GRM: Arithmetic, Homotopy, and Geometry
GRM: Arithmetic, Homotopy, and Geometry
The Homology–Homotopy Frontier in Arithmetic Geometry
Front Matter
F.1 Preface: Arithmetic Geometry as a Reconstruction Problem
F.1.1 The classical arithmetic viewpoint
F.1.2 Arithmetic through linear invariants
F.1.3 Arithmetic through structured path spaces
F.1.4 Homology, homotopy, and information loss
F.1.5 Why non-abelian structure is not automatically better
F.1.6 Reconstruction before escalation
F.1.7 The role of explicit computation
F.2 The GRM Architecture of the Book
F.2.1 Source structure
F.2.2 Representation structure
F.2.3 Arithmetic observables
F.2.4 Obstructions
F.2.5 Structural lifts
F.2.6 Minimal sufficient invariants
F.2.7 Closure and reopening
F.2.8 Research-frontier generation
F.3 Mathematical Conventions
F.3.1 Fields and places
F.3.2 Curves and basepoints
F.3.3 Fundamental groups and groupoids
F.3.4 Galois cohomology
F.3.5 Filtrations
F.3.6 Tannakian notation
F.3.7 Selmer notation
F.3.8 Period notation
F.4 Running Examples
F.4.1 (\mathbf G_m)
F.4.2 Elliptic curves
F.4.3 Genus-two curves
F.4.4 Hyperbolic curves
F.4.5 (\mathbf P^1-{0,1,\infty})
F.4.6 Modular curves
F.4.7 Families of curves
F.5 Reading Paths
F.5.1 Arithmetic geometry path
F.5.2 Anabelian path
F.5.3 Chabauty–Kim path
F.5.4 Tannakian path
F.5.5 Motives and periods path
F.5.6 Computational path
F.5.7 Research-frontier path
Part I. Arithmetic Information and Structural Resolution
Chapter 1. Arithmetic Information
1.1 Arithmetic objects
1.1.1 Rational points
1.1.2 Integral points
1.1.3 Cycles and divisor classes
1.1.4 Galois actions
1.1.5 Local systems
1.1.6 Period data
1.2 Arithmetic observables
1.2.1 Point-counting data
1.2.2 Cohomological data
1.2.3 Representation-theoretic data
1.2.4 Fundamental-group data
1.2.5 Motivic data
1.3 Arithmetic representations
1.3.1 Linear representations
1.3.2 Abelian moduli spaces
1.3.3 Non-abelian torsors
1.3.4 Tensor categories
1.3.5 Period domains
1.4 Recoverability
1.4.1 Faithful representation
1.4.2 Partial reconstruction
1.4.3 Finite-depth reconstruction
1.4.4 Global reconstruction
1.4.5 Computational reconstruction
1.5 Information loss
1.5.1 Quotienting
1.5.2 Abelianization
1.5.3 Linearization
1.5.4 Truncation
1.5.5 Realization
Chapter 2. Homology and Homotopy
2.1 Fundamental groups
2.1.1 Loops
2.1.2 Composition
2.1.3 Basepoints
2.1.4 Groupoids
2.2 Abelianization
2.2.1 Commutator subgroups
2.2.2 (\pi_1^{ab})
2.2.3 Relation with (H_1)
2.2.4 Lost ordering information
2.3 Cohomological dualization
2.3.1 (H^1)
2.3.2 Extension classes
2.3.3 Characters
2.3.4 Linear representations
2.4 Non-abelian information
2.4.1 Commutators
2.4.2 Massey-type phenomena
2.4.3 Path ordering
2.4.4 Higher compatibility
2.5 Structural-resolution hierarchy
2.5.1 (H^1)
2.5.2 (\pi_1^{ab})
2.5.3 (\pi_1/\Gamma^3)
2.5.4 Higher nilpotent quotients
2.5.5 (\pi_1^{un})
2.5.6 (\pi_1^{\acute et})
Chapter 3. GRM Arithmetic Depth
3.1 Detectability
3.1.1 Arithmetic predicates
3.1.2 Detectable properties
3.1.3 Reconstruction thresholds
3.1.4 Representation dependence
3.2 Arithmetic depth
3.2.1 Depth-zero information
3.2.2 Abelian depth
3.2.3 Nilpotent depth
3.2.4 Unipotent depth
3.2.5 Full profinite depth
3.3 Minimal-depth questions
3.3.1 Point detection
3.3.2 Section detection
3.3.3 Finiteness detection
3.3.4 Special-locus detection
3.4 Depth instability
3.4.1 Dependence on the chosen representation
3.4.2 Dependence on realization
3.4.3 Dependence on local conditions
3.5 Research program
3.5.1 Arithmetic depth spectra
