Constraint-Governed Holonomic Jet Transport
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Constraint-Governed Holonomic Jet Transport
Geometry of Jets, Differential Relations, Integrability, and Global Obstruction Theory
Governing problem
Given manifolds X and Y, admissible source and target structures, and a differential relation
R ⊂ Jᵏ(X, Y),
determine whether there exists a native global map
F: X → Y
such that:
jᵏF(x) ∈ Rₓ for every x ∈ X,
while preserving integrability, mixed interactions, boundary conditions, singular strata, transport compatibility, and all surviving globalization residue.
Part I. Reconstructing the Foundations
1. The derivative hierarchy reconsidered
1.1 Gradient, Jacobian, and Hessian as low-order projections
1.2 Why matrices are representations rather than native objects
1.3 From numerical differentiation to geometric transport
1.4 Derivative order versus information type
1.5 Scalar-output and vector-output distinctions
1.6 Directional, multilinear, and tensorial derivative data
1.7 Derivatives as infinitesimal transition laws
1.8 Why determinants do not constitute complete differential information
1.9 Failure of “higher derivative equals larger matrix”
1.10 The passage from differential calculus to jet geometry
2. Invalid or incomplete fundamentals
2.1 The gradient is not primitive
2.2 The metric hidden inside the gradient
2.3 The Hessian is not intrinsically a matrix of second partials
2.4 The connection hidden inside second covariant differentiation
2.5 Coordinate partial derivatives versus invariant derivatives
2.6 Equality of mixed partials and its unstated regularity assumptions
2.7 The Jacobian matrix without declared source and target carriers
2.8 The determinant as destructive scalar collapse
2.9 Local nonsingularity versus global invertibility
2.10 Pointwise algebraic admissibility versus realizability
2.11 Formal derivative data versus derivatives of an actual map
2.12 Local closure falsely promoted to global closure
2.13 Pairwise analysis falsely substituted for n-ary interaction
2.14 Smooth interior analysis falsely substituted for boundary closure
2.15 Globalization falsely identified with gluing or certification
3. The native triadic carrier
3.1 Source carrier, transport law, target carrier
3.2 The irreducible transport triad
TₓX ──dFₓ──→ T_{F(x)}Y
3.3 Source admissibility cone
3.4 Target admissibility cone
3.5 Transport compatibility between admissible cones
3.6 Map, variation, and response as a three-part structure
3.7 Why the derivative cannot be separated from its domain and codomain
3.8 Mixed variation as an irreducible interaction carrier
3.9 Triadic structure in constrained optimization
3.10 Triadic structure in control and kinematics
3.11 Triadic structure in nonlinear PDEs
3.12 Exact conditions under which dyadic projections are recoverable
3.13 Polarization as licensed reconstruction
3.14 Mixed-term extinction as a theorem, not an assumption
Part II. Differential Geometry Required Before Jet Theory
4. Manifolds, bundles, and local representations
4.1 Smooth manifolds and coordinate charts
4.2 Tangent vectors as derivations
4.3 Cotangent vectors and differentials
4.4 Vector bundles and bundle maps
4.5 Tensor bundles
4.6 Symmetric and alternating tensor powers
4.7 Sections and local trivializations
4.8 Coordinate transition functions
4.9 Natural versus coordinate-dependent constructions
4.10 Pullbacks and pushforwards
4.11 Immersions, submersions, and embeddings
4.12 Rank strata and singular differential behavior
5. Metrics, duality, and gradient construction
5.1 The differential (df) as the native first derivative
5.2 Musical isomorphisms ( \flat ) and ( \sharp )
5.3 Metric conversion of covectors into vectors
5.4 Gradient fields on Riemannian manifolds
5.5 Pseudo-Riemannian gradients
5.6 Degenerate metrics and failed gradient reconstruction
5.7 Gradient flows and metric dependence
5.8 Natural gradient and information geometry
5.9 Projected gradients on constraint manifolds
5.10 Gradient discontinuities and nonsmooth replacements
6. Connections, curvature, and Hessian construction
6.1 Why repeated differentiation requires a connection
6.2 Affine connections
6.3 Covariant differentiation
6.4 Torsion
6.5 Curvature
6.6 Parallel transport
6.7 Holonomy
6.8 The covariant Hessian
Hess∇ f = ∇df
6.9 Hessians of maps between manifolds
6.10 Dependence of the Hessian on connection choice
6.11 Intrinsic Hessians at critical points
6.12 Hessian restriction to submanifolds
6.13 Second fundamental form
6.14 Curvature terms generated by derivative commutation
6.15 Higher covariant derivatives and noncommutativity
Part III. Geometry of Jets
