Paper VIStatements and proofs
Statements and proofs
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
The transfer-semigroup realization
Packet blocks of a semigroup
Refinement defects and the Duhamel identity
Markov seeds and stochastic transfer
Markov lumping as exact refinement
Conditional expectation as a stochastic projector
Bayesian conditioning as a normalized nonlinear operator
Exact Bayesian refinement and information-loss packets
- Theorem 8.1: Exact effective-likelihood factorization Read proof
- Theorem 8.2: Bayesian refinement criterion Read proof
- Remark 8.3: The one-packet boundary
- Theorem 8.4: Bayesian information-loss packet Read proof
- Corollary 8.5: Nested Bayesian packet tower Read proof
- Corollary 8.6: Canonical coarsest likelihood-resolving partition Read proof
- Numerical certificate 8.7: Bayesian refinement
- Boundary 8.8: Finite exact theorem
Bayesian flows, conjugacy, and hidden states
Fractional powers as spectral packet transfer
- Theorem 10.1: Fractional packet calculus Read proof
- Theorem 10.2: Poisson subordination Read proof
- Theorem 10.3: Fractional intertwining theorem Read proof
- Corollary 10.4: Fractional semigroup refinement Read proof
- Corollary 10.5: Fractional refinement towers
- Boundary 10.6: Unbounded and nonnormal generators
- Boundary 10.7: Non-self-adjoint fractional powers
Green operators and the Poisson seed
Finite-state Feynman–Kac transfer
Exact elimination of latent packets
Survival and hazard semigroups
One common operator language
Numerical certificates
Remaining conjectural extensions
Packet resolved semigroups
- Definition E2.1: Packet resolved transfer system
- Definition E2.2: Packet blocks
- Theorem E2.3: Semigroup and packet convolution Read proof
- Proposition E2.4: Block differential system Read proof
- Theorem E2.5: Laplace resolvent and packet transform Read proof
- Corollary E2.6: Resolvent expansion over packet paths Read proof
- Theorem E2.7: Reducing packet criterion Read proof
- Theorem E2.8: Dyson expansion over packet paths Read proof
- Corollary E2.9: Termination on an acyclic transfer graph Read proof
Generator defects and functional calculus
Refinement towers and limiting evolution
- Lemma E4.1: Completeness of the bounded inverse limit Read proof
- Theorem E4.2: Exact projective limit semigroup Read proof
- Remark E4.3
- Definition E4.4: Reconstructed fine evolution
- Theorem E4.5: Weighted summable defect completion Read proof
- Corollary E4.6: Contractive summable defect completion Read proof
- Corollary E4.7: Generator defect criterion Read proof
- Theorem E4.8: Asymptotic closure criterion Read proof
- Remark E4.9
- Example E4.10: Summable defects do not force a semigroup limit
- Proposition E4.11: Range leakage criterion Read proof
- Theorem E4.12: Generator control of range leakage Read proof
- Corollary E4.13: A usable closure condition for refinement towers Read proof
Finite Markov transfer and lumping
- Theorem E5.1: Finite Markov semigroup Read proof
- Definition E5.2: Canonical lumping obstruction
- Theorem E5.3: Descent through an aggregation map Read proof
- Corollary E5.4: Coordinate form of the lumping obstruction Read proof
- Theorem E5.5: Characterization of exact discrete lumping Read proof
- Corollary E5.6: Continuous time lumping Read proof
- Theorem E5.7: Finite horizon lumping defect Read proof
- Corollary E5.8: Stationary mass descends Read proof
Conditional expectation and martingale packets
Gaussian transfer and Hermite packets
Bayesian transfer and evidence packets
Bayesian coarsening and finite sufficiency
- Theorem E9.1: Exact two stage Bayesian update Read proof
- Theorem E9.2: Characterization of fixed coarse sufficiency Read proof
- Theorem E9.3: Coarsest fixed sufficient partition Read proof
- Definition E9.4: Within packet information loss
- Proposition E9.5: Exact local distortion Read proof
- Theorem E9.6: Approximate coarse Bayesian rule Read proof
- Definition E9.7: Relative information
- Theorem E9.8: Exact information decomposition Read proof
- Corollary E9.9: Coarse data processing and equality Read proof
- Corollary E9.10: Information created by a Bayesian update Read proof
- Corollary E9.11: Information criterion for local sufficiency Read proof
- Remark E9.12
Bayesian flows and hidden Markov recursion
- Theorem E10.1: Bayesian flow and replicator equation Read proof
- Theorem E10.2: Evidence cumulants and score monotonicity Read proof
- Corollary E10.3: Concentration on the maximal score packet Read proof
- Proposition E10.4: Simplex preservation Read proof
- Theorem E10.5: One step filter stability Read proof
- Example E10.6: Beta binomial transfer
Fractional functional calculus
- Theorem E11.1: Fractional packet calculus Read proof
- Theorem E11.2: Balakrishnan formula in finite dimension Read proof
- Theorem E11.3: Exact fractional naturality Read proof
- Theorem E11.4: Fractional defect bound Read proof
- Remark E11.5
- Standard fact E11.6: Stable subordination
- Theorem E11.7: Poisson subordination Read proof
- Remark E11.8
Green operators and finite difference transfer
- Theorem E12.1: Exact nodal Green reconstruction Read proof
- Corollary E12.2: Discrete Green quadrature Read proof
- Proposition E12.3: Discrete maximum bound Read proof
- Theorem E12.4: Second order Poisson solution error Read proof
- Theorem E12.5: Low mode eigenvalue control Read proof
- Corollary E12.6: Heat and fractional mode transfer Read proof
Finite state Feynman Kac transfer
Schur Feshbach elimination
Memory modes and controlled compression
- Theorem E15.1: Polynomial exponential memory of a finite hidden system Read proof
- Corollary E15.2: Optimal selection for the certified primary block bound Read proof
- Corollary E15.3: Normal hidden dynamics Read proof
- Corollary E15.4: Pole order and hidden Jordan depth Read proof
- Corollary E15.5: Finite auxiliary state realization Read proof
- Theorem E15.6: Volterra stability under kernel compression Read proof
- Corollary E15.7: Trajectory bound for modal truncation Read proof
- Remark E15.8
Survival, moments, and competing risks
- Theorem E16.1: State dependent killing as survival Read proof
- Theorem E16.2: Survival and hazard identities Read proof
- Theorem E16.3: Laplace transform and moments Read proof
- Proposition E16.4: Residual life as a restarted semigroup Read proof
- Theorem E16.5: Competing risk packets Read proof
- Remark E16.6
Transfer paths and product-topology limits
- Theorem R1.1: Packet convolution Read proof
- Proposition R1.2: Packet resolvent convolution Read proof
- Corollary R1.3: Minimum transfer order Read proof
- Theorem R1.4: Several level telescoping identity Read proof
- Corollary R1.5: Summable generator defects Read proof
- Definition R1.6: Projective limit space
- Theorem R1.7: Exact projective limit semigroup Read proof
- Corollary R1.8: Coordinate generators Read proof
Reversible coarse dynamics
Prior-restricted sufficiency and filtering
- Proposition R3.1: Sufficiency relative to a prior disintegration Read proof
- Lemma R3.2: Normalization bound Read proof
- Theorem R3.3: Exact information loss formula Read proof
- Corollary R3.4: Oscillation bound Read proof
- Corollary R3.5: Covariance identity Read proof
- Theorem R3.6: Exact coarse hidden Markov filter Read proof
- Theorem R3.7: One step filter defect Read proof