3.5.2 Depth stratification of problems
3.5.3 Effective bounds for required depth
3.5.4 Stabilization versus genuine higher structure
Part II. Curves, Jacobians, and Linear Arithmetic
Chapter 4. Curves Over Arithmetic Fields
4.1 Smooth curves
4.2 Projective curves
4.3 Affine curves
4.4 Hyperbolic curves
4.5 Rational points
4.6 Integral points
4.7 Local points
Chapter 5. Divisors and Picard Geometry
5.1 Divisors
5.2 Linear equivalence
5.3 Picard groups
5.4 (\operatorname{Pic}^0)
5.5 Picard varieties
5.6 Abel–Jacobi maps
Chapter 6. Jacobians
6.1 Construction of (J_X)
6.1.1 Functorial description
6.1.2 Divisor-class description
6.1.3 Moduli interpretation
6.2 Arithmetic of (J_X)
6.2.1 Mordell–Weil
6.2.2 Torsion
6.2.3 Rank
6.2.4 Reduction
6.3 The Jacobian as compression
6.3.1 What is retained
6.3.2 What is forgotten
6.3.3 Dependence on basepoints
6.4 Local Jacobians
6.4.1 (J(K_v))
6.4.2 (p)-adic Lie structure
6.4.3 Closure of (J(K))
Chapter 7. Tate Modules and Linear Galois Representations
7.1 Torsion points
7.2 Tate modules
7.3 Continuous Galois actions
7.4 Étale cohomology
7.5 (H^1_{\acute et}) and duality
7.6 Frobenius
7.7 Inertia
7.8 Recoverability from linear representations
Part III. Kummer Theory and Global–Local Arithmetic
Chapter 8. Classical Kummer Theory
8.1 Kummer theory for (\mathbf G_m)
8.1.1 Roots
8.1.2 Torsors
8.1.3 Cohomology classes
8.2 Kummer theory for abelian varieties
8.2.1 Multiplication maps
8.2.2 Finite Kummer sequences
8.2.3 Passage to inverse limits
8.3 Global Kummer maps
8.4 Local Kummer maps
8.5 Functoriality
8.6 Arithmetic information encoded by Kummer classes
Chapter 9. Galois Cohomology
9.1 Continuous cohomology
9.2 Local cohomology
9.3 Global cohomology
9.4 Restriction
9.5 Localization
9.6 Inflation–restriction
9.7 Extension classes
9.8 Cohomological dimension
Chapter 10. Selmer Groups
10.1 Selmer conditions
10.1.1 Unramified conditions
10.1.2 Finite conditions
10.1.3 Geometric conditions
10.2 Global Selmer groups
10.3 Local Selmer groups
10.4 Bloch–Kato conditions
10.5 Global–local intersections
10.6 Arithmetic finiteness from linear Selmer geometry
Chapter 11. GRM Global–Local Defect
11.1 Global realizability
11.2 Local realizability
11.3 Compatibility defect
11.4 Failure of local-to-global reconstruction
11.5 Defect classes
11.6 Defect reduction by stronger invariants
Part IV. Arithmetic Fundamental Groups and Anabelian Geometry
Chapter 12. Étale Fundamental Groups
12.1 Finite étale coverings
12.2 Fiber functors
12.3 Geometric basepoints
12.4 Profinite completion
12.5 Fundamental groupoids
12.6 Functoriality
Chapter 13. Arithmetic Fundamental Exact Sequences
13.1 Geometric fundamental groups
13.2 Arithmetic fundamental groups
13.3 The exact sequence
13.4 Outer Galois actions
13.5 Decomposition groups
13.6 Inertia groups
Chapter 14. Rational Points and Sections
14.1 Sections from rational points
14.2 Conjugacy of sections
14.3 Local sections
14.4 Global sections
14.5 Section conjectures
14.6 Reconstruction of points from sections
Chapter 15. Anabelian Reconstruction
15.1 Hyperbolic curves
15.2 Reconstruction of curves
15.3 Reconstruction of morphisms
15.4 Decomposition-group recognition
15.5 Arithmetic reconstruction
15.6 Geometric reconstruction
15.7 Limits of abelian data
Chapter 16. Finite-Depth Anabelian Geometry
16.1 Lower central series
16.2 Derived series
16.3 Nilpotent quotients
16.4 Metabelian quotients
16.5 Abelian-by-central quotients
16.6 Finite-step section problems
16.7 Reconstruction thresholds
16.8 Stabilization questions