7. Jets as finite local transition geometry
7.1 Contact equivalence of maps at a point
7.2 Zeroth jets: values
7.3 First jets: values and differentials
7.4 Second jets: values, first derivatives, and second derivatives
7.5 Higher-order jets
7.6 Coordinate descriptions of jets
7.7 Intrinsic definitions through equivalence classes
7.8 Jet spaces (J^k(X,Y))
7.9 Jet bundles over source and target
7.10 Source, target, and projection maps
7.11 Fibres of the jet bundle
7.12 Affine-bundle structure of successive jet orders
7.13 Infinite jets (J^\infty(X,Y))
7.14 Germs versus finite jets
7.15 Information retained and discarded at finite order
8. Holonomic and formal jet fields
8.1 Jet prolongation of a map
jᵏF: X → Jᵏ(X, Y)
8.2 Holonomic sections
8.3 Formal sections
8.4 Semiholonomic and nonholonomic jets
8.5 The contact distribution
8.6 Contact forms
8.7 Characterization of holonomic sections
8.8 Why arbitrary jet fields are not integrable
8.9 Formal solutions versus genuine solutions
8.10 Jet-level compatibility between neighboring points
8.11 Local reconstruction from compatible jets
8.12 Nonuniqueness of reconstruction
8.13 Jet interpolation
8.14 Whitney extension problems
8.15 Singular and stratified jet spaces
9. Higher derivatives as n-ary carriers
9.1 (k)-th derivatives as symmetric multilinear maps
9.2 Native arity of higher variations
9.3 Diagonal evaluations versus mixed evaluations
9.4 Polarization identities
9.5 Loss incurred by directional-only sampling
9.6 Symmetric tensor decomposition
9.7 Higher derivative contractions
9.8 Higher-order Taylor transport
9.9 Faà di Bruno structure
9.10 Rooted-tree representations of composition
9.11 Higher covariant derivatives
9.12 Curvature-induced failure of full symmetry
9.13 Higher jets in the presence of torsion
9.14 Multijet spaces and interactions among several points
9.15 Jet transversality and genericity
Part IV. Differential Relations
10. Differential equations as geometric subsets
10.1 Differential relations
𝓡 ⊂ Jᵏ(X, Y)
10.2 Equations, inequalities, and inclusion relations
10.3 Linear and nonlinear relations
10.4 Open and closed relations
10.5 Regular and singular relations
10.6 Under-determined, determined, and over-determined systems
10.7 Scalar PDEs as jet submanifolds
10.8 Systems of PDEs
10.9 Differential-algebraic relations
10.10 Rank constraints
10.11 Convexity and positivity relations
10.12 Contact and noncontact constraints
10.13 Boundary differential relations
10.14 Differential inclusions
10.15 Relations on stratified spaces
11. Admissibility geometry
11.1 Pointwise admissibility
11.2 Differential admissibility
11.3 Holonomic admissibility
11.4 Boundary admissibility
11.5 Global admissibility
11.6 Constraint cones in jet fibres
11.7 Eigenvalue cones for Hessian equations
11.8 Positivity and ellipticity cones
11.9 Rank-stratified admissibility
11.10 Orientation-preserving relations
11.11 Symplectic and contact relations
11.12 Causal and hyperbolic cones
11.13 Control-accessibility constraints
11.14 State-dependent admissibility
11.15 Admissibility under singular limits
12. Constraint transport
12.1 Transport of admissible source variations
12.2 First-order transport condition
dFₓ(Aₓ) ⊆ B_{F(x)}
12.3 Second-order transport of feasible accelerations
12.4 Transport of tangent cones
12.5 Transport of normal cones
12.6 Transport across boundaries and corners
12.7 Transport between singular strata
12.8 Compatibility with group actions
12.9 Gauge-covariant transport
12.10 Conservation laws as transport constraints
12.11 Nonlocal constraints invisible to finite jets
12.12 Constraint propagation under composition
12.13 Constraint propagation under inversion
12.14 Constraint propagation under quotienting
12.15 Residue created by failed transport
Part V. Local Integrability
13. Elementary compatibility conditions
13.1 When a vector field is a gradient
13.2 Closed one-forms and local potentials
13.3 Poincaré lemma
13.4 When a matrix field is a Jacobian
13.5 Row-wise curl-free conditions
13.6 Mixed-partial compatibility
13.7 When a symmetric matrix field is a Hessian
13.8 Third-order compatibility conditions
13.9 Higher-order symmetry constraints
13.10 Local exactness versus global exactness
13.11 Compatibility on nonsimply connected domains
13.12 Regularity needed for derivative interchange
14. Frobenius integrability
14.1 Distributions
14.2 Involutivity
14.3 Lie brackets
14.4 Integral manifolds
14.5 Frobenius theorem
14.6 Pfaffian systems
14.7 Differential ideals
14.8 Constraint distributions in control theory
14.9 Nonholonomic constraints
14.10 Bracket generation
14.11 Local accessibility
14.12 Integrable versus nonintegrable geometric structures
