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For coupled width q=2, n=4, and full sites 0,1,3,4, every shortest-query root with compatible commuting actual-prefix projectors has a negative complete direct invoice throughout κ in [0.00330720,0.004]. This covers all stage ranks, positive child allocations and binary tree shapes, using the fixed #6726 native source and separately paid stage idles. Ordinary work is optimistically free.
The exact outward evaluator checks 62,007,750 pattern/allocation cases after reducing 267 necessary inventories to nine minima and six reflection classes. The largest upper bound is -0.06504536308798 per physical port. A concave tangent envelope excludes every allocation with at least 128 leaves. The proof is independent of the concurrent four-stage equality obstruction and deliberately includes an unrealizable rank-20 profile.
Expected PASS_REGISTERED_RESEARCH_PACKAGE. Complete normal and optimized evaluations, independent prefix and literal-histogram checks, source authentication, separate source-aware proof review and committed-byte/mode checks passed. A read-only Git-free export also passed the full optimized allocation evaluator with empty PATH and all file bytes/modes unchanged.
No claim covers new retirements, fragmentation, cross-stage fusion, other sites, larger roots or changed native schedules. No new κ. No physical word or selected-frontier change; the five inherited framework assumptions remain. Changed retirements or coupled schedules require a new complete invoice. Credit zxv #6726 for the native/caller source, thu-nmrc #6717 and Ali Romman for retained geometry, Cinna Davis and the preceding supplier lineage.
Prepared for @CubeJerry with substantial OpenAI Codex assistance. No human/proof-kernel acceptance, selected cold replay or full-history preflight. Base 7c49fe3. Exact dependency pins: ORIGINS.json. Exact sources, proof and reviews. Published complete tree equals the tested tree.
Selected-baseline source bindings and metadata passed the required orientation check; this was not a construction cold replay.