slogdet_specialization races factor-sharing rewrites inside the specialize equilibrium #2477

slogdet_specialization and the factorization-creating rewrites run in the same specialize equilibrium, and the winner on a det node depends on node visit order. When slogdet_specialization reaches the det before reuse_decomposition_multiple_solves has created the factor, the det becomes an SLogDet and det_of_matrix_factorized_elsewhere has nothing left to match, so the matrix is factored twice. Output order alone flips the outcome:

import pytensor
import pytensor.tensor as pt

A = pt.dmatrix("A")
logdet = pt.log(pt.abs(pt.linalg.det(A)))
b1, b2 = pt.dvector("b1"), pt.dvector("b2")
x1 = pt.linalg.solve(A, b1, b_ndim=1)
x2 = pt.linalg.solve(A, b2, b_ndim=1)

pytensor.function([A, b1, b2], [logdet, x1, x2])  # LUFactor and SLogDet: A factored twice
pytensor.function([A, b1, b2], [x1, x2, logdet])  # LUFactor only, det shares the factor

A fix that does not depend on ordering is the missing counterpart rewrite: track SLogDet and share an existing factorization of its input into (sign, logabsdet). Cholesky gives sign = 1 and logabsdet = 2 * sum(log(diag(L))); LU needs the pivot parity for the sign. Either firing order then converges to one factorization.