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

Overview

The paper LorentzCapacity established that the Lorentz factor emerges from the O7 saturation parameter $\eta_n^\tau$ under the Born–Infeld admissibility budget, but left the explicit construction of the temporal projection operator $\mathcal{P}_\tau^{\mathrm{Q5b}}$ as an open problem (Lemma 1). The present paper closes this gap.

The key observation is that three structural inputs — the Lorentzian temporal splitting from Q5b, the Born–Infeld unit capacity-flow sphere, and the Weil filling law — together uniquely determine the map. The explicit formula $\mathcal{P}_\tau^{\mathrm{Q5b}}\,\Sigma_n^{\mathrm{tot}} = \sqrt{1 - B_n^2}$ is proved, and the uniqueness result shows that each of the three inputs is individually necessary: removing any one leaves the formula undetermined.

Scope statement. The main construction and uniqueness theorem are proved unconditionally under hypothesis [U]. This closes Lemma 1 of LorentzCapacity and completes the flat-space Lorentz identification chain.

Main result

The temporal residual map is determined explicitly:

$\mathcal{P}_\tau^{\mathrm{Q5b}}\,\Sigma_n^{\mathrm{tot}} = \sqrt{\Sigma_n^{\mathrm{tot}}\bigl(2-\Sigma_n^{\mathrm{tot}}\bigr)} = \sqrt{1-B_n^2}$

Consequence: $\eta_n^\tau = B_n/\mathcal{P}_\tau^{\mathrm{Q5b}}\Sigma_n^{\mathrm{tot}} \to \beta\gamma$ and $\Phi(\eta_n^\tau) \to 1/\gamma$.

Key contributions

Explicit formula: $\mathcal{P}_\tau^{\mathrm{Q5b}}\Sigma_n^{\mathrm{tot}} = \sqrt{1-B_n^2}$ (proved).
Uniqueness: the three inputs (Lorentzian splitting, BI sphere, Weil filling law) are individually necessary — removing any one leaves the formula undetermined (proved).
Failure analysis: explains the numerical failure of all three candidates tested in LorentzCapacity.
Lemma 1 closed: Lemma 1 of LorentzCapacity proved unconditionally (under [U]).

Structure

Setup and the open problem — recap of the LorentzCapacity framework and the precise statement of Lemma 1.
The three structural inputs — Lorentzian splitting (Q5b), BI capacity sphere, and Weil filling law.
Construction of the map — derivation of the explicit formula from the intersection of the three constraints.
Uniqueness theorem — proof that each input is individually necessary.
Failure of the three prior candidates — retrospective analysis of the numerical tests in LorentzCapacity.
Consequences — Lemma 1 closed; $\Phi(\eta_n^\tau) \to 1/\gamma$ unconditionally; flat-space baseline for LCII.

Dependencies

O7: definition of $\eta_n^{\mathrm{O7}}$, $B_n$, $\Sigma_n^{\mathrm{tot}}$, and the reduced filling model.
Q5b: BFS-to-Carnot–Carathéodory convergence; Lorentzian temporal splitting.
Q11: co-metric closure fixing the Lorentzian signature $g^{\mu\nu} = 2\eta^{\mu\nu}$.
Q10: spectral saturation regime and pre-saturation analysis.
Q8: structural constraints on the projective capacity budget.
U1: hypothesis [U] (universality of the filling law).
Born–Infeld paper: mobility budget $c_{\mathrm{BI}}$ and the unit capacity-flow sphere.
LorentzCapacity: open problem (Lemma 1) closed by this paper.
PTO: principal symbol operator formalism used in the construction.

References

Jérôme Beau. Temporal Residual Map from the Completed Principal Symbol: Construction, Uniqueness, and the Lorentz Identification. Preprint. 10.5281/zenodo.20316526

Relation to the Cosmochrony program

TempProj is the second paper of the Lorentz Capacity sub-programme. It closes the open construction identified in LorentzCapacity and provides the flat-space baseline for the non-homogeneous extension LCII. Together with LorentzCapacity, it completes the derivation of the Lorentz factor from the projective capacity framework, showing that special-relativistic kinematics is not an external postulate but a structural consequence of the spectral admissibility programme.