Finite-refinement evasion of the sonic horizon

Prove or refute that transport along a sufficiently refined path of many small re-gauging steps reaches every subsonic endpoint admissibly, thereby determining whether the sonic horizon is solely a finite-step phenomenon.

Background

The paper establishes an exact sonic horizon for two-step transport: when the endpoint loses admissibility, the conformal factor tends to zero and the frequency dilation diverges. For equal collinear steps, this occurs at the inverse golden ratio Mach number. The authors conjecture that this degeneration can be avoided by replacing a finite large-step path with sufficiently many smaller steps connecting the same subsonic endpoints. The continuum limit supports this possibility because its frequency factor remains finite, but the authors leave the claim unresolved for arbitrary finite refinements.

References

Transport along a sufficiently refined path (many small steps) reaches every subsonic endpoint admissibly; the horizon eq:horizon is a finite-step phenomenon. This is consistent with the continuum limit of \S\ref{sec:curvature}, in which the frequency factor integrates to the finite value $\gamma(u_b)/\gamma(u_a)$, but a proof for arbitrary finite refinements is open.

— A transport geometry of acoustic analogies:exact holonomy of source re-attribution and its observable consequences  (2608.12031 - Sharma, 12 Aug 2026) in Conjecture 1, Section 3.4, subsection “Degeneration of transport: a sonic horizon in analogy space”