These are proposed syntheses from the Second Brain, not new results or established proof dependencies. Only papers already present in this public catalog are linked here.
Proposed reading route
Classification beyond fixed invariants
The fixed-invariant manuscript constructs E₀ classification inside families sharing a full Elliott invariant and comparison radius. The trace-dynamics manuscript codes homeomorphism conjugacy into cocycle conjugacy. Both illustrate the role of descriptive complexity after familiar invariants have been fixed.
Keep separate: E₀ bireducibility, universality for Polish group orbit relations, and failure of classification by countable structures are distinct claims. Fixed first-order theory is an additional assertion in the fixed-invariant family, not a consequence of every Borel coding.
2 additional source notes are outside this catalog edition.
Proposed reading route
Dynamical lengths, spectra, and Cartan pairs
The Cartan-pair draft organizes five length coordinates and their invariance properties. The AF-spectrum draft classifies realizable values and explicitly imports an exclusion theorem from a companion. The primitive-ideal and Fell–Pedersen drafts examine how topological and rank data constrain dynamical lengths.
Keep separate: Track absolute algebra invariants, pair invariants, and groupoid invariants separately. The Cartan abstract identifies a low-complexity case in which the twist can affect one coordinate. The earlier Fell/Pedersen rank paper and its dynamical-length companion have separate identities.
2 additional source notes are outside this catalog edition.
Proposed reading route
Ideals, bases, and reconstruction
The critical-ideal manuscript constructs ideals from errors of basis expansions. The convergence manuscript builds Polishable groups from ideals. The reconstruction paper asks what those groups remember about the starting ideal.
Keep separate: Keep equality, isomorphism of ideals, Rudin–Blass comparisons, and Katětov equivalence separate. The critical-ideal result up to isomorphism and the reconstruction result up to Katětov equivalence answer different questions.
2 additional source notes are outside this catalog edition.
Proposed reading route
Positive identities and order extension
These drafts connect positivity constraints with algebraic and algorithmic structure. The positive-normal-identity draft states undecidability in fixed free groups; the universal-subgroup draft transfers bi-order extension problems; the common-multiple draft gives a positive theorem for abelian-by-nilpotent groups.
Keep separate: The common-multiple theorem concerns nilpotent coefficient groups and nonnegative coefficients. It does not supply an algorithm for the free-group problems. Arithmetic interpretation in free lattice-ordered groups is another kind of definability statement.
2 additional source notes are outside this catalog edition.
Proposed reading route
Symmetry, concentration, and boundedness
The library contains concentration and Lévy stabilizer constructions, formulas for universal minimal flows, automatic continuity for a homogeneous poset, and a separate collection on quantitative boundedness of Polish groups. Their shared objects are symmetry groups with additional topological or metric structure.
Keep separate: Distinguish amenability, extreme amenability, whirly amenability, concentration, and Property (OB). Record the topology and whether an assertion concerns arbitrary abstract homomorphisms or only continuous actions.
Proposed reading route
Uniform lifting and perturbation
The completely positive correction draft controls a correction using a uniform defect. The recursive-subhomogeneous stability draft states a control function independent of pullback length and matrix sizes. The Banach-lattice semiprojectivity draft formulates almost contractive lifting through ideal quotients.
Keep separate: These are comparisons of proof organization across different categories. A Banach-lattice homomorphism and a C*-algebra *-homomorphism satisfy different requirements. Separate fixed parameters, uniform constants, and dependencies on domain or target.
1 additional source note is outside this catalog edition.
