Towards an Einstein--Dirac Unification from the Projective Dynamic Logo Framework

This record presents a theoretical physics manuscript that develops an explicit Einstein--Dirac scheme within the Projective Dynamic Logo (PDL/LDP) framework. PDL models physical reality as a network of minimal logical closures on finite signed graphs, deriving particles and coupling constants from coherence constraints rather than from a presupposed spacetime manifold or fundamental fields. Within this setting, the electron at rest is associated with a minimal $(4,6)$ closure, the proton is described as a tightly constrained hierarchical composite with a finite active surface, and several dimensionless constants such as , α, kB and an effective gravitational coupling are reinterpreted as coherence ratios fixed by this discrete architecture. The work has two main components. On the Dirac side, it constructs a PDL-inspired representation of Dirac spinors as ⊗, where carries the spin-1/2 structure of the $(4,6)$ block and encodes two internal coherence-cycle modes associated with positive- and negative-energy branches. In this representation, the Dirac matrices acquire a natural PDL form, and an effective Dirac--PDL equation is formulated for the electron in the PDL proton Coulomb field, with mass mₑ^ and fine-structure constant determined by coherence. Its nonrelativistic limit reproduces the previously established Schr\"odinger/Pauli hydrogen Hamiltonian with , giving an explicit Dirac--Schr\"odinger compatibility result within PDL. On the Einstein side, the article summarises a coherence-based construction of an emergent metric and an effective gravitational coupling from relational density and a small leakage parameter amplified from nucleonic to macroscopic scales. A coherence tensor is introduced to collect mass-like, spin-related, orbital and leakage-induced contributions, and an Einstein-type relation Gμν[gᵉᶠᶠ]=κ_\, is proposed in which both the effective metric gᵉᶠᶠμν and the coupling κ_ are fixed by the same relational architecture as mₑ^, and . As a concrete application, the gravitational redshift of hydrogenic levels in a weak coherence potential Φcoh~ - M/r is derived and shown to coincide with the standard weak-field general-relativistic expression when written in terms of gᵉᶠᶠ₀₀. The resulting scheme is not yet a full derivation of Einstein--Dirac dynamics from discrete PDL evolution. It provides a constrained target architecture and several nontrivial compatibility conditions—Dirac--Schr\"odinger correspondence, consistency of the hydrogenic spectrum with , and weak-field redshift with —that any future, fully unified Einstein--Dirac construction within the PDL programme will have to satisfy, with the hydrogen atom serving as a minimal test system.