probe: per-query D/F choice (D-MHB-2) and algebraic recipe table × percentile profiles (D-ART-1)

.claude/board/STATUS_BOARD.md

@@ -1,3 +1,11 @@
+## D-ART — Algebraic recipe table over concepts and terminals (2026-10-08)
+
+Board entry: `entries/2026-10-08-algebraic-recipe-probe.md`. A linear canonicaliser and a static guarded recipe table reduce (concept, transformation chain, terminal) to the minimal work; affine transforms of `CrossPowerSums`; projected order statistics of `m`-fold families.
+
+| 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-PHT — Phasors without transcendental calls (2026-10-08)
 
 Board entry: `entries/2026-10-08-phasor-trig-probe.md`. Native `sin_cos`, a `u32`-turns phase LUT, CORDIC and complex recurrence on Wankel apex coordinates and coherent interference, plus an amplitude-bounded exact early exit.
@@ -13,6 +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-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-algebraic-recipe-probe.md

@@ -0,0 +1,183 @@
+# 2026-10-08 — Algebraic recipe table × percentile profiles × Wankel phasors (D-ART-1)
+
+**Status:** MEASURED. Probe only; no production code changes.
+- **Probe:** `crates/lance-graph-mask-risc/examples/algebraic_recipe_probe.rs`.
+- **Revision:** lance-graph `origin/main` 7f63ae9d.
+- **Host:** Intel Xeon @ 2.8 GHz, 4 vCPU, `avx2` build, release, `debug = 0`, rustc 1.98.1. Medians of 5–31 runs.
+
+```text
+CARGO_PROFILE_RELEASE_DEBUG=0 cargo run --release -p lance-graph-mask-risc --example algebraic_recipe_probe
+```
+
+## Question
+
+Given a concept (a shape family and its parameters), a chain of transformations and a requested terminal, how much geometric and trigonometric work can a small, guarded recipe table remove before anything is evaluated? And does it know when it cannot?
+
+## Inventory (read, not assumed)
+
+| item | state |
+|---|---|
+| `PowerSums`, `CrossPowerSums` (ndarray `simd_masking_ops.rs`, re-exported in `simd`) | **SHIPPED** as integer folds (`n`, `Σ`, `Σ²`, `Σxy`) with `checked_merge`. **MISSING:** any shift, scale or affine method; any mean/variance terminal on these types. |
+| `MomentsU32` (ndarray `hpc/statistics.rs`) | **SHIPPED**, unsigned, with `mean`/`variance`. **SEMANTICALLY DIFFERENT** from `PowerSums`; not interchangeable. |
+| Float terminals over power sums | **SHIPPED** consumer-side in lance-graph `crates/jc/src/stats.rs` (ANOVA, Pearson, covariance, regression). |
+| `GroupPowerSumsI32`, `GroupCrossPowerSumsI32` | **SHIPPED** as mask-risc terminals (tiled path only). **PARTIALLY AVAILABLE** end to end: no Quack `Agg`/`GroupAgg` variant reaches them. |
+| `RollingFloor`, `EmpiricalShape` (ndarray `hpc/rolling_floor.rs`) | **SHIPPED**, integer. The location-scale quantile transform `Q = μ + (σ/σ₀)(Q₀ − μ₀)`, Euclidean floor division, clamped to `u32`. Nearest-rank floor `rank_per_10000`; this probe uses that convention. |
+| Arithmetic expression IR / algebraic simplifier | **MISSING.** Quack's typed IR is Boolean predicates only. The rewrites that exist (SQL three-valued logic, survivor-gate hoisting, ternlog fusion) are all Boolean. The planner's ten `OptimizerRule`s all return `Unchanged`. DataFusion is in its grace period. |
+| Rotation / complex / phasor type | **MISSING** in ndarray `hpc` (only `GivensRotation`, no apply). D-PHT-1's LUT is a probe; ndarray PR #343 (`hpc::phase`) is open. |
+| NARS inference recipes | **SEMANTICALLY INCOMPATIBLE.** Not touched; mathematical identities are not inference rules. |
+
+## What the probe holds
+
+- **Concepts:** a regular `m`-fold family `c + r·e^{i(θ + 2πk/m)}`, with either a Wankel centre `e·e^{i3θ}` or a fixed centre. No points are stored.
+- **Canonicaliser:** one left fold of the chain.
+  - Rotations stay exact `u32` turns; `R(a)R(b) = R(a + b)` is a wrapping add.
+  - A negative similarity becomes a half turn plus a positive scale.
