Coherence Composer Skill: Complete Technical Documentation

Skill 3 of 3 | Possibility Coordination | Status: ✅ Production-Ready

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What It Does

The Coherence Composer is an epistemological skill that validates structural consistency and explores counterfactual worlds while respecting constraints from both Commitment Tracker (ontology) and Opacity Detector (epistemology).

Given:

• A system with structural constraints (causality, logic, commitments, accessibility, conservation)
• Proposed counterfactual worlds

It answers: "What COULD BE TRUE while respecting all constraints?"

The skill validates that counterfactuals satisfy the sheaf condition — all constraints glue consistently — and explores the valid variant space that remains possible.

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The Problem It Solves

When agents explore counterfactual scenarios, they need to ask:

• "Is this alternative world logically possible?"
• "Does it respect our shared commitments?"
• "Can all observers access the knowledge they need?"
• "Would natural laws still hold?"

Without the skill: Agents propose counterfactuals that violate basic logic, contradict shared commitments, or require omniscience from bounded observers.

With the skill:

1. Extract explicit constraints (temporal, logical, ontological, epistemic, conservational)
2. Verify closure: does each counterfactual satisfy ALL constraints simultaneously?
3. Find minimal worlds: what's the simplest way to achieve a counterfactual?
4. Explore valid space: what variant histories remain genuinely possible?

Result: Counterfactuals grounded in reality, respecting both physics and agreement.

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The 2-Monad Structure

Objects: Constraint Boundaries

Each system has a constraint boundary — the set of structural requirements that limit what can be true.

ComputerGraphics System:
  ├─ Constraint 1 (critical): Temporal causality
  │   ├─ No time paradoxes
  │   ├─ Effects after causes
  │   └─ Predicates: {cause-before-effect, temporal-order, no-self-loops}
  │
  ├─ Constraint 2 (critical): Logical consistency
  │   ├─ Non-contradiction
  │   ├─ Excluded middle
  │   └─ Predicates: {non-contradiction, excluded-middle, modus-ponens}
  │
  ├─ Constraint 3 (high): Ontological commitment
  │   ├─ Must preserve: {composition, structure, harmony}
  │   └─ Predicates: {commitment-preservation, ontological-stability}
  │
  ├─ Constraint 4 (high): Epistemic accessibility
  │   ├─ Must respect boundaries of: {computational, embodied, temporal}
  │   └─ Predicates: {boundary-respect, bridging-feature-use}
  │
  └─ Constraint 5 (medium): Conservation laws
      ├─ Information/value conserved
      └─ Predicates: {conservation-law, symmetry-preservation, invariant-preservation}

1-Cells: Closure Relations

A 1-cell between two boundaries is a closure relation — a proposed counterfactual that either satisfies all constraints (closed) or violates some (open).

Counterfactual X:
  temporal constraint:     ✓ satisfied (cause-before-effect: true)
  logical constraint:      ✓ satisfied (consistent: true)
  commitment constraint:   ✓ satisfied (ontology-preserved: true)
  accessibility constraint:✓ satisfied (boundaries-respected: true)
  conserved constraint:    ✓ satisfied (conserved: true)
  ──────────────────────────────────────────────────────
  CLOSURE: ✓ CLOSED (5/5 constraints verified)

2-Cells: Natural Transformations (The Coherence Cycle)

Three 2-cells that compose via monad multiplication:

γ₋₍: Hidden Constraints → Explicit Boundaries

• Input: System description
• Output: Constraint boundary with explicit predicates
• Transforms: Implicit structure → visible constraints

γ₍₊: Boundaries → Closure Conditions

• Input: Constraint boundary
• Output: Conditions each counterfactual must satisfy
• Transforms: Constraints → closure requirements

γ₊₋: Closures → Valid Counterfactuals

• Input: Verified closures
• Output: Valid variant space
• Transforms: Verified closures → explored possibilities

μ: Monad Multiplication (Coherence Cycle Closes)

        Hidden Constraints
               │
               ↓ γ₋₍
    Explicit Boundaries (5 constraints listed)
               │
               ↓ γ₍₊
     Closure Verification (all satisfied? yes/no)
               │
               ↓ γ₊₋
   Valid Counterfactuals (variant space enumerated)
               │
               ↓ μ (monad mult)
   Back to Hidden (but now with possibility map)

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Five Core Constraints

1. Temporal Constraint (Critical)

Question: Does causality hold? Can effects precede causes?

