No Scalar Companion: A Second No-Go Theorem for Mechanically Sourced Gravity in Logarithmic Superfluid Vacuum

A companion paper showed that a single-component logarithmic superfluid vacuum gives electromagnetism but not gravity, and pointed to the obvious fix: add a second field with a gapless, compressible sector. This paper takes that fix seriously and shows it fails, at least for any scalar companion coupled at leading order. There are only four ways such a field B can couple to the condensate A at lowest order — through the density, the energy-momentum, the current, or by direct conversion between the two fields — and I work through each. Every one fails, and for a reason I can put a number on. Density coupling breaks the equivalence principle outright. The logarithmic nonlinearity has a scale anomaly, E/N = e_n − b ln(N/N₀), so a soliton's inertial mass and its density charge simply aren't proportional. I check the anomaly numerically to a few parts in 10⁷ and measure the spread across vortex windings, e₃ − e₀ = 2.22b, which also reproduces the soliton spectrum of the collective-gravity paper as a byproduct. Energy coupling is the interesting near-miss. It gives the right light bending — I find γ = −α, which is exactly +1 at the logarithmic background value α = −1 — but it destroys its own medium. The Planck-dense background, once the coupling is fixed to reproduce Newton's constant, polarizes into a runaway self-attraction of B with |c_B²|/c² = π(m_P/m_B)², and there is no free parameter left to tune it away. Derivative coupling avoids that trap by construction, but the same shift symmetry that protects it also means static matter produces no source at all: a total-derivative identity leaves nothing for the field to respond to. Conversion coupling turns out to be a Josephson term. It gaps the very sector it was supposed to keep gapless, and the enthalpy it can hand to the flow is locked to the square of the conversion rate the flow demands. The two requirements collide at a radius r_max = 9ℏ²/(32GMm²), around 10⁻⁷⁴ m for a solar mass — some eighty-odd orders short of any relevant scale, and independent of the second field's density. One thing is behind all four failures: the vacuum's background density is Planckian, and that enormous density turns each coupling against the job it was meant to do. Whatever sources gravity in these models, it is not a linearly coupled scalar field.