gasket-witness-weights

• Reparametrises the gasket residual R(level) by the witness z instead of by the multiplier pair (s,t): every off-diagonal collinear pair of G_level is (sz, tz) with gcd(s,t) = 1 and z a positive integer vector, uniquely.
• Checks the free-digit automaton B(s,t) is a constrained tensor square of a single-coordinate carry automaton C(s,t): T = S (x) S - U (x) U - V (x) V + W (x) W, so B needs no four-tuple state graph.
• Checks the box construction P_level(s,t) = #{z >= (1,1) : z_1 + z_2 <= (3^level-1)/(2 max(s,t)), sz, tz in G_level} against B(s,t) and against the tensor iteration at multipliers where a forward build of B blows up.
• Checks the witness weight w = z_1 + z_2 is never below 4, that the weight layers scale exactly as R_{3w}(level) = R_w(level-1), and that the weight-four layer is counted by the no-adjacent-ones set F_level with |F_level| = Fib(level+1) - 1.
• Checks every multiplier pair above (3^level-1)/10 contributes exactly 4 ordered collinear pairs.
• Checks the golden ceiling M_level(z) <= Fib(level+1) - 1 on every coprime direction in a box, at every level to 40, with z = (1,3) the only attainer, and on six families chosen to favour a breach at every level to 45.
• Carries the second moment E(level) and the residual R(level) to level 17 by grouping all 3^level points by primitive direction, and checks both R/3^level and R/phi^(2 level) fall at every level through 17.
• Rebuilds the ray automaton in the direction coordinate: a multiplier word is a word over the increments {0, z_2, -z_1} summing to zero, so M_level(z) + 1 counts the closed paths of a carry automaton whose states live in [-z_1/2, z_2/2].
• Checks the branch structure that turns the golden ceiling into a theorem: out-degree at most two, branch states in one residue class mod 3, and the two successors of a branch state differing by q/3 where q is the unique member of {z_1, z_2, w} divisible by 3.
• Checks the proved cases over the box and over the de-duplicated union of the six adversarial families, names the directions left over and runs them to level 60, and verifies nine explicit Fibonacci certificates in exact integer arithmetic.
• Checks the renewal criterion Sum_{j>=2} f_j Fib(level+1-j) <= Fib(level-1) on the first-return counts of every occupied direction of the box.
• Solves the golden potential in exact Q(sqrt5) arithmetic: u(0) = 1 and phi u(c) = sum of u over the successors of c for every live c != 0, read off as U(z) = sum of u over the successors of the start state other than itself.
• Checks the criterion U(z) <= phi^-2, which implies M_level(z) <= Fib(level+1) - 1 at every level, on the box, the six adversarial families and a stress list of high v_3 directions, and confirms every solution against both inequalities of the criterion rather than only against the linear system.
• Checks the occupancy residue law: writing q = 3^k q1 for the coordinate divisible by 3 and p for the other, a direction carrying mass at any level has q1 = p mod 3, so the congruence alone empties a large share of the directions with 3 | z1 z2.
• Checks the short first returns: no first return has length between 2 and v_3(q); f_2 is nonzero only at {1,3} and f_3 only at {1,9}, {1,12}, {3,10}, {4,9}, each equal to 1.
• Checks the degree potential: pi(c) = 1 where a live state has out-degree two, phi^-1 where it has out-degree one, pi(0) = 1, is a super-solution of the golden criterion, and sweeping it under the same operator gives a decreasing chain of exact upper bounds on U; the sweep runs over consecutive depths, so the depth reported is the least one that works.
• Checks the burst identity that blocks every valuation-graded potential: U is phi^-(k-1) times the sum of u over the 2^(k-1) burst-floor states, all of valuation 0, while u(p) = phi^-1 at the valuation-0 state p.
• Checks the golden partition bound at v_3(q) = 1: U <= phi^-1 (1 - phi^-max(t,2)) with t = v_3(q1 - p), hence U <= phi^-2 on the arithmetic class t <= 2.
• Checks the two automaton-free restatements of the criterion: Sum_level (M_level + 1) phi^-level <= phi^4, and Sum_m phi^-l(m) <= phi over the multipliers m of the direction, where l(m) is the number of base-3 digits of (z1 + z2) m.
• Checks the route that dies: the state maximum G(level) = max_c N(c,level) does not obey G(level) <= G(level-1) + G(level-2).