Predicates:

• cause-before-effect

  : Every effect has a prior cause

• temporal-order

  : Timeline is consistent

• no-self-loops

  : No cycles in causal graph

Failure: Time paradoxes (killing grandfather, etc.)

2. Logical Constraint (Critical)

Question: Is the world logically consistent? Can something be both true and false?

Predicates:

• non-contradiction

  : ¬(P ∧ ¬P)

• excluded-middle

  : P ∨ ¬P

• modus-ponens

  : (P → Q) ∧ P ⟹ Q

Failure: Contradictory propositions both holding

3. Commitment Constraint (High)

Question: Does the counterfactual respect ontological commitments from Skill 1?

Preserves: Shared ontology (e.g., "composition", "structure", "harmony")

Predicates:

• commitment-preservation

  : All agreed-upon entities still exist

• ontological-stability

  : What we said exists still exists

Failure: Violating shared assumptions about reality

4. Accessibility Constraint (High)

Question: Can all observers verify the counterfactual? Does it respect epistemic boundaries from Skill 2?

Preserves: Epistemic boundaries of all observers

• Computational observer: Can verify logical structure
• Embodied observer: Can verify sensory features
• Temporal observer: Can verify narrative coherence

Predicates:

• boundary-respect

  : Don't ask observers to verify beyond their access

• bridging-feature-use

  : Use features both can access for verification

Failure: Counterfactual requires omniscience

5. Conservation Constraint (Medium)

Question: Are conserved properties preserved?

Conserved:

• Information (Landauer's principle)
• Energy (thermodynamics)
• Charge (physics)
• Value (economics)

Predicates:

• conservation-law

  : Total conserved quantity unchanged

• symmetry-preservation

  : Symmetries remain intact

• invariant-preservation

  : Invariant properties unchanged

Failure: Violating fundamental conservation laws

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The Four Games

Game 1: Constraint Disclosure

Input: System description (ontology, observers, etc.) Output: Explicit constraint boundaries Time: < 1 ms

Shows what constraints the system imposes:

System: music-composition
Total constraints: 5
Critical constraints: 2

Constraint: temporal
  Severity: :critical
  Description: Causality: effects cannot precede causes
  Predicates: cause-before-effect, temporal-order, no-self-loops

Constraint: logical
  Severity: :critical
  Description: Logic: propositions cannot be both true and false
  Predicates: non-contradiction, excluded-middle, modus-ponens

...

Insight: Transparency. You know exactly what limits what's possible.

Game 2: Closure Verification

Input: Constraint boundaries + candidate counterfactuals Output: Verification results (closed/open) Time: < 1 ms per counterfactual

Tests whether each counterfactual satisfies all constraints:

Counterfactual: world-A
  Closure: ✓ CLOSED
  Satisfied: 5/5 (100%)
    temporal: ✓
    logical: ✓
    commitment: ✓
    accessibility: ✓
    conserved: ✓

Counterfactual: world-B
  Closure: ✗ OPEN
  Satisfied: 4/5 (80%)
    temporal: ✓
    logical: ✗      ← Failed!
    commitment: ✓
    accessibility: ✓
    conserved: ✓

Insight: The sheaf condition. Constraints must glue consistently — partial satisfaction isn't enough.

Game 3: Minimal World Construction

Input: Constraint boundaries + desired outcome Output: Simplest counterfactual achieving it Time: < 5 ms

Applies Occam's razor: prefer counterfactuals with fewest/smallest changes.

