The component exponent

• 2026-08-31 [Proved] The component count of a two-letter magic word has an exact closed form on 59 of the 105 letter pairs at base 2, dim = 2, namely 9 of the 15 pairs of distinct symmetry classes and all 6 pairs inside one class: 2^(k-m) with k the last unit place and m the unit count for a unit against a domino (16 pairs), 2^(number of diagonal letters) for a unit against a diagonal (8), 2^(level-r) with r the terminal run for two dominoes of unlike orientation (4), 2^k with k the last diagonal place for a domino against a diagonal (8), 2^(n-j) with n the number of full letters and j their terminal run for a domino against the full tile (4), 1 for the gasket class against the full tile (4), and 1 inside a class except 2^level inside the diagonal class (15); proved by the zero-contact cut on the first four families, again by a rank-1 telescope on the domino against the diagonal, by the row-block argument on the domino against the full tile, and by the contact split with the fill >= 3 line on the rest; exhaustive per pair on all 32766 words of length at most 14 against the representation and all 254 words of length at most 7 against the drawn cells, zero mismatches. The other 46 pairs are closed by the suffix recursion below. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Proved] The zero-contact cut: since contacts multiply, a suffix has h = v = 0 as soon as it holds one letter with h = 0 and one with v = 0, adjacent copies of that suffix tile can never merge, and comp(A_w) = fill(A_prefix) * comp(A_suffix) at the last such suffix; 49420 of all 54240 words of length at most 4 admit the cut with zero mismatches. Its scope is part of the law: the other 4820 words carry nonzero contact in one direction at every suffix, and every word over a domino and the full tile is among them, so that family needs the row-block argument instead. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Proved] The row-block law, and the one non-trivial exponent found: over a domino and the full tile the filled set is a product of a row set with the whole column range, |R| = 2^n at n full letters, each row of R is a full line, and two rows of R are adjacent exactly inside a block of 2^j rows at j the terminal run of full letters, so comp = 2^(n-j); at equal letter frequencies the exponent is (log 2)/2 against per-letter values 0, 0 and a fill ceiling of (3/2) log 2, strictly between them, which no other named pair achieves. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Proved] On every one of those 59 pairs, along any word in which both letters occur with positive frequency, the component growth rate exists, is a function of the letter frequency vector alone, and is read off the closed form; existence is earned from the formula and not assumed. The hypothesis is load-bearing and the same statement without it is false: over {3, 6} at frequency vector (1, 0) the constant word 3^level has rate 0, the word carrying the diagonal letter at the square places has rate log 2, and the word carrying it at the powers of 2 has upper rate log 2, lower rate (log 2)/2 and no limit, the last two having orbit closures that are uniquely ergodic but not minimal. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Proved] The constant word is a degenerate probe for the component exponent: comp(A_(c^level)) = comp(A_c)^level with comp(A_c) = 2 exactly on the diagonal class, so the one linear functional exact on constant words is Phi(f) = (f_6 + f_9) log 2, and it fails at every interior frequency on 7 of the 9 named class pairs, holding only for a unit against a diagonal and for the gasket class against the full tile; the failure is a wrong shape rather than a wrong coefficient, since the true exponent extends to no vertex. The caveat travels with the result: on five of the seven, 28 of the 32 letter pairs, the exponent equals the fill exponent and so saturates the trivial ceiling comp <= fill, carrying nothing the order-blind fill law did not already give. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Verified] The Thue-Morse run structure that makes the exponent exact: no three equal letters in a row, so every terminal run has length at most 2 and every prefix rate is within 2/level of its limit; exactly level/2 of each letter at every even length, not merely in the limit; over all lengths to 2^20 the terminal run of one letter takes the value 0 on 524288 prefixes, 1 on 349526 and 2 on 174762; the first 2^20 letters hold 349525 complete runs of length 1 and 349525 of length 2 with one unfinished run at the cut; and the run-boundary word t_n xor t_(n+1) is the period-doubling word on all 1048575 terms. Witness: lab/rs/magic-words.
