Entanglement = geometry, exactly — the kernel-checked Ryu-Takayanagi equality and the forced emergence direction

Two new kernel-checked results in the constraint-first programme: (1) the EXACT Ryu-Takayanagi equality — srank (maximallyEntangled D) = D, entanglement equals the cut area D^m as an equality, not the usual leading-order bound; (2) the FORCED emergence direction — logDepthForced + meraEmergence prove logarithmic depth and exponential circumference are forced simultaneously, so the balanced binary tree (discrete hyperbolic geometry) is the unique optimum. Proven in Lean 4 + Mathlib (pacland-proof/PaclandProof/QM/MERA.lean), no sorry/axiom, reproducible via lake build. The continuous limit (tree ≃ H²) is the Svarc-Milnor step, named as the frontier, not proven here. Site: https://pacland.morpheus.nz — Repo: https://github.com/robmitchellengineering-hub/pacland

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-09
DOI
https://doi.org/10.5281/zenodo.24077011
Primary Topic
Black Holes and Theoretical Physics
Type
preprint
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preprint

Entanglement = geometry, exactly — the kernel-checked Ryu-Takayanagi equality and the forced emergence direction

Rob Mitchell
Zenodo (CERN European Organization for Nuclear Research)
Black Holes and Theoretical Physics
preprint

Entanglement = geometry, exactly — the kernel-checked Ryu-Takayanagi equality and the forced emergence direction

Rob Mitchell
preprint en

Abstract

Two new kernel-checked results in the constraint-first programme: (1) the EXACT Ryu-Takayanagi equality — srank (maximallyEntangled D) = D, entanglement equals the cut area D^m as an equality, not the usual leading-order bound; (2) the FORCED emergence direction — logDepthForced + meraEmergence prove logarithmic depth and exponential circumference are forced simultaneously, so the balanced binary tree (discrete hyperbolic geometry) is the unique optimum. Proven in Lean 4 + Mathlib (pacland-proof/PaclandProof/QM/MERA.lean), no sorry/axiom, reproducible via lake build. The continuous limit (tree ≃ H²) is the Svarc-Milnor step, named as the frontier, not proven here. Site: https://pacland.morpheus.nz — Repo: https://github.com/robmitchellengineering-hub/pacland

Zenodo (CERN European Organization for Nuclear Research)
Black Holes and Theoretical Physics
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