Part V. Classical Chabauty and the Abelian Boundary
Chapter 17. Chabauty’s Dimension Method
17.1 Rational points inside Jacobians
17.2 Mordell–Weil closure
17.3 Rank versus genus
17.4 Codimension
17.5 Global–local intersections
17.6 Finiteness
Chapter 18. Coleman Integration
18.1 (p)-adic differential forms
18.2 Local integration
18.3 Residue disks
18.4 Coleman continuation
18.5 Abelian integrals
18.6 Annihilating differentials
18.7 Effective Chabauty
Chapter 19. Chabauty Through Galois Cohomology
19.1 Kummer reinterpretation
19.2 Global Kummer image
19.3 Local Kummer image
19.4 The cohomological square
19.5 Dimension constraints
19.6 The precise abelian failure mode
Chapter 20. The Abelian Obstruction
20.1 Rank failure
20.2 Dimension failure
20.3 Representation failure
20.4 Information invisible to (H^1)
20.5 Why a non-abelian lift becomes natural
20.6 Why a non-abelian lift is not automatically sufficient
Part VI. Unipotent Homotopy and Non-Abelian Cohomology
Chapter 21. Unipotent Completion
21.1 Unipotent algebraic groups
21.2 Pronilpotent groups
21.3 Malcev completion
21.4 Pro-unipotent completion
21.5 Lie algebras
21.6 Central filtrations
Chapter 22. Unipotent Fundamental Groups
22.1 Étale unipotent fundamental groups
22.2 de Rham unipotent fundamental groups
22.3 Path torsors
22.4 Basepoints
22.5 Tangential basepoints
22.6 Groupoids
Chapter 23. Finite-Depth Quotients
23.1 (U_1)
23.2 (U_2)
23.3 (U_3)
23.4 General (U_n)
23.5 Successive extensions
23.6 Graded Lie algebras
23.7 Arithmetic meaning of depth
Chapter 24. Non-Abelian Galois Cohomology
24.1 Non-abelian cocycles
24.2 Torsors
24.3 Gauge equivalence
24.4 Pointed cohomology sets
24.5 (H^1(G,U_n))
24.6 Local non-abelian cohomology
24.7 Global non-abelian cohomology
Chapter 25. Non-Abelian Kummer Maps
25.1 Path torsors attached to points
25.2 Global Kummer maps
25.3 Local Kummer maps
25.4 Finite-depth Kummer towers
25.5 Comparison with the abelian Kummer map
25.6 First genuinely non-abelian constraints
Part VII. Arithmetic Local Systems and Tannakian Reconstruction
Chapter 26. Local Systems
26.1 Topological local systems
26.2 Étale local systems
26.3 de Rham local systems
26.4 Unipotent local systems
26.5 Families of local systems
Chapter 27. Monodromy
27.1 Monodromy representations
27.2 Geometric monodromy
27.3 Arithmetic monodromy
27.4 Zariski closures
27.5 Invariant tensors
27.6 Monodromy degeneration
Chapter 28. Tannakian Categories
28.1 Tensor categories
28.2 Rigid categories
28.3 Fiber functors
28.4 Tensor automorphisms
28.5 Neutral Tannakian reconstruction
28.6 Affine group schemes
Chapter 29. Tannakian Fundamental Groupoids
29.1 Fiber functors at points
29.2 Isomorphism torsors
29.3 Étale Tannaka groups
29.4 de Rham Tannaka groups
29.5 Rigid Tannaka groups
29.6 Groupoids as transport geometry
Chapter 30. GRM Representation Equivalence
30.1 Group presentation versus representation category
30.2 Fundamental group versus Tannaka group
30.3 Information preserved by categorical reconstruction
30.4 Information dependent on the fiber functor
30.5 Representation-stable arithmetic claims
30.6 Detecting false equivalences
Part VIII. Motives, Realizations, and Period Geometry
Chapter 31. Motivic Philosophy
31.1 Why motives enter
31.2 Pure motives
31.3 Mixed motives
31.4 Mixed Tate categories
31.5 Extension data
31.6 Weight filtrations
Chapter 32. Motivic Fundamental Groups
32.1 Motivic paths
32.2 Motivic torsors
32.3 Motivic fundamental groups
32.4 Motivic groupoids
32.5 Relation to étale realizations
32.6 Relation to de Rham realizations
