[flyingviper]

This source-bound theorem rules out a large family of productive-helper/native optimizations on the round-three word: arbitrary Clifford charts and native maps cannot lower the complete invoice while every contact retains its original projection dimension and owned-center tariffs remain fixed. Quadratic forms and nonnested transitions are included. The result holds separately in both orientations for all 0 <= C < 1; it does not improve kappa.

The proof combines the standard Fourier-rank metric with incremental concavity. All 358 zero-entrance helpers have rigid residual ranks. Each of the other 50 has a distinct paid-only source-geodesic wire, whose extra tariff covers its entire extra helper credit. The independent checker reconstructs every source incidence, including old-zero edges, verifies all native ranks and main-only adapters, and binds 1,016 path occurrences per orientation to the original source. This gives a concrete stopping rule: change contact heights, owned costs, the scalar graph or helper contracts before extending this search.

The package is research/productive-helper-dimension-floor. It is independent of PR #5301 and uses one live input from canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf. The proof credits the source word, earlier owned/path floors (#5060, #5136, #5140, #5283), the inverse-image Clifford grammar and the classical subspace metric. Independent agents checked the mathematics, source paths and final package; this is not independent human or proof-kernel acceptance.

Validation at head 108428fd916ab5c08fbd8acb40583104c433488e:

• Registered Python 3.12 normal and optimized isolated replays pass, with eight corruption controls and all 18,459 chart-pair calibrations in dimensions one through three.
• Metadata/source pins, committed bytes and full source-history preflight pass.
• Official export integrity and Git-free optimized replay with an empty PATH pass.

python3 -B scripts/research_package.py --package research/productive-helper-dimension-floor

No new physical word is emitted; new phase, compiler, precision, supplier and outer admission are not claimed. The source and selected frontier are unchanged. Prepared by flyingviper with substantial OpenAI Codex assistance and independent agents; Apache-2.0.