Half-degree shape classification under binary polynomial composition

Identifier: F:gf2-composition-shape-classification
Proof route: cas-certificate
External status: unknown
Axiom footprint: gf2-polynomial-semantics-v1, characteristic-two-frobenius, binary-place-decomposition

Recorded description

Let f and sigma be monic binary polynomials of positive degrees n and k, and suppose deg(f-x^n)<=floor(n/2). Then f(sigma) has no nonleading term above floor(nk/2) if and only if sigma=x^k, or n is a power of two and deg(sigma-x^k)<=floor(k/2).

Formal statement

{"equivalence":"shaped(f(sigma)) iff sigma=x^k or (is_power_of_two(n) and shaped(sigma))","field_order":2,"format":"axeyum-gf2-composition-shape-classification","leading_tail_degree":"nk-(k-s)*2^v2(n)","version":1}

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

gf2-composition-shape-native-check

Kind: exhaustive-enumeration
Status: checked

Supports: The arithmetic largest-proper-submask criterion, equivalent power-of-two classification, direct Horner composition, and independence from the shaped source tail.

Checker command

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

Evidence notes

The universal step is the distinct-degree Frobenius expansion recorded in ADR-0566. The 868 map/degree and 5580 varied-source cells are mutation controls, not a replacement for that argument.

Provenance

{
  "date": "2026-08-20",
  "established_by": "Axeyum gf2-lemire lane",
  "source": "direct characteristic-two polynomial algebra"
}