Local Constitutive Equivalence Between COCM and the VFT Optical Sector

This technical note establishes a local constitutive equivalence between the infrared deformation sector of COCM and the optical deformation sector of VFT. Taking as its starting point the open constitutive question identified in Dziabura (2026) — the microscopic origin of the VFT optical control parameter β_opt — we show that there exists a unique radius r* at which the COCM gradient invariant J maps constitutively to β_opt, at fixed orbital phase and in the subcritical domain J < 1/4. The equivalence fails globally due to inequivalent radial profiles and symmetry-breaking classes. As a corollary, once J is fixed microscopically by Bridge 3, the identification of β_opt requires no free parameters. Added §3.4: Geometric interpretation of the matching radius r*, including Proposition X.6 (characterisation of r* via the profile ratio function Phi(r) and its relation to the photon sphere). No changes to the main results. Updated §3.4: Proposition X.6 refined with explicit decomposition r* = r_ph − |δr|, |δr| ~ r_S, confirming geometric association with the photon sphere without exact identity. Closing sentence added. Internal consistency of §3 verified. v1.3: Three formal precision corrections. Lemma X.2: condition (3) now explicitly marked as a hypothesis, not a derived consequence, with equivalence to Phi’(r*)≠0 stated. Lemma X.3: uniqueness qualified as unique in the near-horizon branch; second solution in outer branch declared and excluded by near-field condition. Corollary X.4: M_C0 added as observational input; clarified that no additional input from Bridge 3 beyond J is required. No changes to main results.