The Shnoudi Holographic Tensor-Network Construction An Explicit Construction with Independent Computational Cross-Checks on Mathieu-Group Combinatorial Geometry, with a Precisely Scoped Verification of Ryu–Takayanagi Entropy and the Entanglement First Law

@misc{shnoudi2026shnoudi,
  author       = {Shnoudi, Ibrahim Amin},
  title        = {{The Shnoudi Holographic Tensor-Network Construction An Explicit Construction with Independent Computational Cross-Checks on Mathieu-Group Combinatorial Geometry, with a Precisely Scoped Verification of Ryu–Takayanagi Entropy and the Entanglement First Law}},
  year         = {2026},
  month        = oct,
  date         = {2026-10-09},
  publisher    = {Life of Research},
  howpublished = {Preprint},
  url          = {https://lifeofresearch.com/preprints/93},
  abstract     = {This note constructs an explicit tensor network from combinatorial data associated with the Mathieu group M11 and the extended binary Golay code, and tests it against two structures from the holographic tensor-network literature: the Ryu–Takayanagi minimal- cut entropy formula and the entanglement first law. Terminology is fixed at the outset: “holographic” is used throughout in the restricted tensor-network / quantum-information sense established by Pastawski, Yoshida, Harlow, and Preskill, not as a claim of an AdS/CFT duality, and “correspondence” denotes a constructive map from a combinatorial substrate to an entanglement structure, not a physical duality between two theories. The construction proceeds in four stages, each cross-checked computationally against an independent method within this note — brute-force state-vector computation, exact discrete GF(2) algebra, or control experiments — though not, it should be stated plainly, reproduced by an independent researcher’s separate implementation. First, a sparse, vertex-transitive substrate is obtained as the Cayley graph of M11; its ball-growth is exponential over a substantial radius range, a signature compatible with, though neither necessary nor sufficient to establish, negatively curved geometry, and this note does not claim to establish hyperbolicity. Second, an elementary six-qubit perfect tensor is constructed by a canonical procedure — independent of any arbitrary choice of eigenbasis — from the [[5,1,3]] quantum code; its stabilizer generators are confirmed both as an exact discrete algebraic fact (GF(2) rank 6) and by direct numerical action on the explicit state vector, with a maximum residual, at double-precision numerical accuracy, of 1.2×10⁻¹⁶. Third, a stabilizer-formalism contraction algorithm, validated against brute-force state- vector computation on small networks, is applied to build a 25-node, 98-qubit network from genuine M11 Cayley-graph edges. An exhaustive enumeration — not a sample — of every node-induced connected boundary region (every dangling qubit of each node in a connected node-subset, taken together) at sizes 2 through 10, 22,049 regions in total, establishes that the Ryu–Takayanagi formula holds exactly, with zero deviation in every single case; the same data show that this exact equality does not hold in general for arbitrary, non-contiguous regions, where deviations of up to six units were observed among five hundred fifty-six sampled cases at sizes up to 19, node-induced regions above size 10 not having been exhaustively enumerated. A control experiment, repeated on three independently generated random 4-regular graphs carrying the identical perfect tensor, finds deviations of two units even on connected regions, and a direct structural comparison — confirmed by two independent girth computations — finds the M11 substrate triangle- free with girth exactly 4, where all three controls contain several triangles each (girth 3) — reported as a candidate explanation, not a proven mechanism, since the two graph families differ in several other structural respects as well. Fourth, the extended binary Golay code — independently reconstructed and verified against all four defining properties of the classical uniqueness theorem (self-duality, doubly-evenness, rank 12, minimum distance 8) — is shown, via its natural CSS construction, to give a zero-rate stabilizer state that is not absolutely maximally entangled but achieves maximal entanglement entropy on every one of two hundred sampled regions of size up to ten, with a fraction declining toward the half- system point; this behavior follows an exact closed-form relation to the classical code’s information-set rank structure. Finally, on a small sub-network where exact state-vector computation remains tractable, the entanglement first law is confirmed to the correct leading order on two independent tests; this note states explicitly that this confirms a general fact of quantum relative entropy, not evidence specific to holography, M11, or the Golay code. Four distinct epistemic categories are maintained throughout — definition, mathematical proof, computational verification, and physical interpretation — and no claim in one category is permitted to borrow the certainty of another},
  language     = {en},
  copyright    = {Creative Commons Attribution 4.0}
}