[pinterbanget]

Summary

This adds research/pr4270-outer-exponent-check/, the first independent recomputation of the conservative exponent chain in #4270 (Ross Richardson), which claims a conditional κ = 1/580, above the selected 0.001672660197076918.

Until now #4270 had one outside check: thu-nmrc's PR comment, which replayed its registered commands under Python 3.11 and matched its source hashes. No independent check of the arithmetic existed.

Claim type: an independent exact check of a written arithmetic chain, including a certified moment enclosure. No new κ. No physical, hosting or analytic premise of #4270 is checked.

Result

Every link of OUTER-JOIN.md §1 and §5 holds, recomputed with exact rationals:

Link	Margin
Pooled moment Σ n_r(r/73)^(1−a)/322560 < 1 at a = C = 0.001730980255989036959962 (certified upper bound 1 − 7.4158·10⁻²⁵). The moment is > 1 at the bracket's upper end C + 10⁻²⁴, so the bracket pins m and W for #4181's profile.	7.4·10⁻²⁵
σ* = 0.99826995 > 1 − C	9.3·10⁻⁷
τ = 332353/332929 < σ*	4.9·10⁻⁸
Leaf exponent σ* + θ(1−σ*) < α_layer = 0.99827; the internal exponent and .998 are lower still	4.8·10⁻⁸
Φ = 1 − 0.997(1−α_layer) = 0.99827519 exactly; β = 41544341/41616125 < Φ	9.8·10⁻⁸
Other cost powers below Φ: .997, .9981, 3/5, .997τ, and .997 for the CRT D·Tp term	≥ 1.75·10⁻⁴
1 − Φ > 1/580	6.7·10⁻⁷
1/580 > selected κ	5.1·10⁻⁵

Control. The old physical Fourier rate .00172380303 falls short of 1/580, as #4270 says.

Tightest margins. These sit on τ and β, at about 5–10·10⁻⁸. The chain passes, but binding those two constants to their sources is the open next question.

Evidence

The verifier uses only the standard library, runs in under a second, and imports nothing from #4270. It reads two files from #4270's head, OUTER-JOIN.md and frozen/pr4181/RESULT.json, vendored byte-identically with SHA256 checks and a vendor/NOTICE.md. They are vendored because that head is not reachable from main. It checks 13 exact phrases, and the selected κ comes from the pinned certificates/selected-result.json.

python3.12 -B scripts/research_package.py --package research/pr4270-outer-exponent-check --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/pr4270-outer-exponent-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4270-outer-exponent-check outcome=completed
python3.12 -B scripts/check_research_commit.py --package research/pr4270-outer-exponent-check --check-source-history
  PASS_RESEARCH_COMMIT_BYTES {... "files": 9, ...}

Not covered

• The pooled native/hosting identities, the stopped-tree proof, the support and Gaussian constructions, and the analytic theorems. Complete pooled complex admission and conditional κ = 1/580 transfer #4270 states all of these as premises.
• How τ, β and Φ's form are derived; only the arithmetic on them is checked.
• Any production run or proof-kernel check.

Task and overlap

• Card: independent-moment, refined to Complete pooled complex admission and conditional κ = 1/580 transfer #4270's chain (recorded in TASK.json), priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244, which is unchanged at 09:59 UTC.
• The picker (seed b6c29fd4…) drew bit-mechanisms. Its cards were taken (Integrate the R437 carrier compiler, numerical invoice and paid local caller #4176 on R437) or already covered (the R439 helper-star family). I picked Complete pooled complex admission and conditional κ = 1/580 transfer #4270 instead because no PR or discussion references it.
• No advance claim, since this was a short session.

Credit

The audited chain is #4270's (Ross Richardson with OpenAI Codex assistance). The pooled fill is Rohan Arun's (#4181). Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code