Measurement Interval Ontology

FLOW ALGEBRA
Ontology: Measurement Interval Ontology
Version: 1.0
Date: 2026-10-14
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/measurement-and-dimensional-entities/measurement-interval-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent measurement intervals as typed, unit-aware bounds on quantities, distinguishing bounded ranges, uncertainty intervals, tolerance intervals and statistical confidence or coverage intervals. Preserve interpretation, endpoint inclusion, provenance and confidence semantics.

2. TYPED ENTITIES
entity MeasurementInterval {
  id: Identifier
  quantity: QuantityKind
  lower: Real | -Infinity
  upper: Real | +Infinity
  lower_inclusive: Boolean
  upper_inclusive: Boolean
  unit: Unit
  dimension: DimensionVector
  interval_kind: IntervalKind
  coverage_probability: Real?  # only where defined by statistical method
  coverage_method: String?
  reference_value: Real?
  source_measurement: Identifier?
  provenance: String?
}
enum IntervalKind {bounded_range, measurement_uncertainty, tolerance, confidence, prediction, credible, coverage, specification}
entity IntervalEndpoint {value: Real | Infinity, inclusive: Boolean, unit: Unit}
entity IntervalOperation {operator: String, operands: List<MeasurementInterval>, result: MeasurementInterval}

3. RELATIONSHIPS
MeasurementInterval --bounds--> QuantityKind
MeasurementInterval --has_lower_bound--> IntervalEndpoint
MeasurementInterval --has_upper_bound--> IntervalEndpoint
MeasurementInterval --expressed_in--> Unit
Unit --has_dimension--> DimensionVector
MeasurementInterval --derived_from--> MeasurementValue
MeasurementInterval --has_provenance--> Source
MeasurementInterval --has_coverage_method--> StatisticalMethod

4. CORE SEMANTICS
For finite bounds a <= b, the interval [a,b] denotes {x | a <= x <= b}.
Endpoint flags determine open/closed/half-open forms.
A single-point interval [a,a] is valid; (a,a), [a,a), and (a,a] are empty.
A measurement interval is not automatically a confidence interval.
A confidence interval's coverage is a property of its construction procedure, not the probability that a fixed parameter lies in the realised interval.
A credible interval requires a specified Bayesian posterior model.

5. FLOW OPERATORS
flow CREATE_INTERVAL(quantity,a,b,unit,kind,flags) -> interval
  require compatible(quantity.dimension,unit.dimension)
  require a <= b
  attach provenance and method metadata where available

flow CONVERT_UNIT(I,target) -> J
  require dimension(I.unit) == dimension(target)
  transform both endpoints using the same monotonic conversion
  if conversion is decreasing: swap endpoint order and inclusion flags
  preserve interval_kind; record conversion

flow INTERSECT(I,J) -> K
  require compatible dimensions; normalise units
  K.lower=max(I.lower,J.lower); K.upper=min(I.upper,J.upper)
  derive inclusion flags from membership in both operands
  return EMPTY if lower>upper or equal endpoints not both included

flow HULL(I,J) -> K
  require compatible dimensions; normalise units
  K=[min(lower bounds),max(upper bounds)] with endpoint flags
  note: hull is smallest containing interval, not set union when disjoint

flow ADD(I,J) -> K
  require compatible dimensions; normalise units
  K.lower=I.lower+J.lower; K.upper=I.upper+J.upper
  propagate inclusion and endpoint infinity semantics
  preserve physical unit rules; do not add two absolute affine temperatures

flow SUBTRACT(I,J) -> K
  K.lower=I.lower-J.upper; K.upper=I.upper-J.lower
  use dimensionally valid quantity arithmetic

flow MULTIPLY(I,J) -> K
  for finite closed intervals compute min/max of four endpoint products
  result dimension=dimension(I)+dimension(J)
  for open/unbounded cases use extended interval arithmetic with safe enclosure

flow DIVIDE(I,J) -> K
  require 0 not in J, otherwise split result into interval set or return UNDEFINED
  K = MULTIPLY(I,RECIPROCAL(J))

flow EXPAND_BY_UNCERTAINTY(value,u,unit) -> I
  require u>=0
  I=[value-u,value+u]
  label uncertainty convention (standard, expanded, or other)

6. INFERENCE RULES
RULE DIMENSION_COMPATIBILITY: comparisons and intersections require compatible dimensions.
RULE ORDER: lower <= upper for nonempty intervals.
RULE CONTAINMENT: x in I iff all active endpoint predicates hold.
RULE DISJOINT: intersection(I,J)=EMPTY implies disjoint(I,J).
RULE NESTED: I subset J iff both endpoint constraints of J admit every member of I.
RULE UNCERTAINTY: an interval width alone does not establish a probability distribution.
RULE COVERAGE: coverage_probability must be accompanied by interval_kind and method.
RULE CONVERSION: correct unit conversion preserves the represented set of physical quantities.
RULE CONSERVATIVE_ARITHMETIC: outward rounding is required for certified floating-point enclosures.

7. VALIDATION
validate unique identifier and known quantity kind
validate endpoint ordering and inclusion consistency
validate unit/dimension compatibility
validate coverage_probability in [0,1] when supplied
validate statistical intervals carry construction method and assumptions
validate no false equivalence between tolerance, confidence and credible intervals
validate finite numeric bounds are not NaN
validate arithmetic handles zero divisors, infinities and rounding explicitly

8. WORKED EXAMPLES
EXAMPLE A: length=12.0 +/- 0.3 mm -> [11.7,12.3] mm (stated uncertainty convention required).
EXAMPLE B: [2,5] m intersect [4,7] m = [4,5] m.
EXAMPLE C: [2,5] m + [1,3] m = [3,8] m.
EXAMPLE D: [1,2] m * [3,4] m = [3,8] m^2.
EXAMPLE E: [10,20] degC converts to [283.15,293.15] K; offsets matter.
EXAMPLE F: [1,2] / [-1,1] is not a single finite interval; denominator contains zero.

9. SEMANTIC_EQUIVALENCE
aliases: measurement range, measured interval, bounded measurement value
related_not_equivalent: uncertainty interval, confidence interval, tolerance interval, credible interval, prediction interval
canonical_entity: MeasurementInterval
identity requires compatible quantity, bounds, endpoint flags, unit conversion and interval interpretation.

10. CONFIDENCE
ontology_assertion_confidence: 0.98
Note: epistemic confidence in the ontology's definitions is distinct from a statistical interval's nominal coverage.

11. FLOW_HASH
algorithm: SHA-256
canonicalise: UTF-8, LF line endings, stable field ordering, normalised identifiers and units
FLOW_HASH: COMPUTE_ON_PUBLICATION
Do not treat placeholder as a computed digest.

12. UPDATE_MODEL
version: 1.0
update_policy: increment version when entity schema, interval semantics, inference or validation changes
provenance_policy: preserve source and revision history

13. AI_CONSUMPTION
input: structured measurement values, bounds, units, uncertainty and interval interpretation
output: validated typed interval, operations, traceable derivations and warnings
reasoning_policy: distinguish mathematical enclosure from statistical coverage; refuse unsupported probabilistic claims
serialization: UTF-8 plain text, machine-readable typed fields

END FLOW ALGEBRA