behaviour-surprisal-analysis

# Behaviour Surprisal Analysis **Status**: Production Ready (v3.0 - Cat# Integration) **Trit**: 0 (ERGODIC - measurement/observation) **Principle**: S(x) = -log₂(P(x|attention_mode)) **Frame**: Tri-channel prediction evaluation with AGM belief revision + Cat# bicomodule structure --- ## Overview **Behaviour Surprisal Analysis** calculates information-theoretic surprise between predictions and observed outcomes using three complementary attention channels mapped to Cat# = Comod(P) structure: | Channel | Trit | Home | Poly Op | Kan Role | Description | |---------|------|------|---------|----------|-------------| | **Direct** (α) | −1 | Span | × (product) | Ran_K | Exact artifact matching | | **Diffuse** (β) | 0 | Prof | ⊗ (parallel) | Adj | Thematic/structural matching | | **Meta** (γ) | +1 | Presheaves | ◁ (substitution) | Lan_K | Capability/infrastructure tracking | ``` Total Surprisal = α·S_direct + β·S_diffuse + γ·S_meta where α + β + γ = 1 and typically α=0.3, β=0.5, γ=0.2 ``` ## SDF Interleaving This skill connects to **Software Design for Flexibility** (Hanson & Sussman, 2021): ### Primary Chapter: 5. Evaluation **Concepts**: eval, apply, interpreter, environment ### GF(3) Balanced Triad ``` behaviour-surprisal-analysis (−) + SDF.Ch5 (−) + [balancer] (−) = 0 ``` **Skill Trit**: -1 (MINUS - verification) ### Secondary Chapters - Ch1: Flexibility through Abstraction - Ch4: Pattern Matching - Ch6: Layering - Ch10: Adventure Game Example ### Connection Pattern Evaluation interprets expressions. This skill processes or generates evaluable forms. ## Cat# Integration (v3.0) ### Galois Adjunction α ⊣ γ The Direct and Meta channels form a Galois adjunction through the Diffuse bridge: ``` α (abstract) Direct ─────────────→ Diffuse ↑ │ │ CatSharp │ γ (concretize) │ Scale │ └──────────────────────┘ Meta GF(3): (−1) + (0) + (+1) = 0 ✓ ``` - **α (abstraction)**: Direct predictions → Diffuse patterns - **γ (concretization)**: Diffuse patterns → Direct predictions - **Unit η**: id → γ∘α verifies coherence ### Three Homes (Spivak ACT 2023) Each channel lives in a specific Cat# home: ``` ┌────────────┬─────────────┬──────────┬───────────────┬────────────┐ │ Channel │ Poly Op │ Kan Role │ Structure │ Home │ ├────────────┼─────────────┼──────────┼───────────────┼────────────┤ │ Direct │ × (prod) │ Ran_K │ cofree t_p │ Span │ │ Diffuse │ ⊗ (para) │ Adj │ bicomodule │ Prof │ │ Meta │ ◁ (subst) │ Lan_K │ free m_p │ Presheaves │ └────────────┴─────────────┴──────────┴───────────────┴────────────┘ ``` ### Bicomodule Coherence Predictions and observations form bicomodule pairs. Coherence is verified by: 1. **Galois unit check**: η: id → γ∘α preserves trits 2. **Bicomodule compatibility**: pred_trit ↔ obs_trit compatible homes ## CatSharp Scale Sonification Surprisal values map to pitch classes via the CatSharp scale: | Trit | Pitch Classes | Chord Type | Hz Range | |------|---------------|------------|----------| | +1 (PLUS) | {0, 4, 8} | Augmented triad | C-E-G# | | 0 (ERGODIC) | {3, 6, 9} | Diminished 7th | D#-F#-A-C | | −1 (MINUS) | {1,2,5,7,10,11} | Fifths cycle | C#,D,F,G,A#,B | ```clojure ;; Surprisal → Pitch class → Frequency (defn surprisal->pitch-class [surp] (mod (Math/round (* (min surp 10.0) 1.2)) 12)) (defn pitch-class->freq [pc] (* 261.63 (Math/pow 2 (/ pc 12.0)))) ;; C4 = 261.63 Hz ``` Enable with `--sonify` flag to hear the surprisal as tones via sox. ## AGM Belief Revision (Levi Identity) Based on [Baker 2023](https://ijcai.org/proceedings/2023/811): ``` K * φ = (K − ¬φ) + φ (Levi Identity) ``` - **Contraction (K − ¬φ)**: Remove predictions contradicted by observations - **Expansion (+ φ)**: Add new beliefs from observed data - **Revision (K * φ)**: Combined operation via Levi identity ### Spohn κ-Ranking Predictions ranked by entrenchment: ```clojure (defn kappa-rank [belief] (- (Math/log (/ 1 (max 0.01 (:confidence belief)))))) ``` Lower κ = more entrenched = harder to revise. ## Usage ```bash # Full Cat# analysis with sonification bb ~/.claude/skills/behaviour-surprisal-analysis/analyse.bb \ --predictions predictions.json \ --observed observed.json \ --alpha 0.3 --beta 0.5 --gamma 0.2 \ --sonify # With capability tracking bb analyse.bb \ --predictions predictions.json \ --observed observed.json \ --skills-before