14.13 Torsion as an integrability residue
14.14 Curvature as a transport residue
15. Prolongation and hidden equations
15.1 Prolonging a differential relation
15.2 Derived compatibility equations
15.3 Projection from prolonged relations
15.4 Equations generated by differentiation
15.5 Hidden algebraic constraints
15.6 Hidden rank conditions
15.7 Hidden boundary conditions
15.8 Constraint closure under repeated prolongation
15.9 Finite versus infinite prolongation
15.10 Prolongation towers
15.11 Consistency of successive jet levels
15.12 Formal integrability
15.13 Involutive systems
15.14 Symbol spaces
15.15 Characteristic varieties
16. Spencer and compatibility theory
16.1 Symbol complexes
16.2 Spencer differential
16.3 Spencer cohomology
16.4 Obstructions to formal integrability
16.5 Compatibility complexes
16.6 Differential syzygies
16.7 Gauge identities
16.8 Noether identities
16.9 Overdetermined systems
16.10 Finite-type systems
16.11 Formal exactness
16.12 Deformation complexes
16.13 Infinitesimal automorphisms
16.14 Rigidity and flexibility
16.15 Residue classes generated by nonexact compatibility complexes
17. Cartan–Kähler and involutive completion
17.1 Exterior differential systems
17.2 Integral elements
17.3 Polar spaces
17.4 Cartan characters
17.5 Cartan’s test
17.6 Involutive tableaux
17.7 Analytic existence
17.8 Cartan–Kähler theorem
17.9 Singular integral elements
17.10 Prolongation to involution
17.11 Cauchy data and noncharacteristic conditions
17.12 Local solution dimensions
17.13 Failure modes outside analytic categories
17.14 Local existence without global continuation
Part VI. Flexibility, Rigidity, and the (h)-Principle
18. Formal solutions and genuine solutions
18.1 The formal solution space
18.2 The holonomic solution space
18.3 Inclusion of genuine into formal solutions
18.4 Weak homotopy equivalence questions
18.5 Parametric families
18.6 Relative solution problems
18.7 Approximation by holonomic sections
18.8 Local flexibility
18.9 Global rigidity
18.10 Boundary-conditioned flexibility
19. The (h)-principle as a scoped transport theorem
19.1 Open differential relations
19.2 Ample relations
19.3 Convex integration
19.4 Corrugation and oscillatory correction
19.5 Holonomic approximation
19.6 Immersion and embedding examples
19.7 Underdetermined PDEs
19.8 Flexible geometric structures
19.9 Failure of the (h)-principle
19.10 Rigidity from curvature, topology, or conservation
19.11 Why formal solvability does not universally imply global solvability
19.12 Why the (h)-principle cannot be generalized into a global closure law
19.13 Residue remaining after formal-to-holonomic deformation
19.14 Boundary and singularity restrictions
19.15 Quantitative versus qualitative flexibility
Part VII. Global Obstruction Theory
20. Local solutions on covers
20.1 Open covers and local solution families
20.2 Restrictions to overlaps
20.3 Transition maps
20.4 Pairwise compatibility
20.5 Triple-overlap compatibility
20.6 Higher-overlap coherence
20.7 Descent data
20.8 Effective and ineffective descent
20.9 Why pairwise agreement is insufficient
20.10 Local representatives and gauge equivalence
20.11 Failure of uniqueness on overlaps
20.12 Failure of continuation across singular sets
21. Sheaves, cohomology, and obstruction classes
21.1 Sheaves of local solutions
21.2 Presheaf compatibility
21.3 Germs and stalks
21.4 Čech cocycles
21.5 Coboundaries
21.6 First cohomology and gluing obstruction
21.7 Higher cohomology and higher coherence
21.8 Sheaf cohomology
21.9 Hypercohomology of compatibility complexes
21.10 Obstruction towers
21.11 Deformation and obstruction spaces
21.12 Derived solution spaces
21.13 Local systems
21.14 Nonabelian cohomology
21.15 Stack-valued descent
22. Monodromy, holonomy, and loop residue
22.1 Analytic continuation
22.2 Monodromy representations
22.3 Branch points and multivalued continuation
22.4 Parallel transport around loops
22.5 Holonomy groups
22.6 Flat connections with nontrivial holonomy
22.7 Curvature and infinitesimal holonomy
22.8 Discrete monodromy
22.9 Path dependence
22.10 Transport groupoids
22.11 Loop-space representations
22.12 Failure to return to the original carrier
22.13 Residue from noncontractible cycles
22.14 Global phase and geometric phase
22.15 Gauge holonomy
23. Topological obstructions
23.1 Fundamental group
23.2 Homotopy classes of maps
23.3 Degree and winding number
23.4 Homology and cohomology constraints
23.5 Characteristic classes
23.6 Euler class
23.7 Stiefel–Whitney classes
23.8 Chern classes
23.9 Pontryagin classes
23.10 Obstructions to sections
23.11 Obstructions to trivialization
