# Divisor avatars

- 2026-08-28 [Proved] The Avatar Theorem: for `x = prod p_i^(a_i)` with `p_1 < ... < p_dim` and every `a_i >= 1`, the `dim`-axis design with `f_w = e_w(a_1 - 1, ..., a_dim - 1)` has `P(n) = d(x^n)` at every `n >= 0`, by the substitution `a_i n + 1 = b_i n + (n + 1)` and `prod (b_i t + 1) = sum_w e_w(b) t^w` at `t = n/(n+1)`; the map is injective by Newton's identities and surjective onto the `f_0 = 1` signatures whose fill splits completely into linear factors over `Q`, and a design exists iff `sum_i (a_i - 1) <= dim`; the first ten colossally abundant numbers `2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720` have avatars while the first without one is `21621600`; on the seven `dim = 3` ladders `x = 30, 60, 120, 180, 240, 360, 900` to `n = 20`, all 131 of the 140 powers exceeding 5040 satisfy Robin's inequality, the largest ratio `R(14400) = 1.5732599059` against `e^gamma = 1.7810724180`, margin `0.2078125121`, strictly decreasing in `n` on every ladder, which proves nothing about Robin beyond them; `sigma(m)/m < sigma(N)/N` for every `m < N` defines a superabundant number, not a highly abundant one. Witness: divisor-avatars.
- 2026-08-28 [Proved] The sigma-hunt is closed negatively for polynomial census laws: no census law of a design at odd side `2n + 1` equals `sigma_k(x^n)` for `k >= 1`, `x > 1`, since that grows at least like `x^(kn)` while every such law is a polynomial in `n` of degree at most `dim`, and none equals `sigma(x^n)/x^n`, strictly increasing and bounded hence not constant; at `x = 1` both collapse to the nine constant identities `O(n) = 1 = d(1^n)`; over the nine observables (fill, voids, surface, touched vertices, edges, faces, Euler characteristic, components, cycle rank) on all 22 least-mask representatives of the 256 base-2 3D designs counted through side 21, exactly eight non-fill strict divisor avatars survive, only for voids, Euler characteristic and components; `E` and `R` vanish at `n = 0` while `d(x^0) = 1`, so they are never strict identities, and the four graph laws for codes 30 and 126 hold only for `n >= 1`; if `sigma` is a design observable at all it lives among geometrically growing counts. Witness: divisor-avatars.
- 2026-08-28 [Verified] The eight non-fill divisor avatars come from topology, not measure: the empty design's voids `(2n+1)^3 = d(900^n)`; Euler characteristic and components both `(n+1)^3 = d(30^n)` for `{000}`; both `(n+1)^2 = d(6^n)` for `{000, 100}`; components `n + 1 = d(2^n)` for the three-corner path `{000, 100, 010}`; both `n + 1 = d(2^n)` for the square face `{000, 100, 010, 110}`; plus nine constant identities `O(n) = 1 = d(1^n)`, components for codes `23, 27, 31, 61, 63, 111, 127, 255` and Euler for 255; the statement is for the canonical least-mask representatives, since cube symmetry does not preserve the odd/even origin under coordinate reversal; codes 15 and 27 share a weight signature and a fill law, so eight orbits carry seven weight signatures. Witness: divisor-avatars.