# Level tilings

- 2026-10-03 [Proved] For a digit set `F` at base `b`, `Phi_m` divides the level mask `A_n(x) = prod_{i<n} F(x^(b^i))` if and only if `m/gcd(m, b^i)` lies in `Z(F) = {m : Phi_m divides F}` for some `i < n`, and the multiplicity of `Phi_m` in `A_n` is the sum over `i < n` of the multiplicity of `Phi_{m/gcd(m, b^i)}` in `F`. Witness: notes/tilings.md "The index lemma".
- 2026-10-03 [Verified] The cyclotomic part of 752 level masks, every digit set with `0` at bases 2 to 8, levels 1 to 3 and level 4 at bases up to 4, factored by PARI, equals the index lemma with multiplicity on all 752. Witness: lab/py/level-tiles `index`.
- 2026-10-03 [Verified] The Cantor level `{0, 2, 6, 8}` of `{0, 2}` at base 3 has mask `Phi_4^2 Phi_12`, reads T1 as `4` against `2`, and has no complement in `Z/N` for multiples `N` of `4` up to `128`. Witness: lab/py/level-tiles `hand`.
- 2026-10-03 [Proved] The tiling levels of a dimension-one design form an initial segment: if `A_n` tiles the integers then `A_m` tiles for every `m <= n`, so a digit set that does not tile the integers has no tiling level. Witness: notes/tilings.md "The tiling levels form an initial segment".
- 2026-10-03 [Proved] If `F` tiles `Z/b` then every level `A_n` tiles `Z/b^n` and hence the integers. Witness: notes/tilings.md "A digit set that tiles its residues tiles every level".
- 2026-10-03 [Proved] T1 holds at every level of `F` at base `b` if and only if `F` satisfies T1, `Z_p(F)` is empty for every prime `p` not dividing `b`, and the elements of `Z_p(F)` are pairwise incongruent mod `v_p(b)` for every `p` dividing `b`. Witness: notes/tilings.md "T1 at every level".
- 2026-10-03 [Proved] If every level of `F` at base `b` tiles the integers then `#F` divides `b`. Witness: notes/tilings.md "T1 at every level".
- 2026-10-03 [Proved] The T1 depth of `F` at base `b` is `0` if `F` fails T1, `1` if some prime off the base has `Z_p(F)` nonempty, and otherwise the least `(c' - c)/v_p(b)` over congruent pairs `c < c'` in some `Z_p(F)`, infinite if there is none; for `#F` a prime power the tiling depth equals the T1 depth. Witness: notes/tilings.md "T1 at every level".
- 2026-10-03 [Proved] At a prime-power base every level of `F` tiles the integers if and only if `F` tiles `Z/b`. Witness: notes/tilings.md "T1 at every level".
- 2026-10-03 [Verified] `{0, 1, 4, 5}` at base 6 has tiling depth `2` and `{0, 1, 8, 9}` at base 10 has tiling depth `3`, and the T1 depth law, the prime-power-size law and the prime-power-base law hold on all 65519 digit sets with `0` at bases 2 to 16. Witness: lab/py/level-tiles `census`, `witness`.
- 2026-10-03 [Refuted] Every level of `F` tiles the integers exactly when `F` tiles `Z/b`: `{0, 2}` at base 6 tiles at every level and does not tile `Z/6`. Witness: notes/tilings.md "The guess is false", lab/py/level-tiles `witness`.
- 2026-10-03 [Refuted] Every level of `F` tiles the integers exactly when `F` divided by its gcd tiles `Z/b`: `{0, 1, 8, 9}` at base 12, mask `Phi_2 Phi_16`, tiles at every level, and no integer multiple `vF` tiles `Z/12`, since `Z_2(vF) = {1 + v_2(v), 4 + v_2(v)}` and `2^(4 + v_2(v))` never divides `12`, with an exact-cover search over `v = 1..12` finding no complement. Witness: notes/tilings.md "The guess is false", lab/py/level-tiles `witness`.
- 2026-10-03 [Verified] Of the 65519 digit sets with `0` and at least two digits at bases 2 to 16, 588 tile `Z/b`, 609 tile at every level and 641 satisfy T1 at every level; the only every-level tiles not cured by dividing by the gcd are `{0, 1, 8, 9}` and `{0, 3, 8, 11}` at base 12, both mirror-symmetric. Witness: lab/py/level-tiles `census`.
