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

[carcango]

Summary

This draft PR proposes the candidate 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 attached here:

Source repository · ZIP release asset

ZIP SHA-256: 8e84a693a41c8fe211b67bcdd3539691b35ba409e6d3a0910875925b4a2d4ffb

Changes

• Adds the 1.21418* candidate row to constants/3a.md, conservatively marked as a limit value pending review of the written theorem.
• Records the complete mask, tilts, quotient rate, cardinalities, and reconstructed exponent in the table.
• Updates the 3a README cell and Recent progress section.
• 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 the PR is deliberately opened as a draft and independent review is invited.

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.

This result does not invalidate #146. That PR provides an independently Lean-formalized controlled-carry result and should remain in the chronological history. If #146 merges first, this branch should retain its rows and update 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.

[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".