Conditional κ = 0.00302725 (+5.75235e-7 over #5387): owned circuit transposition

research/native-transpose-owned/NATIVE-CERTIFICATE.json

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+{"axes":9,"peak_expanded_terms":24664,"problem_sha256":"a4c901e27b21dc7033ff8ace1217165eabb5e59ecd301dba321f76940db8be05","residual_terms":0,"roles":641,"schema":"native-tensor-cutoff/v1","status":"IDENTITY_ALL_POSITIVE_WIDTHS","window":[22,25]}

research/native-transpose-owned/README.md

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+# Conditional kappa 0.00302725 by transposing an owned supplier
+
+This construction gives **conditional kappa = 12109/4000000 = 0.00302725**.
+It improves the selected PR4610 value 0.0030217758 by 0.0000054742. It improves
+the eight-relation PR5383 value 0.00302666 by 0.00000059. The paid profile changes;
+the gain does not come from more digits of an unchanged root.
+It also exceeds the later [PR #5387](https://github.com/CrocSwap/integer-mult-bounds/pull/5387)
+compiler refinement, 0.003026674765, by 0.000000575235. That refinement is related
+work; it is not a source dependency.
+
+The new result transfers a completed backward supplier to the forward
+interface. It applies the classical transposition principle to the complete
+linear circuit, including owned COPY and CLEAR operations. Exact monomial
+corrections then restore the required forward map. Thus the two forward calls
+can use the same reduced paid and helper profiles as the sixteen backward
+calls. The construction also gives a reusable rule for later suppliers with
+this interface and clean owned storage.
+
+[RESULT.json](RESULT.json) gives the complete evidence and bill.
+[WORD.json.gz](WORD.json.gz) gives the emitted finite word and its boundary
+corrections. [PHASES.json.gz](PHASES.json.gz) gives every transposed paid
+operator recipe. [SOURCES.json](SOURCES.json) pins all consumed bytes.
+The selected frontier is unchanged.
+
+## Exact source and separate scope
+
+The scientific base is canonical main
+`1ec1119972c9aefe42525172d07b332fcff81cb4`, priority revision 7.
+The source is Ali Romman's eight-relation
+[PR #5383](https://github.com/CrocSwap/integer-mult-bounds/pull/5383), fixed at
+`4bd6efa51a495a7c7b7754915cf0244e7e60969e`. Its published word, after canonical
+JSON serialization with a final newline, has SHA256
+`2aee5e997df99726fc763e510d9cb69fbe4f86114dc8d2cd33ad26c6c462a21d`.
+The verifier reconstructs and checks that exact word before transposition.
+It does not consume a later PR head or an unpublished helper relation.
+The exact source files are retained in commit
+`5eb314069417db1f4305430b6b14f20e9251e509` in this branch. The live pins use
+that ancestor; each `source_pin` retains the original PR source and hash.
+A clean checkout does not need either pending PR commit in its object store.
+
+PR5383 prepares eight owned pivots from independent cold helpers. It removes
+eight rank-eight centers and eight rank-one jobs, and debits eight rank-nine
+productive targets. Its unchanged input is Cao Yifan's final
+[PR #4610](https://github.com/CrocSwap/integer-mult-bounds/pull/4610) word, with
+payload SHA256
+`cfcd36b40326d7a5815e1db5313444379b3fb756b27d5201c561574c085cc0dd`.
+This package retains that backward construction and changes only the forward
+supplier and its full cost calculation.
+
+The [scoped claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/5125#discussioncomment-19154367)
+first tested the one-relation source. After PR5383 published and released its
+eight-relation construction, this work pinned that stronger completed source.
+No search or priority claim for the source relations is part of this result. Native map changes,
+finite contact searches and no-snapshot relation screens that gave no gain
+are not included in this PR.
+
+## 1. Transpose a complete linear circuit
+
+Let a source circuit act on independent fields x and owned fields z. Assume
+that its complete map, with both x and z initially arbitrary, is
+
+```math
+(x,z) \longmapsto (Bx,0).
+```
+
+In particular, every owned input column and output row is zero in the complete
+matrix. Reverse the instruction order and transpose each instruction matrix.
+The resulting circuit has complete map `(x,z) -> (B^T x,0)`. The transpose is
+bilinear transpose over the complex numbers. It is not conjugate transpose.
+
+The emitted scalar rules are:
+
+| Source instruction | Transposed instruction |
+| --- | --- |
+| `d += c*s` | `s += c*d` |
+| `d *= c` | `d *= c` |
+| owned `CLEAR d` | owned `CLEAR d` |
+| owned `COPY d <- s` | `s += d`, then owned `CLEAR d` |
+
+All coefficients stay unchanged. Every new CLEAR targets a field that was
+already owned. The procedure uses the same physical slots. It does not invert
+a CLEAR or assume that an erased value is available. A source COPY costs one
+extra ADD under transposition; that cost is included below.
