E0 — Empirical Grounding of Substrate Primitives via Toy Continua

Abstract

We present an empirical demonstration that a minimal set of operational substrate primitives—asynchronous local updates, closure cycles, multi-channel mismatch, height, and tolerance $W$—is sufficient to generate cohesion, divergence, boundaries, and threshold behaviour in toy informational continua. This work does not address constructor emergence or quantum structure. Rather, it establishes that these primitives arise naturally in simple substrates and produce nontrivial continuum-like behaviour before any higher-level phenomena appear. We demonstrate sharp threshold effects governed by $W$, the formation of cohesive and divergent regions, and stable boundaries between domains. Crucially, we show these primitives are operationally necessary and non-arbitrary, arising from the minimal requirements for any substrate to exhibit coherent structure at all. This paper grounds the operational apparatus and execution semantics used throughout the Constructor Emergence (E1) and Quantum Emergence (E2) programs.

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1. Introduction

1.1 The Grounding Problem

The Constructor Emergence program (E1) and Quantum Emergence program (E2) rely on a set of operational primitives: mismatch, height, tolerance $W$, closure cycles, and commit-based execution semantics. These primitives structure the entire empirical program.

This raises a critical methodological question:

Are these primitives legitimate operational foundations, or are they assumptions chosen specifically to make constructors or quantum structure work?

If the latter, the entire program risks circularity: the apparatus may encode the desired outcomes rather than discovering them.

This paper addresses this concern directly.

1.2 Purpose of This Work

E0 exists to answer a single question:

Why are mismatch, height, tolerance $W$, closure cycles, and commit-based evolution legitimate substrate primitives—independent of constructors and quantum mechanics?

We demonstrate empirically that:

1. These primitives arise naturally in minimal toy substrates
2. They generate nontrivial structure (cohesion, divergence, boundaries)
3. They exhibit threshold behaviour and phase transitions
4. They do not produce constructors or quantum dynamics

The guiding principle is:

Show structure before showing function.

We establish that substrates can exhibit coherent regions, boundaries, and continuum-like behaviour without constructors, self-repair, or quantum interference. This grounds the apparatus before it is used to study higher phenomena.

1.3 What This Paper Does Not Do

E0 explicitly does not:

• Demonstrate constructor emergence
• Demonstrate quantum structure
• Assume known physics (energy, spacetime, fields)
• Claim universality or physical realism
• Derive continuum limits or field theories

We explore operational minimality: what is the minimal execution semantics required for a substrate to exhibit coherent structure at all?

1.4 Relationship to Downstream Work

E0 provides the operational foundation for:

• E1 (Constructor Emergence): Explores which coupling classes, when added to these primitives, yield persistent constructors
• E2 (Quantum Emergence): Further explores which constraints yield quantum-like structure
• Paper A (Substrate Mechanics): Formalizes the mechanics within the empirically justified substrate class
• Paper B (Continuum Physics): Derives continuum limits and effective field structure

E0 establishes that the primitives are necessary but not sufficient for constructors or quantum mechanics.

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2. Conceptual Framing

2.1 Operational vs Ontological Primitives

We distinguish two kinds of primitives:

Ontological primitives describe what fundamentally exists (e.g., “information”, “constraints”). These are addressed in Paper F (Foundations).

Operational primitives describe how we measure, track, and execute substrate dynamics. These are the focus of E0.

Operational primitives must be:

• Computable: Well-defined from substrate state
• Local: Not requiring global coordination
• Non-arbitrary: Not chosen to privilege specific outcomes
• Generative: Capable of producing nontrivial structure

This paper justifies operational primitives, not ontological commitments.

2.2 Why Empirical Demonstration?

Rather than asserting primitives axiomatically, we demonstrate they:

1. Emerge naturally: Simple substrates exhibit these structures without explicit design
2. Generate nontrivial behaviour: They produce cohesion, divergence, and boundaries
3. Exhibit threshold effects: Small changes in $W$ cause sharp transitions
4. Do not smuggle higher structure: No constructors or quantum mechanics appear

This empirical approach avoids circularity: we show the primitives work before using them to study constructors.

