ogdoad

Ogdoad is a pure-Rust library for Clifford algebras, quadratic forms, and related arithmetic over exact, finite, local, transfinite, and game-adjacent coefficient systems. Optional PyO3 bindings expose selected concrete backends without entering the generic math core.

The library supports degenerate metrics. In characteristic two it keeps the quadratic and polar data independent:

e_i^2             = q_i
e_i e_j + e_j e_i = b_ij

An optional upper-triangular a_ij records ordered contraction for a general bilinear metric. Generic product code obtains signs through Scalar::neg and does not special-case the characteristic.

Scope

A Clifford scalar is a commutative ring. Arbitrary partizan games form only an abelian group under disjunctive sum, so Ogdoad is not a Clifford algebra over all games. The scalar pillar includes field-like game subclasses such as finite nimbers and represented surcomplex numbers; genuinely game-theoretic constructions live in the separate games pillar.

Representation limits are part of the API:

Nimber(u128) is F_(2^128), not the algebraic closure of F_2.
Surreal stores finite-support Conway-normal-form expressions, not arbitrary surreal series.
valued and local models have explicit precision caps and do not silently claim exact field laws outside them.
Ordinal::nim_mul, nim_pow, checked_inv, and checked_sqrt preserve the supported Kummer-tower boundary through Option; the Scalar multiplication wrapper panics if that checked boundary is ignored.
fixed-width mathematical payloads use u128 or i128; usize is reserved for dimensions, indices, and ABI hooks.

Architecture

path	role
`src/scalar/`	coefficient traits and exact, finite, valued, global, surreal, and ordinal backends
`src/clifford/`	metrics, blades, multivectors, products, versors, spinors, and geometric-algebra constructions
`src/forms/`	quadratic-form classification, Witt/Brauer theory, Springer and local--global arithmetic
`src/forms/integral/`	lattices, discriminant forms, codes, theta series, genera, neighbors, and Weyl bridges
`src/games/`	impartial, partizan, misere, loopy, thermographic, Witt--FIFO/Brown, octal-certificate, Hackenbush, and game-exterior constructions
`src/py/`	optional per-backend PyO3 bindings; scalar worlds never mix at runtime
`src/linalg/`	crate-private shared linear algebra
`grundy/`	unpublished expression-language workspace crate using only Ogdoad's public API
`formal/`	separately pinned Lean 4 development
`writeups/`	current mathematical papers and BibTeX sources

Public pillars re-export their children shallowly. Each source pillar has an AGENTS.md with its local invariants; docs/README.md indexes the current project documentation.

Use

Rust:

cargo add ogdoad
cargo run --example tour

Python 3.9 or newer:

python -m maturin build --profile dev -i python
python -m pip install --force-reinstall --no-deps target/wheels/ogdoad-*.whl

import ogdoad as og

# q and b are independent in characteristic two.
A = og.NimberAlgebra(q=[og.Nimber(2), og.Nimber(3)], b={(0, 1): 1})
e0, e1 = A.gen(0), A.gen(1)
assert e0 * e1 + e1 * e0 == A.scalar(og.Nimber(1))

# Exact finite-support surreal monomials.
S = og.SurrealAlgebra(q=[og.omega(), og.epsilon()])
assert (S.gen(0) * S.gen(1)) ** 2 == S.scalar(og.Surreal.from_int(-1))

# Checked game constructors preserve their proof and validation boundaries.
arena = og.WittFifoArena(diagonal=[True], polar=[0], input=1)
assert arena.quadratic_value and arena.grundy(state_budget=100_000) != 0

selector = og.BrownSelector(q4=[3], brown_polar=[0], input=1)
outcome = selector.outcome_class(state_budget_per_follower=100_000)
assert og.BrownSelector.decode_outcome(outcome) == selector.residue

code = og.OctalCode([2])
certificate = og.GuySmithCertificate.compute(code, 1, 2, term_budget=16)
assert certificate.heap_grundy(10**30) == 1

The Python layer monomorphizes a documented slice of the Rust backends. It does not provide a runtime-tagged any-scalar algebra. Its typed report surface includes finite quadratic modules and Nikulin criteria, extraspecial and Heisenberg--Weil objects, function-field Brauer--Wall classes, Niemeier data, finite-field Witt decompositions, lexicode turning games, conformal-algebra accessors, represented ordinal finite-subfield degrees, checked Witt--FIFO and Brown constructors, and sealed Guy--Smith periodicity certificates. Python repr delegates to canonical Rust rendering where the core provides it.

Mathematical status

The papers under writeups/ form one current research suite:

paper	result
`transfinite_arf.tex`	classification over perfect Artin--Schreier-surjective characteristic-two fields and its full-nimber specialization
`goldarf.tex`	quadratic-refinement realization in normal play, Gold specialization, Brown selector, and game-exterior obstruction
`thermo_newton.tex`	thermic regrading under Norton multiplication and its separation from Newton tropicalization
`linking_affine.tex`	proved reductions and exact remaining obstruction for isolated-dummy FIFO linking
`excess.tex`	four-arm reduction and exact remaining selected-order problem for transfinite nim excess

The last two universal claims remain open. Their concise statements and sharp proof boundaries are in docs/OPEN.md. Lean checks named ingredients and several end-to-end finite constructions, not the open propositions; see formal/README.md.

Verification

cargo fmt --all --check
cargo test --workspace
cargo clippy --workspace --all-targets -- -D warnings
RUSTDOCFLAGS="-D warnings" cargo doc --no-deps --workspace
(cd formal && lake build --wfail)
npm ci
python scripts/check_writeups.py

docs/VERIFY.md records the optional Python-binding gates, exactness contracts, certificate boundary, and claim classes. Contributions should identify mathematical claims as standard/cited, implemented and tested, proved here, or open; see CONTRIBUTING.md.

Ogdoad is licensed under AGPL-3.0-or-later.

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