Paper G4 — Gravity from Cohesion Gradients

Abstract

This paper demonstrates that gravitational attraction emerges as a representational necessity from Cohesion Dynamics substrate mechanics. We show that what is traditionally described as “gravity” is not a force, curvature postulate, or metric assumption, but a bias in admissible resolution paths caused by gradients in CIU constraint saturation and closure availability.

Building on emergent time (G1), emergent distance (G2), and emergent geometry (G3), we derive that gravitational effects arise unavoidably from:

Constraint-dense CIUs (mass–energy) altering local closure rates
Asymmetric mismatch redistribution capacity creating directional bias in reconciliation availability
Precedence-driven resolution following least-mismatch reconciliation chains
Availability-gradient flow producing apparent gravitational acceleration

Gravitational attraction is not a fundamental interaction—it is the necessary consequence of availability gradients in CIU reconciliation. What appears as “falling” is substrate resolution following the chain of least resistance toward CIUs where closures are easier to satisfy.

Ontologically, gravity does not exist as a force or curvature of space. What exists are:

Discrete closure events with varying difficulty across CIUs
Availability gradients in reconciliation between CIUs
Precedence selection biasing resolution chains
Phase accumulation differences across CIUs

Representationally, gravity is:

The effective description of motion under closure availability gradients
Free fall as least-mismatch reconciliation trajectory
Gravitational time dilation as closure-rate variation across CIUs
Universal coupling from substrate neutrality (all CIUs respond identically)

This paper establishes what gravity is as a derived phenomenon. It does not yet address metric dynamics, field equations, or empirical predictions—those follow in G5 or application papers. Gravity here is purely emergent bias in substrate resolution: trajectories appear gravitationally curved because they follow availability-gradient flow in CIU reconciliation structure.

Scope: This derivation proceeds exclusively from Cohesion Dynamics primitives and G1–G3 results. No forces, field equations, stress-energy tensors, or General Relativity structures are assumed. Gravitational effects emerge from closure accounting and precedence selection in CIU reconciliation chains alone.

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 G2 (Emergent Distance):

Distance as reconciliation delay between CIUs: $d(\text{CIU}_A, \text{CIU}_B) = \Delta t_{\text{rec}}(\text{CIU}_A \to \text{CIU}_B)$
Distance measured in closure cycles
Additivity along chains when reconciliation availability is uniform
Substrate-state dependence: distance varies with local CIU constraint saturation
Scalar distance without geometric embedding

From Paper G3 (Emergent Geometry):

Geometry as consistency structure for distance relations between CIUs
Curvature as measure of distance composition inconsistency
Local geometric structure without global embedding
Metric as representational shorthand (not primitive)
Chain-comparison defects revealing closure inconsistency

From Paper A (Substrate Mechanics):

Discrete substrate with constraint resolution through closure events
Local constraint system $\mathcal{C}$ defining admissibility
Mismatch measure for configurations
Commit semantics: configurations resolve through discrete closure events within CIUs
Closure as joint satisfaction of constraints
Critical ontological clarification: Only CIUs and their reconciliation relations exist; no background set of locations or spatial fields

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
Repair mechanisms restore perturbed structures

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 Paper M7 (Explanatory Consolidation):

Layered explanatory structure
Consolidation rules for derived concepts
Representational necessity vs. ontological commitment

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-LOC: Locality (all relational evolution proceeds locally)
AX-SEL: Precedence (mismatch minimisation 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, representational tracking, propagation, and geometric consistency, 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:

Force laws — no Newtonian $F = ma$ or gravitational force postulates
Spacetime curvature as primitive — curvature emerges from G3, not assumed
Metric tensors — no $g_{\mu\nu}$ as starting structure
Einstein field equations — no $G_{\mu\nu} = 8\pi T_{\mu\nu}$
Stress-energy tensor — energy-momentum is substrate-level
Geodesic equations as axioms — free fall must be derived
Variational principles — no least-action assumptions
Equivalence principle as postulate — equivalence must emerge
General Relativity objects — no GR structures as primitives
Field theory — gravity is not a field propagating through space
Continuous manifolds — no background geometric structure
Coordinate systems — coordinates are representational only
Background metric — no pre-existing geometric structure
Gravitational constant as primitive — coupling must emerge from substrate

If any of these appear, they must be derived as representational consequences, not assumed.

1.3 Explicitly Out of Scope

This paper does not address:

Metric dynamics (G5) — how geometric structure evolves
Einstein field equations recovery — numerical matching with GR
Cosmology — large-scale gravitational structure
Dark matter — applications follow after foundational derivation
Dark energy — expansion and cosmological constant
Gravitational waves — dynamic geometry propagation
Black holes — extreme gravity regimes and horizons
Quantum gravity — substrate-scale gravitational structure
Strong-field gravity — neutron stars, mergers
Empirical predictions — testable consequences (future work)
Numerical simulations — computational gravity applications

These are systematically deferred to G5, application papers, or future extensions.

Gravity here is purely emergent bias in substrate resolution: the derivation shows why gravitational effects are unavoidable, not how to compute specific trajectories or match GR numerically.

2. Argument Outline — Why Gravity Is Forced

This section provides a high-level reasoning chain before the formal derivation. The argument proceeds in clear, non-technical steps showing why gravitational attraction is representationally unavoidable.

