Categorical Rewriting: Triad 4 (World Transformation)

Status: Design Phase Trit Assignment: See GF(3) Balance section Featured: Yes (completes Triad architecture) Verified: Pending implementation

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Overview

Triad 4 synthesizes Categorical Rewriting + Graph Grafting + DisCoPy to enable dynamic world transformation — converting abstract moves from Glass Bead Game into concrete world mutations.

This completes the 4-step cycle:

Triad 1: Foundation (Foundations, axioms, base types)
   ↓
Triad 2: Molding (Glass Bead Game + Validation + ACSet storage)
   ↓
Triad 3: Hopping (World navigation + arbitrage + interleaving)
   ↓
Triad 4: Rewriting (Categorical transformation of worlds) ← YOU ARE HERE
   ↓
[Loop back to adapted foundation]

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Components: GF(3)-Balanced Triad

Component A (+1): discopy-operadic-move-generation (PLAY)

Role: Generate abstract categorical moves from one world state to another

What it does:

• Uses DisCoPy (string diagrams) to represent world transformations as morphisms
• Operadic composition of multiple simultaneous moves
• Generates "counterfactual moves" as Span Profunctors
• Output: Abstract move algebra ready for concrete application

Core Operations:

# String diagram representation of world transformation
move = Box("world_transition",
           Ty("state_A"),
           Ty("state_B"))

# Operadic composition: combine multiple moves
combined_move = move @ other_move  # horizontal composition
threaded_move = move >> another_move  # vertical composition

# Span profunctor (counterfactual generation)
span = Span(apex_world, left_projection, right_projection)
# represents: "what could happen if we move like this?"

Algorithm:

Input: Current world state W_A
Output: Abstract moves to reachable worlds

1. Extract ACSet structure from W_A
2. Identify constraint-respecting morphisms
3. Generate Span Profunctors (move families)
4. Operadically compose into compound moves
5. Return high-value moves (epistemic arbitrage ranking)

Mathematical Foundation:

• String diagrams from DisCoPy (Baez, Ong)
• Operadic composition (May, Markl)
• Span category (pullback-based moves)
• Profunctors (parametrized move families)

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Component B (0): graph-mutation-engine (ERGODIC)

Role: Apply abstract moves to concrete ACSet structures, maintaining consistency

What it does:

• Translates DisCoPy morphisms into DuckDB mutations
• Validates that mutations preserve constraints
• Executes graph grafting (inserting/deleting/rewriting subgraphs)
• Maintains GF(3) conservation across mutations

Core Operations:

-- Example: Harmonic rewriting in music-topos world
-- Move: "insert V chord where ii was expected"

BEGIN TRANSACTION;

-- 1. Identify subgraph to graft
SELECT * FROM harmonies
WHERE position = 'measure_4' AND degree = 'ii'
INTO subgraph_to_replace;

-- 2. Validate constraints before mutation
CHECK (gf3_balance(subgraph_to_replace) = 0);
CHECK (reafference_loop_closes(subgraph_to_replace));

-- 3. Delete old subgraph (maintain referential integrity)
DELETE FROM transitions WHERE source_id IN (SELECT id FROM subgraph_to_replace);
DELETE FROM harmonies WHERE position = 'measure_4';

-- 4. Insert new subgraph with new relationships
INSERT INTO harmonies (position, degree, ...)
VALUES ('measure_4', 'V', ...);

-- 5. Re-establish transitions
INSERT INTO transitions (source_id, target_id, ...)
VALUES (...);

-- 6. Verify conservation
ASSERT SUM(trit) = 0 (mod 3);

COMMIT;

Algorithm:

Input: Abstract move M, ACSet world W_A
Output: Modified ACSet world W_B or FAIL

1. Parse M as DisCoPy diagram
2. Identify subgraph(s) targeted for rewriting
3. Validate GF(3) balance in target subgraph
4. Validate reafference loops close in target
5. Execute graph grafting (DuckDB transactions)
6. Verify output ACSet W_B satisfies all constraints
7. If valid: return W_B
   Else: ROLLBACK and signal constraint violation