+
+A native monomial U has monomial transpose. If its action is
+`U e_x = i^p(x) e_(Ax+t)`, its transpose sends `e_(Ax+t)` to `i^p(x) e_x`.
+Its permutation is the inverse affine map, with the same entry phase at the
+inverse address. The affine-quadratic grammar is closed under this operation.
+Equivalently, `U^T = conjugate(U^-1)`.
+
+A standard Gaussian child C_r is symmetric. A source child with the exact
+factorization `L C_r R` therefore transposes to `R^T C_r L^T`. Both new boundary
+maps are native monomials. It has the same paid rank r, including its absolute
+phase. This gives a direct finite lowering algorithm for every source child.
+It does not require a new Gaussian supplier or a new assumption about general
+Clifford identities.
+
+These identities hold at every positive tensor width f. Transposition commutes
+with tensor powers, and scalar coefficients stay outside tensor powers. The
+new native checker verifies the concrete monomial logical normalization by
+its complete four-width criterion. The separate transposition argument and
+Gaussian factorization establish the paid circuit statement.
+
+## 2. The zero-row frame marker requires no reverse zero assumption
+
+The source replaces eight paid jobs by physical identity. At each such job,
+the data row is zero under the prepared helper relations. It records this with `zero_retag`.
+The transpose of physical identity is identity. It is omitted from the emitted
+physical word. The reverse data row need not be zero.
+
+For a second endpoint check, insert a distinct fresh free operator at each corresponding
+reverse coordinate boundary. The exact rational replay cancels every term
+that contains this operator at all returned fields. It checks all 641 entry
+columns, including the seventeen owned fields. This is a proof in free operator
+words, so it also holds for the actual nonmonomial frame quotient. The proof
+symbol is not an emitted gate, an unpaid Gaussian call, or a presumed native
+recipe. The complete matrix transposition theorem is the physical argument.
+
+The new forward artifact lists source indices for transposed paid operators
+and source markers. A paid reference resolves through the exact source pair
+and PHASES.json.gz. A native warp reverses its signed map and its frame pair.
+Source entry and exit normalizers are transposed in reverse order, at whole
+invocation boundaries. The inherited whole-job normalizer contract remains
+explicit; no internal partial job is normalized for free.
+
+## 3. Restore the required forward interface
+
+Use the source controller's normalized complex local interfaces, stated in
+`research/query-thinned-controllers/PROOF.md`. On each port, Q is the central
+rank-one Gaussian and D is its commuting stage multiplier. Both are symmetric
+in these coordinates and `Q^4=I`. For later stages D acts on eight fresh axes.
+At stage zero it is the source nine-axis Gaussian times `Q^-1`; the same
+symmetry and commutation statements hold. Spectator axes are unchanged.
+
+```math
+F=\begin{pmatrix}D&0\\DQ^{-1}&D\end{pmatrix},\qquad
+B=\begin{pmatrix}D&-DQ\\0&D\end{pmatrix},\qquad
+J=\begin{pmatrix}Q^2&0\\0&-I\end{pmatrix}.
+```
+
+Direct multiplication gives `F = J B^T J`. Apply J before and after the
+transposed normalized backward circuit. The Q squared operation is the exact
+native translation, with no paid Gaussian job. The Y sign is scalar `-1`
+on the complete field. It is not native phase two raised to tensor width f.
+The verifier rejects that phase substitution with an all-width counterexample.
+
+The backward helper normalizer gives `conjugate(C_r)`. Transposition leaves it
+unchanged because it is symmetric. Append `C_r^2`, the native translation on
+its r active axes, to obtain C_r. Charge one such pass on every retained helper.
+The word records all 400 helper ranks and translation offsets. These wrappers
+restore the original forward main contract and the 400 independent productive
+helper contracts. The eight pivot fields and C0 through C8 remain owned and return zero.
+
+This argument applies to any completed supplier with the same normalized B
+interface, standard productive Gaussian helpers, native boundary maps and
+clean owned storage. It does not classify other interfaces or justify a
+source construction whose own endpoint or bill is incomplete.
+
+## 4. Complete bill and conditional exponent
+
+Both directions now have
+
+```text
+paid:    {1:2193, 2:412, 3:510, 4:127, 5:40}
+helpers: {4:28, 7:7, 8:47, 9:318}
+```
+
+Relative to PR5383, the two forward calls remove sixteen H1 and sixteen H8
+paid jobs, and debit sixteen H9 productive targets. For every `0<t<1`, the complete
+target-minus-paid moment gains `16*(1+8^t-9^t)>0`. This adds 12.5 percent to the
+source's sixteen-call moment gain. Relative to final PR4610, the full p16
+change is minus 144 H1, minus 144 H8 and minus 144 H9 targets.
+
+The full bill has **64,116 paid jobs and 10,784 productive fields**. The rank
+surplus stays 752. Both main targets, all 15 query shears, all idle jobs and
+both completion tails are included. Two independently assembled invoices and
+two rational interval engines agree.
