Large Signal ModelReading edition

55. From polyhedral to curved certificates

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Core papers

Paper ITyped operator systems, seed rank, and exact packet coordinates
Paper IIRefinement, weighted transfer, and integral separation
Paper IIITyped transfer, nonlinear packets, and obstruction theory
Paper IVCyclic analytic functions, reciprocal families, and spectral zero data
Paper VMultiresolution Signal Calculus
Paper VITransfer semigroups and memory
Paper VIIQuantum Seeds and Effective Dynamics
Paper VIIIPacket Operator Algorithms, Exact Compilation, and GPU Realization
Paper IXEuler Congruences
Paper XUniversal Residual Seeds
Paper XITame and Wild Packet Geometry for Local Langlands
Paper XIIPatching Support, Hecke Packets, and Conditional Automorphic Completion

Books and monographs

Special FunctionsSpecial Functions: Function families and their structure
Proof Path TheoryCertificates, compression, discovery fields, and mathematical memory
Komorebi SpaceCoherent realizations, obstructions, and descent
Realization BoundariesFinite Grammar, Infinite Groups, and the Limits of Reconstruction
Komorebi MethodologyMathematical Discovery, Realization, and the Discipline of Proof
Quantum and CircuitsQuantum and Circuits: Arithmetic, symmetry, and catalysis
Large Signal ModelRealization, refinement, and predictive closure
Machine-Native ReasoningEvidence, methods, and accountable revision
The K-COS MethodStructured Gaussian Quadratic Forms, Spectral Compression, and Certified Extreme-Tail Evaluation

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Large Signal Model

Part I. Realization and predictive closure (14)

1. Three objects that must remain distinct
2. Exact realization and atlas consistency
3. Coordinate transport and one underlying operator
4. Packets, refinement and transfer blocks
5. Predictive closure is the central contract
6. Constructing the smallest closed linear refinement
7. Joint closure of prediction and Bayesian inference
8. Memory is an exact alternative to adding state
9. Statistical refinement needs its own theorem
10. Bayesian coarsening and the information-loss packet
11. What a typed operator language must guarantee
12. A bounded infinite-resolution bridge
13. A complete small signal example
14. Connection to observable learning

Part II. Learning, stability and coherent expansion (13)

Part II. Learning, stability and coherent expansion
15. From hidden state to an accessible predictive state
16. What finite data identify
17. Why finite agreement is not unrestricted discovery
18. A complete output-only identification example
19. A single propagation law for typed defects
20. Truncating memory without losing the prediction bound
21. Nonlinear packet truncation with an explicit remainder
22. Learning several charts without inventing several worlds
23. Refinement cannot manufacture missing information
24. A growing operator grammar with coherent evidence
25. The finite learned-realization theorem
26. Connection to controlled prediction

Part III. Controlled stochastic realization (15)

Part III. Controlled stochastic realization
27. Actions and observations are different inputs
28. The state is a function of possible future experiments
29. Recovering the operators from an observable probability table
30. Probability validity is not a rank condition
31. Packets can reconstruct a belief without being beliefs
32. Action coverage is a mathematical requirement
33. Learning the coordinate space under bounded noise
34. Noisy operator reconstruction in a fixed core
35. Conditioning a prediction can amplify its error
36. A finite experimental route to the probability table
37. A filtering defect with a packet interpretation
38. A complete two-action rational example
39. Integrated controlled stochastic realization
40. From realization to admissible domains

Part IV. Certified probability domains and robust learning (16)

Part IV. Certified probability domains and robust learning
41. The object to be certified
42. Finite tests give an outer hierarchy, not automatic closure
43. A finite certificate with a positive lift
44. Robustness is a quantified invariance statement
45. Fitting inside a certified domain
46. A norm that measures probability mass before conditioning
47. Whole observation laws can be stable without rare-event margins
48. Conditioning still has an unavoidable cost
49. Searching for a domain from observable operators
50. Selection and noise do not invalidate an exact witness
51. Why finite predictive rank need not mean finitely many positive states
52. A finite approximation with an explicit price
53. Repeated proposal and verification with a confidence budget
54. Integrated certificate-carrying realization
55. From polyhedral to curved certificates

Part V. Curved domains and continuous observations (15)

Part V. Curved domains and continuous observations
56. A matrix certificate for a curved domain
57. The earlier disk obstruction has an exact finite matrix witness
58. An observation density is not a branch probability
59. A finite certificate over an entire observation interval
60. Learning from integral observations rather than point probabilities
61. Finite moments cannot validate an unrestricted density class
62. Exact observation bins preserve their own event law
63. Averaging transformations has a covariance defect
64. A valid density approximation with a resolution bound
65. Full continuous record laws inherit the stability argument
66. A complete continuous-observation example
67. A declared template can be recognized without being rediscovered
68. The combined continuous polynomial sector
69. From continuous semantics to discovery

Part VI. Certificate discovery, approximation and sequential evidence (11)

Part VI. Certificate discovery, approximation and sequential evidence
70. An ellipsoidal certificate with an orientation test
71. Geometry can be proposed from the operators themselves
72. Approximate an unknown density without assuming its exact degree
73. Keep approximation bias separate from coefficient error
74. Evidence from dependent observations
75. Actions need coverage, not only more observations
76. What a residual can prove about misspecification
77. Comparing a model with a process outside its representation
78. An integrated theory of construction and correction
79. From exact geometry to robustness

Part VII. Robust curved certificates and noisy geometry (12)

Part VII. Robust curved certificates and noisy geometry
80. A margin that has probability meaning
81. Why exact boundary preservation can have zero noise radius
82. An explicit repair and the price of its slack
83. Noisy invariant forms are proposals, not certificates
84. Finite exact checking on a continuum
85. A finite uncertainty box and the role of convexity
86. What a search can guarantee
87. Selection from data does not remove the uncertainty quantifier
88. A complete perturbed-operator construction
89. The prediction cost needs a common comparison domain
90. From deterministic robustness to data confidence

Part VIII. Data-supported domains and smaller repairs (12)

Part VIII. Data-supported domains and smaller repairs
91. A smaller repair can be bracketed rather than guessed
92. Confidence from dependent observations
93. From observable moment regression to operator sets
94. A nonlinear observable state with a curved moment domain
95. A fully rational dependent-data reference experiment
96. Structured uncertainty has an observable error bound
97. Long-horizon marginal control is not full-record control
98. A conditional data-to-domain theorem
99. Two distinct completed reference sectors
100. Scope and research boundary
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55. From polyhedral to curved certificates

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55. From polyhedral to curved certificates

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A finite positive lift is one sufficient certificate language. Proposition 46 shows that a low-dimensional model can require a nonpolyhedral domain. Part V represents that same model exactly through a curved matrix domain. A failed finite-generator search is therefore not a general invalidity theorem.

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