A fixed lensing growth relation, ∆Σ = 2/3 ∆µ, in scalar tensor cosmology. An HDC-CBC interpretation

In the effective field theory (EFT) of dark energy the modifications of gravitational lensing () and of structure growth () are usually treated as two independent functions. We show that a single, parameter-free relation, , holds exactly along one locus of scalar–tensor gravity: in the quasistatic sub-horizon regime with luminal gravitational waves () the ratio of deviations is , and the value corresponds to the condition on the EFT functions (Planck-mass run rate equal to braiding), equivalently the Lagrangian condition . On this locus the growth, lensing and slip deviations are locked (to first order) in the fixed proportion — a triple lock — and we exhibit an explicit ghost-free, gradient-stable realisation of the locus; the correlational sector’s own quantisation is established exactly to be ghost-free and single-mode (its complete map onto this covariant realisation, via the still-generic , remains open). Within reduced Horndeski the value selects uniquely, making the relation a falsifiable, cross-observable discriminant for Euclid; we quantify its detectability and confront the three legs against current growth and lensing data, which do not exclude it (an indicative consistency, not a likelihood fit). Throughout, and denote the Planck-mass-normalised (scalar-sector) functions , ; the locked deviations are and , and the universal running is separately measurable (e.g. from gravitational-wave propagation). A direct hi_class Boltzmann integration confirms the scalar-sector to a few across the sampled sub-horizon range and to at , the bare ratio being . We keep truth ahead of advocacy throughout, separating what is established (the master ratio, the locus, the triple lock, the stable realisation) from what is assumed (a minimal single-generator coupling, stated as an explicit postulate rather than derived). Finally we give the locus a physical interpretation in the Hypothesis of Correlational Disequilibrium and Correlated Bubble-Cosmos (HDC–CBC), in which a single coherence field sources both potentials and the factor reflects the traceless weight of the scalar sector.