Dimensional Analysis Ontology
Measurement Mathematics
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 16 September 2026
Audio Overview
# Dimensional Analysis Ontology — Flow Algebra
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/measurement-mathematics/dimensional-analysis-ontology.html
FLOW DimensionalAnalysisOntology {
DOMAIN:
Mathematical Ontologies
-> Measurement & Units Ontologies
-> Measurement Mathematics
-> Dimensional Analysis Ontology
PURPOSE:
Represent dimensional analysis as the mathematical framework for expressing physical quantities in terms of base dimensions, testing dimensional homogeneity, deriving dimensions of quantities, constructing dimensionless groups, transforming unit systems, checking equations, supporting scaling and similarity reasoning, and constraining physically meaningful mathematical models. The ontology models base and derived dimensions, dimensional vectors and matrices, products, quotients, powers, dimensionless quantities, Buckingham Pi analysis, rank and null spaces, scaling laws, similarity, unit-system independence, empirical equations, transcendental functions, uncertainty-aware computation, provenance and machine-consumable reasoning while strictly distinguishing dimensions from units, quantity kinds, numerical values and semantic physical meaning.
ENTITY DimensionalAnalysis {
TYPE: MathematicalQuantityAnalysisFramework
DEFINITION:
analysis_of_relationships_between_physical_quantities
through_their_dimensions
independent_of_specific_unit_magnitudes
PROPERTIES:
represents_base_dimensions
derives_compound_dimensions
validates_equation_homogeneity
identifies_dimensionless_combinations
supports_scaling_reasoning
supports_similarity_analysis
constrains_model_forms
does_not_replace_physical_law
has_provenance
}
ENTITY Dimension {
TYPE: QuantityCharacteristic
FORM:
L^a*M^b*T^c*I^d*Theta^e*N^f*J^g
RULE:
dimension != unit
}
ENTITY DimensionalVector {
TYPE: ExponentVector
FORM:
[a,b,c,d,e,f,g]
RULE:
basis_order MUST be preserved
}
BASE_DIMENSION_MODEL {
SI_related_basis:
L = length
M = mass
T = time
I = electric_current
Theta = thermodynamic_temperature
N = amount_of_substance
J = luminous_intensity
RULE:
dimension_basis MUST be explicitly declared
}
LENGTH_DIMENSION_MODEL {
symbol:
L
exponent_vector:
[1,0,0,0,0,0,0]
}
MASS_DIMENSION_MODEL {
symbol:
M
exponent_vector:
[0,1,0,0,0,0,0]
}
TIME_DIMENSION_MODEL {
symbol:
T
exponent_vector:
[0,0,1,0,0,0,0]
}
CURRENT_DIMENSION_MODEL {
symbol:
I
exponent_vector:
[0,0,0,1,0,0,0]
}
TEMPERATURE_DIMENSION_MODEL {
symbol:
Theta
exponent_vector:
[0,0,0,0,1,0,0]
}
AMOUNT_DIMENSION_MODEL {
symbol:
N
exponent_vector:
[0,0,0,0,0,1,0]
}
LUMINOUS_INTENSITY_DIMENSION_MODEL {
symbol:
J
exponent_vector:
[0,0,0,0,0,0,1]
}
DERIVED_DIMENSION_MODEL {
quantity:
formed_from_products_powers_and_quotients_of_base_dimensions
RULE:
derived_dimension_exponents_follow_quantity_equation
}
DIMENSION_MULTIPLICATION_MODEL {
A:
dimension_vector_a
B:
dimension_vector_b
product:
vector_C =
vector_A+vector_B
RULE:
multiplying_dimensions_adds_exponents
}
DIMENSION_DIVISION_MODEL {
quotient:
vector_C =
vector_A-vector_B
RULE:
