Paper IIIStatements 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
Finite coordinates and typed transfer
- Boundary 1.1: Final theorem denominator
- Definition 2.1: Packet graph and scalar edge system
- Lemma 2.2: Declared support
- Theorem 2.3: Typed interface annihilation
- Theorem 2.4: Collapse of a matched edge pair
- Boundary 2.5: Current fiber scope
- Definition 3.1: Composable word and execution operator
- Theorem 3.2: First broken interface
- Theorem 3.3: Noncomposable words vanish
- Definition 3.4: Recursive word expansion
- Theorem 3.5: Transfer power equals recursive word expansion
- Remark 3.6
- Lemma 3.7: One word is one endpoint block
- Theorem 3.8: Finite operator expansion from words to walks
- Corollary 3.9: Endpoint coefficient walk sum
- Corollary 3.10: Reachability obstruction at fixed length
- Corollary 3.11: Orbit support obstruction
- Theorem 4.1: Exact frame transport is coherent
- Theorem 4.2: Every finite canonical path telescopes
- Definition 4.3: Canonical exact coordinate transport
- Theorem 4.4: Identity, inverse, and cocycle for canonical transport
- Theorem 4.5: Canonical coordinate naturality
- Corollary 4.6: Canonical triangle holonomy is trivial
- Boundary 4.7: Refinement is not reduction
Connections and obstruction witnesses
- Definition 5.1: Noncanonical connection normalized at each chart
- Theorem 5.2: Checked flatness consequences
- Proposition 5.3: The minimal API admits nontrivial holonomy
- Boundary 5.4: Why the minimal defect and loop are distinct
- Definition 5.5: Normalized reversible connection
- Definition 5.6: Calibration and flatness
- Remark 5.7: A connection on a chart graph
- Definition 5.8: Cocycle defect
- Definition 5.9: Triangle holonomy
- Theorem 5.10: Identity between defect and holonomy
- Corollary 5.11: Canonical transport has zero connection defect
- Definition 5.12: Connection transport along a path
- Proposition 5.13: Flat pair transport collapses to endpoints
- Remark 5.14
- Definition 6.1: Gauge change
- Theorem 6.2: Gauge covariance
- Theorem 6.3: Flat connections are gauges of the canonical pair transport
- Boundary 6.4: Flat does not mean calibrated
- Definition 7.1: Solution and obstruction packet for a candidate
- Theorem 7.2: Exact logical force of a local obstruction
- Boundary 7.3: Candidate-specific means candidate-specific
- Definition 7.4: Holonomy obstruction and packet residuals
- Theorem 7.5: Exact reconstruction and detection by obstruction blocks
- Corollary 7.6: Witness supported in one packet
- Proposition 7.7: Gauge covariance when packet projectors are preserved
- Boundary 7.8: Packet patterns are not absolute
- Proposition 8.1: Packet path expansion for different matrices
- Boundary 8.2: Changing packet types along a chain
- Remark 8.3: Checked specialization
- Example 9.1: A scalar triangle
- Example 9.2: A transfer walk with two steps
- Proposition 10.1: Exact naturality for arbitrary functions
- Boundary 10.2: Where nonlinear defects can enter
- Definition 11.1: Finite grammar for operation trees
- Theorem 11.2: Input locality for operation trees
- Boundary 11.3: Exact force of locality
- Definition 11.4: Finitary colored operation signature
- Definition 11.5: Typed operation tree and evaluation
- Formalization target 11.6: Finite multilinear tree expansion
- Formalization target 11.7: Operation-tree defect propagation
- Boundary 11.8: Analytic and geometric exclusions
- Boundary 12.1: No scope promotion by prose
Nonlinear packet coordinates
- Definition 16.1: Nonlinear packet coordinate map
- Theorem 16.2: Exact nonlinear coordinate theorem Read proof
- Corollary 16.3: Dynamical equivalence Read proof
- Proposition 16.4: Change of packet coordinates Read proof
- Example 16.5: A mixed quadratic map
- Remark 16.6
- Definition 17.1: Packet jet
- Theorem 17.2: Exact reconstruction of derivatives Read proof
- Proposition 17.3: Jet norm bound Read proof
- Corollary 17.4: Packet Taylor formula Read proof
- Theorem 17.5: Packet Fa `a di Bruno formula Read proof
- Proposition 17.6: Number of labeled composition terms Read proof
- Corollary 17.7: Structural pruning Read proof
- Example 17.8: The second order composition law
- Remark 17.9
- Theorem 18.1: Polynomial tree expansion Read proof
- Corollary 18.2: Quadratic Catalan expansion Read proof
- Theorem 18.3: Packet labeled tree reconstruction Read proof
