Supplied cyclic-cover construction: verification status

Origin: external subscription review attributed by the user to Astra; exact model ID and effort unverified. The initial native panel did not receive it.

On M10(1,1,1,7), put E_C=V(2)⊕V(8), F_C=V(4)⊕V(6), and W_Q=ker Phi_5(T*). The operator A=T*+(T*)^-1 acts by a=(sqrt(5)-1)/2 on E and b=-(sqrt(5)+1)/2 on F. For a nonzero integral z0=e0+f0 in W, z_n=A^n z0=e_n+f_n is integral, |e_n| tends to zero, and f_n-z_n=-e_n. Rational vectors cannot lie wholly in either irrational eigenspace, since A is rational and its scalar eigenvalues are irrational.

If E is Hodge-isometric along all relevant paths, integral transport gives d_Z(C(g)f_n)≤|e_n|. Compact lattice spacing makes this an equality initially for sufficiently small |e_n|. A parallel section is Lipschitz with arbitrarily small allowed constant. The required logarithmic-time multiplicative increase would then exceed |e_n|, giving a contradiction.

Checked here: genus and zero orders; all nontrivial deck-character dimensions; EKZ's stated exponent formula giving 2/5; elementary cyclotomic and projection algebra. The two primary cyclic-cover papers were inspected at the relevant dimension/stratum/exponent statements and predate the received-date cutoff.

Remaining independent checks: marked-orbit and finite-cover construction; continuous strong-irreducible factorization satisfying the corollary; global Hodge-isometry on E; recurrent compact-return choice and all exact quantifiers. No complete independent certification of this stronger counterexample is claimed. It targets the direct-sum corollary, not a single strongly irreducible block. No counterexample to a main weak-mixing theorem follows from this construction.