\paragraph{Test matrices.} The calibration suite consists of 27 matrices of order $5$ with independently known verdicts: $3\times3$ matrices on or near the boundary of Cain's criterion, embedded block-diagonally, weakly coupled or bordered; the family $B(h)$ of Example~\ref{ex:bh} at $h=4,5,6$, bordered; vertex and off-vertex matrices of the supplied Pell-core subset-sum test instances; perturbations of the circulant core $J$ of Section~\ref{sec:w1}, including Example~\ref{ex:tangential}; matrices with negative definite symmetric part; matrices with a non-D-stable $3\times3$ principal submatrix; and a family with symmetric principal minors. The boundary atlas uses segments $t\mapsto(1-t)A_s+tA_u$ from an exactly certified D-stable $A_s$ to an exactly certified non-D-stable $A_u$ in four classes: \emph{random} ($A_s=-(MM^{\mathsf T}+I)+W-W^{\mathsf T}$ with small random integer matrices $M,W$, which is diagonally stable, and a random $A_u$); \emph{near-circulant} ($D_0(J_q-\frac1{100}I)$ with a random positive diagonal $D_0$ and a circulant $J_q$ moved from a D-stable to an unstable parameter); \emph{sign pattern} (negative diagonal, $a_{i,i+1}>0$, $a_{i+2,i}<0$ cyclically, random magnitudes); and \emph{coNP gadgets} (Pell cores with one attachment, supplied in the archive). A \emph{certified bracket} is a subsegment of relative width at most $10^{-3}$ whose endpoints are certified D-stable and non-D-stable independently of the criterion.

\paragraph{Calibration.} All 27 calibration matrices are decided correctly, with 27 certified verdicts and 0 disagreements.

\paragraph{Boundary atlas.} In a first design (70 segments, 69 certified brackets), bisection used a multistart growth-rate sampler; the criterion found real positive critical points at 23 points that the sampler had labelled stable, and an exact Routh count confirmed strict D-instability at every one of them. In a second design (473 segments, 472 certified brackets, 3 of them certified after the run by a rigorous exclusion with a larger budget) bisection used the criterion's own verdict (compiled parameter homotopy), and every bracket endpoint was certified independently, in the direction the criterion predicted. Of the 541 certified brackets, the stable endpoints carry 363 exact diagonal Lyapunov certificates and 178 rigorous exclusions (Appendix~\ref{app:exclusion}); the unstable endpoints carry 536 exact Routh counts, and 5 are not Hurwitz. With the 23 corrected points this gives 1105 certified instances concentrated at the boundary, with no unresolved disagreement.

\paragraph{One genuine disagreement.} In one further bracket of the second design the implementation reported no critical point at an endpoint that is in fact strictly D-unstable (exact Routh certificate at $d=(1,1,1,10^8,6\cdot10^7)$): a Schur-complement face of a coNP-gadget matrix places the critical point at spread $\gtrsim10^8$, beyond double-precision path tracking. The independent certification refused the bracket, so it is not among the certified brackets. The implementation now runs an exact face test before any stability certificate: for each index set $F$ it evaluates an exact Routh count at $d_F=10^k$ ($k\le12$), which can only certify instability. The defect is in the floating-point implementation, not in Theorem~\ref{thm:explicit}, whose critical point exists by Theorem~\ref{thm:coverage}.

\paragraph{Invariance and $n=3$.} The Lagrange verdict is the same for three exhaustions (log barrier with unit and random weights, reciprocal) on 23/24 Hurwitz calibration matrices; the exception is a path-tracking miss of the reciprocal system at spread $5\cdot10^4$; direct constrained minimization of the reciprocal exhaustion on $C_A$ recovers the missed critical point. The criterion reproduces Cain's exact $3\times3$ criterion on 60/60 random matrices embedded as $\diag(A_3,-1,-2)$.

\paragraph{Births.} Of the 538 distinct brackets certified during the runs, 192 have an exact face certificate at the unstable endpoint (an exact Routh count $>0$ at $d_F=10^k$, $k\le12$, for some index set $F$) and 346 do not; we call them face and interior births. Face births occur in the random (152 of 177) and coNP-gadget (40 of 40) classes; all near-circulant (181) and sign-pattern (140) births are interior births.

\paragraph{Census.} On the 532 certified brackets of the validation runs (pilot runs excluded) the compiled homotopy found a real positive critical point at every one of the 527 Hurwitz unstable endpoints (the other unstable endpoints are not Hurwitz) and none at any of the 532 stable endpoints. On the 24 Hurwitz calibration matrices it found one for every non-D-stable matrix except Example~\ref{ex:tangential}, which branch (c) decides, and for no D-stable matrix.