# 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.