% Options for packages loaded elsewhere \PassOptionsToPackage{unicode}{hyperref} \PassOptionsToPackage{hyphens}{url} \PassOptionsToPackage{dvipsnames,svgnames,x11names}{xcolor} % \documentclass[ 11pt, ]{article} \usepackage{amsmath,amssymb} \usepackage{iftex} \ifPDFTeX \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{textcomp} % provide euro and other symbols \else % if luatex or xetex \usepackage{unicode-math} % this also loads fontspec \defaultfontfeatures{Scale=MatchLowercase} \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1} \fi \usepackage{lmodern} \ifPDFTeX\else % xetex/luatex font selection \fi % Use upquote if available, for straight quotes in verbatim environments \IfFileExists{upquote.sty}{\usepackage{upquote}}{} \IfFileExists{microtype.sty}{% use microtype if available \usepackage[]{microtype} \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts }{} \makeatletter \@ifundefined{KOMAClassName}{% if non-KOMA class \IfFileExists{parskip.sty}{% \usepackage{parskip} }{% else \setlength{\parindent}{0pt} \setlength{\parskip}{6pt plus 2pt minus 1pt}} }{% if KOMA class \KOMAoptions{parskip=half}} \makeatother \usepackage{xcolor} \usepackage[margin=1in]{geometry} \usepackage{longtable,booktabs,array} \usepackage{calc} % for calculating minipage widths % Correct order of tables after \paragraph or \subparagraph \usepackage{etoolbox} \makeatletter \patchcmd\longtable{\par}{\if@noskipsec\mbox{}\fi\par}{}{} \makeatother % Allow footnotes in longtable head/foot \IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}} \makesavenoteenv{longtable} \setlength{\emergencystretch}{3em} % prevent overfull lines \providecommand{\tightlist}{% \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \setcounter{secnumdepth}{-\maxdimen} % remove section numbering \ifLuaTeX \usepackage{selnolig} % disable illegal ligatures \fi \IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}} \IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available \urlstyle{same} \hypersetup{ pdftitle={Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere}, pdfauthor={Jay Thrussell hanks\{Independent researcher. Correspondence: jaythrussell@gmail.com.\}}, colorlinks=true, linkcolor={blue}, filecolor={Maroon}, citecolor={Blue}, urlcolor={blue}, pdfcreator={LaTeX via pandoc}} \title{Riemannian three-manifolds as hypersurfaces of Euclidean four-space: the realizability operator, its symbol, and its linearisation at the round sphere} \author{Jay Thrussell\thanks{Independent researcher. Correspondence: jaythrussell@gmail.com.}} \date{October 2026} \begin{document} \maketitle \emph{This paper supersedes, for local questions, the author's earlier mathematics papers 1, 2 and 4 (Thrussell 2026a, b, d). Appendix C lists every statement that changes. A companion paper treats the global questions.} \emph{The verification scripts for this paper (tests 325--327 and 329--344, their library and their run log) are deposited with it at Zenodo, DOI \href{https://doi.org/10.5281/zenodo.23765515}{10.5281/zenodo.23765515}. The author's earlier work is archived under concept DOI \href{https://doi.org/10.5281/zenodo.21264118}{10.5281/zenodo.21264118}. Appendix V indexes every displayed result against the script that reproduces it, and discloses the use of large-language-model systems.} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{abstract}{% \section{Abstract}\label{abstract}} For a Riemannian three-manifold \((\Sigma, g)\) with negative Einstein endomorphism \(\mathbb M = -G\), the Gauss equation of a hypersurface of \(E^4\) reads \(\mathbb M = \mathrm{adj}\,K\). On \(\{\det\mathbb M > 0\}\) it is solved identically by \(\tilde K = \sqrt{\det\mathbb M}\,\mathbb M^{-1}\). Realizability with nondegenerate second fundamental form is therefore one equation, \(F(g) = \mathrm{Cod}(\tilde K(g)) = 0\). This reduction is classical (Beez, Killing, Thomas), and the zero set of \(F\) has been characterized intrinsically (Rivertz; Agaoka and Hashinaga). We study \(F\) as a nonlinear operator on metrics. \begin{itemize} \tightlist \item \textbf{The symbol.} At every nondegenerate point the principal symbol of \(F\) has rank two. Its kernel is the diffeomorphism directions plus the normal direction \(K\): exactly the symbol of the linearised immersions, at every covector and on both branches. So \(F\) is injective modulo the linearised immersions everywhere, and is nowhere elliptic modulo diffeomorphisms alone. \item \textbf{Transverse-traceless tensors.} The symbol is injective on transverse-traceless tensors at \(\xi\) iff \(\xi^{\mathsf T}(\sigma_1g - K)\xi \neq 0\). So the transverse-traceless slice serves exactly where the first Newton tensor is definite, and a slice adapted to \(K\) serves everywhere. \item \textbf{The count.} The nine Codazzi components satisfy four identities, which leaves five conditions per point, at every nondegenerate shape operator. Cartan's test holds with equality at order three, in exact arithmetic, at sixteen shape operators, the umbilic one included, and so off a proper real-algebraic set. There the formal solutions of the symbol are exactly the linearised immersions. \item \textbf{Meagreness.} On a closed three-manifold the zero set of \(F\), which contains the realizable metrics, is nowhere dense in the nondegenerate domain. \item \textbf{The round sphere.} On transverse-traceless tensors the linearisation is \(\mathrm{Cod}\circ\tfrac12(-\Delta + 2)\), with singular values \(n(n^2-1)/\sqrt2\) at level \(n\). These are Branson's multipliers, and they give the bound \(\lVert DF\,h\rVert^2 \geq 288\lVert h\rVert^2\). The straight path along a kernel direction leaves the realizable set at second order, while the radial lift stays in it. \item \textbf{Shells.} For metrics foliated by round 2-spheres we solve in closed form the first-order problem for deformations of the radial coefficient by a shell harmonic. We write down the metrics of all hypersurfaces swept by round 2-spheres: the radial coefficient is \(H^2 + (\alpha + \Phi\,b\cdot\hat n)^2\), with \(b\) a free tilt of the spheres' 3-planes. So the \(\ell = 1\) first-order direction is realized exactly by a tilt, with no in-plane drift of the centres. An in-plane drift realizes it together with a shear of the shell labels. \item \textbf{The rest.} We treat the degenerate locus through two polynomial forms of the equation, one of them of the bidegree of Rivertz's equalities. We give Gauss--Codazzi as the flatness of two real \(\mathfrak{su}(2)\) connections, the connection form of Lawn and Roth's two-spinor characterization; on the nondegenerate domain, realizability is equivalent to a single parallel spinor. \end{itemize} Every displayed result is reproduced by a script; Appendix V indexes them. \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{introduction}{% \section{1. Introduction}\label{introduction}} \textbf{The question.} Which Riemannian three-manifolds are hypersurfaces of Euclidean four-space? Isometric embedding has two classical regimes. In the \(C^1\) category, codimension one already suffices locally for every metric, and globally whenever there is a short immersion (Nash 1954; Kuiper 1955). In the \(C^2\) category the second fundamental form exists and is constrained by the intrinsic curvature through the Gauss and Codazzi equations. This paper is about the rigid regime, in the one dimension where the constraint takes a special form. \textbf{Why dimension three.} For a hypersurface \(\Sigma^n \subset E^{n+1}\) the Gauss equation \(R_{abcd} = K_{ac}K_{bd} - K_{ad}K_{bc}\) has \(n^2(n^2-1)/12\) components and the second fundamental form has \(n(n+1)/2\). At \(n = 2\) there is one equation, \(\det K = \kappa\), for three unknowns. The problem is then a single Monge--Ampère equation for a height function, determined rather than overdetermined: \begin{itemize} \tightlist \item analytic metrics always realize locally (Janet 1926; Cartan 1927); \item smooth metrics realize locally under sign conditions on the curvature (Lin 1985, 1986); \item local realization can fail, but only by delicate constructions (Pogorelov 1971 at finite regularity; Nadirashvili--Yuan 2002, smooth metrics with no \(C^3\) realization). \end{itemize} At \(n = 3\) the counts agree, six and six. The Gauss equation is then an algebraic system for \(K\) that can be solved in closed form, and the Codazzi equation becomes a condition on the metric alone. For \(n \geq 4\) the Gauss equation overdetermines \(K\). Codazzi then follows from Gauss when the second fundamental form has rank at least four (Thomas 1936), so the criterion carries no separate content (§10). \textbf{The reduction.} In dimension three the Gauss equation contracts, with no loss, to a statement about the Einstein tensor. Writing \(\mathbb M = -G\) as an endomorphism, \[\mathbb M = \mathrm{adj}\,K, \tag{1.1}\] the adjugate of the shape operator (§2). Where \(\det\mathbb M > 0\) the tensor \[\tilde K = \sqrt{\det\mathbb M}\;\mathbb M^{-1} \tag{1.2}\] satisfies \(\mathrm{adj}\,\tilde K = \mathbb M\) for every metric, realizable or not. When \(\det K \neq 0\), the second fundamental form of any realization is \(\pm\tilde K\). Realizability with nondegenerate second fundamental form is therefore the vanishing of the \textbf{realizability operator} \[F(g) = \mathrm{Cod}\big(\tilde K(g)\big), \qquad \mathrm{Cod}(K)_{abc} = \nabla_aK_{bc} - \nabla_bK_{ac}. \tag{1.3}\] \(F\) is a third-order nonlinear operator on metrics. Its values lie in the nine-dimensional bundle of tensors antisymmetric in their first two indices. \textbf{What is classical.