A new π* kernel that canonicalizes pure-integer counting
expressions and FAILS CLOSED on any input whose result isn't a
non-negative sp.Integer. Tighter domain than algebra-symbolic@v1,
which already accepts the same input surface but happily returns
symbolic / negative / non-integer outputs.
Distinguishing feature versus algebra-symbolic@v1:
algebra-symbolic@v1: binomial(n, k) → "binomial(n, k)" (symbolic
passthrough)
combinatorics@v1: binomial(n, k) → PiStarError (fail-closed
on free-symbol output)
algebra-symbolic@v1: binomial(Rational(1,2), 3) → 1/16 (rational)
combinatorics@v1: binomial(Rational(1,2), 3) → PiStarError
(output not Integer)
Boundary kept explicit: binomial(-3, 2) = 6 IS accepted because the
output is an integer 6. The fail-closed rule is on output shape
(Integer ≥ 0), not input range. Documented as
test_generalized_binomial_negative_args_accepted_when_integer.
Output format: plain decimal literal (b"10", b"5040"). Composes
with arithmetic@v1 for byte-identical agreement with the rational
route (b"10/1") so the multi-modality witness (#000028) can pin
equivalence-class agreement when both routes fire on the same
question.
Allowed surface (via SymPy primitives): binomial, factorial, ff /
rf (falling/rising), catalan, bell, partition, stirling, plus
arithmetic compositions over those primitives
(3*binomial(5,2) + factorial(4) = 54).
Coverage:
- 43 unit tests including binomial symmetry C(n,k)=C(n,n-k),
Pascal's rule C(n,k)=C(n-1,k-1)+C(n-1,k), the C(n,k) =
factorial(n)/(factorial(k)·factorial(n-k)) identity,
fail-closed paths (symbolic/negative/non-integer/relational/
parse), round-trip idempotence, composition with arithmetic@v1.
- 10 syntax + 12 semantics bench fixtures, 100% pass.
- bench/batteries/base.py PHASE_1_CARRIERS gains "combinatorics".
- Makefile bench-5s-combinatorics target.
All gate on pytest.importorskip("sympy") so a sympy-less suite
stays green. Full make test: 1537 passed / 28 skipped.
Sequencing rationale honored: this kernel lands FIRST so that
#000033 (claim-pack pillar VII for combinatorics) can bind its
records to the tighter integer kernel from day one — avoids
rebind churn on pi_star_ref fields.
russell@unturf.com
7b7ac3867d