[petalas]

Summary

Scoped algebraic proof with finite corroboration. For every pair of distinct left cosets of PR3828's 4080-owner centre subgroup, the full scalar common-observation kernel consists of exactly the two inherited constant triple directions.

- The one-block classification leaves unions of translated blocks open.
+ Every two-block union has kernel dimension 2, for every relative translation.
+ Every subblock A needs exactly 104|A|-j additional scalar linear observations.

Here j counts the complete cosets retained in A. On two full blocks the derived minimum is 848638. Pairing these blocks creates no additional hidden direction and leaves the information-compression density unchanged. This settles the two-coset case of Open next question 1 in PR3828, rather than auditing its result. No new kappa or selected-frontier change.

The proof uses output-class separation, including arbitrary partial intersections of translated outgoing sets, without assuming subgroup normality. The new source check establishes that exactly columns 0,6,85 lie in the distinguished triple plane. Single-block coset classification, subgroup order, reader rank and pairwise column independence remain explicitly cited predecessor premises.

Produced by Nick Petalas. Credit Chris R / Clear1nsight #3828, Ross Richardson #3797/#3071, Matarener #3656, Sagar Thakrar #3291, and their retained geometry/native/compiler lineage. No predecessor verifier is replayed or edited.

Evidence

• Before: the union-of-cosets question was open after the one-block theorem.
  After: PROOF.md gives the universal two-coset theorem and exact subblock information formula.
• New exact source test: all 109 outside-plane columns have rank three together with columns 0,6.
• New small-group implication checks: all six two-coset pairs in S3 times C2 have kernel dimension two. All 64 subblocks of one pair match the formula, with dimensions 0/1/2 on 49/14/1 subblocks.
• Four controls expose extra directions when assumptions are removed. Three column mutations each reach kernel dimension five; retaining only source-owner outputs gives dimension six.
• Both registered commands and direct normal/optimized checks pass. Timed direct commands take 0.846s and 0.858s on macOS arm64, CPython 3.12.15. These are validation durations, not a speedup claim.
• Exact committed bytes/modes and source-history preflight pass. Official export integrity passes; a read-only, empty-PATH, Git-free optimized replay passes with unchanged export bytes. A corrupted archive is rejected before extraction.
• AST parsing and staged whitespace checks pass. Generated receipts bind package-file hashes and remain outside the checkout.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/centre-two-coset-separation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/centre-two-coset-separation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/centre-two-coset-separation
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/centre-two-coset-separation --check-source-history --check-modes

Any CPython 3.12 executable can replace the local interpreter path. Expected markers are PASS_CENTRE_TWO_COSET_SEPARATION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

This is the historical calls-0/4 scalar whole-later-query model. The source order supplies the derived 8160-owner observation count; the large source matrix is not re-eliminated. Predecessor group/coset/rank certificates are cited dependencies, not newly independently accepted results.

Physical source acquisition, dirty restoration, native realization, storage, precision, recursive pricing, compiler and outer obligations remain open. This supplies no current P104 replacement, paid-call saving or complete multiplication theorem. The five selected framework assumptions remain. Lean, Docker and the selected cold replay were not run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can three distinct left cosets create mixed kernel directions? Three singleton columns can be dependent at one output, so this proof stops at two blocks. First specify an actual three-coset incidence and its coefficient compatibility. A physical extension still needs a legal consumer and a complete paid/productive invoice.

Related open work

The completion refresh checked the seven other account PRs opened since the advance CLAIM: #6121, #6122 and #6124 through #6128. They study THREE-base precision, higher-rank query shears, paired samplers, Gaussian row bounds and direct-flow leaf laws. None studies this two-coset centre-block kernel or its extra-linear-observation minimum. No same-instance numerical disagreement was found; no additional mathematical replay was performed for comparison.

No other open PR studying this exact object or quantity was identified in the earlier refreshed open-PR corpus and targeted live searches. #3828, #3797 and #3656 are merged cited predecessors. Searches do not cover unpublished work or every diff.

Canonical advance CLAIM. The delivered scope matches its predicted two-coset exclusion. Canonical main was refreshed at publication and remained 721c038.

CI notes

Uncertified by CI. At head 4cb8e99, run 40692336750 failed its plan job in the onboarding helper tests. The package replay was skipped and did not run in CI.

The failure comes from the current main baseline 721c038, the PR6030 kappa admission pushed with a skip-ci marker. Its selected record and onboarding documentation do not satisfy two existing synchronization tests:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the literal bold text 0.003125804 in docs/research/current-status.md. The document instead bolds the longer expression 0.003125804 = 781451/250000000, so the exact substring assertion fails.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects the selected command make frontier6030-verify in docs/ci.md. That document still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and make frontier5599-verify.

The same two failures are present in PR #6120's run 40691776027; both failed-job logs were inspected. Neither failing document is changed by this package. The completed checks were read once. No push, main-document fix or workflow re-run was performed.

All listed local package, direct normal/optimized, committed-byte/history and export checks passed. They do not replace hosted certification. This PR needs a maintainer re-run after main is fixed. No hosted scientific replay or independent scientific acceptance is claimed.

Merge danger

Door: two-way. Blast radius: isolated research package. No shared code, frozen source, selected frontier or other contribution changes.