Bind optimal unbiased reversible rounding to the selected first native MOVE

research/selected-native-unbiased-rounding/NOTICE

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+New work: Ali Romman with substantial OpenAI Codex assistance. No subagents.
+Apache-2.0.
+
+DreamingOfClouds, PR6030, supplies the selected word and its retained P104
+lineage through PR5907, PR5827, PR5811, PR5786 and earlier work.
+https://github.com/CrocSwap/integer-mult-bounds/pull/6030
+
+Ali Romman with OpenAI Codex assistance, PR6509, supplies the deterministic
+conserved-sum center family, phase-covariant choice and nearest-grid proof.
+The new package randomizes opposite centers and binds a stochastic interface.
+https://github.com/CrocSwap/integer-mult-bounds/pull/6509
+
+Nick Petalas, PR6438, supplies the prior ensemble question and a different
+optimal planar construction. PR6418 gives the adaptive-reuse bound; zxv's
+PR6066 supplies the earlier reuse warning. Ali Romman's PR6526 supplies the
+whole-word relative stochastic contract not met by this fixed-grid map.
+https://github.com/CrocSwap/integer-mult-bounds/pull/6438
+https://github.com/CrocSwap/integer-mult-bounds/pull/6418
+https://github.com/CrocSwap/integer-mult-bounds/pull/6066
+https://github.com/CrocSwap/integer-mult-bounds/pull/6526
+
+Cinna Davis, 1337isnot1337, PR4088 and PR2226, supplies the consumed exact
+native representations. All source notices remain unchanged. The native
+compiler retains the work of icekylinx, DreamingOfClouds, vendor531,
+Rohan Arun and the preceding Gaussian lineage.
+https://github.com/CrocSwap/integer-mult-bounds/pull/4088
+https://github.com/CrocSwap/integer-mult-bounds/pull/2226
+
+Tekalign's reversible lifting, JoyboyBrian's rounded ports, Arham Tahir's
+numerical work and the PR4743/PR5485 lineage are predecessors of PR6509.
+Chris R / Clear1nsight and the credited PR4017 contributors supply a different
+source-bound deterministic native circuit with dirty scratch.
+https://github.com/CrocSwap/integer-mult-bounds/pull/5485
+https://github.com/CrocSwap/integer-mult-bounds/pull/4017
+
+Lattice parity, nearest-grid error and conditional martingale orthogonality
+are established mathematics. No global literature priority is claimed.

research/selected-native-unbiased-rounding/PROOF.md

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+# A retained-seed native Gaussian interface
+
+This is an all-integer construction with a finite source check. It does not
+change kappa. It supplies zero conditional mean error, bounded absolute error
+and exact data return with retained seeds. It does not supply a reversible bit
+circuit, a tape bill or a uniform relative error coefficient below one.
+
+## Source and new scope
+
+The selected PR6030 word has 22,803 events. Its first MOVE is zero-based event
+9896, `["MOVE",236,3,7,false]`. Frame 3 is empty; frame 7 has basis [21].
+The Boolean is a private-field flag. The native map at width one is the
+canonical frame T_[21] on 512 complex coordinates. The unchanged Clifford
+compiler produces a rank-one recipe L C R. The separate coordinate oracle
+checks all 512 columns, including global phase. SOURCES.json pins both sources
+and the word. Source assertions remain enabled in both registered modes.
+
+The deterministic conserved-sum center family and nearest-grid property are
+prior work in [PR6509](https://github.com/CrocSwap/integer-mult-bounds/pull/6509).
+The new interface randomizes opposite centers, uses one fresh bit per complete
+layer, binds the map to this selected MOVE, and proves its conditional moments,
+tensor bound, seed cost and reachable relative-error limit.
+[PR6438](https://github.com/CrocSwap/integer-mult-bounds/pull/6438) gives the prior
+question of optimal unbiased reversible ensembles for other rational operators.
