# C_3a — Gyarmati–Hennecart–Ruzsa sum–difference constant: certified lower bound 1.187326… **Result.** A capped **sparse-digit** construction, evaluated at its **large-deviation limit**, certifies ``` C_3a >= 1.187326127925948… (rigorous interval lower endpoint) ``` improving the published record **1.1835129324** ([MI2026], PR #95) by **+0.0038131955**, and safely below the proven upper bound `4/3` and the digit-lemma ceiling `5/4`. ## The construction Sparse digit alphabet (a difference-rich motif: dense at the top, gapped at the bottom) ``` A = [0, 8, 9, 12, 16, 17, 18, 20, 21, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] (m = max A = 44) B = 2*max(A)+1 = 89, cap ratio rho = E_pi[p] ≈ 10.7041. ``` For each depth `d`, `U_d = { sum_i a_i B^i : a_i in A, sum_i a_i <= floor(rho*d) }`. The GHR2007 digit lemma gives `C_3a >= theta(U_d) = 1 + log(|U_d-U_d|/|U_d+U_d|)/log(2 max(U_d)+1)` for **every** `d`, and `theta(U_d)` climbs to the limit ``` L(A) = 1 + (I_D - I_S)/log(2 max A + 1), ``` `I_D, I_S` the exact growth rates of `|U_d-U_d|, |U_d+U_d|`. Because each `theta(U_d)` is a valid lower bound and `theta(U_d) -> L(A)`, we have `C_3a >= L(A)` (the same limit-as-lower-bound principle that certifies Zheng's 1.173077 in the ladder). `L(A)` is rigorously lower-bounded by explicit Gibbs tilts — `I_D >= H(pi)` for a feasible law `pi`, `I_S <= 2 rho s + log sum_y exp(-s y)` — with all comparisons done in interval arithmetic (zero floats decide). See `verify.py` for the full self- contained argument and `gsd_3a_cert.json` for the certificate data. ## Why this is the frontier of the method (and where the wall is) Writing `theta - 1 = a(r-1)` with `a = log|U+U|/log(2 max U+1)` and `r = log|U-U|/log|U+U|`, the lemma's own hypothesis `|U-U| < 2 max U+1` forces `a*r < 1`, and the ceiling `5/4` is reached only at `a=3/4, r=4/3`. The capped-digit family lands at **`a ≈ 3/4` (already extremal) but `r ≈ 5/4`** — the difference-set richness `r` is the sole bottleneck. General product measures (arbitrary weight-caps) and block / Markov re-encodings do **not** raise `r` (they re-encode to single-digit integer alphabets already inside this family); `r` tops out near the Hennecart–Robert–Yudin value `1.271` only at uselessly small `a`. So the i.i.d.-cap ceiling is `≈ 1.1875`, and closing the remaining gap to `5/4` would require a genuinely new mechanism (equivalently, disproving Ruzsa's `h=4` inequality). ## Verify ``` pip install mpmath python3 verify.py ``` Ends with `RESULT: ALL PASS — certified C_3a >= L_lower > previous record`. The script recomputes the interval lower bound `L_lower` from `gsd_3a_cert.json`, checks feasibility and the lemma's `d