Paper M7 — Layered Definitions and Explanatory Weight

Abstract

This paper establishes a formal framework for managing definitions, primitives, and explanatory objects across the Cohesion Dynamics (CD) research programme. As the programme has expanded through A-series (substrate mechanics), M-series (mechanisms), B-series (quantum recovery), and G-series (geometry and gravity), important explanatory objects are being repeatedly re-instantiated in prose rather than reused as named, layered definitions. This leads to increasing wordiness, loss of formal sharpness, drift in meaning, and difficulty in later formalisation.

M7 addresses this by introducing:

1. A layered definition system that categorises every definition into exactly one explanatory layer
2. Explicit criteria for explanatory weight determining when concepts should be promoted to reusable definitions
3. A definition lifecycle (pull and push) for maintaining definitional coherence
4. A Definition Stewardship Duty as a first-class editorial procedure

This paper introduces no new physical claims, axioms, or empirical results. It is a meta-structural consolidation paper that systematises how the programme expresses and organises its theoretical objects. M7 enables the programme to become more precise, navigable, and publication-ready while preserving falsifiability and avoiding definition sprawl.

Epistemic role: Meta-structural framework — this paper provides programme-level infrastructure for managing definitions and explanatory objects across all series.

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1. Scope and Dependencies

1.1 Assumed Results

This paper assumes without re-derivation:

From Paper A (Substrate Mechanics):

• Discrete substrate with finite alphabet $\Sigma$ and locations $V$
• Local constraint system $\mathcal{C}$ defining admissibility
• Mismatch measure $M(v;X)$ for configurations $X$
• Commit semantics and tolerance $W$ as structural primitives

From Papers M1–M6 (Mechanism Papers):

• Constructive viability and mismatch (M1)
• Constraint dynamics and emergent constructors (M2)
• Modes as emergent structures (M3)
• Phase, path, and coherence structure (M4)
• Constructor emergence (M5)
• Tolerance window constraint programme (M6)

From B-series (as examples of reuse pressure):

• Quantum state representation and recovery work demonstrating repeated definitional needs

From G-series (as examples of reuse pressure):

• Geometric and gravitational derivation work demonstrating heavy structural object reuse

1.2 What This Paper Does NOT Do

M7 does not:

• Introduce new axioms or substrate primitives
• Modify existing axioms or foundational commitments
• Make physical, empirical, or ontological claims
• Force papers to use specific definitions
• Collapse ontology into representation
• Create new R-series reference material (though it enables future extraction)

1.3 Relationship to Other Series

• F-series (Foundations): Unaffected — ontological commitments remain unchanged
• A-series (Substrate): Unaffected — substrate mechanics specification unchanged
• M-series: M7 is a meta-layer that systematises how mechanisms are expressed
• B-series: Benefits from reduced verbosity and clearer object reuse
• G-series: Directly benefits due to heavy structural reuse (paths, delays, gradients)
• R-series: May host extracted definition registries in future work (out of scope for M7)
• E-series: Unaffected — empirical methods remain unchanged

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2. Argument Outline: Why This Mechanism Matters

2.1 The Problem: Definition Drift and Restatement

As the CD programme has evolved across multiple series, a recurring pattern has emerged:

Important explanatory objects are repeatedly described in prose rather than being named, defined once, and reused.

Examples include:

• Closure cycles — described differently in M4, B-series papers, and G-series papers
• Propagation delays — restated across G1, G2, G3 with slight variations in terminology
• Tolerance regimes — characterized repeatedly across M4, M6, and B-series
• Admissibility conditions — reformulated in different contexts rather than referenced

This creates several problems:

1. Wordiness: Papers become longer as they re-explain previously established concepts
2. Drift: Subtle differences in phrasing lead to genuine or apparent semantic drift
3. Navigation difficulty: Readers cannot easily identify what is primitive vs derived
4. Formalisation barriers: Later consolidation requires reconciling multiple versions
5. Review friction: Reviewers struggle to assess whether claims are new or previously established

2.2 Why This Matters for Programme Coherence

The CD programme is explicitly structured as a multi-series research programme with different epistemic roles (F, A, M, B, G, R, E). This structure is powerful because it:

• Separates ontological commitments from derivations
• Enables systematic empirical falsification
• Allows reviewers to evaluate different types of claims appropriately

However, this structure only works if definitional dependencies are clear and stable.