+
+## Pair map
+
+Use units of a positive grid spacing h. Let x,y be Gaussian integers, s=x+y,
+d=x-y. Choose c(s)=0 when Re(s)+Im(s) is even. Otherwise choose the sign of
+Re(s) when Re(s) is odd, or i times the sign of Im(s) when Im(s) is odd.
+The sign is never taken at zero. Then Re(c)+Im(c) has the parity of
+Re(s)+Im(s), and c(is)=i*c(s). For sigma in {-1,+1}, define
+
+```text
+d' = i*(d-sigma*c(s)) + sigma*c(s)
+F_sigma(x,y) = ((s+d')/2, (s-d')/2).
+```
+
+The numerators are even Gaussian integers: d and s agree componentwise
+modulo two, and rotation exchanges the components of d-sigma*c. The parity
+condition gives d'=s modulo two. The complex sum s is preserved exactly.
+The inverse uses the same s, seed and center, with -i in place of i. Thus
+each fixed seed gives a bijection of the entire Gaussian integer pair lattice
+with an exact two-sided inverse. No input-size bound is needed.
+
+Multiplication of both inputs by i multiplies s,d,c,d' by i. Every fixed-seed
+map and inverse therefore commute with all four Gaussian unit phases. These
+properties retain the prior conserved-sum mechanism; they are not claimed new.
+
+## Exact stochastic optimum and one-bit cost
+
+The ideal pair operator is `C=[[1+i,1-i],[1-i,1+i]]/2`. It sends d to i*d.
+The local error is `sigma*((1-i)c/2,-(1-i)c/2)`. A fair seed independent of
+the complete past therefore gives conditional mean zero. Its squared Hilbert
+norm is h^2*|c|^2: zero for even total parity and h^2 for odd total parity.
+The inverse ensemble has the same mean and error statement for an input
+independent of its seed. Inversion of the seed-dependent forward output is
+used for exact return, not as a fresh-noise step.
+
+For odd total parity, all four real components of the ideal output are
+half-integers. Every grid-valued output has squared error at least
+4*(h/2)^2=h^2. Both outcomes attain this lower bound. For even parity the
+ideal output is integral and both outcomes are exact. The construction is
+therefore pointwise optimal among all unbiased grid-valued random outputs,
+including nonreversible ones. This is an error optimum, not a machine-cost
+optimum.
+
+For N/2 disjoint pairs, use the SAME fresh bit on the entire layer. Given
+the input, each error still has mean zero. Disjoint coordinate support gives
+exact squared error k*h^2 for each seed, where k is the number of odd pairs.
+This is the pointwise minimum total error. The pair errors need not be
+independent; they can be fully correlated.
+
+When k>0, any distribution attaining the minimum has support on nearest
+grid points. One half-integral real component must take its two neighboring
+integers with probabilities one half. Output entropy is at least one bit.
+A seed that deterministically produces it has entropy at least one bit.
+The construction attains this lower bound for a layer. No minimum of f bits
+is claimed for an arbitrary complete f-stage implementation.
+
+## Actual native map and tensor stages
+
+The source monomials L,R are Gaussian-unit signed permutations. They preserve
+the grid and norm. Use L F_sigma R and the literal inverse
+R^-1 F_sigma^-1 L^-1. Its mean is LCR; its squared error is k*h^2, with k
+measured after R. Sum preservation is in the transformed coordinates:
+S L^-1 y = S R x, where S adds each pair. Arbitrary physical-coordinate pair
+sums are not asserted invariant.
+
+For f>=1 and N=512^f, apply a native stage on each address digit, with a fresh
+independent fair bit at each stage. Keep the f bits and reverse the actual
+stage order for inversion. The exact stage operators A_j are deterministic
+unitaries. For each completed stage error delta_j,
+
+```text
+E(delta_j | all earlier data and seeds) = 0,
+||delta_j||^2 <= (N/2)*h^2.
+```
+
+The final error is the sum of these defects transported by the later ideal
+unitaries. Conditional mean zero makes distinct summands orthogonal in L2.
+This requires no independence of their values. Hence
+
+```text
+E(output | input) = (T_[21])^(tensor f) input,
+E(||output-ideal||^2 | input) <= f*N*h^2/2,
+||output-ideal|| <= f*h*sqrt(N/2) for every seed word.
