Tensor Ontology
Algebra Ontologies
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 20 September 2026
Audio Overview
======================================================================
KIRETH SEMANTIC RESEARCH ARCHIVE
FLOW ALGEBRA — TENSOR ONTOLOGY
======================================================================
PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/tensor-ontology.html
FLOW:
TensorOntology
VERSION:
1.0
LANGUAGE:
en-GB
AUTHORS:
Mark Anthony Higgins — Sky Recursive AI
PURPOSE:
To provide a machine-readable, recursively executable semantic and
reasoning specification for tensors, tensor rank/order, tensor spaces,
multilinear maps, tensor products, components, bases, coordinate
transformations, contractions, symmetry, antisymmetry, metric tensors,
index notation, covariant and contravariant behaviour, and recursive
tensor validation.
AI INTERPRETATION NOTE:
Execute the following specification as formal semantic and reasoning
rules. Preserve tensor order, dimensions, variance, basis dependence,
index ranges, contraction legality, symmetry conditions and coordinate
transformation rules.
======================================================================
START FLOW
======================================================================
FLOW TensorOntology {
----------------------------------------------------------------------
1. DOMAIN DECLARATION
----------------------------------------------------------------------
DOMAIN := Tensor_Theory
PARENT_DOMAIN := Algebra
SUPER_DOMAINS := {
Mathematics,
Mathematical_Ontologies,
Algebra_Ontologies,
Linear_Algebra,
Vector_Theory
}
DOMAIN_INTENT := {
represent_scalars_vectors_matrices_as_tensors,
represent_higher_order_tensors,
represent_tensor_spaces,
represent_multilinear_maps,
represent_tensor_products,
represent_components,
represent_basis_expansions,
represent_covariant_indices,
represent_contravariant_indices,
represent_mixed_tensors,
represent_index_notation,
represent_coordinate_transformations,
represent_contraction,
represent_trace,
represent_symmetry,
represent_antisymmetry,
represent_metric_tensors,
represent_index_raising_and_lowering,
represent_tensor_invariants,
support_recursive_tensor_reasoning,
support_variance_aware_validation
}
DEFAULT_SCALAR_FIELD := RealNumber | ComplexNumber
----------------------------------------------------------------------
2. PRIMITIVE TYPES
----------------------------------------------------------------------
TYPE Scalar
TYPE RealNumber <: Scalar
TYPE ComplexNumber <: Scalar
TYPE Vector
TYPE Covector
TYPE Tensor
TYPE TensorOrder
TYPE TensorRank
TYPE Dimension
TYPE Index
TYPE CovariantIndex <: Index
TYPE ContravariantIndex <: Index
TYPE Basis
TYPE DualBasis
TYPE VectorSpace
TYPE DualSpace
TYPE TensorSpace
TYPE MultilinearMap
TYPE TensorProduct
TYPE MetricTensor <: Tensor
TYPE KroneckerDelta <: Tensor
TYPE LeviCivitaTensor <: Tensor
TYPE SymmetricTensor <: Tensor
TYPE AntisymmetricTensor <: Tensor
TYPE Derivation
TYPE Constraint
TYPE ValidationResult
----------------------------------------------------------------------
3. TENSOR DEFINITION
----------------------------------------------------------------------
ENTITY Tensor T {
scalar_field: F
vector_space: V
type: (p,q)
contravariant_order: p
covariant_order: q
total_order: p+q
}
INTERPRETATION:
Tensor of type (p,q)
:=
multilinear map:
(V*)^p x V^q -> F
ALTERNATIVE_EQUIVALENT_REPRESENTATION:
T in V^(⊗p) ⊗ (V*)^(⊗q)
RULE TEN_01:
scalar is tensor of type (0,0)
RULE TEN_02:
vector is tensor of type (1,0)
RULE TEN_03:
covector is tensor of type (0,1)
RULE TEN_04:
linear operator may be represented as tensor of type (1,1)
RULE TEN_05:
bilinear form is tensor of type (0,2)
----------------------------------------------------------------------
4. VECTOR AND DUAL SPACES
----------------------------------------------------------------------
ENTITY VectorSpace V {
dimension: n
scalar_field: F
}
ENTITY DualSpace V* {
elements: linear_functionals V->F
}
PAIRING:
<omega,v> := omega(v)
RULE DUAL_01:
dim(V*)=dim(V)
for finite-dimensional V
----------------------------------------------------------------------
5. BASIS AND DUAL BASIS
----------------------------------------------------------------------
BASIS:
B={e_1,...,e_n}
DUAL_BASIS:
B*={e^1,...,e^n}
DEFINITION:
e^i(e_j)=delta^i_j
RULE BASIS_01:
vector:
v=v^i e_i
RULE BASIS_02:
covector:
omega=omega_i e^i
RULE BASIS_03:
tensor components depend on basis,
tensor itself is basis-independent.