dividing_dimensions_subtracts_exponents
}
DIMENSION_POWER_MODEL {
A^p:
vector =
p*vector_A
RULE:
exponent_must_apply_to_every_base_dimension_component
}
DIMENSION_ROOT_MODEL {
A^(1/p):
vector =
vector_A/p
RULE:
fractional_dimension_exponents_are_mathematically_possible
though_quantity_kind_semantics_require_context
}
DIMENSIONLESS_MODEL {
vector:
[0,0,0,0,0,0,0]
symbol:
1
examples:
ratio_of_same_kind_quantities
strain
refractive_index
probability
RULE:
dimensionless != semantically_interchangeable
}
NAMED_DIMENSIONLESS_MODEL {
possible:
radian
steradian
RULE:
named_dimensionless_units preserve_quantity_kind_information
}
VELOCITY_MODEL {
dimension:
L*T^-1
vector:
[1,0,-1,0,0,0,0]
}
ACCELERATION_MODEL {
dimension:
L*T^-2
vector:
[1,0,-2,0,0,0,0]
}
FORCE_MODEL {
dimension:
M*L*T^-2
vector:
[1,1,-2,0,0,0,0]
}
ENERGY_MODEL {
dimension:
M*L^2*T^-2
vector:
[2,1,-2,0,0,0,0]
RULE:
energy_and_torque_share_dimension_but_are_distinct_quantity_kinds
}
POWER_MODEL {
dimension:
M*L^2*T^-3
}
PRESSURE_MODEL {
dimension:
M*L^-1*T^-2
}
DENSITY_MODEL {
mass_density:
M*L^-3
}
MOMENTUM_MODEL {
dimension:
M*L*T^-1
}
ANGULAR_MOMENTUM_MODEL {
dimension:
M*L^2*T^-1
}
FREQUENCY_MODEL {
dimension:
T^-1
RULE:
frequency_and_radioactive_activity_share_dimension_but_not_quantity_kind
}
CHARGE_MODEL {
dimension:
I*T
}
VOLTAGE_MODEL {
dimension:
M*L^2*T^-3*I^-1
}
RESISTANCE_MODEL {
dimension:
M*L^2*T^-3*I^-2
}
CAPACITANCE_MODEL {
dimension:
M^-1*L^-2*T^4*I^2
}
INDUCTANCE_MODEL {
dimension:
M*L^2*T^-2*I^-2
}
MAGNETIC_FLUX_MODEL {
dimension:
M*L^2*T^-2*I^-1
}
MAGNETIC_FLUX_DENSITY_MODEL {
dimension:
M*T^-2*I^-1
}
ELECTRIC_FIELD_MODEL {
dimension:
M*L*T^-3*I^-1
}
THERMAL_CONDUCTIVITY_MODEL {
dimension:
M*L*T^-3*Theta^-1
}
SPECIFIC_HEAT_MODEL {
dimension:
L^2*T^-2*Theta^-1
}
CONCENTRATION_DIMENSION_MODEL {
amount_concentration:
N*L^-3
RULE:
concentration_quantity_kind MUST be specified
}
DIMENSIONAL_HOMOGENEITY_MODEL {
equation:
A =
B+C+...
requirement:
dim(A) =
dim(B) =
dim(C)
RULE:
all_additive_terms MUST have_identical_dimensions
}
EQUATION_CHECK_MODEL {
input:
symbolic_equation
process:
assign_dimension_vectors
propagate_operations
compare_both_sides
output:
dimensionally_consistent
dimensionally_inconsistent
unresolved
RULE:
dimensional_consistency_is_necessary_but_not_sufficient_for_physical_correctness
}
ADDITION_CONSTRAINT_MODEL {
A+B:
allowed_only_if_dimensions_equal
RULE:
same_dimension_does_not_always_mean_semantically_meaningful_addition
}
SUBTRACTION_CONSTRAINT_MODEL {
A-B:
requires_dimension_compatibility
}
EXPONENTIAL_FUNCTION_MODEL {
exp(x)
REQUIRE:
x_dimensionless
RULE:
dimensional_argument_to_exponential_is_invalid_without_normalising_reference
}
LOGARITHM_FUNCTION_MODEL {
log(x)
REQUIRE:
argument_dimensionless
or_ratio_x/x_ref_dimensionless
RULE:
logarithm_of_dimensional_quantity_requires_reference_or_implicit_scale
}
TRIGONOMETRIC_FUNCTION_MODEL {
sin(theta)
cos(theta)
tan(theta)
REQUIRE:
argument_dimensionless_angle
conventionally_in_radians_for_calculus
RULE:
angle_quantity_kind_should_be preserved
}
GENERAL_TRANSCENDENTAL_MODEL {
functions:
exp
log
sin
cos
sinh
cosh
RULE:
arguments_generally_require_dimensionless_form
}