+  - Translations stay lazy, each tagged with the rotation and scale in force; trigonometry is spent only if a terminal reads the translation.
+  - The class drops back down `Affine ⊃ Similarity ⊃ Rotation ⊃ Identity` when scales multiply to 1 and rotations cancel.
+  - It is linear in the chain length, so it terminates; there is no search and no cycle.
+- **Recipe table:** `static [[[Rid; 9]; 4]; 2]` over (shape, linear class, terminal). The key only proposes. Guards (`m ≥ 3` for isotropy, class, translation present) run at evaluation, and a failed guard falls through to `Materialize`.
+- **Oracle:** materialises the `m` vertices and applies the raw chain to each point, independently of the canonical form.
+
+## Results
+
+### 1. The Wankel concept under `R(+1/8) · T(5, −3) · R(−1/8) · S(2) · S(0.5)`
+
+Canonical form: the identity plus one lazy translation (3 rewrites). The oracle spends 10 transcendental calls and materialises 3 vertices for every query.
+
+| query | recipe | trig | vertices | remaining work |
+|---|---|---|---|---|
+| A count | Const | 0 | 0 | `3` |
+| B Σ positions, centroid | CenterPhasor | 2 | 0 | the centre phasor `3θ`, plus one rotation of the lazy translation |
+| C centred Σ\|z − c\|², centred Σ(x−x̄)(y−ȳ) | IsoInvariant | 0 | 0 | `3R²`, `0` |
+| D Σ\|z\|², translated | OriginNorm | 2 | 0 | centred invariant + `3\|c'\|²` |
+| D′ Σ\|z\|² under `R · S(−1.5)`, untranslated | OriginNorm | 0 | 0 | `3s²(R² + e²)` |
+| E apex 1 | VertexPhasor | 3 | 0 | one vertex phasor |
+| E′ x-quantile, p = 0.5 | SectorQuantile | 3 | 0 | centre, translation, one cosine |
+| F collision with (110, 0), d = 5 | Materialize | 10 | 3 | refused: no summary decides it |
+
+All equal the oracle.
+
+### 2. Guards, 36,000 random (concept, chain, terminal) cases
+
+- m ∈ {2, 3, 4, 5, 8, 16, 256}; chains of 0–4 rotations, signed scales, exact zero scales, translations and general linear maps.
+- All equal the oracle within 1e-9 of the compared values plus an absolute floor of 1e-13 of the case's Σ|p|² + m (for results that cancel to zero).
+- A translation applied after a zero scale survives it: `S(0) · T(5, −3)` has centroid (5, −3).
+
+| recipe | cases | trig/case | oracle trig/case | exactness |
+|---|---|---|---|---|
+| Const | 4,000 | 0.17 | 51.4 | real identity |
+| CenterPhasor | 8,000 | 0.99 | 51.4 | real identity |
+| IsoInvariant | 7,000 | 0.16 | 58.2 | real identity |
+| OriginNorm | 3,500 | 0.64 | 58.2 | real identity |
+| SectorQuantile | 4,000 | 1.99 | 51.4 | real identity |
+| VertexPhasor | 4,000 | 2.19 | 51.4 | real identity |
+| Materialize | 5,500 | 38.5 | 38.4 | reference |
+
+The nonzero trig on Const and IsoInvariant is the canonicaliser leaving the similarity form (a general linear map materialises `L` once).
+
+**Refusals, measured:**
+- `m = 2` under `diag(2, 1)`: the isotropic formula gives 500.00, the truth is 777.16. The `m ≥ 3` guard refuses, and the recipe falls back to `Materialize`.
+- A shear `[1 1.5; 0 1]` gives Σ|z − c|² = 637.50, not the rotation invariant 300.00. The class is `Affine`, and `‖L‖_F²` carries it.
+
+### 3. Populations: affine chains over `CrossPowerSums`
+
+T3∘T2∘T1 integer affine maps (composed `A = [0 −2; −1 −2]`, `t = (20, 4)`). Coordinates are in [−100, 100].
+
+| rows | A: materialise each step + ndarray fold | A2: composed map per row + scalar fold | B: ndarray fold + 3 summary steps | C: ndarray fold + 1 composed step |
+|---|---|---|---|---|
+| 1,000 | 12.0 | 3.6 | 4.8 | 4.8 |
+| 65,536 | 18.8 | 3.7 | 4.9 | 4.8 |
+| 1,000,000 | 20.5 | 3.9 | 5.4 | 5.7 |
+
+ns per row. All four are **bitwise equal** (`i128`). A warm composed summary step costs **13.7 ns**, independent of the row count.