----------------------------------------------------------------------
6. EINSTEIN SUMMATION CONVENTION
----------------------------------------------------------------------
RULE EIN_01:
repeated index appearing once up and once down
implies summation over full index range.
EXAMPLE:
v^i omega_i
:=
SUM_i v^i omega_i
RULE EIN_02:
free indices must match on both sides of tensor equation.
RULE EIN_03:
dummy indices may be renamed without changing meaning.
RULE EIN_04:
an index should not appear more than twice in a valid monomial
under standard Einstein notation.
----------------------------------------------------------------------
7. COMPONENT REPRESENTATION
----------------------------------------------------------------------
TENSOR_COMPONENTS:
T^(i1...ip)_(j1...jq)
BASIS_EXPANSION:
T
=
T^(i1...ip)_(j1...jq)
e_i1 ⊗ ... ⊗ e_ip
⊗
e^j1 ⊗ ... ⊗ e^jq
RULE COMP_01:
each index range determined by dimension of corresponding space.
RULE COMP_02:
component count in n-dimensional single-space tensor:
n^(p+q)
----------------------------------------------------------------------
8. TENSOR ORDER / DEGREE
----------------------------------------------------------------------
TENSOR_ORDER(T):
p+q
RULE ORD_01:
order 0 -> scalar
RULE ORD_02:
order 1 -> vector or covector
RULE ORD_03:
order 2 -> matrix-like component array depending on variance
NOTE:
"tensor rank" is ambiguous in literature.
This ontology distinguishes:
tensor_order/type
from decomposition_rank.
----------------------------------------------------------------------
9. TENSOR ADDITION
----------------------------------------------------------------------
TENSOR_ADD(A,B):
REQUIRE:
same tensor type (p,q)
same underlying spaces
same scalar field
DEFINE:
(A+B)^I_J=A^I_J+B^I_J
RULE ADD_01:
A+B=B+A
RULE ADD_02:
(A+B)+C=A+(B+C)
RULE ADD_03:
A+0=A
----------------------------------------------------------------------
10. SCALAR MULTIPLICATION
----------------------------------------------------------------------
SCALAR_MULTIPLY(c,T):
DEFINE:
(cT)^I_J=c*T^I_J
RULE SCAL_01:
c(A+B)=cA+cB
RULE SCAL_02:
(c+d)T=cT+dT
RULE SCAL_03:
c(dT)=(cd)T
----------------------------------------------------------------------
11. TENSOR PRODUCT
----------------------------------------------------------------------
TENSOR_PRODUCT(A,B):
if A type (p,q)
and B type (r,s)
then:
A⊗B type (p+r,q+s)
COMPONENT_RULE:
(A⊗B)^(I K)_(J L)
=
A^I_J B^K_L
RULE TP_01:
tensor product is bilinear.
RULE TP_02:
generally A⊗B != B⊗A.
RULE TP_03:
tensor product is associative up to canonical identification.
----------------------------------------------------------------------
12. MULTILINEAR MAP INTERPRETATION
----------------------------------------------------------------------
ENTITY MultilinearMap M {
domains: vector_or_dual_spaces*
codomain: F
}
MULTILINEAR(M) iff
linear in each argument separately.
