Magic Words

• Regenerates the order-sensitivity section of connectivity: the two sensitivity tables, the minimal witness, the order-blind laws, the rational-series representation, the component growth families, and the component exponent along a word with its closed forms on all 105 letter pairs, its frequency comparison, its named-word value with a certificate, and its boundary witnesses.
• A word is an ordered list of base-2, dim 2 codes folded by the Kronecker product, first letter outermost, of length L; bit i of a code is residue corner i in row-major order, so code 3 is the top row and code 5 the left column. The empty word is the one filled cell.
• Observables are fixed once and printed once: fill; the main-diagonal count; boundary, the number of filled cells with a void or exterior 4-neighbour; components, 4-connected; holes, the bounded 4-connected components of the complement; the Euler characteristic N - A + Q over cells, face-adjacent pairs and full 2 x 2 blocks; the anti-diagonal profile, filled cells by i + j; and the row and column contact counts.
• The exposed-face reading 4N - 2E of the race is a second observable under the same word "boundary" and is printed beside the first, because it is order-sensitive at length 2 where the cell count is not.
• Length 3 runs over a ten-code library. The library is every code of fill 2 or 3, the 15 non-empty codes less the four one-cell codes and the full tile; the study also scans all 3003 ten-code subsets and prints how many reproduce the page's length-3 row, which is one.
• The rational series is rebuilt from scratch by Hankel-basis elimination in exact rational arithmetic over i128 fractions in lowest terms, checked-multiplied so an overflow panics rather than wraps. Prefixes are taken in breadth-first order and rows are read on all 241 suffixes of length at most 2; the basis words, the matrices, lambda and gamma are outputs, never inputs.
• The representation is then checked against direct geometry on all 54240 words of length 1 to 4 over the 15 codes and on 120 words of length 5 to 7 drawn at a fixed seed, for all four observables.
• The growth section prints the component count of (15^(L-1), 6), (15^(L-1), 3) and (7^(L-1), 6) at L = 2..10 against the closed forms 2 * 4^(L-1), 2^(L-1) and 2 * 3^(L-1), each proved from the contact law before it is printed, and confirms the matrix product predicts every one of them.
• 2 * 4^(L-1) is the largest component count any subset of the 2^L grid can have: one cell per component is an independent set, and a grid with a perfect matching has no independent set above half its cells. The study attains the bound exhaustively at lengths 2 to 4 and through the representation at length 5.
• kappa, the number of components meeting a contact position and so still able to merge with a neighbouring copy, is printed on (15^(L-1), 3) and maximised exhaustively at lengths 1 to 4.
• The cocycle section takes the rate rather than the count. It prints the per-letter constant-word data that pins the frequency functional Phi(f) = (f_6 + f_9) log 2, then the closed forms of the seven pair families, each checked on every word of length at most 14 over the pair against the representation and on every word of length at most 7 against the drawn cells.
• The zero-contact cut is checked where its hypothesis holds and counted where it does not, over all 54240 words of length at most 4, so the law ships with its own scope.
• Thue-Morse is read exactly: equal letter counts at every even length, the longest terminal run, the distribution of the terminal run over every length to 2^20, the run census, and the identity of the run-boundary word with the period-doubling word.
• The rate table prints, per family, the limit, the constant-word prediction, the fill rate, and the exact prefix rate at L = 2^20 under both letter readings, and asserts the prefix rate sits within 2/L of the limit.
• The simplex boundary is printed as three words over (3, 6) at one frequency vector, with rates 0, 1 and none, so the positive-frequency hypothesis of every rate above is visible rather than assumed.
• The last section closes the other 46 letter pairs. Five families are checked the same way, on every one of the 2^13 words over each pair against the representation and every word of length at most 7 against the drawn cells; four of them are one zero-contact rule read four times and the fifth is the gasket against a domino, whose count is a sum of prefix fills rather than a power of 2, which is why the sweep evaluates the rule rather than an exponent.
• The frequency ledger is re-derived from the closed forms pair by pair, never copied: how many of the 105 pairs saturate the fill ceiling, how many refute Phi, and how many sit strictly between the constant-word values and the ceiling.
• The Thue-Morse value is printed with its certificate. Counts along a gasket-and-domino word are astronomical, so the study works with log2 comp factored through the largest term of the sum, which is exact to the last printed digit; every rate is a labelled float, every saturation extremum small enough to name is an exact rational, and the certificate is asserted at every length from 4 to 2^14 on all 16 pairs and both letter readings.
• The invariant cone is printed in exact rationals and checked against the raw matrices step by step, together with the three reasons it does not give the rate: the wrong direction of the orbit, the observation functional sitting on the boundary of the dual cone, and the matrix-norm exponent along 3^inf being a different number from the component exponent.
• The boundary frequency is printed twice, once over (3, 6) where a diagonal letter carries it and once over (3, 7) where no diagonal letter exists, and the second sweep prints a whole block of lengths so the accumulation set is visible as an interval rather than as two sampled points.
• Every geometric figure is cross-checked against mrlymath::bang::magic cell for cell on all 225 words of length 2 and all 3375 of length 3, so two independently written renderers stand behind every count.
• Block reduction gets the same treatment: a periodic word rendered flat by the study is compared cell for cell with Tensor::fractal of its one-period composite, on six cases at periods 2 and 3 and lengths to 6.
• Structural laws are asserted, not the headline counts: the study exits nonzero if a product law, a closed form, the matching bound, the representation or the factory cross-check fails.

