Proof Roadmap

Every pill links to its proof page; sources are one click away. Hover to focus a node and its neighbours.

cascade

CA The Panteley–Loría Cascade Theorem and Its Route to Proposition IV.1 interfaced
RA Attraction to the Inner Set of a Nested Pair interfaced
RC The El-Hawwary–Maggiore Reduction Step interfaced
RSF Set Stability of a Compact Target from the Uniform Semi-Attractor — the Finish of Theorem 6 interfaced
RST Convergent Subsequences and Escape Sequences for Set Stability sealed
RU The Uniform Semi-Attractor from Relative Asymptotic Stability sealed
SD Passivity and Null-Space Damping for the Redundant Free-Flying Manipulator interfaced

BLY Positive-Definiteness and Coercivity of the Block Lyapunov Candidate sealed
CC Convergence to the Origin for the Coupled Attitude–End-Effector Block interfaced
CD Energy Dissipation of the Coupled Block interfaced
CF The Comparison-Function Library sealed
CFE The Coupled-Block Vector Field and Its Energy Function interfaced
CS A Class-K∞ Lower Bound for the Coupled Block Energy sealed
DGI The Determinant Identity of the Coordinated Transform sealed
GI Invertibility of the Coordinated Transform via Its Circumcentroidal Jacobian interfaced
RS The Rayleigh Sandwich for Quadratic Lyapunov Functions sealed
SCP Vanishing of the Skew-Symmetric Quadratic Form in Characteristic Other Than Two sealed
SMD The Smallest Singular Value as a Nonsingularity Criterion sealed
SR Injectivity of the Stiffness Residual sealed
TVC The Time-Varying Comparison Bound interfaced

BL Barbalat's Lemma sealed
CCM Asymptotic Stability of the Coupled Block for a Concrete Positive-Definite Model sealed
CL Global Asymptotic Stability of the Centre-of-Mass Tracking Loop interfaced
CLM Global Asymptotic Stability of the Centre-of-Mass Loop for a Concrete Positive-Definite Model sealed
KC Three Classical Lemmas in Khalil's Own Forms: Comparison, the Class-K Sandwich, and Integral Barbalat sealed
KS Lyapunov's Direct Method for Autonomous Flows interfaced
LC The Lyapunov Comparison Lemma sealed
LI LaSalle's Invariance Principle for Flows interfaced

A1 Rockafellar–Uryasev inf-form planned
A2 Coherence axioms planned
A3 Gaussian closed-form CVaR wall
A4 Affine-of-Gaussian is Gaussian sealed
A5 Atomic-tail SAA planned
A6 Sample-count concentration planned
A7 `r_fail = −1` vs `0` selection-invariance wall
AB The Bounded-Actuation-Noise Ultimate Bound interfaced
B1 Linear covariance propagation wall
B2 Belief-ellipsoid extent sealed
B3 CVaR/chance-constraint on the state cost sealed
B4 Standoff-inflation / target-switch-gating actually improve robustness wall
C1 Actuation noise ⟹ bounded closed-loop error sealed
C2 Vanishing-derate transient ⟹ convergence sealed
C3 σ_crit singularity-threshold cascade wall
C4 The Rayleigh sandwich bridges σ_crit wall
C5 Confidence-gated derate improves robustness wall
CVI The Discrete Rockafellar–Uryasev Form of Conditional Value-at-Risk sealed
CVS State-Cost CVaR — the Inf-Form Instantiated at the Guidance State Cost sealed
D1 Reconstruction-confidence metric wall
E1 Probabilistic margin / self-occlusion planned
GT The Gaussian Tail Integral sealed
VAR The Subadditivity Failure of Value-at-Risk sealed

KD König's Second Theorem and the Inertially Decoupled Centre-of-Mass Row sealed
SDD The Augmented Inertia Decouples the Self-Motion Coordinate sealed