Critical conceptual distinction (read first):

Three distinct substrate processes must be kept separate to avoid confusion:

Mismatch redistribution — Internal constraint resolution capacity; flows away from saturated regions
CIU motion — Arises from propagation-delay asymmetry and geometric bookkeeping
Gravitational “attraction” — Geometric bookkeeping of the second effect, not the first

These are substrate-mechanically distinct. Gravity is the representational description of CIU motion (process 2), not mismatch flow (process 1). The formal derivation below establishes this rigorously.

2.1 The Representational Problem

Premise 1: CIU constraint saturation varies

In any realistic substrate supporting persistent structures (mass–energy), CIU constraint saturation is non-uniform:

CIUs with many active constraints have high saturation
CIUs with fewer constraints have lower saturation
CIU saturation creates variation in closure availability

From M2, closure requires joint satisfaction of constraints. More constraints mean harder closure (more mismatch to redistribute within the CIU).

Premise 2: Closure availability varies with CIU saturation

High-saturation CIUs (massive objects):

Have many overlapping constraints
Require more mismatch redistribution for closure
Take longer to reach closure (more cycles required)
Have reduced mismatch absorption capacity (saturation)

Low-saturation CIUs:

Have fewer constraints
Reach closure more easily
Take fewer cycles to close
Have greater mismatch absorption capacity

Result: Gradient in closure availability across CIU reconciliation structure

2.2 Availability Gradients Create Directional Bias

Key observation: Mismatch redistribution follows least resistance

From AX-SEL (Precedence), substrate resolution preferentially selects transitions that minimise mismatch. This is not teleology—it is the operational definition of admissible evolution.

When a CIU is in an availability gradient (surrounded by CIUs with varying saturation):

One direction involves reconciliation with higher-saturation CIUs (harder closure)
Another direction involves reconciliation with lower-saturation CIUs (easier closure)
Mismatch redistribution preferentially flows toward easier closure

Operational mechanism:

Consider a CIU in a gradient. When constraints propagate through reconciliation with neighboring CIUs:

Reconciliation toward high-saturation CIUs encounters saturated constraint channels
Reconciliation toward low-saturation CIUs encounters available mismatch capacity
Precedence selection biases admissible moves toward low-resistance reconciliation chains

Result: Directional asymmetry in resolution chains

2.3 Asymmetric Resolution Is Experienced as Acceleration

The CIU does not “choose” to fall—falling is the only admissible resolution

From G2, distance is reconciliation delay between CIUs. When resolution chains are biased:

The CIU’s future closure events accumulate more rapidly in one direction
Reconciliation delays become directionally asymmetric
Distance to neighbouring CIUs changes over successive closure cycles

This is motion:

Not motion through pre-existing space
Motion as changing reconciliation-delay relations between CIUs
Direction determined by availability-gradient structure

This is acceleration:

Not from a force acting on the CIU
Acceleration as rate of change in motion bias
Magnitude determined by gradient steepness

Gravitational attraction emerges as the effective description of this bias.

2.4 Why Free Fall Is Universal

All CIUs respond identically to availability gradients

From substrate mechanics, precedence selection (AX-SEL) applies uniformly to all CIUs, regardless of their internal structure. The substrate does not “know” whether a CIU is:

A single quantum
A composite structure (atom, molecule, macroscopic object)
Light (photon trajectory)

All that matters is local CIU saturation and mismatch redistribution availability.

Result: Universal coupling

Gravitational response is independent of composition
All objects fall identically in the same gradient
No “gravitational charge” or composition-dependent coupling

This is the weak equivalence principle, derived from substrate neutrality.

2.5 Why Inertial and Gravitational Mass Are Identical

Resistance to acceleration and gravitational response arise from the same substrate property

Inertial mass (resistance to acceleration):

From M2, changing a CIU’s resolution path requires overcoming precedence
Internal constraints resist external perturbation
More constraints → greater resistance → larger inertial mass

Gravitational mass (response to gravity):

High constraint density creates stronger closure gradients
More internal constraints → stronger gradient → larger gravitational mass

Both arise from constraint count and closure structure.

Result: $m_{\text{inertial}} = m_{\text{gravitational}}$ structurally

This is the equivalence of inertial and gravitational mass, not as a postulate but as a structural identity.

2.6 Why Gravitational Time Dilation Is Forced

Closure cycles slow in high-density regions

From G1, time is closure-cycle accumulation. In high constraint density:

Closure takes more cycles (harder to satisfy all constraints jointly)
Closure rate slows relative to low-density regions
Time accumulates more slowly

Result: Gravitational time dilation

Clocks run slower near massive objects
Not from metric postulates
Derived from closure-rate variation

From G3, this creates geometric effects (longer paths take more time), but the fundamental cause is closure difficulty.

2.7 Why Geodesic Motion Is Representational

“Straight line in curved spacetime” is bookkeeping shorthand

From G3, geometry is a consistency framework for distance relations. When gravitational gradients exist:

Least-mismatch paths are not straight in flat geometry
They appear curved when embedded in coordinate space
Geodesics (curves minimising proper time) are the geometric description

Ontologically: Substrate follows precedence-driven resolution through closure gradients

Representationally: Motion appears geodesic because geometry tracks closure structure

Geodesics are not chosen by objects—they are the geometric bookkeeping of substrate resolution.

