--- name: behaviour-surprisal-analysis-773039c936e0 description: Behaviour Surprisal Analysis version: 1.0.0 --- # 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 ``` ## 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) --- Source attribution: plurigrid Original project: https://github.com/plurigrid/asi/blob/293ce9f76e3abcd5ba18db284688bc0d9ac6691d/plugins/asi/skills/behaviour-surprisal-analysis/SKILL.md License: MIT License information: https://opensource.org/license/mit Original upstream instructions; downloaded from Dave AI. ## Upstream license notice MIT License Copyright (c) 2025 SkillCreator.ai Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. # Dave AI resource metadata Creator / publisher: plurigrid Original project: https://github.com/plurigrid/asi/blob/293ce9f76e3abcd5ba18db284688bc0d9ac6691d/plugins/asi/skills/behaviour-surprisal-analysis/SKILL.md Attribution basis: Source repository record Fields of work: Data & Analytics, Education & Research Roles: Data Analyst, Data Scientist, Researcher Specialties: API Tools: Sox Country applicability: Country not specified Classification basis: Source catalog labels Instruction SHA-256: 97f121f4acf4dfa16554ac4a8e0cb8bc4321b3e20ea74d15f2a24f635ccb4b83