Kronecker words

• 2026-08-28 [Proved] Fill, side, density and the main-diagonal count of a mixed Kronecker word are order-blind at every word length, the first three as products of per-factor quantities and the diagonal by diag(A (x) B) = diag(A) (x) diag(B), exhaustive on all 15^3 words of length 3 at base 2, dim = 2; the whole anti-diagonal profile is order-sensitive on 99 of 105 multisets at length 2 over the 15 non-empty codes and on 204 of 210 at length 3 over the ten-code library of every code of fill 2 or 3, its peak on 23 of 105 and its support on 27 of 105, minimal witness the one-cell codes 1 and 2 with profiles (0,1,0,0,0,0,0) against (0,0,1,0,0,0,0); take the whole profile, never one coefficient. An earlier length-3 reading of 110 of 112 is withdrawn, not confirmed: 112 is the multiset count of an eight-code library that was never recorded, so the figure names no sweep anyone can rerun. Witness: lab/rs/magic-words.
• 2026-09-19 [Proved] A decoration on a letter folds into a code: the fill swap renders the complementary code at every base and side, and at base 2 and odd side the half turn of the rendered tile is the identity and a quarter turn is the render of the transposed corner rule, because side - 1 is even. Witness: magic.md.
• 2026-09-19 [Proved] A unit letter is a property of the code and the side together: at odd side 2k - 1 a base-2 letter fills sum over its corners of k^(zeros) (k - 1)^(ones), so fill 1 forces one corner with every coordinate odd and k = 2, giving c8 as the only letter of fill 1 at side 3 and a smallest fill of 4 at side 5. Witness: mrlymath::formulas::counting, magic.md.
• 2026-09-19 [Proved] No periodic word carries irrational letter frequencies, a word of period p having every letter frequency in (1/p)Z, so at an irrational frequency vector the control for a schedule is the value of the frequency functional and never a periodic word. Witness: magic.md.
• 2026-09-19 [Proved] A mosaic whose palette is the empty cell and one design is a magic pair M (x) c, since the block at a mask site is the palette cell the mask value indexes, so a census over both families counts it once on the word side. Witness: mrlycore::cell::mosaic, mrlymath::bang::magic.
• 2026-09-19 [Proved] Magic meets special at a plain tiling and nowhere else: every mask site of a special carries a block, so a special is a magic pair exactly when all its blocks agree, which is full(f) (x) c at a constant mask or at a design c fixed by the quarter turn. Witness: mrlymath::two::geometry::special, mrlymath::bang::magic.
• 2026-09-19 [Proved] The fill of a mosaic is linear in the mask histogram, fill = sum_i h_i(M) fill(P_i) over disjoint blocks, collapsing to the product law of a word only at a palette of the empty cell and one design, so block reduction and the Kronecker law are given up the moment a palette holds two designs. Witness: mrlycore::cell::mosaic.
• 2026-09-19 [Proved] A special is fill-invariant, fill(special(M, c)) = f^dim fill(c) at a mask of side f in any dimension: an orientation permutes the entries of a cell, so every palette copy carries the fill of c, and the mask lays f^dim disjoint blocks. Witness: magic.md, spot-checked by mrlymath::two::geometry test special_rotations_preserve_sum and mrlymath::three::geometry test orientations_preserve_sum_and_shape.