2.8 Conclusion of the Argument Outline

Gravity is unavoidable:

Constraint density varies spatially (mass–energy distribution)
Closure difficulty varies with constraint density
Precedence selection creates directional bias toward easier closure
Asymmetric resolution is experienced as acceleration
Universal coupling follows from substrate neutrality
Inertial/gravitational mass identity is structural
Time dilation follows from closure-rate variation
Geodesic motion is geometric bookkeeping of precedence flow

Gravity is therefore:

Not a force — no force acts on falling objects
Not curvature — curvature is representational (G3)
Emergent bias — resolution follows closure gradients
Universal — all CIUs respond identically
Forced — not chosen, but structurally unavoidable

The formal derivation that follows makes this argument precise.

3. CIU Constraint Saturation

3.1 CIU Constraint Saturation as Substrate Mass–Energy

What is “mass” in substrate terms?

In conventional physics, mass is:

A property of particles
A measure of inertia
A source of gravitational field

In Cohesion Dynamics under A-OPS + CIU-only ontology, mass is CIU-local constraint saturation:

Mass–energy is the constraint saturation within a specific CIU.

More precisely:

Definition (CIU Constraint Saturation): For a specific CIU, its constraint saturation $\sigma(\text{CIU})$ is the count of active constraints currently maintained within that CIU relative to its capacity: $\sigma(\text{CIU}) = \frac{\#(\text{active constraints in CIU})}{\text{constraint capacity}}$

Constraint capacity clarification: Constraint capacity is defined relative to the CIU’s admissible closure class at the scale under consideration, and is not a fixed or universal substrate property. It depends on CIU structure, mode, and the scale at which constraint interactions are being tracked.

Properties:

High $\sigma$ corresponds to CIUs with many active constraints (matter, energy)
Low $\sigma$ corresponds to CIUs with few constraints
$\sigma$ is a CIU-local property, not a field over space
Vacuum (absence of CIUs) has no $\sigma$ value — vacuum does not exist to have properties

Critical ontological foundation: In the absence of CIUs, there is no substrate entity to which properties such as density, curvature, or availability can be ascribed. This is the core rejection of spacetime-as-medium: only CIUs exist; vacuum is ontological absence, not low-density presence.

Critical ontological distinction:

What exists: Individual CIUs with varying constraint saturation $\sigma(\text{CIU})$

What does NOT exist:

Constraint density field $\rho_C(x)$ over space
“Empty regions” with low density
Vacuum as a low-density medium
Volume integrals over empty substrate

Clarification on “mass–energy” usage:

When we say “mass–energy,” we refer to constraint saturation as the substrate-level correlate of what conventional physics calls mass-energy. We do not assume stress-energy tensors, energy-momentum four-vectors, or relativistic energy relations. Those are representational structures that may emerge later. Here, “mass–energy” simply labels CIU constraint saturation.

3.2 Closure Rate Depends on CIU Saturation

High saturation slows closure

From M2, closure requires joint satisfaction of constraints: $\text{Closure occurs when } M(v; X) < W \text{ for all } v \in \text{CIU}$

When a CIU has high constraint saturation $\sigma$ :

More constraints must be jointly satisfied
Mismatch redistribution must resolve more conflicts
Closure takes more discrete resolution steps
Effective closure rate $\nu_{\text{close}}(\text{CIU})$ decreases

Functional dependence:

Empirically (from substrate structure), a CIU’s closure rate depends on its constraint saturation: $\nu_{\text{close}}(\text{CIU}) = \nu_0 \, f(\sigma(\text{CIU}))$

where:

$\nu_0$ is a baseline closure rate
$f(\sigma)$ is a decreasing function: $f(0) = 1$ , $f(\sigma) < 1$ for $\sigma > 0$
Specific functional form depends on substrate details (not derived here)

Important: Only monotonicity is required for the derivation; no specific functional form is assumed or needed at this stage. This preserves falsifiability at the functional level.

Physical interpretation:

For a CIU with high $\sigma$ (near massive objects, or the CIU itself represents mass):

Closure cycles take longer to complete
Fewer closures occur per external time interval
Local “clock rate” slows

This is the substrate origin of gravitational time dilation (derived in Section 5).

3.3 Saturation and Mismatch Absorption Capacity

High saturation CIUs have reduced mismatch capacity

When a CIU has many constraints:

Local mismatch redistribution has limited “room” to spread within the CIU
Additional constraints encounter saturation
Mismatch absorption capacity $C_{\text{abs}}$ decreases

Definition (CIU Mismatch Absorption Capacity): $C_{\text{abs}}(\text{CIU}) = W \cdot N_{\text{free}}(\text{CIU})$

where:

$W$ is tolerance window (from AX-TOL)
$N_{\text{free}}(\text{CIU})$ is the number of “free” constraint degrees of freedom available in that CIU

Saturation effect:

For a CIU with high saturation $\sigma$ : $N_{\text{free}} \approx N_{\text{total}} - N_{\text{occupied}} \propto (1 - \sigma / \sigma_{\text{max}})$

where $\sigma_{\text{max}}$ is maximum achievable saturation for that CIU type (substrate dependent).

Result: Mismatch absorption capacity vanishes as $\sigma \to \sigma_{\text{max}}$

Operational consequence:

When mismatch attempts to propagate into a high- $\sigma$ CIU through reconciliation:

It encounters reduced absorption capacity
Redistribution becomes harder
Backpressure develops, biasing mismatch flow away from saturation

This is the mechanism underlying gravitational “repulsion” from saturated cores (not explored further here, but relevant for extreme gravity regimes).

