# 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.