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