} The determination of \(K\) by the metric is the rigidity theorem of Beez (1876) and Killing (1885): a hypersurface of \(E^{n+1}\) whose second fundamental form has rank at least three is locally rigid. Thomas (1936) characterised Riemannian spaces of class one algebraically. His paper already contains, in other notation: \begin{itemize} \tightlist \item the double-adjugate identity \(\mathrm{adj}\,\mathrm{adj}\,K = (\det K)K\) (his (9.3)); \item the clearing of the square root by a factor \((\det\mathbb M)^{3/2}\) (his §10); \item the continuity of the sign field on simply connected domains (§9); \item the sufficiency of his conditions over simply connected spaces of type at least three (§11). \end{itemize} Further work: \begin{itemize} \tightlist \item Allendoerfer (1939) extended rigidity to higher codimension, and Berger, Bryant and Griffiths (1983) sharpened the generic picture. \item Vilms (1977) identified the curvature operator on \(\Lambda^2\) and its determinant condition. \item Fodor (2019) reached the same sign structure through the logarithm of the curvature operator. \item In the literature on metrics of embedding class one, the reduction to Gauss-then-Codazzi goes back to Eiesland (1925). Karmarkar (1948) gave the eliminant for spherically symmetric metrics, and Pandey and Sharma \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{1981} \tightlist \item showed that it needs a nondegeneracy hypothesis, the analogue of \(\det\mathbb M > 0\). \end{enumerate} \item Rivertz (1999) found, beyond the sign condition, six polynomial equalities in the curvature and its first derivative that a hypersurface metric must satisfy. Agaoka and Hashinaga (2022) proved that they suffice. A simply connected three-manifold whose curvature determinant \(\lvert\tilde R\rvert\), which is \(\det\mathbb M\) up to a positive factor, vanishes nowhere is locally a hypersurface of \(\mathbb R^4\) iff \(\lvert\tilde R\rvert > 0\) and the six equalities hold. They solve Gauss by \(2\times2\) minors divided by \(\sqrt{\lvert\tilde R\rvert}\), which is (1.2) in components. That is an intrinsic characterization of the zero set of \(F\) on \(\mathcal U\). They do not study the operator: its symbol, involutivity, linearisation or degenerate locus. Proposition 8.1 gives a polynomial form of \(F\) with the bidegree of Rivertz's equalities. \end{itemize} In Lorentzian signature the tensor (1.2) appears in Bryan, Ivaki and Scheuer (2019), with a Minkowski ambient and negative curvature. Their result: \(\sqrt{\det\mathcal E}\,\mathcal E^{-1}\) is Codazzi iff the universal cover embeds in \(\mathbb R^{3,1}\) as a locally convex spacelike hypersurface. They also show that the Einstein volume is monotone along the cross curvature flow of Chow and Hamilton (2004). The identity (1.1) is the case \(k = 2\) of the classical fact that the Einstein tensor of a hypersurface of flat space is minus its second Newton transformation (Reilly 1973). Equivalently, it is the intrinsic character of the even-order elementary symmetric functions (Labbi 2007). Codazzi tensors on space forms are classified locally by Ferus (1981); see also Derdziński and Shen (1983), Derdziński's appendix in Besse (1987), and Merton (2013). \textbf{What this paper does.} We take the reduction as given and study the operator \(F\) on the space of metrics: its symbol and the slices on which it is injective, how overdetermined it is, the size of its zero set, its linearisation at the round sphere, its behaviour on a non-round family where it can be solved completely, its degenerate locus, and a chiral form. Along the way we write down the metrics of all hypersurfaces swept by round 2-spheres. This shows that a sphere in \(E^4\) can realize an \(\ell = 1\) change of its radial coefficient by tilting as well as by translating. Results credited in the text to \emph{an independent machine-assisted audit (October 2026)} are that audit's own; Appendix V describes it. \textbf{Main results.} Throughout, \(\mathcal U\) denotes the open set of metrics with \(\det\mathbb M > 0\). \begin{itemize} \tightlist \item \textbf{Theorem A (§3, the reduction).} \begin{itemize} \tightlist \item The gate \(\det\mathbb M = (\det K)^2 \geq 0\). \item The Gauss equation is solved by construction, and the branch is \(\tilde K = \mathrm{sign}(\det K)\,K\). \item \(\mathbb M > 0\) exactly on the locally convex branch. \item Global sufficiency on simply connected \(\Sigma\). The round \(\mathbb{RP}^3\) shows that the hypothesis is needed. \item A complete, two-ended, non-convex example on the indefinite branch. \end{itemize} \item \textbf{Theorem B and Corollary 4.2 (§4, the symbol).} \begin{itemize} \tightlist \item At every nondegenerate point and every covector \(\xi \neq 0\), \(\sigma(\xi)\) has rank exactly two, with kernel the gauge directions plus \(\mathrm{span}\{PKP\}\), \(P\) the projection off \(\xi\). \item That kernel is exactly the symbol of the linearised immersions \(\mathcal L_Xg - 2\varphi K\). So \(F\) is injective modulo the immersions at every covector, on a slice adapted to \(K\), and is nowhere elliptic modulo diffeomorphisms alone. \item The symbol is injective on transverse-traceless tensors iff \(\xi^{\mathsf T}T_1\xi \neq 0\), where \(T_1 = \sigma_1 g - K\) is the first Newton tensor. So the TT slice serves where \(T_1\) is definite: everywhere on the convex branch, and on the indefinite branch iff \(k_1 + k_2\) and \(k_2 + k_3\) share a sign. \end{itemize} \item \textbf{Theorem C (§4, involutivity).} \begin{itemize} \tightlist \item The nine components of \(F\) satisfy four identities for every metric. They are independent, and \(F = 0\) is exactly five conditions per point, at every nondegenerate shape operator. \item The symbol passes Cartan's test at order three with equality, \(78 = 55 + 19 + 4\), and the first prolongation adds no integrability condition. So \(F = 0\) is involutive. This is proved in exact arithmetic at sixteen shape operators, the round sphere's included, and so holds off a proper real-algebraic subset. \item There the formal solutions of the symbol are exactly the linearised immersions. The general analytic solution depends on four functions of three variables: the immersion itself. \end{itemize} \item \textbf{Theorem D (§5, meagreness).} On a closed three-manifold the zero set of \(F\), which contains the realizable metrics, is closed with empty interior in \(\mathcal U\), on both branches, and so nowhere dense. \item \textbf{Theorem E (§6, the round sphere).} \begin{itemize} \tightlist \item \(\ker DF_\gamma\) is the conformal plus Lie directions, a sum that is not direct. \item On transverse-traceless tensors, \(DF_\gamma = \mathrm{Cod}\circ\tfrac12(-\Delta + 2)\), and \(\lVert DF_\gamma h\rVert^2 = \tfrac12\langle h, (-\Delta + 3)(-\Delta + 2)^2h\rangle \geq 288\lVert h\rVert^2\). \item At level \(n\) the singular values are \(n(n^2 - 1)/\sqrt2\). These are the multipliers of Branson's \(P_3\) on functions, level by level. \end{itemize} \item \textbf{Remark 6.1 and Proposition 6.3 (§6).} The regularised count of transverse-traceless modes is \(\zeta(0) = 10 = \dim SO(4,1)\). Along the straight conformal path, \(F\) leaves its zero set at second order, while the radial lift stays in it. \item \textbf{Theorems F and G, Proposition 7.4 (§7, a worked non-round family).} \begin{itemize} \tightlist \item On shell metrics, for deformations of the radial coefficient by a shell harmonic, the first-order problem is solved completely: kernel \(\ell \in \{0,1\}\) for that family. \item The metrics of hypersurfaces swept by round 2-spheres are written down exactly. Their radial coefficient is \(H^2 + (\alpha + \Phi b\cdot\hat n)^2\): an in-plane speed tied to the chart drift, plus a free tilt. The tilt realizes the \(\ell = 1\) kernel exactly, with no drift. The drift realizes it together with an \(\ell = 1\) shear of the shell labels. \item Nested shells are a future-directed timelike curve in \(\mathbb R^{1,3}\). \item At second order on the round base, the \(\ell = 1\) obstruction is pure \(\ell = 2\). It is removed by the tilt or by the drift, on interior shells. Exact checks on three non-round bases show the same structure. \end{itemize} \item \textbf{§8, the degenerate locus:} two polynomial forms of \(F\), one of the bidegree of Rivertz's equalities, and what they see there. \item \textbf{§9, the chiral form:} Gauss and Codazzi as the flatness of two \(\mathfrak{su}(2)\) connections, and realizability as a single parallel spinor. \item \textbf{§§10--12 (not load-bearing):} why \(n = 3\), regularity, and higher dimensions. \end{itemize} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{preliminaries-and-conventions}{% \section{2. Preliminaries and conventions}\label{preliminaries-and-conventions}} \hypertarget{hypersurfaces}{% \subsection{2.1 Hypersurfaces}\label{hypersurfaces}} Let \(x : \Sigma^3 \to E^4\) be an immersion with unit normal \(N\), induced metric \(g\) and second fundamental form \(K_{ab} = -N\cdot\partial_a\partial_bx\), signed so that the unit sphere with outward normal has \(K = \gamma\). The structure equations of the flat ambient space are \[\text{(Gauss)}\quad R_{abcd} = K_{ac}K_{bd} - K_{ad}K_{bc}, \qquad \text{(Codazzi)}\quad \mathrm{Cod}(K) = 0. \tag{2.1}\] Conversely, on a simply connected \(\Sigma\) a pair \((g, K)\) satisfying both is realized by an immersion, unique up to a rigid motion (the fundamental theorem of hypersurfaces; Mardare 2003, 2005 for low regularity). We say that a metric \(g\) of class \(C^k\), \(k \geq 3\), is \textbf{realizable} if it admits an isometric immersion of class \(C^3\) into \(E^4\). Then \(K\) is \(C^1\) and both equations hold classically. Below this regularity the questions change (§11). \hypertarget{newton-tensors}{% \subsection{2.2 Newton tensors}\label{newton-tensors}} With \(\sigma_k\) the elementary symmetric functions of the eigenvalues of the shape operator, the Newton transformations are \[T_0 = g, \qquad T_1 = \sigma_1g - K, \qquad T_2 = \sigma_2g - \sigma_1K + K^2.\] By Cayley--Hamilton, \(T_2 = \mathrm{adj}\,K\) in dimension three. In an orthonormal eigenframe \(K = \mathrm{diag}(k_1, k_2, k_3)\) the transformation \(T_1\) is diagonal with entries the pairwise sums \(k_2 + k_3\), \(k_1 + k_3\), \(k_1 + k_2\). Likewise \(T_2\) has the pairwise products. \hypertarget{the-gauss-equation-in-dimension-three}{% \subsection{2.3 The Gauss equation in dimension three}\label{the-gauss-equation-in-dimension-three}} Contracting (2.1) gives \(\mathrm{Ric} = \sigma_1K - K^2\) and \(R = \sigma_1^2 - \lvert K\rvert^2 = 2\sigma_2\), so \[G = \mathrm{Ric} - \tfrac12Rg = -(\sigma_2g - \sigma_1K + K^2) = -T_2.\] Writing \(\mathbb M\) for the endomorphism \(-G\) (indices raised with \(g\)), \[\mathbb M = T_2 = \mathrm{adj}\,K. \tag{2.2}\] In dimension three the Riemann tensor is determined by the Ricci tensor, so (2.2) is equivalent to the full Gauss equation, not merely a contraction of it. \hypertarget{the-adjugate}{% \subsection{2.4 The adjugate}\label{the-adjugate}} For a \(3\times3\) symmetric \(K\): \[\det\mathrm{adj}\,K = (\det K)^2, \qquad \mathrm{adj}\,\mathrm{adj}\,K = (\det K)\,K. \tag{2.3}\] For invertible \(K\) the derivative of the adjugate is \[D\,\mathrm{adj}_K(X) = \det K\,\big[(\mathrm{tr}\,K^{-1}X)\,K^{-1} - K^{-1}XK^{-1}\big]. \tag{2.4}\] In an eigenframe it acts on diagonal \(X\) by the matrix \(\partial(k_jk_l)/\partial k_i\), with determinant \(2k_1k_2k_3\), and on the off-diagonal pair \((ij)\) by \(-k_l\), \(l \neq i, j\). Hence \[\det D\,\mathrm{adj}_K = -2(\det K)^2,\] and \(D\,\mathrm{adj}_K\) is invertible iff \(\det K \neq 0\), whatever the signs of the \(k_i\). In particular \(\mathrm{adj}\) is a local diffeomorphism of symmetric matrices near every invertible \(K\). \hypertarget{the-operator-and-its-domain}{% \subsection{2.5 The operator and its domain}\label{the-operator-and-its-domain}} For \(k \geq 2\) and \(\alpha \in (0,1)\) let \[\mathcal U^{k,\alpha} = \{g \in C^{k,\alpha}\ \text{metrics on}\ \Sigma : \det\mathbb M(g) > 0\ \text{everywhere}\},\] an open subset (on compact \(\Sigma\)). For \(k \geq 3\), (1.2) and (1.3) define a smooth map \[F : \mathcal U^{k,\alpha} \to C^{k-3,\alpha}(\Lambda^2T^*\Sigma\otimes T^*\Sigma).