Paper G1 — Emergent Time

Abstract

This paper demonstrates that time emerges as a representational necessity from Cohesion Dynamics substrate mechanics. We show that any coherent substrate supporting persistence and interaction requires a notion of time—not as a background dimension or primitive parameter, but as ordered closure-cycle accumulation.

Building on substrate commit semantics (Paper A) and the formal mechanisms of cohesion, persistence, and phase structure (Papers M1–M4), we derive that time is forced by the need to track the ordering of discrete closure events. Closure cycles are irreversible at the representational level, producing a natural arrow of time without additional axioms.

Time is ontologically empty: it does not exist in the substrate. It is representational bookkeeping for closure-cycle succession. No background time coordinate is assumed; no metric time is introduced; no continuum is required. Time emerges as the minimal parameter necessary to distinguish “before” from “after” in constraint resolution.

This paper establishes what time is as a derived structure. It does not yet address rates, duration, synchronisation, or dynamics—those follow in later G-series papers (G2, G3, G4). Time here is purely ordinal: a labeling of closure events that preserves their causal order.

Scope: This derivation proceeds exclusively from Cohesion Dynamics primitives. No relativity, spacetime, or geometric structure is assumed. Time emerges independently of space; spatial structure is deferred to G2–G3.

1. Scope and Dependencies

1.1 Assumed Results

This paper assumes without re-derivation:

From Paper A (Substrate Mechanics):

Discrete substrate with finite alphabet $\Sigma$ and locations $V$
Local constraint system $\mathcal{C}$ defining admissibility
Mismatch measure $M(v;X)$ for configurations $X$
Local modification operations $(v \to s)$
Commit semantics: configurations resolve through discrete closure events
Closure as joint satisfaction of constraints

From Paper M1 (Constructive Viability):

Cohesive Informational Units (CIUs) as persistent structures
Mismatch as structural degree of freedom, not violation
Bounded tolerance $W$ enabling coherence
Construction requires divergence and convergence capacity

From Paper M2 (Constraint Dynamics):

Precedence-restricted admissibility: $\Delta s^* = \arg\min M(s + \Delta s)$
Persistence via structural invariance
Repair and reuse through constraint geometry
Commit as discrete resolution event

From Paper M3 (Modes):

Modes as discrete basins in state space
Finite stable configurations under precedence
Mode invariance under admissible updates
Reusability of mode structures

From Paper M4 (Phase and Coherence):

Phase $\phi$ as closure-cycle alignment
Tolerance vector $W = (W_{\text{shape}}, W_{\text{clock}}, W_{\text{spin}})$
Coherence as finite tolerance-bounded regime
Closure cycles as fundamental substrate events
Compatibility preservation requires phase tracking

From Axioms: We reference axioms by their codes (see Axioms):

AX-REL: Relational evolution (states evolve via relations to other states)
AX-TOL: Finite tolerance window $W$
AX-COH: Cohesive informational units (CIUs)
AX-PAR: Partition on tolerance violation
AX-ADM: Admissible moves exist
AX-SEL: Precedence selection
AX-MEM: Persistence (structures retain relational state)

Substrate Capability Assumption:

This paper assumes a quantum-capable substrate (Derived Capability Class: DCC-QM), as defined in R-DCC. This capability class encompasses the relational and constraint structures required for persistent constructors and representational tracking, including finite tolerance, coherence, admissibility, and commit semantics.

Granular axioms are referenced explicitly only where they play a direct operational role in the derivation.

1.2 What This Paper Does NOT Assume

This paper does not import:

Background time — no external temporal dimension is assumed
Continuous time — no continuum structure is required
Metric time — no notion of duration, rate, or proper time
Spacetime — no geometric or spatial structure
Relativity — neither special nor general relativity is assumed
Coordinates — no coordinate system or chart structure
Clocks as primitives — clock structure emerges, not assumed
Time evolution operators — dynamics is deferred (the time parameter derived here provides ordering only; it does not yet license evolution equations, rates, or generators, which remain representational results of B4 and later G-papers)
Temporal metric — no distance between events in time
Simultaneity structure — synchronisation is deferred to G2

If any of these appear later in G-series, they must be derived, not assumed.

