Bounded Chaos v0.0

Five rules, zero ceremony.

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.

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.
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.
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).
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.
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. ----------

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