Information-Causality Constraints on Geometric Registration: A Companion Paper to the Conformal-Primary Measurement

The conformal-primary framework presented in our main companion manuscript reads quantum measurement as registration of sub-strate excitations into the metric overlay. Collapse nonlocality is attributed to the absence of primitive metric distance at the substrate level. This note states the correlation strength problem in registration form and proves the conditional results that follow from the standard positive-state structure. We show that no-signaling is equivalent to the existence of reduced registration descriptions, and that it follows from local completeness once a composition kernel exists. We show that a Popescu-Rohrlich box cannot be represented by a positive registration state with sharp local effects and commuting spacelike algebras. The same hypotheses give the Tsirelson bound, |S|≤2√2, through a sum-of-squares identity. The entropic route through information causality gives the same bound under stated axioms. Section 6.4 gives a worked registration construction based on a conformal positive-frequency kernel. It includes the momentum-space kernel, the Kallen-Lehmann microcausality argument, the Fock representation, the involution, local identity effects, and the positive registration state. Reflection-positive Euclidean substrate data supply the spectrum condition by reconstruction. A Gaussian generalized-free substrate action with positive Kallen-Lehmann density gives an explicit branch, and an interacting substrate can extract the same branch when its registration operator has a positive spectral density with conformal scaling. The result is a precise target for the measurement program: derive this positive spectral branch and the composition kernel from conformal substrate dynamics.