[aliromman]

A Gaussian native child can accept a fine grid and return a pointwise minimum-variance unbiased coarse-grid output, with an exact inverse from its retained seed and residue. For N=512^f, a=r*f active axes and grid ratio 2^k with k>=ceil(a/2), the codec uses exactly 2*k*N retained bits and one fresh uniform seed of k+ceil(a/2) fair bits. Both information counts are sharp in the stated classes. This is a numerical interface and representation codec; no new complete kappa, physical reversible circuit or selected assumption discharge is claimed.

Review the proof, quotient construction and complete direct invoice, executable codec, generated certificate, verification and source pins. Reviewed head: 5f7c893b6af9a75cb19ad22c6db06ed22eea6f54. Base: canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030.

The interface is A=L C_r R with the actual Gaussian-unit monomial phases. Input x represents h*x/2^k; output q represents h*q. Given a fresh uniform integer u below D=2^(k+ceil(a/2)), every real component is rounded by

q_j = floor((n_j+u)/D), where n = 2^ceil(a/2) A x.

One seed serves the entire field; coordinate errors need not be independent. The exact conditional mean is the ideal native output. The mean squared error is the sum of the scalar adjacent-grid variances and is at most N*h^2/2. An integer-valued random output with the same means cannot have a smaller sum. This is not a minimum deterministic-error claim.

The explicit Gaussian quotient is Z[i]^N / (2^k A^-1 Z[i]^N). Integral triangular transforms reduce it to pi-adic coordinate widths 2*k-a+2*weight, totaling 2*k*N bits. For every fixed seed and coarse output, exactly 2^(2*k*N) fine-grid inputs occur. A byte-packed residue code labels them, and a literal inverse recovers the input. Finite arbitrary dirty storage cannot reduce the required retained-label capacity. This optimum is for the advertised full fine grid; it is not a hardware clean-bit minimum or a lower bound on correlated caller states. The seed lower bound concerns fixed-length fair-bit sampling, not entropy or expected variable-length cost.

The selected word has 3,453 MOVE events and 2,886 distinct directed edges. Its complete frame ledger passes. The rank census is independently checked by Pauli/tableau composition and by 18-bit matrix inversion/composition with finite-span ranks. For each of the eight present ranks, the unchanged compiler, separate coordinate oracle and new interpreter agree on all 512 columns of the first representative. The other 2,878 edge recipes are not instantiated here.

At f=1, k=5 matches the h/32-to-h grid ratio in PR6499 and PR6506. Each full-grid codec uses 5,120 retained bits and 6–9 seed bits. One slot per MOVE totals 17,679,360 retained bits and 21,683 separate fresh seed bits. These are uncompressed proposed slots, not a complete invoice or joint entropy lower bound. The predecessors' error constants are not imported. At larger tensor widths the rank-eight class needs k>=4*f under this codec; fixed k=5 is not an all-width admission.

Validation uses CPython 3.12.13 and NumPy 2.3.4:

python3.12 -B scripts/research_package.py --package research/native-coarse-grid-residue-codec
python3.12 -B scripts/check_research_commit.py --package research/native-coarse-grid-residue-codec --check-source-history

Normal and optimized registered runs pass. Checks cover 25,184 complete quotient endpoints, 16 retained-bit witnesses, 36 separate dense-tensor oracle fixtures, 432 small seed round trips, 4,096 source columns in each of two interpreters, 128 signed native round trips, 64 arbitrary coarse-output/residue endpoints and exact moments over all 1,920 seeds. All four seeds of a 262,144-coordinate tensor fixture pass with a 524,288-bit label. Eleven semantic/domain controls and three actual source-corruption rejections pass. The source compiler/oracle run unchanged with assertions enabled in a normal subprocess in both modes.

PASS_NATIVE_COARSE_GRID_RESIDUE_CODEC
PASS_REGISTERED_RESEARCH_PACKAGE research/native-coarse-grid-residue-codec outcome=completed
PASS_RESEARCH_COMMIT_BYTES

Thirteen committed package/input files, source ancestry and official export integrity pass. A read-only, Git-free optimized export replay passes in 21.29 seconds with all 14 exported files and modes unchanged. Hosted CI is pending at publication. No predecessor full scientific suite, selected cold replay, all native instances, independent human acceptance or proof-kernel review ran.

The complete direct round trip pays three native evaluations, 3*a*N/2 butterflies, a*N triangular pair updates, eight tensor monomial passes with f address digits each, and 2*k*N digits in each codec direction. The proof includes copies, shifts, signed widths, buffers and O((a+k+f)*N*(w+a+k+f)) indexed bit work. Sampling/seed access, residue lifetimes and cleanup, reversible gates, physical routing, fixed multitape schedules, accelerated recursive pricing and full caller/outer closure remain open. The residue cannot be erased from the coarse output alone. All five framework assumptions remain unchanged.

Credit Dugongue's PR5815 and Gaussian Smith factorization, Ali Romman's PR5821 execution, and Nick Petalas / @petalas for PR5880's distinct modular overwrite capacity, PR6438's ensemble question and PR6418's reuse analysis. PR6066 and PR6068 retain scalar residue and fair-bit denominator credit. PR6532 supplies the prior same-grid native interface and its relative-error boundary.

DreamingOfClouds / @DreamingOfClouds supplied the selected word. Cinna Davis / @1337isnot1337 supplied PR4088 and PR2226's consumed native representations. NOTICE preserves the full native/compiler, Gaussian, scalar and precision lineage. Classical rounding, Smith factors and counting are not claimed as new. The new contribution is the explicit coarse-period codec, its scoped optima, source bindings and complete direct invoice.

Advance claim. Main and complete public open/closed PR and Discussion metadata were refreshed. A second exact-main README/TASK search found no matching unbiased coarsening/quotient description. The cited work and PR2730/PR3467 mixed-grid/entrance contracts were inspected. No same claimed interface was found; arbitrary diffs and unpublished work remain limits. No other contribution was edited.

Prepared by Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0. Ten files are added only under research/native-coarse-grid-residue-codec/. The next unreserved step is a full caller error and retained-information schedule with its paid physical and all-width costs. No merge, closure or frontier change is requested.