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

.claude/board/STATUS_BOARD.md

@@ -5,6 +5,7 @@ Board entry: `entries/2026-10-08-algebraic-recipe-probe.md`. A linear canonicali
 | D-id | scope | status | gate / falsifier |
 |---|---|---|---|
 | **D-ART-1** | Recipe table: constants, centre phasor, isotropic invariant, origin norm, sector quantile, vertex phasor, summary affine, materialise; PowerSums affine chains; order statistics by generate/analytic/profile LUT/sector permutation; lookup cost | Shipped (probe; `CrossPowerSums` affine transform proposed for ndarray) | 36,000 cases equal the materialising oracle; summary path bitwise equal in `i128`; dropping the isotropy guard, the half-turn fold, the rank reversal, the half-sector split, the population refusal or the general `‖L‖_F²` each fails |
+| **D-ART-2** | Sector permutation on `ndarray::simd::PhaseLut`; all-order-statistics timing; power-sum fold cost by ingredient; cross-check of the shipped `checked_affine` | Shipped (probe; split-accumulator fold proposed for ndarray) | E-LUT within `(r+e)·lerp_error_bound()` on every rank, and `nearest` instead of `lerp` fails it; the five folds give equal sums; the shipped `checked_affine` equals the probe's transform. Entry: `entries/2026-10-08-recipe-probe-phaselut-foldcost.md` |
 
 ## D-PHT — Phasors without transcendental calls (2026-10-08)
 
@@ -20,7 +21,7 @@ Board entry: `entries/2026-10-08-mexhat-bucket-cascade-probe.md`. Popcount stack
 
 | D-id | scope | status | gate / falsifier |
 |---|---|---|---|
-| **D-MHB-1** | Mexican-hat response: direct geometry (D), ring buckets (E), bit-sliced weight planes (F), per-row choice (G), mask-risc lane (B), exact-bound early exit (H) | Shipped (probe; R-MHB-1 proposed) | D = F = G = B = full scan asserted on every centre; H equals the full decision on every query; one zero crossing, one minimum, symmetry; dropping the unvisited-row bound decides 52 / 512 wrongly |
+| **D-MHB-1** | Mexican-hat response: direct geometry (D), ring buckets (E), bit-sliced weight planes (F), per-row choice (G), mask-risc lane (B), exact-bound early exit (H) | Shipped (probe; R-MHB-1 OPEN: blocked on a 2-D window operand mask-risc's IR lacks, and F alone would bring back the 2.5× worst case Q removes) | D = F = G = B = full scan asserted on every centre; H equals the full decision on every query; one zero crossing, one minimum, symmetry; dropping the unvisited-row bound decides 52 / 512 wrongly |
 | **D-MHB-2** | Per-query D/F choice from the exact window count (Q); early exit over D rows (HD) and after a per-query choice (HQ) | Shipped (probe; Q proposed with R-MHB-1) | Q = D for cuts 0, 100, ∞ on every centre; HD/HQ decide as H after the same rows; a D row dropping one cell fails Q. Entry: `entries/2026-10-08-mexhat-df-choice.md` |
 
