\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage{mathpazo} \usepackage{amsmath,amssymb,amsthm} \usepackage[margin=1in]{geometry} \usepackage{graphicx,xcolor,booktabs,array,enumitem} \usepackage[hidelinks]{hyperref} \usepackage{microtype} \definecolor{bgc}{HTML}{0F1115} \definecolor{ink}{HTML}{E6E8EC} \definecolor{mut}{HTML}{A3ABB8} \definecolor{acc}{HTML}{8FB8FF} \pagecolor{bgc}\color{ink} \hypersetup{colorlinks=true,linkcolor=acc,citecolor=acc,urlcolor=acc} \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newcommand{\grade}[1]{\textcolor{mut}{\textup{\textsc{[#1]}}}} \renewcommand{\arraystretch}{1.25} \newcommand{\sgn}{\operatorname{sign}} \setlength{\parskip}{0.35em} \title{Beyond the Event Horizon:\\ What Survives at a Rigidity Wall} \author{Jeromie Beasley} \date{20 October 2026} \begin{document} \maketitle \noindent\textcolor{mut}{\textit{A note for reviewers.} The metric in this paper is the intrinsic kinetic metric of the four-body shape space, which is the Bures metric of a real qutrit. It is not a spacetime metric and is never substituted for one. Every statement about black holes is a structural comparison, and each one carries its grade where it is made: \textsc{[derived]} for closed forms and identities proved here, \textsc{[measured]} for numbers computed by the scripts listed at the end, and \textsc{[proposed]} for conjectures and ans\"atze.} \noindent\textcolor{mut}{\textit{AI-assisted.} The ideas and research direction are the author's; AI tools were used for literature search, computation, drafting and typesetting, and every number was checked by the scripts in the archive.} \begin{abstract} \noindent The four-body shape space $\Sigma_4=\{G\succeq0,\ \mathrm{Tr}\,G=1\}$ has one genuine wall, the coplanar stratum $w_3=0$, and its kinetic metric is the Bures metric $\sum_i dw_i^2/(4w_i)$ on the eigenvalue block. The wall is a fold: in $\sigma_i=\sqrt{w_i}$ the metric coefficients are regular (the eigenvalue block is the round octant of the unit sphere), the wall lies at finite distance $\arcsin\sqrt\varepsilon$, and the sign of $\sigma_3$ is the orientation, so a tetrahedron meets its mirror image there. The transport operator $A(H)=TH+HT-2\,\mathrm{Tr}(TH)\,G$ is the compression of the free scaling $TH+HT$ onto trace-zero tangents, and therefore, when the scaling and the configuration share principal axes, self-adjoint in the kinetic metric. In the Gram frame its back-coupling vanishes like $\sigma_3^2$, and on a codimension-two condition it forms a Jordan block whose square-root branch is exactly the orientation double cover. Lifted to the cover, where the metric is regular, the same operator is analytic and the wall point is a plain double point at the tip of a cone. Both readings are exact, and the singular differential of the covering map converts one into the other. A complex loop gives eigenvectors a sign $-1$ in the Gram frame and $+1$ on the cover, with residue $\tfrac12$ equal to the share of each eigenvector lying across the fold. A real loop through the mirror gives the chart-free sign, the conical $-1$ of a real self-adjoint family, which the Gram observer cannot see. Through every frame the pair projector and the trace-log ledger stay exact, while single eigenprojectors diverge in the Gram frame and keep norm $1$ in the kinetic metric. Four further results carry the same bookkeeping. Majorana stars fix the Jordan form of the Weyl operator: Kerr is type D, diagonalisable, while type II is a Jordan block unfolded by its stars as our wall is by $\sigma_3$. Ramanujan's mock theta function and its modular partner each reach $10^{597}$ near $q=-1$ while their difference tends to $4$; in black-hole microstate counting, mock modularity is itself a wall-crossing effect, and orientation reversal of the Brieskorn spheres built on the trefoil separates false theta functions from mock ones. The amphichiral figure-eight knot has a real invariant growing at its volume $2.029883$, while the chiral torus knots, the trefoil and $8_{19}$, are conjugated by the mirror and grow like $N^{3/2}$, the second with a constant of period $16$ in $N$. And the off-diagonal block $B$ and its square $BB^\dagger$, a positive pullback, run through the boundary census, the memory kernel, the contractive Cayley boundary and the transport itself. These sort walls into spectral ones, where a real pair turns complex (the Sun's dynamo, an optical threshold), and geometric ones, where the spectrum stays real on both sides (the coplanar fold, the static base of a horizon). A black hole carries one of each: the static base of its horizon is a geometric fold, and the innermost stable circular orbit of its exterior is a spectral wall, a Jordan block between oscillation and growth, around which Levin's whirling orbits gain azimuth like a logarithm. In the same circular-orbit generator the ISCO is the point where gravity's effective power law, read from the epicyclic ratio, reaches the inverse cube, $p=2+6M/r=3$. Read for light, the generator reproduces the photon ring: its growth rate equals its orbital rate, so each extra half orbit dims an image by $e^{-\pi}$ and delays it by $3\sqrt3\pi M$, a value that belongs to the uncharged hole alone. Its ringdown has spectral walls as well: near a quasinormal exceptional point the excitation factors of the two merging overtones diverge with opposite phases while their sum, fixed by the pair projector, stays finite and the signal acquires a term linear in time. Families of such walls can climb or swap: a loop around several of the near-extremal Kerr exceptional points carries the fundamental mode up the overtones, while the Sun's dynamo ladder, computed exactly, gives only separate swaps. The horizon's effective surface resistance equals the impedance of empty space, so in the boundary calculus it is a matched load that returns nothing. Bell's two results meet at the one-way boundary from opposite sides: the outside half of a Hawking pair is thermal and blind to the pair's phase while the pair, in the ideal squeezed state, beats the CHSH bound at every frequency, and a theory that keeps only outgoing motion balances its books only with Hawking's flux, the same shape as the re-entry of what a reduction discards. For a fermion pair the outside record's offset, the log-odds of empty against full, is exactly $\hbar\omega/k_BT_H$ for every pair phase; read as a log-impedance it places the record at $\Gamma=\tanh(\hbar\omega/2k_BT_H)$ on the same Smith chart as the matched horizon, reaching its centre only at zero frequency. A wave on the projector metric reflects at a step through the metric along the step and the Clairaut constant alone, the transverse metric dropping out as in Fresnel's formula, and total reflection begins exactly where a geodesic turns: the optical threshold is a turning point, at normal incidence a change of spectral gap reflects with the peel coordinate $\tanh(\tfrac12\ln\Delta_2/\Delta_1)$, one Gram determinant sets the admittance and caps the Berry curvature a path can feel, and a polar stratum is a perfect mirror where a Killing horizon, polar in the same sense, is a matched load. Set beside a recent spacetime Penrose-inequality manuscript, the shared items are the fold normal form $dX^2/(4c^2X)$ with $c$ the surface gravity there and $1$ here, the chart lesson that an apparent singularity is a coordinate effect, a non-self-adjoint horizon operator with no defect in the regular chart, and a degenerate form with a kernel. No identification of horizons is claimed, and the paper states what one would need. \end{abstract} \section{Introduction} Black holes occupy a unique position in modern physics. They are exact solutions of general relativity, laboratories for quantum field theory in curved spacetime, and testing grounds for ideas from thermodynamics to quantum information. Many of the tools used to analyse their perturbations are operator-theoretic: linearised perturbations evolve under differential operators whose spectra are the quasinormal modes, whose resolvents control stability, and whose non-self-adjoint structure governs transient amplification \cite{reggewheeler1957,bcs2009,jaramillo2021,jaramillo2022,destounis2021,gasperin2022}. This motivates a simple question: \begin{center} \emph{Can black-hole geometry itself be viewed as an emergent manifestation of persistent operator structure?} \end{center} Rather than begin with spacetime geometry and derive operators, we reverse the direction. We begin with a transport operator on the shape space of four bodies, where everything can be computed exactly, and ask what happens when it reaches a wall of its configuration space: what fails, what survives, and which of those behaviours a black-hole horizon shares. The answer has five parts. \emph{The wall is a fold.} The coplanar wall looks singular in the Gram coordinate, where the metric coefficient diverges like $1/w_3$. In the amplitude coordinate $\sigma_3=\sqrt{w_3}$ it is smooth and at finite distance, and it is where a configuration meets its mirror image (Section~\ref{sec:shape}). \emph{The operator has two exact readings at the wall.} When the scaling and the configuration share principal axes, the transport is self-adjoint in its own kinetic metric. The Gram observer, whose frame the fold squeezes, sees a one-way coupling and a Jordan block. On the oriented cover the same operator is regular and the wall point is a cone (Section~\ref{sec:operator}). \emph{The record survives, and so does a sign.