probe(recipe): order statistics on the shipped PhaseLut, all-ranks timing, power-sum fold cost (D-ART-2)

[AdaWorldAPI]

Follow-up to D-ART-1 (#1418). It covers three open items and changes only the probe and the board.

1. Sector permutation on ndarray::simd::PhaseLut

Arm E (two fixed orderings plus one cosine) now also runs with every cosine taken from the shipped table (ndarray #343).

m	single order statistic	error on R = 100	stated bound
any of 3, 16, 256	15–19 ns	about 3.6e-5	7.8–8.3e-5

For comparison:

• generating and selecting costs 106 ns at m = 3 and 5 µs at m = 256;
• the bucketed profile table costs 31–34 ns at m = 3;
• the f64 sector arm costs 42–56 ns.

The bound (r + e) · lerp_error_bound() is asserted on every rank. As a disable run, using nearest instead of lerp gives 0.077 and fails it.

2. All m order statistics per instance

The interpolated profile row wins here: 18 / 42 / 493 ns for m = 3 / 16 / 256. Against it:

arm	m = 3	m = 16	m = 256
E with PhaseLut	35 ns	133 ns	1,956 ns
generate and sort	92 ns	432 ns	9,101 ns

3. Where the power-sum fold spends its time

The earlier "fused 3.7 vs shipped 4.8 ns/row" gap was noise between two i128 folds. Isolated by ingredient on 1M rows:

fold	ns/row
i128 accumulators	4.4–7.5
i64 accumulators	0.5–0.9
shipped grouped fold	5.3–8.1

The mask walk and group key add nothing measurable. The lever is the accumulator width. i64 is not exact for full-range i32, so the proposal is a split-accumulator fold in ndarray: PROBE.

The probe's i128 summary transform also agrees bitwise with the shipped CrossPowerSums::checked_affine (ndarray #344).

Board: entries/2026-10-08-recipe-probe-phaselut-foldcost.md, plus a STATUS_BOARD.md row D-ART-2.

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Review	Status	Commit	Review trigger
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