Correspondence Limits

The principal correspondence limits provide consistency checks on the architecture:

1. Ordinary quantum mechanics. On a type-I detector algebra, normal algebraic states and instruments reduce to density operators, POVMs and completely positive trace-preserving maps.

2. Flat-space QFT. When the reconstructed curvature and backreaction are negligible on the region of interest, the locally covariant theory reduces to its Minkowski-space description.

3. Semiclassical gravity. Retaining the constant-subtracted, state-dependent QFT stress tensor on a classical metric while suppressing record and higher-curvature corrections gives, for disjoint classical and quantum sectors,

   $$G_{\mu\nu}+\Lambda_{\rm br}g_{\mu\nu} =8\pi G_R\left(T_{\mu\nu}^{M,\rm dyn} +T_{\mu\nu}^{\rm QFT,dyn}\right).$$

   Here $S_M^{\rm dyn}$ and $\Gamma_{\rm QFT,dyn}^{\rm ren}$ contain disjoint operators; pure-metric local QFT terms have already renormalized $\Lambda_{\rm br}$, $G_R$, and $\Gamma_{\rm high}^{\rm grav}$. If a species is treated fully quantum mechanically, its term is omitted from $S_M^{\rm dyn}$.

4. Classical general relativity. When quantum expectation values admit a classical matter limit and all higher-curvature and record corrections are negligible, the Einstein equation is recovered.

5. Standard cosmology. For the admitted FLRW branch with negligible nonstandard corrections, the background and perturbation equations reduce to the usual Friedmann and gauge-invariant SVT systems.

6. Classical records. Strong decoherence, stable redundant encoding and long persistence make the objective-record description approach an effectively classical event history, without identifying decoherence with selection.

These limits are requirements on successful branches, not definitions imposed on the source substrate.