From Predictive Dynamics to Spectral Geometry
Abstract
In Alain Connes' noncommutative geometry, geometry is encoded by an algebra, a Hilbert space, and a Dirac operator taken as fundamental spectral data. We show that these structures can instead emerge from a more primitive predictive dynamics. A compatible first-order variation on the induced history algebra canonically generates a real even Hodge--Dirac spectral triple, while the algebra alone determines neither its metric scale nor the full Dirac operator. The resulting spectral geometry is an information-reduced description: dynamically inequivalent predictive systems can share the same effective spectral data. Nevertheless, finite geometries become reconstructible after finitely many calibrations, and on periodic three-dimensional lattice approximants the emergent spectral distance converges to a Riemannian metric reconstructed independently from the dynamics. Thus Connes-type spectral geometry may arise not as fundamental structure, but as an effective geometric description of richer underlying dynamics. This provides a concrete setting in which noncommutative geometry need not be fundamental, but may instead arise as a coarse-grained spectral description.
Related
BibTeX
@misc{pelissier2026from,
title = {{From Predictive Dynamics to Spectral Geometry}},
author = {Aurelien Pelissier},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05752672v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Predictive Dynamics to Spectral Geometry
AU - Aurelien Pelissier
PY - 2026
DA - 2026/09/17
PB - HAL
UR - https://hal.science/hal-05752672v1
AB - In Alain Connes' noncommutative geometry, geometry is encoded by an algebra, a Hilbert space, and a Dirac operator taken as fundamental spectral data. We show that these structures can instead emerge from a more primitive predictive dynamics. A compatible first-order variation on the induced history algebra canonically generates a real even Hodge--Dirac spectral triple, while the algebra alone determines neither its metric scale nor the full Dirac operator. The resulting spectral geometry is an information-reduced description: dynamically inequivalent predictive systems can share the same effective spectral data. Nevertheless, finite geometries become reconstructible after finitely many calibrations, and on periodic three-dimensional lattice approximants the emergent spectral distance converges to a Riemannian metric reconstructed independently from the dynamics. Thus Connes-type spectral geometry may arise not as fundamental structure, but as an effective geometric description of richer underlying dynamics. This provides a concrete setting in which noncommutative geometry need not be fundamental, but may instead arise as a coarse-grained spectral description.
ER -