5.  Coinstantiation of Spacetime and Causality

Space is normally pictured as a stage, with cause and effect as things that happen on it. This chapter argues they are not two things at all, but two ways of describing one new structure, and that they appear in the same instant. Neither can come first, because before that instant there was no earlier.

5.1.  Causality as coinstantiated structure

The previous chapter established that modal sublimation is a nontemporal, nonmechanistic transition: the functor $\Phi:\mathbf{S}\rightarrow\mathbf{G}$ maps a precausal configuration directly into a Lorentzian manifold without passing through any intermediate state and without requiring a prior temporal structure in which the transition could unfold. A consequence of this that deserves its own careful treatment is the status of causality itself. Causality is not, in PST, a background structure that was already present when the universe formed and that can then be applied to the event of its formation. Causality is coinstantiated with geometry: it arises at the same moment as the Lorentzian manifold that carries it, because it is constitutively defined in terms of that manifold. There is no moment, however brief or abstract, at which a causal relation exists but spacetime does not. The two are ontologically inseparable.

This point carries weight precisely because it is not obvious. Much of the philosophical and cosmological literature treats causality as prior to or more fundamental than spacetime. In the Kantian tradition, causality is a category of the understanding applied to experience, logically prior to any empirical fact about the world [34]. In causal set theory, the discrete partial order is the primary structure from which the continuum metric is to be recovered [4, 35]. In quantum gravity approaches that proceed via sum over histories, the causal structure of each history is taken as given, and the amplitude is computed over those histories [36]. Even in ordinary language, the question “what caused $X$?” is considered answerable in principle for any event $X$, the presumption being that every event has a cause and that the causal relation is always available to be invoked.

PST disagrees at the foundational level. The precausal substrate, as defined in Chapter 2, does not possess a causal order. It possesses property differentiation and the tension functional. These are not impoverished substitutes for causality; they are genuinely different kinds of structure. The distinction relation $\delta$ is symmetric and irreflexive: it registers that two properties differ from each other without specifying which is prior to the other or whether either depends on the other. The tension functional $T$ assigns a magnitude to a configuration without implying that one configuration succeeds or precedes another. Neither $\delta$ nor $T$ contains the ingredients from which a temporal order could be constructed. A temporal order requires at minimum an asymmetric, transitive relation on a set of events. The precausal substrate has none. Causality is not absent from the substrate in the way that a feature might be absent from a description that was meant to include it; it is absent from the substrate in the sense that the substrate belongs to a logical type to which causal relations do not apply.

5.2.  Mathematical formulation of coinstantiation

Let $(M,g)=\Phi(C)$ be the Lorentzian manifold produced by modal sublimation of a configuration $C\in\mathbf{S}$. The causal structure on $(M,g)$ is the partial order $\leq$ defined pointwise by the lightcone structure of $g$:

$$p\leq q\;\iff\;q\in J^{+}(p,g)$$ (36)

where $J^{+}(p,g)$ is the causal future of $p$ under the metric $g$:

$$J^{+}(p,g)=\{\,q\in M\;\mid \exists\text{ future-directed causal curve }$$ (37)

$$\gamma:[0,1]\rightarrow M,\;\gamma(0)=p,\;\gamma(1)=q\,\}$$

A curve is causal under $g$ if and only if its tangent vector $\dot{\gamma}$ is everywhere non-spacelike, that is $g(\dot{\gamma},\dot{\gamma})\geq 0$ at every point. This definition depends on $g$ in an essential way: the notion of a causal curve, and hence of causal precedence, is meaningless without a metric. Change the metric and the causal structure changes with it; remove the metric and the causal structure does not become unspecified but ceases to exist as a category of object.

The partial order $\leq$ in equation (36) has no pre-image in the precausal category $\mathbf{S}$. Formally, there is no morphism or relation in $\mathbf{S}$ that maps to $\leq$ under $\Phi$, because the objects of $\mathbf{S}$ are configurations equipped only with $\delta$ and $T$. This is not a deficiency of the model; it is a structural theorem: the functor $\Phi$ is not surjective on structure but on objects. It produces a manifold but does not carry into that manifold a pre-existing causal order; the causal order is generated anew by the Lorentzian signature of the metric produced by equation (30). Therefore the following holds:

$$\text{causality}\;=\;f(g)\;=\;f(\Phi(C))$$ (38)

Causality is a derived relation, defined in terms of the metric, which is itself defined in terms of the precausal tension structure via $\Phi$. It is not an independent primitive.

5.3.  The direction of time as a structural consequence

The Lorentzian signature of the metric $(+,-,-,-)$ introduces an asymmetry that has no parallel in a purely Riemannian geometry. In a Riemannian manifold, all directions from a point are spacelike and the notion of a preferred future does not arise. In a Lorentzian manifold, the lightcone at each point divides the tangent space into two timelike halves, one conventionally designated as the future and the other as the past, and a continuous causal vector field on $M$ selects one half globally, providing a time orientation [37, 38].