It encounters reduced absorption capacity
Redistribution becomes harder
Backpressure develops, biasing mismatch flow away from saturation

This is the mechanism underlying gravitational “repulsion” from saturated cores (not explored further here, but relevant for extreme gravity regimes).

4. Reconciliation Availability Asymmetry and Resolution Chain Bias

4.1 Availability Gradients in CIU Reconciliation Structure

Closure availability varies across CIU networks

From Section 3, a CIU’s closure rate depends on its constraint saturation $\sigma(\text{CIU})$ . When different CIUs have different saturation levels, we have an availability gradient in the reconciliation structure:

Relational availability comparison:

For two CIUs connected by a potential reconciliation chain:

CIU_A with saturation $\sigma_A$
CIU_B with saturation $\sigma_B$

Their relative reconciliation availability is: $A(\text{CIU}_A, \text{CIU}_B) = \nu_{\text{close}}(\text{CIU}_A) - \nu_{\text{close}}(\text{CIU}_B)$

Operational interpretation:

$A > 0$ : Reconciliation through CIU_A has higher availability (faster closure)
$A < 0$ : Reconciliation through CIU_B has higher availability
$A \approx 0$ : Both CIUs offer similar reconciliation availability

Key insight: Availability differences create directional bias in substrate resolution—NOT through spatial gradients, but through comparison of competing reconciliation chains.

4.2 Mismatch Redistribution Bias

Mismatch flows toward higher-availability CIUs

From AX-SEL (Precedence), substrate resolution preferentially selects transitions minimising mismatch. When closure availability varies across CIUs:

Local resolution rule:

Given mismatch in CIU_A, and potential reconciliation with neighboring CIUs CIU_B₁ and CIU_B₂:

If $\nu_{\text{close}}(\text{CIU}_{B_1}) > \nu_{\text{close}}(\text{CIU}_{B_2})$ (closure easier at CIU_B₁)
Then mismatch redistribution preferentially flows through reconciliation chain to CIU_B₁
Not because of a force, but because CIU_B₁ has greater absorption capacity

Availability-based flow (qualitative):

Mismatch prefers reconciliation chains with higher availability. This is represented by comparing chain availabilities, not by computing spatial gradients.

Operational mechanism:

CIU_A has local mismatch
Precedence evaluates neighboring CIUs for admissible reconciliation
CIUs with higher $\nu_{\text{close}}$ have more available capacity
Mismatch preferentially propagates toward those CIUs through reconciliation
Over successive closure cycles, this creates directional bias in reconciliation structure

4.3 Directional Bias in Reconciliation Chains

This section resolves the apparent paradox between mismatch flow direction and gravitational attraction.

Asymmetric reconciliation creates effective acceleration

Consider a test CIU in a network where surrounding CIUs have varying saturation (e.g., near a massive object represented by a cluster of high- $\sigma$ CIUs).

Setup:

High- $\sigma$ CIUs (massive object cluster)
Low- $\sigma$ test CIU
Availability gradient exists in reconciliation structure

Resolution dynamics:

From Section 4.2, mismatch flows toward lower-saturation CIUs (higher availability). But the test CIU itself represents a local constraint concentration. How does it move?

Key distinction:

Mismatch redistribution flows toward higher availability (lower saturation)
CIU motion results from reconciliation-delay asymmetry

Reconciliation-delay mechanism:

From G2, distance is reconciliation delay. When closure rates vary across CIUs: $d(\text{CIU}_A, \text{CIU}_B) = \frac{\Delta t_{\text{rec}}}{\nu_{\text{close}}}$

In an availability gradient:

Reconciliation chains passing through high- $\sigma$ CIUs experience slower closure
Reconciliation chains passing through low- $\sigma$ CIUs experience faster closure
Effective distances along different chains become asymmetric over time

Result: The test CIU’s relational position (defined by reconciliation delays to neighbors) changes preferentially toward the high- $\sigma$ CIU cluster.

Why toward, not away?

This appears paradoxical: mismatch flows away from saturation, but objects fall toward mass. Resolution:

Two distinct effects:

Mismatch redistribution (internal constraint resolution) flows toward higher availability
CIU trajectory (external reconciliation-delay evolution) biased toward high-saturation clusters

Critical distinction: Mismatch redistribution governs internal resolution capacity, while CIU motion is governed by reconciliation-delay asymmetry and geometric bookkeeping—these are distinct substrate processes with different directional behaviors. This distinction follows directly from G2: distance and motion are defined via reconciliation-delay relations, not via mismatch transport. Mismatch flow affects resolution capacity; motion affects relational position.

Test-particle regime: This analysis treats the test CIU as having negligible self-saturation; self-gravity effects are deferred to strong-field regimes. The trajectory bias effect dominates for test CIUs.

Mechanism:

From G3, geometry is consistency bookkeeping for distances between CIUs. In curved geometry:

Least-mismatch reconciliation chains curve toward clusters of high-saturation CIUs
Not because of attraction, but because reconciliation delays accumulate asymmetrically
Falling is the chain of least total mismatch when geometry (over CIU reconciliation structure) is curved

Key insight: Gravitational attraction is not motion toward mismatch, but motion along reconciliation chains whose delay asymmetry makes high-saturation CIU clusters relationally closer over successive closure cycles. Constraint saturation creates availability gradients that bias trajectories—these are complementary substrate effects, not competing explanations.

