[pinterbanget]

Summary

This adds research/pr4416-moment-check/, an independent certified check of the strictification inequalities in #4416 (1337isnot1337). #4416 claims a conditional κ = 0.00268647, the highest open claim at the time. Its TENSOR.md §3–4 needs, at saving C = 0.00268648:

alpha = P_tau/T_tau < 1 - 1/10^8,   gamma = P_1/(10^C T_tau) < 4991/5000,   delta = 7688*11/(10^C T_tau) < 2/3,
alpha^6240 + 2 gamma^6240 + 2 delta^6240 < 1.

Claim type: an independent certified finite check. No new κ. #4416's native audit, outer ledger and inherited premises are not checked.

Result

Link	Value	Required
alpha	0.999999984403908… (1 − alpha = 1.56·10⁻⁸)	< 1 − 10⁻⁸
gamma	0.998191712473…	< 4991/5000
delta	0.643824709152…	< 2/3
Normalized bound	exactly 10240625039/10240638976	< 1
Gap	exactly 13937/10240638976 = 1.36095…·10⁻⁶	as stated

• Every link holds. The enclosures are narrower than 10⁻³⁰ and overlap Conditional κ = 0.00268647 via paid target padding after native composition #4416's recorded intervals. The integer comparisons (4991/5000)^390 < 1/2 and (2/3)² < 1/2 and the Bernoulli step are exact.
• The invoice is bound. RESULT.json's strictification invoice equals its native inventory and TENSOR.md: 7,688 targets and 55,998 paid calls, with rank-one moments 131,723 and 131,115 (surplus 608).

Headroom. As the tensor power grows, the construction works for every C with alpha < 1. The limit saving of this invoice is C* = 0.002686489049, just 9.05·10⁻⁹ above the tested saving. #4416's κ therefore uses essentially all of this p7 invoice's headroom: a higher κ on this route needs a different invoice, not a larger power.

Controls. Completing every target to the full ambient (b = 121) gives delta > 1, and the undamped rank-one tail exceeds 1. Both damping choices are needed.

Evidence

• Standard library only; runs in about 90 seconds.
• Exact rationals: ln by atanh after reducing by powers of two, exp by Taylor with a remainder.
• Imports nothing from Conditional κ = 0.00268647 via paid target padding after native composition #4416. Its TENSOR.md and RESULT.json are vendored byte-identically, SHA256-checked, with vendor/NOTICE.md.

python3.12 -B scripts/research_package.py --package research/pr4416-moment-check --check-only
  PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
python3.12 -B scripts/research_package.py --package research/pr4416-moment-check
  PASS_REGISTERED_RESEARCH_PACKAGE research/pr4416-moment-check outcome=completed

Head: 64b4f5457613ada7f0e3f2f4d26f9978d19cdfa7.

Not covered

• The native audit producing T and P. Tightest margin: alpha clears its requirement by only 5.6·10⁻⁹, while one extra rank-one paid call would move it by about 7.6·10⁻⁶. Rebuilding the invoice from the word is the open next question.
• The tensor, completion and splitting proofs, the compiler ranks, the outer rows and every inherited premise.

Task and overlap

• Card: independent-moment, refined, priority revision 7.
• Base: canonical main 12fa8c5c844be21bfefb38a19503fbd867a11244.
• Claimed in Handoffs and requests for independent checks #1166 before starting (comment 19103501). The Conditional κ = 0.0023 (+5.690e-4, +32.87% over #4181): productive helper jobs compiled into a uniform supplier #4359/Refine PR4359: conditional κ = 0.00230509 and smaller explicit template ranks #4405 checks (Independently check #4359's explicit-rank chain: H0/H1, superpotentials, the exit recurrence and 19,895 exact lockstep flows hold (no new κ) #4404, Independently check #4405's refined #4359 ranks and endpoint family: all reproduce; z0 and the exit constant are not levers, the uniform saving is (no new κ) #4410, Rebalance #4359's own ledger at unchanged kappa 0.0023: explicit ranks shrink to 2^268300549 (complex) and 2^14664 (BIT) with all 54 predicates strict (no new κ) #4413) cover the predecessors, not this p7 successor.

Credit

The construction and invoices are #4416's (1337isnot1337, with OpenAI Codex assistance), building on zxv's #4359, #4405, #4412 and #4414, and Jacob Alan Sussman's Theorem 1G. Prepared by Ryanson Jonathan with substantial Claude Code (Claude Opus 5.5) assistance. Apache-2.0.

🤖 Generated with Claude Code