RUN

• uv run --with numpy python research/lab/py/gasket-witness-weights/witness.py
• About 165 seconds on one core, peak under 3GB at level 17; prints one line per law and exits nonzero on the first failure.
• numpy is used only for the level 15 to 17 array sort; every other law is stdlib.

WITNESSES

• 473 coprime pairs below 40: the tensor square reproduces the B(s,t) return counts at every level to 9, zero mismatches.
• Carry states against reachable B states: (365,1094) 729 against 26931, (41,122) 81 against 835, (25,52) 38 against 393, (31,40) 35 against 354.
• 812 coprime pairs: the box construction agrees with B(s,t) at level 9, zero mismatches; it agrees with the tensor iteration at (365,1094), (41,122), (122,123), (1,2460) and (2431,2458) at level 9 and level 12.
• R(level) = 20, 88, 432, 1624, 5512, 15896, 46064, 124928, 335704, 863848 for level 4..13.
• R_{3w}(level) = R_w(level-1) on all 1869 weight layers divisible by 3 across level 5..13, zero failures.
• No witness of weight below 4 at any level to 13.
• R_4(level) = 12, 36, 108, 336, 988, 2596, 6672, 17480, 45720 for level 4..12, each equal to twice the number of ordered coprime non-3-power-ratio pairs drawn from F_level, and |F_level| = Fib(level+1) - 1 at each of those levels.
• Pairs above (3^level-1)/10 number 18, 57, 163, 402, 1019, 2702, 7060, 18607 at level 6..13; every one contributes exactly 4.
• M_level(1,3) = Fib(level+1) - 1 for level <= 40; over 13158 coprime directions with z_1 <= 120 and z_1 <= z_2 <= 240 there is no level <= 40 and no z with M_level(z) > Fib(level+1) - 1, and (1,3) is the sole attainer at level 40.
• Six adversarial families at every level <= 45, coprime counts 16940, 253, 2998, 1499, 1199, 1199, total 24088, zero breaches: binary base-3 pairs below 3^8, no-adjacent-ones pairs below 3^7, (1,t) with t < 3000, consecutive below 1500, (s,3s-1) and (s,3s+1) with s < 1200. The shelf script gasket-ray-machine/scripts/verify.py ships the same six with the binary family at 3^7, 11369 directions, to stay inside its time budget.
• E(level) = 4003372, 11679626, 34050692, 99800950, 292848756 and R(level) = 863848, 2211960, 5549452, 14100688, 35354824 for level 13..17.
• R(level)/3^level peaks at 0.8401158 at level 8 and falls at every level to 0.2737709 at level 17; R(level)/phi^(2 level) peaks at 3.2378233 at level 12 and falls at every level to 2.7724831; the level ratio R(level+1)/R(level) reads 2.5073119 at level 17, below phi^2 = 2.6180339.
• The direction automaton reproduces the gasket-digit automaton on all 947 coprime pairs below 40 at every level to 20, zero mismatches.
• Of the 13158 coprime z with z_1 <= 120 and z_1 <= z_2 <= 240, exactly 218 have a live automaton beyond the start state; every one has out-degree at most two, one branch class, and carries inside [-z_1/2, z_2/2], the largest live set 37 states. The branch argument settles 206 of them, 107 of which have v_3(q) = 1. The twelve left are (1,9), (1,27), (1,81), (1,90), (4,117), (9,73), (9,82), (9,235), (10,81), (13,108), (27,217), (27,226).
• The first three of those twelve are shift rays, closed by Fib(p+2) Fib(q+2) = Fib(p+q+3) - Fib(p+1) Fib(q+1); the other nine carry Fibonacci certificates of denominators 18, 40, 381, 18, 2013, 18, 40, 2013, 34.
