**Stage 1, □◇C (the necessary possibility of change, [§3.1](<3.1 Necessary Possibility of Change.md>)):** three distinct pressures survive without S5, but they do not jointly supply cross-world invariance. The changeless-ground dilemma removes a supposed ground outside transition; the Unswitchable Capacity rules out onset from a capacityless state; the Performative Witness establishes ◇C at every thinking world. The silent, isolated ¬◇C world is not closed by converting worlds into stages of one history. Full □◇C remains with T1 unless an independent type-bridge is supplied.

**Stage 2, □(T∧S∧Φ) at role-resolution (§[§3.2](<3.2 Triconditional Necessity Positive Account.md>)–3.3):** convergent Moves across construction, determinacy, modal vocabulary, enactment, and the null-state floor reach necessary direction, difference, and bearing without S5. These lines are not premise-disjoint; several share the ∅-floor and the constitutive role-identifications. Their force is nevertheless plural: facthood, world-determinacy, modal content, performative indispensability, and null-state incoherence expose different failure points. The role-level conclusion survives any one local cut, while the shared joints remain explicit.

**Stage 3, Reality = Nature ([§3.4](<3.4 Reality and Nature Are One.md>)):** maximality supplies the shortest weld. The residue dilemma supplies a second class-wide line: a residue either makes a difference and instantiates the roles, or makes none and fails the [§1.1](<1.1 Reality.md>) account of the real. The Grounding Trilemma adds a restricted result: no proposed external ground can ground Nature without instantiating the roles, falling within Nature, or ceasing to occupy that grounding office. Because its third horn leaves idle surplus open, it is not a third freestanding proof of R = N; maximality or the residue dilemma must close that remainder. Structural-role co-reference reinforces the identity from reference.

The spine is therefore not uniformly overdetermined. Stage 1 is pressure-tested rather than independently rebuilt; Stage 2 is multiply supported at role-resolution; Stage 3 has two identity-bearing lines with one shared [§1.1](<1.1 Reality.md>) joint openly named, plus a distinct grounding-office closure. That exact distribution is stronger than a flat claim that every result has been proved many times.


---

← [3.5 What the Convergence Shows](<3.5 What the Convergence Shows.md>) | [3.5.2 Robustness](<3.5.2 Robustness.md>) →