\] It is natural: \(F(\phi^*g) = \phi^*F(g)\) for every diffeomorphism \(\phi\). On non-compact \(\Sigma\) the same holds locally. We write \(\mathcal U\) when the regularity is immaterial. \hypertarget{conventions}{% \subsection{2.6 Conventions}\label{conventions}} \begin{itemize} \tightlist \item Curvature: \([\nabla_a, \nabla_b]V^c = R^c{}_{dab}V^d\) and \(R_{ab} = R^c{}_{acb}\). For a \((0,2)\)-tensor, \([\nabla_a, \nabla_b]h_{cd} = -R^e{}_{cab}h_{ed} - R^e{}_{dab}h_{ce}\). \item The unit round metric \(\gamma\) on \(S^3\) has \(\mathrm{Ric} = 2\gamma\), \(R = 6\), \(G = -\gamma\), \(\mathbb M = \mathrm{Id}\) and \(\tilde K = \gamma\). \item \textbf{One labelling of the \(S^3\) spectra.} Scalar harmonics at level \(n \geq 1\) have \(-\Delta\)-eigenvalue \(n^2 - 1\) and multiplicity \(n^2\). Transverse-traceless (TT) tensor harmonics exist for \(n \geq 3\), with rough-Laplacian eigenvalue \(\mu_n = n^2 - 3\) and multiplicity \(2(n^2 - 4)\), so \(\mu = 6, 13, 22, \ldots\). The first scalar level above the constants, \(n = 2\), is the four first harmonics (eigenvalue 3). \item The spectrum of the unit \(S^2\), which governs the shells of §7, is written \(\Lambda = \ell(\ell+1)\) and never shares a glyph with the above. \item A symmetric 2-tensor is \textbf{TT at \(\xi\)} if \(h\xi = 0\) and \(\mathrm{tr}\,h = 0\). This is the pointwise, symbol-level version, a two-dimensional space for each \(\xi \neq 0\). \end{itemize} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{the-algebraic-reduction}{% \section{3. The algebraic reduction}\label{the-algebraic-reduction}} \textbf{Theorem A.} \emph{Let \((\Sigma^3, g)\) be a Riemannian three-manifold of class \(C^k\), \(k \geq 3\), and \(\mathbb M = -G\).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{(Gate.) If \(g\) is realizable then \(\det\mathbb M = (\det K)^2 \geq 0\) everywhere. For constant curvature \(\kappa\), \(\mathbb M = \kappa\,\mathrm{Id}\) and \(\det\mathbb M = \kappa^3\), so a hyperbolic metric admits no realization on any open set.} \item \emph{(Gauss by construction.) On \(\{\det\mathbb M > 0\}\) the tensor \(\tilde K = \sqrt{\det\mathbb M}\,\mathbb M^{-1}\) satisfies \(\mathrm{adj}\,\tilde K = \mathbb M\), so \((g, \tilde K)\) satisfies the Gauss equation, for every metric.} \item \emph{(Branch.) If \((g, K)\) satisfies the Gauss equation at a point with \(\det K \neq 0\), then \(K = \mathrm{sign}(\det K)\,\tilde K\) there. Consequently, near a point where \(\det\mathbb M > 0\), \(g\) is realizable with nondegenerate second fundamental form iff \(F(g) = 0\) there.} \item \emph{(Global sufficiency.) Let \(\Sigma\) be connected and simply connected with \(\det\mathbb M > 0\) everywhere. Then \(g\) is realizable iff \(F(g) = 0\). The realization is unique up to a rigid motion, and for one choice of normal its second fundamental form is \(\tilde K\). For general connected \(\Sigma\) with \(\det\mathbb M > 0\) everywhere, \(F(g) = 0\) gives an immersion of the universal cover that is equivariant under a homomorphism \(\rho : \pi_1(\Sigma) \to \mathrm{Isom}(E^4)\). Then \(g\) is realizable iff \(\rho\) is trivial.} \item \emph{(Convexity.) \(\mathbb M\) is positive definite iff \(\tilde K\) is definite iff a realization is locally strictly convex. \(\mathbb M\) is never negative definite on \(\{\det\mathbb M > 0\}\). If \(\det\mathbb M > 0\) and \(\mathbb M\) is indefinite, then \(\mathbb M\) has signature \((+,-,-)\), and \(\tilde K\) has exactly one principal value of sign opposite to the other two.} \end{enumerate} \emph{Proof.} \(\mathbb M\) is self-adjoint with respect to \(g\), so it diagonalises in an orthonormal frame. \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \item This is (2.2) together with (2.3). \item Let \(\mathbb M\) have eigenvalues \(m_1, m_2, m_3\), real, nonzero, with \(m_1m_2m_3 > 0\); no sign condition on the individual \(m_i\) is needed. Then \(\tilde K\) has eigenvalues \(\kappa_i = \sqrt{m_1m_2m_3}/m_i\), and \[\kappa_j\kappa_l = \frac{m_1m_2m_3}{m_jm_l} = m_i \qquad (\{i,j,l\} = \{1,2,3\}),\] which is \(\mathrm{adj}\,\tilde K = \mathbb M\) in the common eigenframe. \item If \(\mathbb M = \mathrm{adj}\,K\) with \(K = \mathrm{diag}(k_i)\) and \(\det K \neq 0\), then \(m_i = k_jk_l\), \(\det\mathbb M = (\det K)^2\), and \(\kappa_i = \lvert\det K\rvert/k_jk_l = \lvert\det K\rvert\,k_i/\det K\). So \(\tilde K = \mathrm{sign}(\det K)\,K\). If \(g\) is realized near the point with \(\det K \neq 0\), then \(K\) is continuous with \(\det K \neq 0\), so the sign is locally constant and \(\mathrm{Cod}\,\tilde K = \pm\mathrm{Cod}\,K = 0\). Conversely, \(F(g) = 0\) makes \((g, \tilde K)\) a Gauss--Codazzi pair on a simply connected neighbourhood, which the fundamental theorem realizes. \item If \(F(g) = 0\), then \((g, \tilde K)\) is a Gauss--Codazzi pair on \(\Sigma\), by (2). The fundamental theorem gives an immersion with second fundamental form \(\tilde K\), unique up to rigid motion. Conversely, let \(x\) realize \(g\). Then \((\det K)^2 = \det\mathbb M > 0\), so \(\det K\) never vanishes. By (3), \(K = \varepsilon\tilde K\) with \(\varepsilon = \mathrm{sign}\det K\) continuous, hence constant on connected \(\Sigma\). So \(F(g) = 0\), and reversing the normal if \(\varepsilon = -1\) gives second fundamental form \(\tilde K\). \end{enumerate} For general \(\Sigma\), apply this on the universal cover. Deck transformations are isometries of the lifted metric, so by uniqueness each composes the immersion with a rigid motion, and this defines \(\rho\). An immersion of \(\Sigma\) lifts to one with trivial \(\rho\). Conversely, trivial \(\rho\) means the immersion descends. \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{4} \tightlist \item \(m_i = k_jk_l > 0\) for all \(i\) iff the \(k_i\) share a sign. Definiteness of the second fundamental form is local strict convexity. If \(\mathbb M\) were negative definite, \(\det\mathbb M < 0\). If exactly one eigenvalue of \(K\) has the opposite sign, say \(k_1 < 0 < k_2, k_3\), then \(m_1 = k_2k_3 > 0\) and \(m_2, m_3 < 0\). \(\square\) \end{enumerate} \textbf{Remark 3.1 (relation to Beez--Killing and Thomas).} Part (3) is the Beez--Killing determination in closed form: in dimension three the system \(k_jk_l = m_i\) has at most the two solutions \(\pm K\). Part (2) is a different kind of statement. \(\tilde K\) is defined for every metric in \(\mathcal U\) and satisfies Gauss whether or not the metric is realizable. Rigidity says a solution is unique if it exists; part (2) says a canonical candidate always exists. Part (4) is Thomas's §11 in this notation. His continuity argument for the sign field (§9) is not needed here, because (1.2) involves no choice of sign. \textbf{Remark 3.2 (simple connectivity is needed).} The round metric on \(\mathbb{RP}^3 = S^3/\{\pm1\}\) has \(\mathbb M = \mathrm{Id}\) and \(F = 0\). The developing immersion of its universal cover is the unit sphere, and the deck transformation acts on it by \(\rho = -\mathrm{Id} \in O(4)\), which is not trivial. So the round \(\mathbb{RP}^3\) is not realizable, although it satisfies \(F = 0\) and the gate at every point. \textbf{Example 3.3 (the indefinite branch: the Flamm hypersurface).} On \((2m, \infty)\times S^2\) take \[g = \Big(1 - \frac{2m}\rho\Big)^{-1}d\rho^2 + \rho^2\hat\gamma, \qquad \hat\gamma\ \text{the unit round metric on } S^2.\] The hypersurface of revolution \(\{(\rho\,\omega, z) : \omega \in S^2\} \subset \mathbb R^3\times\mathbb R\) with profile \(z^2 = 8m(\rho - 2m)\) realizes it. Its principal curvatures are \[\kappa_{\rm mer} = -\frac{\sqrt{2m}}{2\rho^{3/2}}, \qquad \kappa_{\rm sph} = \frac{\sqrt{2m}}{\rho^{3/2}}\ \text{(twice)},\] so \(\mathbb M = \mathrm{adj}\,K\) has eigenvalues \((2m/\rho^3, -m/\rho^3, -m/\rho^3)\). Thus \(\det\mathbb M = 2m^3/\rho^9 > 0\) while \(\mathbb M\) is indefinite. \begin{itemize} \tightlist \item \textbf{Completeness.} The sheet \(\rho > 2m\) is not complete: the distance to \(\rho = 2m\) is finite. The complete hypersurface is the two-ended surface of revolution with \(\rho(z) = 2m + z^2/8m\), \(z \in \mathbb R\), diffeomorphic to \(\mathbb R\times S^2\). It carries the metric above on each sheet. On the throat \(z = 0\) it has \(\det\mathbb M = 1/256m^6 > 0\), so \(\det\mathbb M > 0\) everywhere. \item \textbf{What it shows.} It satisfies \(F = 0\) and is not convex. So the convex branch is a proper part of the nondegenerate realizable set, and the gate (1) is necessary but not sufficient. \item \textbf{The TT slice.} Its first Newton tensor is \(T_1 = \mathrm{diag}(2k, k/2, k/2)\) with \(k = \kappa_{\rm sph} > 0\), which is definite. So the TT slice serves along it (§4). \end{itemize} \textbf{Remark 3.4 (the degenerate branch).} Where \(\det\mathbb M = 0\) the equivalence in (3) fails. The Gauss equation with \(\det K = 0\) has large solution families, with flat metrics the extreme case (Hartman and Nirenberg 1959). This is the regime of rank at most two, where Beez--Killing rigidity also fails and the deformable hypersurfaces of Sbrana (1909) and Cartan (1916) live. Section 8 treats it. \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{the-principal-symbol}{% \section{4. The principal symbol}\label{the-principal-symbol}} Throughout this section \(g \in \mathcal U\), \(p \in \Sigma\), and we work in a \(g\)-orthonormal frame at \(p\). We write \(K\) for \(\tilde K(g)(p)\), which is invertible, and \(\sigma_1, \sigma_2\) for its elementary symmetric functions. We use the \emph{formal} symbol, which replaces each \(\partial_a\) by \(\xi_a\). It differs from the usual symbol by a power of \(i\), which changes no rank and no kernel. \hypertarget{the-symbol-and-its-kernel}{% \subsection{4.1 The symbol and its kernel}\label{the-symbol-and-its-kernel}} On \(\mathcal U\) the map \(\mathbb M \mapsto \tilde K\) is a local inverse of \(\mathrm{adj}\), so \(D\tilde K = (D\,\mathrm{adj}_K)^{-1}\circ D\mathbb M\). \(\mathbb M\) is of second order in \(g\) and \(\mathrm{Cod}\) adds one derivative. Every other term of \(DF_g\), including the variation of the connection inside \(\mathrm{Cod}\), has order at most two. Hence the third-order symbol at \(\xi \in T^*_p\Sigma\setminus0\) is \[\sigma(\xi) = \sigma(\mathrm{Cod})(\xi)\circ(D\,\mathrm{adj}_K)^{-1}\circ\sigma_2(D\mathbb M)(\xi), \qquad \sigma(\mathrm{Cod})(\xi)L = \xi\wedge L,\quad (\xi\wedge L)_{abc} = \xi_aL_{bc} - \xi_bL_{ac}. \tag{4.1}\] It maps \(\mathrm{Sym}^2T^*_p\), of dimension 6, to \(\Lambda^2T^*_p\otimes T^*_p\), of dimension 9. Fix \(\lvert\xi\rvert = 1\) and let \(P = g - \xi\otimes\xi\) be the projection onto \(\xi^\perp\). For \(Y\) symmetric and supported on \(\xi^\perp\), write \(\mathrm{adj}_2Y = (\mathrm{tr}\,Y)P - Y\), the adjugate of \(Y\) as a \(2\times2\) matrix on the plane. It is an involution of \(\mathrm{Sym}^2(\xi^\perp)\). Put \[Q_\xi = PKP,\] the compression of \(K\) to \(\xi^\perp\). Every \(h \in \mathrm{Sym}^2T^*_p\) splits uniquely as \(h = PhP + \xi\odot v\), with \(\xi\odot v = \tfrac12(\xi\otimes v + v\otimes\xi)\). \textbf{Lemma 4.1 (the three factors).