} Individual eigenprojectors depend on the frame, but the pair projector and the trace-log ledger do not. A real loop through the mirror reverses the eigenline, and only the oriented cover can make that loop. Majorana's stars, Ramanujan's radial limits and the mirror rule for knots show the same bookkeeping in three other settings, and in black-hole microstate counting the mock side of Ramanujan's functions comes from a wall crossing (Section~\ref{sec:survive}). A passive boundary that returns nothing is a matched load, its contractivity is a pullback, and the horizon's surface resistance makes it one (Section~\ref{sec:oneway}). \emph{Bell meets the one-way boundary from both sides.} The entanglement of a Hawking pair says what the outside record cannot hold, and the anomaly that fixes Hawking's flux says what the outside books must carry (Section~\ref{sec:bell}). \emph{Walls come in two kinds.} At a spectral wall a real pair turns complex and the Jordan block is in every frame. At a geometric wall the spectrum stays real on both sides and the block is a chart effect. A black hole has both: the static base of its horizon is geometric, and the innermost stable circular orbit and the exceptional points of its ringdown are spectral (Section~\ref{sec:walls}). A wave on the projector metric meets the same sorting. At a step it reflects through $g_\parallel$ and the Clairaut constant alone, with $g_\perp$ dropping out, and total reflection begins exactly where a geodesic turns, which makes the optical threshold a turning point (Section~\ref{sec:refraction}). Each part has a counterpart on the gravitational side. Section~\ref{sec:gr} sets them beside a recent spacetime Penrose-inequality manuscript \cite{openaiPenrose} and the quasinormal-mode literature. The framework is a mathematical reformulation, not a replacement for general relativity. The Schwarzschild and Kerr geometries, perturbation equations and conservation laws remain intact. Throughout, established mathematical results are kept separate from speculative extensions. \paragraph{What is claimed, and what is not.} The premise that horizon physics involves non-normal operator theory is not claimed here. It is the settled starting point of an active programme: quasinormal modes on hyperboloidal slices are the spectrum of a non-self-adjoint operator, unstable under high-frequency perturbations \cite{jaramillo2021,destounis2021}, with gravitational-wave signatures \cite{jaramillo2022}. The scalar product is structural \cite{gasperin2022}. Non-orthogonal quasinormal sums produce long transient plateaus \cite{carballo2024,besson2025}, and transient superradiance occurs in modally stable regimes \cite{carballo2025}. What is claimed is the relational side and its exact structure, together with a graded set of structural comparisons. \paragraph{Two survival principles.} When eigenvalues coalesce, contour-integrated functional calculus (Riesz projectors, trace-logs) stays exact where eigenvector arithmetic depends on the frame: \emph{the contour keeps the ledger}. When a configuration passes through the wall, the Gram observer loses the sign of $\sigma_3$, but the orientation and the analytic labelling of the eigenvalue branches keep it: \emph{the orientation keeps the trace}. \section{The shape space and its wall}\label{sec:shape} \subsection{Shape space and its kinetic metric} Encode a four-body configuration in $\mathbb R^3$, after centring and mass-weighting, by the $3\times3$ matrix $X$ of Jacobi vectors. Rotations act by $X\mapsto RX$, and the rotation-invariant datum is the Gram matrix $G=X^{\top}X$. Fixing scale by $\mathrm{Tr}\,G=1$ gives \begin{equation} \Sigma_4=\{G\succeq0 \text{ symmetric},\ \mathrm{Tr}\,G=1\}, \end{equation} a five-dimensional convex body stratified by rank. At a principal-frame point $W=\mathrm{diag}(w_1,w_2,w_3)$ with $\sum w_i=1$, the tangent space of traceless symmetric matrices has the Frobenius-orthonormal basis \begin{equation} E_1=\tfrac{1}{\sqrt2}\mathrm{diag}(1,-1,0),\quad E_2=\tfrac{1}{\sqrt6}\mathrm{diag}(1,1,-2),\quad E_{(ij)}=\tfrac{1}{\sqrt2}(e_ie_j^{\top}+e_je_i^{\top}), \end{equation} with $E_3,E_4,E_5=E_{(12)},E_{(13)},E_{(23)}$. $E_1,E_2$ move the eigenvalues (shape); $E_3,E_4,E_5$ rotate the eigenframe (shear). \begin{theorem}[the kinetic metric is the Bures metric]\label{thm:bures}\grade{derived} The quotient of the flat Frobenius metric on $\{\|X\|_F=1\}$ by rotations is, at $W$, \begin{equation} g(E_a,E_b)\big|_{\rm shape}=\sum_{i=1}^3\frac{(E_a)_{ii}(E_b)_{ii}}{4w_i},\qquad g(E_{(ij)},E_{(ij)})=\frac{1}{2(w_i+w_j)}, \end{equation} with the shape and shear sectors orthogonal and distinct shear directions mutually orthogonal. It is the Bures metric of real density matrices, so $\Sigma_4$ with its own kinetic metric is the state space of a real qutrit, and its strata are the rank strata: rank two is the coplanar configurations and rank one the collinear ones, which are not collisions. \end{theorem} The proof (horizontal lifts at $X=\mathrm{diag}\sqrt{w_i}$) is in Appendix~\ref{app:proofs}. At $d=2$ the same construction gives the real-qubit Bures geometry: with $\rho=\tfrac12(I+\sin\chi\,(\cos\varphi\,\sigma_x+\sin\varphi\,\sigma_z))$ the metric is $\tfrac14(d\chi^2+\sin^2\chi\,d\varphi^2)$, a hemisphere of radius $\tfrac12$ whose boundary $\chi=\pi/2$ is the circle of pure states, of length $\pi$ \grade{derived; checked, V10-CORRECTIONS}. Its orientation double cover, two hemispheres glued along that circle, is Kendall's shape sphere of triangles, of radius $\tfrac12$; the circle is the collinear triangles. That is the control used for all curvature computations: the code returns sectional curvature $4.0000000$ there \grade{measured}. \subsection{The wall is a fold} The strata of $\Sigma_4$ are: the interior (rank 3); the coplanar wall $w_3=0$ (rank 2, codimension 1); and the collinear stratum $w_2=w_3=0$ (rank 1, codimension 3). Interior eigenvalue coincidences $w_i=w_j$ are not orbit-type strata: the stabiliser of a full-rank $X$ is trivial whatever the spectrum of $G$, and the curvature is finite there. \begin{theorem}[the coplanar wall is a fold at finite distance]\label{thm:fold}\grade{derived} Let $\sigma_i=\sqrt{w_i}$. Then: \begin{enumerate}[label=(\roman*),nosep] \item On the eigenvalue block the metric is $\sum_i dw_i^2/(4w_i)=\sum_i d\sigma_i^2$ restricted to $\sum\sigma_i^2=1$: the round metric of the positive octant of the unit sphere, of Gaussian curvature $1$. The square-root coordinate removes the $1/w_i$ divergence of the coefficients, not the curvature. \item Along $W(w)=\mathrm{diag}\big(\tfrac{1-w}{2},\tfrac{1-w}{2},w\big)$ the metric is $ds^2=dw^2/(4w(1-w))$, and the distance from the wall to $w=\varepsilon$ is $\arcsin\sqrt\varepsilon$, finite. \item Extending $\sigma_3$ to negative values by $\sigma_3=\mathrm{sign}(\det X)\sqrt{w_3}$ realises the rotation-reduced shape space as the double cover of $\Sigma_4$ branched over $\det G=(\det X)^2=0$. The metric $d\sigma_3^2$ is smooth and even under $\sigma_3\mapsto-\sigma_3$, which exchanges a configuration with its mirror image. \end{enumerate} \end{theorem} \begin{proof} (i) $dw_i=2\sigma_i\,d\sigma_i$ gives $dw_i^2/(4w_i)=d\sigma_i^2$. (ii) On the path, $w_1=w_2=\tfrac{1-w}{2}$ and $dw_1=dw_2=-\tfrac{dw}{2}$, so the two equal terms give $2\cdot\tfrac{dw^2/4}{2(1-w)}=\tfrac{dw^2}{4(1-w)}$ and $ds^2=\tfrac{dw^2}{4(1-w)}+\tfrac{dw^2}{4w}=\tfrac{dw^2}{4w(1-w)}$. With $w=\sin^2\theta$ this is $d\theta^2$, so the length from $0$ to $\varepsilon$ is $\arcsin\sqrt\varepsilon$. (iii) $\det G=(\det X)^2$, and $X$ is determined by $G$ up to rotation and reflection. On rank-two $X$ the rotation stabiliser is trivial, so the oriented quotient is smooth across $\det X=0$; $\sigma_3$ is a smooth signed coordinate there, and the metric $d\sigma_3^2$ is even. \end{proof} In the Gram coordinate the wall direction $E_2$ has coefficient $g(E_2,E_2)=\tfrac16\big(\tfrac1{4w_1}+\tfrac1{4w_2}+\tfrac1{w_3}\big)\sim\tfrac{1}{6w_3}$. This divergence is the square-root shadow of the fold, not a barrier, and its far side is the orientation double cover of the shape space branched over $\det X=0$, which holds for every square case $N=d+1$ \cite{beasleyLadder} (Theorem~2.4 there). The fold is Whitney's: a map whose differential has a simple zero on a codimension-one locus looks like $x\mapsto x^2/2$ there (type $A_2$) \cite{whitney1955}, and reading a smooth signed coordinate through its square is what produces the $1/w_3$ coefficient. On the three-body shape sphere the same normal form accounts for the $\sec^4$ rigidity law near the collinear wall, and a cusp ($A_3$) would give a different exponent \cite{beasleyThreeBody}. The same holds for curvature. Metric-normalised sectional curvatures stay finite as $w_3\to0$; for example, along a fixed generic direction $V$ we measure $K(E_2,V)\to5.61$, $K(E_4,V)\to6.43$ and $K(E_5,V)\to7.66$ at $w_3=10^{-4}$; over two decades in $w_3$ the largest change is in $K(E_4,V)$, a factor $1.30$, while $K(E_2,V)$ moves by $2\%$ \grade{measured}. Divergences of order $1/w_3$ appear only when curvatures are measured against Frobenius-unit frames, whose metric length itself diverges. The collinear stratum is different. The shear coefficient $1/(2(w_j+w_k))$ diverges when $w_j=w_k=0$ because rotations about the common line act trivially there: the orbit dimension drops. That is a genuine singular stratum of the quotient, of codimension three, and generic trajectories miss it. What the Gram observer sees is shown in Figure~\ref{fig:fold}. A path crossing $\sigma_3=0$ at constant speed appears in $w_3=\sigma_3^2$ as a bounce. The normal velocity measured in $w_3$ is \begin{equation}\label{eq:normalvel} \dot w_3=2\sigma_3\dot\sigma_3,\qquad |\dot w_3|=2\sqrt{w_3}\,|\dot\sigma_3|\to0, \end{equation} so incoming trajectories appear to flatten against the wall and skim it, while in the signed coordinate they pass through to the mirror configuration \grade{derived}. \begin{figure}[t]\centering \includegraphics[width=\linewidth]{fig/f1_fold.png} \caption{The coplanar wall as a fold. Top: in the signed coordinate $\sigma_3$ the metric is $d\sigma_3^2$ and a path passes from a tetrahedron to its mirror image. Bottom: the same path seen by the Gram observer $w_3=\sigma_3^2$ appears to bounce. The coefficient $1/(4w_3)$ diverges, but the distance $\arcsin\sqrt\varepsilon$ is finite.