Self-consistency fixed point
The kinematic arm (given a flow, derive the metric via analog gravity) and the dynamic arm (given fluid equations, derive the flow via Euler) converge on the same result: v_ebb = √(2GM/r). This is a fixed-point argument, not circular reasoning.
The self-consistency argument
Not circular reasoning — a fixed-point proof
Kinematics
Given v_ebb(r), the acoustic
metric IS Schwarzschild
Dynamics
Euler + continuity produce
v_ebb = √(2GM/r)
same flow
same metric
Analog gravity (Unruh 1981)
Mathematically rigorous
v·dv/dr = −GM/r²
Convective acceleration
Fixed point
The flow that creates the metric IS the flow
the substrate's equations of motion produce
Exact at linear order
Nonlinear: open problem
Einstein's equations are not postulated — they are the self-consistency condition