TFPT scientific source excerpt Original repository path: documents/archive--paper-latex--tfpt-42/sections/003-subsection-primitive-relative-spectral-dynamics.tex Original file SHA-256: 8683d7459140a06d5e03a06adefd05bf7e6538356e4e4da4397133aa991b8fe4 Inclusive original lines: 1-200 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. 000001 | \subsection{Primitive relative spectral dynamics} 000002 | 000003 | The carrier theorem is not intended to float above an unspecified microscopic dynamics. The 000004 | starting point is therefore the one-sided boundary datum from which the primitive layer, the 000005 | derived closed datum, and the canonical admissibility projector are reconstructed before the later 000006 | closure theorems are invoked. 000007 | 000008 | \begin{definition}[One-sided boundary admissibility datum] 000009 | \label{def:reduced-theorem-datum} 000010 | The one-sided boundary admissibility datum is 000011 | \[ 000012 | \Tbound:=(\mathcal A_+,\mathcal H_+,D_+,J,\Gamma,B_\Sigma) 000013 | \] 000014 | with the following built-in admissibility conditions: 000015 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt] 000016 | \item $Z(\mathcal A_+)$ is spectrally regular and its Gelfand spectrum is a smooth compact oriented manifold with boundary $M_+$; 000017 | \item $B_\Sigma$ is a self-adjoint tangential boundary operator on $\Sigma:=\partial M_+$ compatible with the boundary restrictions of $J$ and $\Gamma$; 000018 | \item in a collar neighborhood of $\Sigma$, the Dirac operator has canonical product form 000019 | \[ 000020 | D_+=\gamma_n(\partial_n+B_\Sigma); 000021 | \] 000022 | \item the boundary collar is reflection-regular so that canonical doubling, the Calder\'on projector, and the admissibility complex are well posed on the same datum. 000023 | \end{enumerate} 000024 | \end{definition} 000025 | 000026 | \begin{remark}[Almost-commutative realizations] 000027 | \label{ass:almost-commutative-factorization} 000028 | When a factorized realization is useful, one may choose 000029 | \[ 000030 | \mathcal H\cong L^2(\widetilde M,S)\otimes \Hint, 000031 | \] 000032 | with $S\to\widetilde M$ the spinor bundle on the oriented double cover and $\Hint$ a finite 000033 | internal Hilbert space. This is an analytic realization choice rather than part of the boundary 000034 | datum. 000035 | \end{remark} 000036 | 000037 | \begin{theorem}[Doubling, deck involution, and boundary polarization] 000038 | \label{thm:primitive-datum-from-reduced-datum} 000039 | \label{thm:primitive-scaffold-from-reduced-datum} 000040 | Let $\Tbound$ be as above. Let 000041 | \[ 000042 | \widetilde M:=M_+\cup_{\Sigma} M_+^{\mathrm{op}} 000043 | \] 000044 | be the doubled manifold and let 000045 | \[ 000046 | \taudbl:\widetilde M\to\widetilde M 000047 | \] 000048 | be its deck involution. Let $P_C$ be the Calder\'on projector of the APS-realized boundary problem 000049 | for $D_+$ on $\Sigma$, and define the boundary polarization operator 000050 | \[ 000051 | \iC:=2P_C-1 000052 | \] 000053 | on the Cauchy-data Hilbert space $\mathcal H_{\partial}$. 000054 | 000055 | Then: 000056 | \begin{enumerate}[label=(\roman*), leftmargin=1.8em, topsep=3pt, itemsep=2pt] 000057 | \item the geometric objects are 000058 | \[ 000059 | M=\widetilde M/\taudbl, 000060 | \qquad 000061 | \Seam=\mathrm{Fix}(\taudbl), 000062 | \qquad 000063 | \Xbulk=\widetilde M\setminus\Seam; 000064 | \] 000065 | \item the doubled operator and its even and odd parts are 000066 | \[ 000067 | D=D_+\cup_{\Sigma}D_+^{\mathrm{op}}, 000068 | \qquad 000069 | D_{\mathrm{ref}}:=\frac12\bigl(D+\taudbl^{\ast}D\taudbl^{\ast}\bigr), 000070 | \qquad 000071 | D_{\mathrm{rel}}:=\frac12\bigl(D-\taudbl^{\ast}D\taudbl^{\ast}\bigr); 000072 | \] 000073 | \item on boundary Cauchy data the induced reflection $R_{\taudbl}$ satisfies 000074 | \[ 000075 | \operatorname{Ran}(P_C)=\ker(R_{\taudbl}-1), 000076 | \qquad 000077 | \ker(P_C)=\ker(R_{\taudbl}+1), 000078 | \qquad 000079 | \iC=R_{\taudbl}. 000080 | \] 000081 | \end{enumerate} 000082 | Hence the primitive scaffold 000083 | \[ 000084 | \Tprim^{\mathrm{kin}} 000085 | := 000086 | (\Xbulk,\widetilde M,M,\Seam,\taudbl,\iC,D_{\mathrm{ref}},D_{\mathrm{rel}},\chiseed) 000087 | \] 000088 | is derived, not assumed. The derived doubled datum may be recorded as 000089 | \[ 000090 | \Tmin^{\mathrm{cl}}:=(\mathcal A,\mathcal H,D,J,\Gamma,\taudbl,\iC), 000091 | \] 000092 | If one further chooses an almost-commutative factorization 000093 | \[ 000094 | \mathcal H_{\mathrm{prim}}\cong L^2(\Xbulk,S)\otimes \Hint, 000095 | \] 000096 | then $S$ and $\Hint$ refer only to that chosen realization and are not additional 000097 | reconstruction outputs. 