23.12 Obstructions to immersion and embedding
23.13 Index-theoretic obstructions
23.14 Cobordism constraints
23.15 Global topology invisible to local jets
24. Boundary, infinity, and noncompactness
24.1 Boundary jets
24.2 Tangency and transversality at boundaries
24.3 Corner compatibility
24.4 Interface and transmission conditions
24.5 Boundary layers
24.6 Asymptotic conditions
24.7 Properness
24.8 Escape to infinity
24.9 Compactness failure
24.10 Concentration and bubbling
24.11 Loss of mass
24.12 Essential spectrum and nonlocal behavior
24.13 Finite-energy constraints
24.14 Boundary obstruction classes
24.15 Interior closure with unresolved boundary residue
25. Singularities and branching
25.1 Critical points
25.2 Critical values
25.3 Singular jet strata
25.4 Fold singularities
25.5 Cusp singularities
25.6 Thom–Boardman strata
25.7 Catastrophe geometry
25.8 Discriminants
25.9 Caustics
25.10 Branch loci
25.11 Resolution and desingularization
25.12 Stratified transversality
25.13 Singular continuation
25.14 Defect measures
25.15 Residue concentrated on singular sets
Part VIII. Globalization as Surviving Residue
26. Rejecting globalization-as-closure
26.1 Local validity does not construct a global carrier
26.2 Pairwise compatibility does not prove higher coherence
26.3 Empty local residue does not prove empty global residue
26.4 Local integrability does not prove global exactness
26.5 Formal integrability does not prove global existence
26.6 Global existence does not prove uniqueness
26.7 Pointwise invertibility does not prove global invertibility
26.8 Local convexity does not prove global convexity
26.9 Local rank stability does not prove global topology
26.10 Local solution continuation may accumulate monodromy
27. The globalization-residue carrier
27.1 Definition of globalization residue
27.2 Comparison residue
27.3 Transition residue
27.4 Cocycle residue
27.5 Commutator residue
27.6 Holonomy residue
27.7 Chronology residue
27.8 Boundary residue
27.9 Singular residue
27.10 Branching residue
27.11 Nonuniqueness residue
27.12 Nonproperness residue
27.13 Liftback residue
27.14 Provenance residue
27.15 Residue persistence under refinement
28. Residue extraction
28.1 Choose local carriers
28.2 Construct local holonomic solutions
28.3 Compare solutions on overlaps
28.4 Transport comparison data around loops
28.5 Evaluate triple and higher intersections
28.6 Detect path dependence
28.7 Separate gauge-equivalent from genuinely distinct data
28.8 Compute boundary mismatch
28.9 Compute singular-stratum mismatch
28.10 Register nonliftback
28.11 Normalize equivalent obstruction classes
28.12 Preserve irreducible residue
28.13 Identify exact vanishing criteria
28.14 Distinguish residue cancellation from residue erasure
28.15 Replay residue under carrier mutation
29. Native global-carrier construction
29.1 Independent construction requirement
29.2 Global maps as sections
29.3 Global sections of associated bundles
29.4 Developing maps
29.5 Covering-space constructions
29.6 Quotients by group actions
29.7 Global variational constructions
29.8 Elliptic and parabolic continuation
29.9 Fixed-point constructions
29.10 Geometric flows
29.11 Global weak solutions
29.12 Moduli-space constructions
29.13 Reconstruction from complete descent data
29.14 Exact liftback from quotient carriers
29.15 Independent verification of the global object
Part IX. Gradient, Jacobian, and Hessian Reintegrated
30. Gradient as metric-mediated first-order response
30.1 Differential versus gradient
30.2 Directional derivatives
30.3 Steepest ascent depends on the norm
30.4 Riemannian gradients
30.5 Natural gradients
30.6 Projected gradients
30.7 Subgradients
30.8 Gradient flows
30.9 Gradient systems with constraints
30.10 Global obstructions to potential representation
31. Jacobian as first-order transport
31.1 Differential as a bundle map
31.2 Coordinate Jacobian matrices
31.3 Rank and constant-rank geometry
31.4 Kernel and image distributions
31.5 Singular values and anisotropy
31.6 Exterior powers and minors
31.7 Jacobian determinant
31.8 Local inverse theorem
31.9 Global inverse conditions
31.10 Covering behavior
31.11 Folding and collisions
31.12 Integrability of matrix fields
31.13 Jacobian ideals and singular loci
31.14 Polynomial Jacobian problems
31.15 Local rank data versus global map structure
32. Hessian as second-order interaction transport
32.1 Hessian bilinear forms
32.2 Hessian eigenstructure
32.3 Mixed curvature terms
32.4 Restriction to feasible directions
32.5 Hessian of a map
32.6 Covariant Hessian
32.7 Second fundamental form
32.8 Degenerate Hessians
32.9 Morse theory
32.10 Hessian equations
32.11 Monge–Ampère structure