When definitions drift or are restated, the programme loses:

• Falsifiability sharpness — unclear whether failures falsify substrate or prose
• Cumulative coherence — later papers cannot cleanly build on earlier ones
• Publication readiness — journals expect crisp, layered definitions

2.3 What M7 Enables

By introducing a formal framework for definition management, M7 enables:

1. Clearer falsification targets — definitions stabilise, making empirical tests more precise
2. Shorter, sharper papers — reuse replaces restatement
3. Navigable theoretical structure — readers can trace dependencies clearly
4. Publication readiness — programme approaches journal-quality precision
5. Formal consolidation pathway — future axiomatisation becomes feasible

M7 is not optional refinement — it addresses a structural issue that would otherwise degrade programme coherence as the work scales.

Important note on application: This framework is not intended to be applied mechanically; it requires editorial judgment and is designed to guide consolidation, not enforce rigidity. The layered system reflects how the programme already operates in practice and systematises existing patterns rather than imposing new constraints.

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3. The Layered Definition System

3.1 Purpose and Principles

The layered definition system provides a structural organising framework for all definitions introduced across the CD programme.

Core principle: Every definition introduced in the programme belongs to exactly one primary explanatory layer, though it may be referenced across layers according to dependency rules.

Layers are hierarchical in dependency but not in ontological status. Lower layers do not “reduce” higher layers; they establish what higher layers are built from.

3.2 Definition of an Explanatory Layer

Definition 3.1 (Explanatory Layer):
An explanatory layer is a structural category within the CD programme characterised by:

1. Definitional scope — a clear boundary of what kinds of objects belong to it
2. Dependency rules — explicit constraints on which other layers it may reference
3. Epistemic role — what explanatory work objects in this layer perform
4. Placement uniqueness — every definition belongs to exactly one layer

Layers are not arbitrary — they reflect the actual dependency structure of the programme.

3.3 Proposed Layer Structure

Based on the current state of the CD programme, we propose four primary explanatory layers:

Layer 0: Ontological Primitives

Scope: What exists at the substrate level, independent of any emergent description.

Examples:

• Information (as ontological primitive)
• Substrate locations $V$
• Alphabet $\Sigma$
• Constraints $\mathcal{C}$
• Tolerance $W$

Dependency rules: Layer 0 objects may reference only other Layer 0 objects or F-series foundational commitments.

Epistemic role: Establish what the theory is ontologically committed to.

Source series: Primarily F-series and A-series.

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Layer 1: Substrate Mechanisms

Scope: Operational dynamics and structural mechanisms acting on substrate primitives.

Examples:

• Mismatch measure $M(v;X)$
• Commit semantics
• Closure cycles
• Precedence selection
• Divergence and partition
• Admissibility conditions

Dependency rules: Layer 1 objects may reference Layer 0 and other Layer 1 objects, but not Layer 2 or Layer 3 objects.

Epistemic role: Define how the substrate operates and evolves.

Source series: Primarily A-series and M-series.

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Layer 2: Derived Structural Objects

Scope: Objects that are derived from substrate mechanisms but do not yet correspond to effective physical observables.

Examples:

• Paths (as admissible realization histories)
• Propagation delays
• Closure-cycle alignment (phase $\phi$)
• Coherence regimes
• Decoherence conditions
• Gradients and flow structures
• Ancestry sets $\mathcal{A}(X^\star)$

Dependency rules: Layer 2 objects may reference Layers 0, 1, and other Layer 2 objects, but not Layer 3 objects.

Epistemic role: Provide structural building blocks for emergent physics without assuming effective frameworks. Layer 2 objects are structural but not yet observational; they describe substrate-level patterns that may later correspond to effective physics.

Source series: Primarily M-series, with some B-series and G-series formalisation.

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Layer 3: Representational Constructs

Scope: Emergent effective descriptions corresponding to observable physics or geometry.

Examples:

• Time (as emergent coordination)
• Distance (as effective metric)
• Spatial geometry
• Metric structure
• Quantum amplitudes
• Hilbert space structure
• Curvature and gravitational field

Dependency rules: Layer 3 objects may reference any lower layer, but Layer 3 definitions are emergent, not primitive.

