Holographic Scrambling as a Residual Law: A DSFL View on Susskind’s Black Hole Paradigm

We recast Leonard Susskind’s black hole paradigm in terms of a single Lyapunov functional on a calibrated Hilbert space. In the Deterministic Statistical Feedback Law (DSFL) framework, one works with a fixed instrument norm and a quadratic ``residual of sameness’’ measuring the mismatch between blueprint data and physical responses. Admissible dynamics are exactly those that contract this residual and admit an intrinsic DSFL clock parametrising the decay.Specialising to a black hole room that factors into interior, near–horizon collar and far radiation, we identify three sectoral residuals in the \emph{same} norm: a scrambling residual, an inside/outside balance residual, and a gravitational residual built from Einstein imbalance. We show that fast scrambling and the Maldacena–Shenker–Stanford chaos bound can be phrased as rate constraints on the scrambling residual; that a DSFL Page time defined by balancing inside and outside budgets coincides, in Haar and random–circuit evaporator models, with the usual entropic Page time; and that the gravitational residual admits a Lyapunov law with quasinormal– mode rate and a DSFL null expansion encoding linearised QNEC/QFC–type focusing. Appendices develop the general DSFL machinery, prove finite–dimensional envelope theorems, and collect numerical audits in simple quantum and near–horizon toy models, as well as optional connections to circuit complexity and quantum error correction that support the single–residual black–hole picture.