Coupling Parameter Determination and Kernel–Transport Coefficient Matching in Reduced Q5 Transport

This paper closes the primary open step of the Bear Claws Theorem (T50–T52), the explicit derivation of the coupling parameter β from Q5 kernel geometry, in two parts. Part I derives β = a²/4 and h₁ = √ (5/2) directly from matrix computation. The recurrence Y³ = 5Y (T46) pins h₁ = √ (5/2) via 2h₁² = 5. Direct multiplication gives Mₙ, Dₙ·Y = ah₁·[1, 0, 1, 0, 0, 0, −1, 0, −1], zero middle row and column, consistent with |8⟩ as the defect bond mediation vertex. The visible-hidden coupling is the (1, 3) entry ah₁. With Tr (YT Y) = 4h₁², β = a²h₁²/ (4h₁²) = a²/4. The observable phase coefficient satisfies μ = −8βh₁ = −a²√10. Part II (Theorem 53) establishes that μ, β, and a are not free parameters; they are uniquely determined by compatibility between the kernel-derived phase extraction pipeline (T26–T29) and the transport-level rotational residual (T48–T49). The reduced phase plane is two-dimensional, so its antisymmetric subspace is one-dimensional; any antisymmetric operator surviving the pipeline is a scalar multiple of R = A = Ed = iσᵧ. Since both constructions reference the same normalized generator with coefficients μ (extraction pipeline) and 1/160 (transport residual), and the sign is fixed by σ_Γ = −1 from T18: μ = −1/160, β = 1/ (1280√ (5/2) ), a² = 1/ (160√10) The value 1/160 is not an imposed normalization; it is the unique coefficient consistent with both the ordered kernel holonomy and the reduced transport evolution. The full chain T17 → T29 → T46 → β = a²/4 → μ = −1/160 → T52 is non-circular. Status: h₁ = √ (5/2), product structure of Mₙ, DₙY, (1, 3) coupling entry, Tr (YT Y) = 4h₁², and β = a²/4 all solid by direct matrix computation. Generator identification R = A = Ed solid, one-dimensionality of antisymmetric sector is linear algebra. μ = −1/160 solid given generator identification. Sign propagation from T18 through T29, structural, full explicit sign-tracking through the projection pipeline, is the natural supporting calculation. Parametrized form of Mₙ, Dₙ on 9, 8, 24 must ultimately be derived from T17 kernel generators to confirm the alternating antisymmetric form is forced rather than assumed. Dependencies: T29, T36, T37, T46, T52.