A Pre-Registered Test of a Void-Fraction Transmission Hypothesis in a p6m Gross–Pitaevskii Supersolid: A Negative Result

A proposed mechanism for gravity treated it as the secondary Bjerknes force between pulsating topological defects in a superfluid vacuum with a triangular (p6m) lattice. The mechanismrested on one hypothesis: that the medium transmits a fixed fraction ζ of a pulsation amplitudeacross each lattice layer, with ζ equal to the void fraction of the densest planar circle packing,1 − π/√12 ≈ 0.0931. We tested this hypothesis, with every condition registered in advance,on a concrete model: a two-dimensional soft-core Gross–Pitaevskii condensate whose groundstate is a p6m supersolid. The channel registered in advance as carrying the pairwise coupling, the quasi-static density channel, is gapless and transparent: its per-layer transmissionis t → 1. The spectrum’s genuine stop bands give in-gap per-layer factors t = 0.36, 0.54 and0.75 at midgap; the closest is 3.9 times the target. The original code was lost. A reconstruction written from the surviving documents reproduces the ground state, the band structureand the verdict; it gives 0.49 for the first gap (5.2 times the target) and agrees within a fewpercent on the other two. The hypothesis is not realized in this model. We also record threeresults that do not depend on the test. (i) p6m symmetry makes the long-wavelength responseexactly isotropic and forbids odd (Hall) viscosity. It does not forbid ordinary shear viscosity.(ii) The time-averaged Bjerknes force between pulsators of different frequencies vanishes, soa Bjerknes gravity needs one common drive. (iii) The relation α_G = ζ^(2n_C), with n_C fixedby the proton’s Compton wavelength, holds for every ζ ∈ (0,1). It is a cautionary identity thatcarries no information about ζ. We close with three physical requirements a Bjerknes gravitywould still have to meet. The negative result concerns one model and one coupling channel;it makes no claim beyond them.