# Macrostep 08 — Spherical backend

## Objective

Implement SmoothLife on a closed sphere with a tested corrected default and one coherent retained `SphereModel::Legacy`, backed by CPU oracles, GPU simulation, and a modern raylib renderer.

## Dependencies

Macrosteps 00–06 complete. Macrostep 07 is not technically required, but execute in index order for a single team.

## Phase 8.1 — Cube-sphere geometry model

### Substep 8.1.1 — Face frames and directions

Define six typed face frames with a single orientation table. For each face-cell center:

```text
direction = normalize(normal + tan(uπ/4)*axis_u + tan(vπ/4)*axis_v)
```

Use configurable `K`, with legacy `K=128` and internal `R=K/2`. Store direction, area, face, and coordinates in CPU structures; upload immutable geometry once.

### Substep 8.1.2 — Cell areas

Compute each spherical quadrilateral area from two spherical triangles or a proven equivalent. Test:

- every area finite/positive;
- face symmetry;
- total area approaches `4πR²` within a recorded discretization tolerance.

### Substep 8.1.3 — Seam mapping

Implement `SphereModel` as one local enum:

- `Corrected` (default): direction-based projection across every edge and corner; materialize and snapshot-test the mapping.
- `Legacy`: original 24 side-gutter transforms and invalid corner gutters as one inseparable part of the historical sphere model.

Do not expose independent seam/normalization compatibility toggles. Neither model samples and renders to the same atlas; both build a distinct initialized padded sampling atlas from current state.

## Phase 8.2 — Spherical neighborhood oracle

### Substep 8.2.1 — Radius conversion

Freeze and implement:

```text
ri_planar = ra/3
widths = 1
r_geo = R*acos(clamp(1-r_planar²/(2R²),-1,1))
cap_area(r) = 2πR²*(1-cos(r/R))
legacy search bound = ceil(2ra)
corrected search bound = complete nonzero softened support
```

Report invalid chord radii instead of producing NaN.

### Substep 8.2.2 — Direct neighborhood

For each active cell:

1. read center direction `a`;
2. visit candidates from the padded atlas/mapping;
3. compute `distance=R*acos(clamp(dot(a,b),-1,1))`;
4. multiply candidate state by candidate spherical cell area;
5. accumulate softened disk/ring weights;
6. normalize according to the selected complete sphere model:
   - `Corrected`: divide each numerator by the per-center sum of the same `cell_area*kernel_weight` terms actually visited;
   - `Legacy`: divide disk by `2πR²(1-cos(ri/R))` and ring by `2πR²[(1-cos(ra/R))-(1-cos(ri/R))]`, regardless of softened boundaries or masked corner samples.

The CPU implementation may be slow and use reduced `K` for exhaustive tests; clarity and seam correctness are primary. Constant-field uniformity is required for `Corrected`; historical deviations near legacy masked corners are expected fixtures.

### Substep 8.2.3 — Dynamics

Both sphere models support only:

- discrete replacement `clamp(S)`;
- fixed smooth update `clamp(A+0.1*(2S-1))`.

Do not expose base relaxation, configurable `dt`, FFT, AB3, or RK4. Future experimental sphere dynamics require a separately named model rather than extending `Legacy`.

## Phase 8.3 — Deterministic initialization

Always clear all active and gutter storage. Implement seeded sphere splats with explicit reset versus overlay operations. Provide face-coded, constant, impulse, and great-circle test initializers. Never preserve undefined initial texture contents.

## Phase 8.4 — GPU direct solver

### Substep 8.4.1 — Atlas resources

Use separate current, padded-sampling, and next `R32F` targets plus immutable direction/area data. Fill gutters via explicit passes or CPU-generated mapping, then run one direct neighborhood pass over active tiles and swap.

### Substep 8.4.2 — Stencil strategy

Start with a bounded GLSL loop specialized/cached by face size and maximum radius. Precompute valid offset metadata where it reduces `sqrt`, branch, or mapping cost without changing semantics. Recompile only on geometry changes, not rule edits.

### Substep 8.4.3 — Parity

First compare each CPU model to Macrostep 00's independent source-frozen `M/N/S/next` fixtures at centers, edges, and masked corners for both update modes. Then compare GPU to its validated CPU model. Initial one-step max-error target is `≤1e-3`; CPU/GPU agreement alone is insufficient.

## Phase 8.5 — Sphere presentation

Generate one indexed cube-sphere mesh with independent UVs per face. Add:

- an explicit `Corrected`/`Legacy` sphere-model selector with a concise semantic comparison;
- transactional model switching: construct the complete seam map, normalization data, padded atlas, and shader resources before commit; preserve active-face state and generation when `K` is unchanged; retain the previous model on failure; never share model-specific caches;
- orbit/arcball camera and time-based autorotation;
- grid overlay;
- all shared palettes;
- optional two-sided legacy red/blue styling;
- active atlas, padded atlas, direction, area, seam, and geodesic-distance views;
- face labels and edge-orientation debug mode.

Keep rendering geometry separate from simulation resolution so mesh tessellation can change without altering state.

## Phase 8.6 — Verification and performance

### Geometry/seam tests

- all 24 directed edge mappings and corner transitions;
- cross-edge-and-back round trip;
- shared-edge direction equality;
- rotational consistency under cube symmetries;
- total area and constant-field neighborhoods;
- no `acos` NaN.

### Resource tests

- no texture feedback;
- all gutters explicitly initialized;
- framebuffer completeness after rebuild;
- repeated `Corrected↔Legacy` switching agrees with a freshly constructed destination model, preserves state/generation, and never mixes seam or normalization caches;
- failed switches preserve the previous working model;
- rule edits do not rebuild geometry.

### Performance target

On the designated GPU, `K=128`, `ra=10` should target p95 step time below 33 ms for both models. If either fails, ship a lower interactive default while keeping `SphereModel::Legacy` selectable and document the measured cost (roughly 165 million candidate iterations per step before optimizations).

## Deliverables

- CPU oracles for `SphereModel::{Corrected,Legacy}`.
- One coherent retained legacy sphere model, not independent compatibility flags.
- GPU sphere backend and raylib sphere renderer.
- Sphere-specific inspection and seam diagnostic tools.

## Exit gate

- Face orientation, area, edge, and corner tests pass.
- Both CPU models match independent source-frozen end-to-end fixtures for both update modes.
- Constant state produces uniform neighborhoods for `Corrected`; `Legacy` analytic/masked-corner deviations match committed fixtures.
- CPU/GPU one-step parity passes at centers, edges, and corners for both models.
- Transactional model-switch lifecycle tests pass without mixed caches or state loss.
- Default sphere preset is rotatable, inspectable, and deterministic.
- No undefined atlas contents, feedback loop, or unclamped `acos` remains.