Renewal Geometry and the Emergence of Lorentzian Spacetime
Abstract
We construct a finite predictive framework in which Lorentzian spacetime and vacuum Einstein dynamics are reconstructed from operational renewal data. Quotienting histories by equality of their conditional futures yields a canonical predictive carrier, a positive history algebra, and an intrinsic word-Gram hierarchy; together these define the Renewal Grand Tensor, from which relational geometry is reconstructed. Two independent finite criteria select three spatial dimensions, while one additional protected signed direction fixes a (1,3) Lorentzian signature. Calibrated renewal statistics then reconstruct the spatial metric, lapse, and shift. A variational structure is reconstructed directly from the finite dynamics. On certified noncollapsed continuum branches, it converges to Palatini--Holst gravity and yields the vacuum Einstein equation, while failure of the certification conditions produces explicit obstructions. The same framework also supports a cutoff-uniform nonsymmetric 3+1 class with arbitrary three-dimensional dependence, stable across an open family of renewal laws sharing the same continuum limit.
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BibTeX
@misc{pelissier2026renewal,
title = {{Renewal Geometry and the Emergence of Lorentzian Spacetime}},
author = {Aurelien Pelissier},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05744772v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Renewal Geometry and the Emergence of Lorentzian Spacetime
AU - Aurelien Pelissier
PY - 2026
DA - 2026/09/21
PB - HAL
UR - https://hal.science/hal-05744772v2
AB - We construct a finite predictive framework in which Lorentzian spacetime and vacuum Einstein dynamics are reconstructed from operational renewal data. Quotienting histories by equality of their conditional futures yields a canonical predictive carrier, a positive history algebra, and an intrinsic word-Gram hierarchy; together these define the Renewal Grand Tensor, from which relational geometry is reconstructed. Two independent finite criteria select three spatial dimensions, while one additional protected signed direction fixes a (1,3) Lorentzian signature. Calibrated renewal statistics then reconstruct the spatial metric, lapse, and shift. A variational structure is reconstructed directly from the finite dynamics. On certified noncollapsed continuum branches, it converges to Palatini--Holst gravity and yields the vacuum Einstein equation, while failure of the certification conditions produces explicit obstructions. The same framework also supports a cutoff-uniform nonsymmetric 3+1 class with arbitrary three-dimensional dependence, stable across an open family of renewal laws sharing the same continuum limit.
ER -