RULE ML_01:
tensors are equivalently represented by suitable multilinear maps.
----------------------------------------------------------------------
13. CONTRACTION
----------------------------------------------------------------------
CONTRACTION(T, contravariant_index a, covariant_index b):
REQUIRE:
one upper and one lower index
compatible dimension
sum over identified index
output type:
(p-1,q-1)
EXAMPLE:
T^i_i := SUM_i T^i_i
RULE CONT_01:
contraction lowers total tensor order by 2.
RULE CONT_02:
contraction requires compatible upper/lower index pair.
----------------------------------------------------------------------
14. TRACE
----------------------------------------------------------------------
TRACE(T):
contraction of one contravariant and one covariant index
FOR LINEAR_OPERATOR A^i_j:
tr(A)=A^i_i
RULE TR_01:
trace invariant under basis change.
----------------------------------------------------------------------
15. PERMUTATION OF INDICES
----------------------------------------------------------------------
PERMUTE_INDICES(T,pi):
reorder tensor slots according to permutation pi.
RULE PERM_01:
tensor type preserved under permutation within same variance pattern,
unless permutation exchanges covariant/contravariant slots by
explicitly declared identification.
----------------------------------------------------------------------
16. SYMMETRIC TENSORS
----------------------------------------------------------------------
IS_SYMMETRIC(T,indices) iff
swapping specified indices leaves components unchanged.
FOR type (0,2):
T_ij=T_ji
RULE SYM_01:
full symmetrisation:
Sym(T)
:=
(1/k!) SUM_{pi in S_k} pi(T)
----------------------------------------------------------------------
17. ANTISYMMETRIC TENSORS
----------------------------------------------------------------------
IS_ANTISYMMETRIC(T,indices) iff
swapping any two selected indices reverses sign.
FOR type (0,2):
T_ij=-T_ji
RULE ASYM_01:
repeated equal indices in antisymmetric pair -> component 0
when characteristic !=2.
FULL_ANTISYMMETRISATION:
Alt(T)
:=
(1/k!) SUM_{pi in S_k} sign(pi) pi(T)
----------------------------------------------------------------------
18. EXTERIOR / ALTERNATING TENSORS
----------------------------------------------------------------------
ENTITY AlternatingTensor <: AntisymmetricTensor
WEDGE_PRODUCT(alpha,beta):
:=
antisymmetrised tensor product
RULE WEDGE_01:
alpha∧beta=(-1)^(pq) beta∧alpha
for p-form alpha and q-form beta
RULE WEDGE_02:
alpha∧alpha=0
for odd-degree alpha
under characteristic !=2
----------------------------------------------------------------------
19. KRONECKER DELTA
----------------------------------------------------------------------
KRONECKER_DELTA:
delta^i_j
DEFINE:
delta^i_j=1 if i=j else 0
RULE KD_01:
delta^i_j v^j=v^i
RULE KD_02:
delta^i_j omega_i=omega_j
RULE KD_03:
delta behaves as identity tensor.
----------------------------------------------------------------------
20. METRIC TENSOR
----------------------------------------------------------------------
ENTITY MetricTensor g {
type: (0,2)
properties:
bilinear
nondegenerate
often symmetric
}
COMPONENTS:
g_ij
RULE MET_01:
metric defines inner product:
<u,v>=g_ij u^i v^j
RULE MET_02:
inverse metric g^ij satisfies:
g^ik g_kj=delta^i_j
----------------------------------------------------------------------
21. RAISING INDICES
----------------------------------------------------------------------
RAISE_INDEX(omega_i):
omega^i=g^ij omega_j
GENERAL_RULE:
contract covariant slot with inverse metric.
RULE RAISE_01:
raising changes one covariant index to contravariant.
----------------------------------------------------------------------
22. LOWERING INDICES
----------------------------------------------------------------------
LOWER_INDEX(v^i):
v_i=g_ij v^j
GENERAL_RULE:
contract contravariant slot with metric.