Epistemic role: Recover known physics as effective descriptions, enabling empirical falsification. Layer 3 objects are representational constructs corresponding to quantities used in effective physical theories.

Source series: Primarily B-series and G-series.

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3.4 Cross-Layer Dependency Rules

To maintain structural coherence, the following dependency rules are mandatory:

1. Acyclic dependencies: A definition in Layer $N$ may only reference definitions from Layer $\le N$
2. No upward reference: Substrate primitives (Layer 0) cannot reference emergent physics (Layer 3)
3. Clear promotion path: If a Layer 2 object becomes identifiable with a Layer 3 observable, this must be explicitly stated
4. No hidden primitives: If a concept is used in multiple layers, it must be defined at the lowest layer where it applies

Violation of these rules indicates one of the following:

• The definition is misplaced
• A hidden primitive has been smuggled in
• The dependency structure needs revision

3.5 Benefits of Layered Structure

The layered definition system provides:

1. Navigability: Readers can trace dependencies downward to foundations
2. Falsifiability: Empirical failures can be localised to specific layers
3. Clarity: The distinction between substrate and emergent physics remains sharp
4. Modularity: Layers can be independently verified or revised
5. Publication readiness: Journal reviewers can assess primitives vs derived results

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4. Explanatory Weight: When to Promote Concepts to Definitions

4.1 The Promotion Problem

Not every concept that appears in a paper should become a reusable definition. Premature definition creation leads to:

• Definition sprawl (too many named objects)
• Bureaucratic overhead (tracking unused definitions)
• Obscured conceptual structure (signal lost in noise)

Conversely, failure to promote important concepts leads to the problems identified in §2: drift, verbosity, and navigation difficulty.

M7 resolves this by introducing explicit criteria for explanatory weight.

4.2 Explanatory Weight Criteria

Definition 4.1 (Explanatory Weight):
A concept has explanatory weight if it satisfies one or more of the following criteria:

1. Cross-paper reuse: The concept is referenced or required in more than one paper
2. Cross-series reuse: The concept appears in papers from different series (e.g., M-series and B-series)
3. Derivational role: The concept is essential for deriving other results (removing it would require restatement)
4. Stability requirement: The concept’s meaning must remain stable across multiple contexts
5. Complexity reduction: Naming and defining the concept measurably reduces complexity in downstream work

Operational rule: If a concept meets any of these criteria, it should be considered for promotion to a reusable definition.

4.3 Promotion Procedure

When a concept is identified as having explanatory weight:

1. Name it clearly: Assign a precise, unambiguous name
2. Layer it correctly: Determine which explanatory layer it belongs to
3. Define it formally: Provide a formal definition referencing only lower-layer or same-layer objects
4. Document dependencies: Use the Dependency DSL to track which papers rely on it
5. Avoid synonyms: If a similar concept already exists, reuse it or explicitly distinguish

4.4 Anti-Patterns to Avoid

The following are not valid reasons for creating definitions:

• Aesthetic preference: “I think this concept should have a name”
• Novelty signaling: “This makes my paper seem more formal”
• Local convenience: “It’s easier to write this way in this paper”
• Premature abstraction: “This might be useful someday”

Definitions should be created only when they serve programme coherence, not author convenience.

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5. The Definition Lifecycle: Pull and Push

5.1 Overview

To maintain definitional coherence across the programme, M7 introduces a two-phase lifecycle for definitions:

• Pull check (reuse): When editing or creating a paper, check whether existing definitions can be reused
• Push check (promotion): When editing or creating a paper, check whether new concepts should be promoted to definitions

These checks are complementary and required — neither alone is sufficient.

5.2 Pull Check: Reusing Existing Definitions

Purpose: Prevent restating concepts that have already been formally defined.

