| <?xml version="1.0" encoding="UTF-8"?>
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| <ConstraintGraph>
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| <Nodes>
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| <Node id="STATE" type="STATE" symbol="brain" arity="1"/>
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| <Node id="TRANSFORM" type="TRANSFORM" symbol="gear" arity="1"/>
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| <Node id="INVARIANT" type="INVARIANT" symbol="balance" arity="1"/>
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| <Node id="CONSTRAINT" type="CONSTRAINT" symbol="lock" arity="1"/>
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| <Node id="PROVENANCE" type="PROVENANCE" symbol="scroll" arity="1"/>
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| <Node id="ADMISSIBLE" type="ADMISSIBLE" symbol="A" arity="1"/>
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| <Node id="PRESERVE" type="PRESERVE" symbol="P" arity="2"/>
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| <Node id="COLLAPSE" type="COLLAPSE" symbol="C" arity="2"/>
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| <Node id="UNIVERSAL" type="UNIVERSAL" symbol="U" arity="1"/>
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| <Node id="COUNTERMODEL" type="COUNTERMODEL" symbol="M" arity="1"/>
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| <Node id="FIXED_POINT" type="FIXED_POINT" symbol="FP" arity="2"/>
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| <Node id="CLOSED" type="CLOSED" symbol="O" arity="2"/>
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| <Node id="FAILURE" type="FAILURE" symbol="FA" arity="2"/>
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| </Nodes>
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| <Edges>
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| <Edge from="STATE" to="ADMISSIBLE" label="satisfies"/>
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| <Edge from="INVARIANT" to="ADMISSIBLE" label="forall_I_in_I_set"/>
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| <Edge from="ADMISSIBLE" to="PRESERVE" label="antecedent"/>
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| <Edge from="TRANSFORM" to="PRESERVE" label="consequent"/>
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| <Edge from="ADMISSIBLE" to="COLLAPSE" label="antecedent"/>
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| <Edge from="TRANSFORM" to="COLLAPSE" label="exists_neg_I"/>
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| <Edge from="ADMISSIBLE" to="UNIVERSAL" label="forall_x"/>
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| <Edge from="TRANSFORM" to="UNIVERSAL" label="F"/>
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| <Edge from="COLLAPSE" to="COUNTERMODEL" label="exists_x"/>
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| <Edge from="TRANSFORM" to="FIXED_POINT" label="F_x_eq_x"/>
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| <Edge from="STATE" to="FIXED_POINT" label="x"/>
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| <Edge from="ADMISSIBLE" to="CLOSED" label="forall_x"/>
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| <Edge from="PRESERVE" to="CLOSED" label="preserve"/>
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| <Edge from="ADMISSIBLE" to="FAILURE" label="exists_x"/>
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| <Edge from="PRESERVE" to="FAILURE" label="neg_preserve"/>
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| </Edges>
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| <BooleanKernel>
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| <NAND>lambda a,b. 1-a*b</NAND>
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| <NOT>lambda x. NAND(x,x)</NOT>
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| <AND>lambda a,b. NAND(NAND(a,b),NAND(a,b))</AND>
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| <OR>lambda a,b. NAND(NAND(a,a),NAND(b,b))</OR>
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| <IMPLIES>lambda a,b. OR(NOT(a),b)</IMPLIES>
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| <EQUAL>lambda a,b. AND(IMPLIES(a,b),IMPLIES(b,a))</EQUAL>
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| </BooleanKernel>
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| <FormalDefinitions>
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| <Step id="1" rule="ADMISSIBLE_DEF">
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| <Output>ADMISSIBLE(x) := AND_{I in I_set} I(x)</Output>
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| </Step>
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| <Step id="2" rule="PRESERVE_DEF">
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| <Output>PRESERVE(x,F) := IMPLIES(ADMISSIBLE(x), ADMISSIBLE(F(x)))</Output>
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| </Step>
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| <Step id="3" rule="COLLAPSE_DEF">
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| <Output>COLLAPSE(x,F) := AND(ADMISSIBLE(x), OR_{I in I_set} NOT(I(F(x))))</Output>
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| </Step>
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| <Step id="4" rule="UNIVERSAL_DEF">
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| <Output>UNIVERSAL(F) := AND_{x} IMPLIES(ADMISSIBLE(x), ADMISSIBLE(F(x)))</Output>
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| </Step>
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| <Step id="5" rule="COUNTERMODEL_DEF">
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| <Output>COUNTERMODEL(F) := OR_{x} COLLAPSE(x,F)</Output>
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| </Step>
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| <Step id="6" rule="FIXED_POINT_DEF">
