Numerical controls, paths and Itô integrals

These checks ask whether a result responds to its numerical controls as it should, and whether the inputs a solver builds for itself are right. They run the code, so use them on trusted code or in the sandbox.

Check	Property	Typical bug it catches
steps.check_step_independence	A quantity at a fixed physical time settles as dt shrinks.	A velocity or rate used without its factor of dt.
steps.check_tolerance_response	An adaptive solver's error falls as its tolerance is tightened.	A boundary value frozen during the Runge-Kutta stages, which the error estimate cannot see.
stochastic.ito.check_ito_integrals	Generated iterated integrals satisfy exact Itô identities.	A wrong term in the formula for I_ijk.
operators.check_path	A built wire or path has the expected segment lengths and closes if it should.	A helper that continues from the wrong vertex.

from physverify.steps import check_step_independence, check_tolerance_response
from physverify.stochastic.ito import check_ito_integrals
from physverify.operators import check_path

check_step_independence(lambda dt: msd_at_T(dt), [0.1, 0.05, 0.02])     # or return (value, standard_error)
check_tolerance_response(lambda tol: max_error(tol), [1e-6, 1e-8, 1e-10, 1e-11], floor=1e-13)
check_ito_integrals(lambda p: generate(p), (10, 40))                    # generate returns (W, I2, I3)
check_path(vertices, segment_lengths=1.0, closed=True)

Notes

• check_step_independence passes a code that is still converging, as long as the changes between steps shrink. It reports value ~ dt^k when they do not. With few Monte Carlo samples the margin is wide, so a small dependence on dt can pass; it is meant for errors of scale.
• check_tolerance_response needs an error from time stepping alone. Use a problem the spatial scheme solves exactly: u = x sin t solves u_t = u_xx + x cos t with u(0, t) = 0 and u(1, t) = sin t, and it is linear in x, so a central stencil has no spatial error. Tighten atol along with tol, or atol sets the error instead. The required slope, 0.5, is below the p/(p+1) expected of an order p method.
• check_ito_integrals cannot tell a formula error from slow truncation at a single series length, so give two lengths a factor of 4 or more apart. An identity still above rtol that has not fallen by 25 % is reported. The isometry test has low power on small samples: with 2000 samples, scaling one I_ijk by 0.8 was not detected by this test, while scaling it by 0.6 was. Use more samples when one index family is in doubt.

Measured false alarms

On correct code simulated for the purpose, with new random numbers each time:

Check	Correct code	False alarms
check_ito_integrals	left-point sums on 100 and 400 substeps, two noises, 2000 samples	2 of 300
check_step_independence	active Ornstein-Uhlenbeck particle, 1000 samples, dt = 0.1, 0.05, 0.02	0 of 300

These figures apply to these settings only.

Where these came from

In the corpus session of 2026-11-06 the first four reproduced findings first made by hand: a mean squared displacement behaving as dt^-2.06, an RKF error following tol^0.18 on a moving boundary (tol^0.92 with fixed boundaries in the same solver), a multi-noise I_iii residual of about 0.08 that did not fall from series length 10 to 20, and a square coil with a segment of length 1.414. On the same repositories they passed Brownian motion, active Brownian particles, the one-noise and Milstein integrals, and a circular loop.