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000103 | \begin{definition}[Operational seed category]
000104 | \label{def:operational-seed-category}
000105 | Let $\Seedop$ denote the category whose objects are tuples
000106 | \[
000107 | \mathfrak S=(\mathfrak A_{\mathrm{loc}},\tau_t,\Theta,\omega,[u_\Sigma],\mathcal D_{\mathrm{coll}})
000108 | \]
000109 | satisfying:
000110 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000111 |     \item $\mathfrak A_{\mathrm{loc}}$ is a quasilocal Euclidean observable net with positive-time
000112 |     reflection $\Theta$;
000113 |     \item $\omega$ is reflection positive and clustering on the admissible local algebra, with finite
000114 |     relative action with respect to the declared reference sector;
000115 |     \item $[u_\Sigma]\neq 0$ is a nontrivial seam class;
000116 |     \item $\mathcal D_{\mathrm{coll}}$ is a first-order elliptic collar generator whose positive-side
000117 |     restriction is reflection regular.
000118 | \end{enumerate}
000119 | Morphisms are unitary intertwiners preserving the displayed structures.
000120 | \end{definition}
000121 | 
000122 | \begin{theorem}[Operational collar completion]
000123 | \label{thm:operational-collar-completion}
000124 | There exists a functor
000125 | \[
000126 | \Cseed:\Seedop\longrightarrow \mathbf{SBord}^{\mathrm{adm}}
000127 | \]
000128 | which assigns to every operational seed $\mathfrak S$ a canonically induced one-sided admissible
000129 | spectral bordism
000130 | \[
000131 | \Cseed(\mathfrak S)=\mathcal B(\mathfrak S)
000132 | \]
000133 | with induced boundary datum
000134 | \[
000135 | \mathfrak T_{\partial}(\mathfrak S)=(\mathcal A_+,\mathcal H_+,D_+,J,\Gamma,B_\Sigma).
000136 | \]
000137 | \end{theorem}
000138 | 
000139 | \begin{proof}[Proof architecture]
000140 | Reflection positivity and the positive-time net reconstruct the one-sided OS Hilbert space on the
000141 | collar. The first-order elliptic collar generator determines the boundary operator $B_\Sigma$,
000142 | while the nontrivial seam class gives the induced spectral-flow class. The resulting one-sided
000143 | tuple satisfies exactly the axioms of \Cref{def:admissible-spectral-bordism-category}.
000144 | \end{proof}
000145 | 
000146 | \begin{definition}[Essentialization of admissible bordisms]
000147 | For every admissible bordism $B$ write $B^{\triv}$ for the maximal direct summand on which all
000148 | primitive load-bearing data vanish:
000149 | \begin{equation}
000150 | \boxed{
000151 | \begin{gathered}
000152 | \text{seam spectral flow}=0,\\
000153 | \text{carrier polarization acts trivially},\\
000154 | \text{determinant degree}=0,\\
000155 | \text{primitive Yukawa type}=0.
000156 | \end{gathered}}
000157 | \end{equation}
000158 | Define the essentialized bordism by
000159 | \begin{equation}
000160 | B^{\textup{ess}}:=B/B^{\triv}.
000161 | \end{equation}
000162 | All minimization data below are evaluated on $B^{\textup{ess}}$ rather than on a presentation that
000163 | may contain contractible spectator summands.
000164 | \end{definition}
000165 | 
000166 | \begin{definition}[Canonical defect filtration]
000167 | \label{def:canonical-defect-filtration}
000168 | For $B\in\mathbf{SBord}^{\mathrm{adm}}$ define
000169 | \begin{equation}
000170 | \mathfrak D(B):=\bigl(d_0(B),d_1(B),d_2(B),d_3(B)\bigr)\in\mathbb N^4.
000171 | \end{equation}
000172 | After replacing $B$ with $B^{\textup{ess}}$, set
000173 | \begin{align}
000174 | d_0(B)&:=|\SF(U_\Sigma)|,\\
000175 | d_1(B)&:=\rankess H_{\mathrm{prim}}^{\mathrm{fin}},\\
000176 | d_2(B)&:=\deg_{\det}^{+}(B)=c_1(L_+)+c_1(L_-),\\
000177 | d_3(B)&:=h_\Sigma^{\red}(B).
000178 | \end{align}
000179 | Here $L_+$ and $L_-$ are the determinant lines on the two boundary-polarized finite blocks before
000180 | they are later named $L_2$ and $L_3$, and $h_\Sigma^{\red}$ is the residual boundary nullity after
000181 | Calderon normal form. The order is lexicographic, but the order is not a preference weighting: each
000182 | coordinate is defined only after the earlier obstruction has been minimized.
000183 | \end{definition}
000184 | 
000185 | \begin{remark}[Why $N_{\rm corner}$ is not a minimization coordinate]
000186 | The four-corner structure is no longer an independent slot in the minimization functional. It is a
000187 | downstream reading of the minimal nontrivial seam class together with the spin lift; schematically
000188 | \begin{equation}
000189 | |\SF(U_\Sigma)|=1\quad\Rightarrow\quad N_{\rm corner}=4.
000190 | \end{equation}
000191 | Objects derived from an earlier obstruction do not belong as free entries of the defect filtration.
000192 | \end{remark}
000193 | 
000194 | \begin{theorem}[Invariance of the defect lexicography]
000195 | Let $\mathcal C$ be the category of nontrivial admissible one-sided spectral bordisms modulo unitary
000196 | intertwiners and contractible stabilizations. Let
000197 | \begin{equation}
000198 | \mathfrak D:\mathcal C\to\mathbb N^k
000199 | \end{equation}
000200 | be the canonical defect filtration. If $F:\mathcal C\to\mathcal C'$ is an equivalence of admissible
000201 | presentations and for every coordinate there is a strictly increasing map
000202 | $\psi_i:\mathbb N\to\mathbb N$ with
000203 | \begin{equation}
000204 | d'_i(F(B))=\psi_i(d_i(B)),
000205 | \end{equation}
000206 | then $B$ is a lexicographic minimizer of $\mathfrak D$ if and only if $F(B)$ is a lexicographic
000207 | minimizer of $\mathfrak D'$.
000208 | \end{theorem}
000209 | 
000210 | \begin{proof}
000211 | Strictly increasing coordinate maps preserve and reflect the first coordinate at which two defect
000212 | vectors differ. Therefore $\mathfrak D(B)<_{\rm lex}\mathfrak D(C)$ if and only if
000213 | $\mathfrak D'(F(B))<_{\rm lex}\mathfrak D'(F(C))$. Hence minimizers are preserved by equivalence.
000214 | Existence follows because $\mathbb N^k$ is well founded under lexicographic order. Uniqueness is
000215 | then reduced to classification of the final minimized stratum by the one-sided boundary datum up to
000216 | unitary equivalence.
000217 | \end{proof}
000218 | 
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}