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \(\sigma_2(D\mathbb M)(\xi)h = -\tfrac12\,\mathrm{adj}_2(PhP)\). It vanishes on \(\{\xi\odot v\}\) and maps \(\mathrm{Sym}^2(\xi^\perp)\) bijectively to itself. \item For every symmetric \(K\), \[D\,\mathrm{adj}_K(X) = KX + XK - \sigma_1X - (\mathrm{tr}\,X)K + (\sigma_1\,\mathrm{tr}\,X - \mathrm{tr}\,KX)\,g,\] and \(D\,\mathrm{adj}_K(\xi\otimes\xi) = \mathrm{adj}_2(Q_\xi)\). \item On symmetric \(L\), \(\xi\wedge L = 0\) iff \(L \in \mathrm{span}\{\xi\otimes\xi\}\). \end{enumerate} \emph{Proof.} (1) The linearised Ricci tensor has symbol \(\tfrac12\big(\xi\otimes h\xi + h\xi\otimes\xi - h - (\mathrm{tr}\,h)\,\xi\otimes\xi\big)\), and the linearised scalar curvature has symbol \(h(\xi,\xi) - \mathrm{tr}\,h\). Substituting \(h = \xi\odot v\) gives zero for both. For \(h = Y\) supported on \(\xi^\perp\) we have \(h\xi = 0\), and the symbol of \(G\) is \(-\tfrac12Y - \tfrac12(\mathrm{tr}\,Y)\,\xi\otimes\xi + \tfrac12(\mathrm{tr}\,Y)\,g = \tfrac12\,\mathrm{adj}_2Y\), and \(\mathbb M = -G\) gives the sign. Bijectivity holds because \(\mathrm{adj}_2\) is an involution. \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{1} \item Differentiate \(\mathrm{adj}\,K = K^2 - \sigma_1K + \sigma_2g\). In a frame with \(\xi = e_1\) and \(K = \begin{pmatrix}k_{11} & k^{\mathsf T}\\ k & Q\end{pmatrix}\), the first row and column of \(D\,\mathrm{adj}_K(e_1\otimes e_1)\) vanish, and the lower block is \((\sigma_1 - k_{11})\mathrm{Id} - Q = \mathrm{adj}_2Q\). \item Contracting \(\xi_aL_{bc} = \xi_bL_{ac}\) with \(\xi^a\) gives \(L = \xi\otimes w\), and symmetry forces \(w \parallel \xi\). \(\square\) \end{enumerate} \textbf{Theorem B (the symbol).} \emph{Let \(g \in \mathcal U\), \(p \in \Sigma\), \(K = \tilde K(g)(p)\) and \(\xi \neq 0\).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{\(\sigma(\xi)\) has rank exactly 2, and} \[\ker\sigma(\xi) = \{\xi\odot v : v \in T^*_p\} \oplus \mathrm{span}\{Q_\xi\},\] \emph{a space of dimension 4.} \item \emph{Let \(T_1 = \sigma_1g - K\) be the first Newton tensor. On \(TT_\xi = \{h : h\xi = 0,\ \mathrm{tr}\,h = 0\}\), \(\sigma(\xi)\) is injective iff \(\xi^{\mathsf T}T_1\xi \neq 0\).} \item \emph{\(Q_\xi\) is a multiple of \(P\) iff \(K\vert_{\xi^\perp}\) is a multiple of the identity. This holds for every \(\xi\) iff \(K \propto g\).} \end{enumerate} \emph{Proof.} (1) By Lemma 4.1(1) the first factor has kernel \(\{\xi\odot v\}\) and image \(\mathrm{Sym}^2(\xi^\perp)\), of dimension 3. The middle factor is invertible because \(\det K \neq 0\) (§2.4). By Lemma 4.1(2) its inverse carries \(\mathrm{adj}_2(Q_\xi) \in \mathrm{Sym}^2(\xi^\perp)\) to \(\xi\otimes\xi\). By Lemma 4.1(3) that is the only direction the last factor kills. So the rank is \(3 - 1 = 2\). The kernel is the set of \(h\) with \(\mathrm{adj}_2(PhP) \in \mathrm{span}\{\mathrm{adj}_2Q_\xi\}\), that is, \(PhP \in \mathrm{span}\{Q_\xi\}\). Finally \(Q_\xi \neq 0\), since \(Q_\xi = 0\) would make \(K = \xi\odot w\), of rank at most 2. \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{1} \item A kernel element is \(h = cQ_\xi + \xi\odot v\). Since \(Q_\xi\xi = 0\), transversality gives \(\tfrac12\big((v\cdot\xi)\xi + v\big) = 0\), so \(v = 0\). Then \(h = cQ_\xi\) is trace-free iff \(c\,\mathrm{tr}\,Q_\xi = 0\), and \(\mathrm{tr}\,Q_\xi = \sigma_1 - K(\xi,\xi) = \xi^{\mathsf T}T_1\xi\). \item If \(K\vert_{\xi^\perp}\) is scalar for every \(\xi\), then for every orthonormal pair \(u, w\) (take \(\xi \perp u, w\)) \(K(u,u) = K(w,w)\) and \(K(u,w) = 0\). \(\square\) \end{enumerate} \emph{Attribution and checks.} The proof of (1) through Lemma 4.1 is the one given by an independent machine-assisted audit (October 2026). The same audit gave the trace form of (2), with the exact counterexample below, and (3). Earlier versions had established (1) only at sample shape operators, and had claimed (2) without the condition. The formula for \(D\,\mathrm{adj}_K\) shows that Lemma 4.1(2) is a polynomial identity, valid for every symmetric \(K\). Lemma 4.1 is checked symbolically; rank and kernel exactly at twelve random rational \(K\), definite and indefinite; the TT statement, in floating point, on random \(K\) and at null covectors of \(T_1\) (\texttt{test\_325} P1--P2, \texttt{test\_326} P1). The kernel is checked again, exactly, in \texttt{test\_332} P3. The formula (4.1) itself is checked against the full nonlinear operator: at a point of a metric with \(g(0) = 1\), the linearisation of \(F\) on \(h = ((\xi\cdot x)^3/6)\,h_0\) is \(\sigma(\xi)h_0\) exactly, on both branches (\texttt{test\_335} P5). \textbf{Corollary 4.2 (the kernel is the immersions'; slices).} \emph{Let \(g \in \mathcal U\), \(K = \tilde K(g)(p)\) and \(\xi \neq 0\).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{\(\ker\sigma(\xi) = \{\xi\odot v\}\oplus\mathrm{span}\{K\}\). This is the symbol at \(\xi\) of the linearised immersions \(h = \mathcal L_Xg - 2\varphi K\), with \(X\) counted one order above \(\varphi\). At a realizable metric these are the actual first-order deformations. So the symbol sequence \((X, \varphi) \mapsto h \mapsto \sigma(\xi)h\) is exact at \(h\), at every \(\xi\) and every nondegenerate \(K\), on both branches. In particular \(F\) is nowhere elliptic modulo diffeomorphisms alone: its symbol kills one direction more, the normal variation.} \item \emph{\(\sigma(\xi)\) is injective on every complement of its kernel. The space \(S_\xi = \{h : h\xi = 0,\ \langle h, Q_\xi\rangle = 0\}\) is one, at every \(\xi\). When \(h\xi = 0\), \(\langle h, K\rangle = \langle h, Q_\xi\rangle\), so \(S_\xi\) is the symbol of the conditions \(\mathrm{div}\,h = 0\), \(\langle h, \tilde K\rangle = 0\). At an umbilic point it is \(TT_\xi\).} \item \emph{\(TT_\xi\) is a complement iff \(\xi^{\mathsf T}T_1\xi \neq 0\) (Theorem B(2)). Call \(g\) \textbf{TT-admissible} at \(p\) if this holds for every \(\xi \neq 0\), that is, if \(T_1\) is definite at \(p\). With principal values \(k_1 \leq k_2 \leq k_3\), the eigenvalues of \(T_1\) are the pairwise sums \(k_i + k_j\), so:} \begin{itemize} \tightlist \item \emph{a TT-characteristic covector exists iff \(k_1 + k_2 \leq 0 \leq k_2 + k_3\);} \item \emph{the convex branch is TT-admissible everywhere;} \item \emph{on the indefinite branch, TT-admissibility holds exactly where \(k_1 + k_2\) and \(k_2 + k_3\) have the same sign.} \end{itemize} \end{enumerate} \emph{Proof.} (1) \(K - Q_\xi = \xi\odot w\) with \(w = 2K\xi - K(\xi,\xi)\,\xi\), so the kernel of Theorem B(1) is \(\{\xi\odot v\} + \mathrm{span}\{K\}\). The sum is direct because \(K\) has rank 3 and \(\xi\odot v\) has rank at most 2. The symbol of \(\mathcal L_Xg\) at \(\xi\) is a multiple of \(\xi\odot\hat X\), and \(-2\varphi K\) is of order zero in \(\varphi\). \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{1} \item A kernel element in \(S_\xi\) is \(h = cQ_\xi + \xi\odot v\) with \(h\xi = 0\), so \(v = 0\) as in the proof of Theorem B(2). Then \(c\lvert Q_\xi\rvert^2 = 0\), and \(Q_\xi \neq 0\). \(S_\xi\) has dimension \(3 - 1 = 2\), the codimension of the kernel. \item is Theorem B(2) and the eigenvalues of \(T_1\). \(\square\) \end{enumerate} The existence criterion in (3) is the audit's. Examples: \begin{itemize} \tightlist \item \(K = \mathrm{diag}(2,2,-1)\) has \(T_1 = \mathrm{diag}(1,1,4)\), so it is TT-admissible. \item The Flamm hypersurface (Example 3.3) is TT-admissible everywhere. \item \(K = \mathrm{diag}(2,3,-3)\) has \(T_1 = \mathrm{diag}(0,-1,5)\), so it is not. At \(\xi = e_1\) the TT tensor \(h = \mathrm{diag}(0,1,-1) = \tfrac13Q_{e_1}\) satisfies \(\sigma(e_1)h = 0\) exactly, while \(\sigma(e_1)\) is injective on \(S_{e_1}\) (\texttt{test\_332} P3). \end{itemize} \textbf{Remark 4.3 (what a TT-characteristic covector is).} \(\xi^{\mathsf T}T_1\xi = \mathrm{tr}(K\vert_{\xi^\perp})\). So \(\xi\) is TT-characteristic iff the second fundamental form restricted to the plane \(\xi^\perp\) is trace-free: the normal curvatures in any two orthonormal directions of that plane sum to zero. When \(T_1\) is positive definite, the hypersurface is \emph{2-convex} in the sense of Huisken and Sinestrari (2009). So TT-admissibility is 2-convexity for one of the two orientations. It is a property of the slice, not of \(F\): at a TT-characteristic covector the kernel direction \(Q_\xi\) happens to be trace-free. \textbf{Remark 4.4 (the York reading).} At each \(\xi\), \(\mathrm{Sym}^2T^*_p = TT_\xi\oplus\{\xi\odot v\}\oplus\mathrm{span}\{P\}\), with dimensions \(2 + 3 + 1\): York's decomposition (York 1973) at the level of symbols. The kernel of \(\sigma(\xi)\) is the gauge part plus \(Q_\xi\). \begin{itemize} \tightlist \item At an umbilic point \(Q_\xi\) is a multiple of \(P\). The kernel is then gauge plus conformal, and the image is \(\sigma(TT_\xi)\). This is the situation at the round sphere (§6). \item Off umbilics the fourth kernel direction turns with \(K\), and \(S_\xi\) turns with it. \(TT_\xi\) remains a complement exactly when \(\xi\) is not TT-characteristic. \end{itemize} \textbf{Remark 4.5 (what the slices are used for).} Meagreness (§5) uses only that \(\sigma \neq 0\), and holds on both branches whatever \(T_1\) does. The round-sphere analysis (§6) works on TT, which serves there. Neither \(TT_\xi\) nor \(S_\xi\) is a subbundle as \(\xi\) varies, so neither gives an estimate by itself; §6 repairs this at \(\gamma\) by augmenting the operator. Off umbilics the natural gauge is \(\mathrm{div}\,h = 0\), \(\langle h, \tilde K\rangle = 0\), whose symbol is \(S_\xi\) at every covector (§13, problem 6). \hypertarget{four-identities-five-conditions-and-cartans-test}{% \subsection{4.2 Four identities, five conditions, and Cartan's test}\label{four-identities-five-conditions-and-cartans-test}} \textbf{Lemma 4.6 (identities of the Codazzi tensor).} \emph{Let \(K\) be a symmetric \(C^2\) tensor on a Riemannian three-manifold and \(C = \mathrm{Cod}\,K\), \(C_{abc} = \nabla_aK_{bc} - \nabla_bK_{ac}\).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{\(C_{abc} + C_{bca} + C_{cab} = 0\).} \item \emph{\(\nabla^a(\mathrm{adj}\,K)_{ab} = B_b(C)\), where \(B_b(C) := K^c{}_b\,C^a{}_{ca} - \sigma_1\,C^a{}_{ba} + K^{ac}C_{abc}\).} \item \emph{\(\sum_{\mathrm{cyc}(d,a,b)}\nabla_dC_{abc} = -\big(R^e{}_{cda}K_{be} + R^e{}_{cab}K_{de} + R^e{}_{cbd}K_{ae}\big)\). The right-hand side vanishes when \(R_{ecda} = K_{ed}K_{ca} - K_{ea}K_{cd}\).} \end{enumerate} \emph{Proof.} (1) The six terms cancel in pairs by the symmetry of \(K\). \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{1} \item Write \(K_{ab;c} = \nabla_cK_{ab}\). From \(\mathrm{adj}\,K = K^2 - \sigma_1K + \sigma_2g\) and \(\nabla_b\sigma_2 = \sigma_1\sigma_{1;b} - K^{cd}K_{cd;b}\), \[\nabla^a(\mathrm{adj}\,K)_{ab} = \sigma_1(\sigma_{1;b} - K^a{}_{b;a}) + K^c{}_b(K^a{}_{c;a} - \sigma_{1;c}) + K^{ac}(K_{cb;a} - K_{ac;b}).