We advance prime-indexed discrete scale invariance (p-DSI) as a falsifiable organizing principle:only rescalings by elements of a prime lattice Λ = {pα : p ∈ P, α > 0} are dynamically privileged.Interleaving ordinary evolution with recursive quantum collapse yields log-periodic correctionsto scaling with arithmetic-locked frequencies ωp,α = 2π/ ln(pα). We derive statistical tests thatseparate prime-fixed spectral combs from strong nulls and outline cross-domain experiments.Brief mentions of abyssal symmetries and a τ -field (chronofluid) are given only for context; ourfocus is the falsifiable p-DSI core.