[aliromman]

This constructive numerical interface runs the actual rounded C0 arithmetic inside the spectator-row layout and proves exact return of its zero padding. It supplies a sufficient rounded-return contract for the released handoff. There is no new κ and no change to the selected frontier. The direct backend is fully charged; it does not supply the accelerated recursive tariff or the missing production global-H compiler export.

The proof and complete cost, receipt, backend, independent physical-coordinate checks, and source manifest are pinned to ecc322ebd18978e5eddb6e140081b07d7aee33b4. Canonical base: 7c49fe32c55de02c2c873655c804b16e4c3fef0a.

For every valid source plan, a complete row is a product of unchanged native C0 blocks. Every such block fixes its exact zero vector under nearest-even arithmetic. Whole-row permutations can move padding, provided their final permutation is restored. No arithmetic mixes a padded row with a live row. This gives exact payload disposal without assuming that a rounded inverse restores a nonzero input.

For grid precision g = P + ceil(f/2) + 1, the inherited completed-call estimate gives complex coordinate error strictly below 2^-P. The source conservative signed width covers every numerator. Sparse inverse exposure occurs before the next original-coordinate Gaussian-unit diagonal, which preserves the error bound. Arbitrary correlations among live inputs are allowed.

For padded volume V' < 2V, the implementation executes V'/8^f native calls and f V'/2 butterflies, including every zero row. It charges 14 f V' native integer operations and 2 f V' real reads and writes each. Explicit indexing, copies, gathers, scattering, padding and restoration add O(n V') word work. Total direct work is O((n+f)V'), with O(V'(w+n)) bits of owned logical storage. This is an indexed reference, not a fixed-tape or constant-workspace implementation.

Validation on CPython 3.12.13, standard library:

python3.12 -B scripts/research_package.py --package research/rounded-spectator-callee
python3.12 -B scripts/check_research_commit.py --package research/rounded-spectator-callee --check-source-history

Both registered normal and optimized runs pass. Each checks 36 fixtures, 15,936 exact padded zeros and 84,608 paid butterflies. Exact Fraction formulas in original coordinates agree with the predecessor reference. Actual nonzero rounding occurs. Rejection controls cover the genuine cross-row residual, wrong axis, early diagonal, invalid stored input, insufficient width and both changed source files before execution.

PASS_ROUNDED_SPECTATOR_CALLEE 36 fixtures; 15936 exact padded zeros; 84608 literal paid butterflies
PASS_REGISTERED_RESEARCH_PACKAGE research/rounded-spectator-callee outcome=completed
PASS_RESEARCH_COMMIT_BYTES

Metadata, nine committed package/input files, source ancestry, and the official export's integrity and two declared inputs pass. The finite fixtures corroborate the written all-size argument; they do not prove it by sampling. Hosted CI is pending. No selected cold replay, Lean, Docker, independent human review or proof-kernel review ran.

JoyboyBrian / @JoyboyBrian supplied the literal arithmetic and all-width error/range bounds in PR #5485, consumed as rounded.py and its proof. zxv / @zxv supplied the planner, sparse exposure and exact reference in PR #6070, consumed as verify.py with its proof. Both executed files are authenticated against the canonical base. Their unchanged source notices retain the native-port, support-router, compiler and numerical lineage. The general zero-fixed-point observation is not claimed as a new theorem.

The next missing step is an accelerated source-owned callee with its complete call order, ownership, stock, rounding cuts and physical cost, or a different proved disposal invariant if arithmetic mixes live and padded rows. The selected PR #6030 and all five framework assumptions remain unchanged. Coordination claim; the final all-state overlap check found no competing implementation of this scoped connection, but did not inspect arbitrary historical diffs or unpublished work.

Prepared by Ali Romman with substantial OpenAI Codex assistance. New work is Apache-2.0.