, lab/rs/magic-words.
- 2026-08-28 [Verified] Any two designs with `fill >= 3` at base 2, `dim = 2` commute and give exactly one component in either order, since every such tile is connected and carries both a top-bottom and a left-right contact, so every pair of adjacent outer copies merges through the inner tile and connectedness of the outer tile merges them all; exhaustive over the four `fill = 3` tiles plus the full tile, all 10 pairs among codes 7, 11, 13, 14, 15. Witness: lab/rs/magic-words.
- 2026-08-28 [Proved] Block reduction: every periodic schedule `(c_1, ..., c_p)^level` equals the `level`-fold self-similar product of its one-period composite tile `A_(c_1) (x) ... (x) A_(c_p)`, of base `prod_i side_i` and fill `prod_i fill_i`, by associativity of the Kronecker product alone, so periodic mixed words carry no new theory and the first genuinely non-stationary behaviour requires an aperiodic word; cell for cell on six test cases at periods 2 and 3 and lengths to 6, the study's flat rendering against `Tensor::fractal` of the composite. Witness: dimensions.md, lab/rs/magic-words.
- 2026-08-28 [Verified] Contact counts of a mixed Kronecker word are exactly multiplicative, `h(A_w) = prod_i h(A_(c_i))` and `v(A_w) = prod_i v(A_(c_i))` for the row and column contact counts, because `L(A (x) B) = L(A) (x) L(B)` on outer columns and rows and the inner product of Kronecker products is the product of inner products, so whether adjacent copies touch is order-blind and decided factor by factor even where the component count is not; induction on length, exact on all `15^3` words with zero mismatches, the correct strengthening of the length-2 boundary theorem. Witness: lab/rs/magic-words.
- 2026-08-28 [Verified] The naive geometric transfer state for mixed products is unbounded: `kappa(A_w)`, the number of components meeting a contact position and so still able to merge with a neighbouring copy, reaches `2^(level-1)` on the family `w_level = (15^(level-1), 3)`, whose product is `2^(level-1)` disjoint full-width rows each meeting both the left and right column, from the full-tile Kronecker power, exact at `level = 2..10`, with exhaustive search over all 15 codes showing `w_level` is a maximiser at `level <= 4` with maxima 1, 2, 4, 8; the same family's component count is exactly `2^(level-1)` too. Witness: lab/rs/magic-words.
- 2026-08-28 [Verified] Order sensitivity of mixed products is exactly the noncommutation of the cocycle matrices: 14 of the 15 pairs of component-matrix classes fail to commute, the only commuting pair being the two zero-contact classes, `fill = 1` and the diagonal pairs, whose matrices have rank 1 with `M_(6,9) = 2 M_(1,2,4,8)`; the vertical and horizontal domino classes carry different matrices, so the representation sees more than the square's full symmetry group; checked on all 15 class pairs. Witness: lab/rs/magic-words.
- 2026-08-31 [Proved] The scale dimension of a magic word over a finite alphabet is a frequency functional: log side and log fill of a prefix are sums of per-letter values from a finite set, so whenever each letter's frequency exists the ratio `log fill / log side` converges to the frequency-weighted average, and in particular every uniquely ergodic word (Thue-Morse, period-doubling, Fibonacci) has a scale dimension equal to that average; open are finite alphabets without letter frequencies and all unbounded alphabets, where frequencies can exist while the dimension oscillates. Witness: magic.md.
- 2026-08-31 [Proved] The fill assumption `lab/py/slice-ladder-controls` states before printing its five staircase dimensions is discharged: the parity-carpet code at odd side `side` fills `E^2 + 2EO = side^2 - ((side-1)/2)^2` with `E = (side+1)/2` and `O = (side-1)/2`, the octagonal fill `3k^2 - 2k` at `side = 2k - 1` already on this ledger for the same rule, so the staircase dimensions stand without the assumption. Witness: magic.md, sequences.md.
- 2026-08-31 [Proved] The hyperoctahedral group acts diagonally through the Kronecker product, `g . (A (x) B) = (g . A) (x) (g . B)`, because reflecting a mixed-radix coordinate reflects every digit at once, so a magic word canonicalises under one shared symmetry applied to all letters and never letter by letter. Witness: magic.md.
- 2026-08-31 [Proved] Every value in `[0, log 8 / log 3]` is the scale dimension of some word over the two letters `carpet(3)` and `c8(3)`: rational carpet frequencies by periodic words, irrational by Sturmian words, endpoints by constant words, all through the frequency functional. Witness: magic.md.
- 2026-09-06 [Proved] The component count of a mixed Kronecker word is a rational series of Hankel rank 4: with `lambda = (1,0,0,0)`, `gamma = (1,1,1,1)^T` and one `4 x 4` integer matrix per code in six classes, `comp(A_w) = lambda M_(c_1) ... M_(c_level) gamma` at every word, by the four transfer laws in the observable frame `(gamma, h, v, phi)` and induction from `(comp, H, V, fill)(A_e) = (1,1,1,1)`; the same induction reads `H`, `V` and the fill off the same matrices. Witness: connectivity.md THE JOINT SPECTRAL RADIUS OF THE COCYCLE, lab/py/jsr-schedules.
- 2026-09-06 [Verified] Euler characteristic, boundary and holes of a mixed Kronecker word are rational series in the word, of Hankel rank 4, 8 and 11: there are `lambda`, `gamma` and one matrix per code with `phi(A_w) = lambda M_(c_1) ... M_(c_level) gamma`, exhaustive on all 54240 words of length at most 4 over the 15 non-empty codes plus 120 seeded words of length 5 to 7, all four observables, zero mismatches, built by Hankel-basis elimination in exact rational arithmetic; the rank is unexplained, staying 4 while the geometric state grows like `2^(level-1)` and the component count reaches `2 * 4^(level-1)` on `(15^(level-1), 6)`, the checkerboard and the largest component count any subset of the `2^level` grid can carry. Witness: lab/rs/magic-words.
- 2026-09-19 [Proved] Only a trailing solid letter is a magnification: `A (x) full(n)` replaces each filled cell of `A` by a solid block while `full(n) (x) A` lays `n^dim` copies of `A` in a grid, so `c15(2), carpet(3)` and `carpet(3), c15(2)` differ cell for cell at equal side and equal fill. Witness: magic.md, mrlymath::bang::magic.
- 2026-09-19 [Proved] A count along a periodic word satisfies a linear recurrence, being a fixed vector times the powers of one integer matrix in the rank-4 representation, so a match against a table of sequences is a recurrence and never an arithmetic fact. Witness: lab/rs/magic-words, connectivity.md.