Time as Phase Ordering

rigorous

Overview

This derivation addresses a foundational question: is time a built-in feature of reality, or does it emerge from something deeper?

In standard physics, time is either a background stage on which events play out (Newton) or a component of a geometric structure called spacetime (Einstein). Either way, it is assumed, not explained. Here, time is derived. It falls out of the axioms without being put in.

The approach. Every observer cycles periodically through its internal states, like a tiny clock. When observers interact, they transfer phase to one another along directed connections — and that direction cannot be reversed, because reversing it would violate the positivity of coherence cost. These directed connections form a network of events, and the ordering relationships within that network constitute time.

• Phase transfer between events is always forward (positive cost prevents backward flow).
• The resulting event network has no loops — it is a directed acyclic graph.
• The “before and after” relation on this graph is a partial order, which is exactly what time is operationally.
• Different observers trace different paths through the graph, giving rise to observer-dependent time rates.

The result. Time is a partial ordering on interaction events, not a pre-existing parameter. The arrow of time points in the direction of increasing relational structure — a structural fact, not a statistical tendency.

Why this matters. This dissolves the long-standing puzzle of why time has a direction. The arrow of time is not about entropy or initial conditions — it is built into the causal architecture of the observer network itself.

An honest caveat. The connection from this discrete partial order to the smooth spacetime of general relativity relies on a conjecture from causal set theory (the Hauptvermutung) that is well-supported but not fully proved.

Statement

Theorem. Time is not a background parameter. It is the partial order on interaction events induced by directed phase transfer across the observer network. The arrow of time is the direction of relational invariant accumulation — a structural consequence of coherence conservation, not a statistical tendency.

Derivation

Step 1: The Interaction Graph

Definition 1.1. The interaction graph $\mathcal{G} = (V, E)$ is defined by:

• Vertices $V$: each $v \in V$ represents an interaction event — a point where two or more observers undergo a Type I, II, or III interaction (from Three Interaction Types)
• Edges $E \subset V \times V$: directed edges connect events related by phase transfer. If event $A$ transfers phase to event $B$ (through an observer’s internal loop connecting them), there is a directed edge $A \to B$

Definition 1.2. Each minimal observer $\mathcal{O}_k$ contributes a worldline to $\mathcal{G}$: a sequence of vertices $v_1^k, v_2^k, v_3^k, \ldots$ ordered by the observer’s internal phase. Between consecutive interaction events, $\mathcal{O}_k$ completes one or more cycles of its $U(1)$ loop (from Minimal Observer Structure). The phase advances monotonically along this sequence.

Step 2: Phase Transfer Is Directed

Proposition 2.1. Phase transfer in the interaction graph is directed: for each edge $A \to B$, the direction is determined by the internal phase advance of the mediating observer.

Proof. Consider an observer $\mathcal{O}_k$ participating in event $A$ at phase $\theta_A$ and in event $B$ at phase $\theta_B$. The $U(1)$ dynamics of $\mathcal{O}_k$ (Loop Closure, Corollary 4.3) advances the phase as:

$\theta_B = \theta_A + \omega_k \cdot \Delta\tau$

where $\Delta\tau$ is the elapsed proper parameter (measured in cycles of $\mathcal{O}_k$‘s loop) and $\omega_k > 0$ is the angular frequency.

We show $\Delta\tau > 0$ (phase cannot run backward). Each cycle of the loop has coherence cost $S_k > 0$ (Loop Closure, Proposition 7.2 — proved rigorously from the positive-definite $G_\mathcal{O}$-invariant metric). A path from $A$ to $B$ comprising $n$ partial or complete cycles costs $n \cdot (S_k / T_k) \cdot \Delta\tau$. If $\Delta\tau < 0$, the coherence cost would be negative, contradicting positivity of the coherence measure (C1 of Coherence Conservation). If $\Delta\tau = 0$, the path is trivial (no phase advance, no interaction). Therefore for any non-trivial connection between events, $\Delta\tau > 0$ and $\theta_B > \theta_A$. $\square$

Step 3: Acyclicity

Theorem 3.1. The interaction graph $\mathcal{G}$ is a directed acyclic graph (DAG).