In PST this asymmetry is not imposed as an additional datum. It is inherited from the asymmetry of the precausal substrate. Asymmetric tension, by definition, is a non-symmetric functional: $T(\bar{C})\neq-T(C)$ generically, where $\bar{C}$ is the complement of $C$ in $D$. Across the functor $\Phi$ this asymmetry survives as the time orientation: the Lorentzian signature itself is fixed by hyperbolic well-posedness of the Goldstone wave equation (Computation 2), and the preferred time direction of the manifold reflects the substrate’s globally fixed modal-sublimation direction $\hat{\tau}\in V_{7}$, locked at threshold crossing by P3. The thermodynamic arrow of time, the cosmological arrow of time, and the distinction between retarded and advanced solutions in electrodynamics are all grounded in this single asymmetry inherited from the substrate.

A technical caveat is required. The complement-asymmetry $T(\bar{C})\neq-T(C)$ (11) is a configuration-level condition; a global time orientation on $M$ requires a continuous nowhere-vanishing timelike vector field on the entire manifold. By Geroch’s theorem [37], a Lorentzian manifold is time-orientable if and only if such a field exists, which is a global topological condition not automatic from local asymmetry. In PST the global character of the precausal configuration $C$ provides the resource: the globally fixed modal-sublimation direction $\hat{\tau}\in V_{7}$ (Chapter 4) gives a precausal structure that $\Phi$ maps to a candidate timelike field on $M$.

The continuity and nowhere-vanishing of this field follow from three structural ingredients.

1. The modal-sublimation direction $\hat{\tau}\in V_{7}$ is fixed globally at threshold crossing (§1.7, P3), so it is a single non-vanishing element of $V_{7}$ rather than a configuration-dependent choice.

2. The substrate-to-spacetime projection $\rho$ underlying $\Phi$ is permutation-equivariant under the substrate’s $S_{D}$ symmetry, by the same permutation-invariance argument that drives the cosmological-principle theorem (§16.2, “Theorem (Cosmological Principle)”). That argument also forces the equidistribution assumption A2 of §18.1.

3. The Mosco-convergence framework of §18.1 delivers $\rho$ in the strong-resolvent topology, which preserves continuity of the projected direction.

Combining (i)–(iii): $\hat{\tau}$ is mapped by $\Phi$ to a globally non-vanishing continuous timelike direction on $M=\mathbb{R}\times S^{3}$; specifically $\Phi(\hat{\tau})=\partial/\partial t$ along the $\mathbb{R}$ factor. Geroch time-orientability is therefore a structural consequence of the modal threshold (P3) plus the substrate permutation invariance plus the Mosco convergence, rather than an additional assumption.

5.4.  The causal order and the degenerate vacuum

A further consequence of coinstantiation concerns the vacuum manifold. As established in Chapter 4, the post-threshold vacuum is the $V_{7}$-vacuum sphere $\mathcal{V}\cong S^{6}$. For the U(1) Goldstone-phase and angular-momentum analysis below, the relevant substructure is the 2-plane slice fixed by $\hat{\tau}$ and any perpendicular unit direction $\hat{\eta}\perp\hat{\tau}$: the projected ring $\mathcal{V}_{\hat{\eta}}\cong S^{1}$. Henceforth in this book, statements involving “$\mathcal{V}\cong S^{1}$” refer to this projection, the U(1)-relevant slice of the full $S^{6}$ vacuum manifold (P3); the perpendicular five additional directions on $S^{6}$ carry the gauge-theoretic content (§14.2). Motion along $\mathcal{V}$ is the only stable state the vacuum admits. In the instantiated geometry, this motion projects under $\Pi$ into persistent non-vanishing stress-energy, which in turn sustains curvature. Crucially, the causal order $\leq$ treats this motion as part of the background structure: the lightcone structure generated by the projected metric encodes the vacuum motion as a baseline curvature rather than as a deviation from flatness. This is why the structural argument holds that orbital motion at all scales does not require an initial condition that specifies it separately: it is a candidate for the ground state of the causal structure. The explicit $\Pi$-mapping connecting this Noether charge to the observed orbital angular momentum of massive bodies is computed in §9.5.

5.5.  Category error and the dissolution of first-cause questions

With coinstantiation established, PST is in a position to address what is perhaps the most persistent question in the philosophy of cosmology: what caused the universe to come into being? The question appears legitimate. It has the grammatical form of a causal question, it points to a definite event (the coming-into-being of the universe), and it invites an answer of the form “$X$ caused the universe”.

The question is not, however, answerable. It is not unanswerable in the sense that the answer is hidden or that the evidence has been destroyed; it is unanswerable because it commits a category error, applying a relation to an entity that belongs to a logical type to which the relation does not apply [39, 40]. To ask what caused the universe is to presuppose that the causal relation exists and is applicable to the event of the universe’s formation. But the causal relation, as shown above, is coinstantiated with the universe: it comes into being with the very event whose cause is being sought. A relation cannot serve as the explanation for the event that constitutes its own first instantiation.

