Paper IIStatements 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
Refinement and residue coordinates
- Definition 3.1: Packet system
- Definition 3.2: Refinement
- Theorem 3.3: Children, parents, and transitivity Read proof
- Definition 3.4: Coordinate change across levels
- Theorem 3.5: Exact coordinate change and coherence Read proof
- Remark 3.6: Refinement and reduction
- Corollary 3.7: Orthogonal Hilbert packets Read proof
- Theorem 3.8: Refinement of operator blocks Read proof
- Corollary 3.9: Support under refinement Read proof
- Theorem 3.10: Common refinement of commuting packet systems Read proof
- Proposition 3.11: Coarsest common refinement Read proof
- Example 3.12: A two level split
- Definition 4.1: Residue selector
- Proposition 4.2: Residue packet algebra Read proof
- Theorem 4.3: Root-of-unity realization Read proof
- Theorem 4.4: Character realization Read proof
- Remark 4.5: Two distinct constructions
- Theorem 4.6: Nested residue refinement Read proof
- Example 4.7: The shift graph of residue packets
- Theorem 4.8: Coprime factorization Read proof
- Corollary 4.9: Dyadic and odd coordinates Read proof
- Example 4.10: Residues modulo twelve
Weighted transfer
- Definition 5.1: Complete multiplicative filtration
- Theorem 5.2: Weighted inverse Read proof
- Theorem 5.3: Block path formula Read proof
- Corollary 5.4: Minimum path weight Read proof
- Example 5.5: A longer path that appears earlier
- Corollary 5.6: Acyclic transfer graph Read proof
- Corollary 5.7: Stability of inversion at finite precision Read proof
- Corollary 5.8: Exact finite precision truncation Read proof
- Example 5.9: A triangular transfer
- Theorem 5.10: Certified weighted inverse Read proof
- Corollary 5.11: Finite precision
- Definition 6.1: Weighted transfer series
- Theorem 6.2: Rationality of the transfer series Read proof
- Corollary 6.3: Recurrence for path counts Read proof
- Corollary 6.4: Several independent weights Read proof
- Example 6.5: A transfer series with one loop
- Theorem 6.6: Automatic finite-graph transfer-zeta rationality Read proof
- Corollary 6.7: Linear recurrence
- Example 6.8: A computed transfer zeta
Integral separation and Smith congruences
- Theorem 7.1: Bézout spectral projector Read proof
- Theorem 7.2: Several coprime spectral packets Read proof
- Corollary 7.3: Resultant denominator bound Read proof
- Example 7.4: A resultant bound that is not exact
- Theorem 7.5: Evaluated B 'ezout spectral projector Read proof
- Definition 7.6: Separation weight
- Theorem 7.7: Projector precision delay Read proof
- Definition 8.1: Lattice packet data
- Theorem 8.2: Gluing module and exact separation weight Read proof
- Corollary 8.3: Total gluing and worst precision Read proof
- Corollary 8.4: Two complementary packets Read proof
- Corollary 8.5: Integral splitting criterion Read proof
- Theorem 8.6: Gluing exact sequence under refinement Read proof
- Example 8.7: Strict growth under refinement
- Remark 8.8: Strict growth can occur
- Theorem 8.9: Necessary and sufficient precision delay Read proof
- Example 8.10: The same rational split with two lattices
- Proposition 9.1: Sign projectors over a lattice Read proof
- Proposition 9.2: Integral character averages away from the group order Read proof
- Proposition 9.3: Odd-adic versus dyadic behavior Read proof
- Standard fact 10.1: Smith normal form over a discrete valuation ring
- Proposition 10.2: Computing a packet weight from an inclusion matrix Read proof
- Theorem 10.3: Smith weighted congruence solve Read proof
- Corollary 10.4: Obstruction and branching data Read proof
- Example 10.5: Three weighted directions
- Theorem 10.6: Concrete Smith-weighted congruence solve Read proof
- Example 10.7
Linear refinement defects
- Theorem 11.1: Power defect identity Read proof
- Corollary 11.2: Power norm bound Read proof
- Corollary 11.3: Norm bound
- Corollary 11.4: Polynomial defect Read proof
- Theorem 11.5: Resolvent defect identity Read proof
- Theorem 11.6: Holomorphic functional calculus defect Read proof
- Theorem 11.7: Exponential defect identity Read proof
- Theorem 11.8: Multilevel defect decomposition Read proof
- Example 11.9: A visible one step defect
- Corollary 11.10: Exact compatibility Read proof