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Prove fixed-accuracy Hopf depth O(N/b²+n) across dirty widths - #79

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GoGoKo699 merged 2 commits into
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research/width-sensitive-bilinear
Oct 3, 2026
Merged

GoGoKo699 merged 2 commits into
mainfrom
research/width-sensitive-bilinear

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@GoGoKo699 GoGoKo699 commented Oct 3, 2026 •

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The unary phase-source compiler gives O(n) fixed-accuracy depth at square-root dirty width, but simply replacing its tail queries by the earlier width-sensitive loader retains an O(n) bank route per layer. This PR supplies the missing selected-bilinear query and proves the full width tradeoff.

For fixed eta, two external clean flags, and the existing literal threshold b >= 17(L+n+7), the same complete real-frame circuit has T = O_eta(sqrt N + N/b), G = O_eta(N), and D_T = O_eta(N/b^2 + n). T-count remains optimal in order. Count and depth now match simultaneously through b <= sqrt(N/n), when the interval is nonempty.

The proof includes a 16-T-layer controlled bilinear leaf using one arbitrary returned dirty helper, two-pass dirty selection of matrix blocks, the four-corner indicator echo, and a coupled block-size/chunk-length allocation. Full-input identities preserve literal phases, actual inverses, and dirty-reference return. The original unary source/error allocation and explicit finite-size fallback retain the exact width threshold.

Five new bounded tests cover native guarded leaves and an eight-wire selected block, exact symbolic complete queries, resource schedules, and negative controls. They do not constitute a scalable native frame emitter. Frontier, verification, continuation, and attribution documents are updated; no new external synthesis premise or generic lookup priority is claimed.

Validation: independent proof/skeptical audits found no blockers. All 471 tests pass in Python 3.11 and 3.13 CI. The existing 466-test suite and five new tests also pass locally. All four exact-receipt suites, reviewer walkthrough, bounded native examples, and resource ledgers pass. All five checks are green on b137912, including rendered presentation. The final documentation correction explicitly states the helper-restoration toggle. The high-precision endpoint and larger-width depth optimality remain open.

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GoGoKo699 merged commit 9ae9685 into main Oct 3, 2026
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