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

Current Zenodo release: version 1.5 (2026-10-26). Official title: Temporal Residual Map from the Completed Principal Symbol: Construction, Uniqueness, and the Lorentz Identification.

Overview

LorentzCapacity reduces the Lorentz-factor limit of the O7 saturation parameter, under a quadratic Born–Infeld budget hypothesis, to the construction of a temporal projection $\mathcal{P}_\tau^{\mathrm{Q5b}}$ (its Lemma 1). The present paper constructs it.

Three inputs play distinct roles: the Born–Infeld unit capacity-flow sphere; the Weil filling law; and the temporal splitting of a supplied rank-four co-metric of the conditional form of Q11, $g^{\mu\nu} = \mathrm{diag}(-A_\tau, 2\lambda, 2\lambda, 2\lambda)$ with $A_\tau$ and $\lambda$ independent and fixed by no source. The explicit formula $\mathcal{P}_\tau^{\mathrm{Q5b}}\,\Sigma_n^{\mathrm{tot}} = \sqrt{1 - B_n^2}$ is proved. The sphere and the filling law alone fix the values of the map at the values of $\Sigma_n^{\mathrm{tot}}$ actually attained, a finite set of points for depths $n \le (q-1)/2$, and only there; the splitting is not necessary for this uniqueness. It identifies the value as the temporal component of the capacity flow, using only the decoupled temporal block: no coefficient value enters, and neither does the signature; the normalisation $g^{\mu\nu} = 2\eta^{\mu\nu}$ is a choice of units.

Scope statement. The construction and the uniqueness theorem are proved under a supplied co-metric (the hypotheses of Q11 Corollary 6.1), the identification of $B_n$ with the spatial capacity flow, which is a hypothesis of the paper, and the Born–Infeld lift; hypothesis [U] enters no step. Reading the normalised load as the velocity of an observer requires [H-cone]: the light-cone speed of the co-metric equals the Born–Infeld bound per ordering step. This is the coupling datum that Q11 leaves open, and no source derives it. That reading also uses the Lorentzian signature of the co-metric. In the window of the theorem the load is at most $1/2 + 1/(2q^2)$; the light cone $B = 1$ is reached by the exact torus load of LC-O3.

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, exactly: $\eta_n^\tau = B_n/\mathcal{P}_\tau^{\mathrm{Q5b}}\Sigma_n^{\mathrm{tot}} = \beta\gamma$ and $\Phi(\eta_n^\tau) = 1/\gamma$. Under [H-cone], with the Lorentzian signature, $\beta$ is the velocity of the observer in units of the light-cone speed, and this is time dilation.

Key contributions

Explicit formula: $\mathcal{P}_\tau^{\mathrm{Q5b}}\Sigma_n^{\mathrm{tot}} = \sqrt{1-B_n^2}$ (proved under the stated hypotheses).
Scalar uniqueness: the BI sphere and the Weil filling law alone fix the values at the attained values of the total residual, and only there; the temporal splitting is not necessary for it (proved).
Identification: the temporal splitting identifies that value as the temporal component of the capacity flow; it does not select it.
Candidate maps: the three candidates tested in LorentzCapacity all fail the quadratic law; the depth proxy uses the normalised depth in place of $B_n$.
Lemma 1 of LorentzCapacity: established under the same hypotheses.
[H-cone] named: the identification of the light-cone speed with the Born–Infeld bound, on which the kinematic reading rests.

Structure

Introduction — the open construction of LorentzCapacity and the scope of each input.
Setup — the supplied co-metric of the form of Q11 Corollary 6.1 with two independent coefficients, the quadratic budget hypothesis, and [H-cone].
Capacity-flow bundle — the unit Born–Infeld sphere and the identification of $B_n$.
Construction and uniqueness — the explicit formula, the scalar uniqueness and the identification of the component.
What each input contributes — the quadratic law is necessary for the uniqueness and the filling law for its closed form, the splitting is not; comparison with the LorentzCapacity candidates.
Lorentz identification — $\Phi(\eta_n^\tau) = 1/\gamma$, read as time dilation under [H-cone] and the Lorentzian signature.
Logical status — status of every input, and of LC-O1 to LC-O3.

Dependencies

LorentzCapacity: the open construction (Lemma 1), the Weil filling law, and the O7 observables $B_n$ and $\Sigma_n^{\mathrm{tot}}$.
Q5b: a co-metric of rank three under [H-L] and [H-lift]; its rank-four extension is open (Q5b-O2).
Q8: $A_H = A_Z = 2\lambda$ with $\lambda$ free for a supplied invariant form; no value of $A_Z$.
Q10: no coefficient value is derived.
Q11: the conditional rank-four form $\mathrm{diag}(-A_\tau, 2\lambda, 2\lambda, 2\lambda)$ (Corollary 6.1), with $A_\tau$ and $\lambda$ independent; the modelling identification $\tau = n\,\Delta\tau$ with a free step; the coupling datum is open (Q11-O1).
U1: hypothesis [U] is open; it is not used.
Born–Infeld paper: the capacity bound [A-cap]; the quadratic budget is a hypothesis of LorentzCapacity.

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 constructs the map left open in LorentzCapacity and provides the flat-space baseline for the non-homogeneous extension LCII. Together with LorentzCapacity, it gives a conditional account of the Lorentz factor as the temporal residual of a bounded capacity budget: conditional on a supplied co-metric, on the identification of $B_n$ and, for the kinematic reading, on [H-cone].