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.