[benji-potamus]

Resolves BINIUS-702 (sub-issue 6 of BINIUS-696; groundwork for Phase A / A′).

A sumcheck over {0, ∞}^n: the ordinary sumcheck with round check R(0) + R(∞) = s. For monomial-coefficient tables u, w, Σ_x u^∞(x) w^∞(x) = <u, w>, and the sumcheck reduces to u^∞(r), w^∞(r) at the challenge point r.

Commits

1. Add sumcheck verifier over the infinity hypercube (binius_ip::sumcheck::infinity)
   • verify and batch_verify, mirroring sumcheck::{verify, batch_verify}.
   • RoundProof::recover_infinity: a_d = s - a_0. The wire format is unchanged (the top coefficient is still the one dropped).
2. Add product prover for the infinity-hypercube sumcheck (binius_ip_prover::sumcheck::infinity::batch_prove)
   • Takes pairs [u, w] of mixed variable counts and binds the top variable each round. It computes R(∞) = <u_1, w_1> and the middle coefficient from the cross terms, and takes R(0) from the claim. The fold is u' = u_0 + r u_1.
   • A claim that is smaller than the current round count contributes its constant claim as R, so there is no padding factor. It closes at the low k coordinates of the point.

Tests

• Round trip over five pairs with mixed sizes (5, 0, 3, 5, 1 vars). Each eval equals the monomial-basis evaluation <u, eq_ind_partial_eval::<InfCube>(r)>, and the batched eval matches.
• A tampered round-0 coefficient is rejected (the final eval mismatches).
• recover_infinity round-trips.

The prover is a standalone function and does not implement SumcheckProver. That way it can't be paired with the boolean-cube batch_verify by mistake.

🤖 Generated with Claude Code

[linear-code]

BINIUS-702