Extrinsic χ Compression and Deformation-versus-χ Model-Selection Notes

The space-cell framework admits at least two candidate geometric realizations. The original realization assigns deformation, strain, connection, orientation, or topology-bearing variables to cells and links. The alternative developed here assigns each ordinary three-dimensional cell an additional internal scalar state χ(x,t), interpreted as compression, event-density elevation, hidden displacement, or local update-rate modulation. The χ variable is not initially treated as a literal fourth spatial dimension. It is a parsimonious candidate state coordinate whose adequacy must be tested rather than assumed.

This report consolidates the mathematical target for χ, the richer distortable-cell comparator, and a shared model-selection program. The principal tests are causal propagation, dense weak-field accumulation, radial source recovery, weak-source superposition, low-momentum isotropy, boundedness, source specificity, path reconstruction, Lorentz compatibility, false-source rejection, and parsimony. Separate audit branches are defined for weak-field geometry (GR-AU), Lorentz/covariance behavior (L-CM), moving-source wakes, event-rate/proper-time scaling, and apparatus-context discrimination. The report preserves both candidates: χ may provide a simpler route from event density to potential-like behavior, while deformation may be required for tensor, shear, topology, chirality, frame-dragging, gravitational-wave, or gauge-like effects.

The current conclusion is model-selection neutrality. Extrinsic χ should be preferred only if it matches or exceeds the distortable realization under shared causal and adversarial controls while using fewer state variables and parameters. Failure of χ would not by itself validate deformation; failure of deformation would not by itself validate χ. Both remain provisional branches of a broader discrete information-theoretic research program.