Collision Geometry of Joint Spectra

Paper IV develops a fixed-arrangement collision geometry for joint spectra of commuting finite-dimensional operators. Using the 228-dimensional Rubik's Cube representation as a canonical testbed, the paper analyzes the nine-point joint spectrum induced by the QT/HT averaging operators. The canonical six-layer spectrum is recovered at alpha = 2/3 as the collision quotient of this fixed arrangement.The paper classifies the resulting finite affine branch arrangement, identifies all interior collision parameters, and establishes that alpha = 2/3 is the unique interior parameter producing the maximal spectral collapse. The paper also separates fixed collision geometry from generator-set deformation. Paper IV studies a fixed joint spectral arrangement under varying affine projections only; deformation of the arrangement itself is developed later through the generator-set moduli theory. Source repository: https://github.com/dooven-prime/rime-lite This work forms Part IV of the Representation-Induced Mechanics and Evolution (RIME) research program. Foundational archive DOI: 10.5281/zenodo.21108197