design-dimers

• Domino tilings of a plane design: a code at base base is the mask of cells (i, j) with bit base*i + j, level n its n-th Kronecker power, and T(n) the number of perfect matchings of the cell graph.
• Counts are abs(det K) with K the black-by-white Kasteleyn matrix: sign (-1)^c on a vertical edge in column c, sign (-1)^h on a horizontal edge (r, c)(r, c+1) with h the empty cells below it in column c. Determinants are exact integers from PARI (gp -q, matdet).
• The control is a brute-force transfer count over a broken row profile, exact integers in Python, and an augmenting-path matching test.
• imbalance checks the black-minus-white count of every code at base 2 through level 6 and base 3 through level 4 against s^n at odd base and fill^(n-1) s at even base, and counts the codes with s = 0.
• census finds the first tileable level of every code with s = 0 at base 2 and 3 through level 4 and at base 4 through level 3, counts the untileable ones by orbit and by a sealed unbalanced component of D_m for m = 1, 2, 3, tries the base-4 codes left at level 4 and by the run certificate for codes whose blocks meet in one direction only, and checks nine late codes at bases 5 and 7: untileable at level 1, unsealed, T(2) by determinant and, at base 5, by brute force.
• search draws dense codes from a fixed seed at bases 5, 6 and 7, each cell filled with probability 0.7, keeps the distinct codes, and counts those that tile first at level 2, and at level 3 among those with fill^3 <= 16000 whose level-2 deficiency falls below fill times the level-1 deficiency; then it walks every base-5 code within Hamming distance 4 of the late orbits, by orbit, and tries each code untileable at levels 1 and 2 at level 3.
• count runs the control on 300 random cell sets, then prints T(n) with its 2-adic split for code 15 at base 2 through level 6 (against the rectangle product formula), code 63 at base 3 through level 4 (against F(3^n+1)^(2^(n-1))), the carpet 495 and its sibling 255 through level 4, brute force at side 9 and below. It caches the counts in its data folder for cross and growth.
• cross weights the edges between the big blocks by x, evaluates the determinant at x = 0..E and interpolates in PARI: the number of tilings by crossing count for the carpet and its sibling at levels 2 and 3 and for code 15 at base 2 at levels 2 to 4, the parity, the mean, the share that respects the blocks, and the square test.
• growth prints log T(n) / fill^n, its increments and their ratios, the Hadamard upper bound from the 16-state exposure recursion at levels up to 400, with the idempotence check, the bracket, the geometric-tail closures and the exact constants.

RUN

• uv run python mrlyprod/research/lab/py/design-dimers/dimers.py runs every verb; one verb by name, e.g. ... dimers.py count.
• Runtimes: imbalance 0.1 s, census about 5 s, search 134 to 160 s, count 49 s (the three 4096-cell determinants), cross 3 s, growth under 1 s with the cache; about 3.5 minutes in all.
• Needs gp on the path; standard library only.

WITNESSES

• dimers.md, "The colour imbalance": the law at every code, the counts 5 and 125 of codes with s = 0 (imbalance).
• dimers.md, "The census of tileable codes": the base-2 and base-3 split 97 and 28, the 28 codes and their 5 orbits, the base-4 counts 5699, 7170, 945, 2253, 4709, 68, the 72 codes in 10 orbits of the run certificate, the 2803 one-direction codes that tile level 1, and the 68 open codes in 9 orbits, the nine late codes with their T(2) (census); the sample counts at bases 5, 6 and 7 (search).
• dimers.md, "Kasteleyn on a design": the 300 agreements, the 291 of the column signs alone, the 64 tileable sets, the zero determinant on the carpet without the hole signs (count).
• dimers.md, "The counts": the table, the control against A004003 and the product formula, the code-63 formula, the 2-adic splits (count).
• dimers.md, "What crosses": the parity at odd and even base, P_2(x), the means, ratios and shares (cross).
• dimers.md, "The growth constant": the lower bounds, the Hadamard limits, the brackets, the increment ratios and the tail closures (growth).