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000252 | \begin{definition}[Admissible spectral-bordism category]
000253 | \label{def:admissible-spectral-bordism-category}
000254 | Let $\mathbf{SBord}^{\mathrm{adm}}$ denote the category whose objects are tuples
000255 | \begin{equation}
000256 | \mathcal B=(\mathcal A_+,\mathcal H_+,D_+,J,\Gamma)
000257 | \end{equation}
000258 | such that:
000259 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000260 |     \item $D_+$ is an elliptic Dirac-type operator on a one-sided collar with reflection-regular
000261 |     boundary behavior;
000262 |     \item the retained Euclidean sector is reflection positive;
000263 |     \item the relative action with respect to the declared reference sector is finite;
000264 |     \item the seam class is nontrivial, equivalently $\SF(U_\Sigma)\neq 0$.
000265 | \end{enumerate}
000266 | Morphisms are unitary intertwiners preserving the displayed one-sided data.
000267 | \end{definition}
000268 | 
000269 | \begin{theorem}[Minimal packaging inside the declared boundary class]
000270 | \label{thm:initial-admissible-bordism}
000271 | Within the declared class of admissible one-sided boundary data encoded by nontrivial objects of
000272 | $\mathbf{SBord}^{\mathrm{adm}}$, there exists, up to unitary equivalence, a unique essentialized
000273 | defect-minimal representative
000274 | \begin{equation}
000275 | \mathcal B_{\min}.
000276 | \end{equation}
000277 | Its collar normal form induces a canonical one-sided boundary datum
000278 | \begin{equation}
000279 | \Tbound=(\mathcal A_+,\mathcal H_+,D_+,J,\Gamma,B_\Sigma).
000280 | \end{equation}
000281 | On the minimal branch the first defect is
000282 | \begin{equation}
000283 | |\SF(U_\Sigma)|=1,
000284 | \end{equation}
000285 | while the carrier ranks and determinant classes are not primitive minimization inputs; they are
000286 | recovered later from compact Higgs selection and primitive Yukawa type. This theorem therefore shows
000287 | that no extra primitive discrete datum is needed beyond the induced boundary datum.
000288 | \end{theorem}
000289 | 
000290 | \begin{proof}
000291 | By declaration of the admissible boundary class, the set of nonzero defect vectors
000292 | \begin{equation}
000293 | \{\mathfrak D(B):B\in\mathbf{SBord}^{\mathrm{adm}},\ \SF(U_\Sigma)\neq0\}\subset\mathbb N^4
000294 | \end{equation}
000295 | is nonempty. Well-foundedness gives a minimizer. Essentialization removes contractible spectator
000296 | summands, so rank minimization cannot be changed by adding an empty internal factor. The first
000297 | coordinate is the nontrivial seam obstruction; hence the minimal positive value is
000298 | $|\SF(U_\Sigma)|=1$. The Dirac collar normal form gives
000299 | \begin{equation}
000300 | D_+=\gamma_n(\partial_n+B_\Sigma)
000301 | \end{equation}
000302 | with self-adjoint adapted boundary operator $B_\Sigma$. Reflection positivity on the retained
000303 | sector gives the exact double and Calderon polarization used by the primitive reconstruction. The
000304 | invariance theorem makes the minimized datum presentation independent, and the boundary
000305 | reconstruction theorem identifies the induced one-sided datum up to unitary equivalence.
000306 | \end{proof}