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 $\hbar$, $\alpha$, $k_B$ and an effective gravitational coupling $\Geff$ 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 $\HDirac \cong \Hcycl \otimes \Hspin$, where $\Hspin$ carries the spin-$\tfrac{1}{2}$ structure of the $(4,6)$ block and $\Hcycl$ 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 $m_e^{\PDL}$ and fine-structure constant $\alphaPDL$ determined by coherence. Its nonrelativistic limit reproduces the previously established Schr"odinger/Pauli hydrogen Hamiltonian with $\alphaPDL$, 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 $\Geff$ from relational density and a small leakage parameter $\etaL$ amplified from nucleonic to macroscopic scales. A coherence tensor $\Ccoh$ is introduced to collect mass-like, spin-related, orbital and leakage-induced contributions, and an Einstein-type relation $\mathcal{G}_{\mu\nu}[g^{\mathrm{eff}}]=\kappa_{\PDL}\,\Ccoh$ is proposed in which both the effective metric $g^{\mathrm{eff}}_{\mu\nu}$ and the coupling $\kappa_{\PDL}$ are fixed by the same relational architecture as $m_e^{\PDL}$, $\alphaPDL$ and $\Geff$. As a concrete application, the gravitational redshift of hydrogenic levels in a weak coherence potential $\Phi_{\mathrm{coh}}\sim -\Geff M/r$ is derived and shown to coincide with the standard weak-field general-relativistic expression when written in terms of $g^{\mathrm{eff}}_{00}$. 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 $\alphaPDL$, and weak-field redshift with $\Geff$—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.