[petalas]

Result

Scoped written proof with exhaustive prime-field evidence; no new κ. For Open next question 2 of apoursh's merged #5663 (research/nonsplit-frobenius-characters): comparing different residue presentations needs a checked field isomorphism and its ground-field determinant.

For monic irreducible f, g of degree d over K=F_q and an isomorphism φ: K[u]/(f) → K[v]/(g), u ↦ α, in power bases:

det_K(φ) = V(α, α^q, …, α^(q^(d-1))) / V(v, v^q, …, v^(q^(d-1))),   V = ∏_(i<j)(x_j − x_i)

• det_K(φ)^2 = disc(f)/disc(g).
• The d isomorphisms u ↦ α^(q^k) have determinants (−1)^(k(d−1))·det_K(φ_0). For odd q and even d, both signs occur d/2 times. Otherwise there is one value.
• On a component L[x_i]/(x_i^(p^h_i)) in Derive nonsplit Frobenius-product determinant characters #5663's basis order, φ⊗id has determinant det_K(φ)^N.

The proof is in the package README. The Vandermonde and discriminant identities are classical, so no literature priority is claimed. apoursh's merged #5668 (research/nonsplit-sl-transport) already builds these field maps by a linear solve and divides by det(Q), computed numerically. This package adds the closed form and the sign rule. It changes neither executable.

Evidence and reproduction

The interpreter used locally was ~/git/integer-mult-bounds/.venv/bin/python3.12 (CPython 3.12), run from the repository root:

python3.12 -B scripts/research_package.py --package research/residue-presentation-determinant
python3.12 -B scripts/check_research_commit.py --package research/residue-presentation-determinant --check-source-history
python3.12 -I -W error -B research/residue-presentation-determinant/verify.py
python3.12 -I -O -W error -B research/residue-presentation-determinant/verify.py

All four pass locally. The marker is PASS_RESIDUE_PRESENTATION_DETERMINANT, and each registered command takes about 17 s. The verifier enumerates every ordered pair of irreducibles over twelve (p,d) domains up to (2,5), (3,4), (5,3), (7,3) and (11,2). That gives 279 irreducibles, 18,157 pairs and 51,300 isomorphisms. For each isomorphism it checks:

• the Gaussian-elimination determinant against the Vandermonde ratio;
• the squared determinant against Sylvester-resultant discriminants;
• the cyclic sign rule;
• det^N for Kronecker blocks with N=2,3,4.

It also runs four mutation controls, and each is detected: unordered roots, ignored Frobenius sign, wrong sign exponent kd, and assumed determinant one. Case digest: df98d666…c4641b.

Limitations

• Executable coverage is prime fields only. The written proof covers every finite K.
• This is not a recognizer, a change to the Derive nonsplit Frobenius-product determinant characters #5663/Connect nonsplit tensor recognition to special-linear transport #5668 checkers, or a physical caller. No multiplication-bound consequence follows.
• No cross-check against Connect nonsplit tensor recognition to special-linear transport #5668's numerically computed det(Q) values was run.
• Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Cross-check Connect nonsplit tensor recognition to special-linear transport #5668's det(Q) field factors against det(φ_j)^(N_j) from the closed form, using each recovered map's image of the reference generator.
2. Extend the formula to non-power bases, such as normal bases, that a real producer might store. No named producer or consumer basis is supplied.

Related open work

No open PR studies residue-field presentation determinants. #5953 (cube-zero local algebra isomorphisms) is a different algebra class, and no numbers are compared. Open PRs #5997–#6003 and #6005–#6011, opened from this account during this work, study different objects (connectors, P104 words, chirps, Pauli pairing, Gaussian transitions, Walsh agreement, char-2 jets, group labels, shear targets, the observer error floor and mixed-point construction). None computes a residue-field determinant.

CI notes

Both hosted checks passed at head 59bfb012e568f25f04eb44c62081d7ca9541bf5e: plan and replay (research/residue-presentation-determinant), in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40585039854. The run's head matches the PR head.