  For a system with a Hamiltonian constraint, we demonstrate that its dynamics
preserves invariant with respect to the choice of the lapse function,
regardless of whether the Hamiltonian incorporates quantum corrections.
Applying this observation to the interior of black-to-white holes, we analyze
its dynamics with different choices of the lapse function. The results
explicitly show that the leading-order expansion of both metrics in Refs. [1,2]
and Refs. [3,4] exhibits identical behavior near the transition
surface,therefore in this sense the black-to-white hole model proposed in Refs.
[1,2] may be interpreted as a coarse-grained version of the solution within the
framework of loop quantum gravity. The black-to-white hole solutions with exact
symmetry between the black hole and white hole regions are constructed by
appropriately fixing the quantum parameters in the effective theory of loop
quantum gravity. This approach circumvents the issue of amplification of mass,
which could arise from a mass difference between the black hole and white hole.
Finally, the black-to-white hole solutions with a cosmological constant are
constructed. The numerical solutions for the interior of black-to-white hole
with a cosmological constant are obtained, and their symmetric behavior is also
discussed.