Procedure:

When drafting or editing a paper:

1. Identify explanatory objects used in the paper
2. Search existing definitions (via formal-definitions registry, axioms, or prior papers)
3. Reuse where applicable: If a definition exists, reference it rather than restating it
4. Document dependency: Use the Dependency DSL to record normative references to defining papers
5. Flag drift: If the existing definition is close but not quite right, flag for potential refinement rather than creating a duplicate

Example:

❌ Incorrect (restatement):
“We define a closure cycle as a sequence of constraint updates that return to the initial state…”

✅ Correct (reuse):
“Using the closure cycle structure defined in Paper M4…“

5.3 Push Check: Promoting New Concepts

Purpose: Identify concepts that should be promoted to reusable definitions.

Procedure:

When drafting or editing a paper:

1. Identify new concepts introduced in the paper
2. Apply explanatory weight criteria (§4.2)
3. Promote if criteria met: If a concept has explanatory weight, name it, layer it, and define it formally
4. Document appropriately: Add to formal-definitions registry or designate as a reusable definition
5. Avoid burying primitives: If a concept is actually a new substrate primitive, it must be explicitly acknowledged (and may require programme-level approval)

Example:

❌ Incorrect (buried primitive):
“…and we assume that propagation delays between regions scale with the gradient of closure-cycle density…”
(This introduces a new structural assumption without naming or justifying it)

✅ Correct (promoted definition):
“We define propagation delay as follows: [formal definition]. This is a Layer 2 derived structural object dependent on closure-cycle accounting (M4) and mismatch gradients (A).“

5.4 Lifecycle Summary

Phase	Question	Action
Pull	Does this concept already exist?	Reuse existing definition
Push	Does this new concept have explanatory weight?	Promote to formal definition

Both checks are mandatory for maintaining programme coherence.

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6. The Definition Stewardship Duty

6.1 Purpose and Scope

The Definition Stewardship Duty is a new editorial duty to be incorporated into paper editing workflows, alongside:

• Programme Management Duty
• Dependency Stewardship Duty
• Axiom Integrity Check

Purpose: Ensure that definitions are reused where applicable, promoted when necessary, and correctly layered.

Scope: Applies to all papers across all series (F, A, M, B, G, R, E).

6.2 When the Duty Applies

Definition Stewardship Duty is triggered when:

• Editing or creating any research paper
• Introducing new concepts or explanatory objects
• Referencing previously defined structures
• Reviewing papers for publication readiness

6.3 Core Responsibilities

When performing Definition Stewardship Duty, the editor or author must:

1. Execute Pull Check: Verify that existing definitions are reused where applicable
2. Execute Push Check: Assess whether new concepts meet explanatory weight criteria and should be promoted
3. Verify layer placement: Ensure definitions are placed in the correct explanatory layer
4. Check for synonyms: Detect and eliminate redundant or duplicate definitions
5. Update dependencies: Use Dependency DSL to track definitional dependencies
6. Document explicitly: Acknowledge the Definition Stewardship check in the editing record

6.4 Operational Checklist

For each paper edit or creation:

• Pull Check: Have I reused existing definitions instead of restating?
• Push Check: Have I promoted concepts with explanatory weight?
• Layer Check: Are all definitions placed in the correct layer?
• Synonym Check: Have I avoided creating duplicates or near-synonyms?
• Dependency Check: Have I documented which definitions this paper depends on?
• Acknowledgment: Have I explicitly acknowledged the Definition Stewardship Duty?

6.5 Acknowledgment Format

At the completion of editing, the editor should include an explicit acknowledgment:

Example acknowledgments:

✅ No issues:
”✅ Definition Stewardship Duty checked — all definitions reused appropriately, no new promotions required.”

✅ Reuse applied:
”✅ Definition Stewardship Duty checked — propagation delay and closure-cycle definitions reused from M4; no new definitions introduced.”

✅ Promotion applied:
”✅ Definition Stewardship Duty checked — ‘gradient flow structure’ promoted to Layer 2 definition; meets cross-series reuse criterion (used in G1, G2, G3).“

6.6 Integration with Existing Duties

Definition Stewardship Duty complements existing duties:

• Programme Management Duty: Ensures series metadata and epistemic roles are correct
• Dependency Stewardship Duty: Ensures cross-paper dependencies are declared correctly
• Axiom Integrity Check: Ensures axiom codes are used correctly, not numbers
• Definition Stewardship Duty: Ensures definitional coherence and reuse

All four duties are required for maintaining programme integrity.

