[Dugongue]

PR #4935 left the physical backward interface open. This adds its actual conjugate-frame repair word: eight owned copies, eight rank 1 transports, 424 scalar ADDs and 16 CLEARs give (X,Y,H) -> (X - F^T Y,Y,H) on all 632 independent inputs, preserving all 408 helpers. The complete profile matches the forward word: {1:2209,2:412,3:510,4:127,5:40}. F is nonsymmetric, so the transpose and conjugated frames are checked explicitly.

The package then instantiates PR #4933’s typed p16 caller with F in forward stages and F^T in backward stages. Both codes are involutions, so all 194 inserted evaluations and 97 optional positive-half inverse evaluations use dyadic coefficients, eliminating the inverse-code division by 13. The recipe retains all 18 FULLs, 15 IDLEs and 15 thin queries. Its recursive delta against the original uncoded caller is -144 rank 8 +144 rank 1 calls. Compared with #4933 this pays additional repairs to obtain the dyadic interface; it is not a claimed paid-moment dominance.

No improved κ or complete Gaussian supplier is proved. The 16 negative-Q banks retain their exact ordered codes. Affordable terminal decoding or recursive coded-job admission, complete stock/precision/tape realization and outer recovery remain open. The selected frontier is unchanged.

Validation: normal and optimized registered Python 3.12 standard-library checks; both 632-input native endpoints; all helper/frame/owned-state conditions; sign and unpaid-transport rejection controls; separate sparse Gaussian-integer replay of all 4,096 backward repair columns including phase; 224 basis recoding checks; 112 physical wrapper sets; the inherited noncommutative controller’s 1,024 entries and explicit typed recipe; source hashes, committed-byte preflight and Git-free export. No Lean, full cold multiplier, second full native interpreter, independent human review or hosted CI result is claimed. Full-history provenance ancestry was not checked in the shallow checkout.

python3.12 -B scripts/research_package.py --package research/native-involutive-dual
python3.12 -I -O -B research/native-involutive-dual/verify.py
python3.12 -B scripts/check_research_commit.py --package research/native-involutive-dual

Credit Rohan Arun #1931 and Ali Romman #1862 for the involution mechanism; Dugongue #4935/#4929 for the forward source and clean code; PR #4933, prepared for Dillon, for phase-correct normalizers, gauge identities and the typed caller; Cao Yifan’s round-three source beyond #4610; Cinna Davis #4694/#4698/#4470; Ryanson Jonathan #4516; zxv, Jacob Alan Sussman and all retained lineage. This is a local dual construction and dyadic instantiation, not a first involution or new controller-core theorem. See ATTRIBUTION.md and NOTICE.

Only research/native-involutive-dual/ changes. Prepared by Dugongue with substantial OpenAI Codex assistance from one agent; no subagents. Canonical main was refreshed to cef4b52. Live public PR metadata, targeted heads and cached/local overlap were inspected; Discussion and local-tree coverage are incomplete, and no exhaustive novelty claim is made.

Late overlap: Cayden Burkart’s PR #4940 at 91b9632 proposes reverse-only thick stages and inverse-mixed thin queries. Its source explicitly leaves physical adapter admission open. That different formal stage assignment is not used here, and neither its optimistic root nor removal of its denominator would alone establish a new κ.