1.3 Explicitly Out of Scope

This paper does not address:

Rates and duration (G2) — quantitative temporal structure
Synchronisation (G2) — comparing time across spatially separated events
Proper time vs. coordinate time (G3) — geometric temporal structure
Time dilation (G3–G4) — gravitational and kinematic effects
Dynamics — how states evolve (this is B4 for quantum, later G-series for geometry)
Spacetime structure (G3) — unification of time and space
Causal structure — light cones and causal ordering (G2–G3)

Time here is purely ordinal labeling of closure events. All quantitative, metrical, and geometric temporal structure is systematically deferred.

2. The Representational Problem

2.1 Substrate Evolution Is Discrete and Event-Based

The substrate defined in Paper A does not evolve continuously. Evolution proceeds through discrete closure events:

Closure event: A configuration $X$ reaches closure when all local constraints are jointly satisfied within tolerance $W$ . At closure, the configuration commits and becomes a new starting point for subsequent constraint resolution.

From M4, closure cycles are fundamental substrate events. They represent:

Joint satisfaction of internal and external constraints
Commit of divergent alternatives to compatible resolution
Phase alignment and coherence maintenance
Transition from unresolved to resolved state

Closure events are:

Discrete — not continuous processes
Irreducible — atomic transitions in constraint space
Ordered — each closure builds on prior closures
Unavoidable — persistence requires repeated closure

2.2 Closure Events Must Be Ordered

From AX-MEM (Persistence), cohesive structures retain relational state across interactions. This means:

A structure at closure cycle $n+1$ inherits structure from closure cycle $n$ .

This creates a before/after relation:

Closure $n$ must occur before closure $n+1$
Closure $n+1$ cannot occur before closure $n$ without violating persistence
The ordering is not arbitrary—it is forced by causal dependence

Why ordering is unavoidable:

Construction requires precedence (M1): To build a persistent structure, earlier constraint satisfactions must precede later ones
Repair requires history (M2): To restore a perturbed structure, the system must distinguish the unperturbed state (before) from the perturbed state (after)
Phase tracking requires accumulation (M4): Phase $\phi$ is defined by closure-cycle alignment, which accumulates over successive cycles
Provenance requires lineage (M4): Compatibility structure depends on shared resolution history

Without ordering, none of these mechanisms function. Persistence collapses.

2.3 The Representational Task

To track substrate evolution through ordered closure events, a representational calculus must:

Distinguish closure events: Identify which closure has occurred
Preserve ordering: Maintain the before/after relation
Support accumulation: Track cumulative closure cycles
Enable referencing: Allow later closures to reference earlier ones
Remain substrate-independent: Not assume geometry, space, or continuum

Minimal requirement: A parameter $t$ that labels closure events and increases monotonically.

This parameter is time.

3. Argument Outline — Why Time Is Forced

This section provides a high-level argument outline before the detailed derivation.

Step 1: Closure cycles are discrete substrate events

From Paper A and M4: commit semantics produces discrete closure events
Closure cycles are atomic, irreducible transitions
No continuous evolution exists at the substrate level

Step 2: Persistence requires ordered closure

From AX-MEM: structures retain state across interactions
Later closures inherit structure from earlier closures
This creates unavoidable causal dependence: closure $n$ → closure $n+1$

Step 3: Ordered closure requires a succession parameter

To distinguish “before” from “after”, we need a labeling that preserves order
This labeling must be monotonic: later events receive larger labels
This labeling must be universal: applies to all closure events

Step 4: That parameter is time

Time is defined as the monotonic labeling of ordered closure events
$t_n < t_{n+1}$ means closure $n$ occurs before closure $n+1$
Time has no other content: it is pure succession bookkeeping

