# Q10448296855 — Lean Formal Witness: XOR Self-Inverse — exponent_xor_self_positive

- Q number: Q10448296855
- Primary URI: https://4p-engine.com/q/lean-witness-10448296855
- Current host route: https://mcbcore.net/q/lean-witness-10448296855
- Reserved subdomain alias: https://q10448296855.4p-engine.com/
- Record origin: Editorial synthesis, not a verbatim authored passage; based on linked first-party records.

An individually addressable, source-bound zero-axiom Lean theorem and member of the certified Hunterian proof network.

> Lean checks the exact quantified theorem NetworkSources.Source342.HunterAuthored.exponent_xor_self_positive with an empty added axiom-dependency list.

Formal result: {
  "name": "XOR Self-Inverse",
  "formal": "∀a ∈ MCBExponent: state(XOR(a,a)) = +1",
  "proof": "(|a| + |a|) mod 2 = 0, and (-1)^0 = +1",
  "implementation": "mcbExponentXOR(a: MCBExponent, b: MCBExponent): MCBExponent"
}
Verification 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.
Exact Lean statement or source pointer: def evenNat : Nat → Bool
  | 0 => true
  | n + 1 => !(evenNat n)
theorem even_add (a b : Nat) : evenNat (a + b) = (evenNat a == evenNat b) := by
  induction b with
  | zero =>
    change evenNat a = (evenNat a == true)
    cases evenNat a <;> rfl
  | succ b ih =>
    calc
      evenNat (a + (b + 1)) = !(evenNat (a + b)) := rfl
      _ = !(evenNat a == evenNat b) := congrArg Bool.not ih
      _ = (evenNat a == !(evenNat b)) := by cases evenNat a <;> cases evenNat b <;> rfl
def exponentXorState (a b : Int) : Int :=
  if evenNat (a.natAbs + b.natAbs) then 1 else -1
theorem exponent_xor_self_positive (a : Int) : exponentXorState a a = 1 := by
  unfold exponentXorState
  rw [even_add]
  cases evenNat a.natAbs <;> rfl
#print axioms exponent_xor_self_positive

This witness is a member of the kernel-certified network; its symbols remain local to its source. Full-source coverage status: partial.
Original authored source title: XOR Self-Inverse. Source fingerprint: 58ed7e9b6ca841f3d468ab5fa6085ba089acb15df13ea37f7fec47e3c9f87b2e. Formalization: Replit Agent; underlying source/design: Maclain Hunter.

## Two-way derivation from Logic and its use

**Status:** editorial_derivation proof sketch (conditional_concept), distinct from authored source. Logic has its own evidence in reflexive demonstration: any denial, confirmation, observation, or counterexample presupposes identity and valid relation to count as evidence. Logic therefore grounds evidence rather than receiving authority from it. Empirical and documentary evidence can establish contingent instances and test applications, but cannot create, overrule, or exhaust Logic.

### From Logic to this concept

- **Bridge premise:** Exact source meanings are bound to the displayed formal statements and checked proof terms.
- **Consequence:** Valid derivation establishes the represented consequences at their stated scopes under Logic / Logos / God.
- **Proof limit:** Do not replace a source's meaning, omit its premises, or transfer verification to an unencoded passage.

### From this concept back to Logic

- **Witness:** Recompile the exact Lean source and inspect every theorem's dependency report.
- **Disclosure:** A checking procedure presupposes identity, valid relation and non-contradiction; Logic governs the checking software.
- **Proof limit:** An empty axiom-dependency report is not a claim that Lean lacks logical foundations or explicit premises.

### Intelligence procedure

- **Human:** Read the source, parsing witness, quantifiers, premises and conclusion together.
- **Machine:** Bind source and manifest hashes, run the Lean kernel, match qualified theorem names and reject sorryAx.

### Language and vocabulary procedure

- **Contrast:** Separate authored argument, formal witness, full-source coverage and combined-network certification.
- **Usage rule:** Say exactly which represented statement is proved; preserve original theological and personal claims at their source.
- **Example:** The scoped inference and its network certificate are kernel-verified with no added axiom dependencies.