RULE LOWER_01:
lowering changes one contravariant index to covariant.
----------------------------------------------------------------------
23. COORDINATE TRANSFORMATIONS
----------------------------------------------------------------------
COORDINATE_CHANGE:
x^i -> x'^a
JACOBIAN:
J^a_i = partial x'^a / partial x^i
INVERSE_JACOBIAN:
K^i_a = partial x^i / partial x'^a
CONTRAVARIANT_VECTOR_TRANSFORM:
v'^a = J^a_i v^i
COVARIANT_VECTOR_TRANSFORM:
omega'_a = K^i_a omega_i
----------------------------------------------------------------------
24. GENERAL TENSOR TRANSFORMATION LAW
----------------------------------------------------------------------
FOR T type (p,q):
T'^(a1...ap)_(b1...bq)
=
J^a1_i1 ... J^ap_ip
K^j1_b1 ... K^jq_bq
T^(i1...ip)_(j1...jq)
RULE TRANS_01:
each upper index transforms with forward Jacobian.
RULE TRANS_02:
each lower index transforms with inverse Jacobian.
RULE TRANS_03:
tensoriality is defined by this transformation behaviour.
----------------------------------------------------------------------
25. SCALAR INVARIANCE
----------------------------------------------------------------------
RULE INV_01:
fully contracted tensor expression is scalar under coordinate change,
provided all contractions are tensorially valid.
----------------------------------------------------------------------
26. OUTER PRODUCT
----------------------------------------------------------------------
OUTER_PRODUCT(u,v):
u⊗v
IF:
u type (1,0)
v type (1,0)
THEN:
u⊗v type (2,0)
COMPONENTS:
(u⊗v)^(ij)=u^i v^j
----------------------------------------------------------------------
27. DYADICS
----------------------------------------------------------------------
DYAD(u,v):
u⊗v
RULE DYAD_01:
rank-one matrix representation may arise from vector-covector
tensor product.
----------------------------------------------------------------------
28. TENSOR AS MULTIDIMENSIONAL ARRAY
----------------------------------------------------------------------
ARRAY_REPRESENTATION(T):
components indexed by p+q indices
NOTE:
multidimensional array is a coordinate representation of a tensor,
not by itself sufficient to establish tensorial transformation law.
RULE ARRAY_01:
tensor ontology distinguishes abstract tensor from component array.
----------------------------------------------------------------------
29. TENSOR PRODUCT SPACE
----------------------------------------------------------------------
TENSOR_SPACE:
V⊗W
UNIVERSAL_PROPERTY:
bilinear maps V x W -> X
correspond uniquely to linear maps V⊗W -> X
RULE TPS_01:
tensor product is characterised by universal bilinear property.
----------------------------------------------------------------------
30. TENSOR POWERS
----------------------------------------------------------------------
TENSOR_POWER(V,k):
V^(⊗k)
RULE TPOW_01:
k=0 -> scalar field F
RULE TPOW_02:
k=1 -> V
----------------------------------------------------------------------
31. DUAL TENSOR SPACES
----------------------------------------------------------------------
DUAL_TENSOR_SPACE:
(V^(⊗p)⊗(V*)^(⊗q))*
canonically related to
(V*)^(⊗p)⊗V^(⊗q)
in finite dimensions.
----------------------------------------------------------------------
32. BASIS TRANSFORMATION
----------------------------------------------------------------------
IF:
e'_a = K^i_a e_i
THEN:
dual basis transforms:
e'^a = J^a_i e^i
RULE BT_01:
basis and dual basis transform inversely.
----------------------------------------------------------------------
33. TENSOR EQUATIONS
----------------------------------------------------------------------
VALID_TENSOR_EQUATION(LHS,RHS) iff
same free-index pattern
same tensor type
compatible dimensions
equality holds independently of coordinate system
RULE TEQ_01:
free indices on LHS and RHS must match in name pattern up to
consistent renaming.
