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Theorem

FPRD-T138

Exact affine sewing across a trace cut

Exact statement

If a prefix uu has weight pp, endpoint yuy_u, multiplier a=3pa=3^p, and guard rur_u, then its exact horizon-ℓ\ell action on a suffix guard is the affine permutation Fu(ℓ)(q)=a−1(q−yu)(mod2ℓ)F_u^{(\ell)}(q)=a^{-1}(q-y_u)\pmod{2^\ell}.

StatusProved by direct modular substitution; internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The theorem identifies the complete continuation action at one cut. A numerical endpoint alone does not determine how every possible suffix guard must be transported.

Definitions

  • Every input in the prefix cylinder is n=ru+2kμn=r_u+2^k\mu.
  • The prefix maps it to Tk(n)=yu+3pμT^k(n)=y_u+3^p\mu.

Hypotheses and scope

  • The suffix has horizon ℓ\ell and requires an input congruent to its guard q(mod2ℓ)q\pmod{2^\ell}.
  • The multiplier is odd, hence invertible modulo every power of two.

Proof or evidence

Substituting yu+aμ≡q(mod2ℓ)y_u+a\mu\equiv q\pmod{2^\ell} and multiplying by a−1a^{-1} gives a unique lift class. Translation and multiplication by a unit are permutations, proving the displayed seam map.

Exact affine sewing proof →

Verification notes

The proof was rederived from the definitions, checked for modulus and inverse scope, and compared with the governed claim. The preserved checker independently reran 370,252 affine/sewing/recovery split instances; those instances corroborate but do not replace the proof.

Limitations

  • The theorem concerns exact compositional state, not acceptance-state minimality.
  • The underlying residue machinery is classical; no novelty claim is made.

Open work

Compare the formulation with the closest parity-vector and 2-adic conjugacy literature before any novelty language.

Notes

The proof is direct modular substitution and uses invertibility of the odd multiplier. The theorem identifies the complete continuation action at one cut; the numerical endpoint alone is not an exact compositional state. No novelty claim is made for the underlying parity-vector formulas.