Constraint Preservation:

• GF(3) balance: Σ trits ≡ 0 (mod 3)
• Reafference closure: action→prediction→sensation→match cycle completes
• Morphism coherence: All diagram compositions commute
• Temporal versioning: valid_from/valid_to intervals maintained

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Component C (-1): semantic-grafting-verifier (COPLAY)

Role: Verify that grafted mutations are semantically sound and exploit arbitrage

What it does:

• Checks that mutated world satisfies predicate logic (Dialectica)
• Verifies "moral correctness" via post-rigorous reasoning
• Measures transformation arbitrage — value gained from move
• Blocks invalid moves; promotes high-value moves

Core Operations:

-- Semantic verification in Lean/Rzk
theorem world_transformation_valid (M : DisCopy.Morphism) (W_A W_B : ACSet) :
  ACSet.satisfies W_A (Constraints.all)  →
  (W_B = apply_move M W_A) →
  ACSet.satisfies W_B (Constraints.all) := by
  -- 1. Structural verification (syntactic)
  have h1 : Morphism.commutes M := by (graph coherence)

  -- 2. Constraint verification (semantic)
  have h2 : GF3.balanced W_B := by (sum of trits)
  have h3 : Reafference.closes W_B := by (loop closure)

  -- 3. Arbitrage calculation
  let arb := epistemic_arbitrage W_A W_B M
  have h4 : arb > 0 := by (prediction error improvement)

  -- 4. Conclude
  exact And.intro h1 (And.intro h2 (And.intro h3 h4))

Verification Strategy (Post-Rigorous):

Level 1 (Syntactic): ✓ Diagram commutes (mechanical check)
Level 2 (Structural): ✓ ACSet constraints preserved (SQL constraints)
Level 3 (Semantic): ✓ Predicates satisfied (Dialectica interpretation)
Level 4 (Epistemic): ✓ Arbitrage positive (prediction error reduced)

If all 4 pass → Move is valid
If any fail → Move is blocked (explain why)

Arbitrage Measurement:

arb(M: W_A → W_B) = (H[W_A] - H[W_B]) × P[W_B correct]

where:
  H[W] = prediction entropy of world W
  P[W correct] = probability predictions for W are accurate

High arb → Move is valuable (reduces uncertainty significantly)
Low arb → Move wastes computational effort
Negative arb → Move increases dissonance (blocked)

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The 4-Component Cycle (Implementation)

Stage 1: Discover Moves (Triad 4A)

Current world W_A
  ↓ [DisCoPy operadic generation]
Abstract moves M₁, M₂, M₃, ... (ranked by arbitrage potential)

Stage 2: Validate Moves (Triad 4B + 4C)

For each move M:
  1. Compute mutated world W_B = apply_move(M, W_A)
  2. Check constraints (GF(3), reafference, coherence)
  3. Measure arbitrage(M)
  4. Filter: keep only valid + high-arbitrage moves

Stage 3: Execute Move (Triad 4B)

Selected move M_best
  ↓ [DuckDB graph grafting transaction]
World W_A → W_B (committed)
  ↓ [Reafference loop]
Agent observes W_B, updates predictions

Stage 4: Learn & Adapt (Triad 3 feedback)

W_B observed
  ↓ [Compare prediction to observation]
Prediction error = ||predicted_W_B - observed_W_B||
  ↓ [Update epistemic model]
Model improves for next move selection

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Example: Music-Topos World Rewriting

Scenario

Current world W_A:
  ├─ Key: C major
  ├─ Measure 4: ii-V-I cadence (classic)
  └─ Harmonic entropy: H=0.78 (fairly predictable)

Goal: Generate interesting harmonic variation that:
  ✓ Preserves key (C major)
  ✓ Maintains voice leading rules
  ✓ Reduces redundancy (H > 0.85)
  ✓ Remains "morally correct" (post-rigorous)

Execution

Step 1: Generate moves (Component A)

Input: Measure 4 context (ii chord, must resolve)

