# Dimers on a design

- 2026-10-03 [Proved] With a cell `(r, c)` black when `r + c` is even, the black-minus-white count of level `n` of a plane code is `s^n` at odd base and `fill^(n-1) s` at even base, `s = sum_((i,j) in F) (-1)^(i+j)`; a code with `s != 0` tiles no level. Witness: dimers.md, The colour imbalance; lab/py/design-dimers, verb imbalance.
- 2026-10-03 [Proved] `T(n) >= T(n-1)^fill`, so the levels a plane code tiles form a ray `n >= n0` or are none, and once some level tiles `theta = lim log T(n)/fill^n` exists and equals `sup_n log T(n)/fill^n`. Witness: dimers.md, Tileability climbs the levels.
- 2026-10-03 [Proved] A component of the mask that no neighbouring small block can reach and whose signed count is nonzero makes every level untileable. Witness: dimers.md, The census of tileable codes.
- 2026-10-03 [Proved] At bases 2 and 3 a plane code tiles some level exactly when `s = 0` and it tiles level 1: at base 3, 97 of the 125 codes with `s = 0` tile every level and the other 28, in 5 orbits, tile none, each by a sealed unbalanced component. Witness: dimers.md, The census of tileable codes; lab/py/design-dimers, verb census.
- 2026-10-03 [Proved] A sealed unbalanced component of level `m`, read as a code at base `base^m`, makes every level untileable. Witness: dimers.md, The census of tileable codes.
- 2026-10-03 [Proved] If the small blocks of a plane code meet in one direction only and no walk of positive length leads from the empty set back to itself under the relation of crossed rows between consecutive blocks, no level tiles. Witness: dimers.md, The census of tileable codes.
- 2026-10-03 [Proved] At base 4, 5699 of the 12869 codes with `s = 0` tile level 1 and 7102 tile no level: 2253 by a sealed unbalanced component of the mask, 4709 more by one of level 2, 68 more by one of level 3, and 72 more in 10 orbits by the run certificate. Witness: dimers.md, The census of tileable codes; lab/py/design-dimers, verb census.
- 2026-10-03 [Verified] The other 68 base-4 codes with `s = 0`, in 9 orbits, tile none of levels 1 to 4 and carry no sealed unbalanced component through level 4. Witness: lab/py/design-dimers, verb census.
- 2026-10-03 [Refuted] Level 1 decides whether a plane code tiles some level: `bang dim 2, base 5, code 19920882` has `s = 0`, no tiling at level 1 and `T(2) = 252236412223488`, by determinant and by brute force. Witness: lab/py/design-dimers, verb census.
- 2026-10-03 [Verified] Nine codes untileable at level 1 and tileable at level 2: base 5 codes `15571455, 19627890, 19757811, 19920882, 32709486, 32715747, 32715771, 32912238` in 7 orbits and base 7 code `136308971855667`. Witness: lab/py/design-dimers, verb census.
- 2026-10-03 [Verified] Among dense codes drawn from a fixed seed, cells filled with probability `0.7`, 27 of 62111 distinct base-5 codes untileable at level 1 without a sealed component tile first at level 2, in 13 orbits, none of 20668 at base 6 and 2 of 6420 at base 7; none of the 48247 codes tried at level 3, those with `fill^3 <= 16000` whose level-2 deficiency is below `fill` times the level-1 deficiency, tiles first there, and the 100824 base-5 orbits within Hamming distance 4 of 15 late orbits hold 16 late orbits and none of 14914 tried at level 3 tiles there. Witness: lab/py/design-dimers, verb search.
- 2026-10-03 [Conjecture] No plane code tiles first at level 3 or later. Witness: lab/py/design-dimers, verb search.
- 2026-10-03 [Conjecture] At even base a plane code that does not tile level 1 tiles no level; at base 4 it is open only for 68 codes, untileable through level 4. Witness: lab/py/design-dimers, verbs census and search.
- 2026-10-03 [Proved] For every set of cells of the square grid with as many black as white cells the number of domino tilings is `abs(det K)`, `K` signed `(-1)^c` on a vertical edge in column `c` and `(-1)^h` on a horizontal edge `(r, c)(r, c+1)`, `h` the empty cells `(r', c)` with `r' > r` inside any box holding the set. Witness: dimers.md, Kasteleyn on a design.
- 2026-10-03 [Verified] The carpet `bang dim 2, base 3, code 495` has `T(n) = 2, 6724, 3862920381083436011392889139781518336` and a 316-digit `T(4)`; its sibling code 255 has `T(n) = 4, 1291616, 1565113733863335194512818740595861896518010511818752` and a 414-digit `T(4)`. Witness: lab/py/design-dimers, verb count.
- 2026-10-03 [Proved] `bang dim 2, base 3, code 63` has `T(n) = F(3^n + 1)^(2^(n-1))` and growth constant `log(phi)/2`. Witness: dimers.md, The counts.
- 2026-10-03 [Conjecture] The odd part of the carpet's tiling count is a perfect square at every level, `T(n) = 2^a q^2` with `a = 1, 2, 10, 128` through level 4. Witness: lab/py/design-dimers, verb count.
- 2026-10-03 [Proved] At every level `n >= 2` of a plane code that tiles some level the number of dominoes crossing between the big blocks is even. Witness: dimers.md, What crosses.
- 2026-10-03 [Verified] On the carpet at level 2 the tilings counted by crossing dominoes have generating polynomial `((x^2 + 2)^4 + x^8)^2`, so `T(2) = 82^2`. Witness: lab/py/design-dimers, verb cross.
- 2026-10-03 [Verified] On the carpet the tilings that respect the big blocks are `1/26.265625` of all at level 2 and `1/924472.759617` at level 3, and a tiling uses on average `5.463415` of 24 crossing edges at level 2 and `20.830336` of 72 at level 3. Witness: lab/py/design-dimers, verb cross.
- 2026-10-03 [Proved] `theta <= lim_n (1/(4 fill^n)) sum_v log d_v` over the cells of level `n`, by Hadamard's inequality on the Kasteleyn matrix, the limit existing because each digit acts on a cell's exposed sides by an idempotent map. Witness: dimers.md, The growth constant.
- 2026-10-03 [Verified] `0.177084 <= theta <= 0.286093` for the carpet and `0.232728 <= theta <= 0.296173` for code 255. Witness: lab/py/design-dimers, verb growth.
- 2026-10-03 [Proved] A plane code whose small blocks never share an edge has `T(n) = T(1)^(fill^(n-1))` and `theta = log T(1)/fill`. Witness: dimers.md, The growth constant.
- 2026-10-03 [Conjecture] The carpet's growth constant lies between `0.184611` and `0.188101`, the geometric-tail closures of its level-4 increment at ratios `3/8` and `0.4676`. Witness: lab/py/design-dimers, verb growth.