Paper G2 — Emergent Distance

Abstract

This paper demonstrates that spatial distance emerges as a representational necessity from Cohesion Dynamics substrate mechanics. Building on the emergent time parameter derived in G1, we show that distance is not a primitive geometric property, coordinate, or background metric, but the minimal scalar required to consistently represent propagation delay between closure events in a cohesive substrate.

Distance arises unavoidably from:

• Finite propagation of constraint influence through the substrate
• Commit-based closure cycles requiring sequential propagation
• Cohesion-limited information transfer preventing instantaneous global updates
• Locality constraints from tolerance-bounded admissibility

We derive that distance is forced as the unique stable bookkeeping quantity for ordering interactions when time (from G1) exists but spatial separation must be representable. Distance is shown to be additive along propagation chains, symmetric in the absence of gradients, and necessarily scalar before geometric structure (G3) emerges.

Ontologically, distance does not exist in the substrate. What exists are:

• Discrete closure events (from G1)
• Finite propagation of admissible state transitions
• Tolerance-bounded constraint resolution domains
• Causal ordering through sequential commits

Representationally, distance is:

• The minimal number of closure cycles required for influence to propagate between regions
• A derived measure of causal separation
• Pure bookkeeping for propagation delay, not substance

This paper establishes what distance is as a derived structure. It does not yet address geometry, angles, dimensions, or curvature—those follow in G3–G4. Distance here is purely scalar propagation delay: a labeling of causal separation that preserves consistency with emergent time. Throughout, distance is treated strictly as representational bookkeeping, not a substrate entity.

Scope: This derivation proceeds exclusively from Cohesion Dynamics primitives and G1 results. No spatial manifolds, coordinate systems, or metric structures are assumed. Distance emerges independently of geometric embedding; geometric structure is deferred to G3.

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1. Scope and Dependencies

1.1 Assumed Results

This paper assumes without re-derivation:

From Paper G1 (Emergent Time):

• Time as emergent ordinal parameter: $t(C_n) = n$ for ordered closure events
• Closure cycles as discrete substrate events
• Temporal ordering of closure events within coherent provenance domains
• Irreversibility of time from commit semantics
• Time as representational bookkeeping for closure succession

From Paper A (Substrate Mechanics):

• Discrete substrate with finite alphabet $\Sigma$
• Local constraint system $\mathcal{C}$ defining admissibility
• Mismatch measure for configurations
• Local modification operations
• Commit semantics: configurations resolve through discrete closure events
• Closure as joint satisfaction of constraints
• Critical ontological clarification: The substrate contains only discrete closure events and constraint relations. There is no background set of “locations” independent of CIUs. What Paper A calls “locations” are representational labels for regions where CIUs may form through admissible closure

From Paper M1 (Constructive Viability):

• Cohesive Informational Units (CIUs) as persistent structures
• Mismatch as structural degree of freedom
• 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
• Local constraint resolution (no global instant reconciliation)
• 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
• Provenance as shared resolution history

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)
• AX-LOC: Locality (all relational evolution proceeds locally)

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, representational tracking, and propagation through adjacency, including finite tolerance, coherence, admissibility, locality, and commit semantics.

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

1.2 What This Paper Does NOT Assume

This paper does not import:

• Spatial manifolds — no background space or embedding structure
• Coordinate systems — no coordinate charts or reference frames
• Metric tensors — no notion of metric distance or inner products
• Geometric structure — no angles, dimensions, or curvature
• Euclidean or non-Euclidean geometry — geometry is deferred to G3
• Spacetime — no unification of space and time yet (G3)
• Speed of light as primitive — propagation rates emerge, not assumed
• Continuous space — no continuum structure is required
• Simultaneity structure — synchronisation across distance is derived, not assumed
• Causal light cones — causal structure requires both time and distance; partial structure emerges here
• General Relativity structures — no GR objects, identities, or equations

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:

• Geometry (G3) — angles, dimensions, curvature, and global consistency of distance
• Metric structure (G3) — quantitative measurement beyond counting closure cycles
• Spatial topology — connectedness, dimensionality, manifold structure
• Direction and orientation — directional structure is secondary to scalar distance
• Gravity (G4) — gravitational effects from cohesion gradients
• Speed of light — propagation rate requires metric structure (G3)
• Spatial dimensions — number and structure of dimensions (G3)
• Triangle inequality in geometric sense — consistency constraints emerge but geometric formulation requires G3
• Synchronisation protocols — practical clock comparison requires developed distance notion and is partially addressed here, fully in G3

Distance here is purely scalar causal separation measured in closure cycles. All geometric, metrical, and directional structure is systematically deferred.