\] The three brackets are \(-C^a{}_{ba}\), \(C^a{}_{ca}\) and \(C_{abc}\). \item The cyclic sum is \([\nabla_d, \nabla_a]K_{bc} + [\nabla_a, \nabla_b]K_{dc} + [\nabla_b, \nabla_d]K_{ac}\). The Ricci identity expands each commutator into two curvature terms, and the terms of the form \(R^e{}_{bda}K_{ec}\) cancel by the first Bianchi identity. With Gauss's curvature, the remaining sum is \[(K^2)_{bd}K_{ca} - (K^2)_{ba}K_{cd} + (K^2)_{da}K_{cb} - (K^2)_{db}K_{ca} + (K^2)_{ab}K_{cd} - (K^2)_{ad}K_{cb} = 0. \qquad\square\] \end{enumerate} Part (2) is the defect form of the classical fact that the Newton tensors of a Codazzi tensor are divergence-free (Reilly 1973). \textbf{Theorem C (the count, and involutivity).} \emph{Let \(g \in \mathcal U\), \(p \in \Sigma\) and \(K = \tilde K(g)(p)\).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{(Identities.) \(F(g)\) satisfies, identically in \(g\):} \begin{itemize} \tightlist \item \emph{the cyclic identity;} \item \emph{\(B(F(g)) = 0\), with \(B\) formed from \(\tilde K\), since \(\nabla^a(\mathrm{adj}\,\tilde K)_{ab} = \nabla^a\mathbb M_{ab} = -\nabla^aG_{ab} = 0\);} \item \emph{the first-order identity \(\sum_{\mathrm{cyc}}\nabla_dF_{abc} = 0\), since \((g, \tilde K)\) satisfies Gauss.} \end{itemize} \item \emph{(Five conditions.) At every point, the four algebraic identities are independent, and they span the left null space of the order-3 symbol map \(\tau_3 : S^3T^*_p\otimes\mathrm{Sym}^2 \to \Lambda^2\otimes T^*\), which has rank 5. So \(F = 0\) imposes exactly five conditions per point. That is two more than the three a metric has modulo diffeomorphisms.} \end{enumerate} \emph{Let \(\mathcal X\) be the set of nondegenerate symmetric \(K\) at which any of the dimensions in (3) differs from the values stated. \(\mathcal X\) is a proper real-algebraic subset, invariant under rotation and scaling. It contains none of the sixteen shape operators examined, among them the umbilic one, which is the round sphere's, and the TT-characteristic \(\mathrm{diag}(2,3,-3)\). At every point where \(\tilde K(g) \notin \mathcal X\):} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \setcounter{enumi}{2} \tightlist \item \emph{(Cartan's test.) Let \(g_q = \ker\tau_q\), and let \(g_{q,k}\) be its elements independent of the first \(k\) coordinates of a generic frame. Then:} \begin{itemize} \tightlist \item \emph{\(\dim g_3 = 55\), \(\dim g_{3,1} = 19\), \(\dim g_{3,2} = 4\);} \item \emph{the prolonged symbol \(\tau_4\) has rank 12;} \item \emph{\(\dim g_4 = 78 = 55 + 19 + 4\).} \end{itemize} \emph{Cartan's inequality \(\dim g_4 \leq \dim g_3 + \dim g_{3,1} + \dim g_{3,2}\) therefore holds with equality, and \(g_3\) is involutive. The left null space of \(\tau_4\) has dimension 15. It is spanned by the twelve prolongations of the algebraic identities and the three functionals \(\xi\wedge C\), which are the symbol of the first-order identity in (1).} \item \emph{(Involutivity.) By (1), every relation among the equations at the first prolongation is an exact identity, so the prolongation adds no integrability condition. With (3), the system \(F = 0\) is involutive at \(p\) (Goldschmidt 1967; see Bryant et al.~1991 and Seiler 2010 for Cartan's test). Its last Cartan character is \(\dim g_{3,2} = 4\). For real-analytic data, its general local solution therefore depends on four functions of three variables (Cartan--Kähler).} \end{enumerate} \emph{Proof of (2).} By Lemma 4.6 the four functionals annihilate the image of \(\tau_3\), so its rank is at most 5 once they are independent. In a principal frame of \(K\) their \(4\times4\) minors include \(\pm k_1k_2k_3\), so they are independent wherever \(\det K \neq 0\). The image of \(\tau_3\) contains the images of \(\sigma(e_1)\), \(\sigma(e_2)\) and \(\sigma(e_3)\), and a \(5\times5\) minor of these is a nonzero rational multiple of a monomial in \(k_1, k_2, k_3\) (\texttt{test\_332} P1). So the rank is exactly 5 at every nondegenerate \(K\). \(\square\) \emph{Scope of (3)--(4), and why the numbers are what they are.} Regard \(g_q\) as homogeneous polynomial tensors of degree \(q\) in \(x\), annihilated by the constant-coefficient operator \(\sigma(\partial)\). By Theorem B, \(\sigma(\xi)\) kills \(\xi\odot v\) and \(K\) itself for every \(\xi\) (\(K - Q_\xi\) is of the form \(\xi\odot w\)). Hence \(g_q\) contains \[M_K^q = \{\partial\odot\omega + \varphi K : \omega\ \text{a homogeneous 1-form of degree } q+1,\ \varphi\ \text{homogeneous of degree } q\}.\] These are the symbols of the linearised immersions \(\delta x = X + \varphi N\), which change the metric by \(h = \mathcal L_Xg - 2\varphi K\). The map \((\omega, \varphi) \mapsto \partial\odot\omega + \varphi K\) is injective for \(q \geq 2\). Suppose \(\partial\odot\omega = -\varphi K\). The linearised curvature of the left side vanishes, so, at the level of constant-coefficient symbols, the Kulkarni--Nomizu product \(\partial^2\varphi \mathbin{\bar\wedge} K\) vanishes. In dimension three, \(A \mapsto A \mathbin{\bar\wedge} K\) has determinant \(2(\det K)^2\) on \(\mathrm{Sym}^2\) (\texttt{test\_326} P4), so \(\partial^2\varphi = 0\), and \(\varphi = 0\) by homogeneity. Then \(\omega\) is a Killing form of degree at least 3, hence zero. So for every nondegenerate \(K\), \[\dim g_q \geq \dim M_K^q = 3\dim S^{q+1} + \dim S^q = (q+2)(2q+5),\] which is \(55, 78, 105, 136, \ldots\) for \(q = 3, 4, 5, 6, \ldots\). The dimensions in (3), and with them \(g_q = M_K^q\), are verified at sixteen shape operators, each in an exact rational frame: \begin{itemize} \tightlist \item several generic ones, on both branches; \item the umbilic one, which is the round sphere's; \item two equal eigenvalues, on both branches; \item the TT-characteristic \(\mathrm{diag}(2,3,-3)\); \item the boundary case \(k_1 + k_2 = 0\); \item a near-degenerate one. \end{itemize} The ranks are computed modulo the prime \(2^{31} - 1\), and that gives the dimensions over \(\mathbb Q\) exactly. A rank modulo \(p\) is at most the rank over \(\mathbb Q\). The lower bound \(\dim g_q \geq \dim M_K^q\) bounds the rank over \(\mathbb Q\) from above. Cartan's inequality \(\dim g_4 \leq \dim g_3 + \dim g_{3,1} + \dim g_{3,2}\) holds in every frame, so the sum over \(\mathbb Q\) is squeezed between 78 and the sum modulo \(p\), which is 78. Equality holds for \(q \leq 5\) at all sixteen (\texttt{test\_332} P2), and, in floating point, for \(q \leq 9\) at two of them (\texttt{test\_325} P3--P5, \texttt{test\_326} P2--P3). \emph{Why \(\mathcal X\) is real-algebraic.} For every nondegenerate \(K\), each dimension in (3) is at least its stated value, so each rank is at most its stated value. For \(g_3\) and \(g_4\) this is the lower bound on \(\dim g_q\). For the flag restrictions it is the same bound on the flag: an element \(\partial\odot\omega + \varphi K\) with \(\omega\) and \(\varphi\) independent of the first \(k\) coordinates is independent of them too, and the map is injective, so in every frame \[\dim g_{3,1} \geq 3\dim S^4(\mathbb R^2) + \dim S^3(\mathbb R^2) = 19, \qquad \dim g_{3,2} \geq 3\dim S^4(\mathbb R) + \dim S^3(\mathbb R) = 4\] (\texttt{test\_344} P4). So \(\mathcal X\) is where some rank falls below its stated value (for the flag restrictions, in every frame): the common zero set of the coefficients, as polynomials in the frame, of the relevant minors. It is closed, and proper because it misses the sixteen. We do not have a proof that \(\mathcal X\) is empty (§13, problem 1). \emph{The hypotheses of (4).} Goldschmidt's criterion needs locally constant ranks, a surjective first prolongation and an involutive symbol. At a point with \(\tilde K(g)(p) \notin \mathcal X\): \begin{itemize} \tightlist \item The ranks of \(\tau_3\), \(\tau_4\) and the flag restrictions are locally constant, because \(\mathcal X\) is closed and \(\tilde K(g)(p)\) depends continuously on the 2-jet of \(g\). So \(g_3\), \(g_4\) and the \(g_{3,k}\) are vector bundles near \(j^3_pg\). Since \(\tau_3\) has rank 5 on all of \(\mathcal U\), by (2), \(\mathcal R_3 = \{j^3g : F(g) = 0\}\) is a submanifold of codimension 5 there. \item The fifteen functionals of (3) span the left null space of \(\tau_4\). \item Each is the symbol of an identity that \(F(g)\) satisfies for \textbf{every} metric on \(\mathcal U\), not only on solutions: the four algebraic identities of Lemma 4.6, their twelve first derivatives, and the cyclic identity \(\sum_{\mathrm{cyc}(d,a,b)}\nabla_dF_{abc} = 0\). The last holds because \(\tilde K\) satisfies the Gauss equation identically (Theorem A(2)), so the curvature term of Lemma 4.6(3) vanishes. The identities are linear in \((F, \nabla F)\), with coefficients in the jet of \(g\). On \(\mathcal R_3\), where \(F = 0\), the functionals therefore annihilate the lower-order part of each prolonged equation. So \(\mathcal R_4 \to \mathcal R_3\) is onto, and the curvature map of the criterion vanishes. \end{itemize} With the involutivity of \(g_3\) from (3), Goldschmidt's theorem gives (4). At every \(K\) tested, then, the formal solutions of the symbol are exactly the linearised immersions, and the count of Theorem C is the count of immersions: three functions for the tangential part and one for the normal displacement. The joint kernel of \(\sigma(\xi)\) over all \(\xi\) is \(\mathrm{span}\{K\}\), the normal direction \(\delta g = -2\varphi K\) (\texttt{test\_325} P5). The identities of Lemma 4.6 are checked exactly at a curved point (\texttt{test\_325} P6). \textbf{Remark 4.7 (earlier counts).} An earlier version reported \(\mathrm{rank}\,\tau_4 = 15\) and Cartan characters \((2,2,1)\). The first came from a multiplicity factor in the prolongation that jet coordinates do not carry. The second stacked single covectors instead of restricting to a flag. Both numbers are wrong: the characters are \((36, 15, 4)\). The conclusion drawn from them, that \(F = 0\) is involutive, is right off \(\mathcal X\), and is proved there by (3)--(4) (Appendix C). \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{meagreness}{% \section{5. Meagreness}\label{meagreness}} \textbf{Theorem D.} \emph{Let \(\Sigma\) be closed, \(k \geq 3\), \(\alpha \in (0,1)\), and \(\mathcal Z = \{g \in \mathcal U^{k,\alpha} : F(g) = 0\}\). Then \(\mathcal Z\) is closed in \(\mathcal U^{k,\alpha}\) and has empty interior. Hence it is nowhere dense, and meagre, in \(\mathcal U^{k,\alpha}\) and in the space of all \(C^{k,\alpha}\) metrics. This holds on both branches.} \emph{Proof.} \(F\) is continuous, so \(\mathcal Z\) is closed in \(\mathcal U^{k,\alpha}\). \emph{\(DF_g \neq 0\) at every \(g \in \mathcal U^{k,\alpha}\).