Phase Space Ontology

FLOW ALGEBRA
PHASE SPACE ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Coordinate and Spatial Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/coordinate-and-spatial-entities/phase-space-ontology-3.html
Version: 1.0 | Date: 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent phase spaces as typed mathematical structures describing the possible instantaneous states of dynamical systems, their trajectories, flows, invariants, and relationships to state spaces and symplectic geometry. Distinguish general dynamical phase spaces from Hamiltonian phase spaces.

2. ENTITY TYPES
PhaseSpace; State; Coordinate; Position; Momentum; Time; DynamicalSystem; VectorField; EvolutionOperator; Flow; Trajectory; Orbit; Equilibrium; InvariantSet; Attractor; Constraint; Observable; EnergyFunction; Hamiltonian; SymplecticForm; Manifold; InitialCondition; Parameter.

3. CORE DEFINITIONS
PhaseSpace(P): a set or manifold P whose points represent admissible instantaneous system states.
State(s,P): s in P.
DynamicalSystem(D,P,T,Phi): evolution Phi defined on its appropriate time-state domain, with Phi(0,s)=s and Phi(t+u,s)=Phi(t,Phi(u,s)) wherever both sides are defined.
Trajectory(s0): t -> Phi(t,s0) on its interval of existence.
Orbit(s0): image of the trajectory of s0.
Equilibrium(s): Phi(t,s)=s for all applicable t.
InvariantSet(A): Phi(t,A) subseteq A for applicable forward times; two-sided invariance requires equality for invertible evolution.
HamiltonianPhaseSpace: a symplectic manifold (P,omega), commonly a cotangent bundle T*Q with canonical coordinates (q,p).

4. MATHEMATICAL STRUCTURES
General autonomous continuous dynamics: ds/dt = F(s), with F a vector field on P.
Discrete dynamics: s_(n+1)=f(s_n), where f:P -> P.
Hamiltonian dynamics (canonical local coordinates): dq_i/dt = partial H/partial p_i; dp_i/dt = -partial H/partial q_i.
Symplectic condition: omega is a closed, nondegenerate differential 2-form; consequently a symplectic manifold has even dimension.
Canonical symplectic form (one convention): omega = sum_i dq_i wedge dp_i.
Conservation of autonomous Hamiltonian: dH/dt = {H,H}=0 along solutions when H has no explicit time dependence.

5. TYPED RELATIONSHIPS
PhaseSpace --CONTAINS--> State
State --HAS_COORDINATE--> Coordinate
DynamicalSystem --HAS_PHASE_SPACE--> PhaseSpace
DynamicalSystem --EVOLVES_BY--> Flow
Flow --MAPS_STATE_TO--> State
InitialCondition --DETERMINES--> Trajectory [when evolution is well posed]
Trajectory --LIES_IN--> PhaseSpace
Trajectory --GENERATES--> Orbit
InvariantSet --SUBSET_OF--> PhaseSpace
Equilibrium --INSTANCE_OF--> State
HamiltonianPhaseSpace --HAS_STRUCTURE--> SymplecticForm
Hamiltonian --GENERATES--> HamiltonianVectorField
Observable --DEFINED_ON--> PhaseSpace
Constraint --RESTRICTS--> AdmissibleStateSet

6. INFERENCE RULES
R1: State(s,P) and Flow(Phi,P) and Phi(t,s) defined => State(Phi(t,s),P).
R2: Equilibrium(s) => Orbit(s)={s}.
R3: InvariantSet(A) and s in A and Phi(t,s) defined for t>=0 => Phi(t,s) in A.
R4: HamiltonianSystem(P,omega,H) and H autonomous and solutions regular => H conserved along each solution.
R5: SymplecticManifold(P,omega) => dim(P) is even.
R6: IdenticalInitialState and deterministic unique evolution => identical trajectories on their common time domain.
R7: Constraint C(s)=0 => admissible states lie in {s in P : C(s)=0}; do not assume this subset is dynamically invariant without proof.

7. SEMANTIC DISTINCTIONS
PhaseSpace is often used synonymously with StateSpace for a dynamical system, but the former frequently emphasises trajectories and geometric dynamics.
ConfigurationSpace Q is not generally identical to Hamiltonian phase space T*Q.
Trajectory is time-parameterised; Orbit is the set of visited states.
PhaseSpace is not necessarily Euclidean, finite-dimensional, symplectic, or Hamiltonian.
Attractor requires a specified dynamical setting and attraction criterion; not every invariant set is an attractor.

8. VALIDATION
V1: Every state references an existing phase space.
V2: Flow composition is checked only where maps and times are defined.
V3: State coordinates obey declared types, units and constraints.
V4: Hamiltonian and symplectic claims require an explicitly declared symplectic structure.
V5: Conservation claims record assumptions and regularity conditions.
V6: Invariance distinguishes forward from two-sided invariance.
V7: Trajectory existence and uniqueness are not inferred without suitable hypotheses.

9. CONFIDENCE AND PROVENANCE
CONFIDENCE: axioms and formal definitions = high when assumptions are explicit; empirical state estimates = evidence-dependent.
PROVENANCE: record source, formalism, coordinate convention, model version, assumptions and observation time.
SEMANTIC_EQUIVALENCE: require compatible state representation, dynamics and constraints before declaring two phase-space descriptions equivalent.

10. FLOW HASH AND UPDATE MODEL
FLOW_HASH: compute from canonicalised ontology identifier, version, entity declarations, typed relationships, inference rules and validation constraints; hash value generated at implementation time, not fabricated.
UPDATE_MODEL: version changes to state schema, dynamics, units, coordinate transforms, assumptions and provenance; revalidate dependent relationships and inferences.

11. AI CONSUMPTION
Input: phase-space specification, states, dynamics, time domain, parameters, constraints and provenance.
Process: type-check -> validate assumptions -> resolve semantic links -> apply guarded inference -> track uncertainty -> produce traceable results.
Output: validated entity graph, reachable or invariant-state claims (where justified), trajectories or symbolic relations, confidence, provenance and version identifiers.
Guardrails: never conflate physical observations with model predictions; do not assert global flow, conservation or uniqueness without hypotheses.

END FLOW ALGEBRA