• 2026-08-31 [Proved] The remaining 46 letter pairs carry exact closed forms too, so all 105 are solved, and the mechanism is a suffix recursion rather than anything spectral: a heavy suffix letter, meaning one of the five connected codes 7, 11, 13, 14, 15 that carry both contacts, leaves the count unchanged because the block graph of A_w (x) A_c is isomorphic to the cell graph; a zero-contact suffix letter, meaning one of 1, 2, 4, 8, 6, 9, leaves isolated cells and collapses the count to a fill; and a domino suffix letter turns the count into the number of maximal runs, whose recursion H(A_wq) = H(A_w) + fill(A_w) at a gasket letter telescopes. Hence comp(A_w) = fill(A_(w_1..p)) at p the last zero-contact place on the 30 pairs of a heavy letter against a light one, giving 3^(g-j), 2^d 3^(g-j), 4^(F-j) and 2^d 4^(F-j) with j the terminal heavy run, and comp(A_w) = 1 + sum of fill(A_(w_1..i-1)) over the gasket places i at or before the last domino place m on the 16 gasket-against-domino pairs, (3,7) and (5,7) among them, the eight column-domino pairs being the transpose of the eight row-domino ones; exhaustive on all 753572 words of length at most 13 against the representation and all 11684 of length at most 7 against the drawn cells, zero mismatches. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Proved] At every interior letter frequency the component exponent exists, is order-blind and equals the fill exponent f_1 log fill_1 + f_2 log fill_2 over the two letters on all 46 of those pairs; existence and order-blindness hence hold on all 105, saturation on 89 of them; the interior hypothesis is used exactly once, to force the terminal run of the heavy letter to be o(level), since a run of eps level would freeze the other letter's count on that block and contradict its positive frequency, and the gasket-against-domino case runs through the sandwich T < comp(A_w) <= 1 + (3/2) T at T = fill(A_(w_1..i*-1)) and i* the last gasket place at or before the last domino place, measured in [1.0004, 2.0000] on six named words with no violation. Nothing is claimed for a word whose letter frequencies fail to exist. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Proved] The two rates the study could not identify are identified exactly: along the Thue-Morse word over any of the 16 gasket-against-domino pairs, (3,7) and (5,7) included, the component exponent is (1/2) log 6 under either letter reading, with the two-sided certificate |log comp(A_(w_1..level)) - (level/2) log 6| <= log 108 + (1/2) log(3/2) < 4.885 at every level >= 4, because the word is cube-free, which caps the sandwich suffix at fill/T in [6, 108], and balanced, which pins the gasket-letter count to within 1/2 of level/2; the study prints the value as 0.895879734614027 nats, a float labelled as such. Measured to level = 2^14 on all 16 pairs and both readings the largest deviation is 4.273459 nats against the certificate's 4.884864. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Proved] The frequency functional Phi(f) = (f_6 + f_9) log 2 is refuted at every interior frequency on 78 of the 105 letter pairs and exact on 27, the 46 new pairs being refutations to a pair; and the exponent saturates the trivial ceiling comp <= fill on 89 of the 105 and falls short on 16, so the domino against the full tile is the unique class pair on the whole alphabet, and not merely among the named 59, whose exponent sits strictly between the constant-word values and the fill ceiling. The comparison is against the value of Phi and never against a periodic word. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Proved] The interior hypothesis is sharp on a pair carrying no diagonal letter, so the pathology is not a property of the diagonal class: over (3, 7) at frequency (1, 0) the constant word 3^level has rate 0, the gasket at the square places gives log 2, and the gasket at the powers of 2 has upper rate log 2, lower rate (log 2)/2 and no limit, with comp pinned to fill(A_(w_1..i*-1)) at i* = 2^k throughout (2^k, 2^(k+1)], so the accumulation set of the prefix rate is the whole interval [(1/2) log 2, log 2] and not its two endpoints. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Proved] A common invariant cone exists and the gasket-against-domino pair is primitive in it, and none of it is needed: phi = (1,2,2,4)^T is a common right eigenvector with M_c phi = k_c phi, so it normalises the row orbit and only that orbit, and in the resulting chart comp/fill = 1 - b - c with the letters acting by N_gasket(b,c) = ((1+b)/3, (1+c)/3) and N_domino(b,c) = ((1+b)/2, 0); the set {0 <= b <= 1, 0 <= c <= 1/2, b + c <= 1} is invariant under both and the length-3 word gasket-domino-gasket maps it strictly inside, vertex images (5/9,1/3), (11/18,1/3), (5/9,1/3), (7/12,1/3) with b + c at most 17/18, three being minimal since domino-gasket sends (1,0) to (2/3,1/3) on the face. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Verified] A by-product and a smoothness split: comp(A_((7,3)^k)) = (6^k + 4)/5, reading 2, 8, 44, 260, 1556, 9332 and checked to k = 8 against both the closed form and the representation, a stationary control whose per-letter rate is (1/2) log 6 again; and since every closed form on the other 89 pairs gives a count of the form 2^i 3^j, the gasket against a domino is the only family whose counts are not smooth, the largest at level = 8 over (3, 7) being 1094 = 2 x 547. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Conjecture] Which words with no letter frequencies carry a component exponent at all, now that all 105 letter pairs have closed forms and interior frequency settles the rate: over (3, 7) the tripling word W_(k+1) = W_k 7^|W_k| 3^|W_k| keeps both letters at lower density 1/4 and still has its prefix rate range over [0.4792, 1.4379] in log 2 units on 1024 <= level <= 4096 with no narrowing, so positive lower density is the wrong hypothesis, but that is measurement and not a proof that the limit fails. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Conjecture] The accumulation set of the saturation comp/fill along Thue-Morse over a gasket-against-domino pair, plausibly the attractor of the two affine chart maps read along the word; the sampled value 0.2325367033 at level = 4096 is a term of an oscillation and is neither a limit nor a maximum, the exact maxima at level >= 5 being 43397/186624 and 151/648 under the two readings. Witness: lab/rs/magic-words, connectivity.md.