000098 | \end{theorem} 000099 | 000100 | \begin{proof} 000101 | Reflection regularity on an exact collar produces the doubled manifold together with the deck 000102 | involution $\taudbl$. The APS-realized boundary problem has a Calder\'on projector $P_C$ whose 000103 | range is the Cauchy-data space of interior solutions. On the doubled collar the deck reflection 000104 | acts on Cauchy data by $R_{\taudbl}$, and the solution space splits into the $\pm1$ eigenspaces of 000105 | this reflection. Therefore 000106 | \[ 000107 | P_C=\frac{1+R_{\taudbl}}{2}, 000108 | \qquad 000109 | \iC=2P_C-1=R_{\taudbl}. 000110 | \] 000111 | The formulas for $D_{\mathrm{ref}}$ and $D_{\mathrm{rel}}$ are the even and odd parts of $D$ under 000112 | the induced involution. 000113 | \end{proof} 000114 | 000115 | \begin{corollary}[No confusion among the three involutions] 000116 | Geometric reflection is carried by $\taudbl$, boundary polarization by $\iC$, and the finite 000117 | carrier split by the restricted involution 000118 | \[ 000119 | \epscar:=\iC|_{\mathcal H_{\mathrm{car}}}. 000120 | \] 000121 | No later theorem may identify these three operators literally; only the displayed restriction and 000122 | transport maps are allowed. 000123 | \end{corollary} 000124 | 000125 | The canonical admissibility projector is built in two transparent stages. The primitive stage 000126 | uses only data already available from $\Tbound$ through the derived boundary polarization $\iC$, 000127 | namely the seam-even selector on Cauchy data, an APS realization that makes the geometric anchor 000128 | self-adjoint Fredholm, and the relative operator $D_{\mathrm{rel}}$. The full stage upgrades this 000129 | primitive projector by the color and determinant projectors that become available only after the 000130 | carrier theorem of \Cref{sec:minimal-carrier-proof} identifies the color factor and after the 000131 | compact bosonic index of \Cref{sec:closure-theorems} fixes the determinant involution. The full 000132 | complex and the upgrade theorem appear once these later inputs are in place; here we close the 000133 | primitive stage. 000134 | 000135 | \begin{definition}[Primitive admissibility complex and primitive projector] 000136 | \label{def:admissibility-complex} 000137 | \label{def:primitive-admissibility-complex} 000138 | Let the primitive Hilbert space carry only the seam factor and the gap sector accessible from 000139 | $D_{\mathrm{rel}}$ alone: 000140 | \[ 000141 | \mathcal H_{\mathrm{prim}}=\mathcal H_{\Seam}\otimes\mathcal H_t, 000142 | \qquad 000143 | \mathcal H_{\mathrm{prim}}^{\mathrm{ext}} 000144 | := 000145 | \mathcal H_{\mathrm{prim}}\otimes\Lambda(\eta_\Sigma,\eta_g), 000146 | \] 000147 | with primitive seam selector 000148 | \[ 000149 | P_{\Seam,+}:=\frac{\mathbf 1+\iC}{2}. 000150 | \] 000151 | Fix $\epsilon_{\mathrm{adm}}>0$ and let $D_{\mathrm{prim}}$ be the APS-realized restriction of 000152 | the primitive sector operator to $\mathrm{Ran}(P_{\Seam,+})$, controlled by $D_{\mathrm{rel}}$ 000153 | relative to the geometric anchor. Define the primitive admissibility penalties 000154 | \[ 000155 | K_\Sigma:=\sqrt{\lambda_\Sigma}\,\frac{\mathbf 1-\iC}{2}, 000156 | \qquad 000157 | K_g:=\sqrt{\lambda_{\mathrm{gap}}}\, 000158 | \mathbf 1_{[0,\epsilon_{\mathrm{adm}}^2/4)}\! 000159 | \bigl(P_{\Seam,+}D_{\mathrm{prim}}^\dagger D_{\mathrm{prim}}P_{\Seam,+}\bigr), 000160 | \] 000161 | the primitive admissibility differential 000162 | \[ 000163 | Q_{\mathrm{prim}} 000164 | := 000165 | \eta_\Sigma^\dagger K_\Sigma 000166 | +\eta_g^\dagger K_g, 000167 | \] 000168 | and the primitive admissibility Laplacian 000169 | \[ 000170 | \Delta_{\mathrm{prim}}:=\{Q_{\mathrm{prim}},Q_{\mathrm{prim}}^\dagger\}. 000171 | \] 000172 | The primitive admissibility projector is the Hodge projector 000173 | \[ 000174 | \Pprim 000175 | := 000176 | \Pi_{\ker\Delta_{\mathrm{prim}}}. 000177 | \] 000178 | \end{definition} 000179 | 000180 | \begin{lemma}[Primitive nilpotency and primitive Hodge projector] 000181 | \label{lem:admissibility-nilpotency} 000182 | \label{lem:hodge-admissibility-projector} 000183 | \label{lem:primitive-admissibility-nilpotency} 000184 | On the primitive scaffold reconstructed from $\Tbound$, 000185 | \[ 000186 | Q_{\mathrm{prim}}^2=0, 000187 | \qquad 000188 | \Delta_{\mathrm{prim}} 000189 | = 000190 | K_\Sigma^2+K_g^2, 000191 | \qquad 000192 | \Pprim 000193 | = 000194 | \operatorname*{s-lim}_{\beta\to\infty}e^{-\beta(K_\Sigma^2+K_g^2)}. 000195 | \] 000196 | In particular $\Pprim$ depends only on $\iC$, the APS realization, and $D_{\mathrm{rel}}$, and 000197 | is therefore reconstructed from the closed datum at the primitive level. Equivalently, 000198 | \[ 000199 | \Pprim=P_{\Seam,+}P_{\mathrm{prim,gap}}, 000200 | \qquad