D-PHT-1: phasors without transcendental calls (probe + board)

.claude/board/STATUS_BOARD.md

@@ -1,3 +1,11 @@
+## 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.
+
+| D-id | scope | status | gate / falsifier |
+|---|---|---|---|
+| **D-PHT-1** | Phasor evaluation: f64/f32 `sin_cos`, LUT nearest/interpolated, CORDIC, recurrence, ndarray `vml`; two-wave closed form; amplitude-bounded early exit with LUT error and exact fallback | Shipped (probe; LUT + integer phase proposed for ndarray) | Wankel invariants to 1e-11; early exit equals the full fold on every detector; dropping the remaining amplitude, the LUT error or the signed cast each fails |
+
 ## D-MHB — Centre-surround response without a raster (2026-10-08)
 
 Board entry: `entries/2026-10-08-mexhat-bucket-cascade-probe.md`. Popcount stacking, quantised DoG and exact early exit against direct window geometry; no wavefield, no raster.

.claude/board/entries/2026-10-08-phasor-trig-probe.md

@@ -0,0 +1,161 @@
+# 2026-10-08 — Phasors without transcendental calls: sin_cos vs phase LUT vs CORDIC vs recurrence (D-PHT-1)
+
+**Status:** MEASURED. Probe only, no production primitive.
+- **Probe:** `crates/lance-graph-mask-risc/examples/phasor_trig_probe.rs`.
+- **Revision:** lance-graph `origin/main` d127f7dc.
+- **Host:** Intel Xeon @ 2.8 GHz, 4 vCPU.
+- **Build:** `x86-64-v3`, release, `debug = 0`, rustc 1.98.1.
+- **Timings:** medians of 5 runs.
+
+```text
+CARGO_PROFILE_RELEASE_DEBUG=0 cargo run --release -p lance-graph-mask-risc --example phasor_trig_probe
+```
+
+## Question
+
+Can two observables that need only phasors avoid every transcendental call in the hot path?
+
+1. **Wankel apex coordinates:** `z_k(θ) = e·e^{i3θ} + R·e^{i(θ + 2πk/3)}`. This is the ideal apex triangle, not the rotor flanks and not sealing.
+2. **Coherent interference:** `I(P) = |Σ A_i e^{iφ_i(P)}|²`, with an amplitude-bounded exact early exit.
+
+The phase is a `u32` in turns, so `3θ` is `phase.wrapping_mul(3)` and wraparound is free.
+
+## Inventory
+
+- **MISSING:** CORDIC, a phase LUT, or a SIMD `sin`/`cos` anywhere in ndarray or lance-graph.
+- **SHIPPED, but not what it looks like:** `ndarray::hpc::vml::vscos`/`vssin` exist, but they call scalar `cos`/`sin` per lane inside an `F32x16` wrapper (`hpc/vml.rs`). That is not a SIMD transcendental.
+
+## Wankel invariants (f64 reference)
+
+All hold, checked on 100,000 angles in ±1000 rad, for (R, e) ∈ {(100, 14), (100, 1e-9), (100, 0), (1, 0.15)}:
+- every apex lies on the housing curve at `θ + 2πk/3`;
+- the triangle side stays `R√3`;
+- the rotor centre stays at distance `e`;
+- threefold symmetry holds.
+
+Maximum error is 5.8e-12 for R = 100. The 3:1 law as `wrapping_mul(3)` equals `3θ mod 2π` on 100,000 phases.
+
+## Results
+
+### Wankel apex: 1M random phases
+
+ns per apex; error in housing units, with R = 100.
+
+| arm | ns | max error | table |
+|---|---|---|---|
+| native f64 `sin_cos` ×2 | 66.8 | reference | — |
+| native f32 `sin_cos` ×2 | 25.4 | 5.1e-5 | — |
+| **L12 LUT, nearest** | **3.1** | 8.7e-2 | 32 KB |
+| L16 LUT, nearest | 4.0 | 5.5e-3 | 512 KB |
+| **L12i LUT, linear interpolation** | **5.6** | 4.6e-5 | 32 KB |
+| L10i LUT, linear interpolation | 5.5 | 5.5e-4 | 8 KB |
+| CORDIC Q30, 24 iterations | 258 | 1.4e-5 | 192 B |
+| ndarray `vml` (4 calls) | 50.9 | 3.4e-5 | materialises 24 MB of arrays |
+
+The 64K run gives the same ordering: f64 61.9, f32 25.2, L12 2.5, L12i 5.8, `vml` 45.6.
+
+### Trajectory, 1M equal steps
+
+| arm | max error |
