The bottleneck of [§2.1.5](<2.1.5 The S5 Bottleneck.md>) holds every candidate to a single condition: grant that it is possible, and you have granted that its possibility is necessary, ◇P → □◇P. Chapter 2 earned that schema but did not apply it; no winner was named and nothing was ruled out. Here, at the close of the convergence, the schema goes to work.

The chapter has independently reached role-level structural necessity without the Euclidean axiom ([§3.5.4](<3. Convergence.md>)), but it has not thereby re-derived the full type-level theorem or □◇C. What follows is therefore a conditional closing sweep over rival necessary grounds, not an S5-only derivation of Nature. S5 can transmit an independently established modal premise; it cannot by itself identify which candidate fills the grounding office.

Take the axiom in its useful contrapositive. A candidate necessary ground G is necessary only if it is possibly necessary (◇□G), so anything that cannot even coherently be possibly necessary cannot be the necessary ground: ¬◇□G ⊢ ¬□G. The question is then only which candidates survive the possibility test.

They do not survive by being few. A sweep that merely lists rivals and knocks them down invites the obvious reply: *what about the one you left off the list?* The elimination here is not a list but a partition. Any candidate ground stands to the Triconditions in exactly one of three ways, and there is no fourth:

> [!info]
> **Core concept.** This closing sweep asks whether anything could be the necessary ground of the world other than Nature—time, space, and persistence together. Instead of listing rivals one by one, it splits the field with no gaps. Any candidate stands to those three conditions in exactly one of three ways: it's the same structure under a new name, it shares none of their structure, or it's those conditions plus some extra. The first is Nature relocated, not a rival. The second can't ground anything in a world made of time, space, and things—and can't even be picked out to name. The third collapses: the extra either does real grounding work, and so is more of the same structure, or does none, and drops out. Every road leads back to Nature.

> [!info]
> **Natural worry.** Any sweep of rivals invites the obvious reply—what about the one you didn't mention? That's why the argument uses a partition, not a list: the three cases are exhaustive by logic, so there's no fourth slot to hide in. The hardest holdout is the classical theist who says God grounds time, space, and things eminently—containing their perfections in a higher, simple way without being temporal, spatial, or a substance. Granted, that ground can be named, so it isn't the second case. But grounding still means being discernible from what you ground in the very act of grounding it, and that discernibility is structure under another name. Contained in a higher mode or not, the structure has to be there to do the work.

> [!lemma]
> **No Rival Ground** · **L8** · *Lemma toward the Necessity of Nature*
>
> Any candidate ground G is either (i) functionally equivalent to T ∧ S ∧ Φ, the same structure under another name; (ii) incommensurable with it, sharing none of its grounding-relevant structure; or (iii) partially overlapping — sharing only some of it, or all of it together with some further remainder. To stand in any relation to the Triconditions at all is to share all of their grounding-relevant structure, none of it, or some of it; the three horns are jointly exhaustive by logic, not by census. Each horn collapses to the Triconditions, so no rival to T ∧ S ∧ Φ can be the necessary ground.

Each horn closes, and closes deductively.

*Horn (i) is a concession, not a defeat.* A candidate that turns out to be the Triconditions under another description has not displaced them; it has arrived at them. The tradition's necessary ground (named as pure act, or being-itself, or that on which all else depends) falls here the moment the description is cashed into what it must *do*: carry succession, hold distinctions apart, bear its own persistence. That is Nature relocated, not a rival to it.

*Horn (ii) is closed by the bottleneck's own core.* To be incommensurable with the Triconditions is to share none of the structure by which anything grounds a temporal, spatial, or physical fact — and so to be unable to ground any such fact, which is to ground nothing in the world we are trying to account for. And a ground incommensurable with modal structure could not be *specified* as a candidate in the first place: to be individuable, describable, and contrastable is already to have Spatiality, Physicality, and Temporality ([§2.1.5](<2.1.5 The S5 Bottleneck.md>)). "Outside the structure" does not name a leaner ground; it names no ground, in the exact sense [§2.1.5](<2.1.5 The S5 Bottleneck.md>) forecloses — there is no *outside* to stand in.

