Replication package for "Finite-perturbation bounds on diffusion path space: closed-form Gram matrices, an exactly solvable supremum, and the cost of dynamical stability"

Abstract

Eleven Python scripts reproducing every numerical claim in the paper"Finite-perturbation bounds on diffusion path space: closed-form Grammatrices, an exactly solvable supremum, and the cost of dynamicalstability". The paper extends the Barankin multi-point bound to the path space of adiffusion. Completing the square on two Cameron-Martin densities convertsthe Gram matrix of tilted path measures into a Feynman-Kac functional,which is exactly solvable for a stationary Ornstein-Uhlenbeck process anda time-averaged quadratic observable. Linear response vanishes identicallyalong constant-force perturbations; the optimal finite tilt sits atz* = 2 + W0(-2e^-2) = 1.593624, the Barankin supremum exceeds the bestsingle-point bound by the parameter-free constant 2/h(z*) = 3.088277, andthe supremum is attained to five decimal places by three test points. Contents: barankin_sup.py the supremum, the ratio 2/h(z*), its independence of the model parameters, the m-saturation table, and the recovery fractions (deterministic) cone_table.py exact bounds and Gram condition numbers on the symmetric and one-sided node sets (deterministic) ou_optimal.py the stationarity condition e^z(2-z) = 2 and z* exactQ.py the finite-window recovery fraction and its split-sample simulation check sampling.py the discrete-sampling factor tanh(x/2)/(x/2), derived as an exact identity and confirmed over eight seeds girsanov_check.py the Gram closed form, checked against the closed form for constant tilts and against the Feynman-Kac side for space-dependent tilts, with Monte Carlo standard errors reported alongside the residuals chi1_cubic.py the exact finite-window susceptibility on the cubic family, by Richardson extrapolation in the perturbation amplitude inaccessible.py the conditioning rates, the node-spacing comparison, and the Nelder-Mead search under two different linear solvers affine.py the affine test-point family and its Gram correlation gyrator.py } the quadrupolar shear on the Brownian gyrator: these chi2_gyrator.py } are expected to diverge, which is the point made in Section 8, and their NaN output is the result rather than a defect stable_blind.py superseded by chi1_cubic.py, retained as a record All scripts are deterministic: each carries an explicit random seed andeach was run twice and compared byte for byte. Captured output for everyscript is in logs/. STATUS.md maps each claim in the paper to the scriptthat produces it and records the limits of what is reproducible, includingone row of Table 4 whose Gram condition number exceeds the reciprocal ofdouble precision and is therefore not portable across linear algebraimplementations. Requires Python 3.11, numpy 2.4, scipy 1.17. Total runtime is about tenminutes on one core.