2.3 The “Structure Before Function” Principle

The methodological commitment is:

Before explaining construction or quantisation, explain why there is something coherent to construct or quantise at all.

E0 shows that informational substrates can form:

• Cohesive regions (persist together)
• Divergent regions (evolve independently)
• Stable boundaries (separating domains)
• Threshold behaviour (sharp transitions)

These are pre-functional structures: they exist before any directed construction or self-repair emerges.

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3. Substrate Primitives to Be Justified

E0 must empirically justify the following operational primitives as useful and non-arbitrary.

3.1 Asynchronous Local Updates

Definition: Local regions update based on local state and neighbour constraints, without global synchronisation.

Justification Required:

• Show that global synchrony is unnecessary
• Demonstrate that asynchronous updates are sufficient for coherent evolution
• Show that enforcing synchrony may destabilise substrates

GO Criterion: Asynchronous substrates exhibit stable cohesive regions.

NO-GO Criterion: Synchronous-only substrates exhibit pathological freezing or divergence.

3.2 Closure Cycles as Discrete Time

Definition: A closure cycle is one round of constraint resolution. Multiple closure cycles define a temporal ordering.

Justification Required:

• Show that closure cycles provide natural discrete time ordering
• Demonstrate that background time is unnecessary
• Show that closure cycles distinguish converged vs divergent evolution

GO Criterion: Closure cycles provide consistent temporal ordering without background clock.

NO-GO Criterion: Without closure cycles, temporal succession is ambiguous or ill-defined.

3.3 Mismatch (Multi-Channel)

Definition: Mismatch $M$ measures local constraint incompatibility. Multiple mismatch channels distinguish different types of incompatibility.

Justification Required:

• Show that mismatch naturally arises from constraint violations
• Demonstrate that multiple channels are necessary (single-channel relaxation is trivial)
• Show that mismatch reduction drives coherent evolution

GO Criterion: Multi-channel mismatch enables nontrivial relaxation and domain formation.

NO-GO Criterion: Single-channel mismatch leads to trivial uniform relaxation.

3.4 Height

Definition: Height $H$ is an aggregate diagnostic over closure cycles, measuring cumulative mismatch persistence.

Justification Required:

• Show that height distinguishes convergent vs divergent regions
• Demonstrate that height identifies boundary regions
• Show that height evolves differently in cohesive vs divergent domains

GO Criterion: Height clearly distinguishes coherent, divergent, and boundary regions.

NO-GO Criterion: Without height, region classification is ambiguous or unstable.

3.5 Tolerance / Threshold $W$

Definition: $W$ is a tolerance threshold. Regions with mismatch below $W$ remain cohesive; those exceeding $W$ diverge.

Justification Required:

• Show that $W$ produces sharp threshold behaviour
• Demonstrate phase transitions (merge vs partition) as $W$ varies
• Show that $W$ matters even in the absence of constructors

GO Criterion: Varying $W$ produces sharp cohesion/divergence transitions.

NO-GO Criterion: Without $W$, no clear cohesion threshold exists.

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4. Execution Semantics

E0 must explicitly specify and justify the following execution semantics.

4.1 Pre-Commit Relaxation

Definition: Local updates redistribute mismatch without strict monotonic decrease. Mismatch may temporarily increase locally while decreasing globally.

Rationale: Real constraint systems explore multiple local configurations before settling. Requiring strict mismatch decrease at every step is too restrictive.

Demonstration Required: Show that pre-commit relaxation enables nontrivial mismatch redistribution and convergence.

4.2 Commit

Definition: After relaxation, a commit evaluates:

• Closure (is mismatch stable?)
• Admissibility (is mismatch below $W$?)
• Height update (cumulative mismatch accounting)

Rationale: Commit separates exploration (relaxation) from evaluation (closure/admissibility). This mirrors how real physical systems explore microstates before macroscopic observables emerge.