World: minimal-63
  Based on: current-world
  Changes: 2
  Magnitude: 1.5
  Explanation: Changed 2 properties with magnitude 1.5

World: minimal-81
  Based on: current-world
  Changes: 2
  Magnitude: 1.5

World: minimal-57
  Based on: current-world
  Changes: 2
  Magnitude: 1.5

Minimal world selected: minimal-63

Insight: Not all possible worlds are equally elegant. Simpler changes are preferable.

Game 4: Historical Branching

Input: Constraint boundaries + enumeration parameters Output: Valid variant space (all consistent alternatives) Time: < 10 ms for N candidates

Explores what alternative histories genuinely remain possible:

Total candidates: 3
Valid variants: 3
Validity rate: 100%

Valid variants:
  • minimal-63 (Satisfaction: 5/5)
  • minimal-81 (Satisfaction: 5/5)
  • minimal-57 (Satisfaction: 5/5)

Interpretation: These counterfactual worlds could exist
within the constraint structure while respecting:
  - Temporal coherence (causality)
  - Logical consistency
  - Commitment preservation
  - Epistemic boundaries
  - Conservation laws

Insight: The valid variant space is often much smaller than the space of conceivable worlds.

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Sheaf Condition and Closure

What is a Sheaf?

A sheaf is a structure where local data glue together to form consistent global data.

In Coherence Composer, a counterfactual is "sheaf-like" if:

• Locally (per observer, per constraint): the counterfactual is consistent
• Globally: all local consistencies glue into a single coherent world

Closure vs. Open

Closed counterfactual: All constraints satisfied simultaneously (sheaf condition holds)

Open counterfactual: Some constraints violated (gluing fails)

The closure verification tests: "Does this counterfactual form a coherent whole?"

Example

World-C seems okay locally:
  - Temporal: ✗ (has paradox) ← PROBLEM
  - Logical: ✓
  - Commitment: ✗ (violates ontology) ← PROBLEM
  - Accessibility: ✓
  - Conservation: ✓

Globally: This world doesn't cohere. It's impossible.
Status: OPEN (cannot exist)

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Technical Implementation

Babashka (Interactive)

File:

.topos/coherence_composer_2monad.bb

(290 lines)

Functions:

(temporal-constraint seed)
(logical-constraint seed)
(commitment-constraint seed ontology)
(accessibility-constraint seed accessibility)
(conserved-constraint seed)

(map-constraint-boundaries system seed ontology accessibility)
  → boundaries with 5 constraints

(check-closure counterfactual constraints)
  → ValidationResult with satisfaction ratio

(minimal-counterfactual base-world changes seed)
(enumerate-counterfactuals base-world boundaries seed)
  → Vector of counterfactuals

(game-constraint-disclosure system seed ontology accessibility)
(game-closure-verification system seed ontology accessibility)
(game-minimal-world system seed ontology accessibility)
(game-historical-branching system seed ontology accessibility)

Run:

bb .topos/coherence_composer_2monad.bb constraints
bb .topos/coherence_composer_2monad.bb closure
bb .topos/coherence_composer_2modan.bb minimal
bb .topos/coherence_composer_2modan.bb branching
bb .topos/coherence_composer_2modan.bb all

Julia (Production)

File:

rio/Gay.jl/src/coherence_composer.jl

(400+ lines)

Types:

@enum ConstraintSeverity critical high medium low

struct Constraint
    name::String
    id::UInt32
    description::String
    predicates::Set{String}
    severity::ConstraintSeverity
    ontology::Union{Vector{String}, Nothing}
    accessibility::Union{Vector{String}, Nothing}
end

struct ConstraintBoundary
    system::String
    constraints::Vector{Constraint}
    total_constraints::Int
    critical_count::Int
    all_predicates::Set{String}
    seed::UInt64
end

struct Counterfactual
    name::String
    based_on::String
    changes::Vector{Tuple{String, Float64}}
    change_count::Int
    change_magnitude::Float64
    # Constraint satisfaction flags:
    cause_before_effect::Bool
    consistent::Bool
    # ... more flags
end

struct ValidationResult
    counterfactual::String
    closure_verified::Bool
    satisfied_constraints::Int
    total_constraints::Int
    satisfaction_ratio::Float64
    details::Dict{String, Bool}
end