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2. The Representational Problem

2.1 Closure Is Local, Not Global

From AX-LOC (Locality) and the constraint structure in Paper A:

Constraint resolution occurs locally with respect to existing cohesion relations. No update requires non-local coordination beyond the current cohesion domain.

This means:

1. When a closure event occurs within a CIU, it does not instantly affect all other CIUs
2. Changes propagate through admissible state transitions from CIU to CIU via reconciliation
3. Global reconciliation would violate locality and require non-local coordination

From M2: Precedence-restricted admissibility ensures that constraint resolution proceeds through local mismatch minimization. There is no mechanism for instantaneous global updates.

From M4: Phase and compatibility tracking require propagation through adjacency relations. Phase alignment cannot jump arbitrary causal separation.

Result: Closure events in different CIUs are causally separated by the finite propagation of constraint influence through reconciliation.

2.2 Propagation Consumes Closure Cycles

From G1, time is defined as the ordinal labeling of closure events. When influence propagates from CIU₁ to CIU₂:

1. Initial closure within CIU₁ occurs at time $t_1$
2. Propagation requires sequential admissible reconciliations through intermediate CIUs
3. Influenced closure within CIU₂ occurs at time $t_2 > t_1$

The difference $\Delta t = t_2 - t_1$ represents propagation delay in closure cycles.

Why propagation delay is unavoidable:

From AX-ADM (Admissible Moves): Each admissible transition is local and finite. There exists no admissible transition that spans arbitrary causal separation in a single closure cycle.

From AX-TOL (Finite Tolerance): Tolerance $W$ is finite, so the maximum “reach” of any single admissible reconciliation is bounded. Influence cannot propagate arbitrarily far per closure cycle.

From AX-REL (Relational Evolution): States evolve via relations to other states. For influence from CIU₁ to reach CIU₂, there must be a chain of relational updates connecting them.

Conclusion: Propagation delay between causally separated CIUs is a necessary consequence of local admissibility and finite tolerance.

2.3 Propagation Delay Creates Ordering That Is Not Temporal Alone

Consider three CIUs that can undergo closure: CIU₁, CIU₂, CIU₃.

Scenario: A closure event occurs within CIU₁ at time $t_0$.

Observation:

• Admissible reconciliation with CIU₂ requires $\Delta t_{12}$ closure cycles
• Admissible reconciliation with CIU₃ requires $\Delta t_{13}$ closure cycles

If $\Delta t_{12} = \Delta t_{13}$, then CIU₂ and CIU₃ can undergo reconciliation at the same time.

Question: Are CIU₂ and CIU₃ at the same “location” or different locations?

Answer from CIU-only ontology: They are distinct CIUs with distinct closure histories but equidistant from CIU₁ in terms of propagation delay. “Location” is representational language for distinguishing CIUs that can undergo admissible closure; it is not a pre-existing background structure.

Representational requirement: We need a parameter that distinguishes:

• Temporal separation: when closure events occur in time
• Spatial separation: which CIUs undergo closure, even if temporally simultaneous

Time (from G1) labels “when.” We need a parameter to label “which CIU” through causal separation.

2.4 The Representational Task

To track substrate evolution through propagation of closure events across distinct CIUs, a representational calculus must:

1. Distinguish causal separation from temporal succession: Closure events can be temporally ordered but involve different CIUs
2. Track propagation delay: The number of closure cycles required for admissible reconciliation to propagate between CIUs
3. Support additivity: Sequential reconciliation through multiple CIUs accumulates delay
4. Enable causal ordering: Determine whether a closure in CIU₁ can affect closure in CIU₂
5. Remain substrate-independent: Not assume geometry, manifolds, or coordinates

Minimal requirement: A scalar parameter $d$ that represents propagation delay for admissible reconciliation between CIUs, measured in closure cycles.

This parameter is distance.