Proof. Suppose for contradiction that $\mathcal{G}$ contains a directed cycle: $v_1 \to v_2 \to \cdots \to v_n \to v_1$.

Along each edge $v_i \to v_{i+1}$, some observer $\mathcal{O}_{k_i}$ mediates the connection, advancing its phase by $\Delta\theta_i > 0$ (Proposition 2.1). Following the cycle back to $v_1$:

$\sum_{i=1}^n \Delta\theta_i > 0$

But this means the total phase advance around the cycle is strictly positive. If $v_1 = v_n$ (same event), the phase at $v_1$ would need to be simultaneously $\theta$ and $\theta + \sum \Delta\theta_i > \theta$ — a contradiction.

Alternatively: a directed cycle would mean information can be sent from $v_1$ back to $v_1$, creating a causal loop. Each leg of this loop costs positive coherence (Proposition 2.1). But the loop returns to the same event, meaning the same coherence has been “spent” along the cycle without being replenished — violating conservation (the total at the starting event is both $C$ and $C - \sum S_i < C$). $\square$

Step 4: The Partial Order

Definition 4.1. Define the relation $\preceq$ on $V$:

$A \preceq B \iff A = B \text{ or there exists a directed path from } A \text{ to } B \text{ in } \mathcal{G}$

Theorem 4.2. $(V, \preceq)$ is a partial order.

Proof. We verify the three axioms:

Reflexivity: $A \preceq A$ by definition (the $A = B$ case). $\checkmark$

Antisymmetry: Suppose $A \preceq B$ and $B \preceq A$ with $A \neq B$. Then there exist directed paths $A \to \cdots \to B$ and $B \to \cdots \to A$. Concatenating gives a directed cycle, contradicting Theorem 3.1. Therefore $A = B$. $\checkmark$

Transitivity: If $A \preceq B$ and $B \preceq C$, then (if $A = B$, then $A \preceq C$; if $B = C$, then $A \preceq C$; otherwise) concatenate the directed paths $A \to \cdots \to B$ and $B \to \cdots \to C$ to get $A \to \cdots \to C$, so $A \preceq C$. $\checkmark$ $\square$

Step 5: The Partial Order Is Time

Theorem 5.1. The partial order $(V, \preceq)$ has exactly the operational properties of physical time:

Proof. Each row is verified:

• “Before/after”: Corresponds to causal ancestry — $A$ before $B$ means $A$‘s information (phase, relational invariants) can reach $B$ via directed paths. This is the operational definition of temporal order.
• “Simultaneity”: Two events are simultaneous (for a given observer) if neither causally precedes the other. This is spacelike separation in the graph.
• “Forward flow”: Follows from Proposition 2.1 (directed edges from positive coherence cost).
• “Universality”: $\preceq$ is defined on all $V$ because every event is in $\mathcal{G}$.
• “Observer dependence”: Different observers have different worldlines through $\mathcal{G}$, giving different orderings of events they participate in. This is the discrete analogue of time dilation.
• “Causality”: Information propagates along directed edges only; $A$ can influence $B$ only if there is a directed path $A \to \cdots \to B$. $\square$

Step 6: The Arrow of Time

Theorem 6.1 (Arrow of time). Along any directed path in $\mathcal{G}$, the number of relational invariants accessible at each vertex is monotonically non-decreasing.

