question-mark

• The Stern-Brocot tree and Minkowski's ? read in the three-slot language of beneath, ## The question mark: a base-2 design whose value is a 2 x 2 matrix product, the map that swaps the continued fraction place slot for the dyadic one, and the two dimensions of every run-length rule of the memory dial at dim 1, widths 2 to 4.
• carry checks v(2n + b) = v(n) M_b on v(n) = (s(n), s(n+1)), s the Stern diatomic sequence, M_0 = [[1,1],[0,1]], M_1 = [[1,0],[1,1]], and the closed form v(n) = (0,1) M_(d_1) ... M_(d_L) over the digits of n most significant first, both exhaustively below 2^span; reads s against the terms A002487 lists, fetched from the public OEIS mirror on GitHub and skipped when offline; prints M_0 M_1 and M_1 M_0; finds the least pair of words of equal length and weight with different s, once on the digits of n without a leading zero and once on all 2^L words of a length, and prints the level-2 cells; and checks the Stern-Brocot row against the Calkin-Wilf row s(n)/s(n+1) under bit reversal of the position, depths 0..depth.
• question computes ? on every node of the Farey subtree of [0,1] to depth depth by the mediant recursion ?((p+p')/(q+q')) = (?(p/q) + ?(p'/q'))/2 and checks, exactly in rationals, the Denjoy formula ?([0; a_1, a_2, ...]) = sum_i (-1)^(i+1) 2^(1 - a_1 - ... - a_i), the tent conjugacy ?(F(x)) = T(?(x)), the branch law ?(1/(a + x)) = 2^(-a)(2 - ?(x)), and the address law ?(node i of row d) = (2i + 1)/2^(d + 1). It then takes every run-length alphabet A at widths 2..4, builds its code, and compares the accepted words of length level with the length-level prefixes of the binary words 0^(a_1 - 1) 1^(a_2) 0^(a_3) ... over A, printing the leading runs of the surplus.
• table first runs the controls and stops on failure: the pressure zero of A = {1,2} against 0.5312805062772051416, the same at 24, 32, 48, 56 modes, A = N against 1, and A = N\{1} at four tail cuts. Then, per run-length code, it prints the alphabet, the polynomial P_A, rho from the transfer matrix of the code under the note's own convention, log_2 rho, the pressure zero dim_CF, and the Holder floor alpha log_2 rho with alpha = log 2 / (2 log phi), asserting on every row that rho is the largest root of P_A, that P_A divides the exact characteristic polynomial of the transfer matrix, and that dim_CF >= alpha log_2 rho.
• P_A is x^m - sum_(a in A) x^(m - a) for finite A with m = max A, and x^(f+1) - 2x^f + (x - 1) sum_(j in F) x^(f - j) for A = N \ F with f = max F, the polynomial cleared from sum_(a in A) rho^(-a) = 1.
• dim_CF is the zero of s -> log lambda(s), lambda(s) the leading eigenvalue of L_(A,s) f(x) = sum_(a in A) (a + x)^(-2s) f(1/(a + x)), collocated on modes Chebyshev-Lobatto points of [0,1] through barycentric interpolation and found by bisection on [0, 1] for finite A and on [0.51, 1.1] for cofinite A. A one-letter alphabet has lambda(0) = 1 exactly and prints 0 without bisection. For cofinite A the sum runs explicitly to a = cut and the tail a > cut is sum_(j <= taylor) f^(j)(0)/j! zeta(2s + j, cut + 1 + x), taylor + 1 terms, Hurwitz zeta on the Taylor coefficients of the interpolant at 0.
