[aliromman]

This completes the determinant-character interface for every supplied purely singular Kronecker sum over every field. Unequal indices, both orientations, arbitrary multiplicities and zero-index blocks are included when both full factor dimensions are positive. The complete SL label group is K*/(K*)^d; over F_q the exact orbit count is gcd(q-1,d). No new kappa is claimed.

Proof, construction, separate literal oracle, exact evidence, source pins, and scope. Contribution head: 618fb04433927ade44ed5b507f8d30f9ede2bde5.

Each right index e contributes (e,-e-1) and each left index contributes (-e-1,e). The integer d is the gcd of same-orientation index differences and opposite-orientation sums e+f+1. Multiplicities change the explicit slice character but not this count. The proof gives all off-diagonal Hom spaces, their determinant-neutral triangular form, characteristic-safe lifts of every GL2 action, and a constructive converse. The executable returns literal stabilizers with any prescribed slice action and corrects a supplied GL tensor map to an SL map with the same tensor image. Over an arbitrary field, a required d-th root is an input witness; universal root extraction is not claimed.

This answers the mixed singular question released by apoursh PR #5672, exact head 81c0577b83cdafa73346da8ade1261b0b202e45f, from Nick Petalas PR5250. It extends the theorem, not only its small examples: R1+R2 has one finite-field SL orbit, R1+L1 has three over F4, and R1+L2 has four over F5.

python3.12 -B scripts/research_package.py --package research/mixed-singular-kronecker-characters

Expected: PASS_MIXED_SINGULAR_KRONECKER_CHARACTERS and PASS_REGISTERED_RESEARCH_PACKAGE. Python 3.12 and the standard library suffice. Normal and optimized checks pass: all 256 Hom cases of indices 0..3 over F2/F3/F4/F5; 98,066 literal solutions and 41,744 invertible stabilizers in twelve full mixed equation domains; 6,684 lifts and 6,684 prescribed-action reconstructions across nineteen model/field pairs; all 468 determinant triples, with 330 witnesses and SL corrections and 138 exclusions. Eight mathematical controls, twelve malformed-domain controls and 1,474 permutation determinant checks pass. Exact committed bytes, official export integrity, Git-free optimized replay and actual frozen-source corruption rejection pass.

The finite executable restricts indices to 0..5, combined multiplicities to 1..3, and both full factor dimensions to sixteen. The all-field result is the written proof. Regular factors, arbitrary-tensor extraction and a physical multiplication consumer remain open. Native maps, ownership, movement, restoration, storage, precision and the full recursive bill are unpriced. No independent human review, proof-kernel acceptance or selected-frontier change is claimed. Hosted CI is separate; the shared main planner regression reported in Discussion1200 remains unrelated to these local replays. Full-history preflight is unavailable in this shallow checkout.

Canonical base and final refresh: df0137efd35e11b4445734bf612bdd4c7ebc330c, priority7, selected conditional kappa .00310045. Canonical claim. Credit apoursh for the retained one-type theorem and symmetric-power implementation, Nick Petalas PR5250, David Ganz's A030/A026/A029 atlas, and the classical Kronecker/symmetric-power lineage. The three source files and their notices are retained byte for byte. New mixed proof, witnesses and independent equation oracle: Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0.