+
+- **f64:** the same identity is not bitwise. Max relative difference 1.5e-14; 1 of 6 fields bitwise equal.
+- **Moments are not sufficient:** {0, 3, 3} and {1, 1, 4} on `y = 0` share `n, Σx, Σy, Σx², Σy², Σxy` = (3, 6, 0, 18, 0, 0). Their maxima are 3 vs 4, and a collision with (4, 0) is false vs true. The table maps Population × {quantile, vertex, collide} to `Materialize` for every class.
+- **Group keys:** with the key `x ≥ 0` and the map `x + 50`, pushing the fold below the transform is wrong (group sizes 2,052 vs 3,059). With a key on an untouched column (id parity), it is exact.
+
+### 4. One projected order statistic per instance
+
+**Symmetry, measured.**
+- `Q(θ + 1/m) = Q(θ) = Q(−θ)` to 2.6e-15 on 20,000 phases.
+- Ties between `cos(θ + 2πk/m)` occur only at `θ ≡ 0 mod 1/(2m)` turn. So inside each half-sector the rank → vertex map is **fixed**, and a phase-conditioned percentile profile collapses to two permutations of `m` entries plus one cosine (arm E).
+- 1/3 turn is not representable in `u32` turns (2^32 mod 3 = 1). Threefold symmetry holds in reals, and to one ulp-turn in `u32`.
+
+**Arms.** Error is the max |error| on R = 100 over 200,003 phases × every one of the `m` ranks (the oracle sorts all `m` projections once per phase). Timings are ns per instance at 1M instances; 65,536 instances agree within 10 %.
+
+| m | A generate + select | B analytic (m = 3) | C profile LUT, 64 buckets | E sector permutation + 1 `cos` | C table | E table | C error, 64 buckets | C error, 63 buckets | E error |
+|---|---|---|---|---|---|---|---|---|---|
+| 3 (Wankel) | 108 | 39 | 34 | 50 | 780 B | 24 B | 1.3e-2 | **1.4** | 2.0e-13 |
+| 16 | 383 | — | 10.6 | 42 | 4.2 KB | 128 B | 4.7e-4 | 3.1e-1 | 2.1e-13 |
+| 256 | 5,037 | — | 10.9 | 42 | 66.6 KB | 2 KB | 7.7e-6 | 1.9e-2 | 2.0e-13 |
+
+- B is one `sin_cos`, ±120° by constants, `cos 3θ = 4c³ − 3c`, and a three-element sorting network.
+- E's time is two `f64` cosines: the value and the Wankel or fixed centre.
+- A query that needs no profile (Σ|z − c|²) costs 1.2–3 ns.
+
+### 5. Lookup and rewrite cost
+
+- **Recipe lookup over 4,096 random keys:** static array 1.2 ns, `match` 2.0 ns, `HashMap` (SipHash) 27 ns.
+- A `HashMap` whose hasher sends every key to one bucket still returns the right recipe (106 ns): structural key equality, not the hash, decides.
+- **8-step chain:** canonicalising costs 49–72 ns; canonicalise + lookup + evaluate Σ|z − c|² costs 62–98 ns, against 301–496 ns for the oracle (3 vertices × 8 raw steps). Ranges are two runs.
+
+### 6. 1-D affine quantile
+
+`Q(aX + b)` is `b + a·X[rank]` for `a ≥ 0` and `b + a·X[n − 1 − rank]` for `a < 0`. It equals the sorted oracle on 2,000 random cases.
+
+## Disable runs
+
+Each disable ran after the commit, turned red, and was restored afterwards.
+
+| disable | effect |
+|---|---|
+| isotropy guard `m ≥ 3` → `true` | case 0, `m = 2` under a linear map: recipe ≠ oracle |
+| negative scale not folded into a half turn | case 6, `m = 16`, Σ positions sign-flipped |
+| 1-D quantile rank reversal for `a < 0` dropped | `a −2, b 3, p 2270` ≠ sorted oracle |
+| sector permutation without the half-sector split | case 3, `m = 4` quantile ≠ oracle |
+| Population × quantile mapped to a summary | the table assertion fires |
+| `‖L‖_F²` of a general map treated as a similarity's | case 2, `m = 3` Σ\|z − c\|² 43,271 vs 136,940 |
+| translation multiplier not updated by later scales | case 11, `T · S(0) · R`: Σ positions ≠ oracle |
+| the first version's `scale / scale_at` translation factor | case 14, `S(0) · T`: Σ positions (0, 0) vs the oracle's translated sum |
+
+## Findings
+
+1. **The terminal decides the representation, and most terminals need no vertices.**
+   - Count, centred second moments and an untranslated Σ|z|² need no phase at all.