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3. Argument Outline — Why Distance Is Forced

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

Step 1: Closure is local

• From AX-LOC and M2: Constraint resolution occurs locally within CIUs
• No global instant reconciliation is possible
• Influence must propagate through admissible reconciliations between CIUs

Step 2: Reconciliation is finite and sequential

• From AX-ADM and AX-TOL: Admissible reconciliations have bounded reach
• Tolerance $W$ is finite, limiting reconciliation per closure cycle
• Influence propagates through chains of local reconciliations

Step 3: Propagation delay is measurable in closure cycles

• From G1: Time labels closure events ordinally
• Delay between closure in CIU₁ and influenced closure in CIU₂ is $\Delta t$
• This delay is invariant (does not depend on re-description)

Step 4: Delay accumulates additively

• Sequential reconciliation through CIU₁ → CIU₂ → CIU₃ adds delays
• $d(\text{CIU}_1, \text{CIU}_3) = d(\text{CIU}_1, \text{CIU}_2) + d(\text{CIU}_2, \text{CIU}_3)$ along reconciliation chain
• Additivity is unavoidable consequence of sequential closure

Step 5: Scalar separation emerges

• Any consistent representation of reconciliation delay must collapse to a scalar
• Direction and orientation are secondary (require geometry from G3)
• Distance is the minimal invariant encoding reconciliation delay

Step 6: Distance is symmetric in the absence of gradients

• Reconciliation from CIU₁ to CIU₂ consumes same closure cycles as CIU₂ to CIU₁
• Asymmetry requires cohesion gradients (G4)
• Base case: $d(\text{CIU}_1, \text{CIU}_2) = d(\text{CIU}_2, \text{CIU}_1)$

Conclusion: Distance emerges as the only stable invariant encoding reconciliation delay between CIUs’ closure events. It is representational bookkeeping for causal separation, not ontological substance.

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

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4. Derivation — Distance from Reconciliation Delay

4.1 Finite Reconciliation Reach

Proposition 4.1.1 (Bounded Reconciliation): In any single closure cycle, constraint influence can propagate at most a finite bounded reach through admissible CIU reconciliations.

Justification:

1. From AX-ADM (Admissible Moves):

   • Admissible transitions affect a CIU and its immediate relational neighbors
   • There exists no admissible transition that spans arbitrary causal separation simultaneously

2. From AX-TOL (Finite Tolerance):

   • Tolerance $W$ is finite
   • For reconciliation to be admissible, it must preserve coherence within tolerance
   • This bounds the “reach” of any single reconciliation to CIUs within finite tolerance

3. From AX-LOC (Locality):

   • All relational evolution proceeds locally
   • Non-local coordination is not available
   • Reconciliation must proceed through adjacent or tolerance-connected CIUs

Result: Define reconciliation step as the maximum causal separation that can be bridged in one closure cycle while preserving admissibility.

Let $\ell_{\max}$ be this maximum step size. Then:

$$\ell_{\max} < \infty$$

This is not a postulate — it follows necessarily from finite tolerance and local admissibility.

Operational interpretation:

• Influence cannot “jump” arbitrarily far between CIUs
• Reconciliation requires sequential closure cycles through intermediate CIUs
• The substrate imposes a natural reconciliation limit per closure cycle

Note on $\ell_{\max}$: $\ell_{\max}$ is not a universal constant. It is a local bound induced by tolerance and admissibility, potentially varying across substrate regimes. Its role here is only to establish finiteness, not magnitude.

4.2 Propagation Paths and Delay

Definition 4.2.1 (Reconciliation Chain): A reconciliation chain from CIU₁ to CIUₙ is a sequence of CIUs:

$$\chi(\text{CIU}_1, \text{CIU}_n) = \{\text{CIU}_1, \text{CIU}_2, \ldots, \text{CIU}_{n-1}, \text{CIU}_n\}$$

such that each consecutive pair $(\text{CIU}_i, \text{CIU}_{i+1})$ can undergo admissible reconciliation (their closures can mutually cohere within tolerance $W$).

Definition 4.2.2 (Chain Length in Closure Cycles): The chain length $L(\chi)$ of reconciliation chain $\chi$ is the number of closure cycles required for admissible reconciliation to propagate from CIU₁ to CIUₙ along chain $\chi$:

$$L(\chi) = |\chi| - 1$$

where $|\chi|$ is the number of CIUs in the chain.

Proposition 4.2.3 (Multiple Chains): In general, there exist multiple reconciliation chains between CIU₁ and CIUₙ, potentially with different chain lengths.