• The six adversarial families overlap: their 24088 coprime members are 23435 distinct directions, of which 717 lie in the box and 22718 are new. Of the 22718: 20945 have zero mass, 1693 fall to the branch argument, 3 are shift rays, and 77 are left to the enumeration. Those 77 hold at every level to 60, zero breaches, worst ratio to the ceiling below 0.1516.
• The renewal criterion holds on all 218 occupied directions of the box at every level to 46; f_1 = 1 everywhere and f_2 = 1 only at (1,3).
• M_level(z) <= D_level(w) on all 829 coprime directions with z_1 <= 30, z_1 <= z_2 <= 60 at level 12; the bound is far weaker than the ceiling, D_24(w) = 4196351, 1683971, 613817, 228519 at w = 4, 10, 28, 82 against Fib(25) - 1 = 75024.
• The state maximum at (1,9) runs 1, 1, 1, 2, 4, 6, 9, so G(4) = 4 > G(3) + G(2) = 3; the recursion fails for 8 directions of the box.
• The golden potential is nonnegative and satisfies both criterion inequalities on all 218 occupied directions of the box; 214 have U <= phi^-2 and the four failures are the shift rays (1,3), (1,9), (1,27), (1,81), each with U = phi - 1 exactly.
• U takes 57 distinct values over the box, in order phi - 1 on the four shift rays, 2 - phi at (1,12), (3,10), (4,9), then (9 phi - 14)/2 at (1,90), (9,82), (10,81).
• Over 36037 directions - the box, the six families and a stress list of high v_3(q) directions - 1995 are occupied and 1987 satisfy U <= phi^-2; the 8 failures are exactly the shift rays (1,3^j) for j = 1..8, no direction has U in the open interval (2 - phi, phi - 1), the maximum among the passing directions is 2 - phi attained only at (1,12), (3,10), (4,9), and the largest live set is 256 states.
• Of the 13158 coprime box directions, 6566 have 3 dividing neither coordinate and carry no mass at any level; 6374 more have 3 dividing a coordinate and still carry none; 218 are occupied.
• Adversarial cross-check on all 218: f_1 = 1, the exact partial sum Sum_{j=2}^{46} f_j phi^-j from the independent first-return enumeration is dominated by phi^-1 U from the linear solve, and U <= phi^-1 at every direction, with equality only on the shift rays.
• Of the 36037 directions, 20193 have 3 | z1 z2 and 15556 of those match q1 = p mod 3; all 1995 occupied directions match, zero violations, so the congruence empties 4637 directions with no automaton built.
• No occupied direction has a first return of length between 2 and v_3(q), zero failures over 1995.
• The burst identity U = phi^-(k-1) Sum_m u(q1 m) over the 2^(k-1) burst-floor states, and u(p) = phi^-1 whenever 2p <= q, hold exactly on all 1995.
• The degree potential is a super-solution on 1968 of the 1995 occupied directions, 1902 of which have no double branching and 66 of which do, so the tool reaches strictly past the branch case.
• Sweeping it settles U <= phi^-2 on 1966 directions - least depth 1 on 1804, 3 on 101, 4 on 44, 5 on 11, 6 on 6 - and leaves 29: the 8 shift rays (1,3^j) and (1,756), (1,2196), (1,2214), (1,2268), (1,2430), (9,2188), (10,2187), (13,1080), (13,3267), (27,730), (27,2188), (28,729), (28,2187), (40,1053), (81,2188), (82,2187), (91,2214), (121,3159), (243,2188), (244,2187), (819,2539).
• 757 occupied directions have v_3(q) = 1; the bound U <= phi^-1 (1 - phi^-max(t,2)) holds at every one, is attained exactly at (1,12) and (3,10), and 512 of them fall in the class t <= 2 where it gives U <= phi^-2 with zero failures.
• The two automaton-free restatements agree with the linear solve on 111 directions with z1 < 40, z2 < 120, checked in exact Q(sqrt5) arithmetic.