# The tile monoid

- 2026-08-31 [Proved] Factorisation of a 0/1 tile is unique once the ordered side profile is named: if `A (x) B = A' (x) B'` with `A, A'` of side `m` and `B, B'` of side `n`, all non-empty, then cutting the composite into an `m x m` array of `n x n` blocks reads `A` off as the 0/1 indicator of the non-zero blocks and `B` as any one of them, since every non-zero block equals `B` and `B` is not the zero tile, so `A = A'` and `B = B'`; the non-empty tiles under the Kronecker product are therefore a monoid graded by side, cancellative and atomic, whose block test decides factorability at a named shape in `O(N^2)` of exact integer comparison, the 0/1 hypothesis being load-bearing since over the rationals `A (x) B = (kA) (x) (k^-1 B)`, and the block reading being the 0/1 shadow of the Van Loan and Pitsianis rearrangement. This is a rediscovery and is cited, not claimed: it is Lemma 2.4 and section 2 of Voet and De Novellis, Identifying Kronecker product factorizations, arXiv:2510.25292, for binary matrices under equality. Witness: lab/rs/code-factorisation, magic.md, arXiv:2510.25292.
- 2026-08-31 [Proved] The ordered side profile is not recoverable from the composite, so the tile monoid has no unique factorisation, and the failure is not axis-separable: `[6]{(0,0),(2,2)}` is both `c1 (x) c257.base3` and `c17.base3 (x) c1`, all four letters of prime side and hence irreducible with differing multisets, and the tile is not a rectangle; the mechanism is an infinite family rather than a side-6 accident, since `I_m (x) I_n = I_mn = I_n (x) I_m` and `E_m (x) E_n = E_mn = E_n (x) E_m` at every pair of sides by the symmetry of `(nm - 1) - x = (n - 1 - i) m + (m - 1 - j)` in `m` and `n`, the same digit identity the diagonal action already carries, so taking `m` and `n` distinct primes gives four irreducible letters at every side with two distinct prime factors; the existence of shape-distinct factorisations is Example 2.5 of Voet and De Novellis, and new here are the axis-separable refutation and the I/E family. Witness: lab/rs/code-factorisation, magic.md, arXiv:2510.25292.
- 2026-08-31 [Proved] Neither the length nor the side multiset of a factorisation is an invariant of the composite, first at side 12 and at no smaller side: `[12]{(0,0),(3,3)}` reads as three irreducible letters of sides `2, 2, 3` and as two of sides `3, 4`, the side-4 letter `[4]{(0,0),(3,3)}` being irreducible because its one candidate cut has two unequal blocks, while every side below 12 is a prime power, where factorisation is unique, or a product of two distinct primes, where every letter has prime side; the short reading needs a letter of composite side, which no plane code is, so length and the side multiset are invariants for free inside the magic-word submonoid generated by prime-side letters and fail in the full monoid, and the alphabet of the words is not the alphabet of the monoid, 65310 of the 65535 non-empty side-4 tiles being irreducible already and the reducible share falling from `0.343328%` at side 4 to `0.0000221%` at side 6; a factorisation whose sizes are not all prime is Example 2.6 of Voet and De Novellis, and new here are the minimality of side 12 and the alphabet gap. Witness: lab/rs/code-factorisation, magic.md, arXiv:2510.25292.