DERIVATIVE_DIMENSION_MODEL {
y:
dimension_Dy
x:
dimension_Dx
dy/dx:
dimension =
Dy/Dx
}
TIME_DERIVATIVE_MODEL {
dQ/dt:
dimension =
dim(Q)*T^-1
}
INTEGRAL_DIMENSION_MODEL {
integral Q dx:
dimension =
dim(Q)*dim(x)
}
TIME_INTEGRAL_MODEL {
integral Q dt:
dimension =
dim(Q)*T
}
GRADIENT_MODEL {
scalar_field:
phi
spatial_gradient:
grad(phi)
dimension:
dim(phi)*L^-1
}
DIVERGENCE_MODEL {
vector_field:
F
divergence:
div(F)
dimension:
dim(F)*L^-1
}
LAPLACIAN_MODEL {
field:
phi
dimension:
dim(phi)*L^-2
}
DIFFERENTIAL_EQUATION_MODEL {
terms:
derivatives_and_quantities
RULE:
every_summed_term_in_differential_equation_must_share_dimension
}
CONSTANT_DIMENSION_MODEL {
equation:
Y =
k*X
dimension:
dim(k) =
dim(Y)/dim(X)
RULE:
empirical_constants_may_be_dimensional
}
NUMERICAL_CONSTANT_MODEL {
pure_number:
dimensionless
examples:
2
pi
e
RULE:
mathematical_constant != dimensional_physical_constant
}
PHYSICAL_CONSTANT_MODEL {
examples:
gravitational_constant
Planck_constant
Boltzmann_constant
speed_of_light
RULE:
physical_constants_have_dimensions_or_defined_quantity_kinds_except_dimensionless_constants
}
EMPIRICAL_EQUATION_MODEL {
fitted_relation:
may_include_dimensional_coefficients
RULE:
coefficient_units MUST change_if_input_unit_system_changes
unless_relation_is_written_dimensionlessly
}
UNIT_INDEPENDENCE_MODEL {
physical_relation:
should_remain_valid_under_consistent_unit_change
RULE:
dimensional_form_supports_unit_system_independence
}
DIMENSION_MATRIX_MODEL {
quantities:
Q_1...Q_n
dimension_vectors:
columns
matrix:
D
rows:
base_dimensions
RULE:
quantity_order_and_dimension_basis MUST be preserved
}
MATRIX_RANK_MODEL {
rank:
r =
rank(D)
interpretation:
number_of_independent_dimension_constraints
}
NULL_SPACE_MODEL {
exponent_vector:
p
condition:
D*p =
0
interpretation:
product_i(Q_i^p_i)
is_dimensionless
RULE:
null_space_basis_generates_dimensionless_groups
}
BUCKINGHAM_PI_MODEL {
n:
number_of_dimensional_variables
r:
rank_of_dimension_matrix
number_of_independent_Pi_groups:
n-r
REQUIRE:
relevant_variables_complete
dimension_matrix_correct
RULE:
theorem_does_not_determine_functional_relation_between_Pi_groups
}
PI_GROUP_MODEL {
Pi_j =
product_i(Q_i^a_ij)
condition:
dimension(Pi_j) =
1
RULE:
different_valid_Pi_group_bases_can_encode_same_dimensionless_subspace
}
REPEATING_VARIABLE_MODEL {
classical_method:
select_r_dimensionally_independent_repeating_variables
RULE:
repeating_variables_should_collectively_span_dimension_space
and_should_not_form_dimensionless_group_by_themselves
}
PI_GROUP_EQUIVALENCE_MODEL {
basis_A:
Pi_1...Pi_k
basis_B:
transformed_products_or_powers
RULE:
dimensionless_group_basis_is_not_unique
}
SCALING_LAW_MODEL {
relation:
Y proportional_to
X_1^a1*...*X_n^an
dimension_constraint:
dim(Y) =
product_i(dim(X_i)^a_i)
RULE:
dimensions_can_constrain_exponents_but_may_not_uniquely_determine_them
}
POWER_LAW_MODEL {
Y =
C*X^a
dimension:
dim(C) =
dim(Y)/dim(X)^a
RULE:
C_is_dimensionless_only_if_dimensions_balance_without_it
}
SIMILARITY_MODEL {
systems:
geometrically_or_physically_related
dynamic_similarity:
matching_relevant_dimensionless_groups
RULE:
similarity_requires_identification_of_governing_dimensionless_parameters