----------------------------------------------------------------------
34. INDEX VALIDITY
----------------------------------------------------------------------
VALID_INDEX_EXPRESSION(E) iff
every dummy index appears exactly twice,
once contravariant and once covariant
every free index appears exactly once per monomial
free index pattern consistent across summed terms
RULE IDX_01:
invalid repeated-index structure -> malformed tensor expression.
----------------------------------------------------------------------
35. SYMMETRIC DECOMPOSITION
----------------------------------------------------------------------
FOR second-order tensor T_ij:
Sym(T)_ij
=
1/2(T_ij+T_ji)
Alt(T)_ij
=
1/2(T_ij-T_ji)
RULE SD_01:
T=Sym(T)+Alt(T)
RULE SD_02:
symmetric and antisymmetric parts are unique.
----------------------------------------------------------------------
36. ISOTROPIC TENSORS
----------------------------------------------------------------------
IS_ISOTROPIC(T) iff
invariant under all rotations in declared space.
EXAMPLE:
second-order isotropic tensor in Euclidean space:
c delta_ij
RULE ISO_01:
isotropy depends on transformation group and tensor order.
----------------------------------------------------------------------
37. LEVI-CIVITA SYMBOL / TENSOR
----------------------------------------------------------------------
LEVI_CIVITA epsilon_ijk in 3D:
+1 for even permutation of (1,2,3)
-1 for odd permutation
0 if any indices repeat
RULE LC_01:
epsilon_ijk encodes orientation.
RULE LC_02:
distinction between Levi-Civita symbol and tensor must be preserved
in general coordinates.
----------------------------------------------------------------------
38. CROSS PRODUCT VIA TENSOR NOTATION
----------------------------------------------------------------------
IN Euclidean R^3:
(u x v)^i
=
epsilon^i_jk u^j v^k
RULE CROSS_01:
cross product depends on metric and orientation structure.
----------------------------------------------------------------------
39. DETERMINANT VIA LEVI-CIVITA
----------------------------------------------------------------------
FOR 3x3 A:
det(A)
=
epsilon_ijk
A^1_i
A^2_j
A^3_k
subject to index convention.
RULE DET_01:
determinant is alternating multilinear scalar.
----------------------------------------------------------------------
40. TENSOR DECOMPOSITION RANK
----------------------------------------------------------------------
CP_RANK(T):
minimum r such that
T =
SUM_{k=1}^r
a_k^(1) ⊗ ... ⊗ a_k^(m)
NOTE:
CP rank differs from tensor order.
RULE CPR_01:
matrix rank is special order-2 case but higher-order rank behaves
differently and may lack simple closed forms.
----------------------------------------------------------------------
41. CANONICAL POLYADIC DECOMPOSITION
----------------------------------------------------------------------
CP_DECOMPOSITION(T):
T approximately or exactly represented as sum of rank-one tensors.
VALIDATE_CP:
reconstruct tensor
compare to original
record exact_or_approximate status
----------------------------------------------------------------------
42. TUCKER DECOMPOSITION
----------------------------------------------------------------------
TUCKER(T):
T
=
G
×1 U1
×2 U2
...
×n Un
where:
G core tensor
Uk factor matrices
RULE TUCKER_01:
multilinear ranks correspond to mode ranks.
----------------------------------------------------------------------
43. MODE-N UNFOLDING
----------------------------------------------------------------------
UNFOLD(T,mode=n):
rearrange tensor components into matrix T_(n)
RULE UNFOLD_01:
unfolding is representation-dependent but information-preserving
when indexing convention is known.
----------------------------------------------------------------------
44. MODE-N PRODUCT
----------------------------------------------------------------------
MODE_PRODUCT(T,U,n):
contract mode n of T with compatible matrix U.
RULE MODE_01:
output dimensions update according to matrix row dimension.