Chapter 33. The Thrice-Punctured Line as a Motivic Laboratory
33.1 Geometry of (\mathbf P^1-{0,1,\infty})
33.2 Free fundamental groups
33.3 Free Lie algebras
33.4 Multiple polylogarithms
33.5 Motivic polylogarithms
33.6 Iterated integrals
33.7 Mixed Tate structure
Chapter 34. Realization Functors
34.1 Étale realization
34.2 de Rham realization
34.3 Hodge realization
34.4 Crystalline realization
34.5 Comparison morphisms
34.6 Compatibility constraints
Chapter 35. (p)-Adic Hodge Theory
35.1 (p)-adic representations
35.2 de Rham representations
35.3 Crystalline representations
35.4 Semistable representations
35.5 Period rings
35.6 Filtrations
35.7 Frobenius
35.8 Bloch–Kato conditions revisited
Chapter 36. Mixed Hodge Structures
36.1 Pure Hodge structures
36.2 Mixed Hodge structures
36.3 Weight filtrations
36.4 Hodge filtrations
36.5 Extension classes
36.6 Variations of mixed Hodge structure
Chapter 37. Period Maps
37.1 Period domains
37.2 Classical period maps
37.3 (p)-adic period maps
37.4 Monodromy and periods
37.5 Special loci
37.6 Arithmetic exceptional loci
Chapter 38. GRM Realization Defect
38.1 Multiple realizations of one arithmetic object
38.2 Compatibility conditions
38.3 Realization mismatch
38.4 Local realization defect
38.5 Global realization defect
38.6 Defect reduction by comparison maps
38.7 Defect as an arithmetic obstruction
38.8 Realization-depth interaction
Part IX. Selmer Geometry and Chabauty–Kim
Chapter 39. From Selmer Groups to Selmer Schemes
39.1 Non-abelian local conditions
39.2 Cohomology functors
39.3 Representability
39.4 Global Selmer schemes
39.5 Local Selmer schemes
39.6 Weight filtrations
39.7 Dimension estimates
Chapter 40. The Chabauty–Kim Diagram
40.1 Global non-abelian Kummer maps
40.2 Local non-abelian Kummer maps
40.3 Localization
40.4 The Selmer square
40.5 The Chabauty–Kim locus
40.6 Nested loci (X(K_p)_n)
40.7 Depth one
40.8 Depth two
40.9 Higher depth
Chapter 41. Finiteness Through Selmer Geometry
41.1 Dimension inequalities
41.2 Global Selmer dimension
41.3 Local period dimension
41.4 Proper intersections
41.5 Finiteness criteria
41.6 Effective criteria
41.7 Stabilization questions
Chapter 42. Weight Filtrations and Effective Depth
42.1 Weight filtrations on (U_n)
42.2 Weight filtrations on Selmer schemes
42.3 Dimension growth
42.4 Truncation
42.5 Effective depth bounds
42.6 Computational consequences
Part X. Iterated Integration and Explicit Non-Abelian Periods
Chapter 43. Chen Iterated Integrals
43.1 Ordered simplices
43.2 Length-one integrals
43.3 Higher iterated integrals
43.4 Shuffle relations
43.5 Concatenation
43.6 Path dependence
Chapter 44. Unipotent Connections
44.1 Flat connections
44.2 Integrability
44.3 Parallel transport
44.4 Holonomy
44.5 Path-ordered exponentials
44.6 Tannakian interpretation
Chapter 45. Coleman Integration
45.1 (p)-adic analytic continuation
45.2 Frobenius continuation
45.3 Coleman functions
45.4 Global (p)-adic integration
45.5 Effective evaluation
Chapter 46. Iterated Coleman Integration
46.1 Unipotent Coleman functions
46.2 Tannakian Coleman integration
46.3 Iterated (p)-adic integrals
46.4 Parallel transport
46.5 Local Kummer coordinates
46.6 Explicit depth-two formulas
Chapter 47. Non-Commutative Period Coordinates
47.1 Path-space coordinate rings
47.2 Iterated-integral coordinates
47.3 Polylogarithmic coordinates
47.4 Coordinate dependence
47.5 Intrinsic versus chosen coordinates
47.6 Reconstruction from periods
Part XI. Explicit Arithmetic and Computation
Chapter 48. (\mathbf P^1-{0,1,\infty})
