SRMF as a Lyapunov Functional

This note states the assumptions and lemmas under which Compitum’s self‑regulating mapping function (SRMF) behaves as a Lyapunov functional for our discrete‑time routing dynamics.

The claims are matched by property‑based tests; file paths are provided for direct inspection.

Setting and Assumptions

• Positive‑definite metric: The SPD matrix M = L L^T + δ I is strictly PD for δ>0.

  • Code: src/compitum/metric.py (metric_matrix), tests: tests/invariants/test_invariants_metric.py.

• Two‑timescale learning: Low‑rank factor L updates every update_stride steps via a backtracking line search; controller updates each step.

  • Code: src/compitum/router.py (stride guards), tests: tests/invariants/test_invariants_control_sy.py::test_two_timescale_metric_update_stride.

• Controller update: Drift integral anti‑windup and clipping ensure bounded trust radius r∈[0.2,5.0] and non‑increase of V_ctrl = integral^2 when d_star = 0.

  • Code: src/compitum/control.py, tests: tests/invariants/test_invariants_control_sy.py::test_lyapunov_decays_under_zero_error.

• Surrogate energy (learning objective): For whitened residual z = x−μ, define E(L;z) = 0.5 β_d ||L^T z||^2.

  • Descent step with backtracking ensures E_{k+1} ≤ E_k.

  • Code: src/compitum/metric.py::batch_update_spd, tests: tests/invariants/test_invariants_lg.py::test_metric_update_line_search_non_increase.

Lemma A (Learning Descent)

Given z≠0 and step size selected by backtracking line search as implemented, the update to L satisfies E_{k+1} ≤ E_k.

Evidence: tests/invariants/test_invariants_lg.py::test_metric_update_line_search_non_increase.

Lemma B (Zero‑Error Lyapunov Decay)

For controller state (drift_integral, r), with d_star=0 for T steps, the Lyapunov candidate V_ctrl = drift_integral^2 is non‑increasing.

Evidence: tests/invariants/test_invariants_control_sy.py::test_lyapunov_decays_under_zero_error.

Lemma C (Two‑Timescale Isolation)

Before hitting update_stride, L remains unchanged; at stride, an update may occur subject to Lemma A. This isolates controller dynamics from learning updates between strides.

Evidence: tests/invariants/test_invariants_control_sy.py::test_two_timescale_metric_update_stride.

Composite Potential and SRMF Link

Define a composite potential along the routed trajectory with fixed model selection and embedding (no switching):

V_total(k) = E(L_k; z) + γ V_ctrl(k), with γ>0.

Under Assumptions above, between successive stride boundaries,

• Controller step: V_ctrl(k+1) ≤ V_ctrl(k) when the observed error proxy d_star is small (zero in the strict Lemma B); more generally the integral term damps drift.

• Learning step at stride: E(L_{k+1}) ≤ E(L_k) by Lemma A.

Hence at the sequence of stride boundaries, V_total is non‑increasing and bounded below by 0. In practice, distance in the certificate (the negative of the distance component) acts as an observable of E.

Evidence: tests/invariants/test_invariants_srmf_lyapunov.py::test_distance_decreases_over_updates.

Practical Takeaway

• The free‑energy‑style SRMF surrogate decreases under our update rules, while the controller Lyapunov candidate decreases under zero or small drift. This realizes SRMF as a Lyapunov functional for the discrete Compitum updates under bounded observer assumptions.

• See also statistical monotonicity and uncertainty properties: tests/invariants/test_invariants_statml.py.