+   - Sum and centroid need one centre phasor; a vertex needs one phasor; an order statistic needs one cosine.
+   - Only collision falls back to materialisation, and the table refuses it rather than approximating.
+2. **The centred second moment of any regular family with `m ≥ 3` is `(m r²/2) · ‖L‖_F²` for every linear map `L`, not only rotations.**
+   - It is phase-free even under shear.
+   - `m = 2` is a rank-one dyad, and the guard is load-bearing.
+3. **The percentile profile is two permutations, not a table.**
+   - Ranks only change at `1/(2m)` turn, so the phase-conditioned profile reduces to a fixed vertex per (half-sector, rank).
+   - A bucketed LUT is fast (10 ns) but must align its bucket edges with the tie points: 63 buckets put a kink inside a bucket and the error jumps from 1e-2 to 1.4 (m = 3).
+   - For m = 3 no table pays: the analytic arm B is within 1.15× of the LUT, and exact.
+4. **Summary transforms are exact and O(1), but the fold of the base population is the floor.**
+   - Folding once and transforming the summary removes every materialisation (2.5–4× against A).
+   - A fused per-row composed transform with a scalar fold (A2, 3.7 ns/row) is faster than the shipped grouped `masked_group_cross_power_sums_i32` without a transform (4.8 ns/row).
+   - The recipe's real value is the warm path: one fold, any number of affine queries at 14 ns each.
+5. **Hashing costs more than many of the recipes it selects.** Σ|z − c|² is one multiply-add, and a SipHash lookup is 27 ns. A static array indexed by enum discriminants is 1.2 ns and cannot collide.
+
+## Recommendations
+
+| item | verdict | correctness contract | measurement |
+|---|---|---|---|
+| Affine transform of `CrossPowerSums` (`S' = AS + nt`, `M' = AMAᵀ + AStᵀ + tSᵀAᵀ + nttᵀ`), checked `i128` | **ADOPT** (a small ndarray PR, as a method on `CrossPowerSums`) | bitwise equal to materialise-and-fold within the documented widths; refuses when the group key reads a transformed column | 14 ns per summary step against 5–20 ns per row |
+| Isotropic invariant `(m r²/2)·‖L‖_F²` for regular families | **ADOPT as a recipe** where such a concept exists | real identity; `m ≥ 3` guard | 0 trig against 56–64 for the oracle |
+| Sector-permutation order statistic | **ADOPT as the method** for projected order statistics of `m`-fold families | exact up to `f64` rounding (2e-13) | 42 ns, flat in `m`; A is 5 µs at m = 256 |
+| Sector permutation with the phase LUT from ndarray #343 instead of `f64` `cos` | **PROBE** | the LUT's stated bound times `r·ρ` | expected ≈ C's 10 ns; not measured |
+| Phase-conditioned profile LUT | **PROBE** for m ≥ 16 only, bucket count a multiple of 2 per sector | bounded; edges must sit on the tie points | 10 ns, 4–67 KB |
+| Profile LUT for the Wankel triangle (m = 3) | **REJECT** | — | the analytic arm is within 1.15× and exact |
+| Static recipe table + linear canonicaliser as a production layer | **PARK** | — | no arithmetic expression IR exists to host it; building one is not justified until a consumer asks concept + terminal queries |
+| `HashMap`-keyed recipe lookup | **REJECT** | — | 27 ns against 1.2 ns |
+| e-graph or search-based rewriting | **REJECT** for this rule set | — | a single linear fold terminates and is stable |
+| `RollingFloor` / `EmpiricalShape` for deterministic geometry | **REJECT the role** | — | static geometry needs no statistical frame |
+| `RollingFloor` / `EmpiricalShape` for learned, drifting profiles | **PARK** | — | the location-scale transform matches, but no learned-shape workload exists here |
+| NARS inference recipes | untouched | — | — |
+
+## OPEN
+
+- One host class only.
+- The fused arm A2 beating the shipped grouped fold is one measurement; whether the group addressing or the mask walk costs the difference is not isolated.
+- Multiple quantiles per instance (all `m` order statistics) were not timed. E then needs `m` cosines and the permutations, but no sort.
+- Reflection symmetry of a projection along a general direction (the affine `β`) costs one `atan2`. It is counted, not removed.