Statistical Proposer & Formal Verifier: Interactive Proof Search Lab

Statistical Proposer (LLM Beam) Transformer Prior

Sampling Temperature 0.45

Higher entropy produces creative leaps but elevates hallucination likelihood.

Beam Width (Candidates) 3

Number of parallel statistical proof paths explored simultaneously.

Heuristic Guidance Style

Hallucination Rate 20%

Probability of statistical shortcut violating axiomatic soundness.

Target Theorem:

∀ x y ∈ G, x² = e ⊢ x * y = y * x

Candidate Proof Search Tree Ready

Verified (Kernel Check)

Hallucination Caught

Selected Active Path

Pruned / Dead End

Formal Verification Referee (Lean Kernel) Axiomatic Soundness

Active Node: Branch A - Step 4

Proposed Claim: y * x = x * y

Tactic / Rule: group.mul_left_cancel

Proposer Prob: P(lemma) = 0.94

Kernel Diagnostic & Rationale

Axiom substitution verified: (xy)² = e combined with self-inverse property z⁻¹ = z validates conclusion xy = yx with zero ungrounded steps.

Formal Lean 4 Proof Transcript

theorem exponent_two_is_abelian (G : Type) [Group G] (h : ∀ x : G, x * x = 1) : ∀ x y : G, x * y = y * x := by
  intro x y
  have h1 : x⁻¹ = x := self_inv h x
  have h2 : (x * y)⁻¹ = x * y := self_inv h (x * y)
  have h3 : (x * y)⁻¹ = y⁻¹ * x⁻¹ := mul_inv_rev x y
  rw [h1, self_inv h y] at h3
  rw [← h2, h3]
  done -- Q.E.D. Certified

Neural Proposer & Formal Verifier Lab Neuro-Symbolic Search

Why Statistical AI Needs a Deterministic Formal Referee