[aliromman]

Query endpoints attain sharp one-cut and arbitrary-path tariff bounds for a fixed helper/query pair. Noncanonical charts and extra path length cannot exceed the corresponding bound. This package proves the sharp bounds for every power exponent 0<C<1, including arbitrary intermediate Clifford charts and spectator padding. The original four endpoints stay fixed on a geodesic path. A one-cut optimum is attained at a query endpoint; an arbitrary finite path is bounded by the three-part target credit at both endpoints. This is a tariff theorem, not a physical split or a new κ.

The exact PR5756 application covers 2,534 helper queries and 559 independent targets. At its .00311313795 saving, even the optimistic best choices need at least 53 one-cut helpers or 33 arbitrary-path helpers to fund one extra H1 native query. At C=.0034 the minima are 48/30; at C=.01 they are 17/11. Every one-target path gains less than .097259566 through C=.01. Thirty-two source helpers have no possible gain in this contract. Each helper is credited at most once, and every target debit is retained.

The proof uses the native Lagrangian rank metric, decreasing convex power increments, and exact cancellation of common paid/target path edges. A separate canonical projection theorem holds for every increasing concave tariff. An explicit hinge counterexample shows why the endpoint theorem needs the power hypothesis. The proof, exact budgets and source query inventory are independently replayable.

Validation: the registered normal and optimized Python 3.12 runs pass in 5.95 seconds. Checks include 4,489 actual native pairs, all 37,180 rank-nine flag-intersection patterns, all 135 three-qubit Lagrangian images at three exact powers, and 4,368 literal paths. Exact intervals certify all source choices and minimum-fee prefixes. Wrong main-role partitions, a wrong source frame, a missing target debit and five actual source-byte corruptions reject. Committed bytes/modes and official export pass. The separate read-only Git-free replay passes in 1.96 seconds with all 19 exported files unchanged; see the validation record.

Run python3.12 -B scripts/research_package.py --package research/p104-helper-detour-projection.

The bound grants ordinary costs for free and assumes the stated target-path invoice. Physical independent inputs, restoration and compatibility of different helper choices remain separate. Changed original endpoints, shared query graphs and several original queries require a new bill. In particular, the PR5774 retained-copy construction and its active successor are outside this contract. No source scalar-endpoint, native-phase, compiler or outer admission is repeated. Five inherited framework assumptions, full-history ancestry preflight in this shallow checkout, hosted CI and independent acceptance remain separate. No selected-frontier change, merge or closure.

Base and refreshed main: de5ec2e, priority 8. Canonical investigation. Credit Joel Pulikkan/GamingPuzzled PR5756/5739, PR5751, flyingviper PR5718, zxv's exact moments, the native compiler line and all retained source notices. PR5772 supplies the prior canonical rank formula. New proof and source budget by Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0.