[zxv]

PR #4359 specifies conditional κ = 0.0023 using uniform ranks 2^276540184361 (complex) and 2^6552928412 (BIT before cover). This follow-up proves a logarithmic leaf-count bound and retunes the complete consumer so sufficient ranks become 2^465000 and 2^7100 at the same κ target.

The rank exponents decrease by approximately 594,710× and 922,948×. The templates remain astronomical and are specified by the predecessor's integer compiler; their instruction lists, productive widths and complete constants have not been materialized. This is a conditional quantitative improvement, with no new κ or selected-frontier change.

The new estimate is

L(n) = P_n/B_n <= n^t * (1 + log2 n)^(1/q).

It replaces the intermediate exponent and large induction constant in the predecessor estimate. The proof retains the exact source/exit recurrence, source complements, integer batch clearing, native adapters, arbitrary-input restoration and separately owned storage. The logarithmic powers are 1/9 for complex and 1/4 for BIT.

Two configurations are certified:

Configuration	Complex rank / uniform saving	BIT rank / uniform saving before cover
Preserve original supplier savings	2^2352000 / 0.002304	2^529000 / 0.002760
Retain κ = 0.0023 with retuned consumer	2^465000 / 0.0023005	2^7100 / 0.002318

For the second configuration, the exact BIT cover gap remains 10^-5, preserving the prime rule and fully charged fallback moment. The atom conversion gives ordinary saving 0.0023011913797. The outer point uses b=0.0023004, q=0.0023002, and effective saving a=0.0023003. All nine cost savings exceed 0.0023; the smallest margin is 0.000000176998. The pinned inherited ledger function reproduces every row and all 63 predicates pass, alongside 22 added interface inequalities. The README includes the full proof and consumer bill.

Validation on Python 3.12.8/macOS arm64:

• Registered package replay and metadata/source-pin validation pass.
• Four rank certificates, 12 rejecting scientific mutations, four failing preceding grid points, and four rejecting outer controls pass.
• An independent file copy with no Git repository passes -I -O -B; changed-source and changed-receipt controls reject.
• Python 3.14.5 generates the same receipt.

python3.12 -B scripts/research_package.py --package research/log-potential-template-ranks --check-only
python3.12 -B scripts/research_package.py --package research/log-potential-template-ranks

This PR adds only research/log-potential-template-ranks/ against canonical main. Ten byte-identical predecessor inputs are pinned and included so replay does not depend on #4359 merging first. No frozen package, shared checker or selected certificate changes. This does not independently admit the native compiler, delayed-cover implementation, precision/support proofs or full multiplication theorem. No Lean, Docker or historical heavy replay was run.

Predecessor credit: @zxv's #4359 at da637c87c89482bc4c86be36bd0071b3d400dbed; Jacob Alan Sussman's Theorem 1G/#4268 and the retained supplier/outer lineage. @pinterbanget's #4404 separately checks the predecessor's original quantitative chain; it does not review this changed bound. The outer ledger retains Vlado Vojdanovski, Chafik Boukhalfa, Jeremy Kothe and the contributors named in its source.

The final overlap refresh also located parallel #4405 (873d1b3c), which sharpens the old power estimate and proposes a higher κ. At unchanged supplier savings its reported rank exponents are 1,546,084,442 and 42,843,607, compared with this PR's 2,352,000 and 529,000. The logarithmic majorant is complementary; its higher-κ parameter point is not instantiated here.

Open next questions: combine with #4405's endpoint, sharpen the weighted source/exit bound, bound productive width and complete instruction counts, and independently review the new majorant and inherited native/outer premises. Prepared with substantial OpenAI Codex assistance; no independent human review of this follow-up is claimed.