Improve C_3a lower bound to 1.19102809 via masked semigroup digits

[kleinwaks]

Summary

This PR improves the lower bound for the Gyarmati-Hennecart-Ruzsa sum-difference constant to

$$ \boxed{C_{3a}>1.19102809}. $$

This supersedes the earlier value (1.19023813) in the same pull request.

Construction

This construction generalizes Zheng's bounded-digit limit construction by replacing bounded digits with digits restricted to a finite mask. Zheng encodes bounded integer vectors as base-$(2B+1)$ integers, permits every digit in $\{0,\ldots,B\}$, and imposes a linear cap on the total digit sum. Here the complete digit interval is replaced by

$$ M=\langle1518,1524,1587,2024,2032,2116\rangle\cap[0,17032], $$

and the carry-free base is $34065=2\cdot17032+1$.

As described in the proof paper, $M$ belongs to a structured sequence of numerical-semigroup constructions rather than being an unstructured search output. Further exploration of the structure that might improve the bound has not been completed.

For a finite mask $M$ with maximum digit $B$, define

$$ P_+(x)=\sum_{s\in M+M}x^s,\qquad P_-(x)=\sum_{d\in M-M}x^{\kappa(d)}, $$

where $\kappa(d)=\min\{a+b:a,b\in M,\ a-b=d\}$ is the least total digit cost of realizing the difference digit $d$. The accompanying proof establishes

$$ C_{3a}\ge 1+\frac{\log P_-(x)-\log P_+(x)}{\log(2B+1)} \qquad(0\lt x \lt 1). $$

Note that Zheng's bound of $1.173077\ldots$ can be recovered from this by maximizing over $x$ with $P_+(x)=1+x+\cdots+x^{2B}$, $P_-(x)=1+2x+2x^2+\cdots+2x^B$, and $B=5$.

For our mask $M$,

$$ \lvert M \rvert=3121,\qquad \lvert M+M \rvert=18730,\qquad \lvert M-M \rvert=32369. $$

A 384-bit MPFR verifier with directed rounding proves the displayed strict bound. Independent Python and PARI/GP programs reconstruct the discrete data and cross-check the numerical value.

Proof and verification package: https://github.com/kleinwaks/masked-digit-sum-difference-bound

Structural motivation

The record mask emerged from a sequence of increasingly structured numerical-semigroup masks, as discussed in the proof paper. Its six generators form the product grid

$$ \{3,4\}\times\{506,508,529\}. $$

The column semigroup is the simple gluing

$$ \langle506,508,529\rangle =23\langle22,23\rangle+508\mathbb N_0, \qquad 508=21\cdot22+2\cdot23. $$

Its two relation degrees, $11638$ and $11684$, are nearly equal. The $2$-by-$3$ grid supplies early exact relations that create sum collisions and cheap near relations with residuals $2,21,23$ that create low-cost differences. The cutoff $B=17032$ lies after the useful relation layers but before the semigroup conductor $28922$.

Changes

• Adds the new lower-bound row and refernce to constants/3a.md.
• Adds comment about masked-digit generalization of Zheng's bounded-digit construction to constants/3a.md.
• Adds comment about product-grid/simple-gluing structure to constants/3a.md.
• Fixes typo Gyamarti -> Gyarmati in constants/3a.md.

AI-use disclosure

During this work, the author used several large language models. OpenAI's GPT-5.6 Sol model was used extensively in software development, analysis of search results, drafting and revising the proof, and preparation of the verification package. Google's Gemini 3.1 Pro Preview was used in software development and analysis of search results; Google's Gemini 3.5 Flash was used to review the proof; and Anthropic's Sonnet 5 was used in analysis of search results. The author set the research goals, directed the computational work, selected which suggestions to pursue, ran the search and verification software, reviewed the mathematical arguments and outputs, and takes responsibility for all claims and references.

[kleinwaks]

Have updated original pull request with improved bound (1.19023813 -> 1.19102809) based on further exploration of the digit mask structure.

[kleinwaks]

Commenting so you are aware of upcoming improvement and do not have to review multiple similar PRs.

It looks like the bound can be improved further by rigorously controlling carrying instead of using a carry-free base, even with this same record digit mask. However, the optimal mask with controlled carrying might not be the same as the optimal mask with a carry-free base, so a new search will be needed. I am working on that and will update here shortly if the PR is still open, as well as update the proof paper to include the controlled carrying construction.

To summarize: generalizing Zheng's construction to finite digit masks instead of allowing all digits below some bound yielded a large improvement, and now generalizing further to allow carrying instead of requiring a carry-free base yields additional improvement (but how large remains to be seen).