Signature Emergence from Orthogonal Spectral Decomposition: A Toy Model Inspired by Penrose Conformal Boundary

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

michaelallan_bahtaji@tup.edu.ph; michaelallanbahtaji336@gmail.com

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