The research
library.

PTG I: A Metric-Free Transport Framework for Scalar Fields

Graham Fincham, Daniel Hilton

PTG I lays the foundation. Strip a manifold down to the bare minimum: a smooth scalar field (the phase), a cone of allowed transport directions at every point, and a torsion-free c…

PTG II: The Phase-Raychaudhuri Equation and Finite-Parameter Focusing

Graham Fincham, Daniel Hilton

PTG II derives PDT's version of the Raychaudhuri equation, the workhorse identity that controls how bundles of trajectories spread or focus. We do it without a metric. With only a …

PTG III: Transport Capacity, Completeness, and Incompleteness Theorems

Graham Fincham, Daniel Hilton

PTG III turns local focusing into a global obstruction. We define transport capacity, a non-negative measure on hypersurfaces compatible with the transport cone, and a notion of tr…

PTG IV: Transport Horizons and Boundary Geometry

Graham Fincham, Daniel Hilton

PTG IV asks what the failure of completeness leaves behind. Each incomplete congruence ends on a canonical terminal hypersurface, the transport horizon, that carries a limiting tra…

PTG V: Quasi-Local Invariants on Transport Horizons

Graham Fincham, Daniel Hilton

PTG V equips the transport horizon with three quasi-local invariants: a transverse area, a capacity-weighted invariant, and a curvature-type invariant built from the limiting shear…

PTG VI: Transport Entropy and Monotonicity

Graham Fincham, Daniel Hilton

PTG VI builds a single scalar functional, the transport entropy, by weighting the three quasi-local invariants from PTG V. The entropy is non-negative, finite on every transported …

PTG VII: Membrane Evolution on Transport Horizons

Graham Fincham, Daniel Hilton

PTG VII makes the horizon dynamical. The entropy gradient, combined with a shear-dissipating term, defines a canonical vector field on the transport horizon. The resulting first-or…

PTG VIII: Structural Admissibility and Uniqueness of Transport Functionals

Graham Fincham, Daniel Hilton

PTG VIII proves that PDT's transport functionals are not arbitrary choices. The focusing functional, the capacity density, and the entropy weights are uniquely determined, up to fi…

PTG IX: Constitutive Closure and Autonomy of the Framework

Graham Fincham, Daniel Hilton

PTG IX closes the programme. Every admissible focusing functional and capacity density arises from a single constitutive tensor; no other admissible scalar, vector, or tensor funct…

PTG-1: Closure Architecture and the Minimal Phase Manifold

Graham Fincham (Delta Phi IP Ltd); Daniel Hilton (Independent Researcher)

PTG-2: Compact Transport Geometry and Curvature Loading

Graham Fincham (Delta Phi IP Ltd); Daniel Hilton (Independent Researcher)

PTG-3: Closure-Compatible Metric Selection

Graham Fincham (Delta Phi IP Ltd); Daniel Hilton (Independent Researcher)

PTG-4: Sector Structure, Chirality, and Structural Algebras

Graham Fincham (Delta Phi IP Ltd); Daniel Hilton (Independent Researcher)

PTG-5: Defect Mass Scaling and Return-Closure Invariants

Graham Fincham (Delta Phi IP Ltd); Daniel Hilton (Independent Researcher)