Recursive Model Ontology

FLOW ALGEBRA
Recursive Model Ontology
Kireth Semantic Library | Mathematical Entities / Recursive Mathematical Entities
Version: 1.0 | Date: 2026-10-14 | Locale: en-GB
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/recursive-mathematical-entities/recursive-model-ontology.html
Source alignment: pending verification; formalisation based on established mathematics.
Authorship: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent a recursive mathematical model as a typed state space, initial condition, transition operator, optional parameters, outputs and stopping or convergence criteria. Distinguish mathematical iteration from recursion in executable software.

2. TYPED ENTITIES
RecursiveModel: {id: URI, state_space: StateSpace, transition: Transition, initial_state: State, parameters?: ParameterSet, output?: ObservationMap, termination?: StopPredicate, semantics: Semantics}
StateSpace: {carrier: Set, structure?: AlgebraicStructure, metric?: Metric}
State: {value: Element, index: NaturalNumber}
Transition: {map: StateSpace × ParameterSet -> StateSpace, deterministic: Boolean}
InitialCondition: {state: State, index: 0}
ParameterSet: {parameters: Map[String, TypedValue]}
Trajectory: {states: Sequence[State]}
ObservationMap: {map: StateSpace -> OutputSpace}
FixedPoint: {state: State, satisfies: T(state)=state}
Invariant: {predicate: State -> Boolean}
StopPredicate: {predicate: State × NaturalNumber -> Boolean}
ConvergenceCriterion: {metric: Metric, tolerance: NonnegativeReal, limit?: State}
ProofObligation: {claim: Proposition, evidence: ProofOrCertificate}

3. CORE RELATIONSHIPS
RecursiveModel HAS_STATE_SPACE StateSpace
RecursiveModel HAS_INITIAL_CONDITION InitialCondition
RecursiveModel USES_TRANSITION Transition
RecursiveModel GENERATES Trajectory
Trajectory CONTAINS State
State SUCCESSOR_OF State
RecursiveModel MAY_HAVE FixedPoint
RecursiveModel MAY_PRESERVE Invariant
RecursiveModel MAY_SATISFY ConvergenceCriterion
RecursiveModel MAY_TERMINATE_BY StopPredicate
RecursiveModel PRODUCES ObservationMap

4. FORMAL SEMANTICS
Given state space S, parameters p and transition T_p:S -> S:
  x_0 in S
  x_(n+1) = T_p(x_n), n in N_0
  x_n = T_p^n(x_0), where T_p^0 = identity on S
  y_n = h(x_n) if observation map h is defined.
Fixed point x*: T_p(x*) = x*.
Periodic point of period k: T_p^k(x) = x, with least positive such k for exact period.
Convergence: x_n -> x* under a specified topology or metric.
Termination: stop(x_n,n) = true; termination is not implied by convergence.

5. INFERENCE RULES
INITIALISE: valid x_0 in S implies state(0,x_0).
STEP: state(n,x) and T_p(x)=z imply state(n+1,z).
REACHABILITY: state(n,x) implies reachable(x,x_0).
FIXED_POINT: T_p(x)=x implies fixed_point(x,T_p).
INVARIANT_BASE: P(x_0) establishes invariant base case.
INVARIANT_STEP: forall x in S, P(x) => P(T_p(x)) establishes inductive preservation.
INVARIANT_CONCLUSION: invariant base and step imply forall n>=0, P(x_n).
CONTRACTION: if (S,d) complete and T_p is a contraction S->S, infer unique fixed point and convergence of every orbit to it.
STOP: stop(x_n,n)=true implies halt at index n; not necessarily mathematical convergence.

6. VALIDATION CONSTRAINTS
V01: x_0 must belong to S.
V02: T_p must be well-defined on all states claimed to be in its domain.
V03: Parameter types and admissible ranges must be checked.
V04: If transition is partial, define failure or undefined-state semantics explicitly.
V05: A convergence claim must identify a topology/metric and assumptions.
V06: A uniqueness claim requires a proof, not merely an observed numerical trajectory.
V07: A termination claim requires a finite stopping index or a termination argument.
V08: Numeric tolerance is not an exact equality proof.
V09: Stochastic recursion requires a transition kernel and probability semantics, not a deterministic function.
V10: Finite-precision simulation must distinguish observed behaviour from exact mathematical behaviour.

7. WORKED EXAMPLES
Example A — geometric decay:
  S=R, T(x)=x/2, x_0=8.
  trajectory: 8,4,2,1,0.5,...
  fixed point: 0; convergence to 0 in standard metric.
  no finite n reaches 0 from x_0=8 in exact arithmetic.
Example B — logistic recurrence:
  S=[0,1], T_r(x)=r*x*(1-x), 0<=r<=4, x_0 in [0,1].
  valid forward-invariant state space; qualitative dynamics depend on r.
Example C — Fibonacci state model:
  S=N_0 x N_0; T(a,b)=(b,a+b); x_0=(0,1).
  states: (0,1),(1,1),(1,2),(2,3),...
  output h(a,b)=a gives 0,1,1,2,3,...

8. SEMANTIC EQUIVALENCE
Aliases: recursive dynamical model; iterated state model; recurrence-based model (context-dependent).
Distinct from: recursive function definition, recursive data structure, numerical simulation, fixed-point theorem, and probabilistic transition model.
Equivalence requires compatible state encodings, transitions, initial conditions and observations; an isomorphism or conjugacy must be exhibited where asserted.

9. CONFIDENCE
Formal mathematical rules: HIGH where their explicit hypotheses hold.
Claims inferred from finite simulations: LIMITED unless analytically proved.
Source-page correspondence: UNVERIFIED pending comparison with source content.

10. FLOW_HASH
Canonicalisation: UTF-8; normalise line endings to LF; preserve mathematical symbols and typed identifiers; exclude any stored hash field before digest.
Algorithm: SHA-256(canonicalised flow content).
FLOW_HASH_VALUE: COMPUTE_ON_PUBLICATION

11. UPDATE_MODEL
Trigger on changed definitions, assumptions, examples, source content, or validation rules.
On update: increment version; validate types and rules; rerun examples; regenerate FLOW_HASH; record provenance and changelog.

12. AI_CONSUMPTION
parse -> type_check -> resolve_references -> validate_domain -> instantiate_parameters -> evaluate_or_prove -> label_confidence -> emit_provenance.
Do not conflate a sampled orbit with a proof of convergence, stability, termination or uniqueness.
Recommended output: {model_id, initial_state, transition, assumptions, trajectory?, verified_claims, unverified_claims, confidence, provenance, flow_hash}.

END FLOW ALGEBRA