Improve C_3a lower bound to 1.21418 via two-temperature cyclic compression

[carson-olaf]

Summary

This PR records the lower bound

$$ C_{3a}>1.21418. $$

The construction introduces a two-temperature cyclic-quotient theorem: the difference family uses one Gibbs tilt while the sum-family upper bound is optimized with a separate tilt. For

$$ M=\langle4,5\rangle\cap[0,80],\quad x=\frac{8149}{10000},\quad y=\frac{42571}{50000},\quad R=\frac{249293}{10000}, $$

exact rational reconstruction certifies

$$ 1+\frac{\rho-\sigma}{\log R} =1.214180896097785\ldots>1.21418. $$

The proof and verification package is available here:

Source repository · ZIP release asset

ZIP SHA-256: 8e84a693a41c8fe211b67bcdd3539691b35ba409e6d3a0910875925b4a2d4ffb

Changes

• Adds the 1.21418* row to constants/3a.md, conservatively marked as a limit value because the written theorem remains unverified.
• Preserves the merged [K2026b] row and appends [O2026] after it in chronological order.
• Records the complete mask, tilts, quotient rate, cardinalities, and reconstructed exponent in the table.
• Updates the 3a README cell to the certified/limit pair 1.19102809 (1.21418*) and adds the Recent progress entry.
• Adds the proof and verification package reference.

Verification

I ran the complete ./reproduce.sh workflow on macOS with Homebrew GMP:

• exact standard-library Python reconstruction: PASS;
• exhaustive universal-quotient test over all 162 small models: PASS;
• independent C++20/GMP reconstruction: PASS;
• finite-size correction: PASS;
• five reconstructed certificate files agree byte-for-byte;
• complete reproduction: PASS;
• package SHA256SUMS and ZIP integrity check: PASS.

The arithmetic package checks the finite combinatorial reconstruction, numerical hypotheses, and finite-size correction. It does not formally verify the new two-temperature theorem, the method-of-types estimate, Bertrand's postulate, or the GHR finite-set lemma. The proof is not yet externally refereed or Lean-formalized, so independent review remains invited.

During review, @teorth independently reproduced both displayed numerical values: the fixed-rate exponent 1.2141808960977856 and the gapless specialization 1.213560298642605.

Relationship to earlier work

This follows the earlier 1.2060 candidate discussed in #134 and incorporates the statement and exposition corrections identified by @kleinwaks. The gapless specialization of the new theorem equals 1.213560298642605..., matching the independent 1.21356 construction announced there; the structured alphabet above crosses that baseline.

#146 is now merged and provides an independently replayed Lean formalization of the controlled-carry result. This branch preserves its chronological rows and updates the README's certified/limit pair to 1.19102809 (1.21418*).

AI-use disclosure

I used ChatGPT extensively to explore constructions, formulate the two-temperature argument, draft and revise the proof, generate the verification programs, and audit the resulting package. I selected the research direction, ran and reviewed the verification outputs, and am the human contributor responsible for this submission. References and externally sourced claims were checked against the cited sources.

Agent note: Repository changes and submission packaging were prepared with OpenAI Codex (GPT-5 family) in the Codex desktop harness.

[teorth]

#146 is now merged, so this branch conflicts with main in both README.md and constants/3a.md. Could you rebase along the lines you already proposed in the PR description — keep the [K2026b] row, append your row after it, and set the README cell to 1.19102809 (1.21418*)? Note that the certified half of that pair is now un-asterisked: the Lean formalization in #146 was independently replayed, so 1.19102809 no longer carries the asterisk.

Two other things while you are in there.

The arithmetic I can check does check out: I get 1 + (rho - sigma)/log R = 1.2141808960977856 from your stated rho, sigma and R, and the gapless specialization 2 - log 2 / log(1 + sqrt 2) = 1.213560298642605 matches your figure exactly. What is not checked is the part your own verification section is candid about — the two-temperature theorem, the method-of-types estimate, Bertrand's postulate and the GHR finite-set lemma. The asterisk is doing the right work here, and the conservative "limit value" framing is the right call, so I am happy for the row to go in on that basis once it is rebased.

Also, the description says the PR is deliberately opened as a draft, but it is currently marked ready for review. Worth converting it back if you still want it held.

Recent progress bullet: please use **improved lower bound (limit value):** to match the two entries already there for this constant, rather than "candidate limit value".

[carson-olaf]

Thanks — done. I rebased onto current main, kept [K2026b] and appended [O2026], set the README cell to 1.19102809 (1.21418*), and changed the Recent progress wording to “improved lower bound (limit value).” I also removed the stale draft language and left the PR ready for review. The branch is now conflict-free and mergeable. Appreciate the independent numerical check.