- 2026-08-31 [Proved] Two factorisations of one tile admit a common refinement exactly when the union of their cut chains is totally ordered by divisibility, because a cut at `d'` dividing a cut at `d` factors the side-`d` left factor through the side-`d'` one; so unique factorisation holds at every prime-power side, where the divisors are a chain, and fails exactly when the cut set `L(C)` holds two incomparable divisors, checked with zero mismatches against direct enumeration of every irreducible factorisation over all 339795 side-12 plane-code composites, of which 7023 carry two or more factorisations and 2376 carry factorisations of unequal length. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Proved] The tile monoid is not a trace monoid, so no canonicalisation may sort or commute letters: `[2]{(0,0)} (x) [3]{(1,1)} = [6]{(1,1)} = [3]{(0,0)} (x) [2]{(1,1)}` uses four pairwise distinct irreducible letters, which no commutation of a letter pair can produce, and only 11 of the 171 side-6 cross-shape tiles are honest commutations against 160 rewritings. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Proved] Two letters render one tile at one side only at side 3: if a base-2 code and a base-3 code agree cell for cell at one `side` then row `r` equals row `r'` whenever `r = r'` mod 2 or mod 3, and at `side >= 4` those two partitions join the whole row range, so every row and every column agrees and a non-empty constant tile is the full tile; the census is 480 pairs at side 2, 15 at side 3 and the full tile alone at sides 4, 5, 6, 7, 8, 9, 12 and 18, the carpet's side-3 partner is uniquely `c495` of fill 8, and at side 9 the readings separate into fills 65, 72 and 64 on three pairwise distinct tiles, though read as level-2 fractals of the side-3 letter they do not diverge at all, since at side 3 they are one tile. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Proved] The canonical name of a composite is the code together with its ordered side profile, `c<code>(side_1 x side_2 x ... x side_level)`, and the two non-injectivities are different objects that must be disambiguated in order: the render collision belongs to the alphabet alone and tabulates once per base and side, since fixed-shape uniqueness proves the fold never creates one, while the fold collision belongs to the profile; profiles of different length occur, so the tie-break orders profiles by length first, finest before coarsest, then lexicographically, before the diagonal-action class rep breaks what is left. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Verified] The side-6 census: the two shape images are injective at 7665 tiles each, 171 tiles lie in both, so 15159 of the `2^36 - 1` side-6 tiles are reducible once the overlap is removed and 68719461576 are irreducible; of the 171, 121 are axis-separable and 50 are not, 11 are commutations and 160 rewritings, fills run `1:36 2:64 3:32 4:16 6:14 12:8 36:1` over the 171 and `2:16 3:32 6:2` over the 50 with outer-fill signature `(1,1):24 (1,2):8 (1,3):16 (2,3):2`, and the 48 that are neither separable nor commuting are exactly the 48 carrying a one-cell letter in at least one reading and exactly the 48 carrying a one-cell outer factor in at least one reading, three statistics on one set checked as sets rather than as counts, since the two one-cell readings differ elsewhere (`0:23 1:8 2:140` against `0:23 1:60 2:88` over the 171). Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Verified] `121 = 11 x 11` is arithmetic with a checked bijection: the 121 axis-separable side-6 cross-shape tiles are exactly the products `R x C` of the 11 lines that factor in both radix orders, `{0} {1} {0,1} {2} {0,2} {3} {4} {5} {3,5} {4,5} {0..5}`, verified as set equality and not as a count. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Verified] Counting reducible tiles at prime-power side is inclusion-exclusion over the divisor chain, equivalently the series `I = T/(1+T)` on the grading, giving 225 at side 4, 1962675 at side 8, 261121 at side 9, 553402322215537199175 at side 16 and `(2^25 - 1)^2 = 1125899839733761` at side 25, cross-validated in one dimension against exhaustive brute force at `N = 4, 8, 16, 9` reading 9, 63, 1431, 49; nothing new happens at a prime-power side, where the two side-8 shape images of 983025 tiles each meet in exactly the 3375 triple products of base-2 codes, checked as set equality, so 3375 is pure associativity and never stands beside 171. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Proved] One-cell letters commute exactly when `a(n - 1) = b(m - 1)`, giving `gcd(m - 1, n - 1) + 1` singleton pairs per axis and, where no common power exists, `gcd(m - 1, n - 1) + 2` commuting pairs in one dimension, checked at nine side pairs and exceeded only at `(3,9)` at 7 against 4 through the common-power branch; at base 2 against base 3 this gives the 11 commuting code pairs `(1,1) (2,4) (3,7) (4,64) (5,73) (6,84) (8,256) (9,273) (10,292) (12,448) (15,511)`, nine of them a commuting row line against a commuting column line and the other two the diagonal and the antidiagonal, with base-2 codes 7, 11, 13, 14 unpartnered, so the carpet code itself does not commute. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Conjecture] The cut set `L(C)` is closed under gcd, with zero failures over every non-empty subset of a line at `N = 1..20` and over all 339795 side-12 plane-code composites and no proof; it is the one missing structural fact, since with it `L(C)` is a meet-subsemilattice of the divisor lattice and the canonical name closes, and without it there is no counting theorem at non-prime-power side, where 171 and 15159 are enumeration rather than formula. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Conjecture] Two tiles commute under the Kronecker product exactly when they are powers of one common tile or the members at their two sides of one scale-free family, the one-cell case being settled by `a(n - 1) = b(m - 1)` and the general case tested only at `(2,3)` in two dimensions and at ten side pairs in one; relatedly, whether the diagonal and the antidiagonal are the only permutation tiles factoring in both radix orders at every coprime split. The next coprime test needs all `2^25` side-5 codes, so this has to be settled by proof and not by search. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Conjecture] The three-family description of cross-shape collisions at a coprime shape - axis-separable rectangles, tiles with a fill-1 outer factor, and the diagonal pair - is exhaustive at side 6 and untested anywhere else. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Conjecture] An intrinsic description of which tiles are Kronecker products, rather than the block test's algorithm and the published decomposition graph; and what the irreducible letters of composite side do, now that they are known to exist and to be generic, which is the question the plane-code word census could not see. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Conjecture] The diagonal embedding `D -> {(x,x) : x in D}` is an injective, divisor-closed embedding of the one-dimensional digit-set monoid into the tile monoid preserving cut sets and irreducibility, and a diagonal tile is axis-separable only at one cell; so a line sweep is a tile sweep, the whole non-uniqueness phenomenon already lives in one dimension, and it lifts to tiles no rectangle can explain. Witness: lab/rs/code-factorisation.