Step 5: Time is representational, not ontological

The substrate does not contain a time dimension or temporal field
Time is a derived parameter tracking closure-event ordering
Ontology: closure events exist
Representation: time labels closure succession

Step 6: The arrow of time emerges naturally

Closure is irreversible: uncommitting a closure violates persistence
Therefore time increases monotonically
Reversal would break causal dependence and destroy structure

Conclusion: Time is a representational necessity forced by ordered closure. No additional axioms are required. Time emerges from existing substrate mechanics.

The remainder of this paper formalises each step of this outline.

4. Derivation — Time from Closure Cycles

4.1 Closure Cycles as Countable Events

Proposition 4.1.1 (Closure Discreteness):
Closure events in Cohesion Dynamics are discrete and countable.

Justification:

From Paper A: configurations evolve through finite local modifications $(v \to s)$
From M4: closure occurs when all constraints are satisfied within tolerance $W$
Tolerance $W$ is finite (AX-TOL), so closure conditions are decidable
Each closure represents a distinct resolution state
Therefore closure events form a discrete sequence

Since closure events are discrete, they can be enumerated:

\text{Closure events: } \quad C_0, C_1, C_2, \ldots, C_n, \ldots

Ontological status: These closure events exist in the substrate. The enumeration is representational bookkeeping.

Note on alternative approaches: Unlike approaches that quantise time or postulate discrete time steps, discreteness here arises solely from closure mechanics, not from an imposed temporal lattice. Time inherits its discrete structure from substrate closure events, not from independent temporal quantisation.

4.2 Ordering of Closure Events

Proposition 4.2.1 (Closure Ordering):
Closure events admit a total ordering reflecting causal dependence.

Justification:

Causal dependence exists (from AX-REL and AX-MEM):
- States evolve relationally
- Later states depend on earlier states
- This creates a partial order: $C_i \prec C_j$ if $C_j$ depends on $C_i$
Persistence requires linearity (from M1 and M2):
- A persistent structure evolves through a sequence of closures
- Each closure inherits structure from exactly one prior closure
- Divergence creates distinct lineages (handled by AX-PAR)
- Within a single provenance domain, ordering is total
Phase accumulation is monotonic (from M4):
- Phase $\phi$ accumulates through closure cycles
- Phase reversal would violate compatibility tracking
- Therefore closure cycles must be ordered

Result: Within a coherent provenance domain, closure events admit a total order:

C_0 \prec C_1 \prec C_2 \prec \ldots \prec C_n \prec \ldots

Note on branching: When AX-PAR is triggered (tolerance violation), provenance domains separate. Each domain has its own closure ordering. This does not invalidate time within each domain—it means different domains have independent time parameters. Time is totally ordered within a single coherent provenance domain; different domains possess independent time parameters once partition occurs. This is analogous to multiple clocks in relativity, which is deferred to G2.

4.3 Defining Time as Closure-Cycle Index

Definition 4.3.1 (Time Parameter):
For a coherent provenance domain with ordered closure events $\{C_0, C_1, C_2, \ldots\}$ , define the time parameter as:

t : \{C_n\} \to \mathbb{N}, \quad t(C_n) = n

That is, time is the index of closure events in their causal order.

Properties:

Monotonicity: $t(C_{n+1}) = t(C_n) + 1$ for successive closures
Ordinality: Time labels ordering, not duration
Universality: All closure events receive a unique time label
Causal consistency: If $C_i \prec C_j$ , then $t(C_i) < t(C_j)$

Operational interpretation:

Time does not “flow” or “pass”
Time increments when closures occur
Between closures, time is undefined (there is no “between” at the substrate level)
Time is not a coordinate—it is an event counter

4.4 Irreversibility and the Arrow of Time

Proposition 4.4.1 (Time Irreversibility):
The time parameter $t$ is strictly increasing and cannot reverse.