- Corollary 18.4: Tree pruning Read proof
- Example 18.5: Catalan parity packets
- Definition 19.1: Grade respecting operation
- Theorem 19.2: Graded packet convolution Read proof
- Corollary 19.3: Cyclic product formula Read proof
- Theorem 19.4: Packet Maurer Cartan equations Read proof
- Proposition 19.5: Packet equations for L infty operations Read proof
- Definition 19.6: Divided square
- Proposition 19.7: Divided square packet equations Read proof
- Example 19.8: Matrix difference grading
- Remark 19.9
- Definition 20.1: Nonlinear refinement defect
- Theorem 20.2: Exact defect under composition Read proof
- Corollary 20.3: Linear product rule Read proof
- Theorem 20.4: Iterated orbit defect Read proof
- Theorem 20.5: Fixed point stability Read proof
- Corollary 20.6: Several refinement levels Read proof
- Proposition 20.7: Derivative defects Read proof
- Example 20.8: Sharp affine bound
- Definition 21.1: Linear correction datum
- Theorem 21.2: Cokernel obstruction criterion Read proof
- Theorem 21.3: Naturality of obstruction classes Read proof
- Theorem 21.4: Packet decomposition of an obstruction Read proof
- Remark 21.5
- Proposition 21.6: Projectors on the obstruction quotient Read proof
- Theorem 21.7: Relative obstruction exact sequence Read proof
- Corollary 21.8: Coarse vanishing and fine vanishing Read proof
- Example 21.9: A fine obstruction erased by coarsening
- Definition 21.10: Mapping cone
- Theorem 21.11: Cohomological obstruction through the cone Read proof
- Corollary 21.12: Packet obstruction complexes Read proof
- Example 21.13: A two component obstruction
Correction, lifting, and smooth transport
- Theorem 22.1: Orthogonal obstruction projector Read proof
- Theorem 22.2: Right inverse correction theorem Read proof
- Corollary 22.3: Stability with respect to the target Read proof
- Theorem 22.4: Local Lyapunov Schmidt reduction Read proof
- Corollary 22.5: Quadratic reduced obstruction Read proof
- Example 22.6: A purely quadratic obstruction
- Remark 22.7
- Definition 23.1: Depth obstruction
- Theorem 23.2: One step lifting criterion Read proof
- Corollary 23.3: Finite branching count Read proof
- Lemma 23.4: Well defined lift classes Read proof
- Definition 23.5: Lift tree
- Theorem 23.6: Finitely branching lift tree Read proof
- Corollary 23.7: Unique lift branch Read proof
- Theorem 23.8: Complete filtered lifting Read proof
- Theorem 23.9: Naturality of filtered obstructions Read proof
- Example 23.10: Three lifting outcomes
- Proposition 23.11: Maurer Cartan lifting obstruction Read proof
- Standard fact 24.1: Spectral sequence of a finite filtered complex
- Proposition 24.2: Fine obstruction survival Read proof
- Corollary 24.3: Direct packet complexes Read proof
- Example 24.4: A fine class killed at the next page
- Remark 24.5
- Theorem 25.1: Exact local elimination Read proof
- Theorem 25.2: Nonlinear Schur derivative Read proof
- Theorem 25.3: Second Schur derivative Read proof
- Corollary 25.4: Packet jets of the reduced equation Read proof
- Example 25.5: Two retained variables and one hidden variable
- Remark 25.6
- Definition 26.1: Gauge change
- Proposition 26.2: Gauge covariance Read proof
- Theorem 26.3: Spanning tree normalization Read proof
- Theorem 26.4: Cycle gluing criterion Read proof
- Corollary 26.5: Number of independent cycle tests Read proof
- Theorem 26.6: Fundamental holonomy intertwining criterion Read proof
- Corollary 26.7: Scalar holonomy bridge Read proof
- Theorem 26.8: Gauge classification of transports Read proof
- Corollary 26.9: Abelian cohomology class Read proof
- Corollary 26.10: Projective transport Read proof
- Example 26.11: A square loop
- Lemma 27.1: Complete packet commutator identity Read proof
- Theorem 27.2: Canonical packet transport Read proof
- Corollary 27.3: One moving subspace Read proof
- Definition 27.4: Direct rotation of nearby packet systems
- Proposition 27.5: Exact direct packet rotation Read proof
- Lemma 27.6: One step direct rotation estimate Read proof
- Theorem 27.7: Direct rotations converge to packet transport Read proof
- Theorem 27.8: Convergence of sampled transports Read proof
- Corollary 27.9: Continuous holonomy from graph products Read proof
- Example 27.10: A rotating line
- Standard fact 27.11: Curvature of a projected connection Read proof