Radial Participatory Horizons: A Nested Causal-Diamond Framework

Persistent anomalies on the largest angular scales of the cosmic microwave background, including a low- power deficit and alignments among the lowest multipoles, together with an independently observed amplitude excess in extragalactic source-count dipoles, have resisted easy explanation within the standard CDM concordance. This programme, motivated by Wheeler's participatory view of quantum measurement, proposes that the observer's causal diamond acts as an information aperture, imposing a direction-dependent recording kernel on the resolved cosmological sky, while leaving the underlying CDM intact. The three preceding papers in this `participatory-horizon' programme treated the observer's recording kernel as a purely angular object, with no dependence on comoving distance along the line of sight. However, these approximations led to a critical theoretical gap. The L=1 kernel at the amplitude anchored by the proposed CatWISE2020 source-count dipole measurement would couple the CMB temperature monopole into the observed dipole at recombination through the same Gaunt rule proposed to produce the source-count excess, generating monopole-to-dipole leakage of order T₀\, ₁^ src 3. 3\, mK, comparable to the entire observed CMB kinematic dipole. The analogous L=2 leakage is similarly problematic at the phenomenological amplitude required for the associated proposed approach to the Axis-of-Evil alignment. Here, we therefore approach the radial dimension through a new geometric construction. A nested family of observer-centred causal diamonds, weighted by a scale-free measure (R) R^-q, produces a radial kernel kL () whose power-law tail ^1-q is set entirely by q and is independent of the interior response shape fL (the exponent-inheritance theorem) and of the Legendre sector~L. Because no diamond larger than the conformal horizon ₀ exists, the nested-family integration range collapses as ₀ and the kernel is suppressed at the last-scattering surface by factors of 10²--10⁴ below its power-law extrapolation, introducing a `boundary-squeeze' mechanism that clears the monopole-leakage constraint in both the L=1 and L=2 sectors. Anchoring the L=1 amplitude to the related CatWISE measurement previously derived (₁^ src=1. 22 10^-3) and propagating through the CAMB line-of-sight integral, the construction yields additional falsifiable CMB predictions. Further, for the L=2 sector, the same geometry produces a derived -roll-off that would replace the heuristic envelope previously proposed for the Axis-of-Evil approach. This radial upgrade converts the previous angular-only programme into an operator-to-observable bridge at the ISW level, with full-source and exact-estimator validation deferred. Near-term tests, limitations, and implications are discussed.