Paper 2026/858

FRI Soundness Above the Johnson Bound via Threshold Halving

Xinxin Fan, IoTeX Network

Abstract

We prove the first unconditional soundness theorem above the Johnson bound for FRI, STIR, and WHIR — the proximity-testing protocols underlying every deployed STARK, zkVM, and FRI-based system on Ethereum's roadmap. For $\mathrm{RS}[F, L, k]$ with $k = 2^m$ and $L$ admitting a fixed-point-free involution (standard for deployed FRI, in either characteristic), for every $\delta \in (\delta_J,\, 1-\rho)$: $$\varepsilon_{\mathrm{FRI}} \;\leq\; \frac{nR}{|F|} \;+\; \left(1 - \frac{\delta}{2}\right)^{\!q}.$$ Three results. (A) The bound above, via threshold halving from Rothblum–Vadhan–Wigderson (STOC 2013); the protocol, prover messages, and verifier checks are unchanged — only the query parameter is recalibrated. The argument enters the unique-decoding regime after one round, where BCIKS locks the distance, making it immune to any open-zone counterexample. (B) The ${\sim}2{\times}$ query overhead is optimal within the correlated-agreement (CA) framework: $\varepsilon_{\mathrm{ca}}(C, \delta, \delta) = \binom{n}{w}/|F|$, tight for large fields, exponential in the code length, and vacuous at FRI scale. (C) First $p$-dependent list-size bounds: $\mathbb{E}[M] = \binom{n}{w}/p^c$ at all codimension excesses $c \geq 1$; a sub-Poisson Bernstein tail at $c = 2$; and a phase-diagram conjecture at $c \geq 3$ (the deployment regime) predicting a linear $M_{\mathrm{true}} \leq \lfloor (2D-1)/c \rfloor$. Impact for Ethereum. Every deployed and proposed FRI-based system — SP1, RISC Zero, Plonky3, the ${\sim}30$ zkVMs on EthProofs, Stwo (Mersenne31 / circle FRI), the planned post-quantum signature aggregation layer — now has a positive proven soundness floor in the open zone above Johnson, replacing a conjecture whose up-to-capacity form was disproved in late 2025 by Crites–Stewart and independently by Kambiré. The cost is a factor ${\sim}2$ in queries; the bound holds in characteristic 2 via additive folding, on the unit circle via the Stwo coupling, and compiles to non-interactive knowledge soundness via Fiat–Shamir and DEEP. The half-threshold CA bound is formally verified in Lean 4 with Mathlib.