13. A complete small signal example
On this page · Permalink · Source line 475
Consider the rational linear signal dynamics
The spatial archive stores \(x_t\). A second atlas uses the orthogonal Haar matrix
Its coefficients \(c_t=Ux_t\) reconstruct as \(x_t=U^*c_t\). The same dynamics in that atlas are \(ULU^*\), not a separate physical law. Thus coordinate transport preserves the full dynamics.
Now retain only the first spatial coordinate. With \(P=\operatorname{diag}(1,0)\), the retained scalar \(z_t\) and discarded scalar \(q_t\) obey
The exact first-coordinate sequence begins
The compressed one-step matrix predicts \(z_2=1/4\) instead of \(1/2\). The lost return path contributes exactly
There are two exact repairs. One is state refinement: start with \(C_0=(1,0)\) and add \(C_0L=(1/2,1)\). The resulting row space is the full two-dimensional space, so it closes. The other is memory:
since \(q_0=0\). This is the exact same system expressed without retaining \(q_t\) as a separate current-state variable. At \(t=2\), it yields
To see a nontrivial reduced closure, adjoin a third independent coordinate with multiplier \(5/6\). The smallest closed refinement of the first coordinate then contains the first two coordinates, not the third. Exact prediction of the chosen summary does not require reconstructing every component of the enlarged signal.
Finally, low energy is not the same as low predictive importance. For
omitting the second coordinate loses norm \(1/1000\), but changes the next first coordinate by exactly one. The correct certificate is the product of omitted magnitude and transfer amplification.