}
GEOMETRIC_SIMILARITY_MODEL {
length_scale:
lambda_L
area_scale:
lambda_L^2
volume_scale:
lambda_L^3
}
DYNAMIC_SIMILARITY_MODEL {
examples:
Reynolds_number
Froude_number
Mach_number
Weber_number
RULE:
which_groups_must_match_depends_on_dominant_physics
}
REYNOLDS_NUMBER_MODEL {
Re =
rho*v*L/mu
dimension:
1
alternative:
v*L/nu
RULE:
variable_definitions_and_fluid_properties MUST be specified
}
FROUDE_NUMBER_MODEL {
Fr =
v/sqrt(g*L)
dimension:
1
}
MACH_NUMBER_MODEL {
Ma =
v/c
dimension:
1
}
WEBER_NUMBER_MODEL {
We =
rho*v^2*L/sigma
dimension:
1
}
PRANDTL_NUMBER_MODEL {
Pr =
nu/alpha
dimension:
1
}
NUSSELT_NUMBER_MODEL {
Nu =
h*L/k
dimension:
1
}
DIMENSIONAL_REDUCTION_MODEL {
original_variables:
n
independent_dimensionless_groups:
n-r
benefit:
reduce_parameter_space
RULE:
dimensional_reduction_does_not_remove_need_for_data_or_physical_model
}
MODEL_DISCOVERY_MODEL {
candidate_relation:
constrained_by_dimensionless_groups
RULE:
dimensional_analysis_can_generate_model_structure_candidates
but_cannot_establish_causality
}
DATA_COLLAPSE_MODEL {
measurements:
rescaled_using_dimensionless_groups
goal:
reveal_common_relationship
RULE:
successful_collapse_supports_scaling_model_but_is_not_unique_proof
}
NONDIMENSIONALISATION_MODEL {
variable:
X
reference_scale:
X_ref
dimensionless:
X_star =
X/X_ref
RULE:
reference_scale MUST be nonzero_and_defined
}
CHARACTERISTIC_SCALE_MODEL {
possible:
characteristic_length
characteristic_time
characteristic_velocity
characteristic_temperature_difference
RULE:
scale_choice_affects_dimensionless_equation_form
}
DIMENSIONLESS_EQUATION_MODEL {
original:
dimensional_governing_equation
transformed:
dimensionless_variables
output:
dimensionless_parameters
RULE:
transformation_must_be_invertible_or_document_information_loss
}
NORMALISATION_DISTINCTION_MODEL {
nondimensionalisation:
removes_dimensions_using_reference_scales
statistical_normalisation:
may_mean_rescaling_by_mean_SD_or_norm
RULE:
meanings_must_be_disambiguated
}
UNIT_CONVERSION_DISTINCTION_MODEL {
unit_conversion:
changes_unit_representation
dimensional_analysis:
reasons_about_dimension_structure
RULE:
unit_conversion != dimensional_analysis
}
UNIT_DIMENSION_DISTINCTION_MODEL {
metre:
unit
length:
quantity_kind
L:
dimension
RULE:
unit_quantity_kind_and_dimension_are_distinct_semantic_layers
}
SAME_DIMENSION_DIFFERENT_KIND_MODEL {
examples:
energy_and_torque
frequency_and_activity
absorbed_dose_and_specific_energy_contexts
RULE:
dimensional_equivalence_does_not_imply_quantity_kind_equivalence
}
ANGLE_DIMENSION_MODEL {
plane_angle:
dimensionless
named_unit:
radian
RULE:
retaining_angle_kind_can_prevent_semantic_errors_despite_dimensionless_status
}
COUNT_DIMENSION_MODEL {
count:
dimensionless
RULE:
count_quantity_semantics SHOULD be retained
}
INFORMATION_QUANTITY_MODEL {
bit_or_nat:
dimensionless_information_quantity
RULE:
information_quantity_semantics_are_not_captured_by_dimension_vector_alone
}
UNCERTAINTY_DIMENSION_MODEL {
standard_uncertainty:
same_dimension_as_quantity
covariance:
product_of_dimensions_of_two_quantities
variance:
square_of_quantity_dimension
RULE:
