The three limbs are not equally supported, and the unevenness must be located exactly. The independently convergent T- and Φ-claims are stated at *role*: directed ordering and bearing or realization, not yet full temporal succession and causal persistence. □S was stated at a higher resolution (a metric, traversable manifold) and at that resolution the Moves did not force it; they earned *distinguishability* only, and the step to metric extension was carried by the Constitutive Premise rather than by the arguments. Held to the same resolution as its siblings, the necessary S-claim is the role of *difference* (co-existing this-rather-than-that, not yet a *where*) and at that resolution it is overdetermined as fully as □T and □Φ: every Move that earns succession or persistence also earns this-rather-than-that. The metric, traversable richness above that role is contingent physical content, parallel to a particular temporal metric for T and particular fields for Φ, taken up where the proof's silence on physical content is the discipline rather than the gap (Chapter 7). At the role layer the three limbs stand together; at the *type* layer they do not, and the asymmetry locates the seam.

> [!info]
> **Natural worry.** If the necessity of space and of a persisting bearer has to be carried across by a further linking claim, doesn't that make them hold only contingently—true here, maybe not elsewhere? No, and the reason is precise. The linking claim isn't a bridge that might hold in one world and fail in another; it's a structural necessity about any medium that hosts succession at all. And a necessary premise transmits its necessity: if succession is necessary, and necessarily comes bundled with space and a bearer, then space and a bearer are necessary too. The move needs no extra modal axiom—just a necessary link and one valid step. The type-claims end up at full strength, reached by a longer road.

> [!info]
> **Core concept.** This section is the book being honest about an uneven spot. All three conditions—time, space, and persistence—come out necessary, but they aren't reached by equally short arguments. The necessity of succession is pinned in one step. The necessity of spatial difference and of a persisting bearer each take two, borrowing a further linking result proved back in Chapter 2. That's the seam. The key point: it's a difference in how long the derivation runs, not in how strong the conclusion is. None of the steps smuggles in the S5 modal axiom, and none of the three ends up weaker than the others. The extra length costs self-containment, not certainty.

Each limb still owes the step from its generic role to its specific type: directed ordering to succession (T), distinctness to spatial co-existence (S), bearing to a causal-persistent being (Φ). All three residuals close on □◇C, the spatial and persistence steps each leaning additionally on a Stage-2 realizer biconditional.

The temporal type-upgrade closes once the necessary possibility of change (□◇C) has been independently secured. Section [§1.4](<1.4 Three Conditions.md>) fixes the T-Condition as what makes succession possible, so □◇C makes succession available in every world. Section [§3.1](<3.1 Necessary Possibility of Change.md>) supplies no-onset and performative support but does not alone provide the required cross-world invariance; T1 remains the clean source unless another modal bridge is granted. The directed ordering a way carries is thereby temporal in type, and the atemporal rival (a synchronic grounding order, asymmetric and influence-carrying but successionless) is excluded. That closure adds no further S5 premise beyond whichever line supplies □◇C; the holdout who grants the permission but contests the identification (insisting the asymmetric, influence-carrying order is grounded synchronically rather than temporally) is met head-on downstream, not here, and owes two discharges there. First, if the directed order is temporal it must be specified as either an A-series or a B-series, so the holdout may run McTaggart's dilemma (the A-series contradictory, the B-series alone too thin to be time) to deny that the ordering is temporal at all. That dilemma, and the regress on its A-series horn, is taken up at [§4.2.5](<4.2.5 The Unreality of Time.md>), with [§4.2.6](<4.2.6 Block Universe.md>) supplying the B-series limb. Second, because the upgrade rests on □◇C (the *possibility* of change, not its actuality), the time-without-change holdout who points to a frozen region grants only what □◇C already allows: the time-without-change case is taken up at [§4.2.2](<4.2.2 Shoemaker's Time Without Change.md>). Both objections are handled there. What remains downstream is the labor of the identification, not the closure of the upgrade.

