Motivic foundation: A_C, the C-motivic Steenrod algebra engine over F₂[τ]#266
Motivic foundation: A_C, the C-motivic Steenrod algebra engine over F₂[τ]#266JoeyBF wants to merge 4 commits into
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Foundation layer for computing the C-motivic Adams E₂ by deformation. Adds the coefficient ring and the product engine, with no wiring into the resolution engine yet — the mod-τ reduction that the engine resolves comes in a follow-up. - `tau`: F₂[τ] as a small homogeneous scalar (`Tau`). Every structure constant in the motivic world is a single power of τ, so coefficients are a one-integer valuation rather than a polynomial; this type carries the whole τ-tower. - `milnor`: `MotivicMilnorAlgebra` = A_C as a free F₂[τ]-module on the Milnor basis Q(E)P(R). The product is computed two ways — a duality oracle (dualize the coproduct ψ) and the closed-form Kong–Lin Theorem 5.1 (arXiv:2411.12890, ρ = 0) — and the fast path is validated exhaustively against the oracle. It is intentionally not an `Algebra` (that trait is over F_p); it is the F₂[τ] engine the deformation lift builds on. `MOTIVIC_PLAN.md` documents the overall deformation approach; `MOTIVIC_GPU_HANDOFF.md` documents the product engine's batching boundary. Tests: rewrite_tau identities, product associativity, weight-homogeneity, and closed-form-vs-duality agreement over a range. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_013ePtYD7Bt4iPeCtmqtqvZE
The two [`super::ctau`] intra-doc links referenced a module that isn't part of this foundation PR, breaking the docs build. Reword to plain prose. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_013ePtYD7Bt4iPeCtmqtqvZE
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Remove MOTIVIC_PLAN.md (a working design doc, not repo-durable) and cut the GPU handoff from a branch-specific handoff brief down to the two engine facts that actually inform a kernel port (F₂ atomic-XOR output with a host-side τ-power lookup; the mod-τ product is odd-primary-Milnor structured) plus where the batch boundary is (`fill_block` / `ProductBlock`). Drop the prose reference to the removed plan doc. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_013ePtYD7Bt4iPeCtmqtqvZE
- `tau.rs`: `[`fp`](fp)` is a redundant explicit link (the label `fp` already resolves to the crate), which `-D rustdoc::redundant-explicit-links` (implied by the docs job's `-D warnings`) rejects. Drop the explicit target. - Run nightly `cargo fmt` for the `lint` job: the repo's rustfmt.toml enables unstable options (reorder_impl_items, group_imports, format_strings) that stable rustfmt silently skips, so the motivic files were never formatted the way CI checks. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_013ePtYD7Bt4iPeCtmqtqvZE
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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
algebracrate.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.Tauencodes 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:The fast path is validated exhaustively against the oracle over a range. This is deliberately not an
Algebraimpl — 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
Algebraimpl layered on top of this engine in the next PR, so this one stays a self-contained, heavily-tested combinatorial core.Docs
MOTIVIC_GPU_HANDOFF.mddocuments the product engine's batching boundary for a future GPU port.Tests
rewrite_tauidentities, 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.🤖 Generated with Claude Code
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