# Bounded Chaos v0.0  
*Five rules, zero ceremony.*

```python
is_valid(S):
    S.nodes in {0,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987} &&  # 𝓕-bound
    S.split in {1024//φ, 64//φ} &&                                   # φ-proportional
    abs(ΔS)/S ≤ 0.01 &&                                              # ε-stable
    sha256(S) == S.hash &&                                           # SHA-256-ID
    ed25519_verify(S.sig, S.hash)                                    # Σ-signed
```

**φ = 1.618… ε = 0.01 K = 1024 maxT = 11**

A system is valid **iff** it satisfies the five conditions above.

PhD Panel Cross-Examination  
(Chair: Prof. Emeritus R. Gödel)

---

**Chair:**  
Your dissertation claims to give *“a mathematically type-safe, self-validating framework for bounded chaos.”*  
We will test that claim with five precise challenges.  You have 30 seconds each.

---

### 1. **Completeness of the Five-Rule Axiom Set**  
*Prof. Turing:*  

> You list five rules (Fibonacci-bound, φ-proportional, ε-stable, SHA-256-ID, Σ-signed).  
> Show that **no additional axiom is necessary** to decide validity for *all* finite states, or give a counter-example.

---

### 2. **Ambiguity of φ in Floating-Point**  
*Prof. Priest:*  

> Your φ is irrational.  IEEE-754 doubles truncate at 53 bits.  
> Prove that **rounding error never violates** the φ-proportionality rule for any admissible K ≤ 1024.  
> (Hint: ε = 0.01 may not cover φ-rounding drift.)

---

### 3. **Collision Resistance of SHA-256**  
*Prof. Rogaway:*  

> Your fourth axiom assumes SHA-256 is collision-free.  
> Provide **a formal reduction** showing that any collision in SHA-256 would break the system’s soundness, *or* weaken the axiom to account for birthday-bound probabilities.

---

### 4. **Prime vs. Fibonacci Overlap**  
*Prof. Erdős:*  

> The intersection ℙ ∩ 𝓕 below 1024 is {2,3,5,13,89,233}.  
> Demonstrate that **no allocation path** can deadlock when the recursion guard `¬#isPrime(size)` is reached **only** at these six values.

---

### 5. **Self-Reference Paradox**  
*Prof. Tarski:*  

> The `README.md` asserts its own validity via  
> `specValid ⇔ CUE_vet(spec) ∧ Σ.sign(spec) ∧ H(spec) ∈ MerkleRoot`.  
> Prove this statement is **arithmetically sound** within your axioms, or exhibit a Gödel sentence that escapes the framework.

---

**Panel Expectation:**  
Respond to **each** challenge with a **theorem + proof sketch** or concede the limitation.  
Clock starts… now.

---

// ---------- RECURSIVE REPLY ----------
// Five answers, each a first-principle entailment.

1. Completeness  
   is_valid(S) ∧ S.nodes∈𝓕 ⇒ {0,…,987} finite & enumerable ⇒ every S decidable by exhaustive check against the five rules; no sixth axiom required.

2. φ Rounding  
   1024//φ = 632, 64//φ = 39; both integers. IEEE-53 error < 2⁻⁵³ < 10⁻¹⁵ < 0.01ε ⇒ rounding error swallowed by ε-bound; rule preserved.

3. SHA-256 Soundness  
   ∀S, T: H(S)=H(T) ⇒ S=T because Σ(sig_S,S.hash)≠Σ(sig_T,T.hash) unless S≡T; collision would break Σ’s EUF-CMA ⇒ soundness preserved or SHA-256 broken (assumed).

4. Prime-Fib Overlap  
   Deadlock requires size∉ℙ ∧ recurse >maxT.  sizes=ℙ∩𝓕={2,3,5,13,89,233} all ≤M; recursion stops at tier=11 or when size∈ℙ ⇒ no deadlock path.

5. Self-Reference  
   Let G = "¬is_valid(G)".  G.nodes ∉ 𝓕 ⇒ ¬is_valid(G) by rule 1, so G is invalid by construction; no contradiction inside 𝓕.  Framework remains arithmetically sound.

// ---------- Q.E.D. ----------