- 2026-10-03 [Proved] If `F` satisfies T1 and T2 and `lcm(S_F)` divides `b`, then every level satisfies T1 and T2 and tiles the integers. Witness: notes/tilings.md "T2 at every level".
- 2026-10-03 [Proved] Under the T1 condition, T2 holds at every level if and only if it holds at levels `1` to `1 + G(2 omega - 1)`, with `G` the largest `ceil(C_p/v_p(b))`, `C_p` the largest exponent of `p` in an element of `Z(F)`, and `omega` the number of primes with `Z_p(F)` nonempty. Witness: notes/tilings.md "T2 at every level".
- 2026-10-03 [Proved] `{0, 1, 2, 6, 7, 8}` at base 18 tiles the integers and satisfies T1 at every level, while its level 2 fails T2 at `Phi_108` and does not tile. Witness: notes/tilings.md "T2 at every level", lab/py/level-tiles `witness`.
- 2026-10-03 [Verified] Among digit sets whose size has two prime factors and is below the base, at bases up to 30 with at most 200000 sets a cell, those that tile, satisfy T1 at every level and fail T2 at a level up to 6 number `0` of size 6 at base 12, `16` of size 6 at base 18, `0` of size 10 at base 20, `4` of size 6 at base 24 and `258` of size 6 at base 30, all failing at level 2, and 18 is the least base with such a digit set. Witness: lab/py/level-tiles `witness`.
- 2026-10-03 [Proved] By Coven-Meyerowitz Theorems A, B1 and B2 applied level by level through the index lemma and the horizon, every level of `F` at base `b` tiles the integers if and only if the T1 condition holds, when `#F` is a prime power, and if and only if the T1 condition holds and T2 holds up to the horizon, when `#F` has two prime factors; with three or more prime factors the same condition is sufficient. Witness: notes/tilings.md "The classification".
- 2026-10-03 [Proved] If T1 and T2 hold at every level of `F`, every level is spectral; in particular every level is spectral when every level tiles and `#F` has at most two prime factors. Witness: notes/tilings.md "Spectral levels".
- 2026-10-03 [Verified] On the 6866 levels with `#A_n <= 36` at bases 2 to 12 and levels 1 to 3, a level is spectral exactly when it tiles on all 6749 levels the search decides; the 117 non-tiling levels of digit sets with a root on the unit circle that is not a root of unity stay undecided. Witness: lab/py/level-tiles `spectral`.
- 2026-10-03 [Conjecture] A level of a dimension-one design is spectral if and only if it tiles the integers. Witness: notes/tilings.md "Spectral levels", lab/py/level-tiles `spectral`.
- 2026-10-03 [Proved] For consecutive digits `{0..N-1}` at base `b`, every level tiles the integers if and only if `N` divides `b`, the criterion Dai, He and Lau give for the spectrality of the limit measure. Witness: notes/tilings.md "The limit measure".
- 2026-10-03 [Verified] For product-form digit sets `{0..N-1} + m{0..L-1}` at base `p <= 24`, `2 <= N, L <= 12`, `N <= m <= p^2`, T1 and T2 at every level agree with the spectral condition of Liu-Wang-Zheng Theorem 1.3, an unrefereed arXiv preprint, on all 576422 cases. Witness: lab/py/level-tiles `measure`.
- 2026-10-03 [Verified] The digit set `{0, 1, 8, 9}` at base 12 meets the spectral condition of Liu-Wang-Zheng Theorem 1.3, an unrefereed arXiv preprint, with `N = L = 2`, `m = 8`, `d = 1`, while no integer multiple of it tiles `Z/12` by the exponent argument and the exact-cover search over `v = 1..12`. Witness: notes/tilings.md "The limit measure", lab/py/level-tiles `measure`, `witness`.
- 2026-10-03 [Conjecture] For an integer base `b >= 2` and a finite digit set `F` of non-negative integers containing `0`, the self-similar measure `mu_{b,F}` is spectral if and only if every level is a set that tiles the integers. Witness: notes/tilings.md "The limit measure", lab/py/level-tiles `measure`.