+|---|---|
+| f64 complex recurrence, never renormalised | 4.8e-9 |
+| f32 recurrence, never renormalised | 1.95 |
+| f32 recurrence, renormalised every 1,024 steps | 1.2e-2 |
+| integer `u32` phase accumulator + L12i | no drift by construction; 3.8e-5, the table's own error |
+
+The integer step differs from the requested Δθ by 5.7e-10 rad. That is a frequency choice, not accumulation.
+
+### Two waves: 65,536 detectors
+
+| arm | ns | error / peak |
+|---|---|---|
+| two `sin_cos` + `\|Σ\|²` | 80.7 | reference |
+| one `cos(Δφ)`: `I = A1² + A2² + 2A1A2 cos Δφ` | 28.9 | 1.7e-14 (an identity) |
+| one L12i lookup of Δφ | 13.4 | 1.7e-7 |
+
+### 1,000 sources × 65,536 detectors
+
+ns per (source, detector) term:
+
+| arm | ns | max \|ΔI\| / peak I |
+|---|---|---|
+| native f64 `sin_cos` | 45.9 | reference |
+| native f32 `sin_cos` | 36.4 | 3.5e-6 |
+| **L12 LUT, nearest** | **9.95** | 4.5e-4 |
+| L10i LUT, linear interpolation | 12.9 | 6.0e-6 |
+| CORDIC 24 | 156.7 | 6.1e-8 |
+| path length only (one `√`) | 2.36 | the floor every arm pays |
+
+### Amplitude-bounded early exit for `I ≥ T`
+
+Sources are visited in descending amplitude. The bound is `max(0, |S| − R)² ≤ I ≤ (|S| + R)²`.
+
+**Correctness, two arms:**
+- **f64 arm:** asserted equal to the full fold on every detector.
+- **L12 arm:** carries its own per-term error `ε = π/2^12 + 2·f32::EPSILON` in the bound. Any detector still undecided after all terms falls back to the f64 reference.
+
+**Terms evaluated** (of 1,000; mean, p95 in brackets):
+
+| T quantile | uniform A ∈ [0.5, 1] | `A_i = i^-1.5` | L12 fallbacks (uniform / heavy) |
+|---|---|---|---|
+| 50 % | 983 (999) | 122 (550) | 2,679 / 278 |
+| 90 % | 968 (996) | 76 (390) | 961 / 160 |
+| 99 % | 941 (980) | 28 (100) | 124 / 35 |
+
+## Falsifiers and disable runs
+
+Each disable run was made after the commit, turned red, and was restored afterwards.
+
+| disable | effect |
+|---|---|
+| early exit ignores the unevaluated amplitude | `exact early exit disagrees at detector 4` |
+| LUT bound ignores the table's own error | `decided wrongly at detector 7088` |
+| `as u64` instead of `as i64` for a phase difference | two-wave LUT error 2.77 of peak, assertion fires |
+
+**This probe's own first run had two bugs, both caught by its assertions:**
+- `as u64` saturates a negative phase difference to 0, giving a 99 % error.
+- Deciding at `R = 0` through `√` then a square disagreed with the full fold at the threshold detector. The fix decides on `re² + im²` exactly as the fold computes it, and absorbs the `√` rounding in a relative margin of 1e-12 before that.
+
+## Findings
+
+1. **A phase LUT removes the transcendental cost.**
+   - L12i matches f32 `sin_cos` accuracy (4.6e-5 vs 5.1e-5 on R = 100) at about a quarter of the time (5.6 vs 25.4 ns), and is about 12× faster than f64.
+   - Nearest L12 is 20× faster than f64 at 8.7e-4 relative error.
+   - The `u32` turns representation makes `3θ` one `wrapping_mul` and wraparound free.
+2. **CORDIC loses on this CPU.** It is 2.6–4.8× slower than native f64 here (16–30 iterations): a serial, branchy shift-add loop against hardware multiply. It is the right tool only without a multiplier or without table space.
+3. **ndarray `vml` is not a SIMD `sin`/`cos`.** It runs as fast as native calls and, used through arrays, materialises its inputs and outputs.
+4. **Algebra beats lookup where it applies.** For two waves, the phase difference turns two complex phasors into one cosine (2.8×), and a LUT on Δφ makes it 6×.