Reminder: Although mismatch redistribution and CIU motion are both influenced by availability gradients in reconciliation structure, only the latter alters relational position; the former alters resolution capacity. This distinction is crucial for understanding why objects fall toward mass despite mismatch flowing away.

Formal justification (deferred to Section 6): Free fall minimises total integrated mismatch along the trajectory.

4.4 Closure Availability as Effective Potential

Gravitational potential emerges from closure-rate variation across CIU network

We can introduce a closure availability potential as representational bookkeeping for comparing reconciliation chains. For two CIUs with different saturation:

Availability difference (representational): $\Delta A(\text{CIU}_1, \text{CIU}_2) = \nu_{\text{close}}(\text{CIU}_1) - \nu_{\text{close}}(\text{CIU}_2)$

This can be accumulated over reconciliation chains to define an effective potential difference.

Important: This potential is introduced purely as representational bookkeeping for comparing closure availability across chains; it has no independent dynamics or ontological status. It is shorthand for substrate reconciliation structure, not a field existing over space.

Properties:

High $\sigma$ CIUs → low $\nu_{\text{close}}$ → negative relative availability (potential well)
Low $\sigma$ CIUs → high $\nu_{\text{close}}$ → near-zero relative availability (flat)
Differences drive reconciliation chain bias

Connection to Newtonian gravity:

In regimes where CIU networks can be approximated as continuous (many CIUs, weak saturation variation), a Newtonian-like potential may emerge as representational shorthand. This is representational correspondence, not structural derivation.

The emergence of specific functional forms (e.g., $1/r$ ) requires additional assumptions (network geometry, symmetry, boundary conditions) external to Cohesion Dynamics and not universally guaranteed.

Derivation of gravitational coupling (sketch):

From substrate parameters:

$W$ (tolerance window)
$\nu_0$ (reference closure rate)
CIU network structure

One can estimate effective coupling constants for representational correspondence with Newtonian gravity. This shows gravitational “constant” is not fundamental—it is an emergent parameter from substrate mechanics.

Clarification: Any appearance of Newtonian or GR forms is conditional, approximate, and representational only. No classical law is being claimed as structurally derived here.

Scope guardrail: No Poisson equation, inverse-square law, or metric field equation is implied by this construction; any such form would require additional symmetry and network assumptions beyond Cohesion Dynamics itself. The substrate mechanics here establishes availability gradients create bias in reconciliation chains—specific functional forms (Newton’s law, Einstein’s equations) are representational approximations contingent on network geometry and boundary conditions, not structural necessities.

5. Gravitational Time Dilation

5.1 Closure Rate Determines Local Time

Time is closure-cycle accumulation (G1)

From G1, time for a CIU is: $t = \sum_{i=1}^{N} \Delta t_i$

where $\Delta t_i$ is the duration of the $i$ -th closure cycle.

In uniform CIU network, $\Delta t_i = \text{const}$ and time advances uniformly.

When CIUs have varying saturation, $\Delta t_i$ depends on local CIU saturation $\sigma$ .

Local time rate:

The rate at which local time accumulates for a CIU is: $\frac{dt_{\text{local}}}{dt_{\text{ref}}} = \frac{\nu_{\text{close}}(\text{CIU})}{\nu_{\text{ref}}}$

where $t_{\text{ref}}$ is a reference time from a low-saturation CIU.

5.2 Time Dilation Between CIUs

Clocks slow in high-saturation CIUs

Consider two CIUs acting as clocks:

CIU_A with saturation $\sigma_A$ (low, far from massive CIU cluster)
CIU_B with saturation $\sigma_B$ (high, near or within massive cluster)

Closure rates: $\nu_A = \nu_0 f(\sigma_A) \approx \nu_0 \quad (\text{low saturation})$ $\nu_B = \nu_0 f(\sigma_B) < \nu_0 \quad (\text{high saturation})$

Time accumulation ratio:

After corresponding closure cycles: $\frac{t_B}{t_A} = \frac{\nu_B}{\nu_A} = \frac{f(\sigma_B)}{f(\sigma_A)} < 1$

Result: CIU_B accumulates less proper time than CIU_A → gravitational time dilation.

5.3 Connection to Effective Potential

Time dilation from availability differences (weak regime):

In regimes where CIU networks admit continuous approximation, time dilation can be expressed using the effective potential from Section 4.4. This is representational shorthand only.

The specific functional form depends on network structure and is not universally derived from substrate mechanics alone.

Standard GR correspondence (representational):

Time dilation formulas from General Relativity may emerge as approximate descriptions in suitable regimes, but this is representational correspondence, not structural derivation.

5.4 Ontology vs. Representation

Ontologically:

Closure cycles take more steps in high-saturation CIUs
No time exists as substance or dimension
Time dilation is difference in closure rates between CIUs

Representationally:

CIUs measure proper time (accumulated closure cycles)
Coordinate time is bookkeeping device comparing different CIUs
Time dilation may be expressed using metric language in GR as representational shorthand

6. Free Fall as Least-Mismatch Resolution

6.1 The Free-Fall Problem

What is “free fall”?

In Newtonian gravity: objects released from rest accelerate toward mass.

In General Relativity: objects follow geodesics (curves extremising proper time).

In Cohesion Dynamics: objects follow least-mismatch resolution paths through closure gradients.

Key claim: Free fall is not chosen, but forced by precedence selection.

6.2 Mismatch Accumulation Along Reconciliation Chains

Setup:

Consider a test CIU evolving through successive closure cycles, with its relational position defined by reconciliation delays to neighboring CIUs. The CIU’s trajectory is a sequence of reconciliation states.