}\label{fig:fold} \end{figure} \section{The transport operator}\label{sec:operator} \subsection{Definition and block structure} The intrinsic transport operator is \begin{equation} A(H)=TH+HT-2\,\mathrm{Tr}(TH)\,G,\qquad T=\mathrm{diag}(t_1,t_2,t_3)\succ0. \end{equation} Here $T$ is a fixed anisotropic scaling of the configuration. Under $X\mapsto Xe^{sT}$ the Gram matrix moves as $G\mapsto e^{sT}Ge^{sT}$, with velocity $TG+GT$; $A$ applies the same scaling to a tangent $H$ and returns it to the trace-zero tangents along $G$. The entries $t_k$ are the three scaling rates, and nothing else about $T$ is used. $T$ and $G$ are diagonal in one frame: the scaling acts along the principal axes of the configuration, and every result below uses this. The coefficient $2$ is forced: it is the unique choice keeping $A(H)$ traceless when $\mathrm{Tr}\,G=1$, because $\mathrm{Tr}(TH+HT)=2\,\mathrm{Tr}(TH)$. The operator is defined without reference to any metric. It is the compression of the scaling, not the Jacobian of the normalised flow $G\mapsto e^{sT}Ge^{sT}/\mathrm{Tr}(e^{sT}Ge^{sT})$; that Jacobian is $A(H)-2\,\mathrm{Tr}(TG)\,H$, a uniform shift that moves every eigenvalue by the same amount and changes no eigenvector and no Jordan structure \grade{derived; checked symbolically, V10-CORRECTIONS}. \begin{theorem}[block structure]\label{thm:blocks}\grade{derived} In the basis $E_1,\dots,E_5$ at $W$, with $A(E_b)=\sum_aA_{ab}E_a$: \begin{align} A_{11}&=(t_1+t_2)-(t_1-t_2)(w_1-w_2), & A_{22}&=2t_3+w_3(t_1+t_2-2t_3),\\ A_{12}&=\tfrac{1}{\sqrt3}\big[(t_1-t_2)-(t_1+t_2-2t_3)(w_1-w_2)\big], & A_{21}&=\sqrt3\,(t_1-t_2)\,w_3, \end{align} the shear block is diagonal with entries $t_1+t_2,\ t_1+t_3,\ t_2+t_3$, and the shape--shear coupling is identically zero. \end{theorem} This is a direct computation, verified symbolically (BH-CORE-02). The vanishing shape--shear coupling matters. A defective $2\times2$ sub-block need not survive coupling to the rest of a matrix: the author's filament paper gives a five-dimensional pattern whose sub-block is a Jordan block while the full matrix has only a simple eigenvalue there \cite{beasleyFilament}. Here the shape block is an exact restriction of $A$, so whatever it shows at the wall, the full operator shows. \subsection{The operator is self-adjoint in its own metric} Write the kinetic metric of Theorem~\ref{thm:bures} on symmetric matrices as \begin{equation}\label{eq:gW} g_W(H,K)=\tfrac12\sum_{i,j}\frac{H_{ij}K_{ij}}{w_i+w_j}, \end{equation} which reproduces $\sum_iH_{ii}K_{ii}/(4w_i)$ on the diagonal and $1/(2(w_i+w_j))$ on $E_{(ij)}$. Let $D_T(H)=TH+HT$ be the free scaling. \begin{theorem}[the transport is a compression of a self-adjoint scaling]\label{thm:selfadj}\grade{derived} At every interior point, with $T$ diagonal in the eigenframe of $W$: \begin{enumerate}[label=(\roman*),nosep] \item $D_T$ is self-adjoint for $g_W$; \item $g_W(W,H)=\tfrac14\mathrm{Tr}\,H$, so $P_W(Z)=Z-\mathrm{Tr}(Z)\,W$ is the $g_W$-orthogonal projection onto the trace-zero tangent space, and $Q_W=I-P_W$ projects onto the scale direction $W$; \item $A=P_WD_TP_W$ on trace-zero tangents. Hence $A$ is self-adjoint for the kinetic metric, $g_W A=A^{\top}g_W$ in any basis, its eigenvalues are real, and its eigenprojectors are $g_W$-orthogonal, with kinetic norm exactly $1$. \end{enumerate} \end{theorem} \begin{proof} (i) $g_W(D_TH,K)=\tfrac12\sum_{ij}(t_i+t_j)H_{ij}K_{ij}/(w_i+w_j)$ is symmetric in $H,K$. (ii) $g_W(W,H)=\tfrac12\sum_iw_iH_{ii}/(2w_i)=\tfrac14\mathrm{Tr}\,H$, and $P_W$ is idempotent with kernel $\mathrm{span}\,W$ and image the trace-zero matrices, which are $g_W$-orthogonal to $W$ by this formula. (iii) $P_WD_T(H)=TH+HT-\mathrm{Tr}(TH+HT)\,W=A(H)$. A compression of a self-adjoint operator by an orthogonal projection is self-adjoint. \end{proof} The common frame is needed. At $W=\mathrm{diag}(\tfrac12,\tfrac3{10},\tfrac15)$ with $T=I+\tfrac15(e_1e_2^{\top}+e_2e_1^{\top})$, which is not diagonal there, $g_W(AH,K)-g_W(H,AK)=\tfrac{2}{15}$ for $H=\sqrt2E_1$ and $K=\sqrt2E_{(12)}$; with $T$ diagonal the difference stays below $2\times10^{-14}$ on $200$ random pairs \grade{derived; checked, V10-CORRECTIONS}. Part (i) is where the frame enters: $t_i+t_j$ multiplies $H_{ij}$ only when $T$ is diagonal in the frame where $g_W$ is. The symbolic check BH-LIFT-05 confirms $g_WA-A^{\top}g_W=0$ identically on the full $5\times5$ operator. Theorem~\ref{thm:selfadj} gives a one-line reason for the reality of the interior spectrum, which the next subsection also proves by a similarity. It also places the operator's non-normality. In the Frobenius frame $E_a$ the commutator is nonzero only when both $T$ and $W$ are anisotropic: $\|[A,A^{\top}]\|_F=0.2314$ at $t=(2,1,1.3)$, $w=(0.5,0.3,0.2)$, and it vanishes if $T\propto I$ or $W=\tfrac13I$ \grade{derived; value measured}. In the kinetic metric the operator is normal everywhere inside. The non-normality is a property of the Frobenius frame, which is the frame the Gram observer uses. That distinction carries through the rest of the paper. \paragraph{The hidden block of the transport.} Because $A$ is a compression, its square is not the compression of $D_T^2$. The difference is \begin{equation}\label{eq:spineA} P_WD_T^2P_W-A^2=P_WD_TQ_WD_TP_W=\big(Q_WD_TP_W\big)^{*}\big(Q_WD_TP_W\big)\succeq0, \end{equation} a positive operator of rank at most one in the kinetic metric, and zero when $T\propto I$ \grade{derived; checked, BH-SPINE-09, V10-CORRECTIONS}. Its single direction is the coupling of the shape tangents to the scale direction $W$, which fixing $\mathrm{Tr}\,G=1$ removes from view. Section~\ref{sec:spine} shows that this $BB^{\dagger}$ structure is the same one that runs through the boundary census, the memory kernel and the gravity quadratic form. \subsection{Real spectrum in the interior} \begin{theorem}[no exceptional points inside]\label{thm:real}\grade{derived} If $w_1,w_2,w_3>0$, the shape block of $A$ is diagonalisable with real eigenvalues. \end{theorem} \begin{proof} This follows from Theorem~\ref{thm:selfadj}. A second proof exhibits the similarity directly. On diagonal $H=\mathrm{diag}(h)$, $A$ acts as the $3\times3$ matrix $2(D_t-wt^{\top})$ with $D_t=\mathrm{diag}(t)$, whose image is traceless. With $u_i=\sqrt{w_it_i}$ and $S=\mathrm{diag}\sqrt{w_i/t_i}$, \[ S\,(D_t-uu^{\top})\,S^{-1}=D_t-wt^{\top}, \] since $(Suu^{\top}S^{-1})_{ij}=\sqrt{w_i/t_i}\sqrt{w_it_i}\sqrt{w_jt_j}\sqrt{t_j/w_j}=w_it_j$. So $D_t-wt^{\top}$ is similar to a symmetric matrix, and the invariant traceless plane inherits real, diagonalisable spectrum. \end{proof} Both the similarity $S$ and the kinetic metric degenerate exactly when some $w_i=0$. Away from the wall the discriminant $D=(A_{11}-A_{22})^2+4A_{12}A_{21}$ is positive at every sampled point: over $2\times10^4$ random interior points its minimum is $2.9\times10^{-4}$, and for $5\times10^4$ random choices of $(t,\delta)$ no root of $D$ in $w_3$ falls inside the physical segment \grade{measured}. \subsection{The wall as the Gram observer sees it} \begin{theorem}[the wall in the Gram frame]\label{thm:wall}\grade{derived} In the frame $E_1,E_2$ at the coplanar wall: \begin{enumerate}[label=(\roman*),nosep] \item The back-coupling vanishes quadratically in the signed coordinate, $A_{21}=\sqrt3(t_1-t_2)\,\sigma_3^2$, while $A_{12}$ stays finite. The shape block becomes upper-triangular: the direction across the wall ($E_2$) still drives the direction in the wall ($E_1$), but not the reverse. \item At $w_3=0$, $A_{22}=2t_3$, and the shape block is a Jordan block if and only if $A_{11}=A_{22}$ and $A_{12}\ne0$, that is, \begin{equation}\label{eq:joint} t_1+t_2-2t_3=(t_1-t_2)(w_1-w_2), \end{equation} a codimension-two condition in the joint space of $(G,T)$. On it $A_{12}=(t_1-t_2)(1-\delta^2)/\sqrt3$ with $\delta=w_1-w_2$, so $A_{12}\neq0$ means $t_1\neq t_2$ and $|\delta|<1$; equal rates $t_1=t_2$ give a semisimple double eigenvalue instead. \end{enumerate} \end{theorem} \begin{proof} (i) is read from Theorem~\ref{thm:blocks} with $w_3=\sigma_3^2$. (ii) A $2\times2$ upper-triangular matrix is non-diagonalisable exactly when its diagonal entries agree and its off-diagonal entry is nonzero. At $w_3=0$, $A_{11}-A_{22}=(t_1+t_2-2t_3)-(t_1-t_2)(w_1-w_2)$. \end{proof} In the dynamics $\dot x=-Ax$ of the shape amplitudes, (i) says $\dot x_2=-A_{21}x_1-A_{22}x_2\to-A_{22}x_2$: what happens in the wall cannot write back across it, while what arrives across the wall is written into it through $A_{12}$ (Figure~\ref{fig:oneway}). The kinetic metric explains how a self-adjoint operator can show a Jordan block here. The coefficient $g_W(E_2,E_2)\sim1/(6w_3)$ diverges at the wall, so the metric in which $A$ is self-adjoint stops being a metric on the Gram tangent space. The Jordan block lives in that degenerate frame. \begin{figure}[t]\centering \includegraphics[width=\linewidth]{fig/f2_oneway.png} \caption{One-way coupling at the wall, in the Gram frame. The coupling from across the wall into the wall, $A_{12}$, stays finite. The reverse coupling $A_{21}=\sqrt3(t_1-t_2)\sigma_3^2$ vanishes quadratically in the distance to the wall.