*The realizer move, applied to T ↔ (S ∧ Φ).* Directed succession is biconditional with both relational extension and persistent bearing: T ↔ (S ∧ Φ). The realizer move runs across all three terms: each, taken in its real (not thin) reading, requires the others as what makes it real rather than notational. Four lines work the T↔S link explicitly (two in each direction, each refuting a distinct deflationary mimic), and each line already deploys Φ in the work that makes it run. The same four lines deliver T↔Φ: not as a separate step, but as what the lines are doing already. The realizer move is repeated from the extension and persistence pivots at §[§2.6.3](<2.6.3 Extension's Two Conditions.md>) and 2.6.4; the result is cross-confirmation, not assembly. The succession pivot alone already delivers the full triconditional; the three vantages are gathered at [§2.6.5](<2.6.5 Mutual Containment.md>).

*Direction 1: T→S.* Two lines converge on the same result: directed succession analytically requires relational extension.

*First, directed succession is a reaching, and a reaching needs an interval to cross: that interval is relational extension (the traversability line).* A thin reading of directed succession: state A is followed by state B, full stop. But Temporality's Directionality ([§1.4](<1.4 Three Conditions.md>)) has already built more into the relation: the after-state is not merely adjacent to the before-state but is reached by it, with the directional ordering itself traveling from one to the other. The travel here is not yet specifically Φ-causal transmission in the canonical event-to-event sense; that is delivered under its full physical reading at [§2.6.4](<2.6.4 Persistence's Two Conditions.md>) and [§2.4.8](<2.4.8 Exchange Requires Physicality.md>). The travel at the level of this line is the operative work that distinguishes a real reaching from a notational adjacency. The line does not assume Φ as a separate premise; it surfaces Φ as already engaged in the work T requires to be more than notation. Travel of this kind still requires a medium to travel through. Directed succession with no interval between its states amounts to bare replacement under a different name, not directed succession with unusual spatial properties. If the directional reaching genuinely runs from A to B, it must cross something. If it crosses nothing, it is not traveling but replacing. The interval it crosses is relational extension.

*Second, the before-and-after asymmetry has nowhere to reside but the relation between the relata, and that relation is extension (the asymmetry-location line).* A thinner reading: A and B are distinct and A is before B, but the asymmetry is primitive, with nothing to ground it. Ask where the asymmetry resides. Not in A alone (A does not contain the fact that B comes after it). Not in B alone (the mirror problem). Not in a third state floating free (that would need its own grounding). The only remaining location is the relation between them, and a relation with structurally differentiated positions between distinct relata is relational extension. The Bateson -(Bateson, G., 1972) gloss lands in the same place: the causal imprint the after-state carries is a difference that makes a difference, and a difference that makes a difference requires a structure of alternatives among which the signal selected. A structure of alternatives is relational extension.

By PII (the Principle of Identity of Indiscernibles), relata with no relational structure and identical intrinsic properties are one relatum, not two; relata with no relational structure but distinct intrinsic properties are distinguishable but unordered: nothing makes one *after* the other. Directed succession specifically requires a relational carrier for the direction. That carrier is extension.

PII is contested. Black -(Black, Max, 1952) offers the standard two-spheres counterexample: two qualitatively identical iron spheres in an otherwise empty universe, distinct individuals despite intrinsic indiscernibility. The line's PII appeal does not require strong PII (individuation by intrinsic properties alone). It requires only the weaker claim that *relata in directed succession* require relational structure to ground their ordering. Black's spheres themselves stand in a relational structure (each at some distance from the other) which is the kind of structure the line invokes. The line's PII reading is the relational one, which survives Black; strong intrinsic PII is not needed for the line to deliver.

*Direction 2: S→T.* Two lines converge from the other direction: relational extension analytically requires directed succession.

*First, a real interval is one that can be crossed, and a crossing is directed succession (the interval-reality line).* A thin reading of relational extension: positions separated by intervals. But an interval that nothing could cross is notation, not a distance. The reality of an interval (what distinguishes a real extended structure from a coordinate labeling) is its traversability. Traversability is directed succession: a crossing is influence moving from one position to another in ordered fashion. Strip out succession and the *interval* loses the feature that made it real rather than notational.

*Second, distinct positions are distinct only by sustaining “at x, then at y,” and that takes directed succession (the position-individuation line).* A thinner reading: positions are distinct by stipulation; nothing further grounds their separation. Ask what makes position x not identical to position y if nothing could distinguish them by occupying them differently. The answer cannot be intrinsic properties of positions, and whether one takes a relationalist line (positions have no intrinsic properties apart from their relational structure) or a substantivalist line (positions have an intrinsic identity, but that identity does no individuating work without the relational structure that ties it to the rest of the manifold), the individuation of x from y must run through how they stand to one another rather than through any content that distinguishes them independently. The answer must be that the positions sustain distinct possibilities: something could be at x without being at y, and could move between them. Sustaining distinct possibilities requires directed succession as the structure within which *at x then at y* has content. Without succession, the *distinctness* of positions collapses into labels that differ in no operative way.

Which line bears the most weight depends on what the reader resists. The traversability and interval-reality lines are decisive for anyone who grants that real structure must be operative rather than notational. The asymmetry-location and position-individuation lines are decisive for anyone who grants the relata but resists the realizer: they close the remaining gap by locating where the contested feature must reside.

