Paper IStatements 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
Foundations and seed rank
- Definition 1.1: Coefficient and module regimes
- Definition 1.2: Canonical theorem record
- Boundary 1.3: Symbolic scope
- Definition 2.1: Typed operator grammar
- Definition 2.2: Realization on a single space
- Theorem 2.3: Typed realization and module equivalence Read proof
- Definition 2.4: Realized operator algebra
- Proposition 2.5: Action factorization and hull invariance Read proof
- Definition 2.6: Linear seed system
- Definition 2.7: Generated hull
- Definition 2.8: Natural presentation
- Proposition 2.9: Algebra generation and module generation are distinct Read proof
- Boundary 2.10: Operators chosen for a target
- Definition 3.1: Seed evaluation map
- Theorem 3.2: Finite free presentation Read proof
- Corollary 3.3: Characterization by minimal free covers Read proof
- Corollary 3.4: Presentation with one seed Read proof
- Proposition 3.5: Criterion for the annihilator to be an ideal on both sides Read proof
- Theorem 3.6: Completion of a dense free presentation Read proof
- Definition 4.1: Seed rank
- Proposition 4.2: Cyclic characterization Read proof
- Proposition 4.3: Monotonicity under enlargement of the acting algebra Read proof
- Theorem 4.4: Quotient monotonicity Read proof
- Corollary 4.5: Isomorphism invariance Read proof
- Theorem 4.6: Short exact sequence bounds Read proof
- Proposition 4.7: Direct sum bounds for a shared action Read proof
- Theorem 4.8: Direct sum formula for independent actions Read proof
- Proposition 4.9: Tensor product bound Read proof
- Boundary 4.10: Tensor rank is not generally multiplicative
- Proposition 4.11: Monotonicity under extension of scalars Read proof
- Definition 4.12: Witness for seed rank
- Proposition 4.13: Scalar quotient witness Read proof
- Classical input 5.1: Nakayama lemma
- Theorem 5.2: Generator number is detected modulo the Jacobson radical Read proof
- Corollary 5.3: Local algebra formula Read proof
- Classical input 5.4: Wedderburn and Artin decomposition
- Lemma 5.5: Multiplicities in the regular module Read proof
- Theorem 5.6: Semisimple seed rank formula over arbitrary division rings Read proof
- Corollary 5.7: Algebraically closed multiplicity formula Read proof
- Corollary 5.8: Full matrix algebra Read proof
- Corollary 5.9: Multiplicity obstruction Read proof
- Classical input 6.1: Primary decomposition for polynomial modules
- Theorem 6.2: Seed rank for a single operator Read proof
- Corollary 6.3: Invariant factor count Read proof
- Corollary 6.4: Formula for geometric multiplicity Read proof
- Corollary 6.5: Criterion for a cyclic matrix Read proof
Packets and transfer graphs
- Definition 7.1: Finite packet system
- Definition 7.2: Packet support and packet pure vectors
- Lemma 7.3: Finite packet aggregation Read proof
- Definition 7.4: Packet analysis and synthesis
- Theorem 7.5: Exact packet coordinate isomorphism Read proof
- Corollary 7.6: Packetwise equality Read proof
- Corollary 7.7: Universal mapping property of finite packet coordinates Read proof
- Theorem 7.8: Tensor product packet systems Read proof
- Proposition 7.9: Dual packet systems Read proof
- Theorem 7.10: Finite Banach packet isomorphism Read proof
- Definition 7.11: Packet coordinate condition number
- Proposition 7.12: Conditioned coordinate transport Read proof
- Corollary 7.13: Completion preserves a finite packet split Read proof
- Definition 7.14: Unconditional packet sum space
- Lemma 7.15: Completeness of the unconditional packet sum Read proof
- Theorem 7.16: Countable unconditional packet isomorphism Read proof
- Remark 7.17: Uniform boundedness
- Boundary 7.18: Infinite packet sums
- Definition 8.1: Transfer block
- Theorem 8.2: Exact operator reconstruction Read proof
- Theorem 8.3: Packet matrix multiplication Read proof
- Corollary 8.4: Packet powers and paths Read proof
- Corollary 8.5: Adjoint reversal Read proof
- Corollary 8.6: Tensor product transfer blocks Read proof
- Theorem 8.7: Blockwise intertwining equations Read proof
- Theorem 8.8: Full block algebra representation Read proof
- Corollary 8.9: Propagation of packet support Read proof
- Corollary 8.10: Dimension count Read proof
- Proposition 8.11: Block diagonal projection and packet commutant Read proof
- Definition 8.12: Support relation and support space
- Theorem 8.13: Support pattern algebra criterion Read proof
- Corollary 8.14: Bounded support patterns closed under * Read proof
- Theorem 8.15: Preorder support algebra decomposition Read proof
- Definition 9.1: Transfer graph of a family generating the algebra
- Theorem 9.2: Reachability support of the generated algebra Read proof
- Proposition 9.3: Reachability obstruction for generated hulls Read proof
- Definition 9.4: Packet pure seed rank
- Proposition 9.5: Basic packet pure rank bounds Read proof
- Definition 9.6: Packet system surjective along edges
- Theorem 9.7: Source component formula for packet pure seed rank Read proof
- Definition 9.8: Packet propagating system
- Corollary 9.9: Unweighted source component count Read proof
- Boundary 9.10: Mixed support seeds can merge source requirements
Transport and realization
- Theorem 10.1: Coordinate transport Read proof
- Definition 10.2: Packet compatible operator morphism
- Definition 10.3: Morphism of seeded systems
- Theorem 10.4: Naturality of generated hulls and transfer blocks Read proof
- Proposition 10.5: Category of seeded packet systems Read proof
- Theorem 10.6: Packet exact sequence for an invariant subspace Read proof
- Theorem 10.7: Packet descent through an invariant quotient Read proof
- Corollary 10.8: Restriction and quotient of transfer blocks Read proof
- Corollary 10.9: Topological packet exact sequence Read proof
- Theorem 10.10: Dense realization theorem Read proof
- Theorem 10.11: Quotient topology and completed realization Read proof
- Boundary 10.12: Continuous surjections and quotient topology
- Definition 11.1: Realization functor and preservation profile
- Theorem 11.2: Equational realization principle Read proof
- Proposition 11.3: Realization export principle Read proof
- Definition 11.4: Semantic realization criterion
- Definition 12.1: Worked realization interface
- Definition 12.2: Paper I export object
- Boundary 12.3: Exact limits of Paper I
Worked models and calculations
- Proposition 13.1: Typed reachability certificate
- Example 14.1: Worked reduction
- Proposition 15.1: Shift seed
- Boundary 15.2
- Corollary 17.1: Certified graph seed number
- Boundary 17.2: Input from spectral classification
- Construction 19.1: Closure specification for the orbit hull
- Construction 19.2: Table specification for transfer blocks
- Construction 19.3: Exact graph seed rank from a certificate