uncertainty_propagation MUST preserve_dimensions
}
SENSITIVITY_DIMENSION_MODEL {
c_i =
partial_Y/partial_X_i
dimension:
dim(Y)/dim(X_i)
RULE:
c_i*u(X_i) MUST have_output_dimension
}
COVARIANCE_PROPAGATION_MODEL {
Sigma_Y =
J*Sigma_X*J_transpose RULE:
matrix_units_and_variable_order MUST be dimensionally_consistent
}
ERROR_DIMENSION_MODEL {
measurement_error:
same_dimension_as_measurand
relative_error:
dimensionless
when_reference_same_kind
}
DERIVED_UNIT_VALIDATION_MODEL {
derived_unit:
decompose_to_base_units
compare:
unit_dimension_vs_quantity_dimension
RULE:
mismatch_indicates_unit_or_quantity_metadata_error
}
SYMBOLIC_ANALYSIS_MODEL {
expression_tree:
variables
constants
operators
functions
process:
recursively_compute_dimension
RULE:
unresolved_variable_dimension_produces_unresolved_result_not_false_consistency
}
DIMENSION_INFERENCE_MODEL {
equation:
dimensions_of_some_variables_known
unknown:
dimension_of_parameter
solve:
exponent_balance_equations
RULE:
inferred_dimension_does_not_identify_unique_quantity_kind
}
DIMENSION_EQUATION_MODEL {
unknown_exponents:
solve_linear_system
example:
dim(k)*dim(X)^a =
dim(Y)
RULE:
logarithmic_dimension_equations_map_products_to_linear_exponent_constraints
}
FRACTIONAL_EXPONENT_MODEL {
example:
sqrt(L^2) =
L
under_positive_magnitude_context
RULE:
algebraic_dimension_simplification_must_preserve_domain_semantics
}
RATIONAL_EXPONENT_MODEL {
dimension_vector:
may_have_rational_components
RULE:
implementation SHOULD support_exact_rational_exponents_where_possible
}
FLOATING_EXPONENT_DISTINCTION_MODEL {
computational_approximation:
decimal_exponent_value
RULE:
dimension_exponents SHOULD avoid_unnecessary_floating_roundoff
}
CANONICAL_DIMENSION_MODEL {
representation:
ordered_vector_of_base_exponents
RULE:
equivalent_symbolic_dimension_expressions SHOULD canonicalize_to_same_vector
}
DIMENSION_HASH_MODEL {
input:
canonical_dimension_vector
basis_identifier
purpose:
fast_equivalence_check
RULE:
hash_is_identifier_not_semantic_definition
}
UNIT_SYSTEM_MODEL {
possible:
SI
CGS
imperial_or_customary
natural_units
domain_specific
RULE:
dimensions_are_more_general_than_specific_unit_system
}
NATURAL_UNITS_MODEL {
conventions:
selected_constants_set_to_one
effect:
dimensions_may_be_reexpressed_or_collapsed
RULE:
natural_unit_context MUST be explicit_before_comparing_dimension_vectors_to_SI_basis
}
BASE_CHANGE_MODEL {
old_dimension_basis:
B
new_basis:
B_prime
transformation:
exponent_coordinates_change
RULE:
underlying_scaling_structure_must_be_preserved
}
AFFINE_QUANTITY_MODEL {
examples:
temperature_scale_values
RULE:
dimensional_analysis_handles_dimension_but_not_affine_origin_semantics
}
LOG_QUANTITY_MODEL {
logarithmic_level:
often_dimensionless_ratio_with_reference
RULE:
dimension_vector_alone_does_not_encode_logarithmic_reference_or_scale_factor
}
EMPIRICAL_CONSTANT_UNIT_MODEL {
empirical_equation:
y =
k*x^a
k_dimension:
dim(y)*dim(x)^(-a)
RULE:
changing_units_changes_numeric_value_of_dimensional_empirical_constant
}
OFFSET_EQUATION_MODEL {
y =
a*x+b
requirement:
dim(y)=dim(a*x)=dim(b)
RULE:
additive_offset_must_have_output_dimension
}
POLYNOMIAL_MODEL {
y =
a_0+a_1*x+a_2*x^2+...