Moves generated:
  M₁: ii → V → I (original, H_change = -0.01, arb = low)
  M₂: ii → IV → I (substitution, H_change = +0.12, arb = medium)
  M₃: ii → V7♯11 → I (chromatic, H_change = +0.25, arb = high)
  M₄: ii → ♭VI → I (modal interchange, H_change = +0.18, arb = high)

Sorted by arbitrage: [M₃, M₄, M₂, M₁]

Step 2: Apply move M₃ (Component B)

-- Graft chromatic V7♯11 into measure 4
UPDATE harmonies
SET chord_root = 'G', chord_type = 'V7♯11'
WHERE position = 'measure_4_beat_2';

UPDATE transitions
SET voice_leading_quality = 'chromatic'
WHERE source_id = (SELECT id FROM harmonies WHERE position = 'measure_4_beat_1')
  AND target_id = (SELECT id FROM harmonies WHERE position = 'measure_4_beat_2');

Step 3: Verify (Component C)

Checks:
  ✓ GF(3): 0 + 1 + (-1) = 0 (balanced)
  ✓ Reafference: Prediction loop closes (all voices resolve)
  ✓ Constraints: ii-V7♯11-I is valid in modal theory
  ✓ Arbitrage: H increases from 0.78 → 0.89 (++value)

Result: MOVE ACCEPTED

Step 4: Observe & Learn

Agent plays W_B (with M₃ applied)
Prediction: V7♯11 resolves with chromatic line motion
Observation: (matches prediction ✓)
Prediction error: δ = 0.02 (very low)

Update: "Chromatic V7♯11 resolutions work well"
         (increase weight for future move selection)

---

Integration with Existing Triads

Input from Triad 3 (World-Hopping)

Triad 3 (world-hopping + epistemic-arbitrage) provides:
  ├─ Current best world W_A (found via hopping)
  ├─ Ranking of reachable worlds by arbitrage
  └─ Prediction errors (where model is weak)

→ This tells Triad 4 "which worlds to rewrite toward"

Output to Triad 3 (World-Hopping)

Triad 4 (categorical rewriting) produces:
  ├─ Modified ACSet worlds (grafted structures)
  ├─ Mutation arbitrage scores (value per move)
  └─ Constraint violation reports (invalid moves)

→ This tells Triad 3 "these moves are executable"

GF(3) Conservation

Triads work as balanced triple:

Triad 2 (Glass Bead Game): PLUS (+1) Generate conceptual moves
Triad 3 (World-Hopping):   ERGODIC (0) Coordinate world exploration
Triad 4 (Rewriting):       MINUS (-1) Validate & constrain mutations

Sum: +1 + 0 + (-1) = 0 ✓ BALANCED

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Implementation Roadmap

Phase 1: DisCoPy Integration (1-2 weeks)

✓ Install DisCoPy
✓ Define world-transformation language (ops, boxes, wiring)
✓ Implement operadic composition
✓ Test on small examples (3-5 node ACSet mutations)

Phase 2: Graph Grafting (2-3 weeks)

✓ Map DisCoPy diagrams → DuckDB mutations
✓ Implement constraint checking (GF(3), reafference)
✓ Build graph grafting operations (delete, insert, rewrite)
✓ Test on music-topos schema (measure rewriting)

Phase 3: Semantic Verification (2-3 weeks)

✓ Implement Dialectica interpreter (post-rigorous verification)
✓ Build epistemic arbitrage calculator
✓ Deploy move ranking/filtering
✓ Integrate with Triad 3 feedback

Phase 4: Integration & Deployment (1-2 weeks)

✓ Connect all 4 components in feedback loop
✓ Test end-to-end: discover → validate → execute → learn
✓ Benchmark (move generation rate, validation latency)
✓ Documentation and examples

Total Timeline: 6-10 weeks to production

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Code Examples

DisCoPy: String Diagram for World Move

from discopy import Ty, Box, Diagram, Functor
from discopy.monoidal import Ty as MonoidalTy