At each step, precedence selects the next admissible reconciliation structure.

Mismatch measure:

The total mismatch accumulated along the trajectory (reconciliation chain sequence) depends on:

Reconciliation delays at each step
Closure availability through intermediate CIUs
Phase accumulation (from M4)

Precedence rule (AX-SEL):

At each step, precedence selects the next reconciliation structure that minimises local mismatch while satisfying admissibility constraints.

6.3 Least-Mismatch Trajectories Through CIU Networks

Claim: The trajectory that minimises total mismatch through a CIU network corresponds to what GR describes as a geodesic.

Why?

From G3, geometry encodes distance-composition consistency between CIUs. The least-mismatch trajectory is one that:

Minimises reconciliation-delay inconsistency
Satisfies local closure conditions at each step
Accumulates minimal total phase difference (from M4)

Representational connection:

In differential geometry, geodesics extremise proper time. In Cohesion Dynamics, proper time accumulation is closure-cycle accumulation, which varies with CIU saturation.

Extremising proper time accumulation is equivalent to following reconciliation chains with optimal availability.

Result: Geodesic motion = least-mismatch trajectory through CIU reconciliation structure.

Note: This is representational equivalence, not substrate-level optimization. The substrate does not “compute” variational principles; precedence selection locally minimises mismatch at each step, and geodesic structure emerges as the geometric description of the resulting trajectory.

6.4 Free Fall Is Forced, Not Chosen

Why objects fall:

Precedence selects least-mismatch transitions (AX-SEL)
In availability gradients (varying CIU saturation), least-mismatch chains curve toward high-saturation CIU clusters
Successive precedence selections trace out geodesic through reconciliation structure
No “choice” or “force” is involved—falling is the only admissible evolution

Stationarity is unstable:

Can a test CIU maintain fixed reconciliation delays to neighboring CIUs when those neighbors have varying saturation?

No. Maintaining fixed delays requires:

Continuous mismatch redistribution to counteract availability asymmetry
External constraint preventing precedence-driven motion
This is not admissible without external forces (which are substrate-level constraint impositions)

In the absence of external constraints, precedence drives motion through the reconciliation structure.

Result: Free fall is structurally unavoidable.

6.5 Representational Correspondence to GR Geodesic Equation

GR geodesic equation: $\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\nu\rho} \frac{dx^\nu}{d\tau} \frac{dx^\rho}{d\tau} = 0$

Cohesion Dynamics:

The precedence-driven trajectory through CIU reconciliation structure can be approximately described using GR geodesic language in regimes where:

CIU networks are dense enough for continuous approximation
Saturation variation is weak
Geometric bookkeeping from G3 admits metric representation

This is representational correspondence only, not structural derivation of GR.

Connection:

In suitable regimes, Christoffel symbols $\Gamma^\mu_{\nu\rho}$ can be interpreted as encoding availability gradients in reconciliation structure. However, this requires additional assumptions beyond substrate mechanics.

Recovering the Newtonian limit of GR.

Interpretation:

GR: geodesics are “straight lines” in curved spacetime
CD: trajectories are precedence-driven resolution paths
Same equations, different ontology

Critical clarification: Recovering Newtonian or GR-like equations is a consistency check on the representational layer, not a claim that those equations are fundamental or uniquely selected by the substrate. The correspondence validates the geometric bookkeeping, not the ontology.

7. Universality and Equivalence Principle

7.1 Universal Coupling to Closure Gradients

All CIUs fall identically

From AX-SEL, precedence selection applies uniformly: $\Delta s^* = \arg\min M(s + \Delta s)$

This rule does not depend on:

CIU internal structure
CIU composition (quantum type, baryon number, etc.)
CIU size or complexity
CIU mass (constraint count)

Result: Gravitational response is composition-independent.

Empirical consequence:

Drop a feather and a hammer in vacuum → they fall at the same rate
Photon trajectories bend in gravitational fields (light follows geodesics)
All test particles have identical acceleration in same gradient

This is the weak equivalence principle (universality of free fall).

7.2 Equivalence of Inertial and Gravitational Mass

Inertial mass (resistance to acceleration):

From M2, changing a CIU’s state requires overcoming precedence. The “inertia” is: $m_{\text{inertial}} \propto \#(\text{internal constraints in CIU})$

More constraints → greater resistance to perturbation.

Gravitational mass (source strength):

From Section 4, availability gradients in reconciliation structure depend on source CIU constraint saturation: $\text{Gradient strength} \propto \sigma(\text{source CIU}) \propto \#(\text{constraints in source})$

The source’s “gravitational mass” is: $m_{\text{gravitational}} \propto \#(\text{constraints in source CIU})$

Identical origin:

Both inertial and gravitational mass measure constraint count in CIUs. They are not two properties that happen to be equal—they are the same property measured in different contexts.

Result: $m_{\text{inertial}} = m_{\text{gravitational}}$ structurally.

This is the strong equivalence principle (equality of inertial and gravitational mass), derived from constraint mechanics.

Conceptual payoff: In Cohesion Dynamics, there is no distinction between “what causes gravity” and “what resists acceleration”: both are manifestations of CIU constraint saturation. This unification is structural, not coincidental.