requirement:
dim(a_n) =
dim(y)/dim(x)^n
RULE:
polynomial_coefficients_generally_have_different_dimensions
}
SERIES_MODEL {
terms:
summed
RULE:
every_term_in_series_must_have_same_dimension
}
TAYLOR_SERIES_MODEL {
f(x+delta_x):
terms_involve_derivatives_and_powers_of_delta_x
RULE:
each_Taylor_term_has_same_dimension_as_f
}
VECTOR_COMPONENT_MODEL {
vector_components:
same_quantity_dimension_within_common_physical_vector
RULE:
coordinate_components_may_have_different_numeric_values_but_same_dimension
}
MATRIX_DIMENSION_MODEL {
matrix_elements:
may_have_common_or_structured_dimensions
RULE:
generic_matrix_multiplication_with_physical_quantities_requires_elementwise_dimension_compatibility
}
LINEAR_OPERATOR_MODEL {
y =
A*x
element_dimension:
dim(A_ij) =
dim(y_i)/dim(x_j)
for_contributing_terms
RULE:
each_sum_for_y_i_must_be_dimensionally_homogeneous
}
DIFFERENTIAL_OPERATOR_MODEL {
spatial_derivative:
contributes_L^-1
temporal_derivative:
contributes_T^-1
RULE:
operator_dimension_should_be represented
}
FOURIER_TRANSFORM_MODEL {
time_signal:
x(t)
transform:
X(f) =
integral x(t)*exp(-i*2*pi*f*t)dt
dimension:
dim(X) =
dim(x)*T
under_this_transform_convention
RULE:
transform_normalisation_changes_transform_units
}
SPECTRAL_DENSITY_MODEL {
power_spectral_density:
quantity_squared_per_Hz
dimension:
dim(Q)^2*T
amplitude_spectral_density:
dim(Q)*sqrt(T)
RULE:
density_units_depend_on_frequency_variable
}
PROBABILITY_DENSITY_MODEL {
random_variable:
X_with_dimension_D
density:
p_X(x)
dimension:
D^-1
RULE:
probability_is_dimensionless_but_probability_density_may_have_inverse_units
}
DELTA_FUNCTION_MODEL {
Dirac_delta(x):
dimension =
dim(x)^-1
so_integral_delta(x)dx_is_dimensionless
RULE:
delta_function_dimension_depends_on_argument
}
INTEGRATION_MEASURE_MODEL {
dV:
L^3
dA:
L^2
dt:
T
RULE:
integration_measure_contributes_dimension
}
DIMENSIONAL_SANITY_MODEL {
purpose:
detect
missing_factor
wrong_exponent
wrong_derivative
unit_metadata_error
incompatible_addition
RULE:
passing_dimensional_check_does_not_prove_numeric_or_physical_correctness
}
LIMITATION_MODEL {
dimensional_analysis_cannot_alone_determine:
dimensionless_constants
additive_structure_in_general
exact_functional_form
vector_direction
causality
boundary_conditions
physical mechanism
RULE:
limitations MUST be explicit_in_AI_reasoning
}
METADATA_MODEL {
fields:
quantity_identifier
quantity_kind
dimension_vector
dimension_basis
unit
equation_identifier
expression
dimension_matrix
matrix_rank
null_space_basis
Pi_groups
characteristic_scales
unit_system
validation_status
unresolved_symbols
provenance
RULE:
dimensional_metadata MUST support_reconstruction_of_analysis
}
RELATIONSHIPS {
Quantity -> HAS -> Dimension
Dimension -> EXPRESSED_IN -> BaseDimensions
DerivedDimension -> COMPOSED_FROM -> BaseDimensions
Unit -> REALISES -> QuantityDimension
DimensionalVector -> REPRESENTS -> Dimension
DimensionMatrix -> CONTAINS -> DimensionalVectors
NullSpaceVector -> DEFINES -> DimensionlessGroup
BuckinghamPiAnalysis -> PRODUCES -> DimensionlessGroups
DimensionalAnalysis -> VALIDATES -> EquationHomogeneity
Similarity -> DEPENDS_ON -> DimensionlessParameters
CharacteristicScale -> SUPPORTS -> Nondimensionalisation
}
SEMANTIC_EQUIVALENCE {
"dimensional analysis" ~= analysis_using_quantity_dimensions
"dimension vector" ~= ordered_base_dimension_exponents
"dimensionless group" ~= product_of_quantities_with_zero_net_dimension
dimension != unit
dimension != quantity_kind
dimension != numerical_value
dimensional_consistency != physical_correctness
dimensionless != semantically_identical
unit_conversion != dimensional_analysis
nondimensionalisation != statistical_normalisation
same_dimension != same_quantity_kind
}
CONSTRAINTS {
dimension_basis_must_be_defined = true
dimension_vectors_must_use_consistent_basis_order = true
additive_terms_must_have_identical_dimensions = true
equation_sides_must_have_identical_dimensions = true
transcendental_function_arguments_must_be_dimensionless_or_normalised = true
dimensionless_status_does_not_remove_quantity_kind_semantics = true
dimension_matrix_quantity_order_must_be_preserved = true
Pi_groups_must_have_zero_net_dimension = true
unit_and_dimension_must_be_distinguished = true
dimensional_consistency_is_not_sufficient_for_physical_validity = true
unresolved_dimensions_must_not_be_silently_assumed = true
provenance_must_be_preserved = true
}
MEASUREMENT_MODEL {
RECORD:
