[petalas]

Summary

Four independent normalized common-image full guards have joint passing probability strictly greater than the product of their marginals over every finite field and every ambient dimension m >= 4. This is a scoped written counting proof with exact finite corroboration. It resolves the four-covector case of Open next question 1 in SHIN09-git's #6710.

four-dimensional subspace classes
  -> explicit eight-term joint law
  -> analytic large-field and dimension tails
  -> 56 remaining exact rational comparisons
  -> J_4 > p^4 for every finite field and m >= 4

The new classification, formula, sign proof and verifier were produced by Nick Petalas. Credit SHIN09-git #6710 for the general subspace law, the question and the triple sampling method, ToikY #6701/#6703 for the binary-family and pair precedents, Nick Petalas #4227/#4807 for the earlier marginal/pair laws, and their retained Ryanson Jonathan, Cinna Davis, source527, commdata2338 and Jose Manuel BIT/normalizer/compiler lineage. Binary and ternary sign coverage is already supplied by #6710. No predecessor executable or scientific suite was replayed.

Only research/independent-four-guard-positive/ is added. The selected frontier is unchanged. Advance claim.

Evidence

Before, the all-field independent-family question beyond triples was open. The package supplies the four-row theorem, with a uniform analytic bound at every field order q >= 25 and exact thresholds at the remaining field orders.

CPython 3.12.15, standard library only. Normal and optimized direct checks passed with identical generated hash-bound receipts. Both registered commands passed. The verifier computes all 56 exceptional sign comparisons, all 258 declared subspaces over fields of orders 2, 3, 4, 5 and 7, and an independent affine-plane count in 16 boxes over fields of orders 2, 3, 4 and 5 at m=4. It distinguishes four altered Mobius terms and the independence guess, and rejects four out-of-domain formula inputs. The universal scope rests on the written proof and cited general counting law.

Timed locally, the registered command arguments took 0.91 seconds in normal mode and 0.90 seconds in optimized mode; the complete launcher took 2.06 seconds. The committed-byte/mode/source-history check passes for d2ed1826b500c8260cb33a7f71cb6cdd531837c4, covering all six files and canonical base e54523517d0f03e58352e647c403a5af3e525a57. Both modes also pass from read-only committed Git-archive bytes with an empty PATH and no Git metadata, with receipts identical to the checkout.

From the repository root:

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

Other hosts may substitute their CPython 3.12 interpreter path. The direct success marker is PASS_INDEPENDENT_FOUR_GUARD_POSITIVE; the registered marker is PASS_REGISTERED_RESEARCH_PACKAGE.

Limitations

The theorem requires four independent normalized covectors and uniform complete GL_m headers. It does not cover arbitrary dependent families or restricted header distributions. Five or more independent rows remain open. Physical selection, emission, dispatch, dirty restoration, stock, precision, the complete recursive bill and supplier admission remain unresolved. Independent human and formal proof-kernel review are unclaimed. No new kappa or complete multiplication theorem is claimed. Selected cold replay, Lean, Docker and unrelated historical suites were not run.

Canonical main was refreshed before submission and remains e54523517d0f03e58352e647c403a5af3e525a57. Current claims and all 546 open PR bodies were inspected before commit. The broad overlap helper's sparse local inventory is incomplete and its closed-PR API scan timed out. Separate canonical-main searches and merged-PR searches checked the object, formula shape and computed fractions. A direct recent-account/source-reference query found #6793, which cites #6710 as background but studies paid monomial-header cost, a different model from this full-GL probability theorem.

Open next questions

• Do five or more independent normalized common-image guards beat their product in every dimension? Start with the intermediate subspaces through the all-ones vector in five dimensions. Their full-support counts need additional structure.
• Can an actual BIT caller use this count with a fully paid uniform complete-header producer? First pin the exact request tuple and distribution. Physical generation, selection, restoration and the complete signed bill remain missing.

Related open work

No open PR studying this independent-four full-GL guard probability was found in the refreshed 546-PR body inventory. The direct question and counting predecessors #6710, #6701 and #6703 are merged. The apparent #1809 source-reference hit was inside a hash; its conditional Lean recurrence theorem studies a different object.

The completion refresh inspected the nine other account PRs created since the CLAIM: #7247, #7249, #7250, #7251, #7252, #7253, #7254, #7255 and #7256. None studies this independent-four full-GL guard probability or computes a competing value. No additional mathematical comparison computation was run.

CI notes

CI did not certify this package. At unchanged head d2ed1826b500c8260cb33a7f71cb6cdd531837c4, run 43023633785, plan job 130532724912, failed in Onboarding helper tests. I read the completed checks once and inspected the actual job log. It reports 760 tests, exactly one failure and 24 skipped. The sole failure is test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record, with AssertionError: '`make frontier7000-verify`' not found in the current CI verification document. The dependent package replay job was skipped after planning failed.

The mismatch is already present in canonical base e54523517d0f03e58352e647c403a5af3e525a57: certificates/selected-result.json names make frontier7000-verify, while docs/ci-verification.md still describes #6030 and make frontier6030-verify. I inspected those base files and the failing assertion, and confirmed that both files and tests/test_research_current_status.py are byte-identical between the base and this head. This PR adds only its six isolated package files.

I independently verified the same sole test failure, exact missing-command assertion, 760-test total and 24 skips in all five comparison logs: run 42905094314, run 42903313790, #7247 run 43023514635, #7249 run 43023693429 and #7250 run 43023829856. These comparisons concern the shared CI failure only, not those contributions' mathematical results.

Local normal/optimized checks, registered replay, committed-byte/source-history checks and read-only Git-free exported replays passed as recorded above. Hosted package certification remains unavailable pending the repository-level repair. No fix, retrigger commit, push or workflow rerun was made.

Merge Danger

Door: two-way. Blast radius: package. This adds one isolated proof and opt-in verifier, with no shared-code or selected-result change.