The measured record.

Two comparisons, kept apart on purpose: what VVE does to the dense default path that real code calls, and what it does to the expert O(N²) routine an expert would call instead. Every ratio names its opponent, its routine, and the thread count on both sides.

Cells measured, cells that came back clean

1,107

cells ok — zero timeouts, zero errors

2

runs end to end on independent hardware

1.5–7.2%

spread on every headline ratio between the two runs

Median of at least three trials per cell. Every ratio is computed within a single machine — absolute seconds move between cloud instances of the same SKU, ratios do not. Full decomposition — eigenvalues and eigenvectors — unless a block says otherwise.

Measured against

Reference netlib LAPACK 3.12.0 and system OpenBLAS

dsyev, full decomposition, via LD_PRELOAD · threads 1 vs 1

Comparison A

Against the call your code already makes.

This is the default path: numpy.linalg.eigh, or a direct dsyev, served by whichever LAPACK stack the machine has. VVE is swapped in with LD_PRELOAD and nothing else changes — same script, same call site, same inputs. On structured matrices it detects the band structure the dense path ignores.

vs single-threaded reference netlib LAPACK 3.12.0 dsyev — the algorithmic baseline

Measured against

single-threaded reference netlib LAPACK

dsyev (the default dense path)full decompositionsingle-threadednetlib reference LAPACK/BLAS 3.12.0

Full decomposition, values and vectors, via LD_PRELOAD with zero code changes. Both sides single-threaded.

Speedup vs single-threaded reference netlib LAPACK dsyev — full decomposition, values and vectors, via LD_PRELOAD with zero code changes. Both sides single-threaded. Clustered spectra: wilkinson (odd N) and schrödinger. Both axes log scale.

Speedup vs single-threaded reference netlib dsyev. Full decomposition. Higher is faster.
Family	N = 2,048	N = 4,096	N = 8,192	N = 16,384
wilkinson (odd N)	9.80×	23.5×	45.7×	89.6×
schrödinger	11.1×	16.3×	27.5×	97.7×

Speedup vs single-threaded reference netlib dsyev. Full decomposition. Higher is faster.
Family	N = 2,048	N = 4,096	N = 8,192	N = 16,384
wilkinson (odd N)	9.8×	23.5×	45.7×	89.6×
schrödinger	11.1×	16.3×	27.5×	97.7×
second_diff	10.9×	16.2×	29.8×	101.7×
laplacian1d	11.3×	15.4×	29.0×	102.6×
anderson	7.5×	12.1×	20.4×	64.1×

Five structured families, and the ordering holds across all of them: the more structure the matrix has, the more work the dense path is doing that does not need doing. anderson is random-disorder — the least structured family in the campaign, and still 64× at N = 16,384.

vs single-threaded fully tuned OpenBLAS dsyev — the delivered stack, one core against one core

Speedup vs single-threaded fully tuned OpenBLAS dsyev. Same call, same sizes, same LD_PRELOAD.
Family	N = 2,048	N = 4,096	N = 8,192	N = 16,384
wilkinson (odd N)	1.2×	2.9×	6.2×	12.2×
schrödinger	1.8×	3.2×	4.8×	11.7×
second_diff	1.7×	3.1×	5.1×	11.9×
laplacian1d	1.7×	2.9×	4.8×	12.0×
anderson	1.3×	2.6×	4.2×	9.0×

Full decomposition, N=16,384

9.0–12.2×

vs fully tuned OpenBLAS, one core against one core dsyev (the default dense path) — N = 16,384 — one core against one core

The reference figures above measure the algorithm. These measure the fight. OpenBLAS brings decades of hand-written SIMD to the same call, and the margin still grows monotonically with N, because SIMD is a constant factor and the exponent is not. VVE has written no hand-tuned SIMD; that is headroom, not a result.

Measured against

single-threaded reference netlib LAPACK

dsyev (the default dense path)full decompositionsingle-threadednetlib reference LAPACK/BLAS 3.12.0

second_diff, N = 16,384 — median wall clock for one full decomposition. All three single-threaded.