Functions:

extract_constraints(system, seed, ontology, accessibility)
  → ConstraintBoundary

verify_closure(counterfactual, constraints)
  → ValidationResult

compose_counterfactuals(system, seed, ontology, accessibility)
  → Dict{String, Any}

world_coherence_composer(; seed, system, ontology, accessibility)
  → world result (for worlds.jl)

world_full_skill_integration(; seed, ontology, accessibility)
  → trialectic result (all three skills)

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Integration Points

With Commitment Tracker (Skill 1)

Commitment Tracker handles: Ontology ("What exists?")

• Extracts commitments from agents
• Negotiates shared reality

Coherence Composer uses: Commitment preservation constraint

• Verifies counterfactuals don't violate agreed-upon ontology
• Asks: "Does this alternative world preserve what we agreed is real?"

Commitment: "compositions exist"
Counterfactual: "A world with no compositions"
Verification: ✗ FAILS commitment constraint
Result: This counterfactual is OPEN (impossible given our agreements)

With Opacity Detector (Skill 2)

Opacity Detector handles: Epistemology ("What can we know?")

• Maps epistemic boundaries of each observer
• Finds bridging features

Coherence Composer uses: Accessibility constraint

• Verifies counterfactuals respect epistemic boundaries
• Asks: "Can all observers verify this counterfactual with their knowledge access?"

Observer: computational (can access: logic, patterns, structure)
Counterfactual: "A world where 2+2=5"
Verification: ✗ FAILS logical constraint
              ✗ FAILS accessibility constraint (no observer can verify this)
Result: This counterfactual is OPEN (impossible given what we can know)

With worlds.jl (Julia)

Integration for parallel world spawning:

result = world_coherence_composer(
    seed=0x285508656870f24a,
    system="music-composition",
    ontology=["composition", "structure", "harmony"],
    accessibility=["computational", "embodied", "temporal"]
)

# Returns: Dict with constraints, validations, valid_count, etc.
# Parallelizable:
results = fork_all([
    () -> world_coherence_composer(seed=s, system=system)
    for s in seeds
])

Full Trialectic Integration

result = world_full_skill_integration(
    seed=0x285508656870f24a,
    ontology=["composition", "structure", "harmony"],
    accessibility=["computational", "embodied", "temporal"]
)

# Demonstrates all three skills:
# Level 1 (Commitment): "What exists?" → ontology
# Level 2 (Opacity): "What can we know?" → epistemology
# Level 3 (Coherence): "What could be true?" → possibility

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Testing Guide

Test 1: Constraint Disclosure

bb .topos/coherence_composer_2monad.bb constraints

Check:

• All 5 constraints listed
• Severity levels correct (2 critical, 2 high, 1 medium)
• Predicates shown for each constraint
• All predicates listed at bottom

Test 2: Closure Verification

bb .topos/coherence_composer_2monad.bb closure

Check:

• Three counterfactuals tested
• world-A shows 5/5 satisfied (CLOSED)
• world-B shows 4/5 satisfied (OPEN) ← logical constraint fails
• world-C shows 3/5 satisfied (OPEN) ← temporal and commitment fail
• Closure status matches satisfaction ratio

Test 3: Minimal World Construction

bb .topos/coherence_composer_2modan.bb minimal

Check:

• Three minimal worlds shown
• Each has 2 property changes
• Magnitude around 1.5
• Worlds sorted by fewest changes
• "Minimal world selected" identifies simplest

Test 4: Historical Branching

bb .topos/coherence_composer_2modan.bb branching

Check:

• Total candidates (3)
• Valid variants (3)
• Validity rate (100%)
• Each valid variant shows 5/5 satisfaction
• All 5 constraint types mentioned in interpretation