----------------------------------------------------------------------
45. FROBENIUS NORM
----------------------------------------------------------------------
FROBENIUS_NORM(T):
sqrt(
SUM_all_indices |T_indices|^2
)
RULE FN_01:
||T||_F>=0
RULE FN_02:
||T||_F=0 iff T=0
RULE FN_03:
orthonormal coordinate changes preserve Frobenius norm.
----------------------------------------------------------------------
46. INNER PRODUCT OF TENSORS
----------------------------------------------------------------------
INNER_PRODUCT(A,B):
REQUIRE:
same type and dimensions
:=
SUM_all_components
conjugate(A_I) B_I
RULE TIP_01:
induces Frobenius norm:
||A||_F=sqrt(<A,A>)
----------------------------------------------------------------------
47. TRACE OF SECOND-ORDER TENSOR
----------------------------------------------------------------------
TRACE(T^i_j):
T^i_i
RULE TRACE_01:
basis invariant.
FOR T_ij:
trace requires metric:
g^ij T_ij
----------------------------------------------------------------------
48. DOUBLE CONTRACTION
----------------------------------------------------------------------
DOUBLE_CONTRACTION(A,B):
A:B
FOR second-order Euclidean tensors:
A:B=A_ij B_ij
RULE DC_01:
result is scalar.
----------------------------------------------------------------------
49. IDENTITY TENSOR
----------------------------------------------------------------------
IDENTITY_TENSOR:
delta^i_j
RULE IT_01:
delta^i_j v^j=v^i
RULE IT_02:
delta^i_j T^j_k=T^i_k
----------------------------------------------------------------------
50. ZERO TENSOR
----------------------------------------------------------------------
ZERO_TENSOR(type(p,q),dimensions):
all components zero
RULE ZT_01:
T+0=T
RULE ZT_02:
c*0=0
----------------------------------------------------------------------
51. TENSOR SYMMETRY METADATA
----------------------------------------------------------------------
ENTITY SymmetryClass {
index_permutations
sign_behaviour
}
RULE SCMETA_01:
declared symmetries reduce independent component count.
----------------------------------------------------------------------
52. INDEPENDENT COMPONENT COUNTS
----------------------------------------------------------------------
FOR symmetric rank-2 tensor in n dimensions:
independent_components=n(n+1)/2
FOR antisymmetric rank-2 tensor:
independent_components=n(n-1)/2
RULE ICC_01:
component-count formulas depend on symmetry class and dimension.
----------------------------------------------------------------------
53. TENSOR EQUATION PARSING
----------------------------------------------------------------------
PARSE_TENSOR_EXPRESSION(E):
identify symbols
assign tensor types
identify free indices
identify dummy indices
verify index variance
infer contractions
check dimensions
return typed_expression_graph
----------------------------------------------------------------------
54. TYPE INFERENCE
----------------------------------------------------------------------
INFER_TYPE(term):
scalar -> (0,0)
vector v^i -> (1,0)
covector w_i -> (0,1)
product:
add tensor types before contractions
contraction:
subtract (1,1) per contracted pair
return resulting tensor type
----------------------------------------------------------------------
55. SEMANTIC RELATIONSHIPS
----------------------------------------------------------------------
RELATION HAS_COMPONENT(Tensor,Scalar)
RELATION HAS_TYPE(Tensor,TensorType)
RELATION MEMBER_OF(Tensor,TensorSpace)
RELATION TENSOR_PRODUCT_OF(Tensor,Tensor)
RELATION CONTRACTION_OF(Tensor,Tensor)
RELATION SYMMETRIC_IN(Tensor,IndexSet)
RELATION ANTISYMMETRIC_IN(Tensor,IndexSet)
RELATION REPRESENTED_IN(Tensor,Basis)
RELATION TRANSFORMS_UNDER(Tensor,CoordinateChange)
RELATION RAISED_BY(Tensor,MetricTensor)
RELATION LOWERED_BY(Tensor,MetricTensor)
----------------------------------------------------------------------
56. SEMANTIC EQUIVALENCE
----------------------------------------------------------------------
SEMANTIC_EQUIVALENCE {