• 2026-08-31 [Conjecture] Whether any alphabet of three or more letters, or any other order-sensitive observable, makes the component-style exponent depend on more than the letter frequencies, which is what a non-stationary result would need; at interior frequency on two letters it provably does not, on any of the 105 pairs. Witness: magic.md, connectivity.md.
• 2026-08-31 [Conjecture] Whether the same suffix recursion that closes the 46 reproduces the 59 forms proved by other means, which would collapse the whole table to one lemma, and whether the Euler, boundary and holes series of Hankel ranks 4, 8 and 11 answer to it as well. Witness: connectivity.md.
• 2026-08-31 [Refuted] That a difference between the component growth rate and the frequency-average prediction is a non-stationary result - on the whole settled class the exponent is a function of the letter frequencies alone, so Thue-Morse returns exactly what a periodic word of the same frequencies and almost every Bernoulli word return, and the tree's own stationary controls miss the prediction by the same amount; the difference refutes the frequency functional, not stationarity, and an aperiodic word witnesses nothing here. Witness: connectivity.md, lab/rs/magic-words.
• 2026-08-31 [Refuted] That the component exponent is a function of the letter frequencies for every word whose frequencies exist - over {3, 6} the frequency vector (1, 0) carries the constant word at rate 0, the diagonal-at-squares word at rate log 2, and the diagonal-at-powers-of-2 word with no rate at all; the statement holds only where both letters have positive frequency, and the same correction restores every family's closed-form rate. Witness: connectivity.md, lab/rs/magic-words.
• 2026-08-31 [Refuted] That the top Lyapunov or matrix-norm exponent of the cocycle is the component exponent, and with it every route to chi through a norm theorem, a joint spectral radius or a projective contraction of forward orbits - along 3^inf the largest entry of M_3^level is exactly 2^(level+2) - 2, reading 6, 14, 62, 1022, 262142, 17179869182 at level = 1, 2, 4, 8, 16, 32, so the norm exponent is log 2, while comp(A_(3^level)) = 1 at every level and the component exponent is 0; the observation functional gamma is a fixed vector of both heavy matrices and so sits on the boundary of the dual cone, which is the geometric form of the same obstruction. Witness: connectivity.md, lab/rs/magic-words.
• 2026-08-31 [Refuted] That entrywise positivity in the standard basis decides whether the letter matrices preserve a common cone - neither M_3 nor M_7 is a non-negative matrix, so the test never applied, and none of the 8190 products of length at most 12 is entrywise positive, yet a common invariant cone does exist in the chart normalised by the right eigenvector phi = (1,2,2,4)^T and the pair is primitive in it at length 3. Witness: connectivity.md, lab/rs/magic-words.
• 2026-08-31 [Refuted] That the Thue-Morse word is a named word along which the component exponent fails to exist - it converges, exactly, to (1/2) log 6 on every gasket-against-domino pair with a two-sided certificate; the genuine non-existence witness is the gasket at the powers of 2 at the boundary frequency (1, 0), whose orbit closure is countable and not minimal, so no uniquely ergodic minimal word is implicated. Witness: connectivity.md, lab/rs/magic-words.