cronokirby

(2026-05) 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

2026-05-02

Abstract

We give the first rigorous O(1)/∣F∣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 (2.0×2.0\times larger) and 281.2 KiB on folded RS (3.5×3.5\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)(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 NormKd/Q(Fd(α))≠0\mathrm{Norm}_{K_d/\mathbb{Q}}(F_d(\alpha)) \neq 0 on a class-field extension of Q[−D]\mathbb{Q}[\sqrt{-D}], D=83,860,066,393,667D = 83{,}860{,}066{,}393{,}667, with Hilbert class field of degree ∼3.4×106\sim 3.4 \times 10^6. The problem is settled rigorously over Q\mathbb{Q} at d∈{4,8}d \in \{4, 8\} (exact rational norm at d=4d=4; multi-threaded msolve at d=8d=8) and connects our work to classical questions in cyclotomic number theory and Hecke L-value non-vanishing.