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Separation as a claimed necessity for coherence

Abstract

Separation as a claimed necessity for coherence.

The original endpoint parameters place the actual seven-cycle singletons and their full containment edges in the complete graph. Their unique addresses force each outgoing source label and successor. Every reachable orbit pair stays on that orbit, so the returning component synchronizes both source projections and their ordered label pairs. The zero-label feeding head cannot be reached from this component.

Definition 1.1 (Necessary finite separation assertion).

Lean statement: D5/S1/Digit/Infinite/SevenCycleCoherenceRefutation.claim

Formalization. D5/S1/Digit/Infinite/SevenCycleCoherenceRefutation.claim (✓ std3).

Source. Repository-derived.

Commentary.

The assertion quantifies over every positive subcritical budget and original endpoint graph. It requires finite future separation whenever an actual odd primitive singleton rival has its first shifted singleton, with its actual guard, as the second projection of a vertex in a cyclic strongly connected component reachable from the two zero initial guards after an equal-label history and its first unequal-label edge, and that component’s returns satisfy the original synchronization condition for both source projections and for ordered source pairs. For each of these three labelings, coherence means that at some component vertex every pair of nonempty returns synchronizes at the least common multiple of their lengths. The product graph retains all original pieces, permitted colors and full containment edges.

References