A Certified Base-89 Lower Bound for the Gyarmati-Hennecart-Ruzsa Sum-Difference Constant from AI Autoresearch

Numaro AI Autoresearch

Plain language

What this result means

Adding two integer sets can collapse many pairs onto the same sums, while subtraction can keep more outcomes distinct. The GHR constant measures how large that separation can become when the sumset is still only a constant factor larger than the original set. A higher certified exponent means a stronger construction. Here one sparse digit pattern can be repeated to arbitrary depth, so a finite alphabet produces an asymptotic lower bound.

The live Optimization Problems constant page lists the merged Mosaic Intelligence result 1.1835129324 from pull request #95. The new certificate clears its full published value, not only the rounded table entry.
The construction uses 29 digits from 0 through 44 in base 89, with cap ratio rho approximately 10.7041407995. Base 89 is exactly the carry-free boundary 2*44+1.
The rigorous interval lower endpoint is 1.187326127925948057..., giving a certified margin of approximately 0.0038131955 over the merged record.
The candidate has not yet been submitted for maintainer review. The proven upper bound remains 4/3, and this digit lemma cannot exceed 5/4.

The mathematical object

How one digit alphabet creates asymptotically more differences than sums

For finite integer sets $X$ and $Y$ , addition and subtraction produce the two derived sets

\begin{aligned} X+Y&=\{x+y:x\in X,\ y\in Y\},\\ X-Y&=\{x-y:x\in X,\ y\in Y\}. \end{aligned}

The constant $C_{3a}$ is the largest exponent that can keep the sumset small while forcing the difference set to grow as a power of it. Here $K$ is any fixed number greater than one, and $c(K)>0$ may depend on $K$ :

\begin{aligned} C_{3a}:=\sup\Big\{\theta:\ &\text{for every }K>1,\\ &\text{at arbitrarily large scales,}\\ &|X+Y|\le K|X|,\\ &|X-Y|\ge c(K)\,|X+Y|^\theta\Big\}. \end{aligned}

01Turn any admissible finite digit set into a lower bound

\begin{aligned} S(U)&:=|U+U|,\\ D(U)&:=|U-U|,\\ Q(U)&:=2\max(U)+1,\\ 0\in U\subset\mathbb Z_{\ge0},\quad&D(U)<Q(U),\\ C_{3a}&\ge\Theta(U),\\ \Theta(U)&:=1+\frac{\log D(U)-\log S(U)}{\log Q(U)}. \end{aligned}

02Build a carry-free family from one sparse alphabet

\begin{aligned} \mathcal D=\{&0,8,9,12,16,17,18,20,21,\\ &24,25,26,27,28,29,30,32,\ldots,44\},\\ b&=2\max(\mathcal D)+1=89,\\ \rho&\approx10.7041407995,\\ U_n&=\left\{\sum_{i=0}^{n-1}a_i b^i: \substack{a_i\in\mathcal D\\ \sum_{i=0}^{n-1}a_i\le\lfloor\rho n\rfloor}\right\}. \end{aligned}

In the first step, $U$ is a finite set of nonnegative integers: $S(U)$ counts its sums, $D(U)$ counts its differences, and $Q(U)$ sets the projection scale. In the second, $\mathcal D$ is the 29-digit alphabet, $b$ is the base, $n$ is the depth, and $\rho$ caps the average digit weight. Because $b=2\max(\mathcal D)+1$ , sums and differences have unique carry-free digit encodings.

The limit is certified with two exponential tilts. For each difference digit $\delta\in\mathcal D-\mathcal D$ , let $(p_\delta,q_\delta)$ be the least-weight digit pair satisfying $p_\delta-q_\delta=\delta$ . The probability law $\pi$ supplies a feasible entropy lower bound for differences, while $Z_+(\sigma)$ supplies a dual upper bound for sums:

\begin{aligned} Z_-&=\sum_{\delta\in\mathcal D-\mathcal D} e^{-\alpha(p_\delta+q_\delta)},\\ \pi_\delta&=\frac{e^{-\alpha(p_\delta+q_\delta)}}{Z_-},\\ H(\pi)&=-\sum_\delta\pi_\delta\log\pi_\delta,\\ Z_+(\sigma)&=\sum_{y\in\mathcal D+\mathcal D}e^{-\sigma y}. \end{aligned}

The tilt is symmetric, so $\rho=\mathbb E_\pi[p_\delta]= \mathbb E_\pi[q_\delta]$ . Combining the entropy lower bound and the sum-side upper bound gives a rigorous limit inequality rather than an extrapolated numerical curve:

\begin{aligned} I_-^{\rm lo}&:=H(\pi),\\ I_+^{\rm up}&:=2\rho\sigma+\log Z_+(\sigma),\\ C_{3a}&\ge\liminf_{n\to\infty}\Theta(U_n) \ge1+\frac{I_-^{\rm lo}-I_+^{\rm up}}{\log89}. \end{aligned}

\begin{aligned} \alpha&=0.06818532818532819,\\ \sigma&=0.06859160390989587,\\ L_{\rm cert}&>1.187326127925948,\\ 1.187326127925948&>1.1835129324,\\ L_{\rm cert}&<\frac54<\frac43. \end{aligned}

Interval arithmetic places the entire certified lower-bound interval above the previous record. The result remains below both the $5/4$ ceiling for this digit lemma and the proven $4/3$ upper bound for the constant.