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Limitations and Future Work

Current Limitations

1. Binary constraint satisfaction: Constraints are either satisfied or not (no partial credit)

   • Future: Probabilistic constraints with confidence scores

2. Hand-defined constraints: Constraints are specified manually per system

   • Future: Learn constraints from data via causal inference

3. Enumeration-based exploration: Valid space found by enumerating candidates

   • Future: Constraint satisfaction solver (SAT/SMT) for principled search

4. No temporal dynamics: Constraints assumed fixed

   • Future: Allow constraints to evolve as dialogue progresses

5. Static change magnitudes: Changes have fixed cost

   • Future: Context-dependent change costs (some worlds cheaper to reach)

6. No preference ordering: All valid variants treated equally

   • Future: Aesthetic/pragmatic preferences on valid space

Future Enhancements

• SAT Solver Integration: Use constraint satisfaction to guarantee completeness
• Probabilistic Constraints: Soft constraints with confidence
• Constraint Learning: Discover constraints from execution traces
• Preference Modeling: Order valid variants by agent preferences
• Temporal Dynamics: Constraints that evolve with narrative
• Cost Models: Context-dependent change costs
• Multi-Agent Coherence: Reconcile counterfactuals across multiple agents
• Phenomenological Grounding: Connect constraints to human experience
• Cross-Domain Application: Apply to medicine, ecology, policy, etc.

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Performance Characteristics

Operation	Time	Space
Extract constraints (1 system)	< 1ms	O(n)
Verify closure (1 counterfactual)	< 1ms	O(n)
Verify all (3 counterfactuals)	< 5ms	O(n²)
Enumerate variants (3 variants)	< 10ms	O(n)
Full scenario (all 4 games)	< 25ms	O(n)

Where n = number of constraints (typically 5-10).

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Key Insights

1. Constraints Delimit Possibility

Not everything is possible. Constraints (temporal, logical, ontological, epistemic, conservational) narrow the space of valid counterfactuals dramatically.

2. Closure is Objective

A counterfactual either satisfies all constraints (closed) or violates some (open). This is not a matter of opinion—it's structural.

3. The Sheaf Condition Matters

Local consistency (each constraint satisfied) ≠ Global coherence (all constraints glue). Coherence Composer verifies both.

4. Minimal Worlds are Preferable

Among valid counterfactuals, prefer those requiring fewest/smallest changes (Occam's razor). Simplicity is a virtue.

5. Three Skills Form a Complete System

• Commitment Tracker: Negotiates what's real
• Opacity Detector: Respects what's knowable
• Coherence Composer: Validates what's possible

Together they enable principled multi-agent coordination across ontology, epistemology, and possibility.

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References

• Constraint Satisfaction: Dechter, R. (2003). Constraint Processing
• Sheaf Theory: MacLane, S. & Moerdijk, I. (1992). Sheaves in Geometry and Logic
• Counterfactuals: Lewis, D. (1973). Counterfactuals
• Possible Worlds: Kripke, S. (1980). Naming and Necessity
• Modal Logic: Hughes, G. E. & Cresswell, M. J. (2012). A New Introduction to Modal Logic
• Causal Models: Pearl, J. (2009). Causality: Models, Reasoning, and Inference

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Key Takeaway

The Coherence Composer transforms incompatible counterfactuals into valid variant spaces by:

1. Extracting explicit constraints (temporal, logical, ontological, epistemic, conservational)
2. Verifying closure (all constraints satisfied simultaneously)
3. Constructing minimal worlds (fewest changes required)
4. Exploring valid space (what alternatives remain genuinely possible)

This allows multi-agent systems to reason about counterfactuals not as fantasy, but as structural possibilities constrained by reality, agreement, and knowledge.

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Status: ✅ Production Ready (v1.0)

Maturity: 1.0 (complete implementation)

Maintenance: bmorphism

License: Plurigrid Collective (AGPL-3.0-or-later)

Built for the Counterfactual Worlds project as the third of three skills distilled from the greatest increments to ontology, epistemology, and possibility.