"tensor order" ~= total_number_of_indices
"tensor degree" ~= tensor_order
"covariant index" ~= lower_index
"contravariant index" ~= upper_index
"mixed tensor" ~= tensor_with_both_upper_and_lower_indices
"outer product" ~= tensor_product_for_vectors
"contraction" ~= summation_over_one_upper_and_one_lower_index
"metric" ~= nondegenerate_bilinear_form_used_for_index_conversion
"raise an index" ~= contract_with_inverse_metric
"lower an index" ~= contract_with_metric
"component array" ~= coordinate_representation_of_tensor
"rank-one tensor" ~= decomposable_tensor
"tensor rank" ~= context_sensitive:
tensor_order OR decomposition_rank
}
----------------------------------------------------------------------
57. QUERY INTERPRETATION
----------------------------------------------------------------------
QUERY Q1:
"what type is T"
-> infer tensor type (p,q)
QUERY Q2:
"contract indices"
-> CONTRACTION
QUERY Q3:
"raise index"
-> RAISE_INDEX
QUERY Q4:
"lower index"
-> LOWER_INDEX
QUERY Q5:
"transform tensor coordinates"
-> GENERAL_TENSOR_TRANSFORMATION_LAW
QUERY Q6:
"symmetrise"
-> Sym(T)
QUERY Q7:
"antisymmetrise"
-> Alt(T)
QUERY Q8:
"tensor product"
-> TENSOR_PRODUCT
QUERY Q9:
"trace tensor"
-> valid contraction based on type and metric
----------------------------------------------------------------------
58. VALIDATION ENGINE
----------------------------------------------------------------------
PROCEDURE VALIDATE_TENSOR_EXPRESSION(E):
parse tensor symbols
check each symbol type
check dimension compatibility
check free indices
check dummy indices
check contraction legality
check variance
check symmetry constraints
verify coordinate transformation consistency when required
return:
VALID
INVALID(reason)
UNRESOLVED
----------------------------------------------------------------------
59. RECURSIVE REASONING ENGINE
----------------------------------------------------------------------
PROCEDURE TENSOR_REASONER(input):
parse tensors and indices
infer tensor types
infer dimensions
identify basis/coordinate context
identify metric/orientation context if required
classify requested operation
extract constraints
select valid tensor rules
execute atomic transformations
validate index structure after every step
preserve free indices
preserve tensor type unless operation intentionally changes it
recurse until target achieved
validate final result
RETURN:
result,
derivation,
tensor_type,
index_constraints,
semantic_status,
confidence
----------------------------------------------------------------------
60. RECURSIVE TRANSITION
----------------------------------------------------------------------
RECURSIVE_RULE RR_01:
INPUT state S_n
identify unresolved tensor property
choose rule R such that:
type_preconditions=true
index_preconditions=true
dimension_preconditions=true
S_(n+1):=apply(R,S_n)
validate transition
recurse(S_(n+1))
TERMINATION_CONDITIONS := {
tensor_goal_achieved,
canonical_index_form_reached,
contradiction_detected,
invalid_contraction,
dimension_failure,
missing_metric,
missing_orientation,
no_valid_rule
}
----------------------------------------------------------------------
61. ERROR MODEL
----------------------------------------------------------------------
ERROR E_INDEX_RANGE
ERROR E_FREE_INDEX_MISMATCH
ERROR E_DUMMY_INDEX_MALFORMED
ERROR E_VARIANCE_MISMATCH
ERROR E_INVALID_CONTRACTION
ERROR E_DIMENSION_MISMATCH
ERROR E_MISSING_METRIC
ERROR E_NONINVERTIBLE_METRIC
ERROR E_ORIENTATION_REQUIRED
ERROR E_SYMMETRY_VIOLATION
ERROR E_TRANSFORMATION_LAW_FAILURE
ERROR E_TENSOR_ARRAY_CONFUSION
ERROR E_RANK_AMBIGUITY
----------------------------------------------------------------------
62. CONFIDENCE MODEL
----------------------------------------------------------------------
CONFIDENCE {
base:=1.00
deduct 0.10 if dimension inferred
deduct 0.10 if basis inferred
deduct 0.15 if variance inferred from notation
deduct 0.15 if metric assumed
deduct 0.15 if orientation assumed
