The Pulsation as a Bipartition: Complete Classification of C1-Admissible Dynamical Laws, and a Singleton Obstruction Theorem for Frustration-Based Selection (Projective Dynamic Logo Framework — Document D68)

Abstract

The founding question of the PDL programme — what distinguishes something that exists from something that does not — is answered in axiom C1 by the postulate of a repeatable, non-trivial binary alternation. The corpus has never argued for C1; it has only assumed it. This document supplies the missing argument, quarantined in an appendix as motivation rather than proof, and then determines exhaustively what C1 can and cannot mean. Three results are established as unconditional theorems of C1–C4. First, a C2-admissible sign configuration on a complete signed graph is exactly a bipartition of its vertices, and the pulsation is exactly a switching by a fixed vertex subset: an involution, hence a two-cycle by construction. Second, the pulsation laws compatible with C1 are completely classified. On n entities there are exactly 2(2^n − 2) admissible laws, falling into exactly two families and no third, and generating exactly 2^(n−1) − 1 distinct relational dynamics — in bijection with the non-trivial coherent configurations. In particular, a strictly universal simultaneous inversion of every entity is the trivial element: it is not incoherent but empty, being the exact gauge redundancy already identified as the U(1) global phase in D46. What survives is a partition of the entities into two phase classes, all entities sharing one period and differing only by a binary offset, each changing state exactly once per relational cycle. Third, the set of frustrated triangles is pointwise invariant under every switching, so the classification is independent of frustration and is not a property of the balanced idealisation. Two negative results follow, both first-class. C4 is blind to the pulsation: the frustration count is identical across all 2^(n−1) − 1 candidate bipartitions, so no minimisation of leakage can select one. And the natural refinement that does select — minimising the frustration carried across the pulsating boundary — always selects a cut of size one, that is, a single privileged entity, which is inadmissible in a relational theory. This last statement is proved here as a theorem for all n and all cut sizes; the proof turns entirely on the two-graph parity condition, and is reproduced exactly on the extremal family. A corollary excludes every functional built from frustration and group size alike. Section 10 carries every object of the paper through explicitly on K4, configuration by configuration, so that the results can be checked by hand before any script is run. Three errata are recorded: two in D46, verified against its source, and one in D43, where an arithmetic bridge between simulation and theorem does not close; correcting the latter shows that the excess of 101 which the corpus has carried unexplained is exactly n_K + (Δn+1)² = 76 + 25. A divergence between D46's definition of the pulsation and the one derived here is recorded as an open problem. Five verification scripts, exhaustive up to n = 7, are deposited with this document.