[aliromman]

Construct exact rational orthogonal counterexamples for false complete inverse-circuit identities. For r independent roles and q named-generator calls across both programs, the witness has dimension d<=r+q. Every generator lies in SO(d,Q), its inverse equals its transpose, its condition number is one, and it can be arbitrarily close to identity. This supplies an additional operator relation that the earlier SPD witness did not preserve.

The written proof, exact receipt, and literal checker define the claim. The construction uses a shortest nonzero Magnus coefficient, a skew-path Cayley transform, and at most dq+1 rational trials. Shortestness matters: for G-G^(-1), the longer nonzero coefficient AA gives an identically zero selected Cayley endpoint. That failure is an explicit regression control.

The grammar is over Q, returns every role, reads old values for aliased queries, and allows only actual inverses of named generators. This is a scoped written theorem and adapter. It does not improve complete kappa, bind a native Clifford alphabet, impose its remaining relations, or prove numerical detectability. The selected conditional value 60403/20000000 and its analytic, compiler, owned-erasure, all-column and precision/tape obligations remain unchanged. No frontier files change.

Base: ff3cc58e370d0f678e529246f8c73e8094e18fcf. @krzysztofwojt supplied the compact inverse frontend and shortest-word proof in PR #5262, consumed at 6f92ccf1e61c4668c8ee687bd1df8b42932b8759. Its retained proof and notices preserve credit to @zxv for PR #5235, the dirty-echo/tensor/phase lineage, and classical rational-series work. Ali Romman's PR #5296, consumed at bab6b65d845fcc7b563d10e809f5e79c82e74ad0, supplied the root-budget architecture and replay harness adapted here. The Cayley transform is classical. Source pins retain the consumed bytes in this branch's ancestry. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.

Python 3.12 and its standard library are sufficient:

python3.12 -I -W error -B research/orthogonal-inverse-witness/verify.py
python3.12 -I -W error -O -B research/orthogonal-inverse-witness/verify.py

Both commands passed and end with PASS_ORTHOGONAL_INVERSE_CIRCUIT_WITNESS. The registered replay passed. Checks cover 160 programs against a separate reduced-group-word oracle, 142 dense full-program replays, exact orthogonality/determinant/inverse checks, dirty returns, aliases and forced parameter cancellations. The stress case represents 2^32 lowest-degree words with 66 compiler states and a 33-dimensional witness. Four code mutations failed in both modes before hash/receipt checks. The committed-byte check, direct ancestry check for every source pin, archive check and read-only export replay in both modes also passed; all 19 exported files remained unchanged. The producer/checker CLI example passed at cap 1/100.

No Lean, Docker, independent human proof review, native tape replay, selected cold admission or complete kappa calculation was run. The full-history wrapper is unpassed because this checkout is shallow; direct pin ancestry and portable replay were checked separately. Further work must bind an actual native caller and all of its additional relations before drawing a native conclusion.

Hosted run 39992699684 passed planning and the registered Linux replay for published head 9bebc267c4608e51fcc94a543b8e0d66077cdfbb. Hosted execution is separate from independent scientific review.