Mathematical Nonsense & Formal Logic Diagnostic Lab

Diagnostic Presets:

Expression Input & Axiomatic System Tokens: 5

Stage 1: Lexical & Type Domain Check PASSED

Symbols recognized; operands map to signature signatures.

Stage 2: Well-Formed Formula (WFF) Syntax PASSED

Valid CFG production tree constructed without dangling operators.

Stage 3: Axiom & Domain Consistency FAILED

Multiplicative inverse axiom requires divisor x ≠ 0.

Stage 4: Derivability & Truth Gate REJECTED

Derives ring collapse 0 = 1 in nontrivial field ℝ.

Formal Classification

Axiom Inconsistent (Semantic Contradiction)

The expression is syntactically well-formed, but its deductive closure directly contradicts the core field axioms of the chosen universe.

Abstract Syntax Tree (AST) & Proof Visualizer Interactive D3 Hierarchy

Valid Node

Contradiction/Violation

Type Warning

Root Signature: Equality Relation `(=) : (ℝ, ℝ) → Prop` Pan & hover tree nodes for AST evaluation

Formal Proof & Contradiction Tracer Field Ring Logic

Step-by-step symbolic deduction showing how formal axioms interact with the tested assertion:

Formal Nonsense vs. Falsity Matrix Epistemological Taxonomy

Category	WFF?	Typed?	Status	Archetype
Gibberish	❌ No	❌ No	Ill-Formed	`++ = ×`
Category Error	⚠️ Partial	❌ No	Type Mismatch	`√piano`
Axiom Contradiction	✅ Yes	✅ Yes	Nonsense / Null	`1 / 0 = z`
Falsehood	✅ Yes	✅ Yes	Sensible (False)	`2 + 2 = 5`
Valid Theorem	✅ Yes	✅ Yes	Proven Tautology	`p ∧ (p→q) ⊢ q`

Verification Ledger Snapshot (Auto-computed):