[petalas]

Summary

The fixed five-call reflection border in PR6398 needs exactly two distinct repair cuts when repairs are linear, central-only and independent of the three stage variables. This is a scoped algebraic proof with exact finite evidence, not a new kappa or supplier admission.

At any one of the six cuts, invertibility forces one repair matrix:

required repair = inverse(suffix) * target * inverse(prefix)
B = 1 and B = -1 force different repairs at every cut

This rules out arbitrary rank and arbitrarily long central-only blocks at one location. A diagonal similarity covers every nonzero border parameter over every field of characteristic other than two. Ali Romman's existing two rank-one corrections attain the lower bound; the new contribution is the six-cut obstruction and unique-correction table. The predecessor's native identity, prices and search results are cited premises, not audit deliverables.

Evidence

Before this result, PR6398 exhibited two direct corrections and left different recovery methods open. The new proof excludes every repair confined to one cut, even if its rank or scalar complexity increases.

Both full direct modes print PASS_BORDER_SINGLE_CORRECTION. Each enumerates 36 cut/parameter witnesses with explicit frozen-repair failures, 36 normalized coefficient-table checks, 48 attaining fixtures and five domain/singularity rejection controls. The normal and optimized registered commands took 0.154 and 0.163 seconds locally, respectively, rounded up. Generated receipts bind all six package files by SHA256. Package replay prints PASS_REGISTERED_RESEARCH_PACKAGE; publication preflight prints PASS_RESEARCH_COMMIT_BYTES, including the base's source-history binding.

Exact reproduction from the repository root:

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

Other hosts may substitute CPython 3.12. The verifier uses only the standard library and writes its generated receipt to stdout. The written proof supplies the all-parameter quantifiers; the fixed rational checks are exact corroboration and a bounded-degree coefficient certificate.

Limitations

Four arbitrary input fields, fixed original factors, fixed cut selection and a formal identity modulo S^2=1 are explicit hypotheses. Retuning the word, additional workspace, adaptive cut selection, stage-dependent operations and approximate recovery remain outside the theorem. A concrete central operator with no negative character is outside the obstruction.

No complete tariff optimum, recursive precision transfer, physical realization or selected-frontier update is claimed. Lean, Docker, native/selected cold replay and predecessor root searches were not run. Independent human and formal proof-kernel review are unclaimed. The exact predecessor head is dee2663ba0918be86993fd9d63c8d64b020f4c83; main was refreshed at 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8.

Produced by Nick Petalas. Credit Ali Romman for the border, open recovery question and attaining word; Cinna Davis for caller interfaces, Jacob Alan Sussman for productive recursion, and the retained native/helper/compiler/BIT/analytic lineage identified in PR6398. Advance claim.

Related open work

• Mixed-query root: conditional κ = 0.00328621476572836015146308 (+4.28455% over #6375) #6398 supplies this exact border and its two-correction upper bound; this PR adds a sharp lower bound on distinct repair locations and does not recount its price or native construction.
• Prove exact signed idle completion for overlapping stage outputs (no κ) #6391 studies a different three-stage overlapping-output completion invoice. It does not repair this fixed border, and this result does not establish a tariff comparison with that proposal.
• Prove exact square-zero tensor error and sharp precision (no κ) #6402 proves an abstract square-zero ordinary tensor-error formula and references this border as adjacent work. It concerns approximate endpoint error; this PR concerns exact repair locations. No common numerical quantity is compared.
• Compose clean native merge with mixed root at raw saving 0.0032868 #6401 composes a changed native invoice with PR6398's exact mixed root. It studies raw funding of that different caller, with no claim about this fixed border's repair locations.

Open next questions

• Can a changed two-cut recovery beat the complete price of the existing two rank-one queries? Start with both explicit cut operations and their signed invoice; the location bound does not settle tariff optimality.
• Can a stage-sensitive operation or changed original factor produce a favorable exact caller? Price the middle-stage dependence, inverse access, restoration and cleanup before extending the construction.

CI notes

Both hosted checks, plan and replay (research/border-single-correction), passed at head 121b10ff0ffba6832b805a405e9f3bdd08cdd582 in GitHub Actions run 41526781598. The completed results were confirmed once with gh pr checks 6400 --repo CrocSwap/integer-mult-bounds. The registered replay certifies both normal and optimized verifier commands. Both modes and all package/source-history checks also passed locally. These checks do not constitute independent scientific or proof-kernel acceptance.

Merge Danger

Door: two-way. Blast radius: isolated. Only the new research package is changed; no shared code or selected record is modified.