# Macrostep 03 — CPU planar simulation engine

## Objective

Build the first complete deterministic SmoothLife simulator using the pure core and a conventional CPU FFT. Support headless stepping and all base dimensions before introducing GPU complexity.

## Dependencies

Macrosteps 00–02 complete.

## Phase 3.1 — FFT convolution backend

### Substep 3.1.1 — Transform convention

Implement the default `FftAlgorithm::Standard` CPU path with a standard library FFT. The retained `LegacyPackedUnitary` algorithm is a GPU backend planned in Macrosteps 05–06, with only a small CPU stage oracle for tests. Define and test the standard convention:

1. unscaled forward transform;
2. spectral multiplication;
3. inverse scaled exactly by `1/(Nx·Ny·Nz)`;
4. divide by sampled kernel sum, or pre-normalize kernels once.

Never apply `sqrt(sample_count)` in the standard path; that correction belongs only to `LegacyPackedUnitary`.

### Substep 3.1.2 — Separable N-D transforms

Implement transforms along each active axis with reusable plans and scratch buffers. Start with full complex storage for clarity. If real-to-complex optimization is later added, retain the full-complex implementation as the reference.

Requirements:

- arbitrary validated extents supported by the chosen FFT library on CPU;
- no x/y/z transposition ambiguity;
- cached plans keyed by shape;
- no steady-state allocations;
- serial correctness before Rayon parallelism.

### Substep 3.1.3 — Spectral kernel cache

Build/cache spectra only when shape or kernel geometry changes. Keep disk/ring spectra, sums, and source parameters together so stale kernels cannot be paired with a new config.

## Phase 3.2 — Planar backend pipeline

### Substep 3.2.1 — Backend state

The planar CPU backend owns:

- current and next state;
- `M`, `N`, target, and derivative channels;
- disk/ring kernels and spectra;
- FFT work buffers/plans;
- integrator and AB history;
- generation and timing counters;
- run descriptor and deterministic initializer state.

### Substep 3.2.2 — Euler/discrete step

Implement the complete stage pipeline:

```text
FFT(A)
→ multiply by KDF/KRF
→ inverse + normalize to M/N
→ evaluate S
→ discrete replacement or Euler derivative update
→ clamp and swap
```

Expose each stage through inspection without forcing a copy when the CPU owns the field.

### Substep 3.2.3 — AB3 and RK4

- AB3 performs one derivative evaluation per step and rotates initialized history.
- RK4 performs the full convolution/rule pipeline at each clamped stage and supports both `StageState` and historical `StepOrigin` relaxation references.
- Discrete mode bypasses both methods.
- Parameter commands use Macrostep 00's full invalidation matrix: only presentation/inspection/pause/scheduler edits preserve history; rule, dynamics, timestep, integrator, kernel, historical-option selection, state/initializer, topology, shape, variant, and backend edits invalidate it.

## Phase 3.3 — Simulation command facade

Define a small object-safe facade or enum dispatch with commands such as:

```text
step(count)
pause/single-step handled by app, not backend
reset(initializer, seed)
apply_hot(rule edit)
rebuild_warm(kernel edit)
recreate_cold(shape/variant edit)
load/export restart state
load/export exact checkpoint
inspection(channel)
metrics()
```

Apply warm/cold changes transactionally: build and validate new resources first, then replace the live backend. Preserve the prior valid simulation if rebuilding fails.

## Phase 3.4 — Headless runner

Implement CLI execution that can:

- choose preset/shape/seed/mode/integrator;
- run exact step counts;
- export restart state, exact continuation checkpoints, and inspection channels;
- print checksums, min/max/mean/variance, occupancy histogram, and elapsed time;
- compare two state files with max, L1, and L2 error.

The headless path and interactive path must call the same backend methods.

## Phase 3.5 — Determinism and parallelism

1. Establish single-thread deterministic fixtures.
2. Add Rayon only around independent transform lines or pointwise passes.
3. Ensure reductions affecting normalization use a defined order or precomputed kernel sums.
4. Document whether multi-threaded execution is bit-identical or tolerance-equivalent.
5. Include thread count and CPU backend version in benchmark metadata, not in mathematical run identity unless output differs.

## Phase 3.6 — Verification

### Stage-level tests

- FFT round trip: zero, constant, impulse, checkerboard, and asymmetric matrices.
- FFT convolution against direct `f64` oracle in all dimensions.
- Default kernel-sum checks.
- One-step output for every dynamics/integrator combination, including both RK4 relaxation references.
- Stage snapshots for `M`, `N`, `S`, derivative, and committed state.

### Deterministic scenario tests

Run fixed states for 1, 2, and 10 steps. Compare arrays for CPU serial mode and aggregate metrics for allowed parallel differences. Include representative 1-D, 2-D discrete, 2-D growth, 2-D relaxation, and 3-D presets.

### Allocation tests

Use a counting allocator or benchmark instrumentation to prove no allocation during steady-state Euler/AB/RK stepping after warm-up.

## Performance baseline, not optimization gate

Record release-build timings for:

- 1-D 1024 and 8192;
- 2-D 128² and 512²;
- 3-D 32³ and 64³;
- Euler versus RK4.

These measurements establish later GPU speedup and regression baselines. Correctness is the exit gate; do not distort architecture to hit a premature number.

## Deliverables

- CPU FFT convolution backend.
- Complete base planar simulator in 1-D/2-D/3-D.
- Headless runner, state import/export, metrics, and comparisons.
- Stage-level fixtures and baseline report.

## Exit gate

- FFT convolution matches the direct oracle within `2e-5` on agreed small fields.
- All base one-step fixtures pass in 1-D/2-D/3-D.
- Seed + preset + shape + backend + step count reproduces output.
- RK4 invokes four complete derivative pipelines; AB history startup/reset is deterministic.
- No steady-state heap allocation occurs after warm-up.
- Headless state and metrics can be consumed without raylib.