Large Signal Model statements and proofs · Komorebi

Find a statement anywhere in this work by title, printed number, type or source label. Entries follow the chapter order. Each proof link opens the written proof passage.

Statements by chapter

2. Exact realization and atlas consistency

Proposition 1: Exact realization and consistency Read proof

3. Coordinate transport and one underlying operator

Proposition 2: Coherent transport and operator realization Read proof

4. Packets, refinement and transfer blocks

5. Predictive closure is the central contract

6. Constructing the smallest closed linear refinement

Proposition 8: Minimal linear predictive completion Read proof

7. Joint closure of prediction and Bayesian inference

Proposition 9: Coarsest joint finite closure Read proof

8. Memory is an exact alternative to adding state

Proposition 10: Exact projected memory and a conditional tail bound Read proof

9. Statistical refinement needs its own theorem

Proposition 11: Ideal statistical benefit of refinement Read proof

10. Bayesian coarsening and the information-loss packet

Proposition 12: Bayesian closure and its exact defect Read proof

12. A bounded infinite-resolution bridge

Proposition 13: Strong fixed-horizon refinement limit Read proof

15. From hidden state to an accessible predictive state

Proposition 14: Exact causal delay closure Read proof

16. What finite data identify

17. Why finite agreement is not unrestricted discovery

Proposition 17: Finite-prefix indistinguishability Read proof

19. A single propagation law for typed defects

Proposition 18: Compositional defect propagation Read proof

20. Truncating memory without losing the prediction bound

Proposition 19: Uniform prediction bound for a stable finite-memory truncation Read proof

21. Nonlinear packet truncation with an explicit remainder

Proposition 20: Packet-interaction truncation bound Read proof

22. Learning several charts without inventing several worlds

Proposition 21: Coherent reconciliation and coordinate-stable transport Read proof

23. Refinement cannot manufacture missing information

Proposition 22: The upward-lift obstruction Read proof

24. A growing operator grammar with coherent evidence

25. The finite learned-realization theorem

Proposition 25: Observationally grounded finite linear realization Read proof

27. Actions and observations are different inputs

Proposition 26: Finite instrument closure Read proof

28. The state is a function of possible future experiments

Proposition 27: Minimal finite behavior realization Read proof

29. Recovering the operators from an observable probability table

Proposition 28: Observable table reconstruction Read proof

30. Probability validity is not a rank condition

Proposition 29: Canonical probability cone Read proof

31. Packets can reconstruct a belief without being beliefs

Proposition 30: Positive packet decomposition and its boundary Read proof

32. Action coverage is a mathematical requirement

Proposition 31: Missing-intervention nonidentifiability Read proof

33. Learning the coordinate space under bounded noise

Proposition 32: Finite-table rank separation and subspace stability Read proof

34. Noisy operator reconstruction in a fixed core

Proposition 33: Noisy core-block realization Read proof

35. Conditioning a prediction can amplify its error

Proposition 34: Normalization-aware filtering stability Read proof

36. A finite experimental route to the probability table

Proposition 35: Reset-experiment probability certificate Read proof

37. A filtering defect with a packet interpretation

Proposition 36: Exact normalized closure defect Read proof

39. Integrated controlled stochastic realization

Proposition 37: Conditional observation-based stochastic completion Read proof

42. Finite tests give an outer hierarchy, not automatic closure

Proposition 38: Test-cone hierarchy and a finite stopping certificate Read proof

43. A finite certificate with a positive lift

Proposition 39: Finite positive-lift certificate Read proof

44. Robustness is a quantified invariance statement

Proposition 40: Robust finite-domain certificate Read proof

45. Fitting inside a certified domain

Proposition 41: Certified correction and its error budget Read proof

46. A norm that measures probability mass before conditioning

Proposition 42: Instrument contraction before normalization Read proof

47. Whole observation laws can be stable without rare-event margins

Proposition 43: Observation-tree total-variation bound Read proof

48. Conditioning still has an unavoidable cost

Proposition 44: Sharp event-probability dependence of conditioning Read proof

49. Searching for a domain from observable operators

Proposition 45: Rational finite-certificate search Read proof

51. Why finite predictive rank need not mean finitely many positive states

Proposition 46: A rational rank-three model with no polyhedral probability-domain certificate Read proof

