Kinematic mimicry without dynamical constants: three obstructions to a condensate origin for de Sitter quantum cosmology

The Lorentzian no-boundary path integral predicts unsuppressed cosmological fluctuations — the Feldbrugge–Lehners–Turok (FLT) catastrophe. A recurring hope is that a microscopic substrate, in particular a Bose–Einstein-condensate (BEC) analogue with a finite healing length ξ, might regularize the integral, single out the de Sitter scale, or generate the Bekenstein–Hawking entropy coefficient 1/4. We show, with one theorem and three independent computations, that this hope fails in a sharp and unified way. First, we prove a conjugate reality lock: because the minisuperspace action has real parameters, conjugate saddles carry exactly antisymmetric fluctuation exponents, so no real perturbation-sector modification—higher-derivative or otherwise— can flip the catastrophic sign. The locus a = 0 is exposed not as a dynamical obstruction butas a non-Cauchy (branch-point) surface on which one-sided data is ill posed. Second, we show the unique unitarity-preserving exit is saddle coalescence: moving the boundary to the de Sitter throat q = 1/H2 merges the conjugate pair onto the real axis (Im N = √1 − q0 in H=1 units), restores a regular Euclidean cap, and provably selects the suppressed Bunch–Davies vacuum—but at a scale fixed by gravity (1/H), not by the substrate (ξ). Third, using a gated covariancematrix method (recovering c = 1 in 1+1D), we demonstrate that the area-law coefficient of the Bogoliubov vacuum is non-universal: it drifts with system size and regulator at measured rates dα/d lnL = −0.041 and dα/d ln m = −0.009, approaches 1/4 only logarithmically and only for ξfar above the cutoff, and therefore cannot encode a scheme-independent constant. An apparent “2 × 0.125 = 1/4” coincidence is shown to be an artifact of corner subtraction and a miscount of the condensate’s degrees of freedom. We argue these are three faces of a single diagnostic: a condensate faithfully reproduces the kinematic skeleton of horizon thermodynamics (area law, Bunch–Davies vacuum, √N geometry, and—via induced gravity—the dimensionless 1/4 ratio) but is structurally incapable of generating its one dynamical input, the absolute scale N.