Modal status answers to Reality's constitution rather than to how any particular world is arranged.[^2-1-3-invariance-scope] If that constitution is invariant, what is possible cannot lapse into impossibility and what is impossible cannot become possible: ◇P → □◇P, with bilateral companion ¬◇P → □¬◇P. Both directions matter; the bilateral reading fixed in [§2.1.2](<2.1.2 Bilateral Frame.md>) carries through here.

That conditional marks the work still to be done. The structural ground of possibility *is* what Reality *consists in*, but its invariance across modal evaluation is not assumed here. The Bottleneck ([§2.1.5](<2.1.5 The S5 Bottleneck.md>)) and the Transition Closure ([§2.1.6](<2.1.6 The Transition Closure.md>)) test whether any candidate alternative can fall outside it. This section fixes the ground to which modal status answers and establishes a separate point: the modal frame is necessarily *occupied* on both sides. Occupancy keeps the frame from being vacuous; the later arguments earn its invariance.

> [!argument]
> **Bilateral Self-Sustenance** · *The Positive Direction*
>
> 1. Let ⊤ abbreviate any logical truth.
>
> 1. At every world, ⊤ obtains.
>
> 1. Whatever obtains at a world is possible relative to the fixed domain of real possibilities.
>
> 1. So ⊤ is possible at every world: □◇⊤. Therefore necessarily something is possible: □∃P: ◇P.

The performative version remains available: the act of denying that anything is possible is actual, and therefore possible. But the formal proof needs no move from the content of a supposition to its possibility; ⊤ supplies the positive witness directly.

> [!argument]
> **Bilateral Self-Sustenance** · *The Negative Direction*
>
> 1. Let ⊥ abbreviate any contradiction, Q∧¬Q.
>
> 1. At no world does ⊥ obtain.
>
> 1. So ⊥ is impossible at every world: □¬◇⊥.
>
> 1. Therefore necessarily something is impossible: □∃P: ¬◇P.

The two directions rest on different grounds. Logical truth secures the positive pole; noncontradiction secures the negative pole. Neither direction depends on S5, and neither moves from conceivability to real possibility.

Together they establish one result.

> [!lemma]
> **Bilateral Self-Sustenance of Modal Structure** · **L1** · *Lemma toward Necessary Possibility*
>
> Necessarily something is possible, and necessarily something is impossible: both poles of the modal structure are necessarily occupied, not merely kept from going empty. (□∃P: ◇P) ∧ (□∃P: ¬◇P).

The bilateral frame also governs the derivative statuses. Necessity, contingency, and actuality are not independent additions to possibility and impossibility; each is intelligible only within the same frame. A necessary proposition is one whose failure is impossible; a contingent proposition is possible while its negation is possible; an actual proposition is true of the realized configuration. The lemma does not yet show that the contingent status is occupied in every world, or even supply a particular contingent proposition. It fixes the grammar within which that further question can be asked.

Three limits govern how the lemma may be used. The first is a dependency. The canonical proof depends on the global reading of real possibility defended in §[§2.1.4](<2.1.4 Modal Stability.md>)–2.1.5. Transition Closure does not replace that premise: it supplies a narrower S5-independent closure on the onset of Change under structural accessibility. Chapter 3 rebuilds the necessity of direction, difference, and bearer at role-resolution without S5, but it does not independently recover proposition-wise □◇C across silent worlds (worlds with no thinker to token a denial; [§3.1.3](<3. Convergence.md>)). The dependencies are therefore complementary rather than interchangeable.[^2-1-3-s5-pointers]

The second is a matter of scope. The proof's modal invariance is over the structural conditions that fix modal status, not over the transworld existence of individuals. Neither Barcan formula is load-bearing here, and the quantified modal logic in use neither requires nor forbids constant domains. The closest contemporary view that runs S5 and modal invariance into a substantive quantifier thesis is Williamson's necessitism, which extends necessity to every existent. The present argument shares the invariance and diverges on the scope: necessity attaches to T∧S∧Φ as the structural condition on modal status, not to every individual that occupies it. The divergence point is engaged at [§4.4.6](<4.4.6 Necessitism.md>); the in-chapter scope statement is at [§2.8.5](<2.8.5 Necessity of Nature.md>).

The third concerns grounding. Modal invariance is a shared theorem with the pluriverse account (Lewis's plenitude of concrete possible worlds also yields S5) but the explanatory ground differs. The pluriverse grounds invariance in the completeness of the world-collection; the present account grounds it in what structural incapacity consists in. The Transition Closure ([§2.1.6](<2.1.6 The Transition Closure.md>)) brings out the difference: on the pluriverse view, worlds have their change-incapacity essentially; on the present view, structural incapacity is constitutive. Both groundings are engaged at [§4.4.1](<4.4.1 Modal Realism.md>) (Lewis's plenitude) and [§4.3.9](<4.3.9 Structural Realism.md>) (structural realism).

*What the Lemma establishes.* The lemma establishes non-vacuity, not proposition-wise modal invariance. The frame necessarily contains at least one possibility and at least one impossibility. It does not yet show that whatever is possible is necessarily possible or that whatever is impossible is necessarily impossible. The axiomatic derivation ([§2.1.4](<2.1.4 Modal Stability.md>)), the Bottleneck's fixed-domain defense ([§2.1.5](<2.1.5 The S5 Bottleneck.md>)), and the Transition Closure ([§2.1.6](<2.1.6 The Transition Closure.md>)) earn that stronger invariance claim. Occupancy and invariance therefore remain distinct: this lemma supplies the first; the arguments that follow supply the second.

[^2-1-3-invariance-scope]: The invariance is over real / metaphysical modal status (whether P obtains in Reality) not over what remains ideally conceivable under a primary intension or what is epistemically open. The two-dimensional pressure on the invariance claim is engaged at [§4.1.2](<4.1.2 Two-Dimensional Semantics.md>).

[^2-1-3-s5-pointers]: The Reverse-Lift diagnostic at [§3.5.4](<3. Convergence.md>) marks exactly what Chapter 3 reaches without S5 and where □◇C remains upstream. The objections to global modal invariance are met in Chapter 4: [§4.1.4](<4.1.4 Humean Contingency.md>) takes up Humean contingency, [§4.4.9](<4.4.9 Modal Collapse Objection.md>) handles modal collapse, [§4.4.6](<4.4.6 Necessitism.md>) engages Williamson’s necessitism, and [§4.4.8](<4.4.8 Whiteheadian Process Metaphysics.md>) takes up the refusal of bilateral invariance.


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← [2.1.2 Bilateral Frame](<2.1.2 Bilateral Frame.md>) | [2.1.4 Modal Stability](<2.1.4 Modal Stability.md>) →