52. A finite approximation with an explicit price

Proposition 47: Polyhedral approximation of the rotation model Read proof

53. Repeated proposal and verification with a confidence budget

Proposition 48: Sequential confidence accounting for certified proposals Read proof

54. Integrated certificate-carrying realization

Proposition 49: Conditional learning with an all-horizon probability certificate Read proof

56. A matrix certificate for a curved domain

Proposition 50: Matrix-embedding probability certificate Read proof

57. The earlier disk obstruction has an exact finite matrix witness

Proposition 51: Exact rational matrix witness for the rotation model Read proof

58. An observation density is not a branch probability

Proposition 52: Continuous-observation instrument semantics Read proof

59. A finite certificate over an entire observation interval

Proposition 53: Polynomial Gram certificate for interval observations Read proof

60. Learning from integral observations rather than point probabilities

Proposition 54: Conditional finite-moment identification Read proof

61. Finite moments cannot validate an unrestricted density class

Proposition 55: A valid continuous counterexample to finite-moment identification Read proof

62. Exact observation bins preserve their own event law

Proposition 56: Exact bin coarsening and its posterior Read proof

63. Averaging transformations has a covariance defect

Proposition 57: Exact covariance correction for an averaged congruence Read proof

64. A valid density approximation with a resolution bound

Proposition 58: Positive cell-average approximation Read proof

65. Full continuous record laws inherit the stability argument

Proposition 59: Continuous-record total-variation bound Read proof

67. A declared template can be recognized without being rediscovered

Proposition 60: Exact recognition of a fixed certified template Read proof

68. The combined continuous polynomial sector

Proposition 61: Conditional continuous realization with a finite domain certificate Read proof

70. An ellipsoidal certificate with an orientation test

Proposition 62: Oriented quadratic probability certificate Read proof

71. Geometry can be proposed from the operators themselves

Proposition 63: Conditional operator-only ellipsoid discovery Read proof

72. Approximate an unknown density without assuming its exact degree

Proposition 64: Valid positive polynomial approximation Read proof

73. Keep approximation bias separate from coefficient error

Proposition 65: Certified coefficient error and approximation bias Read proof

74. Evidence from dependent observations

Proposition 66: Time-uniform predictable-residual certificate Read proof

75. Actions need coverage, not only more observations

Proposition 67: Covered adaptive-action evidence Read proof

76. What a residual can prove about misspecification

Proposition 68: Sequential evidence gives a lower error certificate Read proof

77. Comparing a model with a process outside its representation

Proposition 69: Observation-law error without a shared state space Read proof

78. An integrated theory of construction and correction

Proposition 70: Separate structural acceptance, approximation control and sequential rejection Read proof

80. A margin that has probability meaning

Proposition 71: Robust oriented margin Read proof

81. Why exact boundary preservation can have zero noise radius

Proposition 72: Boundary fragility under mass preserving perturbations Read proof

82. An explicit repair and the price of its slack

Proposition 73: Interior repair with an explicit error budget Read proof

83. Noisy invariant forms are proposals, not certificates

Proposition 74: A posteriori proposal error Read proof

84. Finite exact checking on a continuum

Proposition 75: Rational semicircle and Bernstein certificate Read proof

85. A finite uncertainty box and the role of convexity

Proposition 76: Coefficient boxes and convex uncertainty Read proof

86. What a search can guarantee

Proposition 77: Conditional search for strict rational curved certificates Read proof

87. Selection from data does not remove the uncertainty quantifier

Proposition 78: Data-selected geometry with honest coverage Read proof

89. The prediction cost needs a common comparison domain

Proposition 79: Certified repair with a retained comparison budget Read proof

91. A smaller repair can be bracketed rather than guessed

Proposition 80: Convex repair bracketing Read proof

92. Confidence from dependent observations

Proposition 81: Time-uniform regression confidence Read proof

93. From observable moment regression to operator sets

Proposition 82: Sequential observable-moment operator sets Read proof

94. A nonlinear observable state with a curved moment domain

Proposition 83: Structured curved-domain certificate Read proof

95. A fully rational dependent-data reference experiment

Proposition 84: Exact rational outer confidence intervals Read proof

96. Structured uncertainty has an observable error bound

Proposition 85: Parameter-to-law error without a state-estimation term Read proof

97. Long-horizon marginal control is not full-record control

Proposition 86: Uniform marginal and rolling-window stability Read proof

98. A conditional data-to-domain theorem

Proposition 87: Observable structured learning with a covered domain Read proof