smith-cascade

• Builds the even-convention carry matrix M_even at base 3, the odd carry parameter D = 2R + 1, size n = R + 1, from the digit polynomial P = (sum_k C(D-1,k) x^(2k))(1 + D x + x^2), entry P[c + D - 3c'] + P[-c + D - 3c'].
• Computes its 2-adic Smith divisor valuations a_i by min-valuation pivoting modulo 2^256, at every odd D = 3..511.
• Reads off v_2 = sum a_i, the layers L_j = #{a_i >= j}, the nullity L_1 against the Jacobsthal tent min_t |D - t|/2 + 1, t in {2J(k)+1, 2J(k)+3}, the excess X = v_2 - L_1, and a_max.
• Tabulates per octave [2^k, 2^(k+1)) the maxima of L_2, L_3, L_4, L_5, a_max, X and the slack v_2 - ceil(n/3), and the min-of-cones shape of layers 1, 2, 3.
• Cross-checks v_2 against an exact Bareiss determinant at D <= 61, the three largest rows at precision 1024, and the direct pencil det(fill I - 3 M_even) at D = 7.

RUN

uv run python research/lab/py/smith-cascade/smith_cascade.py

About three minutes. Domain is the page domain, odd D = 5..511, 254 rows.

WITNESSES

• research/claims/:225 254 rows; octave maxima L_2 = J(k-2), L_3 = J(k-4), a_max = floor(log_2 D) + 4; L_4 <= 1; cones at layers 1, 2, 3; X octave maxima 6, 6, 7, 8, 10, 17, 28; v_2 <= ceil(n/3) + 9 with slack 5, 5, 5, 7, 7, 9, 9, extremal 255, 257; v_2 <= n at odd D >= 9, equality 9, 15, violations 5, 7; class D = 1 mod 6 84 rows 13..511, v_2 <= n - 3, ratio 7/18 at D = 19; 95 < 99 at D = 511
• research/claims/:281 tent law 255/255 at odd D = 3..511; max a_i <= 9 first fails at D = 127, 12 by 511; #{a_i >= 3} = 1 first fails at D = 175, reaches 5; #{a_i >= 2} <= 5 first fails at D = 183, reaches 21; D = 7 profile {0,0,0,7}, D = 5 profile {0,0,4}
• research/claims/:210 D = 7 has v_2(det) = 7 >= 6 and det(fill I - 3M) != 0

NOTE

• The octave-2 rows D = 5, 7 break both octave laws: max L_2 = 1 against J(0) = 0 and a_max = 7 against 6; the laws hold on octaves 3..8.
• Every non-spike divisor has a_i <= 3, so L_4 <= 1 is the spike alone and L_5 = #{a_i >= 5} is 1, not 0, at 233 of 254 rows; L_5 = 0 only if the spike is excluded.
• v_2/(D-1) at D = 13 is 1/3, not 7/18; the slack 9 is also attained at D = 511.