R-CCC — Continuum Capability Classification

Purpose

This document presents a formal capability-based classification of informational continua.

R-CCC is a non-normative reference document synthesising eliminative results from E0, E1, and E2; it introduces no new axioms, mechanisms, or assumptions.

The classification is derived from, and constrained by, the completed Constructor Emergence Program (E1), Quantum Emergence Program (E2), and the associated formal developments in M1–M5.

The goal is not to speculate about possible continua, but to classify continua by what they can stably support, as determined by eliminative empirical and formal results.

A key result of the Quantum Emergence Program: Quantum-capable continua do not require additional substrate primitives beyond those established in E0; quantum structure emerges from stricter admissibility and compatibility regimes applied to the same primitives.

This classification is intended as:

A synthesis document
A reference framework for Paper C and later work
A structural guide consolidating Constructor and Quantum Emergence results

Guiding Principle

A continuum is classified by capability, not composition.

Two continua belong to the same class if and only if they support the same constructive capabilities under admissible dynamics, regardless of their microscopic implementation.

Capabilities are ordered by strict necessity within the class of operational substrates explored by E0–E2:

Higher classes require all lower-class capabilities
No higher-class capability was observed to emerge without all prerequisite classes

Capability Ordering

The classes below form a strict hierarchy:

C0 \;\subset\; C1 \;\subset\; C2 \;\subset\; C3 \;\subset\; C4 \;\subset\; C5 \;\subset\; C6

Each inclusion is empirically or formally justified within current eliminative results. No skipped layers have been observed.

Class C0 — Non-Constructive Continua

Defining Capabilities

Local state updates
Possible reversibility
No persistence
No binding
No error correction

Structural Character

Dynamics may be rich or chaotic
No accumulation of structure
No memory beyond trivial recurrence

Representative Phenomena

Simple reversible cellular automata
Pure diffusion processes
Phase-less relational graphs

Axiomatic Foundation

Satisfied R-axioms:

AX-REL (Relational Evolution): States evolve via relations to other states
AX-CON (Global Constraint Invariance): Constraints remain invariant

Missing capabilities:

No tolerance-based admissibility (AX-TOL)
No cohesive units (AX-COH)
No constructor capabilities (C-axioms)

Status

Empirically eliminated as constructor-capable
Ruled out in Phases 1–7 of E1

Class C1 — Error-Corrective Continua

Defining Capabilities

Tolerance-gated admissibility
Local error correction
Persistence under bounded perturbation

Required Structure

Admissibility (tolerance)
Partition on violation

Structural Character

Memory exists
Structure can persist
No construction or reuse

Representative Phenomena

Stable patterns with repair
Non-constructive memories

Axiomatic Foundation

Satisfied R-axioms:

AX-REL (Relational Evolution)
AX-CON (Global Constraint Invariance)
AX-MIS (Mismatch): Constraint incompatibility measure exists
AX-TOL (Tolerance): Finite tolerance window $W$ exists
AX-COH (Cohesion): States within $W$ form cohesive units
AX-CONT (Continuum): CIUs co-evolve preserving mutual tolerance
AX-PAR (Partition): Mismatch exceeding $W$ causes partition
AX-LOC (Locality): Evolution proceeds locally

Satisfied C-axioms:

AX-MEM (Persistence): Structures retain internal state

Missing capabilities:

No admissible move selection (AX-ADM)
No precedence-based selection (AX-SEL)

Status

Necessary but insufficient for construction
Established in Phase 8 of E1

Class C2 — Constructor-Capable Continua

Defining Capabilities

Persistent functional processes
Reusability without retuning
Selective interaction

Required Structure

Admissible alternative sets
Deterministic precedence (selection)

Structural Character

Functional organisation
Machines in the Constructor Theory sense
No hierarchy

Representative Phenomena

Simple constructors
Reusable processes

Axiomatic Foundation

Satisfied R-axioms: All R-axioms (AX-REL through AX-LOC)

Satisfied C-axioms:

AX-ADM (Admissible Moves): Non-empty set of mismatch-reducing transitions
AX-SEL (Precedence): Deterministic selection minimizing mismatch
AX-MEM (Persistence): State retention across interactions

Missing capabilities:

No hierarchical composition (AX-META)

Status

First appearance of constructors
Established in Phases 11–12 of E1
This is the minimum class where Constructor Theory applies