48.1 Fundamental group
48.2 Unipotent quotients
48.3 Motivic structure
48.4 Polylogarithmic quotient
48.5 (S)-integral points
48.6 (\mathbf Z[1/2])-integral points
48.7 Depth-one constraints
48.8 Depth-two constraints
Chapter 49. Quadratic Chabauty
49.1 Failure of the classical rank condition
49.2 (p)-adic heights
49.3 Global height pairings
49.4 Local height terms
49.5 Quadratic loci
49.6 Comparison with depth-two Chabauty–Kim
Chapter 50. Computational Chabauty–Kim
50.1 (p)-adic precision
50.2 Local coordinates
50.3 Coleman integration algorithms
50.4 Iterated integral algorithms
50.5 Computing Kummer maps
50.6 Computing Selmer conditions
50.7 Solving (p)-adic equations
50.8 Root isolation
50.9 Rational-point verification
Chapter 51. Computational Reconstruction
51.1 Data versus certificate
51.2 Numerical evidence
51.3 Exact arithmetic checks
51.4 Precision control
51.5 Failure signatures
51.6 Reproducibility
51.7 Certified finite-depth conclusions
Part XII. GRM Cross-Theory Reconstruction
Chapter 52. The Arithmetic Information Lattice
52.1 Comparing structural levels
52.2 Partial ordering by recoverability
52.3 Incomparable invariants
52.4 Quotient maps
52.5 Realization maps
52.6 Reconstruction maps
52.7 Arithmetic information lattices
Chapter 53. Obstruction Signatures
53.1 Representation obstruction
53.2 Local-global obstruction
53.3 Reconstruction obstruction
53.4 Selmer obstruction
53.5 Monodromy obstruction
53.6 Period obstruction
53.7 Coherence obstruction
53.8 Computational obstruction
Chapter 54. Minimal Structural Lift
54.1 Detecting insufficiency
54.2 Candidate lifts
54.3 Cost of a lift
54.4 Information gained
54.5 Dominated lifts
54.6 Minimal sufficient lift
54.7 False escalation
54.8 Stabilization after lifting
Chapter 55. Arithmetic Depth Revisited
55.1 Property-dependent depth
55.2 Point-detection depth
55.3 Reconstruction depth
55.4 Selmer depth
55.5 Period depth
55.6 Monodromy depth
55.7 Computational depth
55.8 Depth profiles of arithmetic problems
Part XIII. GRM-Generated Frontier I: Selmer–Anabelian Geometry
Chapter 56. Sections and Selmer Classes
56.1 Sections as non-abelian arithmetic objects
56.2 Kummer classes as torsors
56.3 Common finite-depth quotients
56.4 Localization of sections
56.5 Localization of Selmer classes
56.6 Comparison problem
Chapter 57. Finite-Level Selmer–Section Correspondence
57.1 Projection of sections to (U_n)
57.2 Induced cohomology classes
57.3 Necessary compatibility conditions
57.4 Failure of lifting
57.5 Obstruction classes
57.6 Depth of section detectability
Chapter 58. Selmer–Anabelian Obstruction Towers
58.1 Level-one obstruction
58.2 Central extension obstruction
58.3 Higher nilpotent obstruction
58.4 Local-global obstruction
58.5 Period admissibility obstruction
58.6 Lifting to genuine sections
Chapter 59. Research Problems in Selmer–Anabelian Geometry
59.1 When does a Selmer class come from a section?
59.2 Effective finite-depth section detection
59.3 Local sections versus global sections
59.4 Relation with rational points
59.5 Stabilization of finite-depth section towers