• obstruction counts, per width, the run-length codes, the codes fixed by both the digit flip and reversal, those among them with no alphabet, and how many of those carry rho > 1, listing every orphan at widths 2 and 3 and the least live one at width 4.
• graph builds, for any dim 1 code, the graph-directed continued fraction form of beneath, ### Every rule is a graph-directed continued fraction set: states the 2^(k-1) words of k - 1 digits, an edge u -a-> v when a copies of the digit opposite to the last digit of u close only allowed windows and leave state v, the labels a >= k folded into one cofinite edge to b^(k-1). It keeps the recurrent states, the states on a cycle, and finds the pressure zero of the matrix operator (L_s F)_u(x) = sum_(u -a-> v) (a + x)^(-2s) F_v(1/(a + x)) by bisection on the log of its largest real eigenvalue, on [0, 1] when every cofinite edge is transient and on [0.51, 1.1] otherwise; a graph whose recurrent states each carry one label prints 0. First it recovers the pressure zero of all 18 run-length codes from the graph form and stops if any gap reaches 1e-12; then it compares the accepted words of code 11892 at length level with the prefixes of the run words over {1,2} without the pair 22, prints that code's recurrent graph, and prints the 19 width-4 orphans with rho > 1, code 11892 first, and the four named codes 7, 23, 54, 127 as parity-constrained continued fraction sets, asserting dim_CF >= alpha log_2 rho on every row.
• subleading continues L_(A,s) to complex s at A = {1,2}, (a + x)^(-2s) = exp(-2s log(a + x)) on the same collocation, and locates the zeros of det(1 - L_s) off the real axis: a scan of abs(det) on a grid of the box, Newton with a central-difference derivative from every local minimum, duplicates dropped, and the winding number of det around the box asserted equal to the number found. The zero s_1 = sigma + i tau of largest real part in 0 <= sigma <= 0.53, 0.2 <= tau <= 14 at modes modes is refined at 60, 100 and 140 modes, the box is widened to tau <= 80 at 100 modes, and the verb prints the period pi/(tau log 2) in octaves and the amplitude ratio per octave 2^(2(sigma - delta_2)) that a term Q^(2 s_1) of the E_2 count contributes, with the alias 2 tau log 2 - 3 pi that integer-octave sampling sees. It then imports census_walk from lab/py/ford-horocycle, walks the E_2 circles to 2^jmax, samples N_2(Q)/Q^(2 delta_2) at per points per octave from 2^jlo, fits a constant alone and a constant plus the wave of s_1 with only the constant, amplitude and phase free, prints both residuals, the observed octave exponents minus 2 delta_2 beside the fitted wave's at every octave, asserting agreement within 4e-3, and a free fit of sigma and tau on the same points; last it runs the control A = {1,2,3} in 0 <= sigma <= 0.71, 0.2 <= tau <= 80 at 100 modes, the winding number asserted, against the m = 3 walk to 2^18.
• The transfer matrix is the note's: state s the last k - 1 digits, edge s -> (2s + c) mod 2^(k-1) when bit 2s + c of the code is set, pinned on the four named codes 7, 23, 54, 127 before any row prints.

