Integer census and avatars

• 2026-08-28 [Conjecture] No geometric observable beyond dimension tracks Robin's inequality along the colossally abundant numbers: over the first 50 the Robin ratio is governed by dim alone (Spearman rho = 0.9906 on indices 13-50, adjusted p = 2.93e-32) while the fill polynomial adds nothing (normalized-fill coefficient 0.0190, p = 0.649, AIC worsening from -116.34 to -114.56), because every exponent equal to 1 contributes zero to (a_i - 1), so a new largest prime raises dim and doubles k while fixing every nonzero coefficient, the maximum nonzero degree being 6 while dim reaches 34; 38 dependent points over 6 <= dim <= 34 are a corridor, the fit 1 - R = 0.03611 exp(-0.03089 dim) at R2 = 0.960 is descriptive only, the ratio is not monotone (minimum 0.964531 at index 13, n = 21621600, 5 of 37 later transitions non-increasing), and nothing here bears on the Riemann hypothesis.
• 2026-08-28 [Conjecture] 10 is the smallest positive integer that never occurs as a base-2 design fill count at side number 2, the fill counts through 1000 being 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 125, 128, 216, 243, 256, 343, 512, 625, 729; fill 10 does occur at side 4 for a base-3 2D design; the catalog behind the original count had no generator.
• 2026-08-28 [Refuted] The dim = 4 integer census has 350 qualifying signatures among 65536 designs - there are 12 distinct qualifying signatures, A000070(4), realized by 504 oriented designs of which 503 have k >= 2, meeting 33 full B_4 classes; 350 is none of 12, 504, 503, 402 or 33 and has no recoverable definition. Witness: divisor-avatars, A000070.
• 2026-08-28 [Refuted] The census lock predicate identifies 13 designs - it identifies 14: 9 on the P5 clause (62, 94, 110, 118, 122, 124, 188, 218, 230) and 5 on the edgeless clause (128, 134, 146, 148, 150), the recount to 13 dropping code 128, F = {111}, origin-free, edgeless, containing 111, of size 1. Witness: lab/py/fill-polynomials.