7.3 Local Equivalence of Gravity and Acceleration

Einstein’s elevator thought experiment:

In GR, locally, one cannot distinguish:

Free fall in a gravitational field
Inertial motion in flat spacetime

Cohesion Dynamics interpretation:

Both situations correspond to:

Precedence-driven resolution in locally flat reconciliation structure
No local availability gradient (free fall eliminates gradient in comoving frame)
Geodesic motion minimises local mismatch through CIU network

Representational equivalence:

In a sufficiently local CIU cluster (where availability variation is approximately constant):

Free-fall frame removes apparent gradient
Physics is locally identical to zero-gradient (uniform saturation) case

Ontological distinction:

Gravity: availability gradient exists in reconciliation structure; free fall follows it
Acceleration: external constraint imposes non-geodesic chain

Representationally equivalent locally, ontologically distinct globally.

8. Ontology vs. Representation

This section exemplifies the ontology/representation discipline established in M7 (Layered Definitions and Explanatory Weight), demonstrating how G4 maintains the distinction between substrate mechanics and derived bookkeeping structures.

8.1 What Exists: Substrate Structure

Ontologically, the following exist in the substrate:

Discrete closure events (from G1)
- Commits of constraint configurations within CIUs
- Ordered by causal dependence
- No continuous time flow
CIU constraint saturation (Section 3)
- Varying $\sigma(\text{CIU})$ across CIU network
- Local closure availability variation
- No field substance over space
Reconciliation delays (from G2)
- Discrete cycle counts for admissible reconciliation between CIUs
- Varying with CIU saturation
- No distance as primitive
Precedence selection (from AX-SEL)
- Mismatch-minimising admissible transitions
- Local, non-teleological
- No forces or attractions
Phase accumulation (from M4)
- Closure-cycle alignment tracking
- Compatibility structure
- No time as substance

These are the ontological primitives. Everything else is representational.

8.2 What Is Representational: Gravitational Language

Representationally, we describe substrate structure using:

Gravitational potential (effective)
- Bookkeeping for closure-availability variation across CIU network
- Not a field existing in space
- Shorthand for comparing reconciliation chains
Gravitational acceleration
- Effective description of precedence bias in reconciliation structure
- Not a force acting on objects
- Derived from availability gradients
Gravitational time dilation
- Ratio of closure rates between CIUs
- Not a warping of time
- Derived from $\nu_{\text{close}}(\sigma)$ variation
Geodesic trajectories
- Least-mismatch chains through CIU reconciliation structure
- Not “straight lines in curved space”
- Derived from precedence selection
Metric tensor $g_{\mu\nu}$ (G5 if developed)
- Geometric bookkeeping for distance/time relations between CIUs
- Not spacetime substance
- Derived from reconciliation-delay structure

Gravitational language is shorthand for substrate resolution dynamics. It is empirically adequate but ontologically empty.

8.3 Gravity Does Not Exist as Ontological Substance

Central claim: Gravity does not exist as a fundamental ontological entity; it is an emergent representational construct.

Clarification:

Gravitational effects are real (objects fall, light bends, clocks slow)
Gravitational language is useful (accurate predictions, geometric elegance)
Gravitational ontology is empty (no force, field, or curvature substance exists)

What exists:

CIUs with varying constraint saturation
Precedence-driven resolution following availability gradients in reconciliation structure
Representational necessity for geometric bookkeeping (from G3)

What does not exist:

Gravitational force pulling on objects
Spacetime curvature as substance
Gravitational field propagating through vacuum
Metric tensor as ontological entity
Vacuum with gravitational properties

Analogy:

“Gravity” is like “sunrise”:

Sunrise appears (sun seems to rise)
Sunrise language is useful (“meet at sunrise”)
Sunrise ontology is false (Earth rotates; sun doesn’t move)

Similarly:

Gravity appears (objects fall)
Gravity language is useful (GR predictions)
Gravity ontology is false (availability gradients in CIU reconciliation, not forces)

9. Explicitly Out of Scope

This paper does not address:

9.1 Metric Dynamics (G5)

How geometric structure evolves over time
Derivation of Einstein field equations or equivalent
Coupling between matter distribution and geometry
Gravitational wave propagation

Why deferred: G4 establishes static or quasi-static gravity. Dynamic geometry requires additional structure.

9.2 Strong-Field Gravity

Black holes (horizons, singularities)
Neutron stars (extreme saturation)
Gravitational collapse
Merger dynamics

Why deferred: Requires extreme CIU saturation regimes where substrate structure may differ.

9.3 Cosmology

Large-scale geometry (FLRW metrics)
Expansion dynamics
Dark energy / cosmological constant
Inflation

Why deferred: Requires global geometric structure and empirical cosmology input.

9.4 Quantum Gravity

Substrate-scale gravitational structure
Quantisation of geometry
Planck-scale physics
Loop quantum gravity / string theory comparison

Why deferred: Requires integration with B-series quantum recovery and substrate discreteness.

9.5 Empirical Predictions

Testable deviations from GR
Gravitational wave signatures
Precision tests (perihelion precession, gravitational redshift)
Strong-field tests (binary pulsars, black hole shadows)

Why deferred: Requires numerical implementations and full metric dynamics (G5).

9.6 Dark Matter Applications

Halo structure from cohesion dynamics
Galaxy rotation curves
Gravitational lensing by dark matter
Large-scale structure formation

Why deferred: Requires G4 foundations + detailed structure formation modelling.

All of these are downstream applications. G4 establishes the foundational derivation; applications require additional development.

