Granting that change is possible comes with a bill. Chapter 1 named the three Conditions and their bearers: Temporality, borne by Time (a real before-and-after); Spatiality, borne by Space (genuinely distinct states); Physicality, borne by Substance (what carries the exchange). The question here is their modal status.

One contract governs the derivation. *Role-resolution* names directed order, relational difference, and real bearing. *Type-resolution* names temporal succession, spatial extension, and causal persistence. The proof may use T, S, and Φ at role-resolution before the type-derivations land, but it may not inherit the richer types from the names alone. Sections §[§2.4.6](<2.4.6 Succession Requires Temporality.md>)–2.4.8 carry that upgrade explicitly.

The Necessary Possibility of Change was established at [§2.3](<2.3 Necessary Possibility of Change.md>): □◇C, the possibility of change is necessary, open in every possible world. The claim now is a necessary conditional: at every possible world, if change is possible there, then Temporality, Spatiality, and Physicality are instantiated there. Formally: □(◇C → T∧S∧Φ).

> [!premise]
> **Structural Requirement** · **P3** · *Premise 3*
>
> At every possible world, if change is possible there, then Temporality, Spatiality, and Physicality are instantiated there. □(◇C → T∧S∧Φ).

The conditional is not a stipulation, and its necessity is not a modal axiom imported from outside. The necessity is *derivative*: the three Conditions are constitutively required by the possibility of change, and a constitutive requirement holds wherever the thing it conditions holds.

Because the dependence is essential rather than accidental, it does not vary across the modal structure: what essence fixes, it fixes at every world (Fine, Kit, 1994): essence is prior to modality, not read off from it. Whether that essence is itself prior to the modal field, or names the shape of a field that is already primitive, is the priority question [§4.1.10](<4.1.10 The Source of Possibility.md>) takes up without treating the committed essence-prior view as already closed; the appeal here is to essence in its local office (fixing the necessity of the conditional), not to essence standing prior to ◇.

Four lines approach the requirement, each starting from a different vocabulary and converging on the same constitutive identification. The shape is parallel to Kripke's *necessary a posteriori* (Kripke, Saul A., 1980): the dependence is secured by reasoning, and once secured it is necessary; the disanalogy is that the discovery here is logical-derivational from change-possibility, not empirical-microstructural.

*Line 1: Modal.* Any possible world at which change is possible already presupposes the Conditions. A plurality of worlds, and the transitions a possible change would range over, require the relation of genuine distinctness (S), a real directional ordering (T), and a pattern of causal-exchange relations as the carrying medium (Φ). The characterization is relational throughout: S names the relation of distinctness, not the relata it holds between; Φ names the causal-exchange pattern, not a substrate that bears the relations (Ladyman, J., and D. Ross, with D. Spurrett and J. Collier, 2007). Modal structure is constituted by T∧S∧Φ, not merely occupied by it; a world that instantiates ◇C instantiates that constitution. Whether this a priori result is prior to and independent of what physics will eventually describe (or requires naturalizing contact with specific sciences) is engaged at [§4.1.5](<4.1.5 Information-Theoretic Foundationalism.md>) (Ladyman-Ross) and [§4.3.9](<4.3.9 Structural Realism.md>) (structural realism).

*Line 2: Conceptual/Essence.* The Conditions are constitutive definitions, not reference-fixing descriptions. There is no gap between concept and essence at the structural level. Whatever would fill the role of causally structured succession *is* Temporality. There is no deeper microstructure the role-specification leaves open; the individual derivations are developed in the subsections below. This is a structural-essence variant of Kripke's natural-kind essentialism (Kripke, Saul A., 1980), Lecture III: the natural-kind machinery does not directly apply at the structural level, but the identity-of-essence-with-role move is the same, and it is what makes the conditional hold of necessity rather than merely in fact.

The two-dimensional objection (that a primary-intension gap may remain even where the secondary/structural identity holds, on which the zombie-style inference from ideal conceivability of a structureless scenario to its possibility would depend) is engaged at [§4.1.2](<4.1.2 Two-Dimensional Semantics.md>). The conceivability-inversion line at [§3.3.4](<3. Convergence.md>) channels that pressure through the necessary-content reading and points downstream to [§4.1.2](<4.1.2 Two-Dimensional Semantics.md>) for the block.

