Motivic foundation: A_C, the C-motivic Steenrod algebra engine over F₂[τ]

ext/crates/algebra/MOTIVIC_GPU_HANDOFF.md

@@ -0,0 +1,42 @@
+# GPU port notes: the C-motivic Milnor multiply (ℂ, p = 2)
+
+The C-motivic product (`multiply_closed`, Kong–Lin Theorem 5.1 at ρ = 0, in
+`motivic/milnor.rs`) is the arithmetic bottleneck of the deformation pipeline and
+the natural GPU-offload target. This note records the two facts that make a kernel
+tractable and where the batch boundary is; the device plumbing (CubeCL toolchain,
+atomic-XOR output, hash-free indexing) mirrors the classical Nassau kernel and is
+not repeated here.
+
+## The batch unit
+
+`MotivicMilnorAlgebra::fill_block(t1, t2)` computes an entire `ProductBlock` — every
+structure constant `Q(E₁)P(R₁) · Q(E₂)P(R₂)` for a pair of topological degrees — in
+one shot, filling independent `OnceLock` cells in parallel. That block is the shape a
+GPU kernel would fill in a single launch: the resolution asks for the same
+degree-pair block repeatedly, so the per-degree-pair block is the batching boundary.
+
+## Two facts that collapse the apparent complexity
+
+1. **The device output stays F₂ atomic-XOR — τ never touches it.** Every surviving
+   matrix pair `(X, Y)` contributes coefficient exactly `1`, and by weight-homogeneity
+   a fixed output monomial `z` carries a single, determined τ-power — the weight
+   difference between `a·b` and `z`. So the kernel computes only **F₂ presence** of
+   each output index (identical to the classical kernel); the τ-power is a host-side
+   weight lookup, not part of the device accumulator. No `Tau` arithmetic on the GPU.
+   (Pin the exact τ-power expression against `multiply_closed` — `bidegree(...)` uses a
+   negated-weight convention.)
+
+2. **The bottleneck products are the mod-τ product, which is odd-primary-Milnor
+   structured at p = 2.** `A_C/τ ≅ 𝔽₂[ξᵢ] ⊗ E(τᵢ)` has the odd-primary dual's shape
+   (with `2ⁱ` powers, not `pⁱ`), and mod τ the X-matrix constraint collapses from
+   `S(X) ≤ R₁` to `S(X) = R₁` — exactly the classical admissible matrix. The only
+   additions over the classical kernel are the second matrix `Y` (the `τᵢ`/Q-part
+   interaction) and the `2ⁱ` weights. The full `A_C` product (with τ, needed only by
+   the Phase-2 lift) needs both matrices plus the τ-power.
+
+## Validation
+
+Validate any kernel bit-for-bit against `multiply_closed`, which is itself checked
+exhaustively against the duality oracle `multiply` in the crate's tests. Start from
+the classical admissible-matrix X enumeration, add the Y enumeration and the weight
+table.

ext/crates/algebra/src/algebra/mod.rs

@@ -18,6 +18,9 @@ pub use field::Field;
 pub mod milnor_algebra;
 pub use milnor_algebra::MilnorAlgebra;
 
+pub mod motivic;
+pub use motivic::{MotivicMilnorAlgebra, Tau};
+
 mod steenrod_algebra;
 pub use steenrod_algebra::{AlgebraType, SteenrodAlgebra};