perf: parallelize DEEP quotient assembly

[gbotrel]

Summary

Step 3 of the prover parallelisation series flagged in the bench report. After step 1 (#7) and step 2 (#8), deep-quotient+fri-commit was the largest remaining serial chunk on the tall shape (~47% of prove wall, ~960 ms of 2.05 s). This PR replaces the per-shift serial column accumulation with a single row-chunked parallel sweep that fuses C_s materialisation, denominator inversion, and the DEEP quotient add:

deepQuotient[x] += (v_s - Σ_k scales_k · cols_k[x]) / (z_s - ω^x)

Three transformations make the fan-out clean:

1. The serial alphaAcc *= pr.alpha chain across columns and shifts is unrolled up front into a per-column scales slice on each shift bundle, so the per-row loop has no cross-column data dependency.
2. Constant (len-1) columns are folded into a single constContrib term per bundle so the per-row loop only iterates real-width columns.
3. Denominators (z_s - ω^x) for the chunk are inverted via ext.BatchInvertE4 — one inversion + 3·chunkLen ext muls per chunk, instead of chunkLen ext inversions.

Drive-by: ω^shift now uses Element.Exp (binary exponentiation) instead of the O(shift) manual mul loop.

Numbers

Median of 3 synth runs (32 cores, KoalaBear + Poseidon2 + blowup 4 + 4 queries), measured on top of step-1 + step-2:

shape	step 1+2	+ this PR	speedup
tall (2²⁰ × 24)	2.05 s	1.36 s	1.51×
wide (2¹⁴ × 1536)	0.288 s	0.285 s	~1×

DEEP phase share on tall drops from 47% → 20%. Wide was never DEEP-bound (8% of step-1+2 wall) — the win shows where DEEP assembly actually dominates.

vs Plonky3

Same shapes / same primitives:

shape	Plonky3	loom (step 1+2+3)	gap
tall	1.56 s	1.36 s	loom 1.15× faster
wide	0.40 s	0.285 s	loom 1.40× faster

loom now beats Plonky3 on both apples-to-apples shapes. (Baseline on main was P3 9.6× ahead on tall, 3.8× on wide.)

Test plan

• go test ./... — all green
• go test -race ./prover/ — no races
• go vet ./... — only pre-existing copylocks warnings
• Verifier-side Verify(...) is exercised end-to-end by every existing prover test, so any algebraic mismatch in the rewrite would surface there
• prover/BenchmarkProveTall and BenchmarkProveWide added so the improvement remains visible in go test -bench (no dependency on the bench/synth command from perf: parallelize LDE / coset-FFT / Merkle pipeline #7)

🤖 Generated with Claude Code

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Note

Medium Risk
Rewrites core DEEP quotient computation in the prove path; invalid algebra would break proofs, though the change is intended to be equivalent to the prior DeepQuotientExt flow.

Overview
Parallelizes DEEP quotient assembly in ComputeDeepQuotient by replacing per-shift construction of full C_s columns plus poly.DeepQuotientExt with shift bundles (deepQuotientBundle) and a single accumulateDeepQuotient pass that adds ((v_s - \sum_k \text{scale}_k \cdot \text{col}_k[x]) / (z_s - \omega^x)) row-wise.

The serial alpha folding chain is precomputed into per-column scales; length-1 columns fold into constContrib so row work skips constants. Row ranges run under parallel.Execute, with BatchInvertE4 on chunk denominators and ω^shift via Element.Exp instead of repeated multiplies.

Adds BenchmarkProveWide / BenchmarkProveTall (synthetic AIR via buildSynthProveInputs) to track wide vs tall prove shapes in go test -bench.

^{Reviewed by Cursor Bugbot for commit 7bbb828. Bugbot is set up for automated code reviews on this repo. Configure here.}

[Copilot]

Pull request overview

This PR parallelizes DEEP quotient assembly in the prover by replacing serial per-shift accumulation with bundled, row-chunked processing and batched denominator inversion. It fits into the ongoing prover parallelization work by targeting the remaining DEEP quotient hot path.

Changes:

• Replaces serial DEEP quotient column accumulation with deepQuotientBundle preparation plus parallel accumulateDeepQuotient.
• Uses BatchInvertE4 per row chunk to reduce extension-field inversion cost.
• Adds synthetic tall/wide prover benchmarks for tracking performance.

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated no comments.

File	Description
prover/prover.go	Reworks DEEP quotient assembly into precomputed bundles and parallel row accumulation.
prover/bench_deep_quotient_test.go	Adds synthetic prover benchmarks for tall and wide trace shapes.

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