Black-Hole Entropy as Residual Boundary Memory

This paper develops a formal bridge between the Physical Residuality Principle (PRP), the Fractal Consistency Law (FCL), and black-hole thermodynamics. The central thesis is that the Bekenstein-Hawking entropy can be reinterpreted as the logarithmic count of admissible residual boundary configurations rather than as a count of abstract binary symbols or ontological nullities. The proposal does not replace the semiclassical area law; it recovers it and gives it a residual-information interpretation. Within the PRP/FCL framework, the logical zero of information theory is not identified with ontological nothingness. A physical bit is a transition between two non-null admissible states of a substrate, and therefore any horizon degree of freedom must be understood as a residual boundary state. The paper defines the residual boundary configuration space of a horizon, derives the area law by imposing holographic saturation, introduces a discrete residual-cell model, and shows that a binary residual alphabet implies an elementary boundary cell of area four Planck areas times the natural logarithm of two. The resulting interpretation is that a black-hole horizon does not count zeros; it counts residual differences compressed into boundary memory. The paper then develops the connection with Landauer erasure, horizon irreversibility, scrambling, Hawking radiation, and possible observational windows through ringdown spectroscopy, analog horizons, and quantum-information simulators. The work is explicitly framed as a theory-building and proof-skeleton manuscript: it provides a coherent bridge from PRP/FCL residuality to black-hole entropy, while identifying the remaining microscopic derivation required to obtain the residual cell area directly from the fractal substrate rather than from holographic saturation.