} Fix \(p\) and \(\xi\) with \(\sigma(\xi) \neq 0\) (Theorem B), and \(h_0 \in \mathrm{Sym}^2T^*_p\) with \(\sigma(\xi)h_0 \neq 0\). Choose a cutoff \(\chi\) near \(p\) and a function \(f\) with \(df(p) = \xi\), and set \(h_\lambda = \chi\cos(\lambda f)\,h_0\), with \(h_0\) extended smoothly. The terms of \(DF_g(h_\lambda)\) cubic in \(\lambda\) come only from the principal part, and equal \(\lambda^3\chi\sin(\lambda f)\,\sigma(df)h_0\). All other terms are \(O(\lambda^2)\). So \(DF_g(h_\lambda) \neq 0\) for large \(\lambda\). If \(\mathcal Z\) contained an open set \(B\), then \(F \equiv 0\) on \(B\) would give \(DF_g = 0\) for \(g \in B\), a contradiction. \(\mathcal Z\) is closed in \(\mathcal U^{k,\alpha}\) with empty interior there. \(\mathcal U^{k,\alpha}\) is open in the space of all \(C^{k,\alpha}\) metrics, since \(\Sigma\) is compact and \(\det\mathbb M\) depends continuously on \(g\) in \(C^2\). Let \(\overline{\mathcal Z}\) be the closure of \(\mathcal Z\) in the full space, and suppose that a nonempty open set \(V\) lies in \(\overline{\mathcal Z}\). Then \(V\) meets \(\mathcal Z \subset \mathcal U^{k,\alpha}\), so \(V\cap\mathcal U^{k,\alpha}\) is a nonempty open subset of \(\mathcal U^{k,\alpha}\) contained in \(\overline{\mathcal Z}\cap\mathcal U^{k,\alpha} = \mathcal Z\), a contradiction. So \(\overline{\mathcal Z}\) has empty interior. \(\square\) \emph{Attribution.} This proof is the audit's, with the last step as a second independent read gave it. It replaces two earlier arguments. The first was local to the round sphere and used a lower bound for \(DF\) on TT tensors. The second appealed to a cokernel, which is not a proof: the zero map has a cokernel and vanishes everywhere. The audit also gave a one-direction proof at \(\gamma\) that needs only \(DF_\gamma h \neq 0\) for one \(h\). The step the proof uses, that the third-order part of \(DF_g\) is the symbol (4.1), is checked against the full nonlinear operator on both branches (\texttt{test\_335} P5). \textbf{Remarks 5.1.} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{Category, not measure.} The statement is about Baire category in a Banach space. There is no canonical measure on metrics, and none is used. \item \textbf{Nothing outside \(\mathcal U\).} Metrics with \(\det\mathbb M = 0\) somewhere, realizable or not, are not covered. Section 8 treats that locus. \item \textbf{No ellipticity used.} Theorem D holds wherever \(\sigma \neq 0\), which is all of \(\mathcal U\), on the TT-characteristic set included. \item \textbf{Meagre is compatible with connected.} A line in the plane is nowhere dense and connected. The realizable metrics on the convex branch of \(S^3\) form a path-connected, contractible set (the companion paper). Theorem C says how large the realizable set is locally, off \(\mathcal X\): the immersions, four functions of three variables, against six for all metrics. \end{enumerate} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{the-round-sphere}{% \section{6. The round sphere}\label{the-round-sphere}} Let \(\gamma\) be the unit round metric on \(S^3\). Then \(\mathbb M(\gamma) = \mathrm{Id}\), \(\tilde K(\gamma) = \gamma\) (the unit sphere with outward normal), and \(T_1 = 2\gamma\) is definite, so the TT slice serves at every point (Corollary 4.2). We use York's \(L^2\)-orthogonal decomposition (York 1973) \[\mathrm{Sym}^2T^*S^3 = TT \oplus \{\mathcal L_X\gamma - \tfrac23(\mathrm{div}\,X)\gamma\} \oplus C^\infty(S^3)\,\gamma, \tag{6.1}\] and the TT spectrum of §2.6: at level \(n \geq 3\), \(-\Delta = n^2 - 3\) with multiplicity \(2(n^2 - 4)\). On \(S^3\) the rough Laplacian preserves TT. \textbf{Theorem E (the linearisation at the round sphere).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{(Kernel.) \(DF_\gamma(2u\gamma) = 0\) for every function \(u\), and \(DF_\gamma(\mathcal L_X\gamma) = 0\) for every vector field \(X\). The kernel is exactly} \[\ker DF_\gamma = \{2u\gamma\} + \{\mathcal L_X\gamma\}.\] \emph{The sum is not direct. The intersection is \(\{2u\gamma : u\) a first harmonic\(\}\), and for a first harmonic \(f\), \(\mathcal L_{\nabla f}\gamma = -2f\gamma\).} \item \emph{(Factorisation.) On TT tensors, \(DF_\gamma = \mathrm{Cod}\circ\tfrac12(-\Delta + 2)\).} \item \emph{(No TT Codazzi tensors.) If \(h\) is TT and \(\mathrm{Cod}\,h = 0\), then \(h = 0\).} \item \emph{(The \(L^2\) identity.) For TT \(h\), \(\lVert\mathrm{Cod}\,h\rVert^2 = 2\lVert\nabla h\rVert^2 + 6\lVert h\rVert^2\). Hence} \[\lVert DF_\gamma h\rVert^2 = \tfrac12\big\langle h,\ (-\Delta + 3)(-\Delta + 2)^2h\big\rangle.\] \emph{At level \(n\) the factor is \(\tfrac12\big(n(n^2 - 1)\big)^2\). So \(\lVert DF_\gamma h\rVert^2 \geq 288\,\lVert h\rVert^2\) on TT, with equality exactly on level 3.} \item \emph{(Closed range.) For \(k \geq 3\) and \(\alpha \in (0,1)\), \(DF_\gamma\vert_{TT} : C^{k,\alpha} \to C^{k-3,\alpha}\) is bounded below, with closed, complemented range.} \end{enumerate} \emph{Proof.} (1) The radial graphs \(x = (1 + u)\hat n\) have induced metric \((1 + u)^2\gamma + du\otimes du\) exactly, because \(\hat n\cdot d\hat n = 0\). So \(F\) vanishes along them, and differentiating at \(u = 0\) gives \(DF_\gamma(2u\gamma) = 0\). Naturality, \(F(\phi^*g) = \phi^*F(g)\), gives the Lie statement. For the intersection, suppose \(2u\gamma = \mathcal L_X\gamma\). Then \(X\) is a conformal Killing field. On \(S^3\) these are the Killing fields plus the gradients of first harmonics, so \(u = 0\) or \(u = -f\) with \(f\) a first harmonic. Obata's identity \(\nabla^2f = -f\gamma\) gives \(\mathcal L_{\nabla f}\gamma = 2\nabla^2f = -2f\gamma\). That the kernel is no larger follows from (6.1), (2) and (3). \(DF_\gamma\) kills the last two summands of (6.1) by what was just shown, and is injective on TT. \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{1} \tightlist \item Three ingredients, for TT \(h\). \end{enumerate} \begin{itemize} \tightlist \item \emph{The commutator.} On the unit sphere \(R^e{}_{bda} = \delta^e{}_d\gamma_{ba} - \delta^e{}_a\gamma_{bd}\), so for trace-free \(h\), \(\gamma^{cd}[\nabla_d, \nabla_a]h_{bc} = 3h_{ab}\). With \(\nabla^ch_{bc} = 0\) this gives \(\nabla^c\nabla_ah_{bc} = 3h_{ab}\). \item \emph{The curvature variation.} As \((0,2)\) tensors, \(\delta\mathrm{Ric} = \tfrac12\Delta_Lh = 3h - \tfrac12\Delta h\), since the Lichnerowicz Laplacian on TT tensors at the unit \(S^3\) is \(\Delta_L = -\Delta + 6\); and \(\delta R = 0\). So \(\delta G = \delta\mathrm{Ric} - \tfrac12(\delta R)\gamma - \tfrac12R(\gamma)h = 3h - \tfrac12\Delta h - 3h = -\tfrac12\Delta h\). For the endomorphism \(\mathbb M^a{}_b = -g^{ac}G_{cb}\), with \(G(\gamma) = -\gamma\) and \(\delta g^{ac} = -h^{ac}\), this gives \(\delta\mathbb M = -h + \tfrac12\Delta h\), which is trace-free. At \(\mathbb M = \mathrm{Id}\) this gives \(\delta\tilde K = h - \tfrac12\Delta h\) as an endomorphism. Lowering the index with \(g = \gamma + h\) adds one more \(h\), so the \((0,2)\) variation is \(\delta K = \tfrac12(-\Delta + 4)h\). \item \emph{The connection.} At the umbilic background the variation of the Christoffel symbols inside \(\mathrm{Cod}\) contributes \(-\mathrm{Cod}\,h\): since \(\mathrm{Cod}_g\,g = 0\) for every \(g\), the variation of \(\mathrm{Cod}_g\) applied to \(\tilde K_0 = \gamma\) is \(-\mathrm{Cod}\,h\). \end{itemize} So \(DF_\gamma h = \mathrm{Cod}\,\tfrac12(-\Delta + 4)h - \mathrm{Cod}\,h = \mathrm{Cod}\,\tfrac12(-\Delta + 2)h\). \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{2} \tightlist \item \emph{First route.} By Ferus (1981) a Codazzi tensor on the round \(S^3\) is locally \(\nabla^2f + f\gamma\). It is trace-free iff \(-\Delta f = 3f\), and then Obata gives \(\nabla^2f = -f\gamma\), so the tensor vanishes. \end{enumerate} \emph{Second route.} Applying \(\nabla^a\) to \(\nabla_ah_{bc} = \nabla_bh_{ac}\) and using the commutator gives \(\Delta h = 3h\). But \(-\Delta \geq 0\) on TT. \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{3} \item Expanding, \(\lVert\mathrm{Cod}\,h\rVert^2 = 2\lVert\nabla h\rVert^2 - 2\int\nabla_ah_{bc}\nabla^bh^{ac}\). Integrate by parts and use \(\nabla^a\nabla_bh_{ac} = 3h_{bc}\) to get \(\int\nabla_ah_{bc}\nabla^bh^{ac} = -3\lVert h\rVert^2\). On TT, (2) and this give \[\lVert DF_\gamma h\rVert^2 = 2\lVert\nabla k\rVert^2 + 6\lVert k\rVert^2 = 2\langle k, (-\Delta + 3)k\rangle, \qquad k = \tfrac12(-\Delta + 2)h,\] and \(-\Delta\) commutes with itself. \item TT is not a subbundle, so the semi-Fredholm theorem for operators with injective symbol does not apply as it stands. Two standard repairs close it. \end{enumerate} \begin{itemize} \tightlist \item \emph{Augment.} \(\mathcal A = (\mathrm{Cod}, \mathrm{div}, d\circ\mathrm{tr})\) is a first-order operator on the full symmetric bundle with injective symbol: the trace symbol removes \(\mathrm{tr}\,H\), the divergence symbol removes \(\xi^aH_{ab}\), and on what is left \(\lVert\xi\wedge H\rVert^2 = 2\lvert\xi\rvert^2\lVert H\rVert^2\). Its kernel is \(\mathbb R\gamma\), by (3). The elliptic estimate for \(\mathcal A\) and compactness give a lower bound for \(\mathrm{Cod} : C^{k-2,\alpha}_{TT} \to C^{k-3,\alpha}\). Since the rough Laplacian preserves TT and \(-\Delta + 2 \geq 8\) there, \(\tfrac12(-\Delta + 2) : C^{k,\alpha}_{TT} \to C^{k-2,\alpha}_{TT}\) is an isomorphism, and (2) gives the lower bound for \(DF_\gamma\vert_{TT}\). \item \emph{Project.} \(\Pi_{TT}\) is a pseudodifferential operator of order zero, bounded on \(C^{k,\alpha}\) for \(\alpha \in (0,1)\), and \((DF_\gamma\vert_{TT})^*DF_\gamma\vert_{TT}\) is elliptic in that calculus. \end{itemize} Complementation is by \(P = DF_\gamma\,(\Pi_{TT}DF_\gamma^*DF_\gamma\Pi_{TT})^{-1}\,\Pi_{TT}DF_\gamma^*\), the inverse taken on TT. By (4), \(\Pi_{TT}DF_\gamma^*DF_\gamma\Pi_{TT} = \tfrac12(-\Delta + 3)(-\Delta + 2)^2\) on TT, so the inverse is a function of the rough Laplacian, of order \(-6\). The factor after \(DF_\gamma\) is then a pseudodifferential operator of order \(-3\), bounded from \(C^{k-3,\alpha}\) to \(C^{k,\alpha}_{TT}\), and \(P\) is a bounded projection of \(C^{k-3,\alpha}\) onto the range. \(\square\) \emph{Checks.} (1)--(3) and (5) are the content of mathematics paper 1 (Thrussell 2026a), Theorem B. Part (2) is checked as an operator identity, not only on eigenmodes. On random tensors that are transverse-traceless to the order that matters at a point, the linearisation of the full nonlinear operator equals \(\mathrm{Cod}\circ\tfrac12(-\Delta + 2)\), exactly; a tensor whose trace has nonzero gradient fails it (\texttt{test\_336} P1). Part (4) is the audit's (P1-X01). It is checked pointwise at level 3, on the constant trace-free tensors of a left-invariant frame of \(S^3 = SU(2)\), where \(\lvert\nabla h\rvert^2 = 6\lvert h\rvert^2\) and \(\lvert\mathrm{Cod}\,h\rvert^2 = 18\lvert h\rvert^2\) (\texttt{test\_336} P2). The augmented symbol of (5) is \texttt{test\_336} P3. The kernel statements of (1) are checked with the full nonlinear operator, exactly, at a point (\texttt{test\_327} P1--P2). \textbf{Remark 6.1 (the multipliers).