Closes the math half of fox's 2026-05-08 roadmap question ("did we
implement the math and logic stuff?"). Until now arborist had logic
KERNELS (Syllogism, Truthtables, Transitivity sub-batteries with
deterministic evaluators) but no math π*'s. SQD whitepaper §14
framed math as "operations over π*-canonical invariant objects:
integers, algebraic expressions, proof states, constraint graphs"
— that framing is now actually implementable.
arithmetic@v1 — exact rational arithmetic (SQD §14.1)
-----------------------------------------------------
Parse arithmetic expression → fractions.Fraction → canonical
"<num>/<den>" in lowest terms. Decimals via Decimal(str(...)) for
exact rational interpretation:
"0.1" → "1/10" (not 0.1±ε)
"0.1+0.2" → "3/10" (the SQD-canonical floating-point question)
"1+2" → "3/1"
"(1+2)*3" → "9/1"
"6/4" → "3/2" (lowest terms via Fraction invariant)
"2**3" → "8/1"
"-(1+2)" → "-3/1"
Rejects: identifiers, function calls, division by zero, non-integer
exponents, boolean literals. PiStarError with explanatory message.
logic-kernel@v1 — propositional CNF canonicalizer (SQD §14.3)
-------------------------------------------------------------
Parse Boolean expression (AND/OR/NOT/XOR/IMPL/IFF + named atoms +
parens) → eliminate IMPL/IFF/XOR → push NOT inward (NNF) →
distribute OR over AND (CNF) → dedupe + sort literals + sort
clauses + drop tautological clauses → serialize.
Equivalence classes preserved:
A AND B ≡ B AND A (commutativity)
(A AND B) AND C ≡ A AND (B AND C) (associativity)
A IMPL B ≡ NOT A OR B (IMPL rewrite)
A IMPL B ≡ NOT B IMPL NOT A (contrapositive)
NOT (A AND B) ≡ NOT A OR NOT B (De Morgan)
NOT NOT A ≡ A (double negation)
A OR (B AND C) ≡ (A OR B) AND (A OR C) (distribution)
A OR NOT A → TRUE (tautology)
A AND A ≡ A (idempotence)
Atom cap: N=8 (256 max clauses). Larger inputs raise PiStarError;
CNF blow-up is exponential in atom count, the cap keeps
canonicalization deterministic in bounded time.
Surface
-------
- arborist/pi_star/arithmetic.py — full implementation (new file)
- arborist/pi_star/logic.py — full implementation (graduates from
stub; preserves the LogicKernelV1 dataclass shape so the registry
key arithmetic@v1 / logic-kernel@v1 stays stable)
- arborist/pi_star/__init__.py — imports arithmetic to auto-register
- bench/batteries/base.py — PHASE_1_CARRIERS adds "arithmetic" + "logic"
- bench/fixtures/5s/syntax-arithmetic-v1.jsonl (12 fixtures)
- bench/fixtures/5s/semantics-arithmetic-v1.jsonl (15 fixtures)
- bench/fixtures/5s/syntax-logic-v1.jsonl (12 fixtures)
- bench/fixtures/5s/semantics-logic-v1.jsonl (18 fixtures)
- Makefile: bench-5s-arithmetic, bench-5s-logic-kernel,
bench-5s-math aggregate
- tests/test_pi_star.py: 30 new tests
- 13 for arithmetic@v1 (decimals, equivalence, rejects, idempotency,
integer exponents, negative results, the SQD 0.1+0.2 case)
- 17 for logic-kernel@v1 (commutativity, associativity, all rewrite
rules, De Morgan, double-neg, distribution, tautology, idempotence,
contrapositive, atom cap, syntax errors, idempotency)
Cross-modality discipline now spans:
text + claim_lattice + memory + code + arithmetic + logic
— six carrier domains, all with real π* implementations.
Two stubs remain (time-series-quantized@v1, tabular-pinned@v1);
neither is needed for math/logic coverage.
Full suite: 1256 passed, 36 skipped (+30 tests, +57 fixtures).
russell@unturf.com
e4ecf89621