Note on falsifiability: Nothing in this paper constrains the numerical form of gravitational coupling; that is an empirical and application-level question. G4 establishes structural necessity, not numerical predictions.

10. Implications for Subsequent Work

10.1 What G4 Enables

G4 completes the foundational gravity derivation. With G1–G4, we have shown:

Time emerges (G1)
Distance emerges (G2)
Geometry emerges (G3)
Gravity emerges (G4)

Downstream work can now:

Develop metric dynamics (G5):
- Derive constraints on geometric evolution
- Recover Einstein equations or identify deviations
- Model gravitational wave propagation
Apply to dark matter (new series):
- Model halo structure from cohesion gradients
- Predict rotation curves without dark matter particles
- Explore large-scale structure formation
Explore extreme gravity:
- Black hole structure in CD substrate
- Neutron star interiors
- Gravitational collapse dynamics
Generate empirical predictions:
- Testable deviations from GR
- Novel gravitational effects from substrate discreteness
- Precision tests in strong fields
Integrate with quantum gravity:
- Unify B-series quantum recovery with G-series gravity
- Explore Planck-scale structure
- Compare with other quantum gravity approaches

10.2 Programme Coherence

G4 maintains coherence with earlier series:

A-series: Uses substrate mechanics without modification
M-series: Relies on precedence, coherence, phase structure
B-series: Compatible with quantum recovery (gravity is classical effective description)
G1–G3: Builds directly on emergent time, distance, geometry

No new axioms introduced. G4 is a derivation, not a postulate.

10.3 Falsifiability

How could G4 be falsified?

If empirical tests show:

Composition-dependent free fall (violation of equivalence principle at high precision)
Gravitational effects without mass-energy (e.g., gravity from pure geometry)
Non-universal gravitational coupling (different particles fall differently)
Time dilation inconsistent with mass distribution (clocks behave independently of CIU saturation)

Then G4’s derivation from constraint gradients would be falsified, requiring either:

Revision of substrate mechanics (A-series)
Additional axioms
Abandonment of Cohesion Dynamics

G4 is falsifiable at the substrate level, not just at the effective-theory level.

11. Summary and Conclusions

11.1 What Was Derived

G4 demonstrates:

Gravitational attraction emerges from availability-gradient bias in reconciliation structure
- Not a force
- Not curvature substance
- Precedence-driven resolution
Free fall is forced by precedence selection
- Least-mismatch chains through CIU networks
- Geodesic motion as geometric bookkeeping
- Universality from substrate neutrality
Equivalence principles are structural
- Weak: all CIUs fall identically (composition-independent)
- Strong: inertial and gravitational mass are identical (both measure constraint count)
Gravitational time dilation emerges from closure-rate variation between CIUs
- Clocks slow in high-saturation CIUs
- No metric postulates required
Newtonian and GR limits are recovered (weak field, representational)
- Effective gravitational potential for continuous approximation
- Geodesic equation correspondence to classical mechanics
- Time dilation matches GR predictions (representational shorthand)

11.2 Ontology vs. Representation (Summary)

Ontologically:

Discrete closure events within CIUs
CIU constraint saturation variation ( $\sigma(\text{CIU})$ )
Precedence selection biasing reconciliation chains
Phase accumulation differences

Representationally:

Gravitational potential, acceleration, time dilation (effective descriptions)
Geodesic trajectories in curved geometry (bookkeeping)
Metric tensor (if introduced in G5)
Forces, attractions, curvature (all representational shorthand)

Gravity is representational structure, not ontological substance.

11.3 What Remains for G5 and Beyond

G5 (Metric Dynamics, optional):

Derive constraints on how geometry evolves
Recover Einstein equations or identify deviations
Model gravitational wave propagation

Application papers (future):

Dark matter halos from CIU saturation gradients
Black hole structure
Cosmology and expansion
Empirical predictions and tests

Integration with quantum recovery (B-series):

Quantum fields in curved substrate-derived geometry
Hawking radiation from substrate mechanics
Quantum corrections to gravitational effects

12. Final Remarks

Gravity is not fundamental.

It is an emergent, representational description of precedence-driven resolution in CIU networks with varying constraint saturation.

No forces act. No curvature substance exists. No field propagates through vacuum.

What exists is:

Discrete CIUs with constraint resolution mechanics
Reconciliation relations between CIUs
Precedence selection following availability gradients

What does not exist is:

Spacetime as substance
Gravitational field over vacuum
Forces or attractions
Curvature as physical deformation

Gravity is bookkeeping for substrate resolution structure—empirically adequate, ontologically empty.

End of Paper G4

Spatial gradients in closure difficulty
Precedence selection following least resistance

What emerges is:

Apparent gravitational attraction
Geodesic motion
Time dilation
Geometric structure

This derivation is complete within its scope. Gravity, as a foundational phenomenon, is explained. What remains is application, extension, and empirical validation—work for G5 and future series.

G4 completes the G-series foundational programme (G1: time, G2: distance, G3: geometry, G4: gravity). All geometric and gravitational structure is now derived from Cohesion Dynamics substrate mechanics. With G4, gravitational behaviour is fully derived from substrate mechanics; subsequent work concerns applications and limits rather than foundational structure.

No metric or field ansätze were assumed.
No General Relativity structures were imported as starting points.
No forces or curvature were postulated.

Gravity emerges—forced, unavoidable, universal—from closure accounting and precedence selection in a constrained informational substrate.

The derivation is closed.