A Tensor-Network Factorization Template for Qudit Projectors with Conjectured Complexity Scaling

The document formalizes the Local Character Projection Identity (LCPI) as a structural template applicable to qudit systems. It introduces representation-complexity metrics for Matrix Product Operator (MPO) realizations, distinguishing between raw parameter counts and gauge-reduced bond dimensions. The central conjecture states that, provided the triadic branching present in the LCPI expansion remains independent after gauge reduction, the gauge-reduced complexity C_gauge scales asymptotically as χlog_d(3), where χ denotes the bond dimension. This yields a fractal dimension D_f = log_d(3). For the qubit case (d = 2), the exponent is approximately 1.585. A proof sketch of the LCPI identity is provided, together with a falsification criterion based on numerical MPO compression and a detailed roadmap toward a rigorous proof of the proposed scaling law. Main sections include: Definition of the LCPI decomposition and a short proof that the identity is exact (Π² = Π). Presentation of the conjectured complexity scaling D_f ≈ 1.585 in the qubit case. Falsification criterion: the conjecture fails if standard MPO compression absorbs the triadic structure into gauge degrees of freedom. Roadmap for proof, requiring derivation of matching upper and lower bounds on C_gauge(χ) together with verification of gauge invariance of the branching. A logarithmic plot (Figure 1) illustrates the predicted separation between the LCPI scaling (solid line, slope ≈ 1.585) and the conventional manifold baseline (dashed line, slope = 1.0). References: The note cites foundational works on tensor network methods by Roman Orús (2014) and Ulrich Schollwöck (2011).