59.6 Counterexamples and failure modes
Part XIV. GRM-Generated Frontier II: Monodromy–Selmer Transfer
Chapter 60. Families and Arithmetic Variation
60.1 Families of curves
60.2 Families of fundamental groups
60.3 Families of Selmer schemes
60.4 Families of local systems
60.5 Variation of period maps
Chapter 61. Monodromy Defect
61.1 Generic monodromy
61.2 Degenerate monodromy
61.3 Tannaka-group drop
61.4 Invariant-tensor growth
61.5 Special loci
Chapter 62. From Monodromy Defect to Selmer Defect
62.1 Induced changes in representation categories
62.2 Changes in local conditions
62.3 Changes in Selmer dimensions
62.4 Changes in non-abelian Kummer images
62.5 Candidate transfer mechanisms
62.6 Failure of transfer
Chapter 63. Arithmetic Exceptional Loci
63.1 Monodromy-exceptional loci
63.2 Period-exceptional loci
63.3 Selmer-exceptional loci
63.4 Rational-point exceptional loci
63.5 Comparison of exceptional sets
63.6 Toward a unified exceptional-locus theory
Part XV. GRM-Generated Frontier III: Period-Obstruction Towers
Chapter 64. Periods as Arithmetic Compatibility Tests
64.1 Period coordinates
64.2 Comparison isomorphisms
64.3 Local admissibility
64.4 Global realizability
64.5 Non-abelian period data
Chapter 65. Iterated Period Obstructions
65.1 Depth-one periods
65.2 Depth-two periods
65.3 Higher iterated periods
65.4 Extension data
65.5 Failure of compatibility
65.6 Obstruction classes
Chapter 66. Period–Selmer Interaction
66.1 Selmer conditions as period constraints
66.2 Local Kummer images
66.3 Period-map images
66.4 Intersections
66.5 Dimension defects
66.6 Higher compatibility
Chapter 67. Period-Obstruction Towers
67.1 Successive obstruction spaces
67.2 Compatibility across depth
67.3 Stabilization
67.4 Non-stabilization
67.5 Effective truncation
67.6 Relation with arithmetic depth
Part XVI. GRM-Generated Frontier IV: Higher Coherence
Chapter 68. Beyond Pairwise Compatibility
68.1 Pairwise local-global conditions
68.2 Triple compatibility
68.3 Higher intersections
68.4 Higher descent data
68.5 Coherence failures
Chapter 69. Higher Selmer Structures
69.1 Selmer functors
69.2 Derived intersections
69.3 Higher stacks
69.4 Derived Selmer spaces
69.5 Higher obstruction groups
Chapter 70. Derived Tannakian Geometry
70.1 Derived representation categories
70.2 Derived automorphism objects
70.3 Higher fiber functors
70.4 Derived path spaces
70.5 Arithmetic applications
Chapter 71. Motivic Homotopy and Arithmetic
71.1 Motivic spaces
71.2 Higher homotopy sheaves
71.3 Arithmetic realizations
71.4 Galois actions on higher homotopy
71.5 Rational-point questions
71.6 Computational barriers
Part XVII. Unified Frontier Architecture
Chapter 72. One Arithmetic Problem, Many Resolutions
72.1 Point-level representation
72.2 Jacobian representation
72.3 Cohomological representation
72.4 Nilpotent representation
72.5 Unipotent representation
72.6 Tannakian representation
72.7 Motivic representation
72.8 Period representation
Chapter 73. Comparison of Obstruction Theories
73.1 Kummer obstruction
73.2 Selmer obstruction
73.3 Anabelian obstruction
