Certified lower bound C_3a ≥ 1.187326 for the Gyarmati–Hennecart–Ruzsa sum–difference constant

This archive is a self-contained certificate that the Gyarmati–Hennecart–Ruzsa sum–difference constant satisfies C_3a ≥ 1.187326127925948… (rigorous interval lower endpoint), improving the previous record 1.1835129324 by +0.0038131955 and staying strictly below the proven upper bound 4/3 and the digit-lemma ceiling 5/4. The lower bound is the **large-deviation limit** of a capped sparse-digit construction: `U_d = { Σ a_i·89^i : a_i ∈ A, Σ a_i ≤ ⌊ρd⌋ }` with a 29-element alphabet A ⊂ {0,…,44} and cap ratio ρ ≈ 10.7041. Each depth d yields an exact GHR digit-lemma lower bound θ(U_d); θ(U_d) increases to the limit L(A) = 1 + (I_D − I_S)/log(2·max A+1), so C_3a ≥ L(A) — the same limit-as-lower-bound principle already accepted in the record ladder (Zheng 2025). L(A) is bounded below rigorously by explicit Gibbs tilts and interval arithmetic. ## Contents - `gsd_3a_cert.json` — certificate data: alphabet, base, cap ratio, Gibbs tilts (α, s), L_lower. - `verify.py` — standalone verifier (Python 3.9+, mpmath only), imports nothing from producer code. - `README.md` — full mathematical statement and the r ≈ 5/4 bottleneck analysis. - `table_row_3a.md` — the record-ladder row. ## Verification pip install mpmath python3 verify.py # ends: RESULT: ALL PASS — certified C_3a >= L_lower > previous record The verifier recomputes L_lower in interval arithmetic, checks the tilted law's feasibility and the lemma's d<q hypothesis, validates the finite-d dynamic-programming counters against brute force, exhibits θ(U_d) climbing, and confirms L_lower > previous record while L_lower < 4/3 and < 5/4. ## Method note The construction and its evaluation were derived and certified with the **Numaro** (numaro.tech) research tooling. All quantities were independently re-computed before release; the engine reproduces the previous record-holder's published d=120 value to 40 decimal digits as a calibration anchor.