Temporal Residual Map from the Completed Principal Symbol

This repository contains the source of the paper Temporal Residual Map from the Completed Principal Symbol: Construction, Uniqueness, and the Lorentz Identification (TempProj of the Lorentz Capacity sub-programme).

Overview

The companion paper LorentzCapacity (Beau2026n) identifies the construction of a temporal residual map $\mathcal{P}_\tau^{\mathrm{Q5b}}$ as the remaining step to close the Lorentz identification from O7 observables, but defers its explicit construction.

The present paper carries out this construction and proves the map is the unique positive function compatible with three inputs simultaneously:

1. (L) Lorentzian splitting — the Q5b completed co-metric $g^{\mu\nu} = 2\eta^{\mu\nu}$ determines a unique timelike direction $\partial_\tau$.
2. (BI) Born-Infeld capacity sphere — the admissible capacity flow vector $\vec{F}_n$ lies on the unit sphere $(F_n^\tau)^2 + |\vec{F}_n^{\mathrm{spat}}|^2 = 1$.
3. (W) Weil filling law — $\Sigma_n^{\mathrm{tot}} = 1 - B_n$ exactly.

Main Result

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

Consequently: $$\Phi(\eta_n^\tau) = \frac{1}{\sqrt{1+(\eta_n^\tau)^2}} \longrightarrow \frac{1}{\gamma}$$

The three inputs are individually necessary: removing any one leaves the formula undetermined. This explains retroactively the numerical failure of all three candidates tested in LorentzCapacity.

Status

• Main theorem (uniqueness + explicit formula): proved (conditional on [U])
• Lorentz factor corollary: proved (conditional on [U])
• Three-input necessity (Proposition~\ref{prop:failure}): proved

Position in the sub-programme

LorCap → TempProj → LCII

TempProj closes the open problem of LorCap. LCII (Beau2026lco2) extends the result to non-homogeneous regimes (gravitational time dilation).

Dependencies

Paper	Role
LorCap (Beau2026n)	Origin of the open problem
Q5b (Beau2026q5b)	Effective operator and co-metric
Q11 (Beau2026q11)	$A_\tau = 2$, co-metric completion
Q10 (Beau2026q10)	$A_H = 2$, spatial isotropy
Q8 (BeauQ8)	$A_z = 2$, central-direction coefficient
U1 (Beau2026u1)	Proves hypothesis [U]
O7 (Beau2026a11)	Defines $B_n$ and $\Sigma_n^{\mathrm{tot}}$
BI (Beau2026c)	Born-Infeld admissibility budget
PTO (Beau2026pto)	Projective temporal ordering (structural context)

Repository Contents

temporal-projection/
├── out/              # Compiled PDF
├── tex/
│   ├── temporal-projection.tex
│   └── cosmochrony-bibliography.bib
├── compile.sh
├── zenodo.json
└── README.md

Citation

J. Beau, Temporal Residual Map from the Completed Principal Symbol: Construction, Uniqueness, and the Lorentz Identification, Preprint, 2026.

Acknowledgements

Portions of the editorial refinement benefited from iterative interactions with large language models used as analytical assistants. All claims, interpretations, and final formulations remain the sole responsibility of the author.