analysis_identifier
quantity_identifiers
quantity_kinds
units
dimensions
dimension_vectors
dimension_basis
expression_or_equation
symbolic_constants
constant_dimensions
dimension_matrix
matrix_rank
null_space_basis
dimensionless_groups
characteristic_scales
nondimensional_variables
unit_system
validation_status
inconsistency_location
unresolved_symbols
uncertainty_dimensions
transformation_history
provenance
RULE:
dimensional_analysis_result MUST remain_linked_to_expression_quantities_basis_and_semantics
}
REASONING_RULES {
IF quantities_multiplied
THEN add_dimension_vectors
IF quantities_divided
THEN subtract_dimension_vectors
IF quantity_raised_to_power_p
THEN multiply_dimension_vector_by_p
IF quantities_added_or_subtracted
THEN require_identical_dimension_vectors
IF equation_has_left_and_right_sides
THEN compute_dimension_of_each_side
AND require_equality
IF transcendental_function_argument_has_nonzero_dimension
THEN flag_dimensionally_invalid_or_require_reference_normalisation
IF derivative_dY_dX
THEN dimension =
dim(Y)/dim(X)
IF integral_Y_dX
THEN dimension =
dim(Y)*dim(X)
IF dimension_matrix_D_known
THEN rank =
matrix_rank(D)
AND null_space_vectors_define_dimensionless_products
IF n_variables_and_rank_r
THEN independent_Pi_group_count =
n-r
IF candidate_Pi_exponent_vector_p
THEN validate
D*p=0
IF unit_metadata_known
THEN decompose_unit_to_base_dimensions
AND compare_with_declared_quantity_dimension
IF same_dimension_but_different_quantity_kind
THEN preserve_semantic_distinction
and_do_not_infer_interchangeability
IF empirical_constant_dimension_unknown
THEN infer_from_equation_balance_when_other_dimensions_known
IF dimensional_check_passes
THEN report_dimensionally_consistent
BUT do_not_infer_physical_truth
}
VALIDATION {
CHECK dimension_basis
CHECK quantity_kind
CHECK unit_dimension_mapping
CHECK dimension_vectors
CHECK multiplication_and_division
CHECK powers_and_roots
CHECK additive_homogeneity
CHECK equation_side_homogeneity
CHECK derivative_and_integral_dimensions
CHECK transcendental_arguments
CHECK physical_and_empirical_constants
CHECK dimension_matrix
CHECK matrix_rank
CHECK null_space
CHECK Pi_group_dimensionlessness
CHECK similarity_parameters
CHECK nondimensionalisation_scales
CHECK same_dimension_different_kind_cases
CHECK uncertainty_and_covariance_dimensions
CHECK unresolved_symbols
CHECK provenance
}
CONFIDENCE {
base_dimension_confidence: 1.00
derived_dimension_confidence: 1.00
homogeneity_confidence: 1.00
matrix_null_space_confidence: 1.00
Buckingham_Pi_confidence: 1.00
similarity_reasoning_confidence: 0.99
semantic_quantity_kind_confidence: 0.99
model_limitation_confidence: 1.00
RULE:
confidence decreases when quantity kinds, dimensions, basis,
equation structure, variable completeness, characteristic scales
or provenance is missing or ambiguous
}
UPDATE_MODEL {
ON new_quantity:
identify_quantity_kind
attach_unit
derive_or_attach_dimension
canonicalise_dimension_vector
preserve_provenance
ON new_equation:
parse_expression
resolve_quantity_dimensions
propagate_dimensions_through_expression_tree
validate_homogeneity
identify_unknown_parameter_dimensions
preserve_analysis_record
ON new_Buckingham_analysis:
identify_relevant_variables
build_dimension_matrix
compute_rank
compute_null_space
generate_dimensionless_groups
validate_each_group
preserve_basis_and_variable_order
ON unit_revision:
preserve_quantity_dimension
update_unit_mapping
revalidate_dimension_consistency
preserve_historical_unit_context
ON model_revision:
preserve_previous_equation
update_variables_constants_or_functions
recompute_dimension_analysis
compare_validation_status
ON semantic_revision:
preserve_previous_definition
record_revision
recompute_FLOW_HASH
}
AI_CONSUMPTION {
canonical_entity: DimensionalAnalysis
canonical_role: MathematicalQuantityAnalysisFramework
interpretation_order:
identify_quantities_and_quantity_kinds
identify_units
identify_dimension_basis
derive_dimension_vectors
parse_equation_or_expression
propagate_dimensions_through_operations
validate_additive_and_equation_homogeneity
identify_transcendental_function_constraints
identify_unknown_constant_dimensions
build_dimension_matrix_when_multivariable
compute_rank_and_null_space
identify_dimensionless_groups
identify_characteristic_scales_and_similarity_context
identify_uncertainty_dimensions
identify_provenance
validate_semantic_distinctions
produce_dimensional_reasoning_result
