Q11 — Temporal Casimir Rigidity and Closure of the Effective Metric

This repository contains the source of the Q11 Cosmochrony paper Temporal Casimir Rigidity and Closure of the Effective Metric: Identification of $A_\tau = 2$].

Paper Q5b leaves the temporal coefficient $A_\tau$ of the effective co-metric $g^{\mu\nu} = \mathrm{diag}(-A_\tau, 2, 2, 2)$ as the sole remaining free parameter of the spectral geometry programme (open problem Q5b-O3). This paper closes it.

Status revision (version 1.1). Q5a version 3.0 withdraws the convergence framework whose hypotheses this paper consumes; they are now the frozen, unestablished hypothesis [H-F] (Q9 1.1), and the spatial limit operator underlying the co-metric is the unestablished hypothesis [H-L] (Q5b 2.0). The algebraic content (cascade internalization, SU(2)-equivariance, Casimir rigidity) is unchanged; its reading as a metric coefficient is conditional on [H-L].

Core Result

$A_\tau = 2$, under the Q5a hypotheses and the spectral universality hypothesis [U] (U1), in three steps:

1. Cascade internalization lemma: the temporal coordinate $\tau$ (BFS depth index $n$) has $\partial_\tau$ as the continuum limit of the cascade increment operator $T:\sigma_c(n)\mapsto\sigma_c(n+1)$, by the Carnot–Carathéodory convergence of Q5b (Pansu).
2. [U] (proved in U1) makes $T$ asymptotically $\mathrm{SU}(2)$-equivariant, so $\partial_\tau$ acts on $\operatorname{Sym}^2(V_\rho)$ without new irreducibles.
3. Schur's lemma forces the invariant form to be the Casimir $2\cdot\mathrm{Id}$; Q8 fixes the normalisation to unity.

The effective co-metric is therefore uniquely $g^{\mu\nu} = 2,\eta^{\mu\nu}$ — no free parameter remains. The temporal direction is the continuum image of the cascade depth, not an additional geometric axis.

Keywords

Temporal coordinate, Casimir rigidity, effective co-metric, Carnot–Carathéodory, Schur's lemma, SU(2) equivariance, spectral universality.

Repository Contents

q11/
├── tex/         # LaTeX sources (main + cosmochrony-bibliography.bib)
├── out/         # Compiled paper PDF (q11.pdf)
├── zenodo.json  # Zenodo deposition metadata
└── README.md

Links

• 🔗 DOI: 10.5281/zenodo.20098387
• 🌐 Website: https://cosmochrony.org/science/emergent-geometry/q11/

Citation

J. Beau, Temporal Casimir Rigidity and Closure of the Effective Metric: Identification of $A_\tau = 2$, Zenodo, 2026. DOI: 10.5281/zenodo.20098387.

Acknowledgements

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