[aliromman]

Every failed complete dirty linear identity over Q or Q(i), with independent operator symbols, has a counterexample using real symmetric positive-definite contractions with spectra in [1/2,1]. The witness dimension is D+1, where D is the largest nonzero coefficient degree. This enabling proof and adapter show that positivity and condition number at most two do not, by themselves, repair a false identity. The complete kappa calculation and selected frontier are unchanged.

Proof, exact receipt, source pins, and literal checker.

The adapter finds a top-degree word through separate exact-length reachable spaces. A symmetric labelled path then gives discrepancy c/8^D. A rational parameter also puts all generators arbitrarily close to I. The discrepancy can be tiny; this is exact algebra, not a floating-point detection guarantee. Every input and returned role remains part of the contract.

This extends zxv's PR #5235, whose proof supplies the compact circuit compiler and classical weighted-automata bridge. Its ordinary witness can be singular; its shifted witness is unipotent. The new result guarantees symmetric positive-definite contractions and a uniform condition bound. It does not impose inverse, commutation or native relations. The separate inverse-aware frontend in PR #5262 is outside this scope and was not audited here.

Base: current main ff3cc58, priority revision 7. Two small source files and the wrong-order example are retained byte-exact from pending predecessor head 48976da. The original proof and attribution are retained with those source files. SOURCES.json records the original locations and binds the exact retained bytes to ancestor 46bdee8 in this PR history. No unmerged upstream object is required by the registered replay. Its notices and wider dirty-echo/tensor credit remain in NOTICE.md. Classical methods retain Stefan Kiefer and Ran Raz / Amir Shpilka credit. New work is by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. No global literature priority is claimed.

With CPython 3.12 and the standard library:

python3.12 -B scripts/research_package.py --package research/spd-dirty-circuit-witness

Expected: PASS_SPD_DIRTY_WITNESS in both modes, followed by PASS_REGISTERED_RESEARCH_PACKAGE.

Both warning-strict normal and optimized registered replays pass: 240 exact expanded-word comparisons, 384 literal counterexample replays, 96 spectral enclosures, fifteen control groups and the pinned four-role wrong-order echo. The echo has an exact discrepancy of 1/512. The (X+Y)^64 stress circuit has 2^64 top words, 132 coefficient states, at most two basis rows per degree and a 65-dimensional witness. Four code mutations are rejected in both modes before receipt checking: shortest-degree selection, missing symmetric edges, wrong multiplication order and omitted dirty outputs.

Source pins, committed bytes and the official export check pass. Both Git-free replays pass with an empty PATH and all 16 exported files unchanged. Canonical main and scoped overlap were refreshed before submission. The first hosted replay failed before mathematical execution because the unmerged source object was absent. The current source-history repair retains every original byte and assertion; its hosted rerun passed both planning and the focused replay at this exact head. The full-history wrapper was not run in this shallow checkout; direct ancestry passed for every declared source commit. No selected cold replay, Lean, Docker, native realization, physical invoice, independent human review or proof-kernel acceptance is claimed. The written theorem covers all degrees in the stated grammar; finite checks corroborate it.