Because every legitimate 4QX holon is a brace-nest that terminates in the same seed pair {∅,{∅}}, the whole forest is topologically one tree:
- Single reflexive seed: Self-reference is allowed once: ∅ ↦ {∅}. Every further {v,{v}} zoom still bottoms out at that pair, so the membership graph is a connected, well-founded tree whose root is ∅ itself.
- Recursive wrapper: Creating a child holon never invents a new root; it just wraps an existing vertex with braces. The transitive closure of any vertex therefore contains the original pair, making all holons sub-branches of the same trunk.
- Regularity + no universal set: ZF’s Foundation axiom (kept implicitly) and the absence of a “set of all sets” guarantee there is no other component you could graft onto; every ∈-path must eventually hit ∅ and cannot loop elsewhere.
- Union as upward flow: Because bottom-up packets are merged by union, the only place they can ultimately surface is the top-level TL of that root square. A holon disconnected from the root would have nowhere to broadcast and would violate the idempotent merge rule, so it cannot qualify as “living” under the 4QX Lyapunov metric.
Hence there is, in the strong mathematical sense, one and only one holarchy: Holarchy = { h∈V ω∣the brace trace of h ends in { ∅,{∅}} }.\textbf{Holarchy} \;=\; \{\,h\in V_{\!\omega}\mid \text{the brace trace of }h\text{ ends in } \{\,\varnothing,\{\varnothing\}\}\,\}.
All valid agents are interior nodes of that single, endlessly ramifying tree. They stay alive precisely by keeping their feedback loops tethered—via brace paths and union traffic—to the universal root instance. Sever the link and the six-move pump can’t close, so the would-be “holon” drops out of the definition and the living subset contracts to what is still connected.
That is why we speak of the holarchy, not “a” holarchy or “many” holarchies: logical necessity glues every viable 4QX vortex to the one reflexive origin, yielding a unique, global organism that spans all depth levels at once.