32.12 (k)-Hessian relations
32.13 Hessian determinant collapse
32.14 Hessian metrics
32.15 Global consequences and limitations of local curvature
33. Higher differential layers
33.1 Third derivatives and variation of curvature
33.2 Higher symmetric tensors
33.3 Covariant derivative commutators
33.4 Curvature derivatives
33.5 Higher-order critical-point classification
33.6 Finite determinacy
33.7 Singularity germs
33.8 Higher-order differential invariants
33.9 Differential operators on jets
33.10 Infinite-order formal geometry
Part X. Composition, Reconstruction, and Liftback
34. First-order composition
34.1 Chain rule as transport composition
34.2 Domain and codomain matching
34.3 Rank propagation
34.4 Kernel propagation
34.5 Image propagation
34.6 Determinant multiplication
34.7 Loss of information under determinant-only composition
34.8 Composition with constraints
34.9 Composition across singular strata
34.10 Composition residue
35. Second-order composition
35.1 Hessian chain rule
D²(G ∘ F)[u, v]
D²G[dF, u, dF, v]
dG(D²F[u, v])
35.2 Outer curvature term
35.3 Inner curvature transport term
35.4 Mixed-source interactions
35.5 Connection corrections
35.6 Curvature and torsion corrections
35.7 Feasible-direction composition
35.8 Failure of scalar curvature summaries
35.9 Exact second-order liftback
35.10 Residue from omitted interaction terms
36. Higher-order composition
36.1 Faà di Bruno formula
36.2 Partition structure
36.3 Rooted trees
36.4 Operadic composition
36.5 Symmetric multilinear contractions
36.6 Higher-order automatic differentiation
36.7 Truncated jet composition
36.8 Jet-group actions
36.9 Formal diffeomorphisms
36.10 Composition-induced obstruction growth
37. Inversion and reconstruction
37.1 Local inversion
37.2 Formal inverse jets
37.3 Recursive inverse coefficients
37.4 Polynomial inversion
37.5 Analytic continuation of local inverses
37.6 Branching of inverse maps
37.7 Covering maps
37.8 Proper local diffeomorphisms
37.9 Global injectivity constraints
37.10 Reconstruction from derivative fields
37.11 Reconstruction ambiguity
37.12 Exact liftback from compressed carriers
37.13 Noninvertible quotient carriers
37.14 Reconstruction debt
37.15 Global inverse residue
Part XI. Variational and PDE Realizations
38. Calculus of variations on jet bundles
38.1 Lagrangians as functions on jet spaces
38.2 Action functionals
38.3 Euler–Lagrange equations
38.4 Higher-order variational problems
38.5 Boundary terms
38.6 Natural boundary conditions
38.7 Noether identities
38.8 Multisymplectic formulations
38.9 Variational bicomplex
38.10 Conservation-law cohomology
38.11 Constraint manifolds
38.12 Singular Lagrangians
38.13 Gauge degeneracy
38.14 Hamilton–Jacobi structures
38.15 Global variational obstructions
39. PDE geometry
39.1 PDEs as submanifolds of jet bundles
39.2 Characteristics
39.3 Elliptic relations
39.4 Hyperbolic relations
39.5 Parabolic relations
39.6 Fully nonlinear equations
39.7 Degenerate equations
39.8 Weak and viscosity solutions
39.9 Entropy admissibility
39.10 Boundary-value problems
39.11 Initial-value problems
39.12 Compatibility at initial-boundary corners
39.13 Shock and singularity formation
39.14 Prolongation of PDE systems
39.15 Global existence residue
40. Hessian and curvature equations
40.1 Laplace equations
40.2 Monge–Ampère equations
40.3 (k)-Hessian equations
40.4 Prescribed curvature equations
40.5 Ricci and Einstein-type geometric PDEs
40.6 Convexity cones
40.7 Admissible branches
40.8 Degenerate ellipticity
40.9 Comparison principles
40.10 Geometric measure constraints
40.11 Blow-up and concentration
40.12 Boundary convexity
40.13 Global solvability obstructions
40.14 Topology of admissible solution spaces
Part XII. Computational Realization
41. Automatic differentiation as jet propagation
41.1 Computational graphs
41.2 Forward mode
41.3 Reverse mode
41.4 Jacobian–vector products
41.5 Vector–Jacobian products
41.6 Hessian–vector products
41.7 Forward-over-reverse
41.8 Reverse-over-forward
41.9 Higher-order jet propagation
41.10 Truncated Taylor algebras
41.11 Sparsity
41.12 Checkpointing
41.13 Numerical stability
41.14 Nonsmooth operations
41.15 Differentiation through implicit solutions
42. Symbolic compatibility analysis
42.1 Differential elimination
42.2 Differential Gröbner bases
42.3 Janet and Riquier completion
42.4 Cartan–Kuranishi prolongation
42.5 Symbol-rank computation
42.6 Detection of hidden equations
42.7 Differential syzygies
42.8 Involution testing
42.9 Singular branching
42.10 Constraint normalization
42.11 Proof certificates for compatibility
42.12 Exact versus numerical integrability
43. Numerical jet methods