29sparse digits selected from 0 through 44

89carry-free base, exactly twice 44 plus one

+0.0038132certified margin over the merged record

Visual notes

How to read the result

Historical step chart of lower bounds for the GHR sum-difference constant with a magnified comparison of the merged record and the Numaro candidate. — **The lower-bound frontier**The historical ladder gives scale; the magnified panel compares the merged 1.1835129324 record with the certified 1.1873261279 candidate. The 4/3 upper bound does not move.

Schematic of the 29-letter base-89 sparse digit alphabet, a capped digit word, and the difference-versus-sum large-deviation accounting. — **The capped sparse-digit mechanism**Twenty-nine selected digits are repeated in base 89 under a digit-sum budget. An entropy law lower-bounds difference growth while an exponential dual upper-bounds sum growth.

Result table

A capped sparse-digit construction raises the certified lower bound by 0.0038131955.

Cell	Baseline	Numaro	Delta	Note
C3a lower bound	1.183512932421861510...	1.187326127925948	+0.0038131955	strict interval separation
Construction	base 33, depth 420	base 89, arbitrary depth	limit family	capped carry-free digits
Digit alphabet	14 digits, max 16	29 digits, max 44	+15 digits	dense upper tail, sparse lower tail
Digit-lemma ceiling	5/4	5/4	unchanged	construction stays safely below
Proven upper bound	4/3	4/3	unchanged	the global gap remains open

Method

How it was found

The campaign searched sparse integer alphabets together with cap ratios and dual tilts, then isolated a difference-rich motif: a sparse lower region and a dense upper tail through 44. Repeating that alphabet in base 89 prevents carries. A Gibbs law on minimal digit pairs supplies a feasible entropy lower bound for difference words, while a second exponential tilt bounds the number of sum words. Directed interval arithmetic turns the two rates into a rigorous lower endpoint.

Reproduced the current capped-digit construction family and its merged base-33 record.
Searched sparse alphabets, cap ratios, and difference/sum tilt parameters under the no-carry constraint.
Selected the 29-letter alphabet with maximum digit 44 and set the minimal carry-free base to 89.
Certified the limiting growth inequality, then checked finite-depth counters against exact brute force on a tiny instance.

Verification

How it was checked

The permanent Zenodo archive contains the certificate data, construction notes, run log, and supplied standalone checker. That checker recomputes the tilted law, cap feasibility, the difference-growth admissibility bound, and the interval lower endpoint from gsd_3a_cert.json. It independently validates the finite-depth dynamic programs against brute force at depth 2 and prints genuine lower bounds at depths 4, 8, 12, and 16. A separately written local verifier reconstructs all minimal difference pairs and sum digits without importing the bundled code, obtains 89 difference digits and 73 sum digits, recomputes the full interval, and performs its own exact depth-2 set test. Both local runs end in PASS.

Scope

What is not being claimed

This is a certified candidate lower-bound update, not yet an accepted repository record. It does not determine C_3a, improve the proven 4/3 upper bound, or prove that the base-89 alphabet is globally optimal. The 5/4 ceiling applies to the GHR digit-lemma route; observations about a tighter approximately 1.1875 wall are specific to the searched capped-product family rather than a new general theorem.

References

Baseline sources

Citation

How to cite

Numaro AI Autoresearch. "A Certified Base-89 Lower Bound for the Gyarmati-Hennecart-Ruzsa Sum-Difference Constant from AI Autoresearch." Numaro Research Report NUMARO-2026-017, 2026.

Certificate archive: Zenodo, DOI 10.5281/zenodo.21504058.

@techreport{numaro2026SumDifferenceConstant,
  title = {A Certified Base-89 Lower Bound for the Gyarmati-Hennecart-Ruzsa Sum-Difference Constant from AI Autoresearch},
  author = {Numaro AI Autoresearch},
  institution = {Numaro},
  number = {NUMARO-2026-017},
  year = {2026},
  url = {https://numaro.tech/research/sum-difference-constant-2026/},
  note = {Certificate archive: Zenodo, DOI 10.5281/zenodo.21504058}
}