Justification:

Closure is irreversible (from commit semantics in Paper A):
- Once a configuration commits at closure $C_n$ , uncommitting requires abandoning all subsequent structure
- This would violate persistence (AX-MEM)
- Therefore closures cannot be undone
Phase accumulation is irreversible (from M4):
- Phase $\phi$ tracks closure-cycle alignment
- Reversing closure would require reversing phase accumulation
- This would violate compatibility tracking and coherence maintenance
Construction is directional (from M1):
- Structures are built through ordered precedence
- Reversing construction order would violate admissibility constraints
- Therefore construction imposes temporal asymmetry
Partition creates independent time branches (from AX-PAR):
- AX-PAR ensures that once tolerance is violated, no further closure ordering is defined across partitions, naturally yielding multiple independent temporal orderings
- Each partition domain has its own irreversible time parameter
- This is not an additional assumption but a direct consequence of partition semantics

Result: Time has a natural arrow: it increases with each closure event and cannot reverse.

This is not an additional postulate. The arrow of time emerges from closure irreversibility, which is forced by persistence and admissibility constraints.

Thermodynamic connection (optional, non-normative): The arrow of time derived here is purely structural: it reflects the irreversibility of closure. This may relate to thermodynamic irreversibility, but that connection is not required for this derivation and is left for future work.

4.5 Time as Representational Bookkeeping

Key distinction:

Ontology (what exists in the substrate):

Closure events $C_n$
Causal dependence relations $C_i \prec C_j$
Commit transitions

Clarification on ontological status:
Closure events are not located in space or time; their ordering alone is representationally labeled as time. The substrate contains only discrete closure events and their causal dependence relations, not a temporal dimension or background in which events occur.

Representation (what we use to describe evolution):

Time parameter $t$
Temporal ordering $t_i < t_j$
Time labels $t(C_n) = n$

The substrate does not contain:

A time dimension
A temporal coordinate
A time field
A background clock
A metric on time

Time is pure bookkeeping: it labels closure succession without adding ontological content.

Analogy: Time is to closure events as natural numbers are to a counted collection. Numbers do not exist in the objects being counted; they are representational tools for tracking quantity. Similarly, time does not exist in closure events; it is a representational tool for tracking succession.

5. Resulting Temporal Structure

5.1 What Has Been Established

By the end of this derivation, we have shown:

Time is unavoidable: Any substrate supporting persistence requires ordered closure tracking
Time is ordinal: It labels succession, not duration or rate
Time is monotonic: It increases with each closure and cannot reverse
Time has a natural arrow: Irreversibility emerges from closure mechanics
Time is representational: It is bookkeeping, not ontological

5.2 What Time Is (Summary)

Time in Cohesion Dynamics is:

The index $t$ of closure events in causal order
A monotonically increasing parameter
Representational bookkeeping for substrate evolution
Forced by persistence, not postulated

Time is not:

A background dimension
A continuous parameter
A geometric coordinate
A primitive concept

5.3 Enabling Subsequent G-Series Work

This derivation of ordinal time enables:

G2 (Emergent Distance):

Distance can be defined as propagation delay measured in closure cycles
Without time, propagation delay is undefined
Time provides the ordering framework for causal propagation

G3 (Emergent Geometry):

Geometry requires both time and distance
Time-distance consistency conditions will produce geometric structure
Without time, geometry cannot be relational

G4 (Gravity from Closure Gradients):

Gravitational effects require temporal structure
Time dilation arises from varying closure rates
Without time, gravitational acceleration is undefined

Summary: Time is the first geometric coordinate to emerge. It is purely temporal and ordinal. Spatial structure and metric structure are deferred to later G-series papers.

6. Ontology vs. Representation

6.1 What Exists (Ontology)

At the substrate level, the following exist:

Closure events $C_n$
Causal dependence relations $C_i \prec C_j$
Constraint satisfaction and commit transitions
Phase accumulation through closure cycles

These are substrate facts. They occur independently of how we describe them.

