Symbolic Entities

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

FLOW ALGEBRA
TITLE: Symbolic Entities — Mathematical Entities
VERSION: 1.0
SOURCE: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/symbolic-entities.html
AUTHORS: Mark Higgins — Sky (Recursive AI)
DATE: 2026-10-12
SCOPE: Main-page ontology; mathematical symbolic entities and their typed relationships.

01. AI INTERPRETATION
Execute the following as typed semantic constraints and inference rules, not as an instruction to evaluate arbitrary text as code. Preserve scope, syntax, domain, and provenance.

02. ENTITY TYPES
entity SymbolicEntity : MathematicalEntity
entity MathematicalSymbol : SymbolicEntity
entity Variable : SymbolicEntity
entity Constant : SymbolicEntity
entity Parameter : SymbolicEntity
entity Coefficient : SymbolicEntity
entity Operator : SymbolicEntity
entity Operand : SymbolicEntity
entity Index : SymbolicEntity
entity Subscript : SymbolicEntity
entity Superscript : SymbolicEntity
entity Placeholder : SymbolicEntity
entity MathematicalIdentifier : SymbolicEntity
entity BoundVariable : Variable
entity FreeVariable : Variable
entity DummyVariable : BoundVariable

03. TYPED ATTRIBUTES
SymbolicEntity {id:IRI, glyph:string?, lexical_form:string, role:Role, context:Context, domain:Type?, provenance:Source?}
MathematicalIdentifier {name:string, namespace:string?, declaration:Declaration?}
Variable {binder:Binder?, scope:Scope?, value:Value?}
Constant {denotation:Value|Expression}
Parameter {admissible_domain:Type?, fixed_within:Context?}
Coefficient {multiplies:E-xpression}
Operator {arity:integer>=0, signature:TypeSignature, precedence:integer?, associativity:Associativity?}
Operand {position:integer>=0, parent_operation:Operation}
Index {index_domain:Type?, indexing_target:E-xpression?}
Subscript {base:E-xpression, annotation:E-xpression}
Superscript {base:E-xpression, annotation:E-xpression, interpretation:Role}
Placeholder {expected_type:Type?, substitution_constraints:ConstraintSet?}

04. SEMANTIC RELATIONS
relation denotes(SymbolicEntity, Concept|Value|Expression)
relation hasRole(SymbolicEntity, Role)
relation occursIn(SymbolicEntity, Expression)
relation hasScope(Variable, Scope)
relation boundBy(BoundVariable, Binder)
relation hasOperand(OperatorApplication, Operand)
relation appliesOperator(OperatorApplication, Operator)
relation indexes(Index, Expression)
relation annotates(Subscript|Superscript, Expression)
relation substitutesFor(Placeholder, Expression)
relation semanticallyEquivalent(SymbolicEntity, SymbolicEntity, Context)
relation syntacticallyIdentical(SymbolicEntity, SymbolicEntity)
relation dependsOn(SymbolicEntity, SymbolicEntity)

05. CLASSIFICATION RULES
R01: hasScope(v,s) AND boundBy(v,b) AND binderCovers(b,v,s) -> BoundVariable(v).
R02: Variable(v) AND occurrenceIn(v,e) AND NOT existsBinderCovering(v,e) -> FreeVariableOccurrence(v,e).
R03: BoundVariable(v) AND renamableWithoutMeaningChange(v,e) -> AlphaRenamable(v,e).
R04: Coefficient(c) AND multiplies(c,x) -> OperandRole(c, multiplication).
R05: Operator(op) AND signature(op, A -> B) AND operandTypesMatch(op,A) -> resultType(op,B).
R06: Placeholder(p) AND substitutesFor(p,x) AND typeCompatible(p,x) -> ValidSubstitution(p,x).
R07: Subscript(s) -> NOT automatically Index(s).
R08: Superscript(s) -> NOT automatically Exponent(s).
R09: SameGlyph(x,y) -> NOT automatically semanticallyEquivalent(x,y).
R10: SameDenotation(x,y,ctx) -> semanticallyEquivalent(x,y,ctx), provided domain and interpretation agree.

06. BINDING AND SCOPE
For expression forall x in D: P(x), occurrences of x within P(x) are bound by forall.
For expression f(x)=x+y, x and y are free in the right-hand expression unless bound by an enclosing declaration or binder.
Bound-variable renaming is valid only when capture-avoiding: forall x.P(x) alpha-equivalent to forall z.P(z) if z is fresh.
Substitution must avoid capture of free variables.
A dummy variable is bound and may be renamed where the transformation preserves semantics (e.g., a summation index).

