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Prove exact two-call cost for the tied width-two transform - #5222
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Mister-Meeseeks merged 2 commits intoOct 23, 2026
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Mister-Meeseeks merged 2 commits into
Mister-Meeseeks merged 2 commits into
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The widened complex register-mixing game left the optimum paid-call count at fixed tied width two unresolved as the address dimension grows. This package proves that the dense transform costs exactly two paid calls for every m>=2 (one for m=1). The proof generates the complete free sector algebra from tied monomials, bounds every one-call program with arbitrary clean helpers, and constructs two simultaneous cross-sector factors with all helpers reset. Every nonzero allowed source direction is covered by tied perfect-matching conjugacy.
The exact PR #4628 engine contract is retained byte-identically with its original provenance; PR #5178 is the width-bound predecessor. The result concerns the widened game. Free mixing, projector expansion and cleanup are finite but not physically priced; the narrower affine engine and selected multiplication frontier are unchanged.
Validation: registered Python 3.12 normal and optimized checks; exact Gaussian-integer gauges in the first two dimensions; independent ordered-tensor/sparse-factor and four-root projector controls; source ancestry; official ten-file export; readonly Git-free cold checks; nine actual source/receipt/scientific mutations rejected in both modes. Independent AI-assisted written proof and source review passed. No numerical solver, Lean or historical suite was run.
Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0 and predecessor attribution retained.