Lemma Dependency DAG & Soundness Engine

Click and drag nodes to inspect premise flow. Direct deduction edges point to derivations.
Soundness Status SOUND
Acyclicity / Loops 0 Cycles
Premise Gaps 0 Missing
Proof Depth (Levels) 4 Layers
Verification Passed: Formal DAG topological sort confirmed. All premises trace back to foundational axioms without circular reasoning or dangling references.
Axiom / Hyp
Intermediate Lemma
Final Q.E.D.
Circular / Gap
Select any node to inspect derivation premises.

Autonomous Theorem Proving & The Verification Bottleneck

Following OpenAI's release of manuscripts addressing hundreds of open mathematical problems, the primary challenge shifting the mathematics community is not merely generating speculative conjectures, but rigorously auditing dependency structures. Machine-generated papers frequently feature intricate chains of deduction spanning dozens of pages. While individual local inferences often look plausible, proofs frequently suffer from subtle non-constructive circularities, undeclared axiomatic shifts, or dangling premises.

Why Formal Directed Acyclic Graphs (DAGs) Matter in Proof Auditing

A mathematically sound proof must form a strict Directed Acyclic Graph (DAG) rooted in accepted axioms, definitions, or proven literature lemmas. Each successive step Sk requires verified premises {P1, ..., Pm} such that:

  • No Circular Dependencies (Acyclicity): If Li requires Lj, then Lj cannot directly or transitively depend on Li. The existence of any cycle invalidates the entire deductive chain.
  • Exhaustive Premise Grounding: Every premise referenced must either be an explicitly declared hypothesis, a known standard axiom (such as ZFC), or a prior derived lemma with its own valid sub-DAG.
  • Inference Rule Validity: Every transition must explicitly state its deductive mechanism—whether through induction, bounded asymptotic squeeze, or proof by contradiction with explicitly checked hypothesis negations.

How This Verifier Audits Manuscript Claims

This workbench executes Tarjan’s strongly connected components algorithm and topological sorting on the uploaded or authored lemma graph. It instantly isolates:

  • Topological Order: Validates that an objective sequence exists to read and verify lemmas sequentially without forward-referencing assumptions.
  • Root Axiom Tracing: Confirms the reachability of the final Q.E.D. conclusion back to explicit starting assumptions.
  • Orphaned & Dangling Steps: Flags nodes that reference nonexistent premise IDs or lemmas whose output is unused in establishing the target theorem.