Komorebi SpaceStatements 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
Views and coherent realizations
- Definition 1.1: View system
- Definition 1.2: Strict Komorebi realization
- Theorem 1.3: Joint determination and the universal property Read proof
- Proposition 1.4: Functorial transport Read proof
- Proposition 1.5: Terminal evaluation Read proof
- Theorem 2.1: Strict gluing Read proof
- Example 2.2: An object cover with no glue
- Theorem 2.3: Fiber and section compatibility Read proof
- Example 2.4: The loop with no coherent choice
The additive obstruction and the space of extensions
- Proposition 3.1: Differential identities Read proof
- Theorem 3.2: Extension criterion Read proof
- Theorem 3.3: Classification of a nonempty extension fiber Read proof
- Example 3.4: Parity and an independent ambiguity
- Proposition 3.5: Exactness at partial sections Read proof
- Theorem 4.1: Semantics preserving correction Read proof
- Proposition 4.2: A certificate of impossibility Read proof
- Example 4.3: A rigid obstruction
Changing what is prescribed
Resolution towers and exact transport
Finite observations and realized transformations
Banach repair and constructed split coordinates
Repair under a change of representation
Multiscale residuals and the limits of reconstruction
- Theorem 10.1: Finite reconstruction and nested remainders Read proof
- Theorem 10.2: Density criterion for norm recovery Read proof
- Theorem 10.3: Dyadic determination and finite coordinate recovery Read proof
- Proposition 10.4: The dyadic support pattern Read proof
- Proposition 10.5: The recovery domain is (c 0 ) Read proof
Nonabelian coherence and relative trivialization
- Definition 11.1: Cocycle
- Proposition 11.2: Gauge action and its groupoid Read proof
- Theorem 11.3: Nonabelian relative extension Read proof
- Theorem 11.4: The nonabelian extension torsor Read proof
- Example 11.5: A (C 2 ) loop with (S 3 ) coefficients Read proof
- Theorem 12.1: Effective cocycle descent Read proof
- Proposition 13.1: The composition topology Read proof
- Theorem 13.2: Stackhood on the composition site Read proof
- Proposition 14.1: Equivalence of section spaces Read proof
- Definition 14.2: Marked realization groupoid
- Proposition 14.3: Existence and invisible automorphisms Read proof
- Theorem 14.4: Invariance of the marked realization problem Read proof
A boundary identification is data
Coherence need not mean removable phase
Graph views, holonomy, and reconstruction
- Proposition 17.1: The path telescope Read proof
- Theorem 17.2: Holonomy classification Read proof
- Proposition 18.1: Exact orthogonal repair Read proof
- Theorem 18.2: Closed range and quantitative exactness Read proof
- Theorem 18.3: Nearest feasible protected repair Read proof
- Theorem 19.1: Amplified holonomy repair Read proof
- Theorem 20.1: The two-stage repair is the joint projection Read proof
- Theorem 20.2: Closed range for the joint defect Read proof
- Theorem 21.1: Motion of the exact projector Read proof
- Theorem 21.2: Stable surjectivity and derived repair Read proof
- Theorem 21.3: Explicit kernel transport Read proof
- Proposition 21.4: Exact motion of the line Read proof
Reading a stability certificate
A geometry of transformations needs its own cost
Exactification without a fixed contraction rate
- Theorem 24.1: Summable movement gives exactification Read proof
- Theorem 24.2: Variable-rate exactification Read proof
- Proposition 24.3: Controlled paths kill the actual obstruction Read proof
- Proposition 24.4: Path independence requires a comparison Read proof
- Proposition 24.5: Cofinal sampling of a decreasing profile Read proof