Counterfactual Operator Geometry: Metric and Algebraic Structure in Intervention-Conditioned Signal Processing

Counterfactual Operator Geometry introduces an ensemble representation for history-dependent signal processing in which controlled alternative histories generate a family of possible present operators. Rather than describing that family only through pairwise metric distances, the work examines its relational and algebraic organization using normalized commutators and approximate joint diagonalizability (AJD). Controlled experiments show that continuation-conditioned operator ensembles can retain distinguishable algebraic organization even when their pairwise Frobenius geometry is matched to numerical precision. Pair-resolved commutator-field summaries achieved 80% continuation classification and AJD-plus-aggregate summaries 75%, against approximately 50% paired-permutation nulls. Residualization against a stronger iterative AJD baseline reduced the commutator-field result to 56% (p = 0.119); the work therefore does not claim information independent of AJD. The technical note additionally introduces APPARITION, an event-driven spectral-processing application of the framework. Rare counterfactual configurations instantiate persistent spectral entities carrying a frozen relational genome. Two rendering interpretations are implemented: MATERIAL, in which present input excites the frozen spectral anatomy while relational descriptors determine its material response, and AUTONOMOUS, in which the same birth-conditioned anatomy self-radiates. The central proposal is that controlled counterfactual branching can provide persistent relational control variables for stateful signal processing: alternate histories determine not merely the magnitude of a present difference, but the structure and identity of a subsequently instantiated signal object. DOI: 10.5281/zenodo.22699673