Hyperspherical Proximity Soundness for Zero Knowledge and Interactive Oracle Proof Systems: A Provably Sound Closed-Form Error Bound for Reed-Solomon Proximity Testing via Hamming Association Schemes, Delsarte's Duality, and Krawtchouk Polynomials in Lean 4 and Comparator.

Current Reed-Solomon proximity tests in Zero-Knowledge proofs rely on spatial heuristics and loose union bounds, forcing protocol engineers to over-pad. We eliminate this padding tax by moving strictly to high-dimensional Hamming space and applying Delsarte linear programming duality. By formulating Hamming association schemes and Krawtchouk polynomial constraints, we derive a closed-form, zero-slack exponential decay bound for out-of-bounds error vectors under random folding. ε(n, q, ρ) ≤ q-n · Φ(ρ, δ) Fully formalized in Lean 4, our soundness theorem provides a rigorous, deterministic guarantee for proximity testing up to the Johnson threshold, removing heuristic guesswork from production cryptographic systems.