ORTSF Delay-Margin Certificate

Rewritten 2026-07-10 against onn_ws/ONN. The earlier version stated $\tau_{\max} = 177\;\mu\text{s}$ "for 3M-node networks" and derived stability from a cohomological Lyapunov function — neither of which is in the source. The old published control numbers ($\text{dt}_{\max} = 52\text{ ms}$, phase margin $28°$) are not regenerable and are dropped. The two results below are the narrower model-level statements supported by the current source.

Statement (delay margin)#

For the constructed linear plant model $G(s) = C(sI + M)^{-1}B$ with $M = L_F + \mu\,\Pi_{H^\perp}$ (poles $-(\lambda_i + \mu)$, dominant pole $p = \lambda_2 + \mu$), and the integrating controller model $C(s)=K/s$, the phase margin is

and that modeled loop has the delay margin

This model-level gate is implemented and tested in the GuardedORTSFPipeline: it computes $\Delta t_{\max}$ from the spectrum, admits commands while $\text{delay} < \Delta t_{\max}$, and falls back to a zero-velocity command at or above the threshold. The guarded path is not the default deployed control path, so this is not evidence of physical-system or deployed-controller stability.

Single-mode delayed recurrence (sufficient, conservative)#

Separately, consider the scalar additive delayed-feedback reduction on one dominant inconsistent mode,

For this recurrence, with $\rho = \max\{|1-\eta\lambda_2|,|1-\eta\lambda_{\max}|\}$, the condition

is sufficient for all characteristic roots to remain inside the unit disk for every integer delay $d\geq1$. This is a delay-independent result for the stated single-mode scalar model. It is not the product-form general feedback small-gain condition, and it does not certify the full interconnected ORTSF loop.

Honest scope#

• The two results use different models. The first is a linearized plant with an integrating controller; the second is an additive scalar delayed recurrence. Neither is a general nonlinear closed-loop theorem.
• The scalar result does not cover coupled modes or the shipped controller. The multimode bound through the actuation operator $B$ is open, and the abstract $K_c$ has not been mapped to the fabric's multi-joint PD gains and interpolation law.
• The guarded path is not the deployed default. Its threshold behavior is tested, but end-to-end deployment and physical validation remain open.
• This is systems completion, not novelty. Both statements use standard control arguments and are presented as scoped engineering certificates.
• The phase-margin formula is not a $\lambda_2$-only bound. The spectrum enters the modeled pole $p$, while the crossover $\omega_c$ also depends on the controller design. No cohomological invariant supplies the margin.
• No numeric bound is claimed as a physical result. The certificate is symbolic and enforced; the only concrete figure in the source is an illustrative $\Delta t_{\max} = 0.49\text{ s}$ in an end-to-end demo.