TUCSL Theory: Bogoliubov–Dressed Horizon Noise, Unimodular Tensorial Completion, and a Projected Einstein–Langevin Correspondence

We develop a controlled extension of the Transactional Unimodular Continuous Sponta-neous Localization (TUCSL) programme from the previously established horizon-projectednoise sector to a canonical tensorial stochastic-gravity completion compatible with unimodularclosure. The starting point is the already fixed covariant kernel backbone selected by bandage ultralocality, strong Markov coarse-graining, covariance, positivity, and no-signalling. We first construct the Bogoliubov-dressednull stress correlator in Rindler space, derive the explicit Wick-contracted form of the horizonkernel, and show that its Fourier transform obeys a thermal fluctuation–dissipation relation.We then incorporate transverse smearing and full q-smearing (r_c, ℓ_0), obtaining an explicitlow-frequency amplitude and an induced effective collapse scale λ_eff (a). This yields a closedrelation between collapse strength and stationary horizon-focusing noise.The main new conceptual step is a no-go and a replacement theorem. We show that thescalar horizon kernel Cf alone cannot determine a full tensorial Einstein–Langevin noisekernel. A naive lifting based on the scalar heat kernel fails the correct tensorial consistencytest. We then construct a canonical minimal completion on the trace-free symmetric tensorbundle over the Euclidean q-domain by projecting onto the unimodularly admissible subspace.The corresponding semigroup yields a tensorial noise kernel that issymmetric, positive, compatible with unimodular closure in the precise Hodge-projectedsense, and reduces to the previously obtained horizon kernel in the local near-horizonlimit. The outcome is not a derivation of the full nonlinear Einstein equations from RTI/QDATalone.Rather, it is a mathematically controlled “projected Einstein–Langevin correspondence” andits canonical unimodular tensorial completion. We formulate precisely what is proved,what remains conjectural, and which further steps would be needed for a full tensorialstochastic-gravity completion of the TUCSLprogramme.