Justification: The substrate’s admissibility structure (defined by tolerance and coherence) may admit multiple routes for reconciliation between CIUs, analogous to multiple paths in a network.

Result: Propagation delay from CIU₁ to CIUₙ is not unique until we specify the reconciliation chain.

4.3 Minimal Reconciliation Delay as Distance

Definition 4.3.1 (Distance): For two CIUs in a cohesive substrate, define distance $d(\text{CIU}_1, \text{CIU}_2)$ as:

$$d(\text{CIU}_1, \text{CIU}_2) = \min_{\chi(\text{CIU}_1, \text{CIU}_2)} L(\chi)$$

That is, distance is the minimal number of closure cycles required for admissible reconciliation to propagate from CIU₁ to CIU₂ over all possible reconciliation chains.

Properties:

1. Non-negativity: $d(\text{CIU}_1, \text{CIU}_2) \geq 0$ (trivial, counting closure cycles)
2. Identity: $d(\text{CIU}, \text{CIU}) = 0$ (no reconciliation delay within a single CIU)
3. Finiteness: $d(\text{CIU}_1, \text{CIU}_2) < \infty$ for CIUs in the same coherent domain (reconciliation chains exist)
4. Discreteness: $d(\text{CIU}_1, \text{CIU}_2) \in \mathbb{N}$ (distance measured in closure cycles)

Operational interpretation:

• Distance is not a coordinate
• Distance is not a metric (no inner product, no continuum)
• Distance is causal separation measured in closure-cycle delay for admissible reconciliation
• Distance is minimal reconciliation time between CIUs

Ontological status: Distance does not exist in the substrate. What exists:

• CIUs with discrete closure events
• Admissible reconciliation relations between CIUs
• Closure cycles consuming time

Distance is representational bookkeeping for minimal reconciliation delay.

Critical clarification: This is not a metric on a fixed graph. The reconciliation structure itself is dynamic, tolerance-gated, and history-dependent. Distance is therefore contextual to the current cohesive regime and the availability of admissible reconciliation chains, not a static combinatorial object over pre-existing locations.

4.4 Additivity of Distance Along Reconciliation Chains

Proposition 4.4.1 (Triangle Consistency): For CIUs CIU₁, CIU₂, CIU₃, if the minimal reconciliation chain from CIU₁ to CIU₃ passes through CIU₂, then:

$$d(\text{CIU}_1, \text{CIU}_3) = d(\text{CIU}_1, \text{CIU}_2) + d(\text{CIU}_2, \text{CIU}_3)$$

Justification:

1. Sequential reconciliation: Admissible reconciliation propagates from CIU₁ to CIU₂ in $d(\text{CIU}_1, \text{CIU}_2)$ closure cycles, then from CIU₂ to CIU₃ in $d(\text{CIU}_2, \text{CIU}_3)$ closure cycles.

2. From G1 (Time Additivity): Time is cumulative. If closure at CIU₂ occurs at $t_1 + d(\text{CIU}_1, \text{CIU}_2)$ and closure at CIU₃ occurs at $t_1 + d(\text{CIU}_1, \text{CIU}_2) + d(\text{CIU}_2, \text{CIU}_3)$, then total delay is:

   $$\Delta t_{13} = d(\text{CIU}_1, \text{CIU}_2) + d(\text{CIU}_2, \text{CIU}_3)$$

3. Minimality preservation: If this chain is minimal for CIU₁ → CIU₃, then:

   $$d(\text{CIU}_1, \text{CIU}_3) = \Delta t_{13}$$

Result: Distance is additive along reconciliation chains. This is not an assumption; it follows from closure-cycle counting and temporal ordering from G1.

Note on triangle inequality: In general, for arbitrary CIU₁, CIU₂, CIU₃ (not necessarily on the same minimal chain):

$$d(\text{CIU}_1, \text{CIU}_3) \leq d(\text{CIU}_1, \text{CIU}_2) + d(\text{CIU}_2, \text{CIU}_3)$$

This holds because:

• The chain CIU₁ → CIU₂ → CIU₃ is one possible reconciliation chain
• Distance is defined as minimal chain length
• Therefore, the minimal chain cannot be longer than this route

Geometric interpretation: Triangle inequality emerges naturally from minimality of reconciliation chains. It is not imported from Euclidean geometry; it arises from chain optimization. Importantly, triangle inequality here is a chain-minimality property over CIU reconciliation; it does not yet imply embeddability in any geometric space. Full geometric consistency is deferred to G3.

