Signature Emergence from Orthogonal Spectral Decomposition: A Toy Model Inspired by Penrose Conformal Boundary
Year: 2026
Type: article
Abstract: Signature Emergence from Orthogonal Spectral Decomposition: A Toy Model Inspired by Penrose Conformal Boundary Michael Allan A. Bahtaji, Ph.D. Department of Physics, Technological University of the Philippines Ayala Boulevard, Ermita, Manila, Philippines, 1000 [email protected]; [email protected] Dedication: Dedicated to Sir Roger Penrose Abstract. This presents a toy model for emergence of Lorentzian signature from a Euclidean-signature family via orthogonal spectral decomposition. Let be a smooth 4-manifold, an internal flow parameter, and with , , and , distinct. The decomposition separates rotation (, compact ) from anisotropic dilation (, positive spectrum), correcting category errors where fluid Jacobian is identified with orthogonal matrix. This show that under -regularization , , there exists such that is Riemannian for and Lorentzian for , with . The effective limit exists in and is away from crossing, with as allowing attachment of conformal boundary à la Penrose , . The flow , , is linear, globally regular in , , with , no finite-time blowup, in contrast to Navier-Stokes where Leray weak solutions and CKN partial regularity remain optimal. The construction is inspired by Penrose's conformal infinity and Etesi's program on emergent Lorentzian metrics, formalized in functional analytic setting , Littlewood-Paley showing no high-frequency cascade, and Yamabe-type elliptic equation for conformal factor at analogous to pressure Poisson . This provides Lean 4 formalization plan defining as structure with proof , preventing ill-typed identity where , , distinct, interval arithmetic enclosure for rigorous crossing, and SPDE extension for ergodic statistical stability of signature. The model does not solve Navier-Stokes, does not equate Penrose triangle defect with curvature, and does not claim as equality; it provides controlled type-safe mechanism for signature emergence. Keywords: Lorentzian signature emergence, orthogonal decomposition, Penrose conformal infinity, Etesi program, spectral flow, regularity, Littlewood-Paley, Yamabe equation, Lean 4 formal verification, interval arithmetic
Author: Bahtaji, Michael Allan
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213495641