Design windows

• 2026-10-03 [Proved] For a plane code that is neither empty nor a boundary code (all filled cells in one boundary line of the box), the k x k blocks of the design's subshift X_F are exactly the k x k render windows, and p(k) is unchanged by the eight symmetries of the square; a boundary code and the empty code give X_F = {0} and the full code X_F = {1}. Witness: windows.md, The objects.
• 2026-10-03 [Proved] For k <= b^n + 1 the k x k render windows of a nonempty plane code are the k x k blocks of the pictures sigma^n(B), B a 2 x 2 render window, so off the boundary codes p(k) is a census of at most 16 pictures of side 2 b^n; and p(k) <= 64 b^2 k^2 for every code and every k >= 3. Witness: windows.md, Counting through the substitution.
• 2026-10-03 [Verified] The substitution census of k x k windows equals a brute-force count on crate renders for every orbit at base 2 at level 7 with k <= 17, every orbit at base 3 at level 6 with k <= 12, and the carpet at level 7 with k <= 16, with 0 disagreements in 108 orbits. Witness: lab/rs/design-windows, scan.
• 2026-10-03 [Verified] The Sierpinski carpet, code 7 at base 2 drawn at side 3 and code 495 at base 3, has window count 2, 10, 40, 74, 152, 242, 344, 442, 544, 650, 872, 1106 for k = 1..12, computed exactly to k = 244. Witness: lab/rs/design-windows, count 3 244 495.
• 2026-10-03 [Conjecture] With N = 3^n, the carpet's window count is 6k^2 + 12(N - 1)k - 10N^2 - 12N + 8 for N + 1 <= k <= 2N + 1 and 2k^2 + (24N - 4)k - 18N^2 - 24N + 4 for 2N + 1 <= k <= 3N + 1, so p(k)/k^2 has lower limit 8 and upper limit 9.6; checked for 2 <= k <= 244. Witness: lab/rs/design-windows, count 3 244 495.
• 2026-10-03 [Proved] The carpet's window count is Theta(k^2), with k^2/36 <= p(k) <= 576 k^2 for k >= 6. Witness: windows.md, The carpet.
• 2026-10-03 [Proved] Recognition lemma: for a plane code neither empty nor a boundary code, if every occurrence of R_m in a picture sigma^m(B), B a 2 x 2 render window, is at one of the four aligned offsets, then p(k) >= k^2 / (4 b^(2m)) for every k >= 2 b^m. Witness: windows.md, Growth.
• 2026-10-03 [Proved] Every plane code at bases 2 and 3 has window count of order 1, k or k^2; the order is k^2 exactly for the 480 codes that carry the recognition certificate at m = 1, the 4 gasket codes and 476 codes at base 3, and k for the 2 diagonal codes at base 2 and the 10 line codes at base 3. Witness: windows.md, Growth; lab/rs/design-windows, kind.
• 2026-10-03 [Verified] The Sierpinski gasket, code 7 at base 2, has window count 4k^2 - 6k + 4 for 1 <= k <= 129, the square count that Allouche and Berthe prove for the triangle form of Pascal's triangle mod 2, which is the gasket sheared. Witness: lab/rs/design-windows, count 2 129 7.
• 2026-10-03 [Verified] Ten orbits at base 3 have a window count that is one quadratic from k <= 2 to k = 82, among them the centre k^2 + 1, the square {0,1}^2 4k^2 - 4k + 2 and the Cantor dust 4k^2 - 8k + 5; the other 81 orbits with quadratic growth are not one quadratic from k <= 3. Witness: lab/rs/design-windows, count 3 82.
• 2026-10-03 [Proved] For a product code F = A x C that is not a boundary code the window count is n_A n_C + z, with n_A, n_C the counts of the non-zero windows of the two line designs and z equal to 1 when either line has an all-0 window; when neither line set is empty, {0} or {b - 1}, the pictures of X_F are the outer products of line pictures. Witness: windows.md, Products and lines.
• 2026-10-03 [Proved] Scaling lemma: if the subshift of a plane code that is neither empty nor a boundary code is cut out by forbidden r x r blocks and b^n + 1 >= r, then p(j b^n) is at least the number of j x j blocks of X_2 for every j, so X_F is not of finite type whenever the number of j x j blocks of its 2-window approximation X_2 is not O(j^2). Witness: windows.md, Not of finite type.
• 2026-10-03 [Proved] At bases 2 and 3 a plane design's subshift is of finite type exactly when the code is empty, full or a boundary code: 10 of the 16 codes at base 2 and 26 of the 512 at base 3; every other code carries a finite witness of fast growth of X_2. Witness: windows.md, Not of finite type; lab/rs/design-windows, kind.
• 2026-10-03 [Proved] The Sierpinski carpet's subshift is not of finite type: the two 2 x 2 blocks 11/00 and 11/10 laid freely on the even grid stay inside its 2-window approximation, and its holes are squares ringed by 1s. Witness: windows.md, Not of finite type.
• 2026-10-03 [Verified] No forbidden list of blocks up to 10 x 10 cuts out the carpet's subshift, read off p(6) = 242, p(24) = 4570 and p(72) = 42970 against the scaling lemma. Witness: lab/rs/design-windows, count 3 244 495.
• 2026-10-03 [Proved] The 2-window approximation of the gasket's subshift is the set of all two-sided histories of rule 90 on one parity class read along antidiagonals, with 2^(2j - 1) blocks of side j, so the gasket's subshift is not of finite type. Witness: windows.md, Not of finite type.
• 2026-10-03 [Proved] Every plane design's subshift at every base is sofic: every code meets the two hypotheses of Theorem 4.5 of Mozes, both rules b x b and property A for a single deterministic substitution; so the 6 codes at base 2 and 486 at base 3 that are not of finite type are strictly sofic. Witness: windows.md, Sofic.
• 2026-10-03 [Proved] As a corollary of the finite-type verdict at bases 2 and 3 and of soficity by Mozes, no Wang set is conjugate to the subshift of any of the 492 codes at bases 2 and 3 that are not of finite type, and each of them is the letter image of a finite Wang set. Witness: windows.md, The Wang face.
• 2026-10-03 [Verified] The 512 codes at base 3 fall into 102 orbits under the symmetries of the square and give 83 distinct window-count sequences to k = 82. Witness: lab/rs/design-windows, count 3 82.