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

Counterfactual Operator Geometry studies history-dependent signal processing through ensembles of present operators generated from controlled alternative histories. The framework distinguishes ordinary pairwise metric geometry from relational/algebraic organization, represented using normalized commutator descriptors and approximate joint diagonalization (AJD). Version 1.1 adds Validation V2, a reconstruction-free confirmatory experiment designed to test the strongest interpretation remaining from v1.0 on naturally generated operator ensembles. Metric summaries alone decode continuation at 82%, while commutator summaries decode it at 88%. However, after pre-specified training-fold linear residualization of the commutator summaries on the metric field, performance falls to 40% (paired-permutation p = 0.956). Thus, although metric and algebraic organization are mathematically non-equivalent and can be separated constructively, v1.1 does not detect incremental continuation information in the algebraic descriptors beyond the metric descriptors under the tested reconstruction-free linear protocol. Post-hoc diagnostics find low-dimensional, strongly aligned metric and algebraic descriptor-summary spaces, motivating—but not establishing—a hypothesis that the dimensionality of persistent history may influence when algebraic organization becomes non-redundant. The note also documents APPARITION, a history-dependent spectral effect developed for Harmonic DAW. Rare counterfactual configurations instantiate persistent spectral entities carrying frozen relational spectral structure, rendered through two ontologies: MATERIAL and AUTONOMOUS. APPARITION is presented as a constructive application of the framework, not as evidence for the Validation V2 theoretical claim. Its commercial implementation source is not included. The public reproducibility package contains the v1.1 confirmatory experiment, frozen numerical results, and separately identified post-hoc diagnostics. Version DOI: 10.5281/zenodo.22711404Previous version: 10.5281/zenodo.22699673 The central concept of Counterfactual Operator Geometry is retained; the empirical interpretation is narrower and better controlled than in v1.0.