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000219 | \begin{definition}[Operational seed defect]
000220 | \label{def:operational-seed-complexity}
000221 | For $\mathfrak S\in\Seedop$, define the pulled-back seed defect by
000222 | \begin{equation}
000223 | \mathfrak D_{\rm seed}(\mathfrak S):=\mathfrak D(\Cseed(\mathfrak S)).
000224 | \end{equation}
000225 | \end{definition}
000226 | 
000227 | \begin{corollary}[Minimal operational seed induces the canonical boundary datum]
000228 | \label{cor:minimal-operational-seed}
000229 | Let $\mathfrak S_{\min}$ be a defect-minimal nontrivial object of $\Seedop$ with respect to
000230 | $\mathfrak D_{\rm seed}$. Then
000231 | \begin{equation}
000232 | \Cseed(\mathfrak S_{\min})=\mathcal B_{\min}
000233 | \end{equation}
000234 | up to unitary equivalence, hence the canonical one-sided boundary datum $\Tbound$ is induced by the
000235 | minimal operational seed.
000236 | \end{corollary}
000237 | 
000238 | \begin{proof}
000239 | The seed defect is the defect filtration of the induced admissible bordism. Thus minimizing over
000240 | $\Seedop$ is equivalent to minimizing the essentialized defect filtration in
000241 | $\mathbf{SBord}^{\mathrm{adm}}$. The invariant-defect theorem shows that the minimizer does not
000242 | depend on the chosen admissible presentation, and the induced boundary datum is $\Tbound$.
000243 | \end{proof}
000244 | 
000245 | \begin{axiom}[Physical operationality]
000246 | \label{ax:physical-operationality}
000247 | The physically realized world is represented by an object of $\Seedop$.
000248 | \end{axiom}
000249 | 
000250 | \subsection{Initial admissible bordisms and induced boundary data}
000251 | 
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.