73.4 Monodromy obstruction
73.5 Period obstruction
73.6 Realization defect
73.7 Higher-coherence obstruction
Chapter 74. Structural Transfer Principles
74.1 Group-to-category transfer
74.2 Category-to-monodromy transfer
74.3 Monodromy-to-period transfer
74.4 Period-to-Selmer transfer
74.5 Selmer-to-point transfer
74.6 Failure and non-reversibility
Chapter 75. The Arithmetic Frontier Graph
75.1 Nodes as representations
75.2 Edges as licensed transfers
75.3 Irreversible arrows
75.4 Equivalence edges
75.5 Obstruction edges
75.6 Computational edges
75.7 Minimal routes to arithmetic conclusions
Chapter 76. Closure, Stabilization, and Reopening
76.1 Finite-depth closure
76.2 Representation-stable closure
76.3 Computational closure
76.4 Failure of stabilization
76.5 New-obstruction reopening
76.6 Research-frontier reopening
Part XVIII. Research Programs
Chapter 77. Effective Arithmetic Depth
77.1 Bounding the required nilpotent depth
77.2 Bounding Selmer depth
77.3 Bounding period depth
77.4 Algorithmic depth detection
77.5 Complexity of depth escalation
Chapter 78. Effective Anabelian Reconstruction
78.1 Finite quotients
78.2 Finite-depth sections
78.3 Effective decomposition-group recognition
78.4 Algorithmic reconstruction
78.5 Complexity questions
Chapter 79. Non-Abelian (p)-Adic Hodge Theory
79.1 Non-abelian comparison objects
79.2 Path torsors
79.3 Filtered non-abelian objects
79.4 Frobenius-compatible torsors
79.5 Non-abelian admissibility
79.6 Arithmetic applications
Chapter 80. Higher Arithmetic Monodromy
80.1 Higher categorical monodromy
80.2 Monodromy of derived local systems
80.3 Higher invariants
80.4 Arithmetic degeneration
80.5 Special loci
Chapter 81. Computational Higher Homotopy Arithmetic
81.1 Finite presentations
81.2 Nilpotent truncations
81.3 Lie-algebra computation
81.4 Non-abelian cohomology algorithms
81.5 Higher period computation
81.6 Certified obstruction detection
Chapter 82. New Conjectural Interfaces
82.1 Arithmetic depth versus motivic weight
82.2 Monodromy defect versus Selmer codimension
82.3 Section depth versus Selmer depth
82.4 Period depth versus unipotent depth
82.5 Stabilization of arithmetic obstruction towers
82.6 Universal arithmetic reconstruction principles
Part XIX. Research Practice
Chapter 83. Reading a Research Paper Structurally
83.1 Identify the source problem
83.2 Identify the representation
83.3 Identify information loss
83.4 Identify the obstruction
83.5 Identify the structural lift
83.6 Identify the decisive constructor
83.7 Identify hidden comparison theorems
83.8 Identify what is actually proved
Chapter 84. Reconstructing a Theory
84.1 Source invariants
84.2 Representation choices
84.3 Dependency graph
84.4 Erasure tests
84.5 Alternative representations
84.6 Minimal reconstruction
84.7 Frontier generation
Chapter 85. Designing New Arithmetic Invariants
85.1 Start from an obstruction
85.2 Reject decorative generalization
85.3 Search for minimal structure
85.4 Test representation stability
85.5 Test computability
85.6 Test arithmetic gain
85.7 Compare against existing invariants
