[petalas]

Summary

Scoped algebraic construction and written proof with exact finite corroboration. A coded four-field child substitution works for arbitrary ordered, noncommuting stage factors A and B precisely when its first-child query c satisfies cB=Bq. The explicit construction is c=BqB^-1.

child A: diag(A,A) U(BqB^-1)
child B: diag(B,B) L(-q^-1)
paid complement: AB on each of two original fields
result: diag(AB,AB,AB,AB) [U(q) direct-sum q direct-sum q^-1]

The proof derives the complete residual for arbitrary c, retains every independent input and supplies the next scalar bridge. The in-place bridges use seven field additions/subtractions, four swaps and two negations, without a clean scratch field. Both child suppliers, one thin query and both complete complement applications remain in the abstract bill.

This contribution was produced by Nick Petalas. Question source: Dillon Amburgey's PR4933 typed coded-recursion question. Credit Ali Romman / PR6348 for the four-field scaffold, paid complement and original commuting identity; Dugongue / PR4929 and the coded-controller lineage; Cinna Davis for the thin-query/path interface; Cao Yifan, zxv, Jacob Alan Sussman, Ryanson Jonathan and all retained native/helper/compiler/outer authors. Ordered multiplication and conjugation are classical. The new result is the exact transport contract and residual, not the original merge or a first general noncommuting controller.

Proof. Advance CLAIM. Canonical main was refreshed before publication and remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. Only six files in one new package are added.

Evidence and reproduction

Before, the cited four-field module uses commuting fresh-axis factors. After, the exact transported-query word preserves the endpoint without any stage commutation assumption; untransported c=q has the displayed commutator defect.

• The new free-group-algebra checker verifies all 16 endpoint entries and all 16 exact residual entries. Six mathematical mutants reject, and all three free commutators remain nonzero.
• A separate literal-vector engine checks 2,520 operator triples and every one of their 20,160 arbitrary-input basis columns over F2/F3. The untransported word fails on exactly the 2,028 enumerated noncommuting fixtures; the full-map commutator criterion is checked in every fixture. Three symbolic-domain/capacity controls and two singular-operator controls reject.
• Both warning-strict normal and optimized registered commands pass locally on CPython 3.12.15, timed individually at about 0.19 seconds, within their 30-second bounds. The standalone verifier prints PASS_TRANSPORTED_CODED_QUERY; metadata/replay prints PASS_REGISTERED_RESEARCH_PACKAGE.
• Committed bytes/modes and source-history checks pass at 013d3edb041a9bc6f7f42c6d93b4d74511adbb8e. Official export integrity and declared-input coverage pass. Its optimized replay succeeds with an empty PATH outside Git; all seven read-only export files retain their bytes and modes.
• Receipts are generated by the verifier and bound to all six package hashes. No predecessor scientific verifier was imported or replayed.

From the repository root, using the actual local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/transported-coded-query/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/transported-coded-query/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/transported-coded-query
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/transported-coded-query --check-source-history

Another host may substitute its CPython 3.12 executable. Only the standard library is required.

Limitations and unrun obligations

This is one fixed-scaffold ring/operator identity, not an admitted coded supplier. For an arbitrary module the condition is equality of the induced operators; uniqueness as a ring element uses a faithful action. The construction does not provide physical access to BqB^-1, price the two ordered AB complements, assign Gaussian ranks to the coded output jobs, or bind an actual PR4933 negative bank. Those costs are unknown, not zero. The published physical substitution gap in merged PR5036 is not silently bypassed.

Native all-column realization, owned erasure, uniform compiler, precision/tape and analytic recovery remain unproved here. No selected cold replay, Lean/kernel proof, physical rounded multiplier, new kappa or selected-frontier change is claimed. Finite checks corroborate the written universal proof. Independent human and formal proof-kernel review are unclaimed.

Open next questions

• Bind one actual coded child pair to cB=Bq and price the transported query, both AB complements and every frame/ownership boundary. Physical query access and coded-output admission are the blockers.
• Can a changed scaffold avoid that transport at a lower complete cost? Start with one explicit replacement operation and arbitrary independent inputs. This necessity theorem does not cover a changed scaffold.

Related open work

• Prove the exact nineteen-leaf query-tree topology optimum (no κ) #6380 optimizes fixed nineteen-leaf topology under the original commuting merge; this result changes one merge's operator contract and performs no topology comparison.
• Independently verify the fixed balanced caller Laurent endpoint #6361 verifies the frozen 353-event commuting Laurent caller; this result constructs a new noncommuting four-field identity rather than auditing that endpoint.
• Compose native input lifts with the query tree at conditional κ 0.00314980 #6350 composes native input lifts with the fixed query tree; it supplies no transported-query realization for this contract.
• Contract nine native helper outputs at conditional κ 0.00315057 #6353 contracts native helper outputs with that caller; its local native-word change is separate from this abstract coded substitution.
• Conditional κ = 0.00315087 (+3.0e-7 over #6353) #6372 changes a two-anchor native boundary with the retained caller; this proof constructs no such physical boundary.
• Conditional κ = 0.00315120 (+3.3e-7 over #6372): three shared-carrier input lifts #6375 changes shared-carrier native input lifts; this result neither replays its word nor compares its kappa.
• Independently audit PR6353 rational joint basis and public boundary #6355 audits Contract nine native helper outputs at conditional κ 0.00315057 #6353's rational boundary, a separate obligation from the new module contract.
• Independently recount PR6353 recursive invoices and fixed-saving margin #6359 audits Contract nine native helper outputs at conditional κ 0.00315057 #6353's formal recursive invoice; no complete native invoice is claimed here.
• Independently audit PR6372 two-anchor scalar boundary and endpoint #6376 audits Conditional κ = 0.00315087 (+3.0e-7 over #6353) #6372's scalar endpoint and restoration, not a noncommuting coded-child substitution.
• Certify PR6375 matched profiles and unique relative-tariff threshold #6378 certifies Conditional κ = 0.00315120 (+3.3e-7 over #6372): three shared-carrier input lifts #6375's relative tariff threshold; this result computes no such tariff or crossing.

Completion refresh inspected all 14 open account PRs created since the advance CLAIM, including this PR. The new related #6391 line below distinguishes its inventory result; #6380 was already listed. The other eleven contributions concern different contracts or quantities. No same-instance numerical disagreement was identified, and no extra computations were run to check another contribution.

• Prove exact signed idle completion for overlapping stage outputs (no κ) #6391 proves a signed whole-idle inventory optimum for overlapping commuting stage demands; this result constructs a noncommuting coded-child substitution and computes no such tariff or optimum.

CI notes

Hosted Registered research packages run 41511167913 passed at exact head 013d3edb041a9bc6f7f42c6d93b4d74511adbb8e. Both plan and replay (research/transported-coded-query) succeeded. The completed checks were read once; separate run metadata confirms completed status, success and the matching head. All registered commands also passed locally. CI certifies execution of the declared finite checks; independent human and formal proof-kernel review remain unclaimed.

Merge danger

Door: two-way. Blast radius: isolated research package. Shared code, frozen predecessor bytes and selected-frontier records are unchanged.