Logical Leakage as Self-Maintained Probability: A Topological Reformulation in the Projective Dynamic Logo Framework

The Projective Dynamic Logo (PDL/LDP) framework represents 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 pre-specified spacetime or fundamental fields. Within this framework, a small but irreducible leakage parameter \(\eta\) arises at the proton level as the minimal fraction of frustrated (violated) triangles compatible with a hierarchical organisation into valence cores, relational sea, and active surface, and reappears in relational expressions for gravitational and thermal couplings. In this note, we show that the leakage functional \(\eta(G)\) can be reinterpreted as an intrinsic non-closure probability: for any finite signed graph \(G\), \(\eta(G)\) coincides with the probability, under a natural uniform measure on vertex triplets, that a randomly selected triangle is violated. At the proton level, \(\eta\) thus acquires the status of a self-maintained probability, determined by the same coherence-optimisation principle that attempts to minimise it, rather than by an externally imposed stochastic postulate. On this basis, we introduce a topology of coexistence on the space of closed configurations, generated by logical compatibility relations, and define a coherence-cost pseudometric that quantifies the minimal additional leakage required to sustain multiple closures within a common neighbourhood. In this setting, the emergent metric previously proposed in PDL/LDP is reformulated as a cost-based distance, and gravitation is interpreted as a large-scale modulation of coherence cost induced by hierarchically filtered leakage. We then distinguish a local logical entropy, associated with the leakage probability within individual configurations, from a global topological entropy, associated with the distribution of effective leakage levels over the configuration space when many structures coexist. This yields a logical entropic paradox: although the optimisation functional drives each structure towards minimal leakage and hence low local entropy, the aggregation of many such nearly optimal closures into rich hierarchical assemblies generically gives rise to high global entropic complexity. Finally, we outline how this probabilistic and topological reformulation interfaces with existing PDL/LDP reinterpretations of Born-type probabilities, gravitational coupling, and Boltzmann’s constant, thereby suggesting that quantum statistics, gravitation, and thermodynamic granularity may be construed as distinct macroscopic manifestations of a single, logically enforced imperfection of closure in a finite relational network.