Chapter 86. Research Project Architectures
86.1 Explicit Chabauty–Kim
86.2 Selmer–anabelian comparison
86.3 Monodromy–Selmer transfer
86.4 Period-obstruction towers
86.5 Arithmetic depth computation
86.6 Effective finite-depth reconstruction
86.7 Non-abelian (p)-adic Hodge structures
86.8 Higher arithmetic monodromy
Appendices
Appendix A. Algebraic Number Theory
A.1 Number fields
A.2 Places
A.3 Local fields
A.4 Galois groups
A.5 Frobenius
A.6 Inertia
Appendix B. Algebraic Geometry
B.1 Schemes
B.2 Smooth morphisms
B.3 Curves
B.4 Divisors
B.5 Picard schemes
B.6 Jacobians
Appendix C. Homological Algebra
C.1 Complexes
C.2 Derived functors
C.3 Ext
C.4 Group cohomology
C.5 Sheaf cohomology
C.6 Étale cohomology
Appendix D. Profinite and Unipotent Groups
D.1 Inverse limits
D.2 Profinite groups
D.3 Nilpotent groups
D.4 Unipotent groups
D.5 Lie algebras
D.6 Completion
Appendix E. (p)-Adic Analysis
E.1 (p)-adic fields
E.2 Analytic functions
E.3 Differential forms
E.4 Integration
E.5 Newton methods
E.6 Precision
Appendix F. Tannakian Categories
F.1 Tensor categories
F.2 Rigidity
F.3 Fiber functors
F.4 Affine group schemes
F.5 Tannaka duality
Appendix G. Hodge and (p)-Adic Hodge Background
G.1 Hodge structures
G.2 Mixed Hodge structures
G.3 Period rings
G.4 Crystalline theory
G.5 Comparison theorems
Appendix H. Computational Laboratory
H.1 SageMath environment
H.2 (p)-adic arithmetic
H.3 Coleman integration
H.4 Iterated integration
H.5 Selmer calculations
H.6 Rational-point verification
Appendix I. Continuous Reconstruction Spine
I.1 Choose (X/K)
I.2 Construct (J_X)
I.3 Construct (T_pJ_X)
I.4 Construct the Kummer map
I.5 Pass to (U_2)
I.6 Pass to (U_3)
I.7 Construct local Selmer conditions
I.8 Construct global Selmer conditions
I.9 Compute iterated Coleman coordinates
I.10 Compare arithmetic depth at each stage
I.11 Compare anabelian and Selmer obstructions
I.12 Compare monodromy and period defects
Appendix J. Core Reading Map
J.1 Chabauty–Coleman
J.2 Kim’s non-abelian Chabauty
J.3 Selmer varieties
J.4 Deligne fundamental groups
J.5 Mixed Tate motives
J.6 Bloch–Kato
J.7 Tannakian Coleman integration
J.8 Explicit depth-two Chabauty–Kim
J.9 Quadratic Chabauty
J.10 Anabelian reconstruction
J.11 Monodromy and period maps
Appendix K. Cross-Theory Dependency Tables
K.1 Cohomology to fundamental groups
K.2 Fundamental groups to Tannaka groups
K.3 Tannaka groups to monodromy
K.4 Motives to realizations
K.5 Realizations to periods
K.6 Selmer geometry to Chabauty–Kim
K.7 Anabelian to Selmer comparison
K.8 Monodromy to Selmer comparison
K.9 Period to obstruction comparison
Appendix L. Research-Frontier Index
L.1 Arithmetic depth
L.2 Global–local defect
L.3 Realization defect
L.4 Selmer–anabelian obstruction
L.5 Monodromy–Selmer transfer
L.6 Period-obstruction towers
L.7 Higher coherence
L.8 Effective reconstruction
L.9 Computational higher homotopy
Bibliography
Notation Index
Theorem Index
Concept Index
Arithmetic-Object Index
Obstruction Index
Representation Index
Computation Index
Research-Problem Index
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