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

[JoeyBF]

This is the foundation layer of a larger effort to compute the C-motivic Adams E₂ page by deformation. The full work is being split into a stack of independently reviewable PRs; this one has no dependencies and touches only the algebra crate.

What it adds

• motivic::tau — the coefficient ring F₂[τ] as a small homogeneous scalar (Tau). Everything in the motivic world is weight-homogeneous, so every structure constant is a single power of τ. Tau encodes that as a one-integer valuation (not a polynomial), and carries the whole τ-tower so it never has to be threaded through the resolution engine.

• motivic::milnor — MotivicMilnorAlgebra, i.e. A_C presented as a free F₂[τ]-module on the Milnor basis Q(E)P(R). The product is computed two independent ways:

  • a duality oracle — dualize the coproduct ψ of the dual algebra A_∗∗ (correctness-oriented, obviously-correct);
  • the closed-form product of Kong–Lin, Product formulas for motivic Milnor basis (arXiv:2411.12890), Theorem 5.1, specialized to the C-motivic point ρ = 0 (fast).

  The fast path is validated exhaustively against the oracle over a range. This is deliberately not an Algebra impl — that trait is over F_p; this is the F₂[τ] engine the rest of the deformation builds on.

What it deliberately does not do

Nothing here wires into the resolution engine. The mod-τ reduction A_C/τ (the connected finite-type F₂-algebra the engine actually resolves, giving the algebraic Novikov E₂) is a small Algebra impl layered on top of this engine in the next PR, so this one stays a self-contained, heavily-tested combinatorial core.

Docs

MOTIVIC_PLAN.md lays out the whole deformation approach (Cτ ↔ algebraic Novikov, the F₂[τ]-Bockstein SS, the three anchors); MOTIVIC_GPU_HANDOFF.md documents the product engine's batching boundary for a future GPU port.

Tests

rewrite_tau identities, product associativity, weight-homogeneity (product weight − τ-power is additive), and closed-form-vs-duality agreement across a bidegree range. cargo test -p algebra --lib motivic — 22 passing.

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[JoeyBF]

Closing — reopened against upstream at SpectralSequences#266 (the docs-build fix for the removed ctau doc-links is included there).

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