Theory of Quantum Entanglement of Particles: Multiscale Graph States, Region Geometry, Quantum Error Correction, and the Effective-Space Hypothesis

Theory of Quantum Entanglement of Particles: Multiscale Graph States, Region Geometry, Quantum Error Correction, and the Effective-Space Hypothesis Abstract This study develops a finite-dimensional, computationally testable framework for examining how multiscale graph structure organizes quantum entanglement and the accessibility of logical information. For the circulant graph family G_N, where N = 2ⁿ, n ≥ 3, and edge offsets are 1, 2, 4, …, N/2, an analytic proof establishes that the binary adjacency matrix across every contiguous block A of length L ≤ N/2 has rank L over F₂. For the associated graph states, this yields S(A) = min(L, N − L). For G₈, the study enumerates 238 proper connected vertex subsets ("tubes") and 32,640 maximal tubings and computes the entanglement entropy of every tube. Explicit bit-flip and phase-dual quantum error-correcting code embeddings are examined under specified Pauli-noise and erasure models, showing that recovery performance depends on the encoding, error model, and decoder class, not on graph-state entropy alone. The dynamical component, M1-Green, models a four-dimensional logical code coupled to eight intermediary bosonic modes and sequential collision-bin output registers. At selected resonantly matched parameter settings and T = 120/K, the computed mutual information I(Ref:R) reaches 3.9414 bits for G₈ and 3.8876 bits for C₈, below the four-bit upper bound. The study also reports a verified finite-step symmetry for C₈ in the tested setup, Petz recovery, and a numerical semidefinite-programming interval of 0.235788–0.245982 for recovery fidelity in a specified 69-dimensional output-space formulation. A one-port Page-like model, a random-unitary scrambling benchmark, and a two-sided mediated channel are treated as distinct models. The accompanying compact archive provides scripts, data tables, figures, and numerical certificates. The effective-space interpretation remains a hypothesis; global recovery optimality and independent experimental validation are not claimed. Keywords: quantum entanglement; graph states; multiscale graphs; circulant graphs; entanglement entropy; graph associahedra; quantum error correction; erasure channels; logical information recovery; collision-bin dynamics; Petz recovery map; semidefinite programming; quantum scrambling; effective geometry.