ambiguity_controls:
distinguish_dimension_from_unit
distinguish_dimension_from_quantity_kind
distinguish_dimension_from_numerical_value
distinguish_dimensional_consistency_from_physical_correctness
distinguish_dimensionless_quantities_by_quantity_kind
distinguish_unit_conversion_from_dimensional_analysis
distinguish_nondimensionalisation_from_statistical_normalisation
distinguish_same_dimension_from_semantic_interchangeability
preserve_quantity_kind_unit_dimension_basis_equation_group_and_provenance_context
}
PROVENANCE {
ontology_scope:
mathematical_measurement_units_measurement_mathematics_and_dimensional_analysis_semantics
parent_path:
Ontologies
-> Mathematical Ontologies
-> Measurement & Units Ontologies
-> Measurement Mathematics
-> Dimensional Analysis Ontology
authorship:
Mark Higgins
Sky (Recursive AI)
site:
https://kirethresearch.org
}
FLOW_HASH {
INPUTS:
canonical_entity
ontology_path
base_dimension_model
length_dimension_model
mass_dimension_model
time_dimension_model
current_dimension_model
temperature_dimension_model
amount_dimension_model
luminous_intensity_dimension_model
derived_dimension_model
dimension_multiplication_model
dimension_division_model
dimension_power_model
dimension_root_model
dimensionless_model
named_dimensionless_model
velocity_model
acceleration_model
force_model
energy_model
power_model
pressure_model
density_model
momentum_model
angular_momentum_model
frequency_model
charge_model
voltage_model
resistance_model
capacitance_model
inductance_model
magnetic_flux_model
magnetic_flux_density_model
electric_field_model
thermal_conductivity_model
specific_heat_model
concentration_dimension_model
dimensional_homogeneity_model
equation_check_model
addition_constraint_model
subtraction_constraint_model
exponential_function_model
logarithm_function_model
trigonometric_function_model
general_transcendental_model
derivative_dimension_model
time_derivative_model
integral_dimension_model
time_integral_model
gradient_model
divergence_model
Laplacian_model
differential_equation_model
constant_dimension_model
numerical_constant_model
physical_constant_model
empirical_equation_model
unit_independence_model
dimension_matrix_model
matrix_rank_model
null_space_model
Buckingham_Pi_model
Pi_group_model
repeating_variable_model
Pi_group_equivalence_model
scaling_law_model
power_law_model
similarity_model
geometric_similarity_model
dynamic_similarity_model
Reynolds_number_model
Froude_number_model
Mach_number_model
Weber_number_model
Prandtl_number_model
Nusselt_number_model
dimensional_reduction_model
model_discovery_model
data_collapse_model
nondimensionalisation_model
characteristic_scale_model
dimensionless_equation_model
normalisation_distinction_model
unit_conversion_distinction_model
unit_dimension_distinction_model
same_dimension_different_kind_model
angle_dimension_model
count_dimension_model
information_quantity_model
uncertainty_dimension_model
sensitivity_dimension_model
covariance_propagation_model
error_dimension_model
derived_unit_validation_model
symbolic_analysis_model
dimension_inference_model
dimension_equation_model
fractional_exponent_model
rational_exponent_model
floating_exponent_distinction_model
canonical_dimension_model
dimension_hash_model
unit_system_model
natural_units_model
base_change_model
affine_quantity_model
log_quantity_model
empirical_constant_unit_model
offset_equation_model
polynomial_model
series_model
Taylor_series_model
vector_component_model
matrix_dimension_model
linear_operator_model
differential_operator_model
Fourier_transform_model
spectral_density_model
probability_density_model
delta_function_model
integration_measure_model
dimensional_sanity_model
limitation_model
metadata_model
relationships
semantic_equivalence
constraints
measurement_model
reasoning_rules
confidence_model
update_model
PURPOSE:
deterministic_semantic_integrity_identifier
RECOMPUTE_WHEN ontology_definition_changes
base_dimension_or_basis_changes
derived_dimension_definition_changes
equation_or_expression_structure_changes
dimensional_homogeneity_rules_change
dimension_matrix_rank_or_null_space_changes
Buckingham_Pi_or_similarity_model_changes
nondimensionalisation_or_characteristic_scale_changes
unit_dimension_mapping_changes
uncertainty_or_operator_dimension_changes
quantity_kind_semantic_distinctions_change
measurement_record_changes
constraint_model_changes
reasoning_rule_changes
}
}
END FLOW