Class C3 — Hierarchical Constructor Continua

Defining Capabilities

Composition of constructors
Multi-scale persistence
Functional encapsulation

Required Structure

Meta-precedence
Scale-extended selection

Structural Character

Hierarchies of constructors
Separation of functional layers

Representative Phenomena

Chemistry-like organisation
Composite machinery

Axiomatic Foundation

Satisfied R-axioms: All R-axioms

Satisfied C-axioms:

AX-ADM (Admissible Moves)
AX-SEL (Precedence)
AX-MEM (Persistence)
AX-META (Meta-Cohesion): Structures participate as units in higher-order relations

New capabilities:

Template-based influence may begin to appear (AX-TPL)

Status

Established in Phases 13–14 of E1

Class C4 — Mode-Bearing Continua

Defining Capabilities

Discrete internal configurations
Long-lived attractors
Degeneracy
Stability under perturbation

Required Structure

Hierarchical stabilisation
Invariant basins of precedence dynamics

Structural Character

State identity becomes meaningful
Discreteness without quantisation

Representative Phenomena

Mode families
Degenerate stable configurations

Axiomatic Foundation

Satisfied axioms: All R-axioms and C-axioms

Emergent structure:

Discrete stable modes emerge from precedence dynamics
Attractors in configuration space
Degeneracy as structural feature

Status

Established in Phase 15 of E1
Formalised in M3

Class C5 — Interferometric Continua (Pre-Quantum)

Defining Capabilities

Preserved route identity
Phase accumulation along distinct paths
Phase-dependent recombination
Interference
Divergent compatible histories: Multiple resolution paths may coexist pre-commit provided joint mismatch remains within tolerance $W$

Required Structure

Topological differentiation
Phase-bearing routes
Recombination under precedence
Commit-based resolution framework

Structural Character

Alternatives combine non-trivially
Interference without amplitudes or Hilbert space
Deterministic branching pre-commit with deterministic resolution at commit

Representative Phenomena

Constructive and destructive interference
Path-sensitive outcomes

Axiomatic Foundation

Satisfied axioms: All R-axioms and C-axioms

Emergent structure:

Route topology from divergence and recombination
Phase as accumulated mismatch along paths
Interference from precedence-based recombination

Status

Established in Phases 17–18 of E1
Minimum pre-quantum continuum class

Class C6 — Quantum-Emergent Continua

Defining Capabilities

Additive composition of alternatives at an amplitude-like level
Commit-based resolution and branch weighting: Commit deterministically resolves incompatible alternatives; a conserved quadratic measure over resolution branches is induced when outcomes are viewed branch-relatively
Non-factorisable joint admissibility: Admissibility and tolerance are evaluated on composite configurations, preventing subsystem separability (substrate origin of entanglement)
Stable spectral and bound-state structure
Strengthened role of $W$ : The same tolerance parameter $W$ governs mergeability vs partition, coherence vs decoherence boundaries, and limits on joint (entangled) structure

Required Structure

Emergent linear composition rules at merge points
Invariant outcome measures (quadratic weighting)
Structured subsystem coupling with joint admissibility constraints
Commit-based divergence and merge semantics

Structural Character

Hilbert space as an effective calculus (emergent representation, not ontology)
Schrödinger-like or path-integral dynamics in the continuum limit
Deterministic substrate dynamics with branching structure

Representative Phenomena

Quantum mechanics as observed
Discrete spectra
Entanglement (from joint admissibility constraints)
Decoherence (from $W$ -based boundaries)

Axiomatic Foundation

Satisfied axioms: All R-axioms and C-axioms

No additional substrate primitives required: Uses the same operational primitives established in E0 (mismatch, height, tolerance $W$ , closure cycles, commit-based execution) under stricter admissibility and compatibility regimes.

Status

Established by Quantum Emergence Program (E2, Phases Q0–Q6)
Quantum structure emerges from E0 primitives without axiom inflation

Capabilities Introduced by Quantum Emergence (E2)

The Quantum Emergence Program (E2, Phases Q0–Q6) systematically explored which additional structural features are required for quantum-mechanical behaviour to emerge from constructor-capable continua.

