Internal proof audit

September 9, 2026. Originating-agent review, with independent C++ and Python arithmetic implementations.

1. Scope and novelty. The universal conjecture was already disproved at q=3. The claimed result is an exact classification of all remaining prime-power cases plus counting estimates. Higher-degree and arbitrary-coefficient classifications are excluded.
2. Characteristic three. The discriminant is -lambda. Frobenius on an irreducible cubic is even, so this discriminant is a nonzero square. Since -1 is a square over F_(q^2), the primitive-constant requirement fails for every power of three.
3. Kummer extension. For p!=3, q^2=1 mod3. A noncube a gives a degree-three field, with alpha^Q=zeta*alpha and zeta primitive of order three. The basis expansion gives trace 3c and subtrace 3c^2-3auv.
4. Parametrization. Odd characteristic gives uv=-2/(9a), with one parameter in F_Q^*. Characteristic two gives uv=0 and two affine lines. Their intersection is the nonprimitive element 1; counting it twice is harmless for primitive counts and valid for the nonnegative weighted sieve.
5. No hidden subfield condition. The parameter values impose both coefficients identically. The primitive constant follows from the primitive norm and the fact that negation preserves generators over even or square-order fields. This is distinct from freeness in an unrestricted larger field.
6. Character theorem. The actual punctures are six disjoint simple zeros plus two poles in odd characteristic, and three simple zeros plus infinity for each even-characteristic line. Zero-orbit collision would require (-2)^3=1, excluded outside characteristic three. A unique simple zero yields a Kummer inertia exponent coprime to every character order. Fu--Wan's tensor induction is therefore geometrically nonconstant, tame, rank one, and weight zero. The domains omit exactly the rational poles, and none of the zeros is rational over F_Q. The resulting bounds are 6sqrt(Q) and 2sqrt(Q) per line.
7. Counting. Each primitive cubic contributes its three distinct primitive conjugates. The constant is automatically primitive. The characteristic indicator has W(e)-1 nontrivial squarefree-order contributions, proving the stated normalized errors.
8. Sieve. The pointwise e-free inequality is valid on multisets. Subtracting theta(ell) times the core count leaves W(k) character orders. The resulting bound is theta(k)[delta(C+B)-W(k)B(s-1+2delta)]. All positivity inequalities are strict.
9. Infinite cutoff. For omega<=15, exact rational inequalities use q>=1511. For 16<=omega<=56, exact primorial comparisons imply q exceeds the table threshold. At omega=57 the direct primorial bound holds, and all later primes exceed 64, proving the tail by induction. No finite table is extrapolated without that induction.
10. Finite witnesses. All 261 prime powers below 1511 outside characteristic three appear exactly once. Base quotient generators have exact order Q-1, proving those quotients are fields. The separate verifier checks gcd(X^Q-X,h)=1 and every prime divisor order test in the cubic quotient. The complete prime support of q^6-1 is independently regenerated from four small cyclotomic factors and checked to exhaust the integer. No probable-prime or numerical root test is used.

Run python3 check.py. The stored results include all 261 witnesses and the 56 finite sieve rows. The generator search.cpp is optional for reproduction; the verifier checks the supplied certificates directly and uses only the Python standard library. The broader preliminary trace/subtrace census was useful in discovery but is not needed to certify this paper's theorem.

Historical priority, outside proof assessment, and journal-level significance remain provisional. The cohomological character bound is cited, not reproved from first principles.