Function prove_public_eval 

pub fn prove_public_eval<A, P, Channel>(
    alloc: &A,
    public_words: &[Word],
    r_j: &[B128],
    r_y: &[B128],
    channel: &mut Channel,
)
where
    A: Allocator,
    P: PackedField<Scalar = B128>,
    Channel: IPProverChannel<B128>,

Proves the public segment’s evaluation claim.

The shift closes over the public segment as a bit matrix, at r_j over the bit within a word and the low coordinates of r_y over the word index. The verifier holds the segment but not its bits, so the claim is stated here and reduced in two steps:

1. a ring-switch onto the segment’s packed form, leaving the claim sum_x P(x) A(x) against the ring-switching indicator;
2. a sumcheck over the packed segment’s own variables, leaving one evaluation of each factor.

Nothing here is committed, so the verifier finishes on its own: it evaluates the packed segment’s multilinear from the words it holds, and the indicator from its succinct formula.

Arguments

• alloc - the allocator the packed segment and the indicator are drawn from
• public_words - the public segment, unpadded
• r_j - the bit-index challenges
• r_y - the word-index challenges, of which the segment spans the low ones
• channel - the prover channel for sending/sampling

Preconditions

• r_y must have at least as many coordinates as the packed segment spans words