+```
+
+The proof holds at every finite width. The second-moment upper bound is not
+claimed sharp for the complete tensor map. A full 512^2 input and all four
+two-stage seed words are also checked against a Kronecker product of the
+independent source oracle. Finite checks corroborate, but do not replace,
+the universal argument.
+
+Freshness is required. RESULT.json gives a four-coordinate example where
+reusing a seed across two rounded axes changes the mean. Literal inversion
+intentionally reuses the retained seeds; it is not a new stochastic forward
+invocation. General adaptive reuse bias retains its prior credit in
+[PR6418](https://github.com/CrocSwap/integer-mult-bounds/pull/6418).
+
+## Reachable relative-error boundary
+
+Input h*R^-1 e_0 is a single Gaussian unit grid entry with norm h. After R,
+exactly one pair has odd parity. Every same-grid output has error at least h.
+Thus no same-grid rule, randomized or otherwise, can have
+`||delta|| <= u*||input||` for all such inputs when u<1. The constructed
+stage attains equality. It satisfies coefficient one for every grid input:
+k odd pairs require input squared norm at least k*h^2, and its error is
+exactly k*h^2.
+
+This witness is reachable at the actual source boundary. A reverse scalar
+row and a separate forward replay of all 9896 preceding events show that
+setting only public input 236 to this vector produces the same vector at
+the first MOVE. That public input starts in the empty frame. RESULT.json
+records the maximum prefix denominator and magnitude. The scalar prefix can
+therefore be exact on the corresponding finer grid. No earlier native error
+is needed. This does not assert one common grid at every cut of the whole word.
+
+The stage does not discharge the relative premise in
+[PR6526](https://github.com/CrocSwap/integer-mult-bounds/pull/6526).
+An absolute-error caller analysis, changed grid or changed encoding is needed.
+No impossibility is claimed for those other interfaces.
+
+## Complete direct bill and remaining obligations
+
+The supplied integer algorithm uses f*N/2 pair calls, 2*f monomial passes
+and f retained bits. A pair uses at most 13 additions/subtractions, four
+exact half shifts, three sign changes and a constant number of parity, sign
+and seed selections. Each monomial pass moves N complex coordinates and uses
+at most 2*N real sign changes. The two 512-entry route tables are built by
+the displayed nine-bit quadratic formulas, with setup included. Inversion
+has the same counts; a forward/inverse round trip doubles them.
+
+If input component magnitudes are at most M, pair outputs have magnitude at
+most 2*M+1 and temporaries at most 4*M+2. After j stages the component bound
+is 2^j*(M+1)-1. With M<=2^(w-1)-1, w+f signed bits suffice for stored outputs,
+and w+f+1 for pair temporaries. Gaussian unit permutations preserve these
+bounds. These are sufficient, not optimal, widths.
+
+Direct indexed work is O(f*N) bounded-word operations, including data movement,
+buffer initialization and routes. Addresses use 9*f bits. Arithmetic and
+address operations give O(f*N*(w+f)) bit work in this indexed model. There
+are at most two working arrays of N Gaussian entries in addition to the live
+input, constant pair scratch, route tables and f retained bits. This is not
+an in-place bit circuit, minimum-space construction or fixed multitape
+schedule. Sampling, random-tape access, entropy recovery, physical cleanup
+and accelerated recursive-child pricing remain open.
+
+Nonlinearity remains. Data inversion does not imply additivity, native-graph
+flatness, dirty compensation after intervening arithmetic or whole-word
+restoration. Only this MOVE and its stated tensor powers are bound. All five
+framework assumptions and the selected frontier remain unchanged. Selected
+cold replay, complete supplier/caller implementation, native builds,
+independent human acceptance and formal proof-kernel review are unclaimed.

research/selected-native-unbiased-rounding/README.md

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+# Unbiased reversible rounding at the selected first native MOVE
+
+The selected PR6030 word's first native MOVE now has a source-bound integer
+grid map with exact retained-seed inversion and zero conditional mean error.
