Relational Emergence of Born's Rule and a Golden-Ratio Active Surface in PDL

We analyze how two fundamental quantum-theoretic ingredients—Born’s rule for spin-$1/2$ measurements and the fine-structure constant governing proton–electron interactions—can be reformulated within the Projective Dynamic Logo (PDL) framework as emergent properties of finite active surfaces embedded in a relational hierarchy. In PDL, physical entities are represented as finite signed graphs selected by coherence and minimality principles, with the minimal $(4,6)$ closure serving as a logical prototype for the electron, and a two-level hierarchy comprising valence cores and a relational sea modelling the proton. Building on earlier work, in which the proton’s active surface $R_{\mathrm{surf}}=\varphi\,r_{\mathrm{val}}/3$ yields a structural relation $\alpha^{-1}\simeq\gamma^2/R_{\mathrm{surf}}^2$ that reproduces the empirical fine-structure constant without continuous fit parameters, we generalize the concept of active surface to a Stern–Gerlach-type spin measurement apparatus. In this construction, spin eigenstates correspond to two stationary sign configurations on the $(4,6)$ block, while quantum superpositions are encoded as weights on mixed triangles that couple the $(4,6)$ structure to two disjoint interface branches representing the apparatus. A weighted coherence cost assigned to these mixed triangles, together with an exponential selection over global graph configurations, induces effective probabilities for the two outcome basins that coincide with Born’s rule for spin-$1/2$ measurements. We additionally introduce an instrumental coupling parameter $\alpha_{\mathrm{spin}}$ that links the apparatus’ internal coherence budget to the extent of its active surface, in direct analogy with the protonic interpretation of $\alpha$. This unified perspective interprets both proton structure and spin measurement as instances of coherence allocation between internal relational hierarchies and finite active surfaces, and it motivates a potential role for the golden ratio as a relational optimization factor. The status of these results is explicitly propositional: they demonstrate that the PDL framework can reproduce both the functional dependence of Born’s rule and the numerical value of $\alpha$, while leaving open the questions of uniqueness and of rigorously deriving the emergence of the golden ratio from the underlying axioms.