 ## D-RPF — Mask × fold over projections of resident bytes (2026-10-08)

.claude/board/entries/2026-10-08-recipe-probe-phaselut-foldcost.md

@@ -0,0 +1,70 @@
+# 2026-10-08 — Recipe probe on the shipped PhaseLut; all order statistics; where the power-sum fold spends its time (D-ART-2)
+
+**Status:** MEASURED. Probe only; no production code changes. Follows D-ART-1.
+- **Probe:** `crates/lance-graph-mask-risc/examples/algebraic_recipe_probe.rs`, sections 3b, 4 (E-LUT arm) and 4b.
+- **Uses:** `ndarray::simd::PhaseLut` (ndarray #343) and `CrossPowerSums::checked_affine` (ndarray #344).
+- **Host:** Intel Xeon @ 2.8 GHz, 4 vCPU, `avx2` build, release, `debug = 0`. Medians of 5–9 runs. The VM is noisy; ranges below are across three runs.
+
+## 1. Sector permutation on the shipped phase table
+
+Arm **E-LUT** is arm E (two fixed orderings plus one cosine) with every cosine taken from `phase_lut_4096()`.
+
+| m | A generate + select | C profile LUT, 64 buckets | E sector, `f64` cos | **E-LUT** | E-LUT error on R = 100 | its stated bound |
+|---|---|---|---|---|---|---|
+| 3 (Wankel) | 106–108 ns | 31–34 | 49–56 | **15.6–18.9** | 3.7e-5 | 8.3e-5 |
+| 16 | 400–718 | 10.6–20.9 | 42–61 | **15.8–29.0** | 3.5e-5 | 7.8e-5 |
+| 256 | 4,918–5,415 | 10.7–14.9 | 41–46 | **14.6–16.0** | 3.5e-5 | 7.8e-5 |
+
+Ranges are over 65,536 and 1M instances. Errors are max |error| over 200,003 phases × every rank.
+
+- The error stays within `(r + e) · lerp_error_bound()`, asserted on every rank. Disable run: `nearest` instead of `lerp` gives 0.077 against the 8.3e-5 bound, and the assertion fires.
+- E-LUT needs a 32 KB shared table plus a 24 B–2 KB permutation, and its cost does not depend on m. For the Wankel triangle it beats both the profile LUT (2×) and the analytic arm B (2.7×).
+
+## 2. All m order statistics per instance
+
+65,536 instances; m = 256 uses 16,384.
+
+| m | A generate + sort | E sector | E-LUT | C profile row |
+|---|---|---|---|---|
+| 3 | 92 ns | 80 | 35 | **18** |
+| 16 | 432 | 352 | 133 | **42** |
+| 256 | 9,101 | 4,671 | 1,956 | **493** |
+
+Max error: E 3e-13, E-LUT 5e-5, C 2e-2 / 7e-4 / 1e-5 for m = 3 / 16 / 256.
+
+When every rank is wanted, the profile row is 2–4× faster than E-LUT: the row is contiguous and needs no permutation lookup. When one rank is wanted, E-LUT is the choice: it is exact up to the table's bound and needs no per-m table.
+
+## 3. Where the cross-power-sum fold spends its time
+
+1M rows, one group, all rows selected. ns per row, three runs:
+
+| fold | ns/row |
+|---|---|
+| plain scalar, `i128` accumulators | 4.4–7.5 |
+| same, with `i64` accumulators | **0.5–0.9** |
+| `i128` + mask-bit walk | 3.9–6.7 |
+| `i128` + group key | 5.3–9.6 |
+| shipped `masked_group_cross_power_sums_i32` | 5.3–8.1 |
+
+All five produce identical sums.
+
+- **The accumulator width is the cost.** `i64` accumulators are about 8× faster than `i128`, presumably because they vectorise. The mask walk and the group key are within noise.
+- **D-ART-1's "fused 3.7 vs shipped 4.8 ns/row" was noise between two `i128` folds.** It is not a defect in the shipped fold's addressing.
+- **`i64` is not exact for the full `i32` range.** One `x²` reaches 2^62, so two rows can overflow it. A faster exact fold therefore needs either known-bounded inputs, or a split such as `x = hi·2^16 + lo` with `u64` partial sums combined into `u128` at the end. ndarray's `masked_group_bounded_power_sums_u8` is the bounded case of this idea for `u8`.
+
+## 4. Shipped `checked_affine`
+
+The probe's own `i128` summary transform now agrees bitwise with `CrossPowerSums::checked_affine` (ndarray #344) on every row count.
+
+## Recommendations
+
+| item | verdict |
+|---|---|
+| Single order statistic of an m-fold family: sector permutation + `PhaseLut` | **ADOPT** as the method (exact to the table's bound, flat in m) |
+| All order statistics, m ≥ 16: interpolated profile row, bucket edges on the tie points | **PROBE** |
+| Wide-input power-sum fold with split `u64` partial sums | **PROBE** in ndarray, with a parity test against the `i128` fold; measured lever ≈ 8× |
+
+## OPEN
+
+- One host class.
+- The split-accumulator fold is not built. The 8× is measured on `i64` accumulators, which are not exact for full-range `i32`.

.claude/board/entries/README.md

@@ -25,11 +25,12 @@ index row, (3) no duplicate entry id. Checks 1 and 2 are deliberately
 opposite directions; the stranding this convention prevents shows up in
 exactly one of them, never both.
 
-277 entries, 2026-08-06 .. 2026-10-08.
+278 entries, 2026-08-06 .. 2026-10-08.
 
 | date | entry id | finding | file |
 |---|---|---|---|
 | 2026-10-08 | `register128-signed-readings` |  | [2026-10-08-register128-signed-readings.md](2026-10-08-register128-signed-readings.md) |
+| 2026-10-08 | `D-ART-2` |  | [2026-10-08-recipe-probe-phaselut-foldcost.md](2026-10-08-recipe-probe-phaselut-foldcost.md) |
 | 2026-10-08 | `rbac-nested-scope-path` |  | [2026-10-08-rbac-nested-scope-path.md](2026-10-08-rbac-nested-scope-path.md) |
 | 2026-10-08 | `rbac-membership-scope` |  | [2026-10-08-rbac-membership-scope.md](2026-10-08-rbac-membership-scope.md) |
 | 2026-10-08 | `rbac-hotplug-socket` |  | [2026-10-08-rbac-hotplug-socket.md](2026-10-08-rbac-hotplug-socket.md) |