Temporal Casimir Rigidity and Closure of the Effective Metric: Identification of Aτ = 2

Paper Q5b leaves the temporal coefficient A_τ of the effective co-metric gμν = diag(-A_τ, 2, 2, 2) as the sole remaining free parameter of the Cosmochrony spectral geometry programme (open problem Q5b-O3). The present paper closes this problem by proving A_τ = 2 under the Q5a hypotheses and the spectral universality hypothesis [U]. The argument proceeds in three steps. First, we establish a cascade internalization lemma: the temporal coordinate τ, identified with the BFS depth index n in Q5b Remark 3.3, has ∂_τ as the continuum limit of the cascade increment operator T σc(n)↦σc(n+1) by the Carnot–Carath\'eodory convergence of Q5b Theorem 3.2 (Pansu ). This identification requires no new hypothesis. Second, the O-series spectral universality [U], proved in U1, implies that T is asymptotically SU(2)-equivariant, so ∂_τ acts on Sym²(V_ρ) without introducing any new irreducible representation. Third, Schur's lemma forces the unique SU(2)-invariant quadratic form on Sym²(V_ρ) to be proportional to the Casimir C_su(2) = 2, and Q8 fixes the normalisation scalar to unity. The effective co-metric is therefore uniquely determined: gμν = 2\,ημν. No free parameter remains. Status revision (version 1.1). Q5a version 3.0 withdraws the convergence framework whose hypotheses this paper consumes; they are now the frozen, unestablished framework hypothesis [H-F] (Q9 version 1.1), and the existence of the spatial limit operator underlying the effective co-metric is the unestablished hypothesis [H-L] (Q5b version 2.0). The algebraic content of this paper (cascade internalization, SU(2)-equivariance, Casimir rigidity) is unchanged; its reading as a metric coefficient is conditional on [H-L]. The temporal direction is not an additional geometric axis; it is the continuum infinitesimalisation of the admissible cascade itself.