The Participatory Modular Hamiltonian: Architecture for the Participatory Horizon Programme - Part I

Persistent large-angle anomalies in observational cosmology including suppressed CMB correlations beyond 60^, excess source-count dipoles, and quadrupole--octopole alignment with the kinematic dipole, appear to share an observer dependent character that primordial mechanisms should not naturally produce. The participatory-horizon programme proposes that these anomalies arise because the observer's causal diamond acts as an information aperture whose geometry constrains which correlations are realised as classical records, without modifying standard CDM cosmology. This paper, the first of a companion triplet, proposes the Participatory Modular Hamiltonian (PMH): the pair (K䃐, M) of the Casini--Huerta--Myers modular Hamiltonian and a local recording map evaluating the pointwise signal-to-noise ratio of coherent cosmological signals against quantum modular-energy noise. The PMH produces two observable channels: an angular filter h² () = 1 - - (+1) /c² with geometric transition scale c² = 12. 56 from conformal horizon geometry, and a radial recording kernel kL (x) = xL (1 - x²) ⁴, where x = /₀ is fractional comoving depth. The boundary order m = 4 equals the spacetime dimension d, derived from the volume Jacobian relating flat and hyperbolic coordinates. Given the recording prescription and the g₁ = 1 closure, both channels are fully determined with zero free parameters when the geometric prediction for c² is adopted. We prove that the modular Hamiltonian is irreducible within the concentric Bisognano--Wichmann weight basis, demoting the earlier nested-diamond construction to an effective approximation. The boundary-squeeze mechanism suppresses monopole leakage at the last-scattering surface by a factor of 2, 300, ensuring the radial kernel does not contaminate the observed CMB dipole. The angular filter resolves the low-power anomaly, shifting the S₁/₂ statistic from the 8. 3rd to the 49. 7th percentile of CDM realisations. The radial kernel addresses the source-count dipole tension, in which observed amplitudes exceed the kinematic expectation by factors of two to four: the CatWISE-normalised kernel is consistent with a universal plateau amplitude A 5--7 across two independent wavebands. Falsifiable predictions include a depth-dependent dipole profile distinguishable from a constant anomaly parameter with forthcoming spectroscopic tomography, and up to 62\% EE reduction at = 2.