Inverse corners and deterministic canonical subsequences

For a continuous nonvanishing zero-index symbol, geometric decay of all sufficiently large top-right inverse corners of its Toeplitz matrices is equivalent to an inner-annulus extension. The proof of the difficult implication uses an exact bordering identity: summable corner increments control the first inverse row. Exponential control gives an analytic row-generating function, and a Hardy-space localization argument forces exponential decay of the appropriate Fourier tail. At every off-range spectral parameter with winding $+1$, inner nonextension consequently forces the normalized logarithmic determinant to attain its canonical value in the limsup; the reflected assertion holds at winding $-1$. Potential theory then proves canonical subsequences for all continuous unit-winding Jordan-range symbols that lack the extension in the winding direction, and rational approximation permits positive-area Jordan curves as well. Conversely, an extension in the direction selected by a nonzero winding produces a strictly positive limiting lower energy deficit. For unit-winding Jordan ranges, this characterizes exactly when a canonical subsequence exists.