Derive closed-form Lyapunov coefficients from Block 4 structure, solve symbolic recursions, verify proof identities, and produce a Lyapunov proof notebook. Use when the user asks for /lyap-closed-form, lyap-closed-form, Block 5, closed-form Lyapunov proof, symbolic recursion for V_k, or updating examples_peppy/{ALGO_NAME}/{ALGO_NAME}_example_lyap.ipynb from examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b4.json.
Build Lyapunov partial-sum sequences from PEPFlow full-PEP dual certificates and verify their rank profile. Use when the user asks for /lyap-define, lyap-define, Block 3, defining the Lyapunov function, constructing V_k partial sums, or generating examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b3.json from examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b2.json.
Identify special vectors spanning Lyapunov partial sums and extract coefficient patterns as functions of k. Use when the user asks for /lyap-vectors, lyap-vectors, Block 4, special Lyapunov vectors, low-rank decomposition of V_k, coefficient pattern extraction, or generating examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b4.json from examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b3.json.
Find tight PEPFlow performance guarantees with explicit full-PEP dual certificates, sparse lambda structure, S matrices, and algebraic proof state. Use when the user asks for /pep-full-proof, pep-full-proof, Block 2, full PEP proof, sparse dual certificates, or generating examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b2.json from examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b1.json.
Create or validate a PEPFlow algorithm setup module, initialize the Lyapunov notebook, and run initial numerical convergence checks. Use when the user asks for /pep-implement, pep-implement, Block 1, implementing a new PEPFlow example, creating examples_peppy/{ALGO_NAME}/{ALGO_NAME}_setup.py, or turning an algorithm description into a PEPFlow setup plus examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b1.json state.