*Horn (iii) is the one a determined rival reaches for:* not another structure and not none, but the Triconditions *plus* a surplus that does the real grounding while T, S, and Φ ride along. This horn closes without appeal to economy. Ask what the remainder does. The question is narrower than it first looks — not whether the remainder does *any* work, but whether it does grounding work *for the world*. A surplus whose only office lies elsewhere — grounding value, say, or the necessary ground's own necessity — is no rival ground of the world at all; it is set aside by the scope of the claim, not defeated by it, and nothing here pretends to rule such a thing out. Of a surplus that does claim a hand in grounding the world, there are two cases. If it does grounding work, if it helps ground the temporal, spatial, or physical facts any world must exhibit, then it must share the structure by which anything grounds such facts, and whatever thus grounds already carries directed order, distinction, and a bearer ([§3.2.1](<3. Convergence.md>)): the remainder is *more* of the Triconditions, not something other than them. Here the determined rival presses in its one serious form — the classical theist who concedes that the ground is specifiable yet holds that it grounds the temporal, spatial, and physical *eminently*, containing their perfections in a higher simple mode without itself instantiating T, S, or Φ. This is the case specifiability cannot reach: an eminent ground is, by hypothesis, individuable and describable, so the horn-(ii) closure leaves it standing ([§2.1.5](<2.1.5 The S5 Bottleneck.md>)). It is closed instead where the book earns the foreclosure. By the Exclusivity of Nature ([§2.8.6](<2.8.6 Exclusivity of Nature.md>)) any candidate ground is either T ∧ S ∧ Φ in the loaded sense or a variant of ∅, with no eminent middle between them. This is because a determinacy that grounds must be discernible from what it grounds in the very act of grounding it, and that discernibility is structure under another name (pressed against classical simplicity at [§5.2.1](<5.2.1 Divine Simplicity.md>), and against the one-simple-being reading at [§2.6.9](<2.6.9 Triunity of Nature.md>)). Contained eminently or instantiated formally, the structure is there to be shared; the eminent mode relocates it rather than dispensing with it. If it does no grounding work, it is no part of the *ground* at all — not because grounds are *individuated* by their effects, a stronger thesis than we need, but because a component that makes no difference to what is grounded earns no place in the ground of it; it is a free-floating concomitant, and an idle addition is not a smaller theory but a longer name for the same one. Either way the surplus collapses back into T ∧ S ∧ Φ. There is no coherent third thing for horn (iii) to be.

So the partition is sealed by logic, and the candidates the tradition actually presses are only instances passing through it. The null state, and the "brute contingency answerable to nothing," fall to ∅ and its incoherence ([§3.2.6](<3. Convergence.md>)). An abstracta-only ground and a necessary being *beyond* Nature fall at horn (ii): neither can bear or contrast anything without borrowing the structure it was meant to replace. A spatiotemporal-only order (T ∧ S, no Φ) and a substance-only order (Φ, no T ∧ S) fall at horn (iii), the missing condition restored by inseparability ([§1.4](<1.4 Three Conditions.md>)). Modal realism's plurality falls at horn (i): each of its worlds is Nature over again, not an alternative to it. None escapes, because there is nowhere outside the three horns for it to go.

If the Chapter 2 argument has independently established that T ∧ S ∧ Φ is possibly necessary—or already necessary at full type-resolution—then it survives this partition while the rival-ground candidates do not. Only with that independent modal premise may the S5 bridge yield □(T ∧ S ∧ Φ). The partition alone establishes at most uniqueness or non-rivalry within the proposed grounding office; it does not establish existence, possible necessity, or necessity of its occupant.

A single qualification keeps the sweep exact, the same one [§2.1.5](<2.1.5 The S5 Bottleneck.md>) records. What the partition closes, and closes deductively, is the field of rival *grounds*: every candidate that could fill the office falls at one of the three horns. The lone position that survives the bottleneck anywhere is not a rival ground at all but a refusal to *ground* an already-granted modal fact — one proposition's modal status held as an ungrounded brute contingency — and so it is not what this partition adjudicates; it is closed nowhere by contradiction and carried, off every line's spine, on cost alone ([§2.1.5](<2.1.5 The S5 Bottleneck.md>)). Under the comparative standard of [§2.9.3](<2.9.3 Total-Theory Comparison.md>) that too is decisive. Given the independently established modal and structural premises, the office stands filled and rival grounds are excluded within the stated scope. S5 is a transmission principle in that result, not a source from which T ∧ S ∧ Φ follows by itself.


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← [3.5.4 S5 Lives Upstream](<3.5.4 S5 Lives Upstream.md>) | [3.6 Necessity's Reversal](<3.6 Necessity's Reversal.md>) →
