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By @CubeJerry with substantial OpenAI GPT-6 Astra Pro assistance. Independent eigenspace and rational invoice review included. No new kappa; source and scope pins are retained in the package.
Independently reproduce the support cap from CrocSwap#6398 and bind the known CrocSwap#6717 prefix floors to every exact quotient site. Keep the carrier mechanism and complete fixed-benchmark invoices unchanged. By @CubeJerry with substantial OpenAI GPT-6 Astra Pro assistance. No new kappa; no claim of a new general stage-rank threshold.
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Replace individual thin-query annihilation by commutation with the whole three-query bridge of pending #6833. Determine the enlarged carrier space and prove that ordinary sector-independent frames do not enlarge transparent transport.
Full-span dense activation has a strictly worse complete invoice than the source for all positive odd arms and 0<κ<1. All 81 central dense masks on the fixed 9|9 root lose at κ=.00337180; the largest exact complete gain upper bound is −42.130440268650981066. This finite claim is only at that benchmark.
The bridge also yields an 18-dimensional quotient with one thin query, and all eighteen native projectors admit local repairs. An exact applicability check of the known bounds in #6398/#6717 nevertheless excludes commuting rank-two stages, uniform rank-three stages and balanced total incidence 54. The retained support theorem and eight prefix floors imply total rank at least 60; this is not a new general rank threshold or an attaining construction.
New relative to #6841/#6845: whole-block carriers, independent exact interval capacities, complete invoice certificates, and a precise quotient-interface obstruction. Disjoint-time packets need not be globally orthogonal. All native, target, thin and idle terms are counted; extra ordinary adapters are optimistically free.
Base: 7c49fe3.
#6833 source: 51f54e3.
Exact Git/SHA256 pins: SOURCES.json, retained/SOURCE-PINS.json and COROLLARY-SOURCES.json.
Reproduce (Python 3.12, stdlib):
Expected: PASS_REGISTERED_RESEARCH_PACKAGE and the independent geometry, price and support/prefix PASS reports.
Executed: eight registered checks, including normal/-O controls; all 13,851 capacities and dual inequalities, 25 source shapes, 18 projector repairs and eight prefix bounds. Independent nullspace and rational-arithmetic implementations agree. Controls detect omitted neutral/dual conditions, false capacity, corrupt dual prices and omitted commutation. Exact 81-case coverage; no randomized search. Fresh export and commit/remote-tree verification pass.
No new κ. Algebraic stage bounds assume characteristic different from two. The mask-price result is scoped; rank 60 is necessary, not known attainable. Larger or noncommuting target representations still require a legal physical interface and complete recursive accounting. All five framework assumptions and the selected frontier remain unchanged.
By @CubeJerry with substantial OpenAI GPT-6 Astra Pro assistance and separate proof/implementation review. Credit DreamingOfClouds (#6832/#6833/#6842), Ali Romman (#6398), thu-nmrc (#6717), #6841/#6845 and retained native/helper lineage. Apache-2.0.
Proof, exact certificates, source pins and open questions:
https://github.com/CubeJerry/integer-mult-bounds/tree/7d34bef131ad06edb833e481d53e3495576d5195/research/p104-block-bridge-carriers