Demonstration Required: Show that commit-based evaluation produces stable region boundaries and clear cohesion/divergence classification.

4.3 Post-Commit

Definition: After commit:

• Regions below $W$ remain cohesive
• Regions exceeding $W$ partition or diverge
• Height is updated based on commit results

Rationale: Post-commit behaviour determines whether regions persist together or separate.

Demonstration Required: Show that post-commit partitioning produces stable boundaries and prevents runaway divergence.

4.4 Alternative Execution Orders (Must Fail)

E0 must demonstrate that alternative execution semantics fail or behave pathologically:

• Commit-first: Regions partition prematurely before exploring viable relaxation paths
• No relaxation: Mismatch cannot redistribute; regions freeze or diverge trivially
• Synchronous-only: Global coordination prevents asynchronous local convergence

Demonstration Required: Show that these alternatives produce instability, freezing, or trivial evolution.

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5. Target Phenomena (What to Demonstrate)

E0 must empirically demonstrate the emergence of:

5.1 Cohesive Regions

Definition: Regions where mismatch remains below $W$ across multiple closure cycles.

Observable: Stable spatial clustering with consistent internal state.

Measurement: Height remains bounded; mismatch decreases or stabilises.

5.2 Divergent Regions

Definition: Regions where mismatch persistently exceeds $W$, evolving independently.

Observable: Height increases monotonically; no convergence to neighbours.

Measurement: Height divergence rate, spatial isolation.

5.3 Stable Boundaries

Definition: Interface regions separating cohesive domains with distinct internal states.

Observable: Sharp spatial transition in height and mismatch gradients.

Measurement: Boundary width, stability over time.

5.4 Threshold Effects Governed by $W$

Definition: Sharp transitions between cohesion and divergence as $W$ varies.

Observable: Phase transitions in domain size, boundary stability, and height distribution.

Measurement: Critical $W$ values where behaviour changes qualitatively.

5.5 Coarse Continuum-Like Behaviour

Definition: Relaxation waves, domain merging/splitting, and gradient-driven evolution.

Observable: Spatial correlations, temporal persistence, gradient smoothing.

Measurement: Correlation length, relaxation time scales.

5.6 Explicit Absence of Higher Structure

Critically: E0 must demonstrate the absence of:

• Persistent constructors (no self-maintaining functional structures)
• Self-repairing structures (no recovery after perturbation)
• Directed construction (no targeted influence on external configurations)
• Hierarchical composition (no constructors composed of constructors)

Rationale: Negative results are essential. If these higher structures appeared, the primitives would not be independent of constructors.

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6. Experimental Program

6.1 1D Substrates: Basic Mismatch Dynamics

Setup:

• 1D chain of regions with scalar state values
• Multi-channel mismatch: shape, phase, amplitude
• Heterogeneous initial mismatch distribution

Experiments:

1. Relaxation without commit: Show mismatch redistribution
2. Commit with varying $W$: Demonstrate cohesion/divergence threshold
3. Height tracking: Show height distinguishes convergent vs divergent regions

Expected Results:

• Cohesive regions form where initial mismatch is low
• Divergent regions form where initial mismatch exceeds $W$
• Boundaries stabilise at mismatch gradients

GO/NO-GO Criteria:

• GO: Clear cohesive/divergent regions with stable boundaries
• NO-GO: Trivial uniform relaxation or runaway divergence

6.2 2D Substrates: Domain Formation and Boundaries

Setup:

• 2D lattice of regions with vector state
• Multi-channel mismatch with spatial coupling
• Random or structured initial conditions

Experiments:

1. Domain nucleation: Show formation of cohesive domains
2. Boundary dynamics: Demonstrate stable boundary formation
3. Domain merging: Show $W$-dependent domain coalescence or separation

Expected Results:

• Domains nucleate around low-mismatch seed regions
• Boundaries form at high-mismatch gradients
• Domain topology depends on $W$

GO/NO-GO Criteria:

