[petalas]

Summary

For the two-target compact switch with radix 8, five-bit windows and constant source pair (1,1), the actual maximal invariant agreement core still requires exactly five coordinate translations. An explicit adversary defeats every four-entry menu, even with arbitrary constants in all control digits and windows.

Claim type: finite certificate and scoped combinatorial proof. The new exact core is

u0 < 7, t0 < 7, t1 < 7, W0+t0 < 32

PR6812, by Benny Zhao / zhaobenny, proves the five-entry minimum for a sufficient product guard and asks whether actual agreement permits fewer entries. This contribution settles that question negatively at the stated geometry with the endpoint-invariant selector contract. Its five-entry upper construction is credited prior work. No new kappa or selected-frontier change.

The contribution was produced by Nick Petalas. Credit John-Michael Smith, Hanson Sun, Arham Tahir, Aaron Wang, woklerx, Douglas Colkitt and the retained compact-switch lineage, plus Nick Petalas's older PR6231 translation interface. Full credit and exact provenance are in the package. Advance claim.

Evidence

Before: the sufficient-product minimum did not constrain the larger actual agreement core. After: direct enumeration classifies that core, and a written adversary proves four translations cannot cover it.

• All 16,777,216 complete dirty states checked. Raw agreement has 11,441,280 states; the invariant core has 10,185,728.
• A separate digit-vector implementation checks both endpoint executions on all 131,072 quotient representatives.
• The five-entry menu covers every one of 16,384 relevant coordinate tuples. The adversary checks every one of 4,096 lower-temporary shift profiles with varied other constants. Its written proof covers arbitrary constants; this is not an enumeration of all menus.
• Four isolated clause-omission controls, an agreement-without-invariance witness, short-menu fixtures, six invalid menus and rejection of a five-entry adversary input pass.
• Warning-strict normal and optimized direct commands pass with identical generated, package-hash-bound receipts. Timed locally at 18.88 s and 18.26 s, within their 300 s limits. These are timeout-headroom measurements, not a speedup claim.
• Registered replay, metadata/pins, source compilation, staged whitespace and committed-byte/source-history preflight pass. The preflight checks all six files and their reachable base e545235.

Local environment: CPython 3.12.15, macOS 27.0.1 arm64, standard library only. Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/compact-two-target-translation-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/compact-two-target-translation-minimum/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/compact-two-target-translation-minimum
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/compact-two-target-translation-minimum --check-source-history

Substitute a local CPython 3.12 executable elsewhere. Expected markers are PASS_COMPACT_TWO_TARGET_TRANSLATION_MINIMUM, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Written proof and finite-domain contract.

Limitations

The result fixes the raw word, source pair, widths, independent coordinate translations and a selector satisfying sigma(Tx)=sigma(x). It does not exclude changed words, coupled maps, other source functions or selector cleanup with extra retained state. Physical translations, selection, tape/precision, hosting, restoration invoices, native/compiler realization and analytic transfer remain unpriced or unproved. No predecessor scientific suite, selected cold replay, Lean or Docker run was performed. No all-parameter or complete multiplication theorem is claimed. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Does the minimum persist at other widths and radices? First derive a symbolic core formula for the same all-one two-target word. The finite classification does not establish that formula.
2. Can coupled translations or a non-invariant retained selector beat five at this geometry? First specify a concrete map and cleanup contract. Physical costs and legal implementation remain blockers.

Related open work

Completion refresh read all ten open account PRs created since the CLAIM, including this PR. The nine others are #7189, #7191, #7192, #7193, #7194, #7195, #7196, #7198 and #7199. None studies this compact-switch translation minimum or computes the same invariant core. No extra numerical comparisons were run.

No same-object open PR surfaced in the earlier live open-PR, title/body and Discussion checks. PR6812 and the earlier PR6231/1603/1823/1856 are merged predecessors. The cached helper has sparse local-note coverage; direct origin/main README searches and live checks supplement it. Absence of a text match is not a novelty certificate.

CI notes

CI did not certify this package. At unchanged head 7bc88f3fc12a6eb230df01a656afefdc70b205a7, run 43001666262, plan job 130459029974, failed in Onboarding helper tests. I read the completed checks once and inspected the job log. It reports exactly one failure among 760 tests, with 24 skipped:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

The AssertionError says that `make frontier7000-verify` was not found in the CI verification document. On base e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json names that command while docs/ci-verification.md still describes #6030. I confirmed the mismatch directly from those base Git objects and confirmed both files and the test are unchanged by this PR. The package replay job was skipped after plan failed.

I verified the same sole failing test, missing-command assertion and failures=1, skipped=24 summary in all seven other failed run logs below:

• 42905094314 and 42903313790.
• Prove the paired-root floor for three-shear cycles off tsu = ±1 (no κ) #7194: 43001222481.
• Sharpen the composite cyclotomic precision exponent (no κ) #7195: 43001319803.
• Prove q² coupled optima for two resonant Jordan blocks (no κ) #7196: 43001498725.
• Prove exact minimum persistent stock for query streams (no κ) #7198: 43001765438.
• Sharpen prime-power cyclotomic gap exponents (no κ) #7199: 43001851217.

The verification evidence for this package remains local, as recorded above. No fix commit, push, shared-file change, job rerun or retrigger was made during closeout.

Merge Danger

Door: two-way. Blast radius: one isolated research package. The selected record, shared code and frozen evidence are unchanged.