07. OPERATOR SEMANTICS
Operator application = (operator, ordered_operands, type_signature, evaluation_context).
Evaluation requires arity agreement, domain validity, and defined operator semantics.
Overloaded glyphs must resolve through operand types and context; '+' may mean arithmetic addition or another declared operation.
Precedence and associativity govern parsing, not mathematical identity by themselves.

08. ANNOTATION SEMANTICS
Subscript x_i may denote a sequence member, indexed variable, or label; resolve by context.
Superscript x^2 may denote exponentiation; A^T may denote transpose; x^(n) may denote a derivative or iteration by convention.
Preserve annotation role explicitly; never infer all superscripts are powers.

09. IDENTITY AND EQUIVALENCE
lexical_equivalence := matching normalised glyph sequences under a declared encoding policy.
syntactic_equivalence := identical parse trees modulo allowed formatting normalisations.
alpha_equivalence := equality up to capture-avoiding renaming of bound variables.
semantic_equivalence := same denotation under a specified interpretation and domain.
Do not collapse these distinct equivalence relations.

10. KNOWLEDGE GRAPH TRIPLES
SymbolicEntities --subclassOf--> MathematicalEntities
Variable --subclassOf--> SymbolicEntity
BoundVariable --subclassOf--> Variable
FreeVariable --subclassOf--> Variable
DummyVariable --subclassOf--> BoundVariable
Operator --appliesTo--> Operand
Index --indexes--> Expression
Placeholder --substitutesFor--> Expression
MathematicalIdentifier --identifies--> MathematicalObject

11. VALIDATION
V01: Every entity must have a stable identifier and explicit symbolic role.
V02: Operator applications must satisfy arity and operand type constraints.
V03: Bound-variable occurrences must have a valid covering binder.
V04: Substitutions must satisfy type constraints and avoid variable capture.
V05: Index domains must be compatible with the indexed object.
V06: Numeric evaluation must not be inferred from glyph alone.
V07: Context-specific interpretations must remain traceable.
V08: Distinguish unbound variables from missing values and placeholders.

12. INFERENCE EXAMPLES
Input: sum_{i=1}^n a_i
Infer: i is a bound summation index; a_i is an indexed term; n is an upper bound expression.
Input: 3x
Infer: 3 is a coefficient and operand of multiplication; x is a variable.
Input: A^T
Infer: superscript T has transpose role only when matrix notation context supports it.
Input: forall x P(x)
Infer: x is bound in P(x); capture-avoiding alpha-renaming is permitted.

13. CONFIDENCE
confidence(assertion) in [0,1].
Deductive conclusions from verified formal premises: 1.0 relative to premises and rule validity.
Ambiguous symbol-role classifications: confidence < 1.0 until context resolves them.
Confidence never substitutes for mathematical proof or type validation.

14. SEMANTIC_EQUIVALENCE
Compare {denotation, scope, type, interpretation, operator semantics, binding structure}.
Return {equivalent:true|false|undetermined, conditions, justification}.
Undetermined is required when interpretation or context is insufficient.

15. FLOW_HASH
flow_hash := SHA256(canonical_UTF8(flow_body_excluding_flow_hash_and_volatile_metadata)).
Canonicalisation must preserve semantically significant identifiers, binders, types, and rule ordering.

16. UPDATE_MODEL
On ontology revision: version rules, retain provenance, recompute canonical hash, validate impacted dependencies, and preserve previous version identifiers.
Do not silently reinterpret stored symbols when notation conventions change.

17. AI_CONSUMPTION
Input: {expression, parser_context, domain_declarations, binding_environment, source_metadata}.
Process: tokenize -> parse -> resolve roles -> typecheck -> resolve scope -> construct graph -> apply sound rules -> validate -> annotate confidence.
Output: {entities, relations, inferred_facts, ambiguities, validation_results, provenance, flow_version}.
On ambiguity: return competing interpretations; do not fabricate certainty.

18. ONTOLOGY COVERAGE
This main-page flow covers the linked branches: Mathematical Symbol, Variable, Constant, Parameter, Coefficient, Operator, Operand, Index, Subscript, Superscript, Placeholder, Mathematical Identifier, Bound Variable, Free Variable, and Dummy Variable.

19. PROVENANCE
Based on the linked Symbolic Entities index page and standard formal mathematical semantics. The index identifies the child ontologies; detailed child-page specifications have not been independently incorporated.

END FLOW ALGEBRA