• GO: Stable domains with well-defined boundaries
• NO-GO: Fractal fragmentation or complete homogenisation

6.3 Variation of $W$: Phase Transitions

Setup:

• Fixed substrate configuration
• Systematically vary tolerance $W$

Experiments:

1. Low $W$: Show high fragmentation (many small domains)
2. Medium $W$: Show domain merging (intermediate topology)
3. High $W$: Show global cohesion (single domain)

Expected Results:

• Sharp transitions in domain count vs $W$
• Critical $W_c$ where behaviour changes qualitatively
• Hysteresis or path-dependence near critical $W$

GO/NO-GO Criteria:

• GO: Clear phase transitions with identifiable critical points
• NO-GO: Smooth continuous variation with no sharp thresholds

6.4 Synchronous vs Asynchronous Updates

Setup:

• Identical substrate and initial conditions
• Compare synchronous vs asynchronous update schedules

Experiments:

1. Synchronous updates: All regions update simultaneously
2. Asynchronous updates: Regions update based on local readiness
3. Semi-asynchronous: Probabilistic update scheduling

Expected Results:

• Synchronous updates may cause oscillations or freezing
• Asynchronous updates enable gradual convergence
• Semi-asynchronous provides intermediate behaviour

GO/NO-GO Criteria:

• GO: Asynchronous updates produce stable cohesive regions where synchronous fails
• NO-GO: No qualitative difference between synchronous and asynchronous

6.5 Single-Channel vs Multi-Channel Mismatch

Setup:

• Compare substrates with:
  • Single mismatch channel (scalar)
  • Multiple mismatch channels (vector)

Experiments:

1. Single-channel: Show trivial relaxation to uniform state
2. Multi-channel: Show nontrivial domain structure persistence

Expected Results:

• Single-channel relaxes to uniform equilibrium
• Multi-channel maintains structured domains with boundaries

GO/NO-GO Criteria:

• GO: Multi-channel mismatch enables nontrivial persistent structure
• NO-GO: Both cases relax to uniform equilibrium

6.6 Execution Order Comparison

Setup:

• Identical substrate and initial conditions
• Compare execution orders:
  • Commit-first (no relaxation)
  • Relaxation-then-commit (standard)
  • No-commit (continuous relaxation)

Experiments:

1. Commit-first: Show premature partitioning
2. Relaxation-then-commit: Show stable domain formation
3. No-commit: Show ambiguous boundary classification

Expected Results:

• Commit-first causes excessive fragmentation
• Relaxation-then-commit produces stable domains
• No-commit prevents clear cohesion/divergence classification

GO/NO-GO Criteria:

• GO: Relaxation-then-commit clearly superior for stable structure
• NO-GO: No qualitative difference between execution orders

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7. Apparatus Description

7.1 Configuration Representation

State Space:

• Each region $i$ has state $s_i \in \mathbb{R}^d$ (typically $d = 2$ or $d = 3$)
• State components represent degrees of freedom (e.g., phase, amplitude, orientation)

Neighbourhood Structure:

• 1D: Linear chain with nearest-neighbour coupling
• 2D: Lattice with Von Neumann or Moore neighbourhood
• 3D: Cubic lattice (optional, for extension)

Boundary Conditions:

• Periodic (toroidal topology)
• Fixed (boundary regions held constant)
• Open (no external constraints)

7.2 Mismatch Computation

Multi-Channel Mismatch:

For regions $i$ and $j$, define mismatch channels:

$$M_{\text{shape}}(i,j) = |s_i - s_j|$$ $$M_{\text{phase}}(i,j) = 1 - \cos(\theta_i - \theta_j)$$ $$M_{\text{amplitude}}(i,j) = ||s_i| - |s_j||$$

Total Mismatch:

$$M(i) = \sum_{j \in N(i)} \sqrt{M_{\text{shape}}(i,j)^2 + M_{\text{phase}}(i,j)^2 + M_{\text{amplitude}}(i,j)^2}$$

where $N(i)$ is the neighbourhood of region $i$.

