feat(scheme): add HyperPlonk multi-point to single-point opening reduction

[MavenRain]

Toward #904

Phase 1 of #904: standalone sumcheck-based reduction that collapses N = 2^k opening claims f_i(z_i) = y_i (same num_vars) to a single point alpha_2 at which every f_i is opened. Follows HyperPlonk Section 3.8.

Not yet wired into the prover/verifier flow; integration is a follow-up that needs to address heterogeneous num_vars across chips.

[hero78119]

Hi @MavenRain thanks for the PR and interests toward this issue

This PR IOP looks quite neat and fit into original issue.
However per recently roadmap change, we plan a more invasively change which aim to enhance original main sumcheck per-chip to main sumcheck for ALL chips. This design make us stay into 1 main sumcheck IOP instead of 1 main sumcheck per chip cascade one multi-point to single opening opening point IOP. I believe your implementation fit latter while we plan to switch to work on former solution.

The newest issue is #1269, and also attach parent issue for reference #1231

[MavenRain]

Hi @MavenRain thanks for the PR and interests toward this issue

This PR IOP looks quite neat and fit into original issue. However per recently roadmap change, we plan a more invasively change which aim to enhance original main sumcheck per-chip to main sumcheck for ALL chips. This design make us stay into 1 main sumcheck IOP instead of 1 main sumcheck per chip cascade one multi-point to single opening opening point IOP. I believe your implementation fit latter while we plan to switch to work on former solution.

The newest issue is #1269, and also attach parent issue for reference #1231

OK, I'll close this since the plan is to move in another direction.