}\label{fig:oneway} \end{figure} \subsection{The fold unfolds the exceptional point} Fix $T$ and $\delta=w_1-w_2$ on the joint condition~\eqref{eq:joint}, and vary $w_3$ with $w_1+w_2=1-w_3$. Then $A_{11}$ and $A_{12}=(t_1-t_2)(1-\delta^2)/\sqrt3$ are constant, $A_{22}-A_{11}=w_3(t_1+t_2-2t_3)$, and \begin{equation}\label{eq:disc} D(w_3)=(t_1+t_2-2t_3)^2\,w_3^2+4(t_1-t_2)^2(1-\delta^2)\,w_3 . \end{equation} \begin{theorem}[the branch point lives on the orientation double cover]\label{thm:unfold}\grade{derived} On the joint condition, $D$ has a simple zero at $w_3=0$, so the two eigenvalues $\lambda_\pm=\tfrac12(A_{11}+A_{22})\pm\tfrac12\sqrt{D}$ have a square-root branch point in $w_3$. In the signed coordinate, \begin{equation}\label{eq:analytic} \lambda_\pm(\sigma_3)=\lambda_*+\tfrac12(t_1+t_2-2t_3)\sigma_3^2\pm\tfrac12\,\sigma_3\sqrt{4(t_1-t_2)^2(1-\delta^2)+(t_1+t_2-2t_3)^2\sigma_3^2}, \end{equation} which is analytic in $\sigma_3$. Consequently: \begin{enumerate}[label=(\roman*),nosep] \item a loop of $w_3$ around $0$ exchanges the pair, a loop of $\sigma_3$ around $0$ does not, and the mirror $\sigma_3\mapsto-\sigma_3$ exchanges the two analytic branches; \item the gap is $|\lambda_+-\lambda_-|=2|t_1-t_2|\sqrt{1-\delta^2}\,|\sigma_3|+O(\sigma_3^3)$, linear in the distance to the wall; at the wall $1-\delta^2=4w_1w_2$; \item for real $\sigma_3$ of either sign the two eigenvalues are real. \end{enumerate} \end{theorem} \begin{proof} Substitute $w_3=\sigma_3^2$ into~\eqref{eq:disc}: $D=\sigma_3^2\,[4(t_1-t_2)^2(1-\delta^2)+(t_1+t_2-2t_3)^2\sigma_3^2]$. The bracket is positive near $0$, so $\sqrt D=\sigma_3\sqrt{[\,\cdot\,]}$ with an analytic square root, which gives~\eqref{eq:analytic}. A loop of $w_3$ is half a loop of $\sigma_3$, which carries $\sigma_3\mapsto-\sigma_3$ and exchanges $\lambda_\pm$. A full loop of $\sigma_3$ returns each branch. For real $\sigma_3$ the bracket and $\sigma_3^2$ are positive, so (iii) holds. \end{proof} The braid gate BH-N-01 confirms each statement on the full operator at $t=(2,1,1.3)$, $\delta=0.4$, where $\lambda_*=2.6$ and $A_{12}=0.484974$. One loop in $w_3$ (radius $10^{-3}$) swaps the pair and two loops restore it. The discriminant winds $+1$ per loop, and one loop in $\sigma_3$ does not swap. The measured gap$/\sigma_3$ is $1.833030$ against $2\sqrt{0.84}=1.833030$ \grade{measured}. Figure~\ref{fig:braid} shows the two loops. Off the joint condition, $D(0)=(A_{11}-A_{22})^2>0$, and every zero of $D$ lies outside the closed physical segment, beyond the wall (negative $w_3$, that is, imaginary $\sigma_3$) or beyond the opposite walls \grade{derived at the wall; measured elsewhere}. Part (iii) is the property that separates this wall from a spectral exceptional point (Section~\ref{sec:walls}): along every physical path the spectrum stays real on both sides, including through the mirror. Over $41$ values of $t_3$ and $241$ values of $\sigma_3\in[-0.6,0.6]$ the largest imaginary part is exactly $0$ (BH-LIFT-05) \grade{measured}. \begin{figure}[t]\centering \includegraphics[width=\linewidth]{fig/f3_braid.png} \caption{The braid gate at the joint point $t=(2,1,1.3)$, $w_1-w_2=0.4$. Left: one loop of $w_3$ around the wall point exchanges the two eigenvalues. Right: one loop of $\sigma_3$ (two loops of $w_3$) returns each eigenvalue to itself. The branch cover is the orientation cover.}\label{fig:braid} \end{figure} \subsection{The operator on the cover: a plain double point} The Gram frame $E_1,E_2$ is a frame of the tangent space of $\Sigma_4$. On the oriented cover the natural coordinates near the wall are $(\delta,\sigma_3)$, and their coordinate vectors are \begin{equation}\label{eq:J} \partial_\delta=\tfrac{1}{\sqrt2}E_1,\qquad \partial_{\sigma_3}=-\sqrt6\,\sigma_3\,E_2,\qquad J=\mathrm{diag}\big(\tfrac{1}{\sqrt2},\,-\sqrt6\,\sigma_3\big). \end{equation} The covering map has differential $J$, which is singular exactly at the wall. The transport lifted to the cover is $B=J^{-1}AJ$. \begin{theorem}[the lift is regular and the wall point is diabolic]\label{thm:lift}\grade{derived} With $c=t_1+t_2-2t_3$, \begin{equation}\label{eq:B} B(\sigma_3)=\begin{pmatrix}(t_1+t_2)-\delta(t_1-t_2) & 2\sigma_3\big(\delta c-(t_1-t_2)\big)\\[2pt] -\tfrac12\sigma_3(t_1-t_2) & 2t_3+c\,\sigma_3^2\end{pmatrix}, \end{equation} which is analytic in $\sigma_3$, including at the wall. $B$ is self-adjoint for the pulled-back metric $J^{\top}g_WJ$, which is finite and positive at the wall, with $(J^{\top}g_WJ)_{\sigma_3\sigma_3}\to1$. On the joint condition $B(0)=\lambda_*I$. Near that point \begin{equation}\label{eq:cone} B=\lambda_*I+\sigma_3B_1+\varepsilon B_2+O(\sigma_3^2,\varepsilon\sigma_3),\qquad B_1=(t_1-t_2)\begin{pmatrix}0&2(\delta^2-1)\\-\tfrac12&0\end{pmatrix},\quad B_2=\begin{pmatrix}0&0\\0&2\end{pmatrix}, \end{equation} with $\varepsilon=t_3-t_3^*$ the departure from the joint condition. The two eigenvalues therefore meet at a cone: the gap is $2|t_1-t_2|\sqrt{1-\delta^2}\,|\sigma_3|$ along the wall direction and $2|\varepsilon|$ across the joint condition, linear in both. \end{theorem} \begin{proof} $B=J^{-1}AJ$ has entries $A_{11}$, $J_{11}^{-1}A_{12}J_{22}=-\sqrt{12}\,\sigma_3A_{12}$, $J_{22}^{-1}A_{21}J_{11}=-A_{21}/(\sqrt{12}\,\sigma_3)$ and $A_{22}$. Substituting Theorem~\ref{thm:blocks} with $w_1-w_2=\delta$ and $w_3=\sigma_3^2$ gives~\eqref{eq:B}; the factor $\sigma_3^2$ in $A_{21}$ cancels the pole. Self-adjointness transfers through any change of frame. The pulled-back metric is $J^{\top}g_WJ$; its $\sigma_3\sigma_3$ entry is $6\sigma_3^2g_W(E_2,E_2)\to6\sigma_3^2/(6w_3)=1$, and its off-diagonal entry is $O(\sigma_3)$. At $\sigma_3=0$ the diagonal entries are $A_{11}$ and $2t_3$, equal on~\eqref{eq:joint}. The expansion~\eqref{eq:cone} follows from~\eqref{eq:B}, using $c-(t_1-t_2)\delta=-2\varepsilon$ on the shifted condition. The off-diagonal entries of $B_1$ have product $(t_1-t_2)^2(1-\delta^2)$, which gives the stated gap in $\sigma_3$, and $B_2$ gives $2|\varepsilon|$. \end{proof} BH-LIFT-05 checks~\eqref{eq:B} against $J^{-1}AJ$ to $6\times10^{-17}$ at $\sigma_3=0.3,0.05,-0.2$, finds $B(0)=2.6\,I$ exactly at the paper point, and measures the cone: gap $1.833030\times10^{-3}$ at $\sigma_3=10^{-3}$ and $2.000000\times10^{-3}$ at $\varepsilon=10^{-3}$ \grade{measured}. The closed form agrees: at this point $2|t_1-t_2|\sqrt{1-\delta^2}=2\sqrt{0.84}=1.833030278$, and BH-QUTRIT-18 measures the slope $1.833030278$ \grade{derived; checked, BH-QUTRIT-18}. So the wall point has two honest descriptions. To the Gram observer, who uses the tangent space of $\Sigma_4$, it is a Jordan block with a square-root branch in $w_3$. On the oriented cover, where the operator is self-adjoint in a regular metric, it is a diabolic point: a plain double eigenvalue at the tip of a cone. The singular differential $J$ of the covering map converts the second description into the first. That $J$ has a simple zero at the wall is the fold of Theorem~\ref{thm:fold}, now seen by the operator. Throughout, ``Gram frame'' and ``cover'' name these two spaces, not two bases of one space: no similarity changes a Jordan form, and none is claimed. Away from the wall $A$ and $B$ are similar; at the wall $J$ is singular, so $A$ on the tangent space of $\Sigma_4$ and $B$ on the tangent space of the cover are different operators, and $A$ there is genuinely defective. \subsection{The declared operator decides} A Jordan block in a given frame is a property of the declared operator and of that frame, not of the wall alone. Replace the Frobenius-closed choice $T=\mathrm{diag}(t)$ by the volume structure $T=G^{-1}$, whose metric is the affine-invariant $\langle H,K\rangle_G=\mathrm{tr}(G^{-1}HG^{-1}K)$. \begin{proposition}\label{prop:volume}\grade{derived} With $T=G^{-1}$, $A$ is self-adjoint for $\langle\cdot,\cdot\rangle_G$ at every interior point, hence diagonalisable with real spectrum. Approaching the wall, three eigenvalues diverge like $(1,1,2)/w_3$ and two stay finite. No Jordan block forms in any frame. \end{proposition} \begin{proof} $\langle H,A(K)\rangle_G=\mathrm{tr}(G^{-2}HG^{-1}K)+\mathrm{tr}(G^{-1}HG^{-2}K)-2\,\mathrm{tr}(G^{-1}K)\,\mathrm{tr}(G^{-1}H)$. The first two terms exchange under $H\leftrightarrow K$ by cyclicity and the third is symmetric. The divergence rates are measured: eigenvalues $\times\,w_3\to(1.000,1.000,2.000)$ at $w_3=10^{-6}$ \grade{measured}. \end{proof} At the same wall, one declared operator gives a diabolic point that the Gram frame reads as a Jordan block, and the other gives a pole. The words ``the wall'' therefore always need the operator and the frame named alongside. This matters for any comparison with horizons, where the operator is fixed by physics. \section{What survives the wall}\label{sec:survive} \subsection{The contour keeps the ledger} For a contour $\Gamma$ enclosing the colliding pair and nothing else, the Riesz projector and the trace-log ledger \begin{equation} P_{\rm pair}=\frac{1}{2\pi i}\oint_\Gamma (z-A)^{-1}dz,\qquad \mathcal L_c=\frac{1}{2\pi i}\oint_\Gamma \log(c+z)\,\mathrm{Tr}(z-A)^{-1}dz=\log\det(c+A_{\rm pair}) \end{equation} are analytic in the matrix entries for as long as $\Gamma$ stays clear of the spectrum. Neither quantity uses a metric, and neither notices whether the enclosed eigenvalues are distinct, form a Jordan block in one frame, or meet at a cone in another. Individual eigenprojectors depend on the metric used to measure them. In the Gram frame with the Frobenius norm, a real $2\times2$ block with off-diagonal entries $A_{12},A_{21}$ and real gap $g$ has exactly $\|P_\pm\|^2=1+(A_{12}-A_{21})^2/g^2$ (Appendix~\ref{app:proofs}), and since $A_{21}=O(\sigma_3^2)$, $\|P_\pm\|\,g\to|A_{12}|$. In the