+One fresh fair bit serves every pair in a complete butterfly layer. Its
+pointwise squared error is the minimum possible grid error: h^2 for each
+odd-parity pair and zero otherwise. Each seed preserves the complex pair
+sum and Gaussian phase covariance in the native pair coordinates.
+
+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 f*N*h^2/2, and an exact literal
+inverse with the retained bits in reverse stage order. This is a construction
+and proof with finite corroboration, not a new kappa.
+
+A reachable first-MOVE input has error/input norm ratio at least one under
+every same-grid rule. Thus this map cannot fill a uniform relative allowance
+u<1. The absolute-error caller join remains open. The [proof](PROOF.md),
+[generated receipt](RESULT.json), [source pins](SOURCES.json) and
+[scope record](TASK.json) state the contracts and limits.
+
+## Source and evidence
+
+Canonical base: 7c49fe32c55de02c2c873655c804b16e4c3fef0a; priority revision 8;
+selected PR6030. Event 9896 is MOVE field 236 from empty frame 3 to frame 7
+with basis [21]. The unchanged compiler and independent coordinate oracle
+agree on all 512 columns. The new interpreter checks real and imaginary basis
+inputs, signed dense inputs and every seed of a full 512^2 tensor input.
+A separate reverse row and literal scalar replay check the first-MOVE witness.
+
+The verifier checks all pairs with real components from -4 through 4, both
+directions, all Gaussian unit phases and signed inputs including 2^1024-scale values.
+It checks seed reuse bias, a wrong inverse seed, a biased fixed seed,
+deterministic floor, a non-covariant center, invalid seeds, the relative-error
+counterexample and source corruption. RESULT.json gives the exact counts.
+
+## Reproduce
+
+Use CPython 3.12 and the declared NumPy 2.3.4 dependency:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/selected-native-unbiased-rounding
+```
+
+Both registered commands use isolated, warning-strict normal and optimized
+modes. Expected markers: `PASS_SELECTED_NATIVE_UNBIASED_ROUNDING` and
+`PASS_REGISTERED_RESEARCH_PACKAGE`. The source compiler and coordinate oracle
+require assertions, so each mode runs them in an unchanged normal subprocess.
+The new verifier's checks remain active under optimization.
+
+## Complete scope and remaining work
+
+The direct implementation uses f*N/2 pair calls, 2*f monomial passes and f
+retained bits. The proof includes setup, arithmetic, routing, sufficient
+integer widths, data movement and O(N) buffer storage. This is an indexed
+integer implementation, not a reversible bit circuit or fixed multitape
+schedule. Sampling, random-tape access, entropy recovery, physical cleanup,
+accelerated recursive tariffs and full caller propagation are unpriced.
+Data inversion does not prove additivity, graph consistency or cleanup after
+arbitrary intervening arithmetic. All five assumptions remain open.
+
+Selected cold replay, predecessor full scientific suites, native builds,
+independent human review and proof-kernel review are unrun. See the PR for
+publication and hosted status. A hosted pass verifies the registered commands,
+not acceptance of the universal proof.
+
+## Open next questions
+
+The next question is a full absolute-error caller join with finite scalar
+records and owned fresh randomness. First bind its completed native defects
+to the selected suffix transfers. A physical supplier and full tape invoice
+are the blockers. No next scope is reserved.
+
+## Attribution and overlap
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance; no subagents.
+[NOTICE](NOTICE) separates the prior deterministic center family in PR6509,
+Nick Petalas's ensemble and reuse results, the selected word, and the consumed
+native compiler/oracle lineage. Their sources remain unchanged.
+
+Public open/closed PR and Discussion metadata was refreshed before the claim.
+A second exact-main README/TASK search found no matching unbiased Gaussian or
+butterfly description. Related PR6509, PR6438, PR6418, PR4017, PR6499 and PR6526
+were inspected. This construction has a different numerical contract and an
+actual selected native binding. Arbitrary diffs and unpublished work remain
+search limits. No novelty certificate is claimed.