The parallel with simpler category errors is instructive. “What is north of the North Pole?” presupposes that the directional relation north-of is defined at the North Pole, but the relation is defined only within the coordinate system that has the North Pole as its boundary. “What was there before the Big Bang?” presupposes that the relation earlier-than is defined at the Big Bang, but that relation depends on the spacetime whose earliest moment is the Big Bang. “What caused the universe?” presupposes that the causal relation is defined for the event of universal instantiation, but that relation depends on the very spacetime that is being instantiated. In each case the error is structural: the question applies a relation in a domain where that relation does not exist.

The dissolution of the category error is not a retreat. PST does not avoid the question by refusing to engage; it dissolves it by exhibiting the precise logical structure that makes it malformed. This dissolution is, moreover, constructive: in place of the unanswerable question, PST offers the answerable one. Why does the universe exist at all? Because the precausal substrate possesses property differentiation, and property differentiation entails asymmetric tension, and once tension exceeds the modal threshold, modal sublimation to instantiated geometry is logically necessary, not contingently triggered. The question Leibniz posed, “why is there something rather than nothing?” [41], receives not an arbitrary answer but a principled one: because the logical possibility of property differentiation is irreducible, and a substrate bearing irreducible property differentiation cannot remain uninstantiated once the threshold condition is satisfied. Existence is not a brute fact; it is the necessary expression of a logically unavoidable modal imbalance.

5.6.  Contrast with competing approaches

Several approaches in the literature address the question of cosmological origins in ways that are instructive to compare with PST.

The Hartle–Hawking no-boundary proposal [36] avoids a boundary in time by making the metric Euclidean in the deep past, eliminating the initial singularity by changing the signature. This is technically elegant but it does not dissolve the category error: the sum over geometries is still computed within a pre-existing mathematical framework equipped with a notion of quantum amplitude, and the question of why that framework is instantiated rather than another is not addressed. The no-boundary proposal also treats causality as part of the background formalism rather than as a derived structure.

Vilenkin’s tunneling proposal [42] posits a quantum tunnelling event from nothing. But “nothing” in this context is not the absence of structure; it is a specific quantum gravitational state with a precisely defined wave function. The transition from that state presupposes both quantum mechanics and the causal structure within which the transition is defined. The proposal is internally consistent but does not escape the regress: it pushes the question back to the prior state and the framework defining the transition.

Causal set theory [4, 232] makes the causal partial order primitive and attempts to derive the continuum from it. It is the sharpest test of this chapter’s claim, because a causal set is exactly what §5.1 said does not exist: a causal relation with no spacetime, a locally finite partial order and nothing else. Its warrant is Malament’s theorem [223], that in a past and future distinguishing spacetime the causal order alone fixes the topology, the differentiable structure and the conformal metric, so that the order plus a volume count recovers $g$. This is in some respects the inverse of PST: where causal set theory starts with $\leq$ and tries to recover $g$, PST derives both $g$ and $\leq$ from $\Phi$. The two are not in contradiction, and the reason should be stated rather than assumed. Malament’s theorem cuts PST’s way: in the continuum the order and the conformal geometry are one structure read twice, which is coinstantiation stated as a theorem. What the discrete orders add is the result of Kleitman and Rothschild [224], that almost every finite partial order is a three-layer object approximated by no manifold at all; the manifold-like orders are a vanishing fraction, and causal set theory needs a dynamics, sequential growth or a sum over orders weighted by a discrete action, to make them typical. In PST’s vocabulary the generic finite order is a combinatorial object that is called causal by its name. It carries a causal relation in the sense of this chapter only when it is the order of a Lorentzian metric, which the sprinkling limit supplies and the generic order lacks; the claim of §5.1 is about the relation, not about the name. Causal set theory faces the question of what determines the particular causal set that becomes this universe, a question it answers by probabilistic sampling over a measure on causal sets. PST does not face this question in the same form: the particular geometry that arises is determined by the tension configuration $C$, which is itself subject to the modal potential.

The position of PST is that all approaches that take causality as a starting point, whether as a background relation, a quantum mechanical framework, or a primitive partial order, are working within the instantiated domain and can at best describe the structure of what has been instantiated. The question of why anything is instantiated at all requires a theory that can speak from outside the instantiated domain. The precausal substrate, and the functor $\Phi$ through which it gives rise to geometry, is that theory.

5.7.  Summary

Spacetime and the causal order (the fact that causes precede effects) do not arise one after the other. They are two aspects of a single transition and cannot exist independently of each other. This has a profound consequence: there is no “before” the universe, because time itself is part of what the transition produces. Asking what caused the universe is a category error, like asking what is north of the North Pole. The Big Bang is not a beginning at a point in time; it is what the transition looks like from inside the geometry it creates.

Nothing new is postulated in this chapter: causality co-emerges with geometry through the same modal-sublimation step, with no separate causal postulate added, and the arrow of time follows as a structural consequence of P2’s asymmetric tension direction $\hat{\tau}$. The causal order itself is interpreted as the partial order on instantiated events induced by $\Phi$, an interpretation consistent with the standard relativistic causal structure on $M$.