Where the Proxies Begin to Diverge

Nonlinearity is the boundary

Every proxy in Part Two depends, one way or another, on superposition. Gravity does not superpose: the field carries energy and energy gravitates, so strong fields source themselves. This is why merger waveforms cannot be obtained from any linearised scheme and require full numerical relativity, and why the GEM circuit picture, the metamaterial lattice, and the fluid picture all simply stop having anything to say about the last orbits of a binary.

A merger of supermassive black holes — including the multiple-body events now being discussed in the literature — is the clean case. Horizons merge, apparent horizons form and vanish, the geometry itself is dynamical. There is no medium description of that. Any statement about such an event framed in constitutive parameters or in a fluid analogy is outside its band by many orders of magnitude.

The other boundaries

The Madelung picture loses its footing as soon as more than one particle is involved, since the configuration space leaves three dimensions. Metamaterial band structure assumes periodicity and linear response, and gravitational wave coupling is far too weak for the proposed lattices. Analogue gravity has no dynamics. Holography has no established de Sitter version. Each of these is a specific, nameable edge, and the value of naming them is that a reader can check whether a given claim sits inside or outside.

The absence of a warning bell

The recurring danger is that none of these approximations announces its own failure. A linearised calculation returns a number in the strong-field regime; it is simply the wrong number. This is why the discipline has to be procedural — stated at adoption — rather than diagnostic after the fact.