*Line 3: Grounding.* Any demand for a ground of the Conditions is itself made from inside the framework the demand seeks to stand outside. Grounding is a directed relation (the T-role) between distinct relata (the S-role) whose dependence genuinely obtains in a fact or realizer (the Φ-role). For atemporal grounding, this last role is dependency-stability, not persistence across a temporal interval. The regress cannot get started. The regress-block is a transcendental constraint on access (the demand for the Conditions' ground already uses the Conditions to pose itself) and rules out the regress from inside; the further question whether a necessary being external to the Conditions could ground them from outside is taken up in the grounding treatment below and at [§5.1](<5.1 Theistic Arguments for a Divine Ground.md>).

*Line 4: Facthood.* The most primitive starting point: something is the case. Its denial (*nothing is the case*) is itself something being the case, hence self-refuting. From facthood alone, T, S, and Φ follow directly: any state of affairs requires identity (S), obtaining (T), and a medium (Φ). The move from that identity-requirement to Spatiality is not the claim that bare non-identity is already spatial, which is exactly the haecceitist's opening; it rests on the Relational Distinctness premise, that genuine positional distinctness just is traversable relational extension, defended openly at [§2.4.7](<2.4.7 Distinction Requires Spatiality.md>). With that premise carried there, the S-clause of Line 4 is earned, not assumed. A fortiori, any world at which change is so much as possible meets the requirement.

The derivation is modal-structural, not mathematical-structural: it runs through what any obtaining state of affairs requires, not through which mathematical structures independently exist. Whether the formal specifiability of T∧S∧Φ collapses this into the Mathematical Universe Hypothesis is engaged at [§4.3.7](<4.3.7 Mathematical Universe Hypothesis.md>); whether the derivation uniquely identifies T∧S∧Φ (only this structure, not other formally consistent alternatives) is engaged at [§4.3.6](<4.3.6 The Unreasonable Effectiveness of Mathematics.md>) and [§4.4.4](<4.4.4 Modal Fictionalism.md>).

The four lines converge, but they share a hinge: the role-to-type identifications defended in §[§2.4.6](<2.4.6 Succession Requires Temporality.md>)–2.4.8. A reader may enter through modal, essence, grounding, or facthood vocabulary; none of those entrances substitutes for defending the common constitutive premises. The convergence broadens access to the argument rather than multiplying the warrant for its shared hinge.

One structural note on ideology — in Quine's technical sense, the primitive predicates a theory takes as basic, not a political program: T, S, and Φ are the book's fundamental primitive structural predicates, not derived from anything more fundamental, though Lines 1–4 give independent paths to the requirement. Whether T∧S∧Φ is one joint three ways or three independent joints is answered at [§2.6](<2.6 Triconditionality of Nature.md>). Whether □ itself is primitive or derived from structural incapacity is addressed at [§2.10](<2.10 Three Logical Strata.md>).

The reverse direction T∧S∧Φ → ◇C is not needed for the proof to the conclusion. It is defended at [§2.4.10](<2.4.10 Sufficiency of Structure.md>) and later read as part of the Modal Identity at [§2.6](<2.6 Triconditionality of Nature.md>).

The Structural Requirement is the necessary conditional □(◇C → T∧S∧Φ). Discharged against the Necessary Possibility of Change □◇C ([§2.3](<2.3 Necessary Possibility of Change.md>)), it delivers the unconditional necessity of the structural Conditions; that step is taken at [§2.5](<2.5 Triconditional Necessity.md>). The proof to the conclusion requires only this forward conditional; the biconditional and the identity it grounds do their downstream work later in the chapter.


- [2.4.1 Constituting Modal Structure](<2.4.1 Constituting Modal Structure.md>)
- [2.4.2 Structural Gap](<2.4.2 Structural Gap.md>)
- [2.4.3 Demand Instantiation](<2.4.3 Demand Instantiation.md>)
- [2.4.4 Something Must Be](<2.4.4 Something Must Be.md>)
- [2.4.5 Forward Conditional](<2.4.5 Forward Conditional.md>)
- [2.4.6 Succession Requires Temporality](<2.4.6 Succession Requires Temporality.md>)
- [2.4.7 Distinction Requires Spatiality](<2.4.7 Distinction Requires Spatiality.md>)
- [2.4.8 Exchange Requires Physicality](<2.4.8 Exchange Requires Physicality.md>)
- [2.4.9 Forward Conditional Necessity](<2.4.9 Forward Conditional Necessity.md>)
- [2.4.10 Sufficiency of Structure](<2.4.10 Sufficiency of Structure.md>)
- [2.4.11 The Frozen-World Test](<2.4.11 The Frozen-World Test.md>)
- [2.4.12 The Masking Test](<2.4.12 The Masking Test.md>)


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← [2.3 Necessary Possibility of Change](<2.3 Necessary Possibility of Change.md>) | [2.4.1 Constituting Modal Structure](<2.4.1 Constituting Modal Structure.md>) →
