A Tensorial Extension of Two-Channel Balance Diagnostics

This preprint develops and validates a tensorial extension of two-channel balance diagnostics. A local two-channel composition is first reduced to a signed imbalance and a scalar amplitude, and the resulting sector-wise amplitudes are then aggregated using trace-free directional dyads. The local (2×2) trace-free channel-space tensor is shown to satisfy the exact identity (A_B=√2DER^2), demonstrating that it contains no independent local directional information beyond the scalar imbalance. The nontrivial extension is instead a three-dimensional spatial tensor whose principal axis summarizes directional coherence across sectors. Six controlled synthetic regimes are examined: independent isotropic fluctuations, common driving, residual directional interaction, a fixed directional axis, bipolar outflow, and a rotating principal axis. After finite-grid mean-dyad subtraction, the fixed-axis regimes yield axis scores of 0.9935, 0.9883, and 0.9930. A rotating axis is recovered with a median windowed score of 0.9966. After removal of a global cubic common mode, the residual-interaction tensor is approximately 16.6 times stronger than the common-driver control and retains an axis score of 0.9887. The results remain stable under sector randomization, temporal null tests, moving-block bootstrap, multiplicative channel noise, alternative positive normalizations, sector weighting, orientation masking, and directional rebinning. This work is presented as a phenomenological diagnostic framework. It does not claim an observational detection or identify the diagnostic tensor with a physical spin-2 field, graviton, metric perturbation, or energy–momentum tensor.