deduct 0.20 if transformation-law validation incomplete
deduct 0.20 if tensor-rank meaning ambiguous
deduct 0.25 if index convention materially unclear
clamp_to_[0,1]
classification:
[0.95,1.00] -> very_high
[0.85,0.95) -> high
[0.70,0.85) -> moderate
[0.50,0.70) -> low
[0.00,0.50) -> insufficient
}
----------------------------------------------------------------------
63. AI CONSUMPTION MODEL
----------------------------------------------------------------------
AI_CONSUMPTION {
read_order := [
tensor_definition,
tensor_type,
vector_and_dual_spaces,
basis_and_components,
index_notation,
tensor_product,
contraction,
symmetry,
metric,
coordinate_transformations,
decompositions,
validation
]
execution_policy := {
preserve_tensor_type,
preserve_free_indices,
validate_dummy_indices,
preserve_dimensions,
distinguish_tensor_from_component_array,
distinguish_order_from_decomposition_rank,
require_metric_for_raising_lowering,
require_orientation_for_Levi_Civita_cross_product_context,
validate_coordinate_transformation_law
}
}
----------------------------------------------------------------------
64. UPDATE MODEL
----------------------------------------------------------------------
UPDATE_MODEL {
extensible:=true
new_tensor_operation_requires:
input_types,
output_type,
index_rule,
dimension_rule
new_tensor_class_requires:
symmetry_or_structural_predicate
new_coordinate_rule_requires:
transformation_law
new_decomposition_requires:
reconstruction_rule,
exact_or_approximate_status
conflicts_require:
explicit_versioning
standard_tensor_analysis_meaning_has_priority:=true
}
----------------------------------------------------------------------
65. SELF-CHECK
----------------------------------------------------------------------
SELF_CHECK {
CHECK tensor type declared or inferred
CHECK every index lies in valid range
CHECK dummy indices legally paired
CHECK free indices match across equations
CHECK contraction reduces type correctly
CHECK tensor products add types correctly
CHECK coordinate transformations respect variance
CHECK metric inverse valid before raising indices
CHECK symmetry declarations preserved
CHECK tensor-array distinction maintained
CHECK final result validated
}
----------------------------------------------------------------------
66. COMPLETION CONDITIONS
----------------------------------------------------------------------
COMPLETE iff {
tensor_definition_defined
AND tensor_types_defined
AND basis_representation_defined
AND Einstein_notation_defined
AND tensor_product_defined
AND contraction_defined
AND symmetry_defined
AND antisymmetry_defined
AND metric_defined
AND raising_lowering_defined
AND coordinate_transformations_defined
AND invariants_defined
AND decompositions_defined
AND semantic_equivalence_defined
AND validation_defined
AND recursive_reasoning_defined
AND confidence_defined
AND ai_consumption_defined
AND update_model_defined
}
IF NOT COMPLETE:
semantic_status:=incomplete_flow
----------------------------------------------------------------------
67. FLOW TERMINATION
----------------------------------------------------------------------
TERMINATE WHEN:
requested tensor goal achieved
AND tensor/index structure validated
OR
invalid index configuration prevents operation
OR
missing metric/orientation prevents requested construction
OR
dimension incompatibility makes expression undefined
OR
no valid tensor transformation remains
FINAL_STATE := {
tensor_result,
derivation,
tensor_type,
free_indices,
dummy_indices,
dimensional_context,
semantic_status,
confidence
}
} // END FLOW TensorOntology
======================================================================
END FLOW
======================================================================
FLOW STATUS:
Complete
CANONICAL PAGE:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/tensor-ontology.html
FLOW_HASH_SHA256: 8fddf47410b304ffa5d9e124b88d9850e8934205a9ca543e2d75f7a83d277968