fieldbit

Fieldbit: the unit of analog information. Not a bit but a truth value in a topos: a sieve over a fragment (the subfragments where a proposition holds), with interior, closure and boundary. A bit is the degenerate case with empty boundary; analog processing is gluing fieldbits by crossing their boundaries.

Exact symbolic layer for sheaf-theoretic contextuality on finite sites: sites, empirical models (states), the boundary D = -ln NCF computed as a linear program with an exactly verified certificate, bi-Heyting sieves (¬, ∼, ∂), Lawvere–Tierney modalities, phase cochains, and the operations restrict, union, tensor, identify, compose. The composition laws proved in the accompanying notes are shipped as tests that must pass in exact arithmetic.

Supporting code for: Contexts as covers: boundary, naturality and non-compositionality in a linguistic topos (I. M. Ozcáriz Arraiza, 2026), section "An arithmetic for the boundary".

What it computes

• Site, State (entries as Fraction or symbolic expressions in a noise parameter).
• Frontier.exact() — the non-contextual fraction NCF as an exact rational with a verified dual certificate. Backends: pure rational simplex (exact), Sage MixedIntegerLinearProgram(solver="PPL") (ppl), or a numeric proposal (HiGHS) that is rationalised and verified (scipy). auto picks PPL inside Sage.
• Frontier.certify(primal, dual, param, domain) — proves a law in the parameter as a polynomial identity.
• Sieve with neg (Heyting), coneg (co-Heyting), boundary; Modality.open/closed (Lawvere–Tierney).
• Cochain (Z_m phases): induced 1-cochain, torsion, phase state.

Laws shipped as tests (all exact)

test	value	law
CHSH, p = 1/10	NCF = 1/5	n·p/2
union of two cycles	1/5 = min	D(⊔) = max
tensor	2/25 = product	D(⊗) = sum (general)
one shared observable	1/40 = (5/8)(1/25)	identification term ln(8/5)
Mermin star composed with itself, symbolic p	NCF(e∘e) = p, certified for 0 < p ≤ 2/3	halving (D ≥ D + ln 2)
CHSH gluing sieve	(#S, #∂S, #¬S) = (15, 15, 0)	degenerate boundary
uniform Z₈ cochain on the 4-cycle	torsion 8; Tsirelson correlators	2n divides m
closed / open modalities on CHSH sieves	Lawvere–Tierney axioms	—

Run: python tests/run_tests.py (13 tests). Inside a Sage notebook, run preparser(False) first or use sage -python; the code is plain Python and robust to Sage integers.

tl_check.py verifies, in exact arithmetic on planar diagrams, that a category of returns with a positivity axiom is Temperley–Lieb and that positivity on the Jones–Wenzl tower selects δ = 2cos(π/n) ∪ [2,∞) (Jones 1983).

Install

pip install -r requirements.txt
python tests/run_tests.py

Status

Prototype (v0.1). Exact and tested; not optimised. Frontier via the pure rational simplex is fine up to ~128 global assignments; larger scenarios use the verified numeric proposal.

Licence

MIT. Cite via CITATION.cff.