4.5 Symmetry in the Absence of Gradients

Proposition 4.5.1 (Distance Symmetry — Base Case): In a uniform substrate (no cohesion gradients, no preferred directions), distance is symmetric:

$$d(\text{CIU}_1, \text{CIU}_2) = d(\text{CIU}_2, \text{CIU}_1)$$

Justification:

1. From uniform admissibility: If the substrate is uniform (no spatial variation in tolerance, phase, or constraint structure), then:

   • Reconciliation from CIU₁ to CIU₂ encounters the same admissibility conditions as reconciliation from CIU₂ to CIU₁
   • Chain lengths are identical in both directions

2. From AX-REL (Relational Evolution):

   • Relational structure is bidirectional (if CIU₁ relates to CIU₂, then CIU₂ relates to CIU₁)
   • In the absence of gradients, reconciliation structure (not in time, but in causal connectivity) has no preferred direction

3. From commit semantics:

   • Time is irreversible (from G1)
   • But reconciliation structure (in uniform substrate) has no preferred direction
   • Therefore, chain lengths are symmetric

Result: Symmetry is the default in uniform substrate. Asymmetry requires additional structure (cohesion gradients, which are deferred to G4).

Clarification on symmetry vs. time reversibility: Distance symmetry does not imply time reversibility. Time irreversibility (from commit semantics in G1) is fully preserved. Symmetry applies only to reconciliation structure, not temporal order.

Note on non-uniform substrates: When cohesion gradients exist (different closure rates, different tolerance regimes), symmetry may break:

$$d(\text{CIU}_1, \text{CIU}_2) \neq d(\text{CIU}_2, \text{CIU}_1)$$

This is deferred to G4. For now, we establish symmetric distance as the base case.

4.6 Distance as Scalar (No Direction Yet)

Key observation: The distance parameter $d(v_1, v_2)$ defined above is purely scalar:

• It encodes propagation delay (a number of closure cycles)
• It does not encode direction, angle, or orientation
• It does not distinguish between different paths of the same length

Why direction is deferred:

To define direction or orientation, we need:

1. Angular structure — requires geometric consistency (G3)
2. Dimensional structure — requires embedding or manifold structure (G3)
3. Coordinate systems — requires geometric framework (G3)

None of these exist yet. Distance here is pure causal separation without directional information.

Analogy: Distance is like knowing “it takes 5 steps to get there” without knowing “which way to go.” Direction requires additional geometric structure.

Result: Distance is scalar propagation delay. Geometric embedding and directional structure are systematically deferred to G3.

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5. Resulting Distance Structure

5.1 What Has Been Established

By the end of this derivation, we have shown:

1. Distance is unavoidable: Any substrate with local closure and finite reconciliation reach requires a notion of causal separation between CIUs
2. Distance is scalar: It measures reconciliation delay without directional structure
3. Distance is additive: Sequential reconciliation accumulates delay along chains
4. Distance is symmetric (base case): In uniform substrates, reconciliation delay is bidirectional
5. Distance is discrete: Measured in closure cycles (natural number)
6. Distance is representational: It is bookkeeping for reconciliation delay, not ontological

5.2 What Distance Is (Summary)

Distance in Cohesion Dynamics is:

• The minimal number of closure cycles $d(\text{CIU}_1, \text{CIU}_2)$ required for influence to propagate through admissible reconciliation between CIUs
• A scalar measure of causal separation
• Representational bookkeeping for reconciliation structure
• Forced by locality and finite reconciliation reach, not postulated

Distance is not:

• A coordinate
• A metric (in the geometric sense)
• A continuous parameter
• An embedding in background space
• A geometric primitive
• A property of vacuum or empty space (vacuum has no CIUs, hence no reconciliation, hence no distance)

5.3 Enabling Subsequent G-Series Work

This derivation of scalar distance enables:

G3 (Emergent Geometry):

• Geometry arises from consistency conditions on distance and time between CIUs
• Curvature emerges when distance relations cannot be globally embedded in flat structure
• Metric tensors become representational shortcuts for distance bookkeeping over CIU reconciliations
• Without distance, geometry cannot be relational

G4 (Gravity from Closure Gradients):

• Gravitational effects arise from uneven closure rates in constraint-dense CIUs
• Distance enables representation of varying reconciliation availability between CIUs
• Geodesic deviation emerges from availability gradients in reconciliation chains
• Without distance, spatial gravitational effects are undefined

Summary: Distance is the second geometric parameter to emerge (after time from G1). It is purely scalar and relational over CIU reconciliation structure. Geometric structure (angles, curvature, dimensions) is deferred to G3.