What about sequences that are not spatial, or structures without time? Three stock objections press: primitive thisness (states individuated with no relational structure); primitive asymmetric dependence (the after-state depends on the before-state with nothing between); *non-spatial* sequences (order without extension). None blocks the result. If states are genuinely distinct, that distinctness consists in more than two labels that fail to match. If the after-state depends primitively on the before-state with nothing between, the dependence relation itself is a relation with structurally differentiated positions (dependent and depended-upon) — exactly the relational carrier the asymmetry-location line shows the direction requires, so the objection supplies rather than blocks the result. If a *sequence* is only logical order (the natural numbers under <, say, or Prior's -(Prior, A. N., 1967) tense-logic operators), Temporality in the operative sense has not yet arrived; what has arrived is a formalism whose propositional relata are ordered by logical operations, itself a relationally extended abstract structure.

The fuller tests belong with [§4.2.4](<4.2.4 Platonism.md>), where four Platonist positions (Frege-Gödel classical Platonism, Shapiro's *ante rem* structuralism, Maddy's naturalized Platonism, and Balaguer's plenitudinous Platonism) are put under load individually, and where Benacerraf's -(Benacerraf, Paul, 1965); -(Benacerraf, Paul, 1973) multiple-reduction and access results close the residue.

*Scope.* The extension required here is not necessarily physical space in the recognizably Newtonian sense; the succession required is not necessarily physical time. Each is the structural condition of which physical space and physical time are this universe's realizers. The claim is framework-invariant: any medium hosting causally structured succession must exhibit relational extension, and any medium hosting real relational extension must exhibit directed succession. Physical spacetime realizes both. Nothing in the argument restricts the realization to that case.

*Lifting to the triconditional.* Each of the four lines argues T↔S, but each line already deploys Φ in its argument: Φ appears not as a separate premise but as part of the operative work T↔S requires to run. The traversability line invokes travel; travel requires a traveler that persists across the crossing (Φ). The interval-reality line invokes traversal across the interval; traversal requires the same. The asymmetry-location line invokes the asymmetry being carried forward by the after-state; that carrying-forward requires Φ-persistence. The position-individuation line invokes positions sustaining distinct possibilities across the succession that makes them distinct possibilities; the sustaining is Φ at the level of position-bearing.

The reverse direction runs the same way. A Φ-persistent being with no Temporality has no succession to persist through; a Φ-medium with no S has no positions to bear; a Φ-relation between cause and effect requires direction (T) and interval (S) to be more than self-relation. The four lines therefore deliver Φ↔T alongside T↔S, as part of the same realizer move read from the T pivot.

The pairwise biconditional T↔S arrives already engaging Φ; the same lines deliver T↔Φ; the cross-entailment Φ↔S follows from the symmetric work from the S pivot established at [§2.6.3](<2.6.3 Extension's Two Conditions.md>). The result is T ↔ (S ∧ Φ): succession cannot stand without both extension and being, and the conjunction cannot stand without succession.

*Duration* presents as temporal vocabulary: the span of an interval. But the result reached above runs in ordinary language before any formal analysis arrives. Duration requires a persistent being (Φ): without something that endures, there is no duration *of* anything, only a labeled interval with no occupant. It requires relational extension (S): without somewhere for the being to be, the interval cannot be located, and an interval without location is not a span but a label. Strip either Condition and the word stops applying. Duration is not a temporal word that happens to imply the others. It is a triconditional word built around the succession pivot: T ↔ (S ∧ Φ) embedded in a single noun.

*What converges.* Four lines, one move, three Conditions. Succession cannot occur without extension to cross, an interval cannot be real without something to cross it, and what crosses must persist to do so.[^2-6-2-succession-pivot-ch4] Three sides of one structural fact: the realizer move read from the succession pivot, with the same fact read from the extension and persistence pivots at §[§2.6.3](<2.6.3 Extension's Two Conditions.md>)–2.6.4.

[^2-6-2-succession-pivot-ch4]: The T pivot, T ↔ (S ∧ Φ) and the four realizer lines (traversability, asymmetry-location, interval-reality, position-individuation), is engaged in Chapter 4. The canonical opponent of the realizer move's real reading is the Humean denial of necessary connection between distinct existences ([§4.1.4](<4.1.4 Humean Contingency.md>)). Sider's stage theory and temporal-parts ontology press on what makes succession real ([§4.2.8](<4.2.8 Stage Theory.md>)), Platonism about non-spatial sequences (Frege–Gödel, Shapiro, Maddy, Balaguer) is the principal order-without-extension opponent ([§4.2.4](<4.2.4 Platonism.md>)), emergent time presses on whether T is fundamental enough to enter the biconditional ([§4.2.3](<4.2.3 Emergent Time.md>)), and Tegmark's mathematical universe asks whether order-without-extension is enough ([§4.3.7](<4.3.7 Mathematical Universe Hypothesis.md>)).


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← [2.6.1 Three Conditions, One Nature](<2.6.1 Three Conditions, One Nature.md>) | [2.6.3 Extension's Two Conditions](<2.6.3 Extension's Two Conditions.md>) →
