PCS: evaluate two Ligerito sumcheck rounds per fold pass

[wu-s-john]

Port lookahead folding into the Ligerito dependency used by BitZ: alternate coefficient-only rounds with a pass that folds two challenges directly from n to n/4 entries. Preserve the existing messages and transcript exactly.

Sources

pinned Flock single-round fused fold and BitZ Ligerito adapter is normative for the target contract. PoC lookahead dispatch is implementation reference only; preserve target field, layout, transcript and security requirements unless this issue explicitly introduces a versioned protocol.

Current and proposed behavior

Pinned Flock already fuses one fold with the next message. The reference additionally stores small lookahead coefficients so the intervening challenge needs no large-buffer scan, then combines the deferred and new folds. Its default applies only at N≥2^14 with enough remaining entries; these thresholds are reference evidence, not mandatory target policy.

Required scope

• Add explicit pending-fold state; flush it before introducing/gluing bases, committing folded values, running OOD evaluations or producing terminal claims. Cover odd-length schedules and minimum tails.
• Maintain the existing polynomial identities, low-bit ordering, GF(2^128) reductions, message encoding, challenge/grinding sequence and complete final PCS acceptance. No reduction in round count, security target or OOD checks.
• Keep a one-round reference path and a diagnostic switch. Integrate through an audited dependency update or a small maintained patch rather than silently changing unrelated Flock algorithms.
• Coordinate OOD state changes with PR feat: add initial OOD sampling to BitZ commitments #76. PCS: compute the first sumcheck message while building the basis #123 may supply round-zero coefficients, but this issue does not require or claim precomputed first-entry lookahead; the PoC root currently supplies no such lookahead.

The reference estimates about 25% less fold-chain traffic by skipping intermediate arrays. This is not a 25% end-to-end latency promise; extra coefficient arithmetic, allocation policy and smaller tails matter.

Done when

• Each message, challenge, final folded polynomial and complete proof matches the ordinary fused path byte-for-byte on even/odd schedules and zero/one/random challenges.
• Introduce/glue/OOD transitions cannot observe unmaterialized pending state.
• Existing corrupted/truncated proof tests continue to reject; golden transcripts cover both dispatch arms.
• Matched timing, allocation and peak-memory measurements on supported architectures justify the engagement policy.

Dependencies and coordination

No prerequisite optimization port. Coordinate with #76.