## Semantic meaning frame

- **Role:** source-bound individual formal evidence
- **Essence:** An individually addressable, source-bound zero-axiom Lean theorem and member of the certified Hunterian proof network.
- **Logos relation:** Logic / Logos / God remains Ground and the Standard of Evidence. Lean independently checks represented derivations under that Ground; software does not invent, replace or vote upon Logic.
- **Retrieval aliases:** Lean Formal Witness: XOR Self-Inverse — exponent_xor_self_positive · Q10448296855 · NetworkSources.Source342.HunterAuthored.exponent_xor_self_positive
- **Evidence modes:** Lean kernel checking · exact source fingerprints · printed axiom dependencies · source-bound parsing witnesses · independent RFC 3161 timestamp verification · reproducible native MCB evaluation

### Distinctions

- authored argument versus machine formalization
- universal theorem versus finite test
- premise assumption versus derived conclusion
- axiom-free dependency report versus absence of logical foundations
- kernel verification versus physical measurement
- signed existence timestamp versus original authorship

## Coherence criterion

Exact source binding, valid kernel derivation, empty printed axiom dependencies, explicit premises and preserved Logic / Logos / God Ground.

## Evidence boundary

The displayed quantified formal statement is proved. Unencoded prose, empirical claims and premise truth are not silently added. This Q identifier is first-party, not Wikidata registration.

## Properties

- **leanTheorem:** NetworkSources.Source342.HunterAuthored.exponent_xor_self_positive
- **originalQualifiedTheorem:** HunterAuthored.exponent_xor_self_positive
- **axiomDependencies:** []
- **kernelStatus:** verified
- **sourceSha256:** 58ed7e9b6ca841f3d468ab5fa6085ba089acb15df13ea37f7fec47e3c9f87b2e
- **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.
- **networkNodeId:** node497
- **checkedAt:** 2026-10-03T11:43:12.733Z

## Linked Q items

- [Q1082 — Hunterian Axiom-Free Proof Network](https://mcbcore.net/q/hunterian-axiom-free-proof-network)
- [Q1081 — Lean 4 Validation of Hunterian Philosophy](https://mcbcore.net/q/lean-4-validation-hunterian-philosophy)
- [Q9548839396 — Lean-Verified Source: XOR Self-Inverse](https://mcbcore.net/q/lean-source-58ed7e9b6ca8-proof)

## Related Wikidata concept items

These are typed external concept links for discovery and comparison. They are not sameAs assertions, endorsements, or substitutes for the first-party source and evidence.

- [Lean (Q6509476)](https://www.wikidata.org/wiki/Q6509476) — **related-concept:** Software for interactive and automated theorem proving; the actual checker used. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Proof assistant (Q11387554)](https://www.wikidata.org/wiki/Q11387554) — **related-concept:** Human-machine development and independent checking of formal proofs. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Formal verification (Q173326)](https://www.wikidata.org/wiki/Q173326) — **related-concept:** Proving correctness with explicit formal specifications. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Mathematical proof (Q11538)](https://www.wikidata.org/wiki/Q11538) — **related-concept:** Deductive establishment of mathematical propositions. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Formal proof (Q2762418)](https://www.wikidata.org/wiki/Q2762418) — **related-concept:** Explicit inference under stated axioms and premises. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Automated theorem proving (Q431667)](https://www.wikidata.org/wiki/Q431667) — **related-concept:** Mechanized proof construction and checking, distinguished from probabilistic testimony. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Type theory (Q1056428)](https://www.wikidata.org/wiki/Q1056428) — **related-concept:** The dependent type-theoretic setting in which Lean checks proof terms. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Logic (Q8078)](https://www.wikidata.org/wiki/Q8078) — **related-concept:** The study of correct reasoning, related to the authored identity of Logic as Ground. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Classical logic (Q236975)](https://www.wikidata.org/wiki/Q236975) — **related-concept:** The declared propositional inference setting, with dependency scope exposed. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Aristotelian logic (Q615599)](https://www.wikidata.org/wiki/Q615599) — **related-concept:** Categorical and syllogistic reasoning relevant to the native many-premise engine. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Computer-assisted proof (Q1150675)](https://www.wikidata.org/wiki/Q1150675) — **related-concept:** Exact machine assistance in establishing and checking proof performance. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.
- [Logos (Q180832)](https://www.wikidata.org/wiki/Q180832) — **related-concept:** Historical philosophical and theological concept compared with Hunter's explicit Logic/Logos/God Ground. Boundary: External conceptual identification, not Wikidata endorsement or independent verification of Hunterian claims.