Formal Verification of the Collatz State Transition Kernel

To prove code mathematically and logically, we do not run it, execute it, or pass test inputs to it. We apply Formal Verification: we express the program as a logical statement and prove that for all valid inputs, the invariant holds true.

Below is the formal, axiomatic mathematical proof of the Collatz state transition kernel (S_t+1 = f(S_t, input, parity)) using Hoare Logic and Inductive Proof.

1. Formal Specification & Definitions

Let the state space be defined as a pair S = (n, v), where:

• n ∈ ℕ₀ is the sequence index.
• v ∈ ℕ is the integer value representing the current sequence state.
• D ∈ { ODD, EVEN } is the decision domain.
• Mod2: ℕ → {0, 1} is a deterministic arithmetic function evaluating parity.

Let P(S_t, ctx) ∈ {0, 1} be the parity evaluation function, returning 1 if and only if the state value is odd, and 0 otherwise.

The state transition function f(S_t, input, ctx) is defined as:

f(S_t, input, ctx) =

  • (n_t + 1, 3v_t + 1)    if P(S_t, ctx) = 1 (ODD)

  • (n_t + 1, v_t / 2)    if P(S_t, ctx) = 0 (EVEN)

2. Invariant Claim to Prove

We claim that for any sequence of inputs of length k ≥ 0, the state machine satisfies three fundamental invariants:

1. State Monotonicity Invariant (I₁): n_t+1 > n_t. The sequence index never regresses.
2. Odd Growth Invariant (I₂): If P(S_t, ctx) = 1, then v_t+1 = 3v_t + 1. S_t+1 is strictly bound to the odd operation.
3. Even Decay Invariant (I₃): If P(S_t, ctx) = 0, then v_t+1 = v_t / 2. S_t+1 is strictly bound to the even operation.

3. Mathematical Proof by Induction

Base Case (t = 0): Genesis

• S₀ = (0, v₀), where v₀ = input₀.
• n₀ = 0 ∈ ℕ₀.
• v₀ is a valid integer > 0.
• Base invariants hold: n₀ = 0 ≥ 0, and lineage originates at genesis input input₀.

Inductive Hypothesis:

Assume for an arbitrary step t = k, the invariants I₁, I₂, I₃ hold for S_k = (n_k, v_k).

Inductive Step (t = k + 1):

Evaluate step transition S_k+1 = f(S_k, input_k+1, ctx_k+1).

Case A: Parity Evaluates to True (P(S_k, ctx_k+1) = 1)

1. By definition of f, n_k+1 = n_k + 1.
2. Since n_k ∈ ℕ₀, n_k + 1 > n_k ⇒ n_k+1 > n_k.
   • I₁ Holds: Sequence monotonically increments.
3. By definition of f, v_k+1 = 3v_k + 1.
   • I₂ Holds: Because arithmetic is deterministic, v_k+1 uniquely maps to v_k via the odd rule.
4. Conclusion for Case A: S_k+1 is committed and arithmetically chained to S_k.

Case B: Parity Evaluates to False (P(S_k, ctx_k+1) = 0)

1. By definition of f, n_k+1 = n_k + 1.
2. Since n_k ∈ ℕ₀, n_k + 1 > n_k ⇒ n_k+1 > n_k.
   • I₁ Holds: Sequence index monotonically increments.
3. By definition of f, v_k+1 = v_k / 2.
   • I₃ Holds: State value deterministically halves.
4. S_k+1 = (n_k + 1, v_k / 2).
   • Conclusion for Case B: System strictly adheres to the even decay rule; no unverified mathematical transition occurs.

By Mathematical Induction, the system invariants (I₁, I₂, I₃) hold for all t ∈ ℕ₀. ■

4. Hoare Logic Verification (Pre/Post-Conditions)

In program logic, we express the execution block using Hoare Triples: {P} C {Q}, where P is the precondition, C is the code command, and Q is the postcondition.

{ Precondition P: state == S_t AND valid_memory(state) }
1. decision, trace = evaluate_parity(state, context);
2. IF decision == ODD THEN
3. next_seq = state.sequence + 1;
4. next_v = (3 * state.v) + 1;
5. state = (next_seq, next_v);
6. ELSE
7. next_seq = state.sequence + 1;
8. next_v = state.v / 2;
9. state = (next_seq, next_v);
10. END IF
{ Postcondition Q:
(decision == ODD   ==> state.seq == S_t.seq + 1 AND state.v == (3 * S_t.v) + 1)
AND
(decision == EVEN  ==> state.seq == S_t.seq + 1 AND state.v == S_t.v / 2)
}

• Proof of Correctness: Lines 2–5 satisfy the left conjunct of Q. Lines 6–10 satisfy the right conjunct of Q. The code is formally sound under Hoare logic.

What This Proof Actually Guarantees

This mathematical proof proves the internal logic of the code itself:

• It proves the algorithm cannot produce an invalid state sequence.
• It proves that a parity failure can never accidentally advance the wrong branch (v_t+1 ≠ 3v_t+1 when P = 0).
• It proves that S_t is immutably linked to S_t-1 via strict, deterministic integer arithmetic.

It does not prove that the sequence will eventually reach 1, that the orbit will infinitely decay, or that the global Collatz loop is fully resolved—because those are macroscopic properties of the integers, not mathematical properties of the algorithmic state transition.

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