On the Combinatorial Selection and Local Uniqueness of the Proton Architecture in the PDL Framework

This paper develops a combinatorial foundation for the proton architecture within the Projective Dynamic Logo (PDL) framework. In PDL, physical reality is modelled as a network of minimal logical closures on finite signed graphs, with the (4,6) block playing the rôle of an elementary electron-like closure. At the composite level, previous PDL work proposed a specific integer architecture for the proton, encoded by the quintuplet (n_u,n_d,r_{\mathrm{val}},R_{\mathrm{sea}},R_{\mathrm{tot}}) = (24,28,930,10087,11017), and used this structure to derive the fine-structure constant, stationary/dynamical fractions, and active surfaces. However, the status of this architecture as a selected or unique configuration remained only partially formalised.In this work, the proton problem is recast as an explicit combinatorial selection problem on integer parameters and signed graphs. A set of structural and phenomenological constraints is formulated (valence multiplicities in tetrads, quasi-completeness of valence cores, stationary versus dynamical fractions, existence and scaling of an active surface, reproduction of the fine-structure constant, and global closure), and translated into algebraic conditions on the integer quintuplet. On this basis, the paper defines the notion of an \emph{admissible proton configuration} and introduces a selection functional S that aggregates fraction fidelity, agreement with the empirical \alpha, coherence quality, and minimality.The main technical result is that the PDL proton quintuplet (24,28,930,10087,11017), together with its associated graph hierarchy, satisfies all admissibility conditions and achieves a high selection score within a bounded domain of integer parameters. The paper then formulates a local combinatorial uniqueness conjecture: within a suitable neighbourhood of this quintuplet in parameter space, and under fixed tolerance thresholds, the PDL configuration is expected to be either the only admissible tuple or the only one realising a strict local maximum of S. A concrete search programme combining analytical reductions and numerical exploration is outlined.These results support the interpretation of the PDL proton architecture not as an arbitrary choice, but as a structurally selected configuration in a non-trivial discrete landscape. Together with the minimality of the (4,6) block, they provide a more rigid structural backbone for the PDL programme, in which both electron- and proton-like closures are constrained by combinatorial and optimisation principles rather than by free parameter tuning.