} The singular values of \(DF_\gamma\) on TT at level \(n\) are \(n(n^2 - 1)/\sqrt2\). The numbers \(n(n^2 - 1)\) are the eigenvalues, level by level, of Branson's third-order conformally covariant operator on functions on \(S^3\), \(P_3 = \sqrt{-\Delta + 1}\,(-\Delta)\) (Branson 1995; Beckner 1993). We record the coincidence of multipliers. We do not claim that \(DF^*_\gamma DF_\gamma\) on TT is an intertwining operator. The \(\zeta\)-function \(\sum_{n \geq 3}2(n^2 - 4)\lambda_n^{-s}\) continues to \(s = 0\) with the value \[2\big[\zeta_H(-2,3) - 4\zeta_H(0,3)\big] = 10\] for \(\lambda_n = n(n^2 - 1)\), for \(\lambda_n = \tfrac12n^2(n^2 - 1)^2\), and indeed for any \(\lambda_n = c\,n^p(1 + a_1n^{-2} + a_2n^{-4} + \cdots)\), a convergent expansion in even powers of \(1/n\). The binomial terms of the continuation would need a pole at \(2k - 2 = 1\) or \(2k = 1\), and there is none. Odd powers do contribute: \(\lambda_n = n^3(1 + a/n)\) gives \(\zeta(0) = 10 + 8a/3 - 2a^3/9\) (\texttt{test\_333}; the two multipliers above are \texttt{test\_336} P4). Ten is the dimension of the conformal algebra \(\mathfrak{so}(4,1)\) of \(S^3\). A heuristic, which we do not make precise: the TT tensors are the trace-free tensors less the image of the conformal Killing operator. That operator's kernel, the conformal Killing fields, has no image and returns with a positive sign. On a closed odd-dimensional manifold the full bundles count zero. Over all \(n \geq 1\) the same multiplicity counts 4, and the \(n = 1\) term \(2(1 - 4) = -6\) makes up the difference. The scalar levels \(n \geq 3\) count \(\zeta(-2) - 1 - 4 = -5\), the constants and first harmonics removed. The determinant is \(\ln\det_\zeta\big(n(n^2 - 1)\big)\vert_{TT} = -14.0371968\ldots\), computed by differentiating the continuation and, independently, from a closed form in Hurwitz zeta values (\texttt{test\_336} P4). \textbf{Proposition 6.2 (conformal directions modulo Lie, and the detector).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{Translating the unit sphere by \(\varepsilon e\) gives, as a radial graph, \(u = \varepsilon\langle e, \hat n\rangle + O(\varepsilon^2)\), a first harmonic. So the four ambient translations are conformal directions that lie in the Lie orbit, and they make the sum in Theorem E(1) non-direct.} \item \emph{On conformal directions, \(\delta R(2u\gamma) = -4(\Delta + 3)u\). Because \(R(\gamma)\) is constant, \(\delta R\) vanishes on the Lie orbit. For \(u \in C^\infty(S^3)\), \(2u\gamma\) lies in the Lie orbit iff \(u\) lies in the span of the first harmonics iff \(\delta R(2u\gamma) = 0\). So \(\delta R\) detects conformal directions modulo the Lie orbit.} \item \emph{(The shell dipole is local.) In hyperspherical coordinates \(\gamma = dr^2 + \sin^2r\,\hat\gamma\), on the punctured region \(0 < r < \pi\), the tensor \(h = 2\lambda\cos\theta\,dr^2\) equals \(2w\gamma + \mathcal L_\eta\gamma\) with} \[w = (\lambda + c\sin r)\cos\theta,\] \emph{for any constant \(c\). The \(c\sin r\) member is a translation. On the \(\lambda\) member,} \[\delta R = 4\lambda(2\cot^2r - 1)\cos\theta,\] \emph{which grows like \(8\lambda\cos\theta/r^2\) at the poles. So \(h\) is a local deformation on punctured shells, annihilated by \(DF_\gamma\) there and not in the Lie orbit. It does not extend to a smooth tensor on \(S^3\).} \end{enumerate} \emph{Proof.} (1) \(\lvert\hat n + \varepsilon e\rvert = 1 + \varepsilon\langle e, \hat n\rangle + O(\varepsilon^2)\), and \(\langle e, \hat n\rangle\) is a first harmonic. \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \setcounter{enumi}{1} \item For \(h = 2u\gamma\), \(\delta R = -\Delta\,\mathrm{tr}\,h + \nabla^a\nabla^bh_{ab} - h^{ab}R_{ab} = -6\Delta u + 2\Delta u - 12u\). Invariance: \(\delta_XR = \mathcal L_XR = 0\). The equivalence follows from the proof of Theorem E(1). \item Write the axisymmetric \(\ell = 1\) generator as the 1-form \(\eta = q(r)\cos\theta\,dr + k(r)\sin\theta\,d\theta\), with \(w = s(r)\cos\theta\), and match components of \(h = 2w\gamma + \mathcal L_\eta\gamma\), where \((\mathcal L_\eta\gamma)_{ab} = \nabla_a\eta_b + \nabla_b\eta_a\). The \(rr\) component gives \(q' = \lambda - s\). The \(\theta\theta\) component, with which the \(\varphi\varphi\) one agrees, gives \(k = -s\sin^2r - q\sin r\cos r\). Substituting both into the \(r\theta\) component, \(k' - q - 2k\cot r = 0\), the \(q\) terms cancel, leaving \(-s'\sin r + (s - \lambda)\cos r = 0\). Finally \(-\Delta\cos\theta = 2\cos\theta/\sin^2r\). \(\square\) \end{enumerate} This corrects the earlier statement that the \(\ell = 1\) kernel splits invariantly as a direct sum of translations, irreducible moduli and Lie directions. The three overlap, and the shell dipoles are local. Theorem E's global kernel is unaffected. Parts (2) and (3) are checked with the full nonlinear operator, exactly, at a point: the detector, its vanishing on a first harmonic, and, on the dipole, \(\delta R = 4\lambda(2\cot^2r - 1)\cos\theta = 5\lambda/26\) at the test point \(\cos r = 3/5\), \(\cos\theta = 5/13\) (\texttt{test\_327} P4). The decomposition in (3) is checked symbolically, for every \(c\), with \(\eta_r = (c\cos r + q_0)\cos\theta\) and \(\eta_\theta = -(c\sin r + q_0\sin r\cos r + \lambda\sin^2r)\sin\theta\) (\texttt{test\_336} P5). \textbf{Proposition 6.3 (a kernel direction, two paths).} \emph{Let \(f = x_0x_1\vert_{S^3}\), a harmonic at level \(n = 3\). The paths} \[g_{\rm conf}(t) = e^{2tf}\gamma, \qquad g_{\rm rad}(t) = e^{2tf}\big(\gamma + t^2\,df\otimes df\big)\] \emph{have the same first variation \(2f\gamma \in \ker DF_\gamma\). The second is the metric of the embedding \(x_t(p) = e^{tf(p)}p\), so \(F(g_{\rm rad}(t)) = 0\) for every \(t\) small enough that \(g_{\rm rad}(t) \in \mathcal U\). For the first,} \[F(g_{\rm conf}(t)) = -t^2\,DF_\gamma(df\otimes df) + O(t^3), \qquad DF_\gamma(df\otimes df) \neq 0.\] \emph{So the straight path along a kernel direction leaves the zero set at second order. The kernel is tangent to the realizable set, here along the radial lift, but the straight line is not.} \emph{Proof.} \(\lvert d(e^{tf}p)\rvert^2 = e^{2tf}(\gamma + t^2df^2)\) because \(p\cdot dp = 0\). Then \(g_{\rm conf} = g_{\rm rad} - t^2e^{2tf}df\otimes df\), and Taylor's theorem about \(g_{\rm rad}(t)\) gives the expansion. That \(DF_\gamma(df\otimes df) \neq 0\) is an exact computation at one point (\texttt{test\_327} P3). There the whole expansion is also checked with the full nonlinear operator: the \(t^0\) and \(t^1\) terms vanish, the \(t^2\) term is \(-DF_\gamma(df\otimes df)\), and the radial lift gives zero through \(t^2\). \(\square\) The expansion is the audit's (XP02). The companion paper gives a global form of the same fact: along \(g_{\rm conf}\) the total absolute curvature has the deficit \(J = 2\pi^2 - \pi^2t^4/12 + O(t^6)\), which certifies non-realizability for small \(t \neq 0\). \textbf{Remark 6.4 (local structure near \(\gamma\)).} Theorem E(5) is the input an implicit-function argument needs: modulo the diffeomorphism action, the realizable set near \(\gamma\) should be a manifold modelled on the conformal directions. We do not carry this out. On the convex branch the companion paper describes the realizable set globally and at finite amplitude: the radial graphs, up to rigid motion. The equivariant construction is listed in §13. \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{a-worked-non-round-family-shells}{% \section{7. A worked non-round family: shells}\label{a-worked-non-round-family-shells}} This section treats metrics foliated by round 2-spheres. There \(F\) can be computed in closed form, and the realizable members can be written down. Everything is local on the region where the shells are nondegenerate, \(\Phi > 0\). \hypertarget{shell-metrics-and-their-realization}{% \subsection{7.1 Shell metrics and their realization}\label{shell-metrics-and-their-realization}} A \textbf{shell metric} is \[g_0 = A(r)^2dr^2 + \Phi(r)^2\hat\gamma, \qquad \hat\gamma\ \text{the unit round metric on } S^2,\] with \(A, \Phi > 0\). Two sectional curvatures control it: \[Q = \frac{A^2 - \Phi'^2}{A^2\Phi^2}\ \ (\text{tangential}), \qquad P = \frac{A'\Phi' - A\Phi''}{A^3\Phi}\ \ (\text{radial}),\] and in the orthonormal frame \(\mathbb M_0 = \mathrm{diag}(Q, P, P)\). We work on the \textbf{positive branch} \(Q, P > 0\). There \(g_0\) is realized by the hypersurface of revolution \(x = (\Phi(r)\hat n, Z(r)) \subset \mathbb R^3\times\mathbb R\) with \(Z' = \sqrt{A^2 - \Phi'^2}\). Its principal curvatures are \[\kappa_{\rm sph} = \sqrt Q\ \text{(twice)}, \qquad \kappa_{\rm mer} = P/\sqrt Q,\] so \(\tilde K_0 = \mathrm{diag}(P/\sqrt Q, \sqrt Q, \sqrt Q)\), and \(T_1\) is positive definite: the TT slice serves along the family. On the round base, \(A = 1\) and \(\Phi = \sin r\), we have \(Q = P = 1\). \hypertarget{first-order-the-radial-coefficient-family}{% \subsection{7.2 First order: the radial-coefficient family}\label{first-order-the-radial-coefficient-family}} Deform the radial coefficient by a shell harmonic: \[A \longmapsto A(1 + \varepsilon Y), \qquad -\hat\Delta Y = \Lambda Y,\ \ \Lambda = \ell(\ell + 1),\] that is, \(h = 2A^2Y\,dr^2\). For \(\ell \geq 1\) the harmonic \(Y\) is singular where the shells degenerate, so the statement is local on \(\{\Phi > 0\}\). \textbf{Theorem F (the shell classification).} \emph{On every base of the positive branch, the linearised residual \(C = DF_{g_0}(h)\) has exactly the components} \[C_{rar} = -\frac{\Lambda - 2}2\cdot\frac{A^3}{\Phi\sqrt{A^2 - \Phi'^2}}\,\partial_aY, \qquad C_{abc} = -\frac{\Lambda - 2}2\cdot\frac{\sqrt{A^2 - \Phi'^2}}{A\Phi P}\big(\hat\gamma_{bc}\partial_aY - \hat\gamma_{ac}\partial_bY\big),\] \[C_{rab} = \mathfrak D(r)\,T_{ab}[Y], \qquad T_{ab}[Y] = \big(\hat\nabla_a\hat\nabla_b - \tfrac12\hat\gamma_{ab}\hat\Delta\big)Y,\] \emph{with \(a, b, c\) angular, all other components zero, and} \[\mathfrak D = -\partial_r\Big(\frac{\sqrt Q}P\Big) + \frac{\Phi'}\Phi\cdot\frac{\sqrt Q}P - \frac{A\Phi'}{\sqrt{A^2 - \Phi'^2}},\] \emph{a function of \(r\) alone that vanishes identically on the round base. Consequently:} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{(Kernel.) \(C = 0\) iff \(\ell \in \{0, 1\}\). At \(\ell = 0\) all three families vanish because \(\partial_aY = 0\) and \(T[\mathrm{const}] = 0\). At \(\ell = 1\) the first two vanish because \(\Lambda = 2\), and the third by Obata's identity on \(S^2\), \(T[Y_1] = 0\).} \item \emph{(Exclusion, for this family.) For \(\ell \geq 2\) the first family alone is nonzero wherever \(\partial_aY \neq 0\), since its prefactor is positive on the branch. So no deformation \textbf{of this family} with \(\ell \geq 2\) is realizable at first order, on any base. Shapes outside the family are not barred: radial graphs are realizable for every \(\ell\) (Theorem E(1)).} \end{enumerate} \emph{Proof.