43.1 Finite differences
43.2 Finite elements
43.3 Spectral methods
43.4 Meshfree derivative reconstruction
43.5 Discrete exterior calculus
43.6 Discrete connections and holonomy
43.7 Jet interpolation
43.8 Structure-preserving integrators
43.9 Constraint stabilization
43.10 Boundary consistency
43.11 Singular-adaptive discretization
43.12 Numerical residue detection
43.13 A posteriori error estimation
43.14 Global continuation algorithms
43.15 Failure of local numerical convergence to certify global structure
Part XIII. Domain Realizations
44. Optimization
44.1 Unconstrained first-order methods
44.2 Riemannian optimization
44.3 Equality constraints
44.4 Inequality constraints
44.5 Tangent and normal cones
44.6 KKT geometry
44.7 Lagrangian Hessians
44.8 Second-order feasibility
44.9 Trust-region methods
44.10 Saddle geometry
44.11 Degenerate critical manifolds
44.12 Global topology of level sets
44.13 Nonconvex obstruction residue
44.14 Optimization on singular spaces
45. Robotics and control
45.1 Configuration manifolds
45.2 Forward kinematics
45.3 Manipulator Jacobians
45.4 Singular configurations
45.5 Velocity transport
45.6 Acceleration transport
45.7 Hessians of kinematic maps
45.8 Nonholonomic constraints
45.9 Controllability distributions
45.10 Lie-bracket generation
45.11 Constraint stabilization
45.12 Global configuration-space topology
45.13 Monodromy in inverse kinematics
45.14 Motion-planning obstruction classes
45.15 Local controllability versus global reachability
46. Computer vision and image geometry
46.1 Image jets
46.2 Scale-space jets
46.3 Gradient features
46.4 Structure tensors
46.5 Hessian detectors
46.6 Affine region adaptation
46.7 Differential invariants
46.8 Optical-flow constraints
46.9 Aperture problem
46.10 Multiview Jacobians
46.11 Singular camera configurations
46.12 Global correspondence residue
46.13 Occlusion and boundary strata
46.14 Topological image structure
47. Machine learning
47.1 Parameter-to-output Jacobians
47.2 Loss gradients
47.3 Hessians and generalized Gauss–Newton operators
47.4 Fisher information geometry
47.5 Neural tangent kernels
47.6 Representation Jacobians
47.7 Rank collapse
47.8 Flat and sharp directions
47.9 Constraint manifolds in parameter space
47.10 Implicit bias
47.11 Local linearization versus global function behavior
47.12 Symmetry and gauge redundancy
47.13 Mode connectivity
47.14 Singular learning theory
47.15 Generalization residue invisible to local derivatives
Part XIV. Domain-Specific Constraint Validators
48. Carrier validator
48.1 Source type declared
48.2 Target type declared
48.3 Map type declared
48.4 Admissible variation carrier declared
48.5 Boundary and singular strata declared
48.6 Coordinate dependence declared
48.7 Metric dependence declared
48.8 Connection dependence declared
48.9 Orientation and measure dependence declared
48.10 No naked matrix accepted as a complete object
49. Holonomicity validator
49.1 Jet data consistency
49.2 Mixed-partial compatibility
49.3 Contact-form annihilation
49.4 Prolongation compatibility
49.5 Symbol compatibility
49.6 Formal integrability
49.7 Actual local reconstruction
49.8 Reconstruction uniqueness
49.9 Regularity requirements
49.10 No formal solution promoted as a genuine solution
50. Triadic interaction validator
50.1 Preserve source–transport–target typing
50.2 Preserve simultaneous variations
50.3 Record mixed derivative terms
50.4 Prohibit unsupported pairwise decomposition
50.5 Permit dyadic projection only with exact reconstruction
50.6 Verify polarization conditions
50.7 Preserve coupling under composition
50.8 Track interaction terms through quotienting
50.9 Track interaction residue after approximation
50.10 No determinant-only replacement of full interaction geometry
51. Differential-relation validator
51.1 Relation domain and order
51.2 Fibrewise admissibility
51.3 Rank-stratum consistency
51.4 Constraint-cone consistency
51.5 Boundary compatibility
51.6 Singular compatibility
51.7 Composition stability
51.8 Prolongation closure
51.9 Constraint transport
51.10 Admissibility under coordinate change
52. Globalization-residue validator
52.1 Local cover specified
52.2 Overlap maps computed
52.3 Triple-overlap compatibility tested
52.4 Loop transport computed
52.5 Holonomy recorded
52.6 Monodromy recorded
52.7 Boundary mismatch recorded
52.8 Singular residue recorded
52.9 Nonuniqueness recorded
52.10 Nonliftback recorded
52.11 Local closure not promoted to global closure
52.12 Global object independently constructed
53. Boundary and tail validator
53.1 Boundary jets
53.2 Corner compatibility
53.3 Asymptotic behavior
53.4 Properness
53.5 Escape to infinity