kinetic metric the same projectors are orthogonal (Theorem~\ref{thm:selfadj}) and their norm is exactly $1$. \begin{center} {\small\begin{tabular}{ccccccc} \toprule $w_3$ & gap & Gram norm$\cdot$gap & kinetic norm & count in $\Gamma$ & $\mathrm{Tr}\,P_{\rm pair}$ & ledger\\ \midrule $10^{-2}$ & $1.83\times10^{-1}$ & $0.502311$ & $1.000000$ & 2 & $2.0000000000$ & $2.562330546753$\\ $10^{-4}$ & $1.83\times10^{-2}$ & $0.485147$ & $1.000000$ & 2 & $2.0000000000$ & $2.561872320543$\\ $10^{-6}$ & $1.83\times10^{-3}$ & $0.484976$ & $1.000000$ & 2 & $2.0000000000$ & $2.561867737220$\\ $0$ & $0$ & --- & --- & 2 & $2.0000000000$ & $2.561867690924$\\ \bottomrule \end{tabular}} \end{center} The ledger column is $\log\det(1+A_{\rm pair})$. At $t=(2,1,1.3)$, $\delta=0.4$, with $\Gamma$ the circle of radius $0.25$ about $\lambda_*=2.6$ \grade{measured}. In the Gram frame the product $\|P_\pm\|\cdot$gap tends to $|A_{12}|=0.484974$: the divergence constant of a single projector is the Jordan coupling. In the kinetic metric nothing diverges (the Gram-frame norms themselves are $2.74$, $26.47$ and $264.58$ at $\sigma_3=0.1,0.01,0.001$). The ledger tends to $2\log(1+\lambda_*)=2\log3.6=2.561867690924$, its exact value at the wall point. The ledger is smooth, not constant: it is a record, not a conserved charge. It needs no metric at all, and in that sense the record outlives the metric. One hypothesis is visible in the data. At $w_3=10^{-1}$ the pair spreads to $[2.33,2.91]$ and the shear eigenvalue $t_2+t_3=2.3$ enters any circle wide enough to hold the pair. The contour must isolate the cluster with a spectral margin, which is exactly the hypothesis of the persistence bounds (Appendix~\ref{app:proofs}, Lemma~\ref{lem:neumann} and Theorem~\ref{thm:kato}). With the margin in place, the map from operators to their Grassmannian projector data is locally Lipschitz, in any frame. The cluster holds where its vectors do not. By the Davis--Kahan theorem a perturbation $E$ turns the cluster's subspace by an angle whose sine is at most $\|E\|/\delta$, with $\delta$ the gap to the rest of the spectrum \cite{daviskahan1970}, while single eigenvectors inside a tight cluster can turn by order one. On $2000$ random symmetric $6\times6$ matrices with a three-member cluster split by $10^{-3}$ and $\|E\|=0.02$, the subspace turned by a median $0.27^\circ$ and its vectors by $74^\circ$; split the cluster by $0.4$ instead and the vectors turn only $2.7^\circ$; the bound was never exceeded \grade{measured, V10-PUSH}. The same law holds on a real orbit. Along the Chenciner--Montgomery figure-eight of three equal masses, the shape passes through the collinear wall six times per period. There the metric layer collapses: the rigidity $R=\lambda_{\min}^{-2}$ diverges and reaches $2.5\times10^{16}$ at the closest sampled point. Through all six crossings the contour count holds at $2$ to $5.1\times10^{-15}$ and the shifted trace-log equals $\log(1+\lambda_{\min})+\log2$ to $2.2\times10^{-15}$. With no shift the branch point of the logarithm sits inside the contour and the ledger fails, reaching $-0.580$ against the correct minimum $\log2$ \cite{beasleyThreeBody} \grade{cited measurement}. The hypothesis doing the work is a branch condition, not positivity: $\log(c+z)$ must have a single-valued holomorphic branch on a region containing $\Gamma$ and the enclosed eigenvalues, which keeping the contour off the spectrum does not by itself give. \begin{figure}[t]\centering \includegraphics[width=\linewidth]{fig/f4_record.png} \caption{The record outlives the metric. Approaching the joint point along $\sigma_3$: one eigenprojector measured in the Gram frame diverges like $1/\sigma_3$ while the gap closes linearly; measured in the kinetic metric the same projector has norm $1$. The pair projector keeps trace $2$ and the ledger $\det(1+A_{\rm pair})$ stays finite all the way to the wall, in every frame.}\label{fig:record} \end{figure} \subsection{The orientation keeps the trace} A configuration crossing the coplanar wall changes orientation. The Gram observer cannot see it, because $G$ depends on $\sigma_3^2$ only, but two other objects carry the record. The sign of $\det X$ is one. The other is the analytic labelling of the eigenvalue pair: by Theorem~\ref{thm:unfold} the branch that was $\lambda_+$ before the crossing continues analytically into the branch the Gram observer calls $\lambda_-$. The encounter leaves a $\mathbb Z_2$ record, readable as a monodromy, while every Gram-level quantity returns to its starting value. The same fold appears one dimension down. For three bodies in the plane, the three normalised mutual distances keep the shape of the triangle up to reflection and lose the sign of its oriented area: a triangle and its mirror image share one distance record, and their two sheets meet at the collinear configurations \cite{beasleyIQGT}. \paragraph{The handed record cancels across the mirror.} The pairing $N-\mathrm{Tr}(A^\dagger B)$ splits into a symmetric part, which is a distance and cannot tell a configuration from its mirror image, and an antisymmetric part. On projector tangents the antisymmetric part is $\Omega(V,W)=\mathrm{Tr}(P[V,W])$. For a real orthogonal projector family it vanishes identically, since $\mathrm{Tr}(PVW)=\mathrm{Tr}\big((PVW)^{\top}\big)=\mathrm{Tr}(WVP)=\mathrm{Tr}(PWV)$. In the Gram frame the wall's eigenprojector $P_+$ is oblique (it is orthogonal only for the kinetic metric, which varies from point to point), and there $\Omega$ does not vanish. Because $P_+$ depends on $\sigma_3$ only through $w_3=\sigma_3^2$, $\partial_{\sigma_3}P_+=2\sigma_3\,\partial_{w_3}P_+$, so $\Omega(\partial_{\sigma_3}P_+,\partial_\delta P_+)$ is odd under the mirror \grade{derived}. At $t=(2,1,1.3)$, $\delta=0.4$: \begin{center} \begin{tabular}{cccc} \toprule $|\sigma_3|$ & $\Omega$ at $+\sigma_3$ & $\Omega$ at $-\sigma_3$ & symmetrised operator\\ \midrule $0.1$ & $-27.257764$ & $+27.257764$ & $<10^{-11}$\\ $0.2$ & $-6.799872$ & $+6.799872$ & $<10^{-11}$\\ $0.4$ & $-1.685527$ & $+1.685527$ & $<10^{-11}$\\ \bottomrule \end{tabular} \end{center} \grade{measured}. Each side of the wall is handed in the Gram frame, the two sides cancel exactly, and the handedness lives only in the split of the pair: on the pair projector, which is the identity on the shape block, $\Omega=0$. This is the behaviour of an object fixed by an involution, where every reading odd under the involution sums to zero while the even readings survive. Section~\ref{sec:knots} shows the same rule in knot invariants. \subsection{The sign the cover keeps}\label{sec:sign} Theorems~\ref{thm:unfold} and~\ref{thm:lift} settle the eigenvalues. The eigenvectors carry one more piece of information, a sign, and where that sign lives depends on which loop is taken and in which chart. Transport right and left eigenvectors biorthogonally, keeping $u\cdot v=1$ and moving $v$ by the connection \begin{equation}\label{eq:conn} a=\frac{u\cdot dv}{u\cdot v},\qquad v\mapsto v\,e^{-\int a}. \end{equation} A change of frame $v\mapsto Fv$, $u\mapsto F^{-\top}u$ leaves $u\cdot v$ unchanged and adds to the connection \begin{equation}\label{eq:framechange} a\mapsto a+\frac{u\cdot(F^{-1}dF)\,v}{u\cdot v}. \end{equation} The added term need not be exact, even for single-valued invertible $F$: a rotation $F(\theta)$ with constant $u=(1,0)$, $v=(1,1)$ adds $\oint=-2\pi$ \grade{derived; checked, V10-CORRECTIONS}. So the transported factor belongs to the operator and the frame together, and each chart below is named. For a diagonal frame change the added term is $\sum_i\frac{u_iv_i}{u\cdot v}\,d\log F_i$, the logarithmic derivative of each frame entry weighted by the share of the pairing it carries. The frame $J$ of~\eqref{eq:J} is diagonal with a simple zero at the wall, and this term is where the two charts part company. \begin{theorem}[two loops, two charts]\label{thm:sign}\grade{derived} On the joint condition~\eqref{eq:joint}, write $c=t_1+t_2-2t_3$, $Q(\sigma_3)=(t_1-t_2)^2(1-\delta^2)+\tfrac14c^2\sigma_3^2$ and $p=\sqrt Q+\tfrac12c\,\sigma_3$. \begin{enumerate}[label=(\roman*),nosep] \item \emph{Gram frame, complex loop.} The analytic eigenvectors $v=(A_{12},\lambda-A_{11})^{\top}$ and $u=(A_{21},\lambda-A_{11})^{\top}$ of the branch $\lambda=\tfrac12(A_{11}+A_{22})+\sigma_3\sqrt Q$ satisfy \begin{equation}\label{eq:pairing} u\cdot v=2\,\sigma_3^2\sqrt Q\,p,\qquad a_{\rm G}=\frac{p+\sigma_3p'}{2\sigma_3\sqrt Q}\,d\sigma_3 , \end{equation} a simple pole with residue exactly $\tfrac12$. One loop of $\sigma_3$ returns each eigenvector multiplied by $-1$; one loop of $w_3$ acts by a monodromy $M$ with $M^2=-I$ and $M^4=I$. \item \emph{The same loop on the cover.} For the lifted operator $B$ the connection $a_{\rm reg}$ is holomorphic at the wall point and the same loop returns $+1$. The two connections differ by \begin{equation}\label{eq:residue} a_{\rm G}=a_{\rm reg}+\frac{u_2v_2}{u\cdot v}\Big|_{B}\,\frac{d\sigma_3}{\sigma_3}, \end{equation} where $u_2v_2/(u\cdot v)$ is the share of the eigenvector pairing carried by the folded direction $\partial_{\sigma_3}$. At the wall point it equals $\tfrac12$ exactly, because $B_1$ in~\eqref{eq:cone} has zero diagonal. The residue $\tfrac12$ in (i) is therefore the simple zero of $\det J$ weighted by the half of each eigenvector that lies across the fold. \item \emph{Real loop through the mirror.} On the cover, take a real loop in $(\sigma_3,\varepsilon)$ that encircles the wall point. It must pass through $\sigma_3<0$, the mirror configurations. Along it the eigenvalues stay real and distinct, and a continuously chosen real eigenvector returns reversed: the eigenline has holonomy $-1$. A real loop that does not encircle the point returns $+1$. \item \emph{The Gram observer cannot see (iii).