Key Results

No new substrate primitives required:

All quantum structure emerges from the same operational primitives established in E0: mismatch, height, tolerance $W$ , closure cycles, and commit-based execution
Quantum capabilities arise from applying these primitives under stricter admissibility and compatibility regimes

Deterministic substrate dynamics:

The substrate remains fully deterministic
Branching occurs when multiple resolution paths satisfy admissibility constraints pre-commit
Commit deterministically resolves incompatible alternatives
What appears as “probabilistic” quantum behaviour emerges from branch-relative outcome measures, not fundamental stochasticity

Refinements of capability, not ontology:

Divergent compatible histories: Multiple paths explored pre-commit when joint mismatch remains within $W$
Commit-based branch weighting: Quadratic measure over branches emerges from how commits resolve alternatives
Joint admissibility: Tolerance and admissibility evaluated on composite configurations (substrate origin of entanglement)
Strengthened $W$ role: Same tolerance parameter governs coherence boundaries, mergeability, and entanglement limits

Empirical Basis

These capabilities were established through systematic eliminative testing across six research phases:

Q0: Interference support and path mergeability
Q1: Phase-like relational structure
Q2: Constraint class interactions
Q3: Amplitude composition rules
Q4: Quadratic outcome weighting
Q5: Non-factorisable subsystem composition
Q6: Discrete spectral structures

All results are reproducible from constraint-based simulations documented in E2.

Significance

The “no axiom inflation” result is central: quantum structure does not require introducing probability, measurement postulates, or new physical laws. It emerges from the same deterministic constraint-resolution framework that underlies all continuum classes, applied with sufficient strictness at the admissibility boundary.

Summary Table

Class	Core Capability	Required Additions	Key Axioms
C0	Local dynamics	None	AX-REL, AX-CON
C1	Persistence	Admissibility	AX-TOL, AX-COH, AX-MEM
C2	Construction	Precedence	AX-ADM, AX-SEL
C3	Hierarchy	Meta-precedence	AX-META
C4	Modes	Hierarchical stabilisation	All C-axioms
C5	Interference	Topological routes + phase	Emergent route structure
C6	Quantum mechanics	Stricter admissibility regimes (E2)	Same as C5 (no new axioms)

Interpretation

Within the class of operational substrates explored by E0–E2, this classification demonstrates that quantum mechanics is not generic.

Given the operational primitives and execution semantics established in E0, quantum-capable continua occupy a narrow region of the space of possible continua, requiring a precise accumulation of structural capabilities.

The existence of physics like ours therefore places strong constraints on the underlying substrate, as indicated by current eliminative results.

Key Insights

Strict Hierarchy: No capability level can be skipped within tested substrate classes
Axiomatic Grounding: Each class corresponds to specific axiom combinations
Empirical Validation: Class boundaries established by eliminative research (E0, E1, E2)
Predictive Power: Framework guides Quantum Emergence Program (E2)

Relationship to Ongoing Work

E0 (Empirical Grounding) establishes operational primitives independent of constructors
E1 (Constructor Emergence Program; see M1–M5) spans multiple internal capability boundaries, with the defining emergence of constructors at C2 and further refinements (repair, hierarchy, modes, interference) across higher levels up to C5
E2 (Quantum Emergence Program, Phases Q0–Q6) has established C6 from C5, demonstrating that quantum structure emerges from E0 primitives without requiring additional axioms
Paper C will narratively integrate this classification into the broader Cohesion Dynamics framework
Papers M1–M5 provide formal mechanisms underlying capability emergence

Usage Guidelines

For Researchers

When proposing new substrate structures or mechanisms:

Identify which capability class is required for your phenomenon
Verify all prerequisite axioms are satisfied
Check empirical results from E1 for consistency
Do not skip capability levels

For Critics

When evaluating claims about emergent structure:

Ask: “What capability class does this require?”
Verify: “Are all prerequisite capabilities demonstrated?”
Challenge: “Can this be achieved in a lower class?”
Demand: “What eliminative evidence supports the classification?”

Closing Remark

Within the scope of operational substrates explored by E0–E2, this classification reframes the study of fundamental physics as a problem of capability stratification.

Rather than asking “what equations describe reality?”, it asks:

What kinds of realities are capable of supporting equations at all?

Given the operational primitives and execution semantics established in E0, the answer constrains not just our universe, but the space of possible physical laws consistent with those primitives.

References

E0: Empirical Grounding of Substrate Primitives
E1: Empirical Classification of Constructor-Capable Substrates
E2: Empirical Classification of Quantum-Capable Substrates (in progress)
M1–M5: Formal Mechanism Papers
Axioms: R-axioms and C-axioms (see Reference: Axioms)
Paper C: Constructor Emergence Narrative (forthcoming)

Status: Reference framework established
Epistemic Role: Synthesis and classification
Primary Value: Structured capability hierarchy grounding emergence research
Usage: Reference for E2, Paper C, and subsequent work