# Define world types
WorldState = Ty("World")
Constraint = Ty("Constraint")

# Define a move: transition from one harmonic context to another
harmonic_move = Box(
    "ii → V7♯11",           # move name
    WorldState @ Constraint,  # input: (current world, constraints)
    WorldState @ Constraint   # output: (new world, constraints)
)

# Compose with another move
resolution = Box(
    "V7♯11 → I",
    WorldState,
    WorldState
)

# Combine moves
compound_move = harmonic_move >> resolution

# Generate alternative by operadic composition
alternative = Box("modal_interchange", WorldState, WorldState)
move_family = harmonic_move @ alternative  # parallel composition

DuckDB: Graph Grafting Transaction

-- Wrap mutations in transaction for atomicity
BEGIN;

  -- 1. Identify target subgraph
  CREATE TEMP TABLE target_nodes AS
  SELECT id FROM harmonies
  WHERE measure = 4 AND beat = 2;

  -- 2. Validate before mutation
  ASSERT (SELECT SUM(gf3_trit) FROM target_nodes) = 0;
  ASSERT (SELECT reafference_loops_close(id) FROM target_nodes);

  -- 3. Remove old edges
  DELETE FROM transitions
  WHERE source_id IN (SELECT id FROM target_nodes);

  -- 4. Update nodes
  UPDATE harmonies SET chord = 'V7♯11'
  WHERE id IN (SELECT id FROM target_nodes);

  -- 5. Add new edges
  INSERT INTO transitions (source_id, target_id, voice_leading)
  VALUES ((SELECT id FROM harmonies WHERE measure=4 AND beat=2),
          (SELECT id FROM harmonies WHERE measure=4 AND beat=3),
          'chromatic_voice_leading');

  -- 6. Verify post-condition
  ASSERT (SELECT SUM(gf3_trit) FROM harmonies) = 0;

COMMIT;

Lean: Semantic Verification

-- Verify move is valid
theorem harmonic_move_valid :
  ∀ (W : MusicWorld) (M : DisCopy.Move),
    GF3.conserved W →
    valid_move M W →
    GF3.conserved (apply_move M W) := by
  intro W M h_conserved h_valid
  -- 1. Move preserves structure
  have h1 : morphism_coherent M := by exact h_valid.coherent
  -- 2. Constraints preserved
  have h2 : GF3.conserved (apply_move M W) := by
    apply_rules [GF3.addition_invariant, morphism_preserves_structure]
    exact h1
  exact h2

---

Success Criteria

Correctness

• All moves respect GF(3) conservation
• All moves preserve reafference closure
• All moves pass post-rigorous semantic check
• No constraint violations in production

Performance

• Move generation: < 100ms for small worlds (< 1000 nodes)
• Move validation: < 50ms per move
• Graph grafting: < 200ms transaction time
• Full cycle (discover → validate → execute): < 500ms

Usability

• DisCoPy diagrams match intuitive move descriptions
• Error messages explain constraint violations
• Arbitrage scores rank moves by value
• Example library covers major patterns

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Related Skills

• glass-bead-game (Triad 2A): Conceptual move generation
• self-validation-loop (Triad 2B): Move validation
• acsets (Triad 2C): ACSet storage
• world-hopping (Triad 3A): World navigation
• epistemic-arbitrage (Triad 3C): Arbitrage measurement
• discopy (foundation): String diagram computation
• categorical-rewriting (this skill): Move execution

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See Also

• DiscoHy - Streaming computation framework (used for move generation)
• Catlab.jl - ACSet computation (used for validation)
• DisCoPy (Baez, Ong) - String diagram library
• Operadic composition patterns
• Graph rewriting literature

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Status: Design complete, ready for implementation Estimated effort: 6-10 weeks Impact: Completes 4-tier world transformation architecture

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Graph Theory

• networkx [○] via bicomodule
  • Universal graph hub

Bibliography References

• category-theory: 139 citations in bib.duckdb

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: 0 (ERGODIC)
Home: Prof
Poly Op: ⊗
Kan Role: Adj
Color: #26D826

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.