- 2026-08-31 [Refuted] That every shape-distinct factorisation of a tile is axis-separable, which would have closed the question with a one-line lemma - `[6]{(0,0),(2,2)}` factors as `c1 (x) c257.q3` and as `c17.q3 (x) c1` with four irreducible letters and is not a rectangle, separability covering 121 of the 171 side-6 cross-shape tiles and none of the diagonal family, and the diagonal and antidiagonal families put a non-separable witness at every side with two distinct prime factors, so the door stays open. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Refuted] That the mechanism of a shape-distinct factorisation is always a side-6 cross-shape collision sitting inside the word - the side-12 tile `[12]{(0,0),(3,3)}` has cut set `{1,2,3,4,12}` with no cut at 6, so no side-6 collision sits inside it, and its two readings differ in length; the earlier statement was read off a sample of words over plane codes and is a property of that universe, not a law. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Refuted] That the length and the side multiset of a factorisation are invariants of the composite - true inside the magic-word submonoid, where every plane code has prime side and every word over it has length exactly the number of prime factors of the side, and false in the full tile monoid, first at side 12, where `[12]{(0,0),(3,3)}` reads at lengths 3 and 2; the plane-code census could not have found the witness, since the short reading needs the irreducible side-4 letter `[4]{(0,0),(3,3)}`, which is no plane code. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Refuted] That the cut set `L(C)` is closed under lcm, and with it the naive reading that any two factorisations refine to a common one - the line `{0,3}` at `N = 12` has `L = {1,2,3,4,12}`, holding 2 and 3 and not 6, and the first failure by mask order at that side is the line `{1,2}` with the same cut set; 132 failures over every line to `N = 20` and 2376 over the 339795 side-12 plane-code composites, against zero failures of gcd closure in both sweeps. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Refuted] That the commuting pairs are the four corner cells plus the scale-free families of row, column, full tile, diagonal and antidiagonal - the cells `[3]{(1,1)}` and `[5]{(2,2)}` commute at side 15, both readings giving `[15]{(7,7)}`, and neither is a corner cell nor a member of any of those families; the four-corner picture is an artifact of `gcd(1,2) = 1` at sides `(2,3)`, the correct criterion for one-cell letters being `a(n - 1) = b(m - 1)`. Witness: lab/rs/code-factorisation, magic.md.
- 2026-08-31 [Refuted] That the reachable literature cannot reach the tile factorisation question, an earlier positioning against graph products - the isomorphism-versus-equality gap is real for graph products, but Voet and De Novellis, arXiv:2510.25292, is a binary-matrix paper working under equality that already contains fixed-shape uniqueness, the prime vocabulary, the shape-distinct factorisation, the non-prime factor sizes and a decomposition graph enumerating every factorisation; the Proved core recorded above is a rediscovery, and the single verbatim quotation the old positioning rested on could not be recovered from its source and is withdrawn rather than carried. Witness: magic.md, arXiv:2510.25292.
- 2026-08-31 [Refuted] The annotation `1125899839733761 = 65535^2` in a draft of the prime-power counts - the integer is right and its name is wrong, `1125899839733761 = (2^25 - 1)^2 = 33554431^2` at side 25, while `65535^2 = 4294836225` is a side-16 term; transcription, not mathematics, and the generator now prints the identity beside the value. Witness: lab/rs/code-factorisation.