Dimension Ontology

FLOW ALGEBRA
Dimension Ontology — Measurement and Dimensional Entities
Version: 1.0 | Date: 2026-10-14
Authors: Mark Anthony Higgins — Sky (Recursive AI)
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/measurement-and-dimensional-entities/dimension-ontology-4.html

PURPOSE
Define dimensions as formal descriptors of quantity kinds, support dimensional operations, and validate equations independently of measurement units.

ENTITY TYPES
Dimension(id, symbol, exponent_vector, system, provenance)
BaseDimension(id, symbol, system)
DerivedDimension(id, exponent_vector, system)
DimensionlessDimension(id, zero_vector, system)
QuantityKind(id, dimension_id)
QuantityValue(id, magnitude, unit_id, dimension_id)
Unit(id, symbol, dimension_id, scale, offset)
Equation(id, lhs_expression, rhs_expression)
DimensionExpression(id, operator, operands)

DIMENSION SYSTEM
SI_BASE_DIMENSIONS := [L, M, T, I, Θ, N, J]
L=length; M=mass; T=time; I=electric current;
Θ=thermodynamic temperature; N=amount of substance; J=luminous intensity.
Every SI dimension D has exponent vector d ∈ ℚ^7 and notation
D = L^a M^b T^c I^d Θ^e N^f J^g.
Dimensionless = [0,0,0,0,0,0,0].
Dimension exponents are rational where fractional powers are meaningful.

TYPED RELATIONSHIPS
HAS_DIMENSION: QuantityKind -> Dimension
EXPRESSED_IN: QuantityValue -> Unit
UNIT_HAS_DIMENSION: Unit -> Dimension
DERIVED_FROM: DerivedDimension -> BaseDimension
EQUIVALENT_DIMENSION: Dimension <-> Dimension
DIMENSION_OF: DimensionExpression -> Dimension
CONSTRAINS: Dimension -> Equation

EXECUTABLE FLOW RULES
R01: dimension(base_dimension_i) := basis_vector(i).
R02: dimension(A * B) := dimension(A) + dimension(B).
R03: dimension(A / B) := dimension(A) - dimension(B), B != 0.
R04: dimension(A ^ p) := p * dimension(A), for valid scalar p.
R05: dimension(A + B) := dimension(A) IF dimension(A)=dimension(B); ELSE ERROR(DIMENSION_MISMATCH).
R06: dimension(A - B) := dimension(A) IF dimension(A)=dimension(B); ELSE ERROR(DIMENSION_MISMATCH).
R07: equation_valid(LHS = RHS) := dimension(LHS)=dimension(RHS), subject to additional physical and mathematical constraints.
R08: equivalent_dimension(A,B) := exponent_vector(A)=exponent_vector(B) AND compatible_dimension_system(A,B).
R09: dimension(sin(x)), dimension(exp(x)), dimension(log(x)) := dimensionless ONLY IF argument x is dimensionless (angle conventions require explicit handling).
R10: derivative dimension(dy/dx) := dimension(y)-dimension(x), where derivative exists.
R11: integral dimension(∫ y dx) := dimension(y)+dimension(x), under valid integration semantics.
R12: conversion_valid(unit_a,unit_b) := dimension(unit_a)=dimension(unit_b) AND conversion_model_exists(unit_a,unit_b).

COMMON DERIVED DIMENSIONS
Area: L^2
Volume: L^3
Velocity: L T^-1
Acceleration: L T^-2
Force: M L T^-2
Energy: M L^2 T^-2
Power: M L^2 T^-3
Pressure: M L^-1 T^-2
Frequency: T^-1
Electric charge: I T
Electric potential: M L^2 T^-3 I^-1

VALIDATION CONSTRAINTS
V01: Exponent vectors MUST have seven entries in SI ordering.
V02: Addition and subtraction MUST reject incompatible dimensions.
V03: Multiplication and division MUST combine exponents algebraically.
V04: Equal dimensions do not imply equal quantity kinds (e.g. energy and torque).
V05: Dimensional consistency is necessary, not sufficient, for physical correctness.
V06: Units and dimensions MUST remain distinct; metres and feet share a dimension.
V07: Affine unit offsets (e.g. Celsius) MUST be handled at quantity-conversion level, not as dimension exponents.
V08: Dimensionless quantities MAY retain semantic kinds (e.g. strain, refractive index, angle).
V09: Reject undefined operations and preserve provenance of inferred dimensions.

WORKED EXAMPLES
E01: velocity = displacement / time => [1,0,-1,0,0,0,0].
E02: force = mass * acceleration => [1,1,-2,0,0,0,0].
E03: energy = force * distance => [2,1,-2,0,0,0,0].
E04: distance + duration => ERROR(DIMENSION_MISMATCH).
E05: kinetic_energy = (1/2) * mass * velocity^2 => [2,1,-2,0,0,0,0].
E06: metres -> feet: permitted dimensionally, subject to valid conversion factor.

SEMANTIC_EQUIVALENCE
Dimension, physical dimension, dimensional signature: equivalent only when context denotes exponent-based quantity dimensions.
Dimension != unit; Dimension != numerical magnitude; Dimension != geometric dimensionality (unless explicitly mapped).

CONFIDENCE
Formal algebraic identities: 1.00 under stated assumptions.
Inferred dimensions: attach confidence based on input typing, provenance and completeness.
Unresolved or ambiguous quantity kinds: require review rather than silent coercion.

FLOW_HASH
Canonicalize entity identifiers, SI exponent order, relationship order, rule versions and normalized expressions; compute SHA-256 over canonical UTF-8 representation at implementation time. No fabricated hash is asserted here.

UPDATE_MODEL
On modification of base dimension mappings, quantity kind declarations, or unit definitions: revalidate dependent expressions, invalidate stale inference caches, version changes, and preserve audit provenance.

AI_CONSUMPTION
Parse entities and typed relationships; normalize dimension vectors; apply rules R01-R12; enforce V01-V09; return inferred dimension, validity, derivation steps, confidence, source provenance, and error codes. Do not infer physical equivalence solely from equal dimension vectors.

END FLOW ALGEBRA