+5. **The bound early exit is a random-walk problem.**
+   - With N comparable amplitudes, `|S| ~ √N·A` while the remaining bound is `~N·A`, so the bound only closes at the very end: 94–98 % of the terms are still needed.
+   - With heavy-tailed amplitudes it saves 8–36× (28–122 of 1,000 terms).
+   - Whether it pays is a property of the amplitude distribution, not of the bound.
+6. **After the LUT, the path length is the next floor.** One `√` per term is 2.36 ns against ~10 ns for the LUT arm. The rest is table access and the multiply-add.
+7. **Incremental trajectories:**
+   - an integer phase accumulator plus a LUT has zero drift by construction;
+   - an f64 recurrence drifts 4.8e-9 in 1M steps, negligible;
+   - an f32 recurrence is unusable without renormalisation, and still 1e-2 with it.
+
+## Recommendations
+
+| item | verdict | benefit | falsifier |
+|---|---|---|---|
+| `u32`-turns phase + 2^12 `(cos, sin)` LUT with linear interpolation | **ADOPT** (separate PR, ndarray, exposed through `ndarray::simd`) | ~4.5× vs f32 `sin_cos`, ~12× vs f64, same accuracy as f32 | max error ≤ 5e-5 relative on 1M phases; `3θ = wrapping_mul(3)` exact |
+| Closed form for two waves, `cos Δφ` | **ADOPT** where exactly two coherent terms meet | 2.8× (6× with LUT) | identity to 1e-14 |
+| Amplitude-bounded early exit | **PROBE**, only for heavy-tailed amplitude sets | 8–36× fewer terms when heavy-tailed; none when uniform | equals the full fold on every detector; dropping R fails |
+| LUT + error-inflated bound with exact fallback | **PROBE** | exact decisions on top of the LUT speed | dropping ε fails; fallback count reported |
+| Integer phase accumulator for trajectories | **ADOPT**, together with the LUT | no drift | error equals the table bound after 1M steps |
+| CORDIC | **REJECT** on CPU | — | 2.6–4.8× slower than native f64 |
+| f32 complex recurrence | **REJECT** without renormalisation | — | 1.95 units after 1M steps |
+| ndarray `vml` sin/cos as a SIMD path | **REJECT the equivalence** | — | per-lane scalar; no speed-up |
+
+## Not done here
+
+- **SIMD gather of the LUT** (AVX2 `vpgatherdd`) was not measured.
+- **The Wankel rotor as a moving mask** was not built: containment, `point_inside_rotor`, rotating aperture × wavefront, visibility.
+- **VSA bind/bundle against complex multiply/add** was not compared.
+- **Log₂(r²) overlap buckets** were not built. D-MHB-1 counterexample 5 already shows that distance buckets cannot see phase, and log buckets are coarser still.
+
+## OPEN
+
+- One host class only.
+- The LUT's place in ndarray, and whether its interpolation should be `f32` or fixed point, are open.

.claude/board/entries/README.md

@@ -25,14 +25,15 @@ 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.
 
-274 entries, 2026-08-06 .. 2026-10-08.
+275 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 | `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) |
+| 2026-10-08 | `D-PHT-1` |  | [2026-10-08-phasor-trig-probe.md](2026-10-08-phasor-trig-probe.md) |
 | 2026-10-08 | `moore-nars16-isa-visible-representation` |  | [2026-10-08-moore-nars16-isa-visible-representation.md](2026-10-08-moore-nars16-isa-visible-representation.md) |
 | 2026-10-08 | `D-MOORE-NARS-0` |  | [2026-10-08-moore-nars-0-recipe-learning-gomoku.md](2026-10-08-moore-nars-0-recipe-learning-gomoku.md) |
 | 2026-10-08 | `D-MHB-1` |  | [2026-10-08-mexhat-bucket-cascade-probe.md](2026-10-08-mexhat-bucket-cascade-probe.md) |