{"id":93,"identifier":"LOR-00093","url":"https://lifeofresearch.com/preprints/93","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","abstract":"This note constructs an explicit tensor network from combinatorial data associated with \r\nthe Mathieu group M11 and the extended binary Golay code, and tests it against two \r\nstructures from the holographic tensor-network literature: the Ryu–Takayanagi minimal-\r\ncut entropy formula and the entanglement first law. Terminology is fixed at the outset: \r\n“holographic” is used throughout in the restricted tensor-network / quantum-information \r\nsense established by Pastawski, Yoshida, Harlow, and Preskill, not as a claim of an AdS/CFT \r\nduality, and “correspondence” denotes a constructive map from a combinatorial substrate \r\nto an entanglement structure, not a physical duality between two theories. The \r\nconstruction proceeds in four stages, each cross-checked computationally against an \r\nindependent method within this note — brute-force state-vector computation, exact \r\ndiscrete GF(2) algebra, or control experiments — though not, it should be stated plainly, \r\nreproduced by an independent researcher’s separate implementation. First, a sparse, \r\nvertex-transitive substrate is obtained as the Cayley graph of M11; its ball-growth is \r\nexponential over a substantial radius range, a signature compatible with, though neither \r\nnecessary nor sufficient to establish, negatively curved geometry, and this note does not \r\nclaim to establish hyperbolicity. Second, an elementary six-qubit perfect tensor is \r\nconstructed by a canonical procedure — independent of any arbitrary choice of eigenbasis \r\n— from the [[5,1,3]] quantum code; its stabilizer generators are confirmed both as an exact \r\ndiscrete algebraic fact (GF(2) rank 6) and by direct numerical action on the explicit state \r\nvector, with a maximum residual, at double-precision numerical accuracy, of 1.2×10⁻¹⁶. \r\nThird, a stabilizer-formalism contraction algorithm, validated against brute-force state-\r\nvector computation on small networks, is applied to build a 25-node, 98-qubit network \r\nfrom genuine M11 Cayley-graph edges. An exhaustive enumeration — not a sample — of \r\nevery node-induced connected boundary region (every dangling qubit of each node in a \r\nconnected node-subset, taken together) at sizes 2 through 10, 22,049 regions in total, \r\nestablishes that the Ryu–Takayanagi formula holds exactly, with zero deviation in every \r\nsingle case; the same data show that this exact equality does not hold in general for \r\narbitrary, non-contiguous regions, where deviations of up to six units were observed \r\namong five hundred fifty-six sampled cases at sizes up to 19, node-induced regions above \r\nsize 10 not having been exhaustively enumerated. A control experiment, repeated on three \r\nindependently generated random 4-regular graphs carrying the identical perfect tensor, \r\nfinds deviations of two units even on connected regions, and a direct structural comparison \r\n— confirmed by two independent girth computations — finds the M11 substrate triangle-\r\nfree with girth exactly 4, where all three controls contain several triangles each (girth 3) —\r\nreported as a candidate explanation, not a proven mechanism, since the two graph families \r\ndiffer in several other structural respects as well. Fourth, the extended binary Golay code \r\n— independently reconstructed and verified against all four defining properties of the \r\nclassical uniqueness theorem (self-duality, doubly-evenness, rank 12, minimum distance 8) \r\n— is shown, via its natural CSS construction, to give a zero-rate stabilizer state that is not \r\nabsolutely maximally entangled but achieves maximal entanglement entropy on every one \r\nof two hundred sampled regions of size up to ten, with a fraction declining toward the half-\r\nsystem point; this behavior follows an exact closed-form relation to the classical code’s \r\ninformation-set rank structure. Finally, on a small sub-network where exact state-vector \r\ncomputation 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 \r\ngeneral fact of quantum relative entropy, not evidence specific to holography, M11, or the \r\nGolay code. Four distinct epistemic categories are maintained throughout — definition, \r\nmathematical proof, computational verification, and physical interpretation — and no \r\nclaim in one category is permitted to borrow the certainty of another","authors":[{"name":"Ibrahim Amin Shnoudi","affiliation":"Jordan University","orcid":""}],"type":"preprint","license":"CC-BY-4.0","keywords":[],"publication_date":"2026-10-09","version":1,"views":13,"downloads":10,"created_at":"2026-10-09 12:21:13","doi":"","source":"","source_url":"","files":[{"id":93,"name":"The_Shnoudi_Holographic_Correspondence-6.pdf","size":586459,"mime":"application/pdf","downloads":10,"download_url":"https://lifeofresearch.com/preprints/93/files/93/The_Shnoudi_Holographic_Correspondence-6.pdf"}]}