Wall clock at N = 16,384
Stack	Seconds	Wall clock
reference netlib LAPACK 3.12.0 dsyev — single-threaded	2,932.9 s	48 min 53 s
fully tuned OpenBLAS dsyev — single-threaded	342.0 s	5 min 42 s
VVE — single-threaded	28.8 s	28.8 s

Forty-nine minutes, or five minutes and forty-two seconds, or twenty-nine seconds. Same matrix, same call, same core.

Measured against

LAPACK dstemr (MRRR), reference netlib LAPACK 3.10.0

dstemr — LAPACK's expert O(N²) tridiagonal eigensolver · threads 1 vs 1

Comparison B

Against the routine an expert would reach for instead.

The strongest objection to Comparison A is that a numerical expert would never let the dense path see a tridiagonal matrix — they would call dstemr and get O(N²) from LAPACK itself. It is a fair objection, so here is that fight, run separately and never blended with the one above. Accuracy is gated on both sides: neither result counts unless its residual and orthogonality both pass.

vs single-threaded LAPACK dstemr (MRRR), full decomposition, accuracy gated on both sides

Speedup vs single-threaded dstemr on clustered spectra. Full decomposition. Higher is faster.
Family	N = 4,096	N = 16,384	N = 32,768	N = 45,056
wilkinson	3.2×	3.8×	4.6×	5.2×
schrödinger	2.6×	3.2×	3.9×	4.9×

Measured against

LAPACK's own expert O(N²) routine, single-threaded

dstemr (MRRR)full decompositionsingle-threadednetlib reference LAPACK 3.10.0, reference BLAS

wilkinson, N = 45,056 — median wall clock for one full decomposition. Both single-threaded.

Wall clock at N = 45,056
Stack	Seconds	Wall clock
LAPACK dstemr (MRRR) — single-threaded	883.3 s	14 min 43 s
VVE — single-threaded	171.1 s	2 min 51 s

The margin vs single-threaded LAPACK dstemr widens monotonically with N — 3.21× at 4,096, 5.16× at 45,056, both sides single-threaded — because VVE holds a flat scaling slope where MRRR's degrades on clusters. The last number in that row is the current floor.

The memory law.

Speed is a comparison. Memory is a law: eigvalsh requires the matrix to exist. The bands-native path does not build one.

Bands-native values-only path, second_diff. Peak RSS against the dense N×N buffer the same problem would require.
N	VVE peak RSS	Dense N×N	Ratio	max \|Δλ\|
8,192	28.7 MB	0.5 GiB	18×	1.78e-15
32,768	31.5 MB	8.0 GiB	260×	1.78e-15
131,072	≈ 53.0 MB	128 GiB	≈ 2,500×	1.78e-15
524,288	≈ 53.0 MB	2,048 GiB	≈ 39,600×	1.78e-15

VVE peak RSS against the dense N×N requirement, log-log. One line is flat; the other is a slope of 2.

Memory figure data
8,192	28.7 MB	0.5 GiB
32,768	31.5 MB	8.0 GiB
131,072	53.0 MB	128 GiB
524,288	53.0 MB	2,048 GiB

Accuracy does not drift across that 64× range. max |Δλ| = 1.78e-15 at every size in the table — machine precision, flat.

Accuracy is the gate, not the trade.

No result in either comparison counts unless it passes an accuracy gate on both sides — residual and orthogonality recorded per cell. Eigenvalue agreement runs at about 1e-15, machine epsilon, flat with N. On tightly clustered spectra agreement loosens to roughly 1e-10, which is a conditioning limit that constrains every eigensolver on that input equally.

~1e-15

eigenvalue agreement, flat with N

max|Δλ| across the bands-native range

1.78e-15

max |Δλ|, bands-native, N = 8,192 → 524,288

0

correctness failures across the campaign

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