PR kind

• fix
• feat

DRAFT -- GPU scorecard pending. Replaces this PR's third attempt (block-krylov). No GPU available right now for this content; opening as a draft to avoid the missing-scorecard auto-close.

Why the previous content is gone

block-krylov was repeat-tested (5 fresh-seed GPU runs, pinned decaying-spectrum regime --data-rank 256 --rank-m 256) and turned out to be a regression versus the retired nystrom-iter attempt on both axes at once:

Splitting a fixed per-part width budget across Krylov depth traded away direct landmark breadth for GEMM-refined columns and lost more than it gained -- the second consecutive confirmation that spending the fixed M budget on extra GEMMs against the full n x n operand burns latency faster than it buys accuracy at this scale, no matter how the GEMMs are arranged.

Checked the premise numerically before writing a third implementation

Before touching GPU time again, two cheap NumPy checks (no GPU needed):

1. Leverage-score-weighted landmark sampling -- ruled out. Computed true leverage scores of this project's own decaying-spectrum fill directly: they come out essentially uniform (std/mean ~0.08, matching a random subspace). The fill's k^-alpha decay is applied to an abstract rank-direction axis that a dense Gaussian mix then spreads evenly across every physical column -- there is no per-column importance signal to weight by on this synthetic data. Any cheap leverage-score proxy (e.g. raw column norm) would just be chasing sampling noise, not real structure.
2. Redundancy-avoiding selection -- confirmed real. Uniform-random landmark draws are wasteful in a different way: by chance they can include near-parallel (redundant) columns. Oversampling a small candidate pool (oversample * w landmarks, still just a memory gather) and greedily picking the least-redundant subset via column-pivoted Gram-Schmidt (RRQR-style) beat plain uniform sampling's column-space recovery in 8/8 trials (mean rel-err 0.120 vs 0.132, tight std bands, no overlap).

Summary (new content)

New transform rrqr-nystrom:

• same 3-way space split {col(A), row(A), row(B)} as every prior attempt on this PR (#156's redundancy proof still holds)
• oversamples oversample * w (default oversample=4) uniform landmark columns/rows per part -- still a pure memory gather, same mechanism as nystrom
• greedily selects the least-redundant w of them via column-pivoted Gram-Schmidt: repeatedly pick the remaining candidate with the largest residual norm after projecting out everything already chosen

Crucially, the pivoted selection runs on the small oversampled pool, not a GEMM against the full n x n operand -- cost is O(n w²), not O(n² w), roughly two orders of magnitude cheaper at n=8192 than nystrom-iter/block-krylov's GEMM-refinement approach. This directly targets the failure mode that sank both previous attempts on this PR, rather than trying a third variant of the same GEMM-refinement idea.

Verified so far (CPU, numpy only -- not a claim of GPU dominance)

test_rrqr_nystrom_beats_plain_nystrom_on_decaying_spectrum reproduces the pre-implementation numeric check as an automated test: pivoted selection from an oversampled pool spans col(A) measurably better than an equal-size uniform draw on decaying-spectrum data. 9 tests total: registration, input validation (including graceful handling of oversample=1), orthonormality, seed determinism, exact low-rank recovery, and honest (non-zero) basis_flops.

CPU-safe validation

$ uv run python -m strategy.smoke        # 3/3 transforms pass
$ uv run --extra test python -m pytest tests/ strategy/tests/ eval/tests/ -q
321 passed, 24 skipped

GPU Result (required for feat PRs only)

Pending -- will attach before marking ready for review. Plan: same pinned decaying-spectrum regime used for both prior honest negative results on this PR (--fill decaying-spectrum --data-rank 256 --rank-m 256), comparing rrqr-nystrom against nystrom (isolates the selection mechanism's contribution) and rsvd, 5+ fresh seeds -- not a single run.

Scope

strategy/transforms.py + strategy/README.md + strategy/tests/test_rrqr_nystrom.py only (Open -- "the main event" zone per CONTRIBUTING.md); new transform, no API change, no protected-path edits.

Checklist

• CPU-safe validation passed
• My commits do not include Co-authored-by footers for coding agents
• GPU scorecard attached (pending -- draft)