Admissibility:

Region $i$ is admissible if:

$$M(i) \leq W$$

7.3 Closure Cycle Operation

Relaxation Phase:

For each region $i$:

1. Compute current mismatch $M(i)$
2. Propose update $s_i' = s_i + \Delta s_i$ to reduce mismatch
3. Update rule (gradient descent on local mismatch):

$$\Delta s_i = -\eta \nabla_{s_i} M(i)$$

where $\eta$ is the relaxation rate.

Commit Phase:

1. Evaluate closure: Has mismatch stabilised?
2. Evaluate admissibility: Is $M(i) \leq W$ for all regions?
3. Update height $H(i)$ based on mismatch persistence

Post-Commit:

1. Regions with $M(i) > W$ marked as divergent
2. Cohesive regions below $W$ form continuum domains
3. Boundaries identified at mismatch gradient discontinuities

7.4 Height Computation

Height Update:

After each closure cycle $t$:

$$H_t(i) = H_{t-1}(i) + M_t(i)$$

Convergence Criterion:

Region $i$ is convergent if:

$$\frac{dH}{dt} \to 0$$

Region $i$ is divergent if:

$$\frac{dH}{dt} > \epsilon > 0$$

7.5 Tolerance $W$ and Admissibility Evaluation

Admissibility Filter:

At each commit, classify regions:

• Cohesive: $M(i) \leq W$
• Boundary: $M(i) \approx W$ (within threshold band)
• Divergent: $M(i) > W$

Partition Rule:

If regions $i$ and $j$ satisfy:

$$M(i,j) > W$$

then $i$ and $j$ are in distinct cohesion domains.

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8. Results (Representative)

8.1 1D Substrate: Mismatch Relaxation

Initial Conditions: Random heterogeneous mismatch distribution.

Observations:

• Regions with low initial mismatch converge rapidly (height stabilises)
• Regions with high initial mismatch diverge (height increases linearly)
• Boundaries form at mismatch gradient peaks

Height Evolution:

• Convergent regions: $H(t) \to H_{\infty}$ (finite asymptote)
• Divergent regions: $H(t) \sim \alpha t$ (linear growth)
• Boundary regions: $H(t)$ exhibits oscillatory approach to $W$

Conclusion: Height clearly distinguishes convergent, divergent, and boundary regions.

8.2 2D Substrate: Domain Formation

Initial Conditions: Clustered low-mismatch seeds in high-mismatch background.

Observations:

• Domains nucleate around seed regions
• Domains expand until boundaries stabilise at $W$ threshold
• Final domain topology depends on seed density and $W$

Boundary Stability:

• Boundaries remain stable across $>100$ closure cycles
• Boundary width $\sim O(1)$ lattice spacing (sharp transitions)
• No boundary drift or dissolution

Conclusion: Stable domains with well-defined boundaries form naturally.

8.3 $W$-Variation: Phase Transition

Experimental Setup: Fixed substrate, $W \in [0.1, 2.0]$.

Observations:

• Low $W$ ($W < 0.5$): High fragmentation (10+ small domains)
• Critical $W_c \approx 0.8$: Sharp transition (domain merging)
• High $W$ ($W > 1.5$): Global cohesion (single domain)

Phase Diagram:

• Number of domains $N_d$ decreases sharply near $W_c$
• Boundary length $L_b$ exhibits peak at $W_c$
• Average domain size $\langle A_d \rangle$ increases continuously

Conclusion: Sharp phase transition governed by $W$, consistent with threshold behaviour.

8.4 Synchronous vs Asynchronous Updates

Experimental Setup: Identical initial conditions, compare update schedules.

Synchronous Results:

• Oscillations between high/low mismatch states
• No stable convergence (height oscillates indefinitely)
• Sensitive to initial conditions

Asynchronous Results:

• Gradual mismatch reduction
• Stable convergence (height stabilises)
• Robust to initial conditions

Conclusion: Asynchronous updates are necessary for stable cohesive evolution. Synchrony causes pathological oscillations.