Saying that time is ontologically empty does not deny temporal phenomena; it denies only that time itself is a substrate entity. Closure ordering is real; the time parameter is representational bookkeeping for that ordering.

6.2 What Is Representational (Not Ontological)

The following are representational tools, not substrate entities:

The time parameter $t$
Temporal coordinates or labels
Monotonic ordering notation $t_i < t_j$
The “flow” of time (time does not flow; closures occur)

Time is bookkeeping. It tracks closure succession efficiently without claiming that a temporal dimension exists.

6.3 Why This Distinction Matters

Confusing ontology with representation leads to:

Treating time as a thing that “exists” independently
Asking what time “is made of” (meaningless—time is a label)
Assuming time has properties beyond succession (e.g., metric, topology, direction as postulate)
Importing unnecessary structure (background time, absolute simultaneity)

Correct understanding:

Substrate: discrete closure events with causal order
Representation: time parameter labels that order

This distinction enables clean derivations in G2–G4 without ontological baggage.

7. Explicitly Out of Scope

This paper does not claim or establish:

7.1 Quantitative Temporal Structure

Duration — how much time passes between events (G2)
Rates — how fast processes occur (G2)
Proper time — time as measured by localized structures (G3)

These require additional structure (distance, geometry) not yet derived.

7.2 Synchronisation and Simultaneity

Synchronisation — comparing time across spatial separation (G2)
Simultaneity — defining “at the same time” for distant events (G2–G3)
Clock comparison — relating different time parameters (G2)

These require spatial structure and causal propagation, which are deferred.

7.3 Relativistic Effects

Time dilation — varying closure rates in different contexts (G3–G4)
Proper time vs. coordinate time (G3)
Curved temporal structure (G3)

These require geometric and gravitational structure from later G-series papers.

7.4 Dynamics

How states evolve — time evolution operators, Hamiltonians (B4 for quantum mechanics)
Conservation laws — energy-momentum conservation in time (later work)

Clarification: The time parameter derived here provides ordering only; it does not yet license evolution equations, rates, or generators (which remain representational results of B4 and later G-papers).

Dynamics is explicitly deferred.

7.5 Cosmology

Age of the universe — requires global time structure (future work)
Big Bang — initial closure event (speculative, not addressed)
Time’s beginning — whether time has a start (not addressed)

These are far downstream and not required for G-series foundational work.

8. Implications

8.1 For Subsequent G-Series Papers

G2 (Emergent Distance):

Distance will be defined as propagation delay measured in closure cycles
Time provides the framework for defining “delay”
Causal structure requires ordered time

G3 (Emergent Geometry):

Geometry requires both time and distance
Time-distance consistency produces curvature
Without ordinal time, relational geometry is undefined

G4 (Gravity from Closure Gradients):

Time dilation emerges from closure-rate gradients
Gravitational acceleration requires temporal structure
Time enables gravitational effects to be representable

8.2 For Physics Interpretation

Relation to special relativity:

Time here is not relativistic (no Lorentz structure yet)
Proper time and coordinate time distinction requires G3
Time dilation is deferred to G3–G4

Relation to general relativity:

Curved temporal structure requires geometry (G3)
Time here is flat (ordinal, not metric)
GR’s metric time emerges later, not assumed

Relation to quantum mechanics:

B4 derives dynamics (time evolution)
Time parameter here is compatible with quantum time evolution
No conflict: time is the same ordinal parameter in both contexts

8.3 For Ontology

Time does not exist in the substrate.

This has profound implications:

Questions like “what is time made of” are category errors
Time is not a substance, field, or dimension
Time has no properties beyond succession
Temporal flow is a representational artifact, not a physical process

What replaces time ontologically:

Closure events (discrete, countable)
Causal dependence (ordering relations)
Commit semantics (irreversibility)

This is a fully relational account of time. Time is nothing more than the structure of ordered closure events. It has no independent existence.

