Input semantics

中文

Input and arithmetic

A ChainWindow represents one fixed degree:

\[ C_{k+1}\xrightarrow{D}C_k\xrightarrow{A}C_{k-1},\qquad AD=0. \]

A has shape m×n and D has shape n×p, using column vectors. Supply ordered, unique nonempty basis identifiers for each space and n finite positive weights. Empty spaces are supported. Invalid dimensions, bases, coordinates, weights, or AD≠0 raise InvalidInput, a ValueError subclass.

from homology_operator import Matrix

matrix = Matrix.from_rows(((1, 1, 0), (0, 1, 1)))
assert matrix.kernel_basis() == ((1, 1, 1),)
assert matrix.solve((1, 0)) == (1, 0, 0)
assert Matrix.from_rows((), ncols=3) == Matrix.zero(0, 3)

Coordinates must be integer 0/1 values; booleans, floats, truncation, and implicit reduction modulo 2 are rejected. solve(b) returns a deterministic solution with free variables set to zero, or None when no solution exists. A valid empty solution is ().

arithmetic	Accepted weights	Geometry
ExactInteger	Arbitrary-precision integers	Exact; objective ratios use Fraction
ExactRational (default)	Integers and fractions.Fraction	Exact rational arithmetic
FloatingPoint	Integers, floats, fractions converted to finite positive binary64	Numerical fsum; no exact geometric optimality certificate

The caller declares weight_semantics and unit. Positive costs do not imply that weights are areas or volumes. Original coordinates determine geometry.