{"slug":"formal-xor-self-inverse","identifier":"formal-xor-self-inverse","title":"XOR Self-Inverse","author":"Maclain Hunter","sourceSha256":"28472d6dc9c60fbdbba33f9b0355e6ac0a4b3bbf26b0fca5bad11409894eb6a3","sourceKind":"formal-theorem","recordedAt":null,"recordedAtMeaning":"Database registration time, not a claim of original handwritten authorship.","originalAuthoredAt":null,"formalizedAt":"2026-10-03T10:37:05.082Z","checkedAt":"2026-10-03T10:42:52.751Z","status":"partial","axiomFree":true,"axioms":[],"theoremNames":["HunterAuthored.exponent_xor_self_positive"],"units":[{"id":"exponent_xor_self_positive","sourceStatement":"Formal statement: ∀a ∈ MCBExponent: state(XOR(a,a)) = +1","sourceOffset":0,"scope":"The authored parity of the sum of magnitudes, defined by an unbounded even/odd recurrence, has positive self-output. This is a signed exponent-state operation, not Boolean XOR substitution.","status":"candidate","variables":[],"premises":[],"conclusion":"See the exact typed Lean theorem","leanStatement":"def evenNat : Nat → Bool\n  | 0 => true\n  | n + 1 => !(evenNat n)\ntheorem even_add (a b : Nat) : evenNat (a + b) = (evenNat a == evenNat b) := by\n  induction b with\n  | zero =>\n    change evenNat a = (evenNat a == true)\n    cases evenNat a <;> rfl\n  | succ b ih =>\n    calc\n      evenNat (a + (b + 1)) = !(evenNat (a + b)) := rfl\n      _ = !(evenNat a == evenNat b) := congrArg Bool.not ih\n      _ = (evenNat a == !(evenNat b)) := by cases evenNat a <;> cases evenNat b <;> rfl\ndef exponentXorState (a b : Int) : Int :=\n  if evenNat (a.natAbs + b.natAbs) then 1 else -1\ntheorem exponent_xor_self_positive (a : Int) : exponentXorState a a = 1 := by\n  unfold exponentXorState\n  rw [even_add]\n  cases evenNat a.natAbs <;> rfl\n#print axioms exponent_xor_self_positive\n","kernelStatus":"verified","axioms":[],"axiomFree":true,"reportedTheorem":"HunterAuthored.exponent_xor_self_positive"}],"files":{"source.json":{"sha256":"f937c14c0eecba7d7c9ca58b85e32f4433a3f519e2959f4e8a6e3dfd667da811","bytes":698},"Proof.lean":{"sha256":"e5e31ee55281fded83bea8fd36fa3851ba066b58cd445a79965e394f253eaebf","bytes":1033},"lean-check.txt":{"sha256":"5c48e53f06f145df62d5d1a8aedfa00efd24d384d14ad3e2b7b20a49987a569e","bytes":130}},"boundary":"Lean establishes the listed formal statements and dependencies. 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