| name | lyap-define |
| description | 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. |
Lyap Define
Quick Start
Read references/workflow.md, then execute it as the authoritative Block 3 workflow.
Start from examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b2.json and preserve the workflow's dual extraction, grouping-code construction, partial-sum verification, and examples_peppy/{ALGO_NAME}/state/{ALGO_NAME}_b3.json output.
Codex Adaptation
- Recreate the PEP context exactly as in the setup module before evaluating expressions.
- Keep interpolation duals and extra named duals separate until the grouping is verified.
- Prefer small, inspectable grouping code that can be stored in JSON and re-executed by
lyap-vectors.
- Before using a generic LDL decomposition for
S, inspect any existing examples_peppy/_references/{ALGO_NAME}/ notebook/module for verified S_guess, S_piece, square-term, or rank-profile constructions; adapt those proof-specific pieces first, and do not treat missing per-step fields in {ALGO_NAME}_b2.json as proof that no known decomposition exists.
- Use the PEPFlow Lyapunov sign convention: define
V_k so the final proof makes V_k nonincreasing (or at least V_k <= V_0) under the chosen residual convention. Do not accept a grouping merely because its symbolic residual is zero; also verify the inequality direction used in the telescoping proof.
- Verify the rank profile numerically and state the tolerance.
- In notebooks, the visible partial-sum construction cell should print ranks one per line as
rank V_k: r, include a blank line after rank V_0, then print the interior-rank consistency check in the same cell; do not print every grouped dual term or add a separate rank-profile dump cell by default.
- Keep workflow artifacts under
examples_peppy/{ALGO_NAME}/state/.
- If ranks are inconsistent, inspect indexing and dual grouping before changing the mathematical ansatz.
Related Skills
Run lyap-vectors next to identify special spanning vectors.
