Update bounded_chaos.md

bounded_chaos.md

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+### **Bounded Chaos (BC) Framework**  
+**First-Principles Specification**  
+
+---
+
+### **1. Root Definition**  
+A system is **Bounded Chaos** if and only if:  
+```  
+∃ U₁₆, φ, ε, K :  
+  ∀ S ∈ System,  
+    Constructible(S, U₁₆) ∧  
+    Entropy_Bounded(S, ε) ∧  
+    State_Compressible(S, φ, K)  
+```  
+Where:  
+- **`U₁₆`**: 16-state universal constructor  
+- **`φ`**: Golden ratio (1.618...)  
+- **`ε`**: Maximum entropy delta per operation (0.01)  
+- **`K`**: Kolmogorov bound (11φ·log|S|)  
+
+---
+
+### **2. Foundational Axioms**  
+
+#### **2.1 Construction Axiom**  
+*"All valid states derive from U₁₆"*  
+```  
+Constructible(S, U₁₆) ≡ ∃ t ∈ ℕ : S = U₁₆^t(∅)  
+```  
+**Requirements**:  
+- U₁₆ implementation must be hash-locked (SHA-256)  
+- All state transitions must prove U₁₆ ancestry  
+
+#### **2.2 Entropy Axiom**  
+*"No operation exceeds ε energy cost"*  
+```  
+Entropy_Bounded(S, ε) ≡ ΔS(S → S') ≤ ε  
+```  
+**Enforcement**:  
+- Hardware: TPM-measured energy bounds  
+- Software: Reject transitions where ∑ΔS > ε  
+
+#### **2.3 Compression Axiom**  
+*"States obey φ-scaled Kolmogorov bounds"*  
+```  
+State_Compressible(S, φ, K) ≡ |K(S)| ≤ 11φ·log(|S|)  
+```  
+**Verification**:  
+- Compile-time proof via Lean/Coq  
+- Runtime check: Reject states exceeding K bits  
+
+---
+
+### **3. Cryptographic Primitives**  
+
+| Primitive | Purpose | Invariant |  
+|-----------|---------|-----------|  
+| SHA-256   | Artifact locking | H(S) = H(S') ⇒ S = S' |  
+| Ed25519   | Signature | Verify(pk, msg, sig) ∈ {0,1} |  
+| CUE       | Validation | Schema(S) ⇒ S ⊨ Axioms |  
+
+**Rules**:  
+1. All system states must include `H(U₁₆ || previous_state)`  
+2. All transitions must be Ed25519-signed  
+3. All configurations must validate against CUE schema  
+
+---
+
+### **4. Enforcement Mechanisms**  
+
+#### **4.1 Proof Pipeline**  
+```mermaid  
+graph TB  
+    A[YAML] -->|CUE| B[Generate]  
+    B --> C[Lean: U₁₆ proofs]  
+    B --> D[Coq: φ proofs]  
+    C --> E[Artifacts]  
+    D --> E  
+    E -->|Hash-Lock| A  
+```  
+
+#### **4.2 Runtime Checks**  
+1. **Energy Monitor**:  
+   ```python  
+   def execute(op):  
+       assert ΔS(op) ≤ ε - global_ΔS  
+       global_ΔS += ΔS(op)  
+   ```  
+2. **State Validation**:  
+   ```rust  
+   fn validate(S: State) -> bool {  
+       S.verify_signature() &&  
+       S.kolmogorov() ≤ 11φ * log(S.size()) &&  
+       S.ancestry.proves(U₁₆)  
+   }  
+   ```  
+
+---
+
+### **5. Irreducible Components**  
+
+| Component | Purpose | Replaceable |  
+|-----------|---------|-------------|  
+| U₁₆       | Construction | No |  
+| φ         | Scaling | No |  
+| ε         | Energy bound | No |  
+| SHA-256   | Locking | Only with stronger hash |  
+| Ed25519   | Signing | Only with stronger sig |  
+
+**Implications**:  
+- Changing any irreducible component requires proving:  
+  ```  
+  ∀ S, new_component(S) ⇒ old_component(S)  
+  ```  
+
+---
+
+### **6. Attack Surface**  
+
+| Vector | Defense | Root Principle |  
+|--------|---------|----------------|  
+| State explosion | K-bound | Compression Axiom |  
+| Energy theft | ΔS ≤ ε | Entropy Axiom |  
+| Signature forgery | Ed25519 | Crypto Primitive |  
+| U₁₆ tampering | Hash-lock | Construction Axiom |  
+
+---
+
+### **7. Minimal Implementation**  
+
+```rust  
+struct BC_Core {  
+    state: Vec<u8>,  
+    prev_hash: [u8; 32],  
+    energy_used: f64,  
+    sig: Ed25519Sig,  
+}  
+
+impl BC_Core {  
+    fn execute(&mut self, op: Operation) {  
+        let ΔS = op.calculate_energy();  
+        assert!(self.energy_used + ΔS <= 0.01);  
+        assert!(op.kolmogorov() <= 11φ * log(self.state.len()));  
+        self.state = U₁₆(op);  
+        self.energy_used += ΔS;  
+        self.sig = sign(op);  
+    }  
+}  
+```  
+
+---
+
+### **8. Recursive Validation**  
+To verify a BC system:  
+1. Check `H(U₁₆)` matches reference  
+2. Validate all `ΔS ≤ ε`  
+3. Verify all `K(S) ≤ 11φ·log|S|`  
+4. Confirm Ed25519 signatures  
+
+```  
+[STATUS: FIRST-PRINCIPLES DOCUMENTATION COMPLETE]  
+```  
+
+This document contains:  
+- 3 axioms  
+- 3 primitives  
+- 4 enforcement mechanisms  
+- 0 examples  
+- 0 implementation details beyond minimal spec  
+
+All higher-level constructs must reduce to these elements.
+
 ### **BDC Framework: Unified Documentation Snapshot**  
 *(All components frozen at this milestone with cryptographic checksums)*