RUN

• CARGO_BUILD_JOBS=4 cargo run --release -p magic-words
• About two seconds; prints only, writes nothing, and holds one 1024 x 1024 raster at a time.

WITNESSES

• connectivity.md the order table: 0, 0, 0, 74, 78, 10 sensitive of 105 multisets at length 2 and 0, 0, 36, 188, 188, 100 of 210 at length 3.
• connectivity.md the denominators: 225 words and 105 multisets with two or more orderings at length 2; 1000 words and 210 at length 3.
• connectivity.md the scope guard: the whole anti-diagonal profile sensitive on 99 of 105 at length 2, its peak on 23 and its support on 27.
• connectivity.md the minimal witness: comp(A_3 (x) A_6) = 4 against comp(A_6 (x) A_3) = 2.
• connectivity.md the fill 2 split: all 7 same-class pairs commute, diagonal against diagonal at 4, all 8 adjacent-against-diagonal pairs 4 against 2.
• connectivity.md the fill 3 and above line: all 10 pairs among codes 7, 11, 13, 14, 15 give one component in either order.
• connectivity.md the order-blind sweeps: the main-diagonal count factors and the contacts multiply on all 3375 words of length 3, and boundary and interior survive the factor swap on all 256 code pairs, all with zero mismatches.
• connectivity.md the series table: Hankel ranks 4, 4, 8, 11 with 6, 6, 9, 10 distinct matrices and basis words e, 3, 5, 15, e, 3, 5, 15, e, 3, 5, 7, 11, 13, 14, (7,14), e, 3, 5, 7, 11, 13, 14, 15, (7,6), (7,7), (11,11).
• connectivity.md the component representation: lambda = (1,0,0,0), gamma = (1,1,1,1)^T, six classes {1,2,4,8}, {3,12}, {5,10}, {6,9}, {7,11,13,14}, {15}, 14 of 15 class pairs failing to commute, and M(6,9) = 2 M(1,2,4,8).
• connectivity.md the growth line: 2 * 4^(L-1) on (15^(L-1), 6), 2^(L-1) on (15^(L-1), 3), 2 * 3^(L-1) on (7^(L-1), 6).
• connectivity.md the closed forms: 59 of the 105 letter pairs, in seven families of 16, 8, 4, 8, 4, 4 and 15 pairs, each exhaustive to length 14 against the representation and to length 7 against the drawn cells.
• connectivity.md the constant-word line: comp(c^L) = comp(c)^L on all 15 codes and comp(c) = 2 exactly at codes 6 and 9.
• connectivity.md the cut law: 49420 of 54240 words of length at most 4 admit the cut, zero mismatches, with h = 0 on codes 1, 2, 4, 5, 6, 8, 9, 10 and v = 0 on codes 1, 2, 3, 4, 6, 8, 9, 12.
• connectivity.md the Thue-Morse reading: terminal run at most 2, exactly L/2 of each letter at even length, terminal-run counts 524288, 349526, 174762 over lengths to 2^20, the period-doubling boundary word, and the prefix rates 1/2, 1/2, 1048575/1048576, 1, 524287/1048576, 0.
• connectivity.md the simplex boundary: 4/7, 8/15, ..., 8192/16383 at the powers of 2 and n/(n+2) at the squares.
• connectivity.md the other forty-six: five families of 16, 8, 4, 2 and 16 pairs, 753572 words against the representation and 11684 against the drawn cells, zero mismatches.
• connectivity.md the ledger: 89 of 105 pairs saturating and 16 falling short, Phi exact on 27 and refuted on 78, and 4 letter pairs strictly between.
• connectivity.md the Thue-Morse value: prefix rates 1.291967463826 and 1.292352803727 in one reading and 1.291291597694 and 1.292183837194 in the other, largest deviation 4.273459 nats against the certificate constant 4.884864, saturation minimum 0.0113766545, the sampled 0.2325367033 at L = 4096, and the exact maxima 43397/186624 and 151/648.
• connectivity.md the sandwich: comp/T in [1.0004, 2.0000] over six named words with no violation.
• connectivity.md the cone: the chart (1/3,1/3) (2/3,0) (5/9,1/3) (14/27,4/9) (41/54,0) (95/108,0) (203/324,1/3), the gasket-domino-gasket vertex images with b + c at most 17/18, 0 of 8190 entrywise-positive products, and max |entry(M_3^L)| = 2^(L+2) - 2.
• connectivity.md the boundary over (3, 7): 0.502367981 at L = 2048 through 0.500219405 at L = 32768, and the block 4097 <= L <= 8192 sweeping 1.00123 down to 0.50073.
• connectivity.md the by-products: (6^k + 4)/5 reading 2, 8, 44, 260, 1556, 9332, the largest count 1094 at L = 8, and the tripling word's range [0.4792, 1.4379].
• research/claims/ the transfer-state row: kappa = 2^(L-1) on (15^(L-1), 3) at L = 1..10 with maxima 1, 2, 4, 8 over all words of length 1 to 4.
• research/claims/ the block-reduction row: six periodic cases at periods 2 and 3 and lengths to 6, matching cell for cell.
• mrlymath::bang::magic, mrlymath::bang::MagicLayer and mrlycore::Tensor::fractal, the crate paths both renderers are checked against.