Paper 49B: Deriving the Critical Angle θcrit from Triadic Closure, VROS Micro-Resonators, and Data-First Breakpoint Estimation

A critical angle becomes a physical prediction only after specifying the state, interaction, loadingand observable to which it refers. This paper develops that distinction for Holosphere coherencetransport using real unit quaternions and the occupied cuboctahedral contact graph. A sharpthree-step geometric result retains the angle π/3: it guarantees a nonnegative reference overlap whenevery rotation increment obeys that bound. Compatible endpoint rotations, however, close around atriangle at any strain; the overlap bound is not a universal closure or failure theorem. For an explicitlystated cosine interaction on the full thirteen-site motif, the equally loaded center–shell equilibriumhas an exact stiffness spectrum and first loses stability at π/2 within the common-axis sector. Freeequal-shell motion has a different separatrix at π. The surrounding six-site octahedral void motif isidentified separately from the occupied cuboctahedron. Exact finite-graph connectivity calculationsdistinguish broken-site clustering from loss of retained connectivity. A count-level observation modeland a nuisance-adjusted information criterion then specify when an experimental breakpoint canactually be estimated. The resulting work supplies reproducible geometric and mechanical resultsand a prospective measurement protocol; it does not assign an unmeasured detector breakpoint orhealing rate.