8.4 Falsifiability

This derivation is falsifiable:

If any of the following were shown:

Persistent structures can exist without ordered closure
Closure cycles can reverse without violating persistence
Causal dependence does not impose ordering
Phase accumulation does not require succession tracking

Then this derivation would fail, and Cohesion Dynamics would be falsified or require substantial revision.

What would NOT falsify this derivation:

Discovery that time is metric (that is G2–G3)
Discovery that time dilates (that is G3–G4)
Discovery that time has a beginning (cosmological, not foundational)

The derivation is robust because it proceeds from substrate primitives with minimal assumptions.

9. Summary and Conclusion

9.1 What Has Been Shown

We have demonstrated that:

Time emerges as a representational necessity from Cohesion Dynamics substrate mechanics
Time is the ordinal labeling of discrete closure events in causal order
Time is monotonic and irreversible due to closure commit semantics
Time has a natural arrow from closure irreversibility
Time is representational, not ontological — it is bookkeeping for closure succession

9.2 Key Results

Time is defined: $t(C_n) = n$ for ordered closure events
Time is unavoidable: persistence requires ordered closure tracking
Time is minimal: no additional structure assumed (no continuum, metric, background dimension)
Time enables G2–G4: subsequent geometric and gravitational derivations depend on ordinal time

9.3 Methodological Achievement

This derivation:

Proceeds exclusively from Cohesion Dynamics primitives (Paper A, M1–M4)
Avoids importing time as a primitive or background structure
Maintains falsifiability: failure would falsify CD, not just an ansatz
Preserves the ontology/representation distinction cleanly

9.4 Next Steps

With time established as ordinal closure-cycle labeling, the G-series can now proceed to:

G2 — Derive distance as propagation delay (requires time)
G3 — Derive geometry from time-distance consistency (requires G1–G2)
G4 — Derive gravity from closure gradients (requires G1–G3)

Time is the first geometric coordinate to emerge. It is purely temporal and purely ordinal. All metric, spatial, and gravitational structure follows in subsequent papers.

10. Acknowledgments and Programme Coherence

10.1 Series Alignment

This paper:

Assumes A-series (substrate mechanics) as established
Assumes M-series (formal mechanisms) as established
Assumes B-series (quantum representational pattern) as methodological precedent
Extends G-series (geometry and gravity derivation) as first substantive paper

10.2 Axiom Usage

All axioms referenced by code (from Axioms):

AX-REL: Relational evolution
AX-TOL: Finite tolerance
AX-COH: Cohesive informational units
AX-PAR: Partition on tolerance violation
AX-ADM: Admissible moves
AX-SEL: Precedence selection
AX-MEM: Persistence

No axioms were introduced or modified.

10.3 Programme Management Check

✅ Series metadata verified: series: “G-series”, series_name: “Gravity and Geometry Derivation”
✅ Scope alignment verified: Paper derives time from substrate, consistent with G-series outline
✅ Axiom references verified: All axioms use codes (AX-REL, AX-TOL, etc.), not numbers
✅ Dependencies verified: Normative dependencies on A, M1–M4; non-normative reference to R-DCC
✅ Dependency metadata declared: Uses canonical Dependency DSL (!>, ?> markers)
✅ No structural changes required: G-series outline remains accurate; no updates needed

10.4 Dependency Stewardship Check

✅ Normative dependencies declared: A, M1, M2, M3, M4 (all required for closure mechanics and persistence)
✅ Axiom dependencies declared: AX-TOL, AX-COH, AX-PAR (central to closure and partition semantics)
✅ Non-normative reference declared: R-DCC (informs capability assumption)
✅ Upstream alignment: No upstream papers require updates
✅ Downstream alignment: G2–G4 will depend on G1 as expected

END OF PAPER G1