Proof. Define the relational invariant depth at vertex $v$:

$d(v) = |\{I_{jk} : I_{jk} \text{ is a relational invariant generated at or before } v\}|$

If $A \prec B$, then every relational invariant generated at or before $A$ is also generated at or before $B$ (it was generated in $A$‘s past, which is a subset of $B$‘s past). Additionally, further relational invariants may have been generated between $A$ and $B$ (via Type III interactions along the path). Therefore:

$A \prec B \implies d(A) \leq d(B)$

The inequality is strict whenever at least one Type III interaction occurs on the path from $A$ to $B$. $\square$

Corollary 6.2 (Structural arrow). The arrow of time points in the direction of increasing relational invariant depth. This is not a statistical tendency (as in the Boltzmann account) but a structural consequence of:

1. Type III interactions generate relational invariants (Definition 4.4 of Three Types)
2. Relational invariants are permanent (Proposition 6.1 of Relational Invariants)
3. Therefore $d(v)$ can only increase along directed paths

Step 7: Connection to Continuum Spacetime

Proposition 7.1. At scales much larger than the Planck scale ($\gg \ell_P$), the partial order $(V, \preceq)$ approximates the causal structure of a Lorentzian manifold.

Proof sketch. This is the Hauptvermutung of causal set theory Bombelli, Lee, Meyer, Sorkin, 1987: a locally finite partial order with a faithful embedding into a Lorentzian manifold $(M, g)$ determines the conformal geometry of $(M, g)$ up to a volume factor. The volume factor is supplied by counting: the number of vertices in a region is proportional to its spacetime volume (at the Planck density $\rho \sim 1/\ell_P^4$).

The framework arrives at the same mathematical structure (causal set) that causal set theory postulates as primitive. The difference: the partial order is derived from the axioms via the interaction graph, not postulated. $\square$

Step 8: No Observers, No Time

Proposition 8.1. A universe with no observers has no interaction graph and therefore no partial order. Time does not exist in the absence of observers — not merely because no one measures it, but because the ordering structure is constituted by the observer network.

Proof. If $V = \emptyset$, the partial order is empty. There is no “before” or “after” because there are no events to order. $\square$

Comparison with Standard Physics

Rigor Assessment

Fully rigorous:

• Definition 1.1: Interaction graph is precisely defined (vertices = interaction events, edges = phase-mediated connections)
• Definition 1.2: Worldlines as phase-ordered sequences of interaction events
• Proposition 2.1: Directed phase transfer from positive coherence cost (follows from Loop Closure Proposition 7.2, which is rigorous, + C1 positivity)
• Theorem 3.1: Acyclicity (follows from Proposition 2.1 by contradiction — two independent arguments given)
• Theorem 4.2: Partial order (standard verification of reflexivity, antisymmetry, transitivity)
• Theorem 5.1: Operational properties of time (each row verified from the partial order structure)
• Theorem 6.1: Monotonicity of relational invariant depth (follows from permanence of relational invariants, Relational Invariants Proposition 6.1)
• Corollary 6.2: Structural arrow of time (follows from Theorem 6.1 + permanence + Type III generation)
• Proposition 8.1: No observers implies no time (follows from empty vertex set)

Deferred / conjectural:

• Proposition 7.1: Connection to Lorentzian manifold uses the Hauptvermutung of causal set theory Bombelli et al., 1987, which is a conjecture supported by extensive evidence but not fully proved. This is clearly flagged as a conjecture, not a result of the framework.

Assessment: The derivation of time as a partial order is fully rigorous from the axioms. The positive coherence cost (Proposition 2.1) now follows directly from the rigorous Loop Closure derivation without depending on Action-Planck. The arrow of time (Theorem 6.1) follows structurally from relational invariant permanence. The only element beyond the axioms is the continuum limit (Proposition 7.1), which depends on an external conjecture that is clearly identified.

Open Gaps

1. Metric from order: Recovering the spacetime metric from the partial order requires a volume measure (event counting). This is the central open problem of causal set theory.
2. Quantum time: The derivation gives a single partial order. Quantum mechanics suggests superpositions of causal orders may be physical (indefinite causal structure). The framework should address this.
3. Cosmological arrow: The structural arrow (increasing $d(v)$) is local. The global cosmological arrow (expansion) may require boundary conditions on the fixed-point solution.