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7. Programme Role

7.1 What M7 Enables

M7 enables the CD programme to:

1. Scale coherently: As the programme grows, definitions remain stable and reusable
2. Publish effectively: Papers become shorter, sharper, and more journal-ready
3. Fail cleanly: Empirical falsification targets substrate claims, not loose prose
4. Consolidate formally: Future axiomatisation or formalisation becomes feasible
5. Onboard reviewers: External readers can navigate the theory structurally

7.2 What M7 Constrains

M7 does not constrain:

• Physical claims or derivations
• Ontological commitments
• Empirical methods
• Freedom to introduce new concepts (subject to explanatory weight criteria)

M7 only constrains how definitions are expressed and reused, not what the theory can claim.

7.3 Failure Modes

If M7 is ignored or poorly applied:

• Definition sprawl: Too many overlapping definitions obscure structure
• Definition drift: Concepts shift meaning across papers, degrading coherence
• Bureaucratic overhead: Excessive formalism without clarity gains
• Rigidity: Inability to introduce new concepts due to over-regulation

M7 is designed to prevent drift while enabling flexibility. It succeeds only if applied with judgment, not mechanically.

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8. Implications for Other Series

8.1 Implications for A-Series

Effect: Minimal direct impact. A-series substrate specification remains unchanged.

Benefit: Future A-series refinements benefit from clearer layer boundaries.

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8.2 Implications for M-Series

Effect: M7 is itself an M-series paper and applies to all M-series work.

Benefit: Mechanism papers become shorter and more reusable; definitional stability increases.

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8.3 Implications for B-Series

Effect: Significant reduction in verbosity; clearer distinction between substrate structure and emergent quantum formalism.

Benefit: B-series papers can reference Layer 1–2 definitions cleanly, making quantum recovery derivations more transparent.

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8.4 Implications for G-Series

Effect: Major benefit due to heavy reuse of structural objects (paths, delays, gradients).

Benefit: G-series papers become significantly shorter; geometric derivation becomes more navigable.

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8.5 Implications for R-Series

Effect: M7 enables but does not require future extraction of definition registries into R-series.

Benefit: R-series could host consolidated definition catalogues in future work.

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8.6 Implications for E-Series

Effect: Minimal direct impact; empirical methods unchanged.

Benefit: Clearer definitions make empirical test targets more precise.

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9. Explicitly Out of Scope

M7 explicitly does not:

1. Introduce new axioms or modify existing axioms
2. Make physical, ontological, or empirical claims
3. Force papers to use specific definitions (only to reuse where applicable)
4. Create a formal definition registry (that is future R-series work)
5. Impose automated validation tooling (tooling may be developed later)
6. Constrain derivational freedom (only definitional expression)
7. Replace proper citation of external works
8. Encode confidence, validity, or epistemic status of definitions

M7 is a procedural and structural framework, not a theoretical claim.

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10. Illustrative Examples

10.1 Example 1: Correct Reuse (Pull Check)

Scenario: A new G-series paper needs to reference closure cycles.

❌ Incorrect approach:
Restate closure cycles in prose within the G-paper.

✅ Correct approach:
Reference M4’s formal definition:
“Using the closure-cycle structure defined in M4, we derive…”

Layer check: Closure cycles are Layer 1 (substrate mechanisms). G-series (Layer 3) can reference Layer 1. ✅

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10.2 Example 2: Correct Promotion (Push Check)

Scenario: Multiple G-series papers use the concept of “propagation delay gradient.”

❌ Incorrect approach:
Describe it differently in each G-paper.

✅ Correct approach:
Promote to Layer 2 definition:
“Definition (Propagation Delay Gradient): The spatial rate of change of propagation delay between adjacent regions, derived from closure-cycle density gradients (M4) and mismatch flow (A).”

Explanatory weight: Cross-series reuse (G1, G2, G3) + derivational role. ✅

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10.3 Example 3: Layer Violation (Incorrect)

Scenario: An A-series paper references “time” as a primitive.

❌ Incorrect approach:
“We assume time is continuous and flows uniformly…”

Layer check: Time is a Layer 3 representational construct. A-series (Layer 0–1) cannot reference Layer 3. ❌

✅ Correct approach:
A-series should reference substrate dynamics, not emergent time. If time is needed, derive it in B-series or G-series.

