Outcome Ontology

FLOW ALGEBRA
OUTCOME ONTOLOGY
Kireth Semantic Library | Entity Graph | Mathematical Entities | Probability Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/probability-entities/outcome-ontology-2.html
Version: 1.0 | 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent an outcome as one possible elementary result of a random experiment, distinguishing the realised outcome from the set of possible outcomes, events, and observed measurements.

2. TYPED ENTITIES
RandomExperiment: process with defined possible elementary results.
Outcome: elementary result omega in a sample space Omega.
SampleSpace: set Omega of all modelled possible outcomes.
Event: measurable subset A of Omega.
Trial: indexed execution of an experiment.
RealisedOutcome: outcome observed on a specific trial.
OutcomeLabel: human-readable representation of an outcome.
OutcomeValue: value assigned by a measurable random variable.
RandomVariable: measurable mapping X: Omega -> S.
SigmaAlgebra: family F of measurable events on Omega.
ProbabilityMeasure: P: F -> [0,1], countably additive, P(Omega)=1.
Observation: recorded evidence, potentially imperfect, about a trial result.

3. CORE SIGNATURES
sample_space: RandomExperiment -> SampleSpace
possible_outcome: RandomExperiment x Outcome -> Bool
member_of: Outcome x SampleSpace -> Bool
realises: Trial x Outcome -> Bool
contains_outcome: Event x Outcome -> Bool
measurable_event: Event x SigmaAlgebra -> Bool
maps_to: RandomVariable x Outcome -> OutcomeValue
probability_of: ProbabilityMeasure x Event -> [0,1]
observed_as: Trial x Observation -> Bool

4. AXIOMS AND INVARIANTS
A1. For experiment E with sample space Omega, possible_outcome(E,omega) iff omega in Omega.
A2. Every completed, well-defined trial has exactly one elementary realised outcome in Omega; incomplete or ambiguous observations are modelled separately.
A3. An event A is a subset of Omega; the event occurs on a trial exactly when its realised omega belongs to A.
A4. A probability measure assigns probabilities to measurable events, not automatically to arbitrary subsets of Omega.
A5. If {omega} is measurable, P({omega}) is the singleton-outcome probability; in continuous models it may be zero despite omega being possible.
A6. Distinct elementary outcomes are mutually exclusive on the same trial, although multiple events may occur simultaneously.
A7. The outcome space depends on the experiment's modelling resolution and need not coincide with a measured value or label.
A8. Observed records may be noisy or missing; observation does not necessarily identify the realised outcome uniquely.

5. MATHEMATICAL SEMANTICS
Probability space: (Omega,F,P).
Elementary outcome: omega in Omega.
Event occurrence: 1_A(omega) = 1 if omega in A, otherwise 0.
Discrete singleton mass: p(omega) = P({omega}), when measurable.
For countable Omega with all singleton events measurable: sum_{omega in Omega} p(omega) = 1.
Random variable: X: (Omega,F) -> (S,Sigma_S), measurable.
Distribution of X: P_X(B) = P({omega in Omega : X(omega) in B}).

6. EXAMPLES
Fair die: Omega={1,2,3,4,5,6}; realised omega=4; event 'even'={2,4,6}.
Two coin tosses: Omega={HH,HT,TH,TT}; HT and TH are distinct ordered outcomes.
Continuous draw: Omega=[0,1] with Borel sigma-algebra and uniform measure; P({0.5})=0 although 0.5 is a possible outcome.

7. TYPED RELATIONSHIPS
Outcome --ELEMENT_OF--> SampleSpace
Outcome --POSSIBLE_RESULT_OF--> RandomExperiment
Trial --REALISES--> Outcome
Outcome --MEMBER_OF--> Event
Event --SUBSET_OF--> SampleSpace
RandomVariable --MAPS--> OutcomeValue
ProbabilityMeasure --ASSIGNS_TO--> MeasurableEvent
Observation --EVIDENCE_ABOUT--> Trial

8. INFERENCE RULES
R1. realises(t,omega) AND omega in A => occurs(A,t).
R2. realises(t,omega) AND omega not in A => not occurs(A,t).
R3. A and B disjoint AND measurable => P(A union B)=P(A)+P(B).
R4. {omega} measurable AND P({omega})>0 => omega has positive singleton probability.
R5. P({omega})=0 DOES NOT imply omega is impossible.
R6. X measurable AND B measurable => {omega:X(omega) in B} is measurable.
R7. Observation alone DOES NOT imply unique realised outcome unless the observation model is injective and reliable.

9. SEMANTIC DISTINCTIONS
Outcome != Event: an outcome is an elementary member; an event is a set of outcomes.
Outcome != SampleSpace: one possible result versus all modelled results.
Outcome != RandomVariableValue: different outcomes can map to the same value.
Possible != PositiveProbability: zero-probability outcomes may still be possible.
RealisedOutcome != Observation: observations may be uncertain or corrupted.

10. CONFIDENCE AND PROVENANCE
CONFIDENCE: mathematical definitions are deductive within declared model assumptions; empirical identification of outcomes requires observation provenance.
PROVENANCE: retain experiment identifier, trial identifier, model version, sample-space definition, observation source, and timestamp when applicable.
SEMANTIC_EQUIVALENCE: outcome labels may be equivalent only under an explicit bijection preserving the relevant experimental structure; coarse-graining is not generally one-to-one equivalence.

11. VALIDATION
V1. Every Outcome reference resolves to a declared SampleSpace.
V2. Every completed Trial resolves to exactly one realised Outcome.
V3. Every Event is a subset of its SampleSpace.
V4. Probability requests require a measurable Event.
V5. Probability measure is nonnegative, normalised and countably additive.
V6. Distinguish outcomes, labels, observed records and random-variable values.
V7. Preserve experiment context across relationship traversals.

12. FLOW HASH / UPDATE MODEL
FLOW_HASH: compute a deterministic cryptographic digest over canonical UTF-8 serialisation of ontology identifier, version, types, relations, axioms and inference rules; exclude the hash field itself. No hash is claimed without computation.
UPDATE_MODEL: changes to sample-space granularity, measurability, observation model or probability assumptions require version increment, validation and provenance retention.

13. AI CONSUMPTION
Parse typed entities and relationship edges; validate context before inferring event occurrence; separate logical certainty from estimated probability; avoid equating zero measure with impossibility; emit explicit assumptions, confidence, provenance and version metadata.

END FLOW ALGEBRA