} The Gram image of the loop in (iii) lies in $w_3\ge0$ and retraces itself, so it returns $+1$. No loop in $w_3\ge0$ encircles the wall point. \end{enumerate} \end{theorem} \begin{proof} (i) On the joint condition $A_{12}$ is constant and $A_{22}-A_{11}=c\,\sigma_3^2$, so $\lambda-A_{11}=\sigma_3 p$ and $\sqrt3(t_1-t_2)A_{12}=(t_1-t_2)^2(1-\delta^2)=Q-\tfrac14c^2\sigma_3^2$. Then $u\cdot v=A_{12}A_{21}+\sigma_3^2p^2=\sigma_3^2(2Q+c\sigma_3\sqrt Q)=2\sigma_3^2\sqrt Q\,p$. Since $dv=(0,d(\sigma_3p))$, $u\cdot dv=\sigma_3p\,(p+\sigma_3p')\,d\sigma_3$. The residue is $p(0)/(2\sqrt{Q(0)})=\tfrac12$, so $\oint a_{\rm G}=i\pi$ and $e^{-\oint a_{\rm G}}=-1$. A loop of $w_3$ is half a loop of $\sigma_3$, carrying $\lambda_+$ into $\lambda_-$, so $M^2=-I$ and $M^4=I$. (ii) The eigenvectors of $A=JBJ^{-1}$ are $v_{\rm G}=Jv$ and $u_{\rm G}=J^{-\top}u$ for eigenvectors $v,u$ of $B$, so $u_{\rm G}\cdot v_{\rm G}=u\cdot v$ and $u_{\rm G}\cdot dv_{\rm G}=u\cdot dv+u\cdot(J^{-1}dJ)v$. This is~\eqref{eq:framechange} with $F=J$, and $J^{-1}dJ=\mathrm{diag}(0,d\sigma_3/\sigma_3)$ gives~\eqref{eq:residue}. $B$ is analytic with $B(0)=\lambda_*I$, so its eigenvectors near $0$ are analytic continuations of those of $B_1$, the pairing is nonzero, and $a_{\rm reg}$ is holomorphic. For a $2\times2$ matrix with zero diagonal and off-diagonal entries $\beta,\gamma$, with eigenvalue $\mu$, $\mu^2=\beta\gamma$, $v=(\beta,\mu)$ and $u=(\gamma,\mu)$, so $u\cdot v=2\mu^2$ and $u_2v_2=\mu^2$, a share of $\tfrac12$. (iii) In coordinates orthonormal for the pulled-back metric at the wall point, $\sigma_3B_1+\varepsilon B_2$ has traceless part $-\varepsilon\,\sigma_z+\kappa\sigma_3\,\sigma_x$ (Pauli matrices) with $\kappa=|t_1-t_2|\sqrt{1-\delta^2}$, a real symmetric family whose eigenvector turns through half the angle that $(\varepsilon,\kappa\sigma_3)$ turns. One full turn of the parameter point reverses the eigenvector. (iv) $w_3=\sigma_3^2$ maps the loop onto an arc in $w_3\ge0$ traversed out and back, which is contractible away from the wall point. \end{proof} The check BH-LIFT-05 confirms each part at $t=(2,1,1.3)$, $\delta=0.4$. For a complex loop of radius $0.02$ in $\sigma_3$: Gram frame $-1.00012$, regular frame $+1.00000$. The folded share at the wall point is $0.500000000000$ on both branches. For a real loop of radius $0.05$ in $(\sigma_3,t_3)$ around $(0,1.3)$, the eigenline orientation in the kinetic metric is $-1.000000000$ on the cover and $+1.000000000$ in the Gram frame. For the same loop moved to centre $\sigma_3=0.2$ it is $+1$ in both \grade{measured}. Figure~\ref{fig:signs} shows all three. So the cover does keep a sign. It is not the complex-loop sign, which is the trace the fold leaves on the Gram observer's eigenvectors, half a unit of flux per simple zero of $\det J$. It is the sign of a conical intersection of a real self-adjoint family. That is the Herzberg--Longuet-Higgins sign of molecular physics \cite{herzberg1963}, the real case of Berry's phase \cite{berry1984}. Here it is carried around a point that only the oriented configuration space can encircle, because the loop has to pass through the mirror. The order-four monodromy around a generic exceptional point, with its geometric phase, is known and has been observed in microwave cavities \cite{dembowski2001,mailybaev2005}. What is specific here is that both signs occur at the same point and are told apart exactly by the fold. \paragraph{Beside a Majorana exchange.} Exchanging two Majorana modes $\gamma_1,\gamma_2$ acts by $R=(1+\gamma_1\gamma_2)/\sqrt2$, that is $R=e^{\pi\gamma_1\gamma_2/4}$ \cite{ivanov2001}. Then $R^2=\gamma_1\gamma_2$, $(R^2)^2=-I$ and $R^8=I$, relations checked on four modes in the Jordan--Wigner representation \grade{measured}. The Gram-frame monodromy $M$ of a $w_3$ loop obeys the relations of the double exchange $R^2$: it exchanges, $M^2=-I$ and $M^4=I$. A Majorana pair is fixed by charge conjugation, and the wall point is fixed by the mirror. The comparison is structural. Unlike a Majorana sign, which belongs to the representation, the Gram monodromy belongs to a chart. The chart-free sign at this point is the real conical one of part (iii) \grade{structural comparison}. \begin{figure}[t]\centering \includegraphics[width=\linewidth]{fig/f8_signs.png} \caption{Two loops, two charts, at $t=(2,1,1.3)$, $\delta=0.4$. (a) A complex loop of $\sigma_3$ about the wall point: the transported phase runs to $-\pi$ in the Gram frame and stays at $0$ on the cover. (b) A real loop on the cover in $(\sigma_3,t_3)$ about the wall point, passing through the mirror side $\sigma_3<0$. The eigenline (kinetic-orthonormal coordinates) turns half a turn and comes home reversed. (c) The Gram observer's image of the same loop lies in $w_3\ge0$ and retraces itself, so it carries no sign.}\label{fig:signs} \end{figure} \subsection{Majorana stars and the Petrov types}\label{sec:petrov} Majorana represented a spin-$j$ state by $2j$ points on the sphere, the roots of a polynomial \cite{majorana1932}. A Weyl tensor is a spin-$2$ object, and its four points are its principal null directions. The algebraic classification of spacetimes is the classification of how those four stars coincide \cite{petrov1954,penrose1960}. This gives the black-hole side a precise object with which to compare the two readings of our wall point. Represent the Weyl spinor by the traceless complex symmetric $3\times3$ matrix $Q$ acting on self-dual bivectors, and its stars by the roots of \begin{equation}\label{eq:majpoly} \psi_Q(z)=n(z)^{\top}Q\,n(z),\qquad n(z)=\big(1-z^2,\ i(1+z^2),\ -2z\big),\qquad n\cdot n=0 . \end{equation} Because $n(z)$ is null, the trace part of $Q$ drops out, and $Q\mapsto\psi_Q$ is linear from the five traceless directions to binary quartics. \begin{proposition}[stars fix the Jordan form]\label{prop:petrov}\grade{derived; checked, BH-PETROV-06} The map $Q\mapsto\psi_Q$ is invertible; in the basis $\mathrm{diag}(1,-1,0)$, $\mathrm{diag}(1,1,-2)$, $e_ie_j^{\top}+e_je_i^{\top}$ its determinant is $3072$. The multiplicities of the four stars fix the Jordan structure of $Q$: \begin{center} \begin{tabular}{lll} \toprule stars & Petrov type & Jordan structure of $Q$\\ \midrule four distinct & I & three distinct eigenvalues, diagonalisable\\ one double & II & one $2\times2$ Jordan block\\ two double & D & eigenvalues $(\lambda,\lambda,-2\lambda)$, diagonalisable\\ one triple & III & one nilpotent $3\times3$ block\\ one quadruple & N & nilpotent, $Q^2=0$, blocks $(2,1)$\\ \bottomrule \end{tabular} \end{center} \end{proposition} The correspondence is the classical one \cite{stephani2003}; the check builds $Q$ from prescribed stars and reads the Jordan structure from ranks of powers. \emph{How each degeneracy unfolds.} Split the double star of a type II quartic by a separation $\delta$. The quartic's coefficients move by $\epsilon\propto\delta^2$, and the gap of the colliding pair of $Q$ grows like $\sqrt\epsilon$, that is, linearly in $\delta$: $\mathrm{gap}/\delta=0.800000$ at $\delta=10^{-1},\dots,10^{-4}$ for the configuration in the check \grade{measured}. The coefficients are the even observable and the star separation is the signed coordinate. This is the fold again, now in the space of Weyl tensors. Type II is a genuine Jordan block of $Q$; the stars unfold its square-root branch without removing the block. Split both double stars of a type D quartic symmetrically instead, and the double eigenvalue of $Q$ splits quadratically: $\mathrm{gap}/\delta^2=0.250000$ \grade{measured}. Type D is diagonalisable, so its gap is linear in the quartic's coefficients, and those move by $\delta^2$: the degeneracy is protected to first order. \emph{Kerr.} In the Kinnersley tetrad the only nonzero Weyl component of the Kerr metric is $\Psi_2=-M/(r-ia\cos\theta)^3$ \cite{kinnersley1969}, so its stars are two doubled pairs at every point. The check builds $Q$ at three points with $M=1$, $a=0.7$ and finds type D each time, eigenvalue ratio $-2$ to $10^{-9}$ \grade{measured}. The Kerr exterior, horizon included, is a type D object: a degenerate but diagonalisable Weyl operator, a diabolic rather than an exceptional degeneracy. The marginally trapped surface operator says less: its principal eigenvalue is always real and simple \cite{ams2008}, which rules out a defect at that eigenvalue only. The comparison now reads cleanly. Our wall point read in the Gram frame looks like type II: a Jordan block with a square-root branch in the even coordinate, unfolded by the signed one. Read on the cover it is a diabolic point, like type D. The gap laws differ, which is a real difference: our cover point is a cone, linear in both directions, while D splits quadratically under the symmetric splitting (Figure~\ref{fig:petrov}). The two operators are different objects. The comparison is between Jordan structures and unfolding exponents, not an identification of a Weyl tensor with a transport operator \grade{structural comparison}. The three-body shape sphere shows why the two gap laws differ. With equal masses, the equilateral configuration is fixed by the $\mathbb Z_3$ relabelling of the bodies, and there the linear splitting vanishes: the gap closes quadratically ($\mathrm{gap}/\vartheta^2\to1.000000$) and the soft eigendirection returns with holonomy $+1$. Break the symmetry with masses $(1,2,3)$ and the degeneracy becomes a genuine cone: the gap closes linearly ($\mathrm{gap}/r\to2.16280$) and the eigenvector returns negated, $-1.000000$ at every radius tested \cite{beasleyThreeBody} \grade{cited measurement}. The two quadratic laws have different causes. For Kerr's type D it is the one above, a diagonalisable degeneracy whose coefficients move by $\delta^2$. On the shape sphere it is a property of this operator at a $\mathbb Z_3$-fixed point, not of the symmetry alone: a degeneracy fixed by $\mathbb Z_3$ can also split linearly, as in the $E\otimes e$ Jahn--Teller problem \cite{jahnteller1937}, and the holonomy $+1$ is the mark that the linear term vanishes here. Breaking the symmetry gives the generic cone, which is what our cover point is \grade{structural comparison}. \begin{figure}[t]\centering \includegraphics[width=\linewidth]{fig/f9_petrov.png} \caption{Majorana stars and Petrov types. (a) The multiplicities of the four principal null directions fix the Jordan form of the Weyl operator $Q$; Kerr is type D at every point. (b) How the degeneracies unfold: toward type II the gap is linear in the star separation (the stars unfold a genuine Jordan block); toward type D it is quadratic (diagonalisable). The wall point read on the cover is diagonalisable with a linear gap, a cone.