Every Move so far has found structural roles wherever something is; the last escape is absolute structural absence: no states, no items, no ordering, no distinction, no bearer. This is not merely an empty world, since an objectless world may still have structure. The target is ∅, the joint absence of T, S, and Φ. If ∅ is possible, the role-level necessity claim fails.

> [!argument]
> **The Null-State Reductio** · *from the impossible empty world to Triconditional Necessity*
>
> 1. Suppose the world is empty after all: nothing runs, nothing is set apart, nothing carries over. That is the null state, ∅: the joint failure of Temporality, Spatiality, and Physicality together (¬T ∧ ¬S ∧ ¬Φ).
>
> 1. But that is no state at all. A state with nothing occurring, nothing distinct, and nothing persisting has no subject left to be a state of, so the null state cannot obtain (¬◇∅).
>
> 1. Ruling out the empty state, though, secures only that *at least one* Condition holds: □(T ∨ S ∨ Φ). It does not yet give all three.
>
> 1. The lift to the full conjunction is carried by the inseparability of the Conditions ([§1.4](<1.4 Three Conditions.md>)): they cannot come apart, so securing any one brings the others. There is no world running on time alone, or standing in space alone, or bearing substance alone. That closes the disjunction to the full conjunction: necessarily, Temporality, Spatiality, and Physicality hold together (□(T ∧ S ∧ Φ)).

*Why the null state is impossible.* The Chapter-2 line reached this verdict (¬◇∅) through heavy modal machinery, its own Null-State reductio. Here it rests on something lighter and earlier: the Chapter-1-level result that *something is the case* ([§3.2.1](<3. Convergence.md>)). Because that line needs no S5, the backstop is the most secure floor in the chapter, the one every other Move can fall back to. It has a second, independent support as well: to so much as assert that *nothing* is the case is itself something being the case, so the denial refutes itself ([§3.2.1](<3. Convergence.md>)).

*Why one Condition can't stand without the others.* This is where a recombination theorist pushes hardest: if T, S, and Φ were distinct existences, ruling out their joint absence would still leave worlds with one present and the other two gone. The reply runs on the inseparability of the Conditions ([§1.4](<1.4 Three Conditions.md>)), which denies the very distinctness recombination needs ([§3.2.2](<3. Convergence.md>)): secure any one and the other two come with it, so no partial world survives. That inseparability is claimed only at the level of the roles the Conditions play: direction, difference, and bearer. Upgrading those roles to their full physical types, namely real succession, extension, and persistence, is the separate work of [§3.5.3](<3. Convergence.md>).

*The subtraction argument.* The sharpest modal case for a possible empty world is not recombination but Baldwin's subtraction argument -(Baldwin, Thomas, 1996) (there might be a world of finitely many concrete things; each of them might not exist; and the non-existence of one need not require another). It is iterated to a world with no concrete objects at all, and pressed further by Efird and Stoneham -(Efird, David; Stoneham, Tom, 2005) to the question whether spacetime itself is among the subtractable concreta. The reply is the role/type distinction already in hand. Grant every premise for *occupants*: any particular object, and any particular metric or manifold, is contingent physical content that subtraction may remove ([§3.5.3](<3. Convergence.md>), Chapter 7). What subtraction cannot quantify over is the structural roles themselves, because they are not further items in the domain of concreta but what makes a domain a domain of distinct, persisting, ordered items at all. Subtract every occupant and the argument reaches empty structure, not nothing: a *where* with nothing in it still instantiates the difference-role, and the inseparability of the roles ([§1.4](<1.4 Three Conditions.md>); step 4 above) carries the other two with it. Push on, as the Efird–Stoneham spacetime question invites, and try to subtract the structure too, and the terminus is not the empty world but ∅ — which [§2.7.2](<2.7.2 Incoherence of ∅.md>) and the reductio above have already shown is not a state that could obtain but a description that fails to describe. The subtraction argument stays valid only while its variable ranges over occupants; extended to the roles, its final step removes the very conditions under which 'a world with n objects' counted as a world, so what it reaches is the incoherent ∅, not a lean possibility that defeats the floor.

The backstop should not be conflated with the distinct claim that changelessness is impossible. An objectless but structured world, a changeless structured world, and absolute structural absence are different targets. The present reductio addresses only the last. Its S5-free contribution is therefore exact: ∅ is not a possible state, and, given role-level inseparability, no possible world lacks all three roles. Full type-level necessity remains subject to the bridge identified at [§3.5.3](<3. Convergence.md>).


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← [3.2.5 Shape of Possibility](<3.2.5 Shape of Possibility.md>) | [3.3 Triconditional Necessity: Negative Account](<3.3 Triconditional Necessity Negative Account.md>) →