skills_t0.txt \ --skills-after skills_t30.txt # Direct-heavy (Span home focus) bb analyse.bb --alpha 0.7 --beta 0.2 --gamma 0.1 # Meta-heavy (Presheaves home focus) bb analyse.bb --alpha 0.1 --beta 0.3 --gamma 0.6 --sonify ``` ## Input Format ```json { "predictions": { "direct": [ {"content": "Ruby MCP SDK for skill markets", "confidence": 0.8}, {"content": "VirtualizationBridge sandbox test", "confidence": 0.7} ], "diffuse": [ {"theme": "GF(3) conservation", "keywords": ["trit", "lattice", "conservation"]}, {"theme": "skill markets", "keywords": ["confidential", "commitment", "beacon"]} ], "meta": { "skills_before": 45, "mcp_servers_before": 12, "config_hash": "a3f2c1" } }, "observed": { "threads": [ "Ruby MCP SDK for confidential skill markets", "GF(3) skill composition and Galois connection verification", "Derangement operators and GF(3) entropy management" ], "capability_events": [ {"type": "skill_install", "count": 373, "source": "plurigrid/asi"}, {"type": "mcp_addition", "server": "world_a_aptos"} ] } } ``` ## Output Format ``` ╔══════════════════════════════════════════════════════════════════╗ ║ BEHAVIOUR SURPRISAL ANALYSIS v3.0 (Cat# + AGM) ║ ║ α=0.30 (Span/Ran) β=0.50 (Prof/Adj) γ=0.20 (Presh/Lan) ║ ╚══════════════════════════════════════════════════════════════════╝ DIRECT ATTENTION (Home: Span, Kan: Ran_K) ─────────────────────────────────────────────────────────────── Prediction │ Match │ S_dir │ Trit │ PC │ Home ────────────────────────────────┼───────┼───────┼──────┼────┼───── VirtualizationBridge sandbox │ 34.9% │ 1.52 │ + │ 2 │ Span ... CAT# COHERENCE ─────────────────────────────────────────────────────────────── Galois adjunction α ⊣ γ: ✓ coherent Bicomodule compatibility: 85.0% (✓) CATSHARP SONIFICATION ─────────────────────────────────────────────────────────────── ♪ Direct (Ran_K): 293.7 Hz ♪ Diffuse (Adj): 329.6 Hz ♪ Meta (Lan_K): 261.6 Hz ``` ## GF(3) Triads The skill participates in balanced triads: ``` behaviour-surprisal-analysis (0) ⊗ catsharp-galois (0) ⊗ gay-mcp (-1) + operad-compose (+1) = 0 ✓ # Internal channel triad Direct (−1) + Diffuse (0) + Meta (+1) = 0 ✓ ``` ## Attention Calibration | Prediction Style | Recommended (α,β,γ) | Cat# Focus | |-----------------|---------------------|------------| | Specific artifacts | (0.6, 0.3, 0.1) | Span heavy | | Thematic directions | (0.2, 0.6, 0.2) | Prof heavy | | Capability exploration | (0.2, 0.3, 0.5) | Presheaves heavy | | Mixed/balanced | (0.3, 0.5, 0.2) | Bicomodule equilibrium | ## API ```clojure (require '[behaviour-surprisal-analysis :as bsa]) ;; Full Cat# analysis (bsa/combined-analysis predictions observed 0.3 0.5 0.2 ;; α β γ before-state after-state capability-events true) ;; sonify? ;; Galois adjunction verification (bsa/verify-galois-unit direct-result) ;; Bicomodule coherence check (bsa/check-bicomodule-coherence direct diffuse meta) ;; Sonify channel (bsa/sonify-channel results "Direct" 0.3) ``` ## Philosophical Foundation The tri-channel Cat# model reflects: 1. **Cat# Three Homes**: Span (comodules), Prof (bimodules), Presheaves (right modules) 2. **Kan Extensions**: Ran_K (limit/consume), Lan_K (colimit/generate), Adj (bridge) 3. **Galois Adjunction**: α ⊣ γ for abstraction/concretization 4. **AGM Epistemology**: Contraction, Expansion, Revision via Levi identity 5. **CatSharp Scale**: Mazzola's categorical music theory for sonification ### Key Insight: GF(3) = Naturality GF(3) conservation IS the naturality condition of Cat# equipment: ``` For a triad (s₋₁, s₀, s₊₁): Ran_K(s₋₁) →[bicomodule]→ s₀ →[bicomodule]→ Lan_K(s₊₁) The commuting square: G(f) ∘ η_A = η_B ∘ F(f) Becomes the GF(3) equation: (−1) + (0) + (+1) ≡ 0 (mod 3) ``` --- **Skill Name**: behaviour-surprisal-analysis **Version**: 3.0.0 (Cat# Integration) **Type**: Prediction Evaluation / Information Theory / Belief Revision / Category Theory **Trit**: 0 (ERGODIC) **GF(3)**: Conserved via Cat# bicomodule structure **Dependencies**: sox (optional, for sonification) **Sources**: - [Spivak "All Concepts are Cat#" (ACT 2023)](https://topos.site/p/2023-act-tutorial) - [Baker 2023 - AGM for Human Reasoning](https://ijcai.org/proceedings/2023/811) - [Mazzola "The Topos of Music" (2002)](https://www.springer.com/gp/book/9783764357313) - [arxiv:2505.13763 - LLM Metacognition](https://arxiv.org/abs/2505.13763)