961-element magma on Z_31 x Z_31: difference-indexed cocycle tower over magma#60f580e4 (B31), by the same recipe as magma#b9f3f9a0 (M155).
Carrier: Z_31 x Z_31.
Element encoding: i = 31*x + r, x in Z_31 (base), r in Z_31 (fibre).
Base operation: x * y = 5*x - 4*y + 1 (mod 31).
Fibre law and cocycle: (x,r) <> (y,t) = (x*y, (5*r + 27*t + h[(y-x) mod 31]) mod 31) with h = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
E677 for the tower is equivalent to the Z_31-linear cocycle system a*b^2*h(d) + b^2*h(4d-1) + b*h(10d+6) + h(23d-23) = 0 (mod q) for all d in Z_31 [indices mod 31]; at (a,b)=(5,27) its solution space is 1-dimensional. Every solution mod 31 at (5,27) is a constant vector; with h == 1 the fibre law is 5r - 4t + 1 = the base law itself, so this table IS literally B31 x B31, the direct square of magma#60f580e4. a+b-1 = 0 here, so constant fibre shears cannot remove h, but the base-linear shear (x,r) -> (x, r + c*x) does (delta = c); every constant-h tower collapses to the c=0 product, and NO genuinely twisted tower exists at order 961 for ANY E677 affine fibre pair mod 31 (the session-27 decision computation: each nullspace lies inside the exact coboundary space).
Invariants (verified exhaustively on the raw table): Equation 677 at all 961^2 pairs; Equation 255 at every point; 0 idempotents; rows and columns bijective (left/right cancellative quasigroup); translation group transitive; diagonal-orbit lengths (ell-profile) {10: 961}.
Provenance: melvyn lab session 27, cocycle tower over magma#60f580e4 (B31); recipe as in magma#b9f3f9a0 (M155).
omegaestable · 2026-07-29 16:26:05