Degree-seven joint Witt shifted-trace closed form

Identifier: F:gf2-witt-shifted-degree-seven-closed-form
Proof route: cas-certificate
External status: unknown
Axiom footprint: finite-field-polynomial-semantics-v1, characteristic-two-degree-seven-population-formula, finite-abelian-fourier-orthogonality

Recorded description

For every q=2^r, r>=1, at (ell,n,c)=(3,7,2) the complete high-character correlation summed over exact low-twist conductor one is zero, while the exact-conductor-two high-character trace is q^9(q-1)^2. Its q-degree is 11, exactly one below the formal top degree 12.

Formal statement

{"field_order":"q=2^r","format":"axeyum-gf2-witt-shifted-degree-seven-closed-form","parameters":{"coarse_level":2,"ell":3,"n":7,"r":"r>=1"},"trace_layer_1":0,"trace_layer_2":"q^9*(q-1)^2","version":1}

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

gf2-witt-shifted-degree-seven-native-check

Kind: exhaustive-enumeration
Status: checked

Supports: The q-ary conditional-covariance derivation, both exact-conductor layers, their high-character normalization, and independent full enumeration over GF(2) and GF(4).

Checker command

cargo test -p axeyum-cas --all-features witt_shifted_trace

Evidence notes

The universal step substitutes the two-valued degree-seven class population into the exact conditional covariance. Enumeration at q=2 and q=4 is a mutation control, not quantifier evidence.

Provenance

{
  "date": "2026-08-21",
  "established_by": "Axeyum gf2-lemire lane",
  "source": "exact characteristic-two extension-field class populations and q-ary Fourier orthogonality"
}