RUN

• uv run python mrlyprod/research/lab/py/question-mark/question.py carry, 0.2 seconds.
• uv run python mrlyprod/research/lab/py/question-mark/question.py question, 1.8 seconds.
• uv run python mrlyprod/research/lab/py/question-mark/question.py table, 29 seconds, almost all of it the six cofinite rows and their tail controls.
• uv run python mrlyprod/research/lab/py/question-mark/question.py obstruction, 0.4 seconds.
• uv run python mrlyprod/research/lab/py/question-mark/question.py graph, 32 seconds, half of it the six cofinite rows of the control.
• uv run python mrlyprod/research/lab/py/question-mark/question.py subleading, 59 seconds, 20 of them the E_2 walk to 2^24 and 15 the control's scan.
• uv run python mrlyprod/research/lab/py/question-mark/question.py all, 122 seconds.
• --span 16, --depth 12, --level 14, --modes 40, --cut 2000, --taylor 4, --jmax 24, --jlo 12 and --per 16 are the dials.
• Prints only, writes nothing, touches the network once, for the A002487 terms, and reads one sibling, lab/py/ford-horocycle/ford_horocycle.py, for the census walk.

WITNESSES

• The lines of beneath, ## The question mark, and the subleading paragraph of apollonian, ## The horocycle.
• 0 mismatches on v(2n + b) = v(n) M_b and on the digit product below 2^16; the 92 listed terms of A002487 agree; M_0 M_1 = [[2,1],[1,1]] and M_1 M_0 = [[1,1],[1,2]]; s(5) = 3 and s(6) = 2 on the words 101 and 110, the least pair without leading zeros, and s(3) = 2 against s(5) = 3 on 011 and 101 with them; level 2 reads 0, 1, 1, 2; 0 mismatches between the Stern-Brocot row and the bit-reversed Calkin-Wilf row through depth 12.
• 0 mismatches on 8193 Farey nodes to depth 12 for the Denjoy formula, the tent conjugacy, the branch law at a = 1..5 and the address law.
• At width 3, code 126, A = {1,2}, level 14: 1220 accepted words, 987 prefixes of ?(E_A), 0 missing, 233 surplus, every one with leading run 00; at code 219, A = N\{1}: 1220, 610, 0, 610, the surplus exactly the words opening on 1.
• The A = {1,2} control reads 0.5312805062772050 against 0.5312805062772051416, gap 1.1e-16, and moves by at most 8.9e-16 between 24 and 56 modes; A = N reads 1.0000000000000000; A = N\{1} reads 0.84088458641455 at every cut from 500 to 4000.
• The table: 18 codes on 11 alphabets, every row passing the three assertions, the same alphabet at two widths printing the same dim_CF.
• Orphans: 2 of 4 symmetric codes at width 2, 3 of 8 at width 3, 53 of 64 at width 4, of which 19 carry rho > 1, the least being code 11892, A = {1,2} with the pair 22 forbidden, at the supergolden rho.
• Graph: the graph form recovers the 18 run-length pressure zeros with a largest gap of 8.9e-16; code 11892 at level 14 has 378 accepted words, 277 prefixes of ?(M) for M = {1,2} without 22, 0 missing, 101 extra with leading runs 0 and 00, and a recurrent graph of 6 states and 10 edges with no cofinite edge; the 19 orphan rows and the 4 named rows of the note, 11892 at 0.416817764433, 48765 at 0.531280506277, 54699 at 0.785953471982, code 7 at 0.798858366966, every row above its Holder floor.
• Subleading: one zero of det(1 - L_s) at A = {1,2} in 0 <= sigma <= 0.53, 0.2 <= tau <= 14, s_1 = 0.457015235231 + 6.958882679527 i, moving by at most 8.9e-16 from 40 to 140 modes, the eigenvalue nearest 1 there 1.000000000000; 33 zeros in the box to tau = 80, the winding number agreeing, the next two 0.428067039 + 78.156951119 i and 0.412635450 + 71.206868515 i; delta_2 - sigma_1 = 0.074265, period 0.6513 octaves, ratio 0.9022 per octave, phase advance 3 pi + 0.2223 per octave; on 193 points of N_2(Q)/Q^(2 delta_2) from 2^12 to 2^24 the constant fit leaves rms 1.64e-02 and the wave of s_1 leaves 7.67e-04, max 4.05e-03, amplitude 0.131210; the twelve octave deviations from j = 12 to 23 are reproduced within 0.0034, +0.0475 against +0.0472 at j = 15 and +0.0124 against +0.0127 at j = 23; the free fit reads sigma = 0.4556, tau = 6.9598; the control A = {1,2,3} has 67 zeros in 0 <= sigma <= 0.71, 0.2 <= tau <= 80, the winding number agreeing, the largest real part at 0.489705291051 + 45.352143150104 i, 0.2160 below delta_3, factor 0.7413 per octave, and its census leaves rms 3.85e-04 to a constant, 3.83e-04 with the wave.