TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex
Original file SHA-256: 1de90a4a7379bef804cc7815d72c0f86bb398a5771e9b00997c4bb15e8815f9e
Inclusive original lines: 166-306
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

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}
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.
000306 | \end{proof}