The Informational Pressure Principle: A Self-Consistent Framework for Emergent Spacetime

We present a mathematically rigorous formulation of the **Informational Pressure Principle (IPP)** as a mechanism for emergent spacetime. By treating the cosmological horizon as a quantum information substrate, we resolve previous inconsistencies regarding dimensional scaling. We define information density \( \rho_I \) operationally via the **Bures metric** and introduce **informational pressure** as the thermodynamic driver of geometric dynamics. Crucially, we resolve the tension between the \( N^{-2/3} \) suppression law and holography by reinterpreting \( N^{2/3} \) as the scaling of **independent geometric channels** on the horizon — distinct from the total entropy \( S_{\max} = N \ln 2 \). This leads to a consistent derivation of **three macroscopic spatial dimensions** for \( N \sim 10^{122} \). The framework yields a well-posed **Poisson equation on \( S^2 \)** and predicts: (i) **black hole clustering** \( w(\theta) \propto \theta^{-1/2} \), consistent with **Chandra CSC 2.0** data, and (ii) a **quantized gravitational wave ringdown frequency spacing** \[\Delta f = \frac{c^3 \ln 2}{16\pi^2 G M} \approx 896~\text{Hz} \cdot \left(\frac{M_\odot}{M}\right).\] All coupling constants are derived from fundamental parameters \( (\hbar, c, G, \Lambda) \) without adjustable free parameters. This work establishes IPP as a self-contained, falsifiable, and observationally grounded framework for the emergence of spacetime.