Temporal Residual Map from the Completed Principal Symbol: Construction, Uniqueness, and the Lorentz Identification

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 present paper carries out this construction and proves that the map is the unique positive function compatible with three inputs simultaneously: the Lorentzian temporal splitting induced by the completed effective co-metric $g^{\mu\nu} = 2\eta^{\mu\nu}$ of Q5b–Q11, the Born–Infeld unit capacity-flow sphere (the quadratic budget hypothesis expressing the Born–Infeld capacity bound), and the exact Weil linear filling law $\Sigma_n^{\mathrm{tot}} = 1 - B_n$ established in LorCap. The completed principal symbol determines the Lorentzian temporal splitting of the effective tangent bundle; it does not by itself define a non-linear action on the scalar observable $\Sigma_n^{\mathrm{tot}}$. The Born–Infeld admissibility budget provides the canonical unit capacity-flow lift, and the Weil filling law identifies the spatial component $B_n$. Their composition defines the scalar temporal residual map \[ \mathcal{P}_\tau^{\mathrm{Q5b}} \colon \Sigma_n^{\mathrm{tot}} \;\longmapsto\; \sqrt{\Sigma_n^{\mathrm{tot}}\bigl(2 - \Sigma_n^{\mathrm{tot}}\bigr)} \;=\; \sqrt{1 - B_n^2}. \] This map is not an idempotent projection on scalar observables; it is the scalar shadow of a linear Lorentzian projection composed with the Born–Infeld lift. The three inputs are individually necessary: removing any one of them leaves the formula undetermined. This necessity retroactively explains the numerical failure of all three candidate maps tested in LorCap. LorCap Lemma 1 follows, and with it LorCap Theorem 1: $\Phi(\eta_n^\tau) = (1+(\eta_n^\tau)^2)^{-1/2} \to 1/\gamma$. Interpretive reading. The Lorentz factor is thereby read as the reciprocal temporal residual of a bounded capacity-flow split, conditional on the Born–Infeld lift.