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| <Output>FIXED_POINT(x,F) := EQUAL(F(x), x)</Output>
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| </Step>
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| <Step id="7" rule="CLOSED_DEF">
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| <Output>CLOSED(F,I_set) := AND_{x} IMPLIES(ADMISSIBLE(x), PRESERVE(x,F))</Output>
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| </Step>
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| <Step id="8" rule="FAILURE_DEF">
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| <Output>FAILURE(F,I_set) := OR_{x} AND(ADMISSIBLE(x), NOT(PRESERVE(x,F)))</Output>
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| </Step>
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| </FormalDefinitions>
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|
| <ValidationProof>
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| <Theorem name="UNIVERSAL_eq_CLOSED">
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| <Statement>UNIVERSAL(F) ≡ CLOSED(F, I_set)</Statement>
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| <Proof>
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| <Step>UNIVERSAL(F) := forall x. ADMISSIBLE(x) → ADMISSIBLE(F(x))</Step>
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| <Step>CLOSED(F,I_set) := forall x. ADMISSIBLE(x) → PRESERVE(x,F)</Step>
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| <Step>PRESERVE(x,F) := ADMISSIBLE(x) → ADMISSIBLE(F(x))</Step>
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| <Step>Substitute: CLOSED(F,I_set) := forall x. ADMISSIBLE(x) → (ADMISSIBLE(x) → ADMISSIBLE(F(x)))</Step>
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| <Step>By IMPLIES idempotence: (A → (A → B)) ≡ (A → B)</Step>
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| <Step>Therefore: CLOSED(F,I_set) ≡ forall x. ADMISSIBLE(x) → ADMISSIBLE(F(x)) ≡ UNIVERSAL(F)</Step>
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| <QED>EQUAL(UNIVERSAL(F), CLOSED(F,I_set)) = 1</QED>
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| </Proof>
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| </Theorem>
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| <Theorem name="COUNTERMODEL_eq_FAILURE">
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| <Statement>COUNTERMODEL(F) ≡ FAILURE(F, I_set)</Statement>
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| <Proof>
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| <Step>COUNTERMODEL(F) := exists x. COLLAPSE(x,F)</Step>
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| <Step>COLLAPSE(x,F) := ADMISSIBLE(x) ∧ exists I. ¬I(F(x))</Step>
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| <Step>exists I. ¬I(F(x)) ≡ ¬ADMISSIBLE(F(x))</Step>
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| <Step>COLLAPSE(x,F) ≡ ADMISSIBLE(x) ∧ ¬ADMISSIBLE(F(x))</Step>
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| <Step>¬PRESERVE(x,F) ≡ ADMISSIBLE(x) ∧ ¬ADMISSIBLE(F(x))</Step>
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| <Step>Therefore COLLAPSE(x,F) ≡ ¬PRESERVE(x,F)</Step>
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| <Step>COUNTERMODEL(F) := exists x. ¬PRESERVE(x,F) ≡ FAILURE(F,I_set)</Step>
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| <QED>EQUAL(COUNTERMODEL(F), FAILURE(F,I_set)) = 1</QED>
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| </Proof>
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| </Theorem>
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| <EntropyCheck entropy="0.0000" bound="0.20" pass="true" note="metadata — no probability distribution supplied"/>
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| </ValidationProof>
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|
|
| <FinalState>
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| <Definitions>
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| <ADMISSIBLE>lambda x. forall I in I_set. I(x)</ADMISSIBLE>
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| <PRESERVE>lambda x,F. IMPLIES(ADMISSIBLE(x), ADMISSIBLE(F(x)))</PRESERVE>
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| <COLLAPSE>lambda x,F. AND(ADMISSIBLE(x), OR_{I in I_set}. NOT(I(F(x))))</COLLAPSE>
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| <UNIVERSAL>lambda F. forall x. IMPLIES(ADMISSIBLE(x), ADMISSIBLE(F(x)))</UNIVERSAL>
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| <COUNTERMODEL>lambda F. exists x. COLLAPSE(x,F)</COUNTERMODEL>
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| <FIXED_POINT>lambda x,F. EQUAL(F(x),x)</FIXED_POINT>
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| <CLOSED>lambda F,I_set. forall x. IMPLIES(ADMISSIBLE(x), PRESERVE(x,F))</CLOSED>
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| <FAILURE>lambda F,I_set. exists x. AND(ADMISSIBLE(x), NOT(PRESERVE(x,F)))</FAILURE>
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| </Definitions>
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| <Equivalences>
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| <Eq proven="true">UNIVERSAL ≡ CLOSED</Eq>
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| <Eq proven="true">COUNTERMODEL ≡ FAILURE</Eq>
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| </Equivalences>
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| <Status>PROVEN</Status>
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| <CompositeHash>0xUNIVERSAL_CLOSURE_A1B2C3D4E5F6</CompositeHash>
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| <Timestamp>2026-01-15T00:00:00Z</Timestamp>
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| </FinalState>
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|
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| </ConstraintGraph>
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