53.6 Concentration
53.7 Singular limits
53.8 Degenerate rank limits
53.9 Worst-fibre behavior
53.10 Global continuation under extreme conditions
53.11 No finite interior sample promoted to universal closure
54. Compression and collapse validator
54.1 Information removed by projection
54.2 Reconstruction map
54.3 Fibre ambiguity
54.4 Rank information loss
54.5 Mixed-term loss
54.6 Boundary information loss
54.7 Global-topology loss
54.8 Determinant collapse
54.9 Spectrum collapse
54.10 Quotient-carrier residue
54.11 Exact conditions for admissible compression
54.12 Prohibition of unledgered collapse
Part XV. Hidden Constraints and New Reframings
55. Hidden constraints generated by the unified theory
55.1 Realizability constraint
55.2 Holonomicity constraint
55.3 Prolongation constraint
55.4 Mixed-interaction constraint
55.5 Rank-stratum constraint
55.6 Coordinate-covariance constraint
55.7 Metric-dependence constraint
55.8 Connection-dependence constraint
55.9 Boundary-compatibility constraint
55.10 Singular-continuation constraint
55.11 Properness constraint
55.12 Global-coherence constraint
55.13 Native-carrier construction constraint
55.14 Exact-liftback constraint
55.15 Residue-persistence constraint
56. Three irreducible levels of validity
56.1 Pointwise algebraic validity
56.2 Local differential realizability
56.3 Global geometric admissibility
56.4 Why success at one level cannot compensate for failure at another
56.5 Pointwise nondegeneracy without integrability
56.6 Local integrability without global exactness
56.7 Global existence without uniqueness
56.8 Global existence without stability
56.9 Stability without boundary closure
56.10 Boundary closure without topological triviality
57. Reframing derivative theory
57.1 Derivatives as transition carriers
57.2 Jets as finite local state geometry
57.3 Differential relations as admissibility geometry
57.4 Integrability as realizability
57.5 Prolongation as hidden-constraint extraction
57.6 Curvature and torsion as commutator residue
57.7 Holonomy and monodromy as loop residue
57.8 Cohomology as overlap residue
57.9 Singularity theory as rank-fracture geometry
57.10 Globalization as residue extraction
57.11 Native global construction as a separate proof obligation
Part XVI. Unified Solution Architecture
58. Canonical problem statement
58.1 Declare (X), (Y), and all geometric structures
58.2 Declare the order (k)
58.3 Declare the relation (\mathcal R\subset J^k(X,Y))
58.4 Declare admissible source variations
58.5 Declare admissible target responses
58.6 Declare boundary and singular conditions
58.7 Declare the required global object
58.8 Declare equivalence and gauge classes
58.9 Declare forbidden information loss
58.10 Declare the exact liftback requirement
59. Canonical prosecution sequence
59.1 Type the source, target, and transport carriers
59.2 Construct the local jet relation
59.3 Validate coordinate covariance
59.4 Extract algebraic constraints
59.5 Prolong the relation
59.6 Compute compatibility conditions
59.7 Test formal integrability
59.8 Construct local holonomic sections
59.9 Compare local solutions
59.10 Transport around overlaps and loops
59.11 Extract globalization residue
59.12 Test boundaries and singular strata
59.13 Construct the native global carrier independently
59.14 Lift local certificates into the global carrier
59.15 Replay every constraint at global scale
60. Failure classification
60.1 Undefined carrier
60.2 Coordinate artifact
60.3 Metric artifact
60.4 Connection artifact
60.5 Algebraic incompatibility
60.6 Nonholonomic formal data
60.7 Prolongation contradiction
60.8 Noninvolutive relation
60.9 Boundary mismatch
60.10 Singular obstruction
60.11 Monodromy obstruction
60.12 Holonomy obstruction
60.13 Cohomological obstruction
60.14 Nonproperness
60.15 Failed global liftback
61. Valid endpoint classes
61.1 Genuine global solution
61.2 Global solution modulo gauge
61.3 Global weak solution
61.4 Global solution on a covering space
61.5 Stratified global solution
61.6 Local solution with exact obstruction class
61.7 Formal solution with explicit integrability residue
61.8 Proved nonexistence from obstruction
61.9 Moduli space of admissible global solutions
61.10 Residue-complete frontier without false closure
Part XVII. Advanced Extensions
62. Groupoids, algebroids, and pseudogroups
62.1 Jet groupoids
62.2 Lie pseudogroups
62.3 Differential invariants
62.4 Lie algebroids
62.5 Integrability of algebroids
62.6 Monodromy groups
62.7 Groupoid holonomy
62.8 Symmetry reduction
62.9 Quotient singularities
62.10 Stack-like solution carriers
63. Derived and homotopical geometry
63.1 Derived intersections
