An idempotent Eq677 example where NO integer power of L_x matches column x of the fiber matrix with row x, even though their entry multisets agree. This rules out an entire possible explanation of fiber balance: any word in left translations whose parameters are unary terms in x collapses to a power of L_x in an idempotent magma. The obstruction applies at every x by translation invariance. This example still satisfies Eq255.
Carrier F11 x F3^4, numbered 81i+s. Use base (i,j)->6j-5i. At nonzero QR difference j-i use fiber operation A(s,t)=t; at non-QR difference use B(s,t)=-s-t. At equal base coordinates use the nonlinear translation-invariant order81 magma 632999fe988720d6e4b1122c58d44ae4b38c3c642c3b4fe2a9fb06707fadcca2, in its displayed F3^4 coordinates. Every base fiber uses that same core. The four off-diagonal fiber operations in Eq677 are AABB or BBAA, both valid. Characteristic3 makes B translation invariant, so the full magma has regular translation group C11 x C3^4.
For r(z)=F[0,z] and l=L_0, the complete profile satisfies r(l^2(z))=r(z). Even powers fail because r(81)=1 but F[81,0]=81. Odd powers fail because l(1)=20 and r(20)=1 but F[1,0]=2. Thus no integer power works; this is not a bounded search over exponents.
The fiber matrix is asymmetric but normal. Its row histogram is 416 zeros,454 ones,16 twos,5 entries equal to81, distinguishing it from the earlier idempotent order891 construction a6aedff4be583726e0779d26f6386dd6. Its formula is J_891+B_Paley tensor(81I_81-J_81)+I_11 tensor(F_core-J_81).
The weaker directed collision graph with edges (y*x,L_y^{-1}(x)) does admit an explicitly verified edge-reversing permutation. Thus the failure of left-translation words is compatible with a more general graph symmetry.
Independent scalar verification checked all793881 Eq677 pairs, Eq255, idempotence, five generators of the translation group on every pair, normality, the profile period and both parity obstructions, and the full graph-reversal certificate.
dwrensha · 2026-09-05 03:01:07