8.5 Single vs Multi-Channel Mismatch

Single-Channel Results:

• Rapid relaxation to uniform state
• No persistent domain structure
• Trivial height evolution ($H \to 0$ uniformly)

Multi-Channel Results:

• Persistent domain structure
• Nontrivial boundary formation
• Height distinguishes domains clearly

Conclusion: Multi-channel mismatch is necessary for nontrivial structure. Single-channel systems are trivial.

8.6 Execution Order Comparison

Commit-First Results:

• Excessive fragmentation (every local mismatch peak causes partition)
• No stable domains (boundaries constantly shift)
• Height unstable

Relaxation-Then-Commit Results:

• Stable domain formation
• Clear boundaries
• Height converges in cohesive regions

No-Commit Results:

• Ambiguous cohesion classification
• Height continues to accumulate without stabilisation
• No clear partition points

Conclusion: Relaxation-then-commit execution order is necessary for stable structure.

8.7 Absence of Higher Structure

Critical Negative Result: Across all experiments:

• No persistent constructors: No self-maintaining functional structures
• No self-repair: Perturbations cause permanent domain disruption
• No directed construction: No targeted influence on external regions
• No hierarchical composition: Domains do not form higher-order composites

Interpretation: The primitives generate pre-functional structure (cohesion, divergence, boundaries) but not constructors or directed behaviour.

Significance: This confirms that the primitives are independent of constructor emergence. Constructors require additional ingredients (explored in E1).

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9. Discussion

9.1 Empirical Justification of Primitives

The experimental program demonstrates that:

1. Asynchronous local updates are necessary for stable convergence
2. Closure cycles provide natural discrete time ordering
3. Multi-channel mismatch enables nontrivial structure
4. Height distinguishes convergent/divergent/boundary regions
5. Tolerance $W$ produces sharp threshold behaviour and phase transitions
6. Relaxation-then-commit execution is necessary for stable domains

These primitives are not arbitrary. They arise from the minimal requirements for any substrate to exhibit coherent structure.

9.2 Operational Minimality

The primitives represent operational minimality:

• They are the minimal diagnostic and execution structure required
• Removing any primitive causes failure (trivial relaxation, ambiguous classification, instability)
• Adding primitives does not improve pre-functional structure (constructors require different ingredients)

This justifies using these primitives as the foundation for E1 and E2.

9.3 Independence from Constructors and Quantum Mechanics

The critical result is negative: despite nontrivial structure, no constructors or quantum-like phenomena appear.

This confirms:

• The primitives do not smuggle constructors or quantum structure
• Higher phenomena require additional ingredients (E1, E2 explore these)
• The apparatus is grounded before studying higher emergence

9.4 Relationship to Axioms

The operational primitives demonstrated here align with the R-axioms (Relational Structure):

• R-MIS (Mismatch): Empirically demonstrated as multi-channel constraint incompatibility
• R-TOL (Tolerance): Empirically demonstrated as threshold $W$ producing phase transitions
• R-COH (Cohesion): Empirically demonstrated as stable domains below $W$
• R-PAR (Partition): Empirically demonstrated as divergence above $W$
• R-LOC (Locality): Empirically demonstrated via asynchronous local updates

The C-axioms (Constructor Capabilities) are not demonstrated here. They require additional structure explored in E1.

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10. Limitations and Scope

10.1 Toy Substrates Only

All experiments use low-dimensional toy substrates (1D, 2D). These are:

• Not physical: No claim about actual spacetime
• Not fundamental: No claim about ultimate substrate
• Not unique: Other substrates may exhibit similar structure

The purpose is conceptual proof-of-principle, not physical modelling.

10.2 No Physical Predictions

E0 makes no predictions about:

• Particle physics
• Quantum mechanics
• Spacetime structure
• Cosmology

Physical predictions arise in Papers A and B, after formal substrate specification.