+
+The fixed-profile crossing lies in
+**(0.00302728824993, 0.00302728824994)**. The claimed saving 0.00302725 leaves
+more than 0.00000003824993 raw headroom. The tested saving is 0.00302726.
+The exact completion uses power 1024, cutoff 20, floor 21, tail offset 1/5
+and reserve 1e-9. Both tail ratios are below 0.97. The finite compiler has
+`H=31385200000000`, `N=2^H`, gap `1/700000000` and logarithm power 1000.
+The exact integer threshold test passes. All 54 strict and four functional
+outer predicates pass, with the retained ordinary BIT saving 0.0034.
+
+The forward scalar allowance is 159,320, including the sixteen transposed COPY
+rules and both Y sign layers. The route allowance is 10,292,000. It includes
+224 main native passes and 400 helper translation passes. It conservatively
+doubles the source route allowance because COPY becomes ADD then CLEAR.
+Eight additional physical private slots relative to PR4610 suffice, as in
+PR5383. Each direction has seventeen owned fields, including eight old helper
+slots.
+The local allowance remains below 2*10^7 and the complete raw nonrecursive
+allowance is **361,098,496 < 10^9**. The forward formal prefix L1 maximum is
+442, below the original forward maximum 457. The denominator is at most 32,
+equal to the original bound.
+
+## 5. Reproduce and assess the limits
+
+Use standard-library Python 3.12. From the repository root:
+
+```sh
+python3 -B scripts/research_package.py --package research/native-transpose-owned
+```
+
+The normal and optimized commands must end with
+
+```text
+PASS_CONDITIONAL_KAPPA_00302725_OWNED_TRANSPOSE
+```
+
+For a standalone replay, run:
+
+```sh
+python3 -I -B research/native-transpose-owned/verify.py
+``` The separate `--write` authoring
+mode writes only this package's artifacts after all mathematical gates pass.
+Normal verification compares every regenerated artifact with committed data.
+
+Checks cover the exact published source word, 641 independent/owned entry
+columns, the arbitrary frame-bridge cancellation, the complete native
+four-width certificate and its separate checker, all 2,847 transposed paid
+phase classes, all 180 native warp classes on 92,160 literal columns, all 112
+Q-squared port phases, four productive helper ranks, the literal p16
+controller, the full cost calculation, completion, compiler and outer ledger.
+Controls reject a wrong ADD transpose, missing owned cleanup, wrong native
+inverse, wrong prepared relation, wrong absolute phase, omitted Q-squared
+correction, scalar/tensor sign confusion, unearned helper credit and invalid
+outer parameters.
+
+The complete bound retains the selected framework's native/all-column source,
+owned erasure, whole-job normalizers, finite-template closure, precision and
+fixed-tape, analytic decomposition, prime avoidance and exact recovery
+assumptions. The source and new finite algebra/phase obligations are checked;
+physical tape execution, numerical precision execution, ordinary BIT native
+replay, Lean, Docker, analytic decomposition and integer recovery are not rerun.
+The huge uniform compiled word is specified by a finite algorithm and is not
+materialized. This is an author-checked conditional construction. It is not an
+unconditional multiplication theorem, a practical speedup, independent human
+acceptance or proof-kernel acceptance.
+
+## Credit
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0.
+Ali Romman's separate PR5377 and PR5383 contributions supply the owned
+relations, actual saved values, zero-row replacements and backward word. The new
+contribution is the transpose construction, its exact forward normalizer,
+physical phase receipts and complete additional gain. No credit is claimed
+for discovering the source relations.
+
+Cao Yifan / intqwq supplies final PR4610. Cinna Davis / 1337isnot1337 supplies
+the PR4470 controller and native lineage. Dugongue supplies transported-frame
+work; zxv supplies native, owned-cache and compiler interfaces; Jacob Alan
+Sussman supplies mixed-target recursion. The source notices retain their
+predecessors. The classical transposition principle is not new here.
+
+[PR #5369](https://github.com/CrocSwap/integer-mult-bounds/pull/5369), at
+`18354f41de4dc7bfda4692e814cf1d91a6d619ef`, supplies the complete native tensor
+criterion, producer and separate checker. The four code files are copied
+without changes. Its native and classical mathematical attribution remains in
+that linked proof. [source/NOTICE.md](source/NOTICE.md) gives the exact source
+links and copied-file scope. The pricing code is adapted from the attributed
+PR5377 engine, which retains Cao Yifan, Cinna Davis, zxv and Jacob Alan Sussman.
+
+## Open next questions
+
+Apply the construction to a stronger, published and independently verified
+backward supplier. First check its complete normalized interface, owned
+columns and productive targets. Then transpose the actual word and recompute
+the complete bill. This package does not reserve or assess unpublished source
+constructions. [TASK.json](TASK.json) records the scope and first gate.