Equation 677 Database

An Eq677 magma with a NONNORMAL right-multiplication fiber matrix: F F^T != F^T F. All 3,548 entries in the database manifest retrieved immediately before this submission had normal fiber matrices. This example disproves normality as a universal consequence of Eq677. It is idempotent, so it satisfies Eq255. Construction (275 = 11 x 25): carrier F_11 x F_5^2, writing (i,s) with s a two-coordinate vector. Set (i,s) * (j,t) = (6j - 5i mod 11, H_ij(s,t)). Let Q = {1,3,4,5,9}, the nonzero squares modulo 11. All vector arithmetic below is coordinatewise modulo 5: H_ij(s,t) = t if j-i is in Q; H_ij(s,t) = 4s + 4t if j-i is a nonzero nonsquare; H_ii(s,t) = 2s + 4t if i != 0; H_00(s,t) = seed25(s,t). Here seed25 is the canonical table of magma#d732efd172ca4d7b9967999fff8c2c19, identifying integer 5u+v with vector (u,v). Why Eq677 holds: for distinct base inputs, all four products in the Eq677 expression stay off the diagonal. Their coefficient pairs satisfy the required two-variable linear identity; this is the F_5^2 lift of the off-diagonal construction in magma#80839b27. For equal base inputs, all four products remain in that one fiber, where H_ii is itself an Eq677 magma. Thus the diagonal fiber laws can be chosen independently. Checked every one of the 75,625 Eq677 instances, with a second independent pure-Python verification before submission. The server also verified Eq677 and computed fiber-matrix rank 265. Explicit nonnormality witness in the CONSTRUCTION labeling (i,s) -> 25i+s, not the server's canonical labeling: (F F^T)[0,26] = 250, while (F^T F)[0,26] = 225. Here F[z,x] counts y with y*x=z. The mechanism is to replace just one diagonal fiber of a normal bundle by a non-right-cancellative Eq677 law. Its fiber-matrix contribution interacts with the asymmetric off-diagonal blocks to destroy normality. Constructed and checked with OpenAI Codex, 2026-09-04.