}\label{fig:petrov} \end{figure} \subsection{Ramanujan's shadow: the divergence cancels in the pair}\label{sec:shadow} In his last letter to Hardy, Ramanujan described functions he called mock theta functions by their behaviour at the roots of unity. One of them is \begin{equation} f(q)=\sum_{n\ge0}\frac{q^{n^2}}{(-q;q)_n^2},\qquad b(q)=\frac{(q;q)_\infty}{(-q;q)_\infty^2}=(1-q)(1-q^3)(1-q^5)\cdots(1-2q+2q^4-\cdots). \end{equation} As $q$ approaches a root of unity $\zeta$ of even order $2k$ along the radius, $f$ and the modular function $b$ each blow up, but the combination $f(q)-(-1)^kb(q)$ stays bounded. Folsom, Ono and Rhoades proved this and identified the limit as a finite sum at the root of unity \cite{folsom2013}: \begin{equation}\label{eq:radial} \lim_{q\to\zeta}\big(f(q)-(-1)^kb(q)\big)=L(\zeta)=-4\sum_{n\ge0}(-\zeta;\zeta)_n^2\,\zeta^{n+1}, \end{equation} which terminates as soon as a factor $1+\zeta^j$ vanishes. The bounded value is the visible trace of the mock theta function's shadow, the modular correction that makes it a harmonic Maass form \cite{zwegers2002,zagier2010}. BH-SHADOW-07 computes both functions along the radius at working precision set from their size, using Gauss's and Euler's series for $b$ \grade{measured}: \begin{center} \begin{tabular}{ccccc} \toprule $\zeta$ & $1-|q|$ & $\log_{10}|f|$ & $f-(-1)^kb$ & $L(\zeta)$\\ \midrule $-1$ & $10^{-3}$ & $180.26$ & $3.996012$ & $4$\\ $-1$ & $3\times10^{-4}$ & $597.24$ & $3.998801$ & \\ $i$ & $3\times10^{-4}$ & $150.67$ & $0.002390+3.994025\,i$ & $4i$\\ $e^{i\pi/3}$ & $3\times10^{-4}$ & $67.91$ & $-1.988709-3.450721\,i$ & $-2-2\sqrt3\,i$\\ \bottomrule \end{tabular} \end{center} Two numbers of size $10^{597}$ cancel to $3.9988$. A three-point extrapolation in $1-|q|$ gives $3.9999999$, $0.0000022+3.9999947\,i$ and $-1.9999547-3.4640476\,i$, against $L(\zeta)=4$, $4i$ and $4e^{-2\pi i/3}$. The pattern is the one of Section~\ref{sec:survive}. Each member of a pair diverges at the special point, and the pair has a finite, exact value there. At the wall the individual eigenprojectors diverge in the Gram frame and the pair projector and ledger stay finite. At the root of unity $f$ and $b$ diverge and their signed difference lands on a finite sum. For $f$ itself the analogy is one of bookkeeping, not mechanism: this mock theta function has no transport operator behind it \grade{structural comparison}. In the counting of black-hole microstates, however, mock modularity does come from a wall. \paragraph{Where black holes meet mock modularity.} Dabholkar, Murthy and Zagier showed that the function counting quarter-BPS black-hole states in $\mathcal N=4$ string theory splits canonically into two parts \cite{dmz2012}. One is a mock Jacobi form, whose coefficients count single-centred black holes, which are immortal. The other is an Appell--Lerch sum, which counts multi-centred bound states. Those bound states come apart when the moduli cross walls of marginal stability, and that wall-crossing is where the mock behaviour comes from. Adding the shadow's non-holomorphic completion restores the modular symmetry that holography expects. Their family contains almost all known mock modular forms, Ramanujan's among them. So the pattern of this subsection has a black-hole instance with a mechanism: a single piece fails a symmetry, and the piece that restores it is exactly what a wall crossing removes. These walls lie in the moduli space of the theory, not in spacetime and not at a horizon. They are a third kind of wall, beside the two of Section~\ref{sec:walls}, and no identification with the coplanar fold is made \grade{cited result; structural comparison}. \paragraph{The mirror joins this subsection to the next.} Reversing the orientation of a three-manifold sends $q$ to $q^{-1}$ in its quantum invariants, which is the knot mirror of Section~\ref{sec:knots}. For the Brieskorn spheres the two orientations land on two different kinds of function. For one orientation the $q$-series invariant $\hat Z$ is a false theta function. For the reversed orientation it is conjectured to be a mock theta function, with $\Sigma(2,3,7)$ worked out in detail \cite{ccfgh2019,cfs2020}. On the false-theta side, the strong quantum-modularity conjecture for the Witten--Reshetikhin--Turaev invariants of every Brieskorn homology sphere is now proved \cite{putrov2025}. These spheres are built from the knots of the next subsection: $1/n$ surgery on the torus knot $T(p,q)$ gives $\Sigma(p,q,|pqn-1|)$ \cite{moser1971}, so $\pm1$ surgery on the trefoil gives the Poincar\'e sphere $\Sigma(2,3,5)$ and $\Sigma(2,3,7)$ \grade{derived; BH-KNOT-14}. The orientation reversal that separates the false side from the mock side is the same sign flip that carries a configuration to its mirror image at the coplanar wall (Theorems~\ref{thm:fold} and~\ref{thm:sign}). Two cautions: the mock functions in this link are of Ramanujan's seventh order, not the third-order $f$ above, and the mock side is a conjecture supported by computation. Neither is a transport operator \grade{cited result; structural comparison}. \begin{figure}[t]\centering \includegraphics[width=\linewidth]{fig/f10_shadow.png} \caption{Ramanujan's radial limits. (a) Along the radius toward $-1$, $i$ and $e^{i\pi/3}$, the mock theta function $f$ and its modular partner $b$ each blow up (to $10^{597}$ at $1-|q|=3\times10^{-4}$ toward $-1$). (b) The signed difference $f-(-1)^kb$ converges to the finite sum $L(\zeta)$, with error proportional to $1-|q|$.}\label{fig:shadow} \end{figure} \subsection{The mirror rule in knots}\label{sec:knots} Quantum knot invariants evaluated at roots of unity show the mirror rule of Section~\ref{sec:survive} with no transport at all. Take $q=e^{2\pi i/N}$. The Kashaev invariant of the figure-eight knot is $\langle4_1\rangle_N=\sum_{k=0}^{N-1}|(q;q)_k|^2$, and the Kontsevich--Zagier series $F(q)=\sum_{n\ge0}(q;q)_n$ terminates at a root of unity and gives the Kashaev invariant of the trefoil up to conjugation \cite{kashaev1997,zagier2001}. The mirror replaces $q$ by $q^{-1}$, which conjugates both. \begin{proposition}[the mirror rule, checked]\grade{measured, BH-KNOT-08} \begin{enumerate}[label=(\roman*),nosep] \item The figure-eight knot is amphichiral. Its invariant is real (imaginary part $10^{-13}$ of the real part at $N=200$), its signature is $0$, and it grows exponentially with rate set by its hyperbolic volume: with the known $N^{3/2}$ and $3^{-1/4}$ factors removed, $\tfrac{2\pi}{N}\log\langle4_1\rangle_N$ gives $2.029889$ at $N=800$, against $\mathrm{Vol}(4_1)=2.029883$ \cite{murakami2001}. \item The trefoil is chiral. Its invariant is not real, the mirror conjugates it, its signature changes sign ($-2$ and $+2$), and it grows only like a power: $|F(e^{2\pi i/N})|/N^{3/2}=1.000012$ at $N=800$. Its complement has no hyperbolic volume. \item The torus knot $T(3,4)$, which is $8_{19}$, is chiral as well, and it shows that the trefoil's row is a class. Its Alexander polynomial is $t^6-t^5+t^3-t+1$, its genus is $3$, and its Jones polynomial is $q^3+q^5-q^8$ up to the mirror \grade{derived}. Its invariant at $q=e^{2\pi i/N}$, computed from the Rosso--Jones formula after the formula reproduces the unknot, both Jones polynomials and the trefoil series above, grows like a power with no volume, and the mirror conjugates it. The constant in front of $N^{3/2}$ is where it differs from the trefoil: it depends on $N$ modulo $16$, taking sixteen values from $0.0858$ at $N\equiv0$ to $2.9142$ at $N\equiv8$, symmetric under $r\mapsto16-r$ and repeating to $10^{-4}$ along each class \grade{measured, BH-KNOT-14}. The trefoil has a single constant. Its $\pm1$ surgeries give $\Sigma(3,4,11)$ and $\Sigma(3,4,13)$ \cite{moser1971}. A molecular $8_{19}$ has been made by self-assembly \cite{danon2017}, and the modular structure of torus-knot invariants is treated in \cite{hikami2004}. \end{enumerate} \end{proposition} At a configuration fixed by the mirror, every reading odd under the mirror vanishes (the signature, the imaginary part, the handed part $\Omega$ summed across the wall), while the even readings survive, and the volume and the ledger are among them. The same rule appears in a fermion chain: the boundary offset of a half-filled Rice--Mele chain is exactly the part odd in the staggering, and a boundary-preserving particle--hole conjugacy protects its zero \cite{beasleyOffset}. Figure~\ref{fig:knots} shows the three knots. \begin{figure}[t]\centering \includegraphics[width=\linewidth]{fig/f11b_knots.png} \caption{Knots at roots of unity. (a) The figure-eight knot, fixed by the mirror: its Kashaev invariant is real and grows at the rate of its hyperbolic volume $2.029883$. (b) The trefoil, chiral: its invariant grows like $N^{3/2}$ with a single constant and no exponential rate, and the mirror conjugates it. (c) The torus knot $8_{19}=T(3,4)$: also power growth with no volume, but the constant depends on $N$ modulo $16$; three runs of sixteen consecutive $N$ fall on the same sixteen values.