Paper 2026/861

Action–Orbit FRI Soundness Above the Johnson Radius: A Rigorous $O(1)/|F|$ Bound on Plain Reed–Solomon, with $2\times$ Smaller STARK Proofs at Ethereum Scale

Xinxin Fan

Abstract

We give the first rigorous $O(1)/|F|$ FRI commit-phase soundness bound for plain Reed–Solomon above the Johnson radius — the central open question in the proximity-gap line, made urgent by the late-2025 disproof of the up-to-capacity conjecture (Crites–Stewart; BCHKS; Diamond–Gruen). The bound is proved via a structural mechanism new to the proximity-gap literature: the action–orbit symmetry on the cyclic FRI evaluation domain (five-line proof, no correlated agreement, no character sums, no list-decoding). The construction is unconditional for sparse adversary inputs; for general inputs it reduces to a single sparse-worst-case dominance conjecture (Q2), consistent with every adversarial construction in the proximity-gap literature, including Arnon–Boneh–Fenzi (ABF) Lemma 6.13, Crites–Stewart, and BCHKS. Deployment consequence. Plugged into the ABF §6.3 deployment instance at 128-bit security, the bound halves the rigorous STARK proof size on plain Reed–Solomon: 79.8 KiB against ABF's smallest deployable candidates — 161.4 KiB on interleaved RS ($2.0\times$ larger) and 281.2 KiB on folded RS ($3.5\times$ larger), both requiring strictly stronger code-class assumptions. Production STARK L2s (Starknet's Cairo pipeline, Plonky3-based rollups, the next generation of zkVMs) ship plain RS today without a rigorous above-Johnson bound for it; our work is the first deployment-scale rigorous construction on plain RS at 128-bit security — no protocol modification, no new hardness assumption, no change of code family. Mathematical depth. A separate universal extension to the $(3k/2, 2k)$ family of arising pencils reduces to a single named open problem in number theory (Q1): the non-vanishing of an explicit norm $\mathrm{Norm}_{K_d/\mathbb{Q}}(F_d(\alpha)) \neq 0$ on a class-field extension of $\mathbb{Q}[\sqrt{-D}]$, $D = 83{,}860{,}066{,}393{,}667$, with Hilbert class field of degree $\sim 3.4 \times 10^6$. The problem is settled rigorously over $\mathbb{Q}$ at $d \in \{4, 8\}$ (exact rational norm at $d=4$; multi-threaded msolve at $d=8$) and connects our work to classical questions in cyclotomic number theory and Hecke L-value non-vanishing.