94 lines
3.5 KiB
Markdown
94 lines
3.5 KiB
Markdown
# Bounded Chaos v0.0
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*Five rules, zero ceremony.*
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```python
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is_valid(S):
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S.nodes in {0,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987} && # 𝓕-bound
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S.split in {1024//φ, 64//φ} && # φ-proportional
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abs(ΔS)/S ≤ 0.01 && # ε-stable
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sha256(S) == S.hash && # SHA-256-ID
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ed25519_verify(S.sig, S.hash) # Σ-signed
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```
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**φ = 1.618… ε = 0.01 K = 1024 maxT = 11**
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A system is valid **iff** it satisfies the five conditions above.
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PhD Panel Cross-Examination
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(Chair: Prof. Emeritus R. Gödel)
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---
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**Chair:**
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Your dissertation claims to give *“a mathematically type-safe, self-validating framework for bounded chaos.”*
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We will test that claim with five precise challenges. You have 30 seconds each.
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---
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### 1. **Completeness of the Five-Rule Axiom Set**
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*Prof. Turing:*
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> You list five rules (Fibonacci-bound, φ-proportional, ε-stable, SHA-256-ID, Σ-signed).
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> Show that **no additional axiom is necessary** to decide validity for *all* finite states, or give a counter-example.
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---
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### 2. **Ambiguity of φ in Floating-Point**
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*Prof. Priest:*
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> Your φ is irrational. IEEE-754 doubles truncate at 53 bits.
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> Prove that **rounding error never violates** the φ-proportionality rule for any admissible K ≤ 1024.
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> (Hint: ε = 0.01 may not cover φ-rounding drift.)
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---
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### 3. **Collision Resistance of SHA-256**
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*Prof. Rogaway:*
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> Your fourth axiom assumes SHA-256 is collision-free.
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> 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.
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---
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### 4. **Prime vs. Fibonacci Overlap**
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*Prof. Erdős:*
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> The intersection ℙ ∩ 𝓕 below 1024 is {2,3,5,13,89,233}.
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> Demonstrate that **no allocation path** can deadlock when the recursion guard `¬#isPrime(size)` is reached **only** at these six values.
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---
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### 5. **Self-Reference Paradox**
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*Prof. Tarski:*
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> The `README.md` asserts its own validity via
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> `specValid ⇔ CUE_vet(spec) ∧ Σ.sign(spec) ∧ H(spec) ∈ MerkleRoot`.
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> Prove this statement is **arithmetically sound** within your axioms, or exhibit a Gödel sentence that escapes the framework.
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---
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**Panel Expectation:**
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Respond to **each** challenge with a **theorem + proof sketch** or concede the limitation.
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Clock starts… now.
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---
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// ---------- RECURSIVE REPLY ----------
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// Five answers, each a first-principle entailment.
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1. Completeness
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is_valid(S) ∧ S.nodes∈𝓕 ⇒ {0,…,987} finite & enumerable ⇒ every S decidable by exhaustive check against the five rules; no sixth axiom required.
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2. φ Rounding
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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.
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3. SHA-256 Soundness
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∀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).
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4. Prime-Fib Overlap
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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.
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5. Self-Reference
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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.
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// ---------- Q.E.D. ---------- |