63.2 Derived critical loci
63.3 Cotangent complexes
63.4 Deformation complexes
63.5 Higher obstruction classes
63.6 (L_\infty)-algebras
63.7 Formal moduli problems
63.8 Derived jet spaces
63.9 Homotopy-coherent descent
63.10 Higher globalization residue
64. Noncommutative and discrete generalizations
64.1 Difference jets
64.2 Discrete differential relations
64.3 Discrete connections
64.4 Discrete holonomy
64.5 Noncommutative derivatives
64.6 Free difference quotients
64.7 Quantum differential calculi
64.8 Graph and network jets
64.9 Hybrid continuous-discrete systems
64.10 Discrete-to-continuum liftback
65. Open structural problems
65.1 Minimal sufficient jet carriers
65.2 Exact criteria for admissible derivative compression
65.3 Quantitative formal-to-holonomic conversion
65.4 Global residue classification for nonlinear PDEs
65.5 Integrability under low regularity
65.6 Boundary-sensitive (h)-principles
65.7 Singular differential relations
65.8 Derived obstruction theory for solution spaces
65.9 Certified computational integrability
65.10 Domain-aware automated constraint discovery
65.11 Exact detection of hidden prolongation constraints
65.12 Triadic and n-ary alternatives to matrix flattening
65.13 Global construction under persistent local residue
65.14 Reconstruction from incomplete differential observations
65.15 A general theory of residue-governed global admissibility
Closing synthesis
Gradient ⊂ first-order scalar jet data
Jacobian ⊂ first-order map jet data
Hessian ⊂ second-order interaction data
Jets = finite local transition geometry
Differential relations = admissible jet geometry
Integrability = realizability of formal data
Global obstruction theory = classification of surviving nonclosure residue
The governing architecture is:
TYPE
→ JET CARRIER
→ DIFFERENTIAL RELATION
→ PROLONGATION
→ INTEGRABILITY
→ LOCAL HOLONOMICITY
→ GLOBALIZATION RESIDUE
→ NATIVE GLOBAL CONSTRUCTION
→ LIFTBACK
→ REPLAY
Glossary
| Term | Definition |
|---|---|
| Admissibility Cone | The set of allowed variations or jets at a point that satisfy the local constraints. Often written as cones in the tangent or jet space. |
| Carrier (Triadic) | The irreducible structure: Source carrier + Transport law + Target carrier. Represented as TₓX ──dFₓ──→ T_{F(x)}Y. |
| Contact Distribution / Contact Forms | The natural distribution on the jet bundle that enforces compatibility. Holonomic sections are integral manifolds of this distribution. |
| Constraint Transport | Propagation of admissible cones: for example, dFₓ(Aₓ) ⊆ B_{F(x)}. Failures produce residues. |
| Differential Relation | A subset 𝓡 ⊂ Jᵏ(X, Y) encoding PDEs, inequalities, or constraints on allowed jets. |
| Formal Section | Any section of the jet bundle (assigns jet data pointwise) that may not come from an actual map. |
| Globalization Residue | The irreducible leftover obstruction after local solutions and transport. The post treats globalization as surviving residue rather than closure. |
| Holonomic Section | A section that is the jet prolongation jᵏF of an actual map F: X → Y. |
| Holonomic versus Nonholonomic | Holonomic = fully integrable to a map. Nonholonomic = constraints that do not reduce to position-level restrictions. |
| Jet (jᵏF(x)) | Finite-order local transition geometry: equivalence class of map germs agreeing up to order k at x. |
| Jet Bundle / Jet Space (Jᵏ(X,Y)) | The space of all k-jets. Successive orders form an affine bundle: Jᵏ → J^{k-1}. |
| (h)-Principle | For open/ample relations, formal solutions can often be deformed to holonomic ones (with scope limits). |
| Prolongation | Lifting a relation or section to higher jet order to expose hidden equations. |
| Residue | Any surviving mismatch (comparison, transition, holonomy, boundary, singular, etc.). |
| Source / Target Admissibility | Constraints on allowable variations in the source or target. |
| Transport Law | How source variations map to target variations under the differential. |
| Triadic Structure | The three-part irreducible nature of maps: source admissibility – transport – target admissibility. |
Additional Important Terms
- Formal Integrability: Consistency across all prolongations (checked via Spencer cohomology).
- Involutivity: Frobenius-type condition for local existence of integral manifolds.
- Stratified Jet Spaces: Jets adapted to singular strata or rank drops.
- Global Obstruction Theory: Uses cohomology, monodromy, characteristic classes, etc., to explain why local holonomic solutions fail globally.
- Whitney Extension / Jet Interpolation: Reconstructing maps from compatible jet data on sets.
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