Measurement, Energy, and the Emergence of Time: A Unified Framework from Observable Dimensionality

We present a dimensional physics framework: constraint geometry (phase-space structure of systems evolving under physical laws) is fundamental; information emerges when measurement extracts classical records from this geometry. High-dimensional systems have complex constraint geometry—modes coupled non-locally, resisting bit-wise decomposition. When geometry reconfigures faster than observation can extract, we have timing inaccessibility. Time accumulates through irreversible commits (h_epsilon: 0 -> >0) that extract information from geometry.

Under quasi-linear entropy scaling, we prove an Observable Dimensionality Bound: D_eff > D_crit^(epsilon) <=> H_dyn(epsilon) > C_obs, where D_crit^(epsilon) = C_obs * tau_e / (alpha * h_epsilon). Here D_eff is effective mode count, H_dyn the registrable (bit-equivalent) geometry reconfiguration rate, C_obs the observer's Shannon-Hartley capacity, tau_e the task timescale, h_epsilon information per mode, and alpha compressibility.

This bound unifies three scales: (i) QM: wavefunction collapse to measurement outcome at cost >= k_B T ln 2 per bit (Landauer); (ii) GR: pre-geometric entanglement commits at horizons, energy flux creates time (delta Q = T delta S), yielding Einstein's equation -- we heuristically recover the Schwarzschild metric from a tensor-network bond-cutting picture; (iii) Consciousness: neural constraint geometry (D_eff ~ 10^5 modes) with sparse behavioral commits (C_obs ~ 10^2 bits/s) operates supercritically (D_eff/D_crit ~ 10^3), dissipating ~ 10^3 x lower power than clocked digital systems of comparable effective dimensionality -- a parameter-robust prediction. Biological systems provide empirical validation as the best-documented macroscopic supercritical regime. As a proof of concept for Standard Model connections, we sketch how a tensor-network bond-cutting picture reproduces area-law confinement and the sign/scale of QCD's running coupling (Section 6), yielding an order-of-magnitude string tension consistent with lattice QCD. QM, GR, and consciousness are instances of information emerging from constraint geometry at different scales.