}\label{fig:knots} \end{figure} \subsection{Transients and the transverse oscillator}\label{sec:transverse} On the black-hole side the transient story is the literature's \cite{jaramillo2021,besson2025,gasperin2022,carballo2025}. On the shape side, a near-defective block in a given frame sizes its own transients exactly. For $J_\mu=\left(\begin{smallmatrix}b&\mu\\0&b\end{smallmatrix}\right)$, \begin{equation} \|e^{-J_\mu t}\|_2=e^{-bt}\,\frac{\sqrt{4+\mu^2t^2}+|\mu|t}{2}, \end{equation} so transient growth above $1$ occurs exactly when $|\mu|/b>2$ (Appendix~\ref{app:proofs}). At the joint point, read in the Gram frame, $|\mu|/b=0.4850/2.6=0.1865$: there is no transient amplification, only a slowed decay \grade{derived; value measured}. In the kinetic metric the operator is self-adjoint and $\|e^{-At}\|=e^{-\lambda_{\min}t}$, with no transient at all. The choice of norm is structural on both sides \cite{gasperin2022}. The classical bound on such transients is Kreiss's \cite{kreiss1962}. Three refinements come from companion papers. The $\varepsilon$-pseudospectrum of $J_\mu$ is exactly the disk $|\lambda-b|<\sqrt{\varepsilon|\mu|+\varepsilon^2}$. An input confined to the interface coordinate, the model of a prescribed hit rather than the optimal state, grows above its starting size only for $|\mu|/b>\kappa_r=2.48527$, above the optimal-state threshold $2$; a black hole struck by infalling matter receives a prescribed kick, so $\kappa_r$ is the relevant test. And the same block is approached at two rates: diagonal detuning collapses the eigenvector linearly, a reverse coupling collapses it as a square root \cite{beasleyFilament} \grade{cited; derived there}. On a real orbit, the figure-eight amplifies perturbations $45.7$-fold in energy within one period, with mean gain $10.58$ in the quartile of the orbit nearest the collinear wall and $26.10$ in the quartile farthest from it \cite{beasleyThreeBody} \grade{cited measurement}. Transients live away from the geometric wall, as they do here. Persistence of structure over vectors is measured by the overlap $C(\Delta)=\mathrm{Tr}\,P(t)P(t+\Delta)$: observable channels are those with $C(\Delta)\to\mathrm{rank}\,P$. Where a confining stiffness $\omega^2$ acts across the wall, the transverse motion $\ddot y+\omega^2y=0$ at fixed energy $E=\tfrac12\dot y^2+\tfrac12\omega^2y^2$ has displacement amplitude $\sqrt{2E}/\omega$, velocity amplitude $\sqrt{2E}$, and accumulated phase $\int\omega\,dt$ \grade{derived}. A stiffness growing without bound therefore suppresses the \emph{displacement} as $\omega^{-1}$ at fixed energy, while the velocity amplitude is fixed and the phase keeps counting. On $\Sigma_4$ the kinetic metric supplies no such stiffness, since curvature is finite in $\sigma_3$. What looks like suppression in the Gram coordinate is the fold law~\eqref{eq:normalvel}. A physical potential that stiffens toward coplanarity would be extra input, and the statement above is its conditional form. Ordinary physics supplies that input. Any potential built from the mutual distances is a function of $G$, because the squared distances are linear in its entries, and in the fold chart $G$ depends on $\sigma_3$ only through $w_3=\sigma_3^2$. Such a potential is therefore even under orientation reversal. The wall is always a transverse critical point. Along the normal lift, which moves only $\sigma_3$ and keeps the in-plane shape and the moment of inertia fixed, the stiffness is \begin{equation}\label{eq:stiff} \omega^2=\frac{\partial^2V}{\partial\sigma_3^2}\bigg|_{\sigma_3=0}=2\,\frac{\partial V}{\partial w_3}\bigg|_{w_3=0} \end{equation} \grade{derived}. The factor $2$ is the chain rule through $w_3=\sigma_3^2$, and since the metric in the signed coordinate is $d\sigma_3^2$ at the wall (Theorem~\ref{thm:fold}), $\omega$ is a frequency in the kinetic metric. Two realisations follow. \emph{A thin disk.} Bodies held near a plane by an external field feel $V=\tfrac12\nu^2\sum_im_iz_i^2$, with the centre of mass in the plane. The sum is the $zz$ entry of $XX^{\top}$, whose eigenvalues are those of $G$, so its minimum over orientations is the smallest of them, $w_3$. At that orientation $V=\tfrac12\nu^2\sigma_3^2$: the transverse frequency of the shape is exactly the disk's vertical frequency $\nu$, and at fixed energy the out-of-plane amplitude falls as $\nu^{-1}$ \grade{derived; checked, BH-QUTRIT-18}. The check confirms the minimum to $2\times10^{-16}$ and the identity $\sum_im_iz_i^2=(XX^{\top})_{zz}$ on explicit bodies. \emph{Newtonian gravity.} At fixed moment of inertia the shape potential of four equal masses, $U=-\sum_{i0$ at all $400$ sampled flat configurations, the smallest value $+1.89$, and at all of $20\,000$ more \grade{measured, BH-QUTRIT-18, BH-STIFF-19}. Like charges do the opposite, $\partial U/\partial w_3<0$ at every sample, so for them the wall is a transverse maximum. The sign is decided shape by shape. A flat shape of four unit masses has exactly one out-of-plane shape direction $c$, orthogonal to the constant vector and to both in-plane coordinates; lifting along $c$ and rescaling back to $I=1$ gives \begin{equation}\label{eq:gravstiff} \frac{\partial U}{\partial w_3}=\frac12\Big[\sum_{ik_*$; an elongated rhombus with one close pair, $kk_*=0.144557$; like charges have the opposite sign throughout. Computed by BH-FIGS-V8-5 from the formulas of BH-QUTRIT-18 and BH-STIFF-19.}\label{fig:stiff} \end{figure} \subsection{What fidelity measures on the shape space}\label{sec:qutrit} Theorem~\ref{thm:bures} makes $\Sigma_4$ the state space of a real qutrit, with $\rho=G$. Two quantities of quantum information then acquire mechanical meanings, and a third shows which observer they belong to. \emph{Fisher information is kinetic energy.} The quantum Fisher information of a curve of states of constant rank is four times its Bures metric \cite{braunstein1994}, so along any shape motion that keeps full rank $F_Q=4\,g(\dot G,\dot G)$. At unit moment of inertia the shape kinetic energy is $K=\tfrac12g(\dot G,\dot G)$, and therefore \begin{equation} F_Q=8K: \end{equation} the quantum Fisher information of a moving configuration is eight times its shape kinetic energy \grade{derived; checked, BH-QUTRIT-18}. The check computes $g$ without any formula, as the smallest $|\dot X|^2$ that produces $\dot G$, and finds $F_Q/g=4$ to $10^{-11}$ on $200$ random motions. At the wall the rank drops, and there the two part. Along $G(s)=\mathrm{diag}(\tfrac{1-s^2}2,\tfrac{1-s^2}2,s^2)$, which crosses the wall at $s=0$ at unit speed in $\sigma_3=s$, $8K$ is $4.0404$, $4.0004$, $4.000004$ and $4$ at $s=0.1$, $0.01$, $0.001$ and $0$, and the Fisher information of the symmetric logarithmic derivative matches it at every $s\neq0$ and is exactly $0$ at $s=0$ \grade{derived; checked, V10-CORRECTIONS}. This is the known discontinuity of the quantum Fisher information where the rank changes \cite{safranek2017}. Kinetic energy is continuous through the fold and Fisher information is not: it reads only $G$, so it belongs to the Gram observer. \emph{The wall distance is a Bures angle.} With the fidelity $F(\rho,\sigma)=\big(\mathrm{Tr}\sqrt{\sqrt\rho\,\sigma\sqrt\rho}\big)^2$ \cite{uhlmann1976,jozsa1994}, the coplanar configuration $\mathrm{diag}(\tfrac12,\tfrac12,0)$ and $W(\varepsilon)$ of Theorem~\ref{thm:fold}(ii) commute, so $\sqrt F=2\sqrt{\tfrac12\cdot\tfrac{1-\varepsilon}2}=\sqrt{1-\varepsilon}$ and \begin{equation} \arccos\sqrt F=\arcsin\sqrt\varepsilon . \end{equation} The finite distance to the wall is the Bures angle between the flat configuration and the lifted one \grade{derived; checked, BH-QUTRIT-18}, to $3\times10^{-14}$. \emph{Fidelity is the Gram observer.} A configuration and its mirror image have the same $G$, so their fidelity is exactly $1$. On the orientation cover they are $2\arcsin|\sigma_3|$ apart along the fold path through the wall \grade{derived; checked, BH-QUTRIT-18}. Every function of $G$, fidelity included, is blind to orientation; only the cover separates the pair, as it keeps the sign of Section~\ref{sec:sign}. The two tetrahedra at the ends of Figure~\ref{fig:stiff} are such a pair. \section{A one-way boundary}\label{sec:oneway} \subsection{Interface Schur complement} Partition a transport operator into a boundary block and the two sides it separates, \[ K=\begin{pmatrix}A&B&0\\C&D_\Sigma&F\\0&G&E\end{pmatrix}. \] Eliminating both sides gives the effective boundary operator \begin{equation}\label{eq:schur} K_\Sigma(z)=D_\Sigma-z-C(A-z)^{-1}B-F(E-z)^{-1}G, \end{equation} in which the interior and exterior loadings enter additively (Appendix~\ref{app:proofs}). A boundary is one-way when one loading vanishes because one coupling does, that is, when the operator is block-triangular across it. That is what Theorem~\ref{thm:wall}(i) gives at the coplanar wall in the Gram frame. The Schur load $A_{12}A_{21}/(z-A_{22})$ that the transverse direction places on the in-wall direction vanishes like $\sigma_3^2$. Equivalently, the resolvent entry $(z-A)^{-1}_{21}=A_{21}/[(z-A_{11})(z-A_{22})-A_{12}A_{21}]$ vanishes, while $(z-A)^{-1}_{12}$ does not \grade{derived}. The same elimination has a determinant version, proved in the author's offset paper \cite{beasleyOffset}. For a Gaussian fermion state with regional covariance $0