Realization BoundariesStatements 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
What a realization must supply
- Definition 1.1: Operator-seed realization
- Definition 1.2: Finite semantic description
- Definition 1.3: Computable copy
- Definition 1.4: Topological generation
- Proposition 1.5: Countable-code bound Read proof
- Definition 1.6: Grammar-relative description complexity
- Boundary 1.7: No uncharged infinite parameters
- Theorem 1.8: Reachability obstruction Read proof
The squarefree ambient system
- Theorem 2.1: The torsion subgroup of the product Read proof
- Proposition 2.2: Diagonal endomorphisms of the direct sum Read proof
- Corollary 2.3: An infinite seed obstruction Read proof
- Theorem 2.4: Topological cyclicity Read proof
- Proposition 2.5: Profinite completion of the direct sum Read proof
- Theorem 2.6: Unique division in the ambient quotient Read proof
Prime spectra and finite descriptions
- Theorem 3.1: Subgroup and isomorphism classification Read proof
- Corollary 3.2: Finite observations leave continuum many possibilities Read proof
- Theorem 3.3: The prime-spectrum criterion Read proof
- Corollary 3.4 Read proof
- Theorem 3.5: Unary selector classification Read proof
- Example 3.6: A computable group outside the unary grammar
Pure preimages and common finite observations
- Definition 4.1: Purity
- Theorem 4.2: Pure-preimage correspondence Read proof
- Theorem 4.3: Ambient root criterion Read proof
- Theorem 4.4: One finite-quotient system for every preimage Read proof
- Proposition 4.5: Finite-target equivalence Read proof
- Proposition 4.6: Structure of an ambient preimage Read proof
The paired-prime groups
Root certificates and finite quotients
Classification, morphisms, and seed rank
- Lemma 7.1: Tail signature Read proof
- Theorem 7.2: Unmarked isomorphism classification Read proof
- Corollary 7.3: Marked classification Read proof
- Example 7.4: The same group with a different marked point
- Theorem 7.5: Complete homomorphism description Read proof
- Theorem 7.6: Torsion-image obstruction Read proof
- Corollary 7.7: Endomorphism ring Read proof
- Corollary 7.8: Automorphisms Read proof
- Theorem 7.9: Seed-rank separation Read proof