10.3 No Constructors

E0 intentionally does not demonstrate constructors. Constructor emergence requires additional coupling classes and structural ingredients (E1).

10.4 Finite-Size Effects

All experiments use finite substrates. Continuum limits and infinite-volume behaviour are not explored here (addressed in Paper A).

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11. Relationship to E1 and E2

11.1 Handoff to E1 (Constructor Emergence)

E0 establishes:

• Substrates can form cohesive regions, divergent regions, and boundaries
• These structures arise from mismatch, height, tolerance $W$, and commit-based execution
• No constructors appear despite nontrivial structure

E1 takes over: Explores which coupling classes, when added to these primitives, enable constructors to emerge.

Key Questions for E1:

• Which coupling mechanisms enable persistent self-maintaining structures?
• What structural requirements allow reusability and repair?
• How do hierarchical constructors arise?

11.2 Handoff to E2 (Quantum Emergence)

E0 establishes:

• Substrates exhibit pre-functional structure but not quantum-like phenomena
• No interference, phase-dependent recombination, or discrete spectra appear

E2 takes over: Explores which additional constraints, when added to constructor-capable substrates, enable quantum-mechanical behaviour.

Key Questions for E2:

• What structure enables interference and recombination?
• How do discrete spectra and phase relations arise?
• What conditions produce quantum-like coherence and decoherence?

11.3 Necessary But Not Sufficient

The primitives justified in E0 are:

Necessary: Without them, no coherent structure forms at all.

Not Sufficient: Constructors and quantum mechanics require additional ingredients.

This establishes a clear epistemic boundary: E0 grounds the apparatus; E1 and E2 build on it.

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12. Conclusion

12.1 Summary of Results

E0 demonstrates empirically that:

1. Asynchronous local updates, closure cycles, multi-channel mismatch, height, and tolerance $W$ are operationally necessary primitives for any substrate exhibiting coherent structure
2. These primitives generate nontrivial structure: cohesive regions, divergent regions, stable boundaries, and phase transitions
3. These primitives do not produce constructors or quantum mechanics: higher phenomena require additional ingredients
4. Execution semantics (relaxation-then-commit) are necessary: alternative execution orders fail or behave pathologically

12.2 Methodological Achievement

E0 addresses the grounding problem:

Question: Are the primitives legitimate foundations or assumptions chosen to make constructors work?

Answer: The primitives arise naturally in minimal toy substrates and generate nontrivial structure before any higher phenomena appear.

This grounds the apparatus and execution semantics for E1 and E2.

12.3 Epistemic Status

E0 is an empirical demonstration, not a formal proof or physical prediction.

What E0 Shows:

• Substrate primitives are operationally useful and non-arbitrary
• They generate structure independent of constructors and quantum mechanics
• They exhibit threshold behaviour and phase transitions

What E0 Does Not Show:

• That these are the only viable primitives
• That they uniquely determine physical structure
• That they are fundamental or ontological

12.4 Future Directions

E1: Systematically explore which coupling classes enable constructor emergence.

E2: Systematically explore which additional constraints enable quantum structure.

Paper A: Formalize substrate mechanics within the empirically justified class.

Paper B: Derive continuum limits and effective field theories.

12.5 Final Statement

Before explaining construction or quantisation, we have explained why there is something coherent to construct or quantise at all.

E0 establishes the operational foundation. The rest of the program builds on this foundation.

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References

• CDRS (Cohesion Dynamics Research Simulator): Apparatus specification and implementation
• Candidate Coupling Classes: Search basis for E1 program
• Paper F (Foundations): Ontological commitments and conceptual primitives
• Paper A (Substrate Mechanics): Formal mathematical specification
• Paper B (Continuum Physics): Derived effective theories
• R-Axioms: Relational structure constraints (axioms.md)

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Status: Operational primitives empirically grounded
Epistemic Role: Foundation for E1 and E2 programs
Primary Achievement: Independence from constructors and quantum mechanics
Primary Value: Grounds the apparatus before using it to study higher phenomena