Add 3DGS 4x4 cognitive shader SoA plan

Author: AdaWorldAPI

.claude/plans/3DGS-4x4-cognitive-shader-SoA-plan.md

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+# 3DGS 4x4 Cognitive-Shader SoA Plan — ndarray
+
+## Goal
+
+Promote the 3DGS math carrier from a narrow 3x3-only view into a 4x4 cognitive-shader-friendly SoA carrier while preserving the correct 3DGS covariance spine.
+
+Important distinction:
+
+```text
+3x3 SPD covariance
+  remains the mathematically correct spatial covariance for each 3D Gaussian
+
+4x4 carrier
+  becomes the homogeneous / temporal / semantic / shader-lane envelope around that covariance
+```
+
+Do not destroy the 3x3 SPD invariant. Lift it into a 4x4 block where needed.
+
+## Why 4x4
+
+The 4x4 shape maps better to:
+
+```text
+homogeneous transforms
+SIMD lane groups
+cognitive-shader-driver BindSpace columns
+Mat4 camera/world transforms
+quaternion / covariance / opacity packing
+(4x4)^4 tensor-block fanout
+```
+
+It also gives a natural bridge between rendering, registration, ultrasound frame fusion, genetics-like transition matrices, and neuronal adjacency blocks.
+
+## Core representation
+
+Add or stabilize a Mat4 carrier under the existing linalg path:
+
+```text
+src/hpc/linalg/
+  mat4.rs
+  spd4.rs
+  block4.rs
+```
+
+Candidate types:
+
+```rust
+pub struct Mat4x4<T> {
+    pub m: [[T; 4]; 4],
+}
+
+pub struct Sym4<T> {
+    pub a00: T, pub a01: T, pub a02: T, pub a03: T,
+                pub a11: T, pub a12: T, pub a13: T,
+                            pub a22: T, pub a23: T,
+                                        pub a33: T,
+}
+
+pub struct Splat4Carrier<T> {
+    pub spatial_sigma_3x3: [T; 6],
+    pub lane4: [T; 4],
+    pub transform4x4: Mat4x4<T>,
+}
+```
+
+## 3x3 to 4x4 lift
+
+Use explicit lifts instead of pretending 3x3 and 4x4 mean the same thing.
+
+```text
+3DGS spatial covariance Sigma3
+        ->
+4x4 homogeneous carrier Sigma4
+        ->
+projection / temporal / semantic / shader lane operations
+        ->
+extract spatial 3x3 or image-space 2x2 when needed
+```
+
+Example lift:
+
+```rust
+pub fn lift_spd3_to_sym4(sigma3: [f32; 6], w_lane: [f32; 4]) -> Sym4<f32>;
+pub fn extract_spd3_from_sym4(sym4: Sym4<f32>) -> [f32; 6];
+```
+
+## Cognitive-shader SoA layout
+
+Prefer 4-lane grouped SoA columns:
+
+```text
+BindSpace4
+  lane0: position / nucleotide / neuron source / feature id shard
+  lane1: covariance / transition / edge weight / local statistic
+  lane2: opacity / expression / activation / confidence
+  lane3: time / phase / semantic role / provenance
+```
+
+Concrete splat columns:
+
+```text
+pos_xyzw[]
+scale_xyz_opacity[]
+quat_xyzw[]
+color_rgba[]
+feature_id_time[]
+certificate_confidence[]
+```
+
+## (4x4)^4 tensor-block fanout
+
+Interpret `(4x4)^4` as a four-level block grammar, not as one giant dense matrix.
+
+```text
+level 0: Mat4 local carrier
+level 1: 4x4 block of carriers
+level 2: 4x4 block of blocks
+level 3: 4x4 graph/tile/neural field super-block
+```
+
+This gives hierarchical locality:
+
+```text
+splat -> block -> tile -> region -> graph domain
+```
+
+## Domain-neutral kernels
+
+Create kernels that do not care whether the 4x4 block is geospatial, ultrasound, genetic, or neuronal.
+
+```rust
+pub trait Block4Kernel<T> {
+    fn score_block(&self, block: &Block4<T>) -> f32;
+    fn contract(&self, parent: &Block4<T>, child: &Block4<T>) -> f32;
+    fn certify(&self, block: &Block4<T>) -> Block4Certificate;
+}
+```
+
+## 3DGS compatibility rules
+
+- Keep `Spd3` as the authoritative spatial covariance.
+- Use `Mat4x4` for transforms and carriers.
+- Use `Sym4` only when the extra lane has defined meaning.
+- Image-space EWA still extracts the proper spatial covariance path.
+- Pillar tests must assert that 4x4 lifts do not break 3x3 invariants.
+
+## Pillar additions
+
+Add substrate probes:
+
+```text
+Pillar-18: 3x3-to-4x4 lift preserves spatial SPD invariants
+Pillar-19: Block4 contraction under HHTL hierarchy
+Pillar-20: 4-lane SoA equivalence against scalar AoS reference
+```
+
+## Acceptance criteria
+
+- 4x4 carrier compiles behind `linalg` or a specific feature gate.
+- `splat3d` can use 4x4 transforms without losing 3x3 covariance correctness.
+- SoA 4-lane layout has scalar reference tests.
+- HHTL cascade can score 4x4 blocks.
+- Pillar tests prove lift/extract invariants.
+
+## Cross-repo link
+
+`lance-graph` should use this as the numeric substrate for the 4x4 cross-domain fanout plans.