Transport Papers T08. Black-Hole Entropy and Scrambling

T08 of the Transport Papers. Version 1.0. 23 pages. Epistemic status: Tier A/B (itemised). T05's counting step enters with two conditional inputs, the counted epoch and the per-cell density. This paper is the derivation programme for that count, with the dynamics and observational record of the boundary modes, at declared tiers. Proved: the stretched singularity is scheme-robust; its slice volume per unit Killing time is exactly 8πM at leading order under the framework's own curvature cutoff; the count N = 8πν MΔt/ℓ_P² reproduces the Bekenstein-Hawking entropy exactly when νΔt = M/2; and the internal cutoff's mass scaling is the unique one from which an admissible timescale of order M reaches the area law. Conditional, on the flagged intrinsic-clock premise: the epoch is the modular period β = 8πM, collapsing two unknowns into the single falsifiable density ν = 1/16π in the 1-nat convention, equivalently the integer-holographic statement that the boundary mode space registers S_BH fermions per modular period, up to the absorbed ln 2: one per four Planck cells of horizon; the naive reading overshoots by exactly the horizon-area coefficient 16π. The transverse sphere emerges by Kostant's principal embedding with principality forced by the flagged indifference premise, the fuzzy count matching the slice geometry and both naive aggregations refuted. The clocks of the boundary modes run at exactly 2πT, so operator spreading saturates the chaos bound by construction, at free-fermion tier. The observational record: the registered no-echo prediction stands consistent across four gravitational-wave catalogues, one confirmed echo train from refutation. Forty-seven exact-arithmetic checks certify the computational identities. Position in the series: Within the series it relies on T00, T01, T03, T04, T05, T06, T07. The series is surveyed in T17, The Transport Programme, whose status table records every paper's tier and check count. Verification: verify_T08.py checks every computational identity in the paper in exact rational arithmetic (47 checks, all passing at deposit). It imports GA.py, the series' shared exact-arithmetic library, deposited alongside so the record is self-contained. Run with a bare Python 3: python verify_T08.py. Files: T08_entropy_scrambling.pdf; T08_entropy_scrambling.tex; verify_T08.py; GA.py; LICENSE-CODE.txt. Licence: the paper (PDF and LaTeX source) is CC BY 4.0. The code files in this record are additionally released under the MIT licence; see LICENSE-CODE.txt. Authorship and disclosure. The founding idea of the series is the author's: that division by zero in a Clifford algebra should be read not as a prohibition but as a transport into a higher-dimensional algebra, and that the transported division need not be accompanied by an inverse operation. The direction of investigation and the critical review of the results are likewise the author's. The mathematical development and the drafting of the text were carried out in extended collaboration with Claude Fable 5 (Anthropic), an AI system.