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6. Ontology vs. Representation

6.1 What Exists (Ontology)

At the substrate level, the following exist:

• CIUs undergoing discrete closure events
• Admissible reconciliation relations between CIUs
• Reconciliation chains through sequential admissible reconciliations
• Closure events within CIUs
• Causal ordering of closure events (from G1)

These are substrate facts. They occur independently of how we describe them. Critically, there is no pre-existing set of “locations” independent of CIUs—only CIUs and the reconciliation structure that emerges through their closures.

6.2 What Is Representational (Not Ontological)

The following are representational tools, not substrate entities:

• The distance parameter $d(\text{CIU}_1, \text{CIU}_2)$
• Scalar separation labels
• Reconciliation delay as “distance” (the term itself)
• Minimal chain counting
• “Location” language (representational shorthand for “which CIU”)

Distance is bookkeeping. It tracks causal separation efficiently without claiming that a spatial dimension, metric, or distance field exists.

6.3 Why This Distinction Matters

Confusing ontology with representation leads to:

• Treating distance as a thing that “exists” independently
• Asking what distance “is made of” (meaningless—distance is a counting measure)
• Assuming distance has properties beyond reconciliation delay (e.g., continuity, embedding)
• Importing unnecessary structure (background space, manifolds, metrics, vacuum paths)

Correct understanding:

• Substrate: discrete CIUs with reconciliation chains and closure events
• Representation: distance parameter labels reconciliation delay

This distinction enables clean derivations in G3–G4 without geometric baggage.

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7. Explicitly Out of Scope

This paper does not claim or establish:

7.1 Geometric Structure

• Angles — angular relationships require geometric consistency (G3)
• Dimensions — number of independent directions (G3)
• Curvature — failure of global flat embedding (G3)
• Metric tensors — quantitative distance measurement beyond counting (G3)

These require additional consistency structure not yet derived.

7.2 Continuous Distance

• Continuum limit — smooth distance fields (G3, if applicable)
• Real-valued distance — distance here is discrete (natural numbers)
• Distance as continuous parameter — no continuum assumed

Continuity requires limit-taking and is deferred.

7.3 Directional Structure

• Vectors — directional quantities require geometric embedding (G3)
• Orientation — preferred directions or frames (G3)
• Coordinate systems — geometric charts and atlases (G3)

Direction is secondary to scalar separation and is deferred.

7.4 Gravitational Effects

• Gravitational attraction — cohesion gradients affect distance (G4)
• Curved spacetime — unified geometric structure (G3–G4)
• Geodesics — least-action or least-mismatch paths (G4)

Gravity requires both geometric structure and gradient analysis, both deferred.

7.5 Propagation Speed as Fundamental Constant

• Speed of light — requires metric structure to define speed (G3)
• Maximal propagation rate — requires quantitative rate definition (G3)
• Lorentz structure — requires spacetime unification (G3)

Propagation rate as quantitative constant is deferred.

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8. Implications

8.1 For Subsequent G-Series Papers

G3 (Emergent Geometry):

• Geometry will be defined as consistency structure for time-distance relations
• Curvature will emerge when distance relations cannot be globally flattened
• Metric tensors will be representational shortcuts for distance bookkeeping
• Without scalar distance, geometric structure cannot be relational

G4 (Gravity from Closure Gradients):

• Gravitational effects will emerge from spatial variation in closure rates
• Distance enables representation of cohesion gradients across space
• Geodesic deviation will arise from distance-dependent closure density
• Without distance, spatial gravitational structure is undefined

8.2 For Physics Interpretation

Relation to special relativity:

• Distance here is not relativistic (no Lorentz structure yet)
• Lorentz structure requires joint treatment of time and distance
• This paper intentionally derives distance alone
• Proper distance and coordinate distance distinction requires G3
• Invariant intervals require spacetime unification (G3)

Relation to general relativity:

• Curved distance structure requires geometry (G3)
• Distance here is “flat” (scalar, no curvature)
• GR’s metric distance emerges later, not assumed

Relation to quantum mechanics:

• Distance is compatible with quantum propagation (wavefunction spread)
• Causal structure from distance supports quantum causality
• No conflict: distance is the same scalar parameter in both contexts

8.3 For Ontology

Distance does not exist in the substrate.

