[aliromman]

The selected PR6030 word's first native MOVE has an optimal unbiased reversible Gaussian grid map. One fresh fair bit serves its entire butterfly layer. Retaining that bit gives an exact data inverse. The construction preserves the exact complex pair sum and Gaussian phase covariance, and attains the smallest possible grid error at every input. It also supplies a reachable counterexample to a uniform relative-error coefficient below one. No new complete kappa or selected assumption discharge is claimed.

Review the proof and complete direct bill, generated certificate, integer map and inverse, independent checks, and source pins. Base: canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. Reviewed head: 9327b8f6b7f94728f188f8a27595bd8b14775446.

The binding is zero-based event 9896, MOVE field 236 from empty frame 3 to frame 7 with basis [21], in the selected source word. The unchanged Clifford compiler and coordinate oracle agree on all 512 columns, including global phase.

For grid spacing h, a layer has squared error exactly kh^2, where k is its number of odd-parity pairs. This is the pointwise minimum among all unbiased grid-valued outputs. If k>0, any distribution attaining that minimum needs at least one bit of seed entropy; the construction attains it. Pair errors may be correlated. For N=512^f and f fresh stage bits, the completed tensor map has exact mean T_[21]^(tensor f), mean squared error at most fN*h^2/2, and an exact inverse using the retained bits in reverse stage order. The tensor variance bound is not claimed sharp.

The relative limit is also source-bound. Set only public input field 236 to i*h*e_21. All 9,896 preceding scalar events can be exact on the h/2 grid, with component magnitudes at most h. The first native input then has norm h, but every output on the h grid has error at least h. Thus this map cannot supply the u<1 relative premise in PR6526. An absolute-error caller join or changed grid/encoding is needed. This is not a general precision impossibility.

Validation uses CPython 3.12.13 and NumPy 2.3.4:

python3.12 -B scripts/research_package.py --package research/selected-native-unbiased-rounding
python3.12 -B scripts/check_research_commit.py --package research/selected-native-unbiased-rounding --check-source-history

Normal and optimized registered checks pass: 26,244 pair directions, 104,976 phase checks, 128 signed wide pairs including 2^1024-scale values, 1,024 real/imaginary native basis inputs and eight dense signed inputs. All four seed words of a 262,144-coordinate tensor fixture have the exact mean and exact data return; its mean squared error is 78,276 grid units squared, below the proved 262,144 bound. Seven semantic controls include a concrete mean change from seed reuse and the actual relative-error witness. Three source corruption controls pass. Source assertions remain enabled in an unchanged normal subprocess in both modes.

PASS_SELECTED_NATIVE_UNBIASED_ROUNDING
PASS_REGISTERED_RESEARCH_PACKAGE research/selected-native-unbiased-rounding outcome=completed
PASS_RESEARCH_COMMIT_BYTES

Thirteen committed package/input files, source ancestry and official export integrity pass. The optimized exported replay also passes in a read-only Git-free directory with empty PATH; all 14 exported files retain their bytes and modes. The initial head passed hosted run 41572017455. Hosted CI for the current head is pending. The first receipt comparison caught a tuple/list mismatch in a control fixture; the producer now emits a JSON array, and both registered modes pass with the same scientific values. Selected cold replay, predecessor full suites, native builds, independent human acceptance and proof-kernel verification were not run.

The direct implementation pays fN/2 pair calls, 2f monomial passes, route setup, data movement, O(N) buffers and f retained bits. The proof includes sufficient signed widths and the complete O(f*N) indexed work bound. It does not supply a reversible bit circuit, sampler/random-tape access, entropy recovery, physical cleanup, a fixed multitape schedule, an accelerated recursive-child tariff or whole-word error propagation. Exact data inversion does not imply additivity, graph consistency or dirty compensation after intervening arithmetic. All five framework assumptions remain.

The deterministic conserved-sum family and nearest-grid classification are prior work in Ali Romman's PR6509 and its proof. This result adds opposite-center randomization, the one-bit conditional contract and this selected source binding. Nick Petalas / @petalas supplied PR6438's optimal-ensemble question and PR6418's seed-reuse analysis, with zxv's PR6066 warning retained. DreamingOfClouds / @DreamingOfClouds supplied PR6030; Cinna Davis / @1337isnot1337 supplied PR4088 and PR2226's consumed representations. NOTICE retains native/compiler, Tekalign lifting, JoyboyBrian port, Arham Tahir and earlier method credit. PR4017 supplies a distinct deterministic native circuit with dirty scratch; no competing cost is asserted.

Advance claim. Main and complete public open/closed PR and Discussion metadata were refreshed before publication; a second exact-main README/TASK search found no matching unbiased Gaussian/butterfly description. PR6509, PR6438, PR6418, PR4017, PR6499 and PR6526 were inspected. No same claimed native stochastic interface was found. Arbitrary diffs and unpublished work remain coverage limits. No other contribution was edited.

Prepared by Ali Romman with substantial OpenAI Codex assistance; no subagents. Apache-2.0. Only the ten files in research/selected-native-unbiased-rounding/ are added. The next unreserved step is an absolute-error caller join with exact scalar records and a paid fresh-randomness interface. No merge, closure or frontier change is requested.

The current head changes only the phase helper: it evaluates the selected unit phase instead of evaluating unused sign cases. This makes the executed helper match the stated at-most-two-sign bound per coordinate. Both registered modes and the sealed optimized export were rerun; all numerical receipts remain unchanged.