Fibrewise Born–Infeld Capacity Lapse and Gravitational Time Dilation

The LorCap note identifies the construction of a temporal residual map $\mathcal{P}_\tau^{\mathrm{Q5b}}$ as the remaining step needed to close the Lorentz identification from O7 observables. The TempProj paper carries out this construction in the homogeneous regime and establishes the uniqueness of the resulting map. The present paper extends this construction to the non-homogeneous fibrewise regime; the homogeneous case is recovered as Corollary cor:homogeneous. In a non-homogeneous configuration a localised spectral occupancy $\epsilon(x)$ reduces the available BI capacity budget at each point $x$ to a local radius $R(x)^2 = 1-\epsilon(x)$. The Weil filling law and the Lorentzian splitting are unchanged. A fibre-by-fibre uniqueness argument forces the fibrewise temporal residual map: \[ \mathcal{P}_{\tau,x}^{\mathrm{Q5b}}\Sigma^{\mathrm{tot}}(x,n) \;=\; R(x)\,\sqrt{f(x,n)\bigl(2-f(x,n)\bigr)}, \] where $f(x,n) = 1-\beta(x,n)$ and $\beta(x,n)$ is the local normalised spatial mobility. Gravitational time dilation is identified as the position-dependence of the temporal residual capacity: \[ \frac{d\tau}{dn}\bigg|_x = R(x)\,\sqrt{1-\beta^2} = \frac{R(x)}{\gamma}. \] The local capacity radius is identified metrically as $R(x)^2 = -2\,g_{\tau\tau}(x)$, where $g_{\tau\tau}(x)$ is the inverse metric temporal component and the factor of $2$ comes from the flat-space reference $g_{\tau\tau}^\infty = -1/2$ (Q11). In the weak-field regime, the Newtonian profile of $g_{\tau\tau}(x)$ established in the Gravity paper gives $R(x) \approx 1 + \Phi_N(x)/c^2$, recovering gravitational time dilation as a consequence of the local BI capacity budget. The open spectral problem is to derive $R(x)$ directly from the local occupancy $\epsilon(x)$ without importing the Einstein equations (LC-O2-O1). Interpretive reading. The purpose of this note is not to rederive the Einstein equation. The infrared Einstein response is supplied by the renormalized spectral entropy functional of the Gravity paper. The purpose is instead to identify the local kinematic meaning of gravitational time dilation in the capacity-flow language: a non-uniform projected configuration reduces the local Born–Infeld capacity radius, and the temporal residual map becomes fibrewise. In summary: special-relativistic time dilation is the homogeneous capacity trade-off between spatial mobility and temporal residual capacity; gravitational time dilation is the fibrewise version of the same trade-off, in which the available capacity radius is reduced by local spectral occupancy.