This has profound implications:

• Questions like “what is distance made of” are category errors
• Distance is not a substance, field, or coordinate
• Distance has no properties beyond propagation delay counting
• Spatial extent is a representational artifact, not a physical entity

What replaces distance ontologically:

• Locations in substrate graph $V$
• Propagation paths through admissible transitions
• Closure cycles accumulating along paths

This is a fully relational account of distance. Distance is nothing more than the structure of propagation delay between locations. It has no independent existence.

8.4 Falsifiability

This derivation is falsifiable:

If any of the following were shown:

1. Influence can propagate instantaneously (violating finite propagation)
2. Propagation delay is not additive along paths (violating sequential closure)
3. Minimal propagation paths do not exist (violating well-defined distance)
4. Distance is not symmetric even in uniform substrates (violating base-case symmetry)

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

What would NOT falsify this derivation:

• Discovery that distance is continuous (that is a G3 continuum-limit result)
• Discovery that distance is curved (that is G3 geometry)
• Discovery that distance is asymmetric in gravitational fields (that is G4 gradient effect)

The derivation is robust because it proceeds from substrate primitives and G1 with minimal assumptions.

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9. Summary and Conclusion

9.1 What Has Been Shown

We have demonstrated that:

1. Distance emerges as a representational necessity from Cohesion Dynamics substrate mechanics and G1 time structure
2. Distance is the minimal scalar required to represent reconciliation delay between CIUs’ closure events
3. Distance is additive along chains due to sequential closure accumulation
4. Distance is symmetric in uniform substrates from bidirectional reconciliation structure
5. Distance is representational, not ontological — it is bookkeeping for causal separation between CIUs

9.2 Key Results

• Distance is defined: $d(\text{CIU}_1, \text{CIU}_2) = \min_{\chi} L(\chi)$ for reconciliation chains $\chi$
• Distance is unavoidable: locality and finite reconciliation reach require causal separation tracking
• Distance is minimal: no additional structure assumed (no geometry, metrics, manifolds, background locations)
• Distance enables G3–G4: subsequent geometric and gravitational derivations depend on scalar distance over CIU reconciliations

9.3 Methodological Achievement

This derivation:

• Proceeds exclusively from Cohesion Dynamics primitives (Paper A, M1–M4, G1)
• Avoids importing distance as a primitive or coordinate structure
• Avoids assuming background locations or vacuum paths
• Maintains falsifiability: failure would falsify CD, not just an ansatz
• Preserves the ontology/representation distinction cleanly
• Respects CIU-only ontology: only CIUs and their reconciliation relations exist

9.4 Next Steps

With distance established as scalar reconciliation delay between CIUs, the G-series can now proceed to:

• G3 — Derive geometry from time-distance consistency over CIU reconciliations (requires G1–G2)
• G4 — Derive gravity from availability gradients in reconciliation chains (requires G1–G3)

Distance is the second geometric parameter to emerge (after time). It is purely scalar and relational over CIU structure. All geometric structure (angles, curvature, dimensions) follows in G3.

Programme milestone: With G1 and G2, Cohesion Dynamics now possesses the minimal representational substrate required for geometry: time labels closure succession, distance labels causal separation between CIUs.

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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 G1 (emergent time) as established
• Extends G-series (geometry and gravity derivation) as second 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
• AX-LOC: Locality

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 distance from substrate and G1, consistent with G-series outline
✅ Axiom references verified: All axioms use codes (AX-REL, AX-TOL, AX-LOC, etc.), not numbers
✅ Dependencies verified: Normative dependencies on A, M1–M4, G1; 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, G1 (all required for closure mechanics, locality, and time)
✅ Axiom dependencies declared: AX-TOL, AX-COH, AX-REL, AX-ADM, AX-LOC (central to propagation and locality)
✅ Non-normative reference declared: R-DCC (informs capability assumption)
✅ Upstream alignment: No upstream papers require updates
✅ Downstream alignment: G3–G4 will depend on G2 as expected

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END OF PAPER G2