Spectral Fixed-Point Structures and Dark-Sector Network Dynamics A Conceptual Framework and a Falsifiable Scaling Hypothesis, D = m*b^3

This paper continues the conceptual programme of P40-P42 on spectral geometry, holographic organisation, exceptional-point topology and asymptotic-safety fixed points. It asks whether a fixed point of the renormalisation-group (RG) flow can act as a selection rule on networks of spectral structures, which we propose as a vocabulary for organisation in a hypothetical dark sector. We make this question precise through formal definitions (Section 4) and one selection conjecture (Conjecture 4.1). We then introduce a single phenomenological hypothesis, the Khánh Spectral Density relation D = m*b^3, in which D is an effective spectral density, m a mass scale and b a dimensionless binding parameter. The cubic exponent is postulated, not derived. The central practical contribution of the paper is therefore methodological: we explain why the relation is not yet testable (the observable D has not been defined independently of m and b), show with synthetic data that naive regressions return the trivial identity or the baryonic Tully-Fisher slope, and specify a pre-registrable test design in which an independently measured observable Y is fitted as Y ∝ M α b γ with explicit falsification criteria for γ = 3. A simulated injection-recovery study (175 galaxies, 1000 trials per setting) shows that the estimator is unbiased and has nominal 95% coverage, but that strong collinearity between log M and log b (r ≈ 0.99, VIF ≈ 68) inflates the uncertainty on individual exponents and makes α and γ almost perfectly anti-correlated (ρ ≈ −0.99). Only the combination 4α + γ is tightly constrained. Any real test must be designed around this fact.