Reading a Stable vs Near-Critical Capital Tensor Fit

This article is a quantitative analysis piece published for research and educational purposes. The Capital Tensor module is descriptive — it characterises the panel's behaviour on the observed window, not what will happen next. Nothing below should be read as guidance.

Every Capital Tensor snapshot publishes the eigenvalue spectrum of the fitted Jacobian A and a single scalar α(A) — the maximum real part of those eigenvalues. The interpretation is taken from linear-systems analysis: when α(A) < 0, the residual M(t) − M* decays in expectation under the fitted dynamics; the panel is mean-reverting on the observed window. When α(A) > 0, the residual grows in expectation; the panel is unstable on the observed window. α(A) ≈ 0 is the borderline case — near-critical, neither clearly mean-reverting nor clearly unstable.

Three readings, three stories. α(A) of −0.05 (per week) means the residual halves every ~14 weeks under the fitted dynamics — a comfortably stable window. α(A) of −0.001 means the residual halves every ~700 weeks — technically negative, practically near-critical. α(A) of +0.01 means residuals grow by ~1% per week under the fitted dynamics — an unstable window where M(t) is drifting away from M* faster than it's pulled back. Both Capital Tensor's UI and the methodology page apply a numerical-zero tolerance of 1e-10 per week so a fit producing α(A) = 1e-15 doesn't accidentally read as 'stable' when it's really borderline.

The most important caveat is that α(A) is a property of the fit window. M* itself is the time-mean over the fitted weeks; A is the linear regression of ΔM(t+1) on (M(t) − M*) over the same weeks. Re-fitting on a different window gives a different M*, a different A, and a different α(A). That isn't a bug — it's what you'd want from a module that measures the structure of the window you handed it. But it does mean two things. First, you should not project the current α(A) into 'this panel will be stable for the next quarter.' That isn't what the number says. Second, comparing α(A) across re-fits requires acknowledging that the windows differ; the module returns the data_through field on every snapshot so the reader can see exactly which weeks each fit was measured against.

Where α(A) earns its keep is in detecting structural transitions. A panel that has been running α(A) ≈ −0.04 for several weeks and abruptly produces α(A) ≈ +0.005 on the latest fit is telling you something about the latest week of capital movement that the equilibrium tensor itself can hide. The Lyapunov panel V(t) makes the same shift visually obvious — the no-policy curve climbs instead of decaying. Whether you act on that information is, as always, a question the module deliberately doesn't answer.

The full Lyapunov derivation, the friction-fitting procedure for Γ, and the policy-simulator integration scheme live on the methodology page at /methodology under the Capital Tensor section.