Design Windows

• Counts the k x k windows of the subshift of every plane code at bases 2 and 3, decides which subshifts are of finite type, and checks both against brute force, for windows.
• A code at base b is a bitmask over the b x b cells of level 1 in row-major order; the substitution sends a 1 to that tile and a 0 to the empty tile.
• L_2, the 2 x 2 render windows, is the closure of the level-1 windows under one substitution step, stopped at the first repeat.
• The census: for k <= b^n + 1 the k x k windows are the blocks of sigma^n(B), B in L_2; each block is named by the quadruple of the names of its two (k-1)-blocks on the diagonal and its two other corners, so equal names are equal blocks and the table size is p(k).
• One code per orbit of the eight symmetries of the square; the census asserts every image of every orbit's code gives the same count for k <= 12.
• A boundary code, all filled cells in one boundary line of the box, has the subshift {0} and prints p = 1, with the render-window count after the word edge.

VERBS

• count <base> <top> [code]: one line per orbit, fill, the size of L_2, the level it settles at, periodic@k at the first k with p(k) <= k^2/2 or above, the counts p 1..top, the exact quadratic when one holds from k <= 3 to top; the carpet's and the gasket's formula checks; the class tallies and the number of distinct sequences.
• scan <base> <level> <top> [code]: renders each orbit's code with mrlyrs::math::bang::factory::create, asserts it equals the study's own sigma^level(1) and that code 7 at base 2 drawn at side 3 is base-3 code 495, counts distinct k x k blocks of the render by their rows, and prints the gaps against count.
• kind <base>: sft empty, sft full, sft boundary, or not-sft with a witness of fast growth of L_j(X_2): blocks (two blocks of side at most 2 freely arranged), lines (a branching window rule on pictures constant along an axis or a diagonal), xor (L_2 is the eight blocks with one corner the sum of its neighbours); then recognised m for the least m <= 3 at which every occurrence of R_m in a picture sigma^m(B), B in L_2, is at one of the four aligned offsets, else unrecognised; it is run only off the empty, full and boundary codes.
• pairs <base> <code>: L_2 as rows ab/cd.

RUN

• bash scripts/cargo.sh cargo run --release -p design-windows -- count 3 82: under 7 seconds.
• count 3 244 495: the carpet to k = 244, about 2 seconds; count 2 129: under a second.
• scan 3 6 12: about 46 seconds; scan 2 7 17 under a second; scan 3 7 16 495: about 8 seconds.
• kind 2 and kind 3: under a second each. Prints only, writes nothing.

WITNESSES

• windows, "The objects": the symmetry check, the 102 and 6 orbits, the carpet as code 495.
• "Counting through the substitution": the settle levels 1, 2, 3 on 6, 75, 21 orbits at base 3; the scan agreement, 0 gaps in 108 orbits and on the carpet at level 7.
• "The carpet": p(k) to 244, L_2 of size 10, formula holds for 2 <= k <= 244.
• "Growth": the recognised 1 certificate on 91 orbits at base 3 and the gasket, unrecognised on the line orbits; the exact quadratics; formula holds for 1 <= k <= 129 for the gasket.
• "Products and lines": the counts of codes 27, 45, 325, 63, 365.
• "Not of finite type": every verdict and witness; the counts p(6) = 242, p(24) = 4570, p(72) = 42970.
• "The census": the base-2 table, 83 distinct sequences, the tallies 26, 10, 476.