[petalas]

Summary

For C=[[1+i,1-i],[1-i,1+i]]/2, prove the exact scale needed to make every axis prefix integral when both the input and complete tensor output are Gaussian integral. Claim type: scoped written proof with exact finite corroboration.

j processed axes:    sharp Gaussian exponent min(j,f-j), pi=1+i
all prefixes:        least binary scale 2^ceil(floor(f/2)/2)
arithmetic wrapper:  scale input; apply all C axes; divide completed output

The proof uses both integral endpoints. Paired-coordinate inputs attain the bound at every cut, including a middle cut of any fixed axis order. This supplies a sharp arithmetic bridge to #7429's intermediate-integrality precondition. No new kappa or selected-frontier change is claimed.

Produced by Nick Petalas. Credit CubeJerry #7429 at 1a555e469b89af70d309bc966f8f813ac11dc887 for the literal positive C leaf and its explicit integrality gap; jmbarrancoml #3091 and alejandrozu for the established C tensor, physical axis method and unrestricted reserve; and Clear1nsight #3382 for the different maximal simultaneous exactness lattice. Classical fractional-ideal and tensor methods retain their prior credit. No predecessor scientific verifier is imported or replayed.

Only six new files under research/native-c-endpoint-prefix-scale/ are added. Advance CLAIM.

Evidence

• Before: Implement the positive rank-one native C tensor leaf with complete literal control and retained workspace accounting #7429 explicitly requires integral outputs at every tensor axis; integral final output alone does not meet that precondition.
• After: the written all-f proof gives the least common scale, matching inputs and the exact arithmetic scaling/unscaling composition.
• Exact local checks pass 164 sharp cut witnesses through f=10, all 65,808 Gaussian 0/1-component arrays through f=3, 180 separately coded rational-prefix comparisons, direct tensor-entry comparisons through f=4, all 153 axis permutations through f=5, and eight named semantic controls. Exactly 968 enumerated inputs have integral final outputs, and all their 3,792 checked prefixes pass the theorem's bounds.
• Warning-strict normal and optimized CPython 3.12.15 runs pass in about 0.25 seconds each. Both registered commands, metadata/source pins, committed-byte/source-history preflight and a read-only Git-free optimized replay with empty PATH pass locally. Normal, optimized and detached receipts agree at the same package bytes. Receipts are generated from results and file hashes; none is hand-written or committed.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/native-c-endpoint-prefix-scale/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/native-c-endpoint-prefix-scale/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/native-c-endpoint-prefix-scale
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/native-c-endpoint-prefix-scale --check-source-history

Another installed CPython 3.12 path can replace the shown interpreter. Expected markers are PASS_NATIVE_C_ENDPOINT_PREFIX_SCALE, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The integral final output is a required promise. Signed-range bounds, literal scale/unscale programs, source admission, ownership, movement, scratch and retained-history costs remain open. This proof supplies no complete physical MOVE/caller, paid/productive invoice, selected assumption discharge or complete multiplication theorem. All five framework assumptions remain. Native-machine and selected cold replays, formal-kernel verification and independent human review were not run. Every registered local command passed; no registered local obligation is deferred to CI.

Canonical main was refreshed before submission and remains e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8, selected #7000. Exact-formula, general partial-product, main-tree, merged-PR and live-claim searches found no same-scope result in the inspected coverage. The final overlap refresh reported 767 open PRs and 7,348 Discussion records with complete pagination and no warnings. Unpublished work and arbitrary unsearched diffs remain outside that coverage.

Open next questions

Install the sharp scale around one literal positive C leaf: supply an actual producer with integral final output, prove sufficient intermediate signed width, and price both exact conversions with movement, owners, scratch and retained history. A favorable complete bill is required before any supplier claim. This contribution stops at the first sharp arithmetic result and reserves no follow-up.

Related open work

• Implement the positive rank-one native C tensor leaf with complete literal control and retained workspace accounting #7429 implements the same selected positive C tensor. This proves a new arithmetic entrance bridge and does not replay or certify that machine implementation.
• Implement both selected Gaussian linear maps as fixed tape programs with every column and complete owner return #7430 implements the two binary monomial maps around the same selected MOVE. Their physical composition remains separate from this central-tensor integrality theorem.
• Lower the current selected quadratic Gaussian phase to a fixed tape table with complete owner return #7426 implements that MOVE's quadratic Gaussian phase. Phase/tensor composition and its complete range and ownership contracts remain unproved here.
• Remove two CNOTs from the selected left binary map (no κ) #7462 shortens the same MOVE's left CNOT word. It changes a monomial gate count, while this result changes an arithmetic integrality precondition; no competing numerical quantity is computed.
• Bound local Walsh packing at Type B stage 9 #7411 prices a Walsh packing change for a Type B idle. This theorem supplies no Walsh conversion tariff or packing improvement.
• Research: paid-copy E8 exact-RAM realization with an attributed DFT corollary #647 supplies an alternative exact-RAM C-tensor construction. Its computational model and recurrence differ; this result supplies no new runtime or exponent comparison.

The completion refresh inspected the six other open PRs from this account created since the CLAIM, #7462 through #7467. The only same-MOVE contribution is #7462, already listed above. The other scopes concern different matrices, helper-subspace optimization or compiler width; no competing integral-prefix scale or numerical disagreement was identified. No extra mathematical computations were run to compare them.

CI notes

At unchanged head 8b0dd714d30e9c1574387587eb6418ae99ac34e1, run 43281155683, the plan job failed in Onboarding helper tests. The log reports 760 tests, one failure and 24 skipped. The exact failed test is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with AssertionError: '`make frontier7000-verify`' not found in docs/ci-verification.md.

The mismatch is present in canonical base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. Both base files were inspected directly. This PR changes only its six new package files.

The same exact test, assertion and FAILED (failures=1, skipped=24) summary were independently verified in run 42905094314, plan job 130140894651, run 42903313790, plan job 130134720799, and run 43279933171, plan job 131398194664. Each log reports 760 tests. This comparison concerns only the shared CI failure, not those contributions' mathematical results.

The package replay job was skipped because planning failed. CI did not certify this package. The local normal/optimized scientific checks, registered metadata/source-pin replay, committed-byte/source-history preflight and read-only Git-free optimized replay passed as recorded above. Hosted package certification remains unavailable pending the repository-level repair. No fix, retrigger commit or job rerun was made. The contribution's head and mathematical scope are unchanged.

Merge Danger

Door: two-way

Blast Radius: isolated

Six new package files. Shared code, frozen predecessors and the selected frontier are unchanged.