The Mathematical Shadow and the Differential Map: A HEF Note on the Topology of Irreversibility across Quantum, Gravity, and Thermodynamics

Modern physics relies on mathematical operations such as partial tracing, objective-collapse hypotheses, and coarse-graining to describe macroscopic irreversibility. However, these operations act as a “mathematical shadow” that obscures the underlying physical cost of recovering the past.

This concept note proposes the Historical Exclusion Front (HEF) framework, which introduces a “differential map” to track the nonlocalization of discarded recovery keys (κ). Irreversibility is interpreted not as the fundamental loss of information nor as mere probabilistic disorder, but as the divergence of the local operational cost required to un-entangle the past.

By introducing the effective number of keyholes (N_eff) and the operational erasure curve V_erase(m), quantum decoherence, gravitationally induced classicality, and thermodynamic entropy are interpreted within a common operational framework as the breakdown of local time-reversal.