Distinctness closes upstream too, but not on □◇C by itself. The step from bare logical distinctness to *spatial-type* difference, namely that the necessary difference is co-existence (a *where*), not merely qualitative non-identity (a *which*), is carried by □◇C together with the realizer move it activates. The *which*-to-*where* move is made directly at [§2.4.7](<2.4.7 Distinction Requires Spatiality.md>): a non-identity across which nothing could move is a difference of contents or of names, not a distinctness of positions, so whatever grounds genuine positional distinctness must supply something crossable, and to supply something crossable is to supply relational extension. Succession itself yields that extension ([§2.6.2](<2.6.2 Succession's Two Conditions.md>)): a directed succession with no interval between its states is bare replacement under another name; if the directional reaching genuinely runs from one state to the next it must cross something, and the interval it crosses is relational extension. That biconditional, T ↔ (S∧Φ), is the realizer move's result, not a stipulation in [§1.4](<1.4 Three Conditions.md>): defended across four lines of argument and framework-invariant, a claim about any medium that hosts succession rather than about a particular geometry or a material bearer, so it holds of a partless or non-material substrate as readily as a corporeal one. The chain that makes the directed ordering temporal in type therefore makes it extended in type in the same step: a directed succession demands a crossable interval, a *where* and not a bare *which*.

Persistence closes on the same pattern: the realizer move works on Φ ↔ (T∧S) across four lines (continuity, identity, persistence, and causation) at [§2.6.4](<2.6.4 Persistence's Two Conditions.md>), so the bearing role upgrades to a causal-persistent being on □◇C together with that Stage-2 biconditional, just as the spatial step rides □◇C together with T ↔ (S∧Φ). The three type-upgrades run together, and the seam between them must be named precisely: it is a difference in the *length* of the derivation, not in the *grade* of the result. The temporal upgrade reaches its type in one step, on □◇C and [§1.4](<1.4 Three Conditions.md>) alone. The spatial and persistence upgrades reach theirs in two: from □◇C the temporal type is fixed, and then a Stage-2 realizer biconditional (T ↔ (S∧Φ) for the spatial step, [§2.6.2](<2.6.2 Succession's Two Conditions.md>); Φ ↔ (T∧S) for the persistence step, [§2.6.4](<2.6.4 Persistence's Two Conditions.md>); each defended across four lines, not a Chapter-1 stipulation) carries that type across to the others. But the biconditional is no contingent bridge that might hold in one world and fail in another; it is a structural-functional necessity ([§2.6.1](<2.6.1 Three Conditions, One Nature.md>)), framework-invariant, a claim about any medium that hosts succession at all. A necessary premise transmits necessity: □T-at-type together with □(T ↔ (S∧Φ)) yields □(S∧Φ)-at-type, and the step imports no S5, only the necessitation of a biconditional under modus ponens. So the spatial and persistence types are secured at the same modal grade as the temporal one, and secured without S5; what the two extra steps cost is not modal force but self-containment: they draw the linking necessity from Stage 2 rather than supplying it themselves. The one permission the upgrades still owe is the necessity of the realizer biconditional itself. That necessity holds on either reading of *consists in* — structural-functional or Finean real definition — since a real definition is necessary if anything is, so the type-upgrades transmit whichever reading is correct. What [§4.1.10](<4.1.10 The Source of Possibility.md>) argues is the further claim that the necessity is structural-functional rather than essence-prior, which the modal monism of [§2.8.6](<2.8.6 Exclusivity of Nature.md>) needs but the type-layer does not wait on.

No type-upgrade adds a further S5 premise beyond whichever line supplies □◇C, and none is weaker in modal grade than another once that premise is in hand. The metric, traversable richness above the role layer stays contingent physical content (Chapter 7); what closes here is the type, not the realization. The convergence is real, the seam is named and located, and at the type layer all three close at □-strength once □◇C is independently available: the temporal upgrade in one step, the spatial and persistence upgrades in two, through a Stage-2 realizer biconditional that transmits necessity intact. These are type-transmission arguments, not independent derivations of □◇C. The asymmetry is one of derivation length, and of where the linking necessity is anchored, not one of conditional versus unconditional closure.


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← [3.5.2 Robustness](<3.5.2 Robustness.md>) | [3.5.4 S5 Lives Upstream](<3.5.4 S5 Lives Upstream.md>) →