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10.4 Example 4: Synonym Detection

Scenario: Paper B2 introduces “coherence regime”; Paper M4 already defined “tolerance-bounded coherence.”

❌ Incorrect approach:
Treat them as separate concepts.

✅ Correct approach:
Identify whether they are synonyms. If yes, use M4’s definition. If no, explicitly distinguish:
“We define coherence regime as distinct from M4’s tolerance-bounded coherence in that…“

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10.5 Example 5: Concrete G-Series Application

This example demonstrates the practical impact of M7 on actual paper writing.

Before M7 (Repeated Prose):

In a hypothetical G-series paper pre-M7:

“To derive the effective metric, we must account for the fact that information propagates through the substrate at finite rates determined by closure-cycle processing requirements. When a constraint update propagates from region $A$ to region $B$, the number of closure cycles required depends on the density of unresolved constraints along the path. We define the propagation delay as the cumulative closure-cycle count along the shortest admissible path. The spatial gradient of this delay structure—how rapidly the delay changes as we move through the substrate—determines the local effective geometry…”

(~100 words restating concepts)

After M7 (Definition Reuse):

Using M7’s layered framework:

“Using the propagation delay structure (Layer 2, defined in M4 §3.2) and closure-cycle density gradients (Layer 2, defined in A §5.1), we compute the effective metric. The spatial gradient of propagation delays determines local effective geometry (G1 §2.3).”

(~35 words, clear layer references)

Impact:

• 65% reduction in word count for this section
• Clear dependency tracing: Readers know exactly where each concept is defined
• Layer verification: All referenced objects are Layer 2 (structural), appropriately used in G-series (Layer 3 derivation)
• No semantic drift: “Propagation delay” has a single, stable definition

This pattern scales across papers: G1, G2, G3 can all reference the same Layer 2 definitions rather than restating them, reducing total programme wordiness while increasing precision.

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11. Conclusion and Next Steps

11.1 Summary

M7 establishes a formal framework for managing definitions across the Cohesion Dynamics research programme. It introduces:

1. A layered definition system (Layers 0–3) categorising all definitions by structural role
2. Explicit criteria for explanatory weight to guide concept promotion
3. A definition lifecycle (pull and push) for maintaining coherence
4. A Definition Stewardship Duty as a first-class editorial procedure

M7 introduces no new physics, axioms, or empirical claims. It is a meta-structural consolidation that makes the programme more precise, navigable, and publication-ready.

11.2 Success Criteria

M7 succeeds if:

• Future papers become shorter without losing content
• Definitions stabilise across series
• Reviewers can easily identify primitives vs derived objects
• Formal consolidation becomes feasible in future work
• Empirical falsification targets substrate, not prose

11.3 Failure Conditions

M7 fails if:

• Definition sprawl obscures rather than clarifies
• Bureaucratic overhead exceeds clarity gains
• Definitional rigidity prevents introducing new concepts
• The duty becomes mechanical rather than thoughtful

M7 is a framework requiring judgment, not a mechanical procedure.

11.4 Future Work

M7 enables but does not require:

1. R-series definition registry: Consolidating Layer 0–3 definitions into reference material
2. Automated tooling: Validating layer placement and detecting synonyms programmatically
3. External formalisation: Translating CD into formal proof systems or type theories
4. Publication preparation: Preparing multi-paper manuscripts with stable definitional structure

These are out of scope for M7 but become feasible once M7 is adopted.

11.5 Adoption Path

M7 should be adopted incrementally:

1. Immediate: Apply Definition Stewardship Duty to all new paper edits
2. Short-term: Retroactively apply to high-reuse papers (M1–M6, A, B1–B5, G1–G3)
3. Medium-term: Extract stable definitions into formal-definitions registry
4. Long-term: Develop automated tooling for validation and synonym detection

M7 is programme maturation, not a theoretical pivot. It prepares CD for long-term publishability and formal consolidation.

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12. Acknowledgments

This paper consolidates insights from cross-series editorial experience, particularly the observation that G-series and B-series papers exhibit significant definitional reuse pressure. The layered structure reflects actual dependency patterns observed across the programme, not abstract classification.

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End of Paper M7