} Write the lapse of the \(r\)-foliation as \(\mathcal N = A(1 + \varepsilon Y)\). The shell metric \(\Phi^2\hat\gamma\) is unchanged, and the shells are umbilic with \(\kappa_{ij} = \tilde\theta q_{ij}\), \(\tilde\theta = \Phi'/(\mathcal N\Phi)\). Three identities of the foliation give the five components of \(\mathbb M_1 = \delta\mathbb M\): \begin{itemize} \tightlist \item the Gauss (Hamiltonian) identity gives \(\mathbb M_1{}^r{}_r = 2Y\Phi'^2/(A^2\Phi^2)\); \item the momentum identity gives \(\mathbb M_1{}^r{}_\theta = -\Phi'\partial_\theta Y/(A^2\Phi)\) and \(\mathbb M_1{}^\theta{}_r = -\Phi'\partial_\theta Y/\Phi^3\); \item the radial Riccati identity gives \(\mathbb M_1{}^\theta{}_\theta = -\cot\theta\,\partial_\theta Y/\Phi^2 - 2YP\) and \(\mathbb M_1{}^\varphi{}_\varphi = -\partial_\theta^2Y/\Phi^2 - 2YP\) (axisymmetric \(Y\); the general case is the same with covariant derivatives). The \(\theta\theta\) and \(\varphi\varphi\) entries carry each other's Hessian components because \(\mathbb M\) acts on the shell block as a \(2\times2\) adjugate. \end{itemize} These are coordinate components. Lowered with \(g_0\), \(\mathbb M_1\) is symmetric, with \((\mathbb M_1)_{r\theta} = -\Phi'\partial_\theta Y/\Phi\) (\texttt{test\_344} P2). Differentiating \(\tilde K = \sqrt{\det\mathbb M}\,\mathbb M^{-1}\) gives, for the endomorphism, \[\delta\tilde K = \sqrt{\det\mathbb M_0}\,\big[S\,\mathbb M_0^{-1} - \mathbb M_0^{-1}\mathbb M_1\mathbb M_0^{-1}\big], \qquad S = \tfrac12\mathrm{tr}(\mathbb M_0^{-1}\mathbb M_1) = \frac{Y\Phi'^2}{A^2 - \Phi'^2} + \frac{\Lambda Y}{2\Phi^2P} - 2Y.\] Lower the index: \(\tilde h_{ab} = \delta(\tilde K_{ab}) = g_{0,ac}\,\delta\tilde K^c{}_b + h_{ac}\tilde K_0{}^c{}_b\), whose metric term is \(h_{rr}\tilde K_0{}^r{}_r = 2A^2YP/\sqrt Q\) in the \(rr\) slot. Then add the connection's variation. The Codazzi tensor has three \(SO(3)\)-covariant parts: \begin{itemize} \tightlist \item The \((r, a, r)\) part collapses through the factorisations \(A^3\Phi'' - A^2A'\Phi' - A\Phi'^2\Phi'' + A'\Phi'^3 = -QPA^5\Phi^3\) and \((A'\Phi' - A\Phi'')\sqrt{A^2 - \Phi'^2} = PA^4\Phi^2\sqrt Q\). \item In the \((a, b, c)\) part the trace term \(S\) cancels. The factor \(\Lambda - 2\) comes from the term \(\Lambda Y/(2\Phi^2P)\) of \(S\) against \(-2Y\). \item The \((r, a, b)\) part sees only the trace-free part of \(\tilde h\) on the shell, which is \(-(\sqrt Q/P)\,T_{ab}[Y]\), and its \((r, a)\) part \(A\Phi'\partial_aY/\sqrt{A^2 - \Phi'^2}\). The contractions with \(\Gamma^c{}_{ra} = (\Phi'/\Phi)\delta^c{}_a\) give the three terms of \(\mathfrak D\). The trace parts, including the term in \(\Gamma^r{}_{ab}\tilde h_{rr}\) with \(\Gamma^r{}_{ab} = -(\Phi\Phi'/A^2)\hat\gamma_{ab}\), cancel. \end{itemize} Reflection \(\varphi \mapsto -\varphi\) kills every component with an odd number of \(\varphi\) indices, so the list is complete. \(\square\) \emph{Three derivations and their checks.} Mathematics paper 1 (Thrussell 2026a) derives Theorem F three ways, with disjoint inputs: \begin{itemize} \tightlist \item the foliation identities above; \item direct linearisation of \(g \mapsto \mathbb M \mapsto \tilde K \mapsto \mathrm{Cod}\); \item explicit embeddings. \end{itemize} All three families are reproduced from the full nonlinear chain, in exact jets, on six base--radius configurations: the round base, three non-round bases (one at two radii) and a constant-ratio base. This is done at \(\ell = 0, 1, 2, 3\) and at two angular points, together with the vanishing of the other five components and the kernel at \(\ell \in \{0, 1\}\) (\texttt{test\_337} P1--P6). The independent audit had rebuilt the chain on a further base, \(A = 1 + r^2\), \(\Phi = r + r^3/10\), at \(\ell = 1, 2, 3\) (P1-X02). The identity \(\lVert T[Y_\ell]\rVert^2 = \tfrac12\Lambda(\Lambda - 2)\lVert Y_\ell\rVert^2\) (Bochner) shows that the third family vanishes exactly at \(\ell \in \{0, 1\}\), by two mechanisms: the gradient at \(\ell = 0\), and Obata at \(\ell = 1\). \textbf{Proposition 7.1 (the prefactor is a curvature invariant).} \emph{On the positive branch,} \[\mathfrak D = \frac{1}{\kappa_{\rm mer}}\,\partial_r\ln\frac{\kappa_{\rm mer}}{\kappa_{\rm sph}}.\] \emph{So \(\mathfrak D(r_0) = 0\) iff the ratio of principal curvatures is stationary at \(r_0\). And \(\mathfrak D \equiv 0\) iff that ratio is a constant \(c\), which is the two-parameter family \(\Phi'^2 = A^2(1 - C_0^2\Phi^{2c})\), with the round base at \(C_0 = c = 1\).} \emph{Proof.} \(\sqrt Q/P = 1/\kappa_{\rm mer}\). The Codazzi equation of the base, \(\kappa_{\rm sph}' = (\Phi'/\Phi)(\kappa_{\rm mer} - \kappa_{\rm sph})\), collapses the three terms. \(\square\) This is checked symbolically on a general base and on the constant-ratio family, and against the full nonlinear chain on five bases, two of them (the round one and \(A = 2\), \(\Phi = 2r - r^2\)) with \(\mathfrak D \equiv 0\) (\texttt{test\_337} P4, P8, P9). \textbf{Remark 7.2 (a quantitative form).} Write \(Z' = \sqrt{A^2 - \Phi'^2}\), and let \(\lvert C\rvert^2 = C_{ijk}C^{ijk}\) be the full contraction. The first family alone gives, on an interior shell, \[\int_{S^2_r}\lvert C\rvert^2\,dA \;\geq\; \frac{\Lambda(\Lambda - 2)^2}2\cdot\frac{A^2}{\Phi^2Z'^2}\int_{S^2}Y_\ell^2\,d\Omega\] (the audit's P1-X05). The two index orders \(C_{rar}\) and \(C_{arr} = -C_{rar}\) each contribute half. This is a bound on the linearised residual, not an exclusion at finite amplitude. \hypertarget{hypersurfaces-swept-by-round-spheres}{% \subsection{7.3 Hypersurfaces swept by round spheres}\label{hypersurfaces-swept-by-round-spheres}} The \(\ell = 1\) kernel of Theorem F is populated by actual hypersurfaces, and in two ways. Let \(x(r, \omega) = x_0(r) + \Phi(r)R(r)(\omega, 0)\), with \(x_0(r) \in \mathbb R^4\), \(R(r) \in SO(4)\) and \(\omega \in S^2\): a one-parameter family of round 2-spheres in \(E^4\). Write \[R^{\mathsf T}x_0' = (a, \alpha) \in \mathbb R^3\times\mathbb R, \qquad R^{\mathsf T}R' = \begin{pmatrix}\rho\times & -b\\ b^{\mathsf T} & 0\end{pmatrix}.\] So \(a\) is the in-plane velocity of the centre and \(\alpha\) its velocity out of the shell's 3-plane, \(\rho\) is a rotation within the 3-plane, and \(b\) tilts the 3-plane toward its normal. Label points of each shell by a Möbius chart \(z = p + iq\) with \[E = \frac{\lvert z - (u_1 + iu_2)\rvert^2 + u_3^2}{2u_3}, \qquad \hat n = \Big(\frac{p - u_1}E, \frac{q - u_2}E, 1 - \frac{u_3}E\Big),\] for functions \(u_1, u_2, u_3 > 0\) of \(r\), and write \(\vec W = (u_1', u_2', u_3')/u_3\). \textbf{Lemma 7.3 (the chart flow).} \emph{\(E'/E = -\vec W\cdot\hat n\) exactly. So the chart flow is a conformal Killing field of \(S^2\) with no \(\ell = 0\) part and no multipole above \(\ell = 1\).} \emph{Proof.} Differentiate \(E\), and use \((p - u_1)/E = n_1\), \((q - u_2)/E = n_2\), \(u_3/E = 1 - n_3\). \(\square\) \textbf{Proposition 7.4 (swept spheres).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{In coordinates \((r, p, q)\) that follow the orthogonal trajectories of the shells, the induced metric is} \[g = \big[H^2 + (\alpha + \Phi\,b\cdot\hat n)^2\big]dr^2 + \frac{\Phi^2}{E^2}\big(dp^2 + dq^2\big), \qquad H = \Phi' + \Phi\,\vec W\cdot\hat n,\] \emph{with \(a = \Phi\vec W\), the chart's rotation absorbed by \(\rho\).} \item \emph{Conversely, every metric of this form, for any functions \(\Phi > 0\), \(\alpha\), \(b\) and chart data \((u_1, u_2, u_3)\), with positive radial coefficient, is induced by such a family. The family is obtained by integrating \(R' = R\Omega\) and \(x_0' = R(\Phi\vec W, \alpha)\), where \(\Omega\) is built from \(b\) and the chart's rotation \(\rho = (u_2', -u_1', 0)/u_3\).} \end{enumerate} \emph{Proof.} \(R^{\mathsf T}\partial_rx\) has three parts: \begin{itemize} \tightlist \item along \(\hat n\): \(\Phi' + a\cdot\hat n\); \item along the 3-plane's normal: \(\alpha + \Phi\,b\cdot\hat n\); \item tangential: \(a - (a\cdot\hat n)\hat n + \Phi\rho\times\hat n\). \end{itemize} The orthogonal-trajectory chart moves \(\hat n\) by minus the tangential part, divided by \(\Phi\). That is a conformal Killing field (a gradient of a first harmonic plus a rotation), so the chart is Möbius at each \(r\). The shells are umbilic in the hypersurface. With respect to the unit normal \(m\) of a shell within the hypersurface, its second fundamental form is \(-(\hat n\cdot m)\,g_S/\Phi\), a multiple of the shell's induced metric \(g_S\) (here \(\hat n\) stands for \(R(\hat n, 0)\)). So the conformal class of \(dp^2 + dq^2\) is preserved along the trajectories, and \((\ln E^{-2})' = 2(a/\Phi)\cdot\hat n\). \begin{itemize} \tightlist \item By Lemma 7.3, \(a = \Phi\vec W\). \item The radial coefficient is the squared length of the normal part of \(\partial_rx\). \item For (2), choose \(x_0\) and \(R\) so that the three parts match the data, and run the computation backwards. The chart flow is \(\partial_r\hat n\vert_{p,q} = -\vec W + (\vec W\cdot\hat n)\hat n - \rho\times\hat n\) with \(\rho = (u_2', -u_1', 0)/u_3\), so that is the rotation for which the \(r\)-lines are orthogonal trajectories (\texttt{test\_344} P1). \(\square\) \end{itemize} \emph{Checks.} An explicit embedding, built by integrating these equations with \(\alpha\) and \(b\) chosen independently of the drift, reproduces the metric to \(10^{-10}\) (\texttt{test\_338} P1). A member is an immersion wherever its radial coefficient is positive, but need not lie in \(\mathcal U\): the flat members of Corollary 7.5(1) have \(\mathbb M \equiv 0\). The polynomial forms \(\mathcal T\) and \(\mathcal R\) of §8 vanish on every member (Proposition 8.1(4)), and \(F = 0\) wherever \(\det\mathbb M > 0\). This is checked exactly, in polynomial form, on a general member with \(\det\mathbb M > 0\) (\texttt{test\_338} P3). \textbf{Corollary 7.5 (two subfamilies, and the first-order picture).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{(Drift.) With \(b = 0\) and \(\alpha = 0\) every shell stays in one 3-plane, so the hypersurface is a piece of that 3-plane and} \(H^2dr^2 + \Phi^2E^{-2}(dp^2 + dq^2)\) \emph{is Euclidean 3-space in non-concentric spherical coordinates. With \(\alpha = s\Phi'\) and \(b = s\vec W\), \(s^2 = k/(1 - k)\) and \(0 \leq k < 1\), it is} \[\frac{H^2}{1 - k}\,dr^2 + \frac{\Phi^2}{E^2}\big(dp^2 + dq^2\big),\] \emph{Szekeres's quasi-spherical metric (Szekeres 1975). Its embedding by shift, rotation and tilt is Buckley and Schlegel's (2020, §VIII); see also Hellaby and Buckley (2020).} \item \emph{(Tilt.) With no drift, \(\vec W = 0\), the metric \([\Phi'^2 + (\alpha + \Phi\,b\cdot\hat n)^2]dr^2 + \Phi^2\hat\gamma\) is realizable for any \(\alpha\) and \(b\). If \(\alpha \not\equiv 0\), its radial coefficient has the \(\ell = 1\) part \(2\alpha\Phi\,b\cdot\hat n\) at first order in \(b\). On the positive branch \(\alpha \neq 0\) automatically, since there \(A_0^2 = \Phi'^2 + \alpha^2 > \Phi'^2\) (§7.1).} \item \emph{(First order.) About the shell metric with \(A_0^2 = \Phi'^2 + \alpha^2\),} \[\frac{\delta A}{A_0} = \frac{\Phi\,(\Phi'\vec W + \alpha\,b)\cdot\hat n}{A_0^2},\] \emph{which is regular where \(\Phi' = 0\). The tilt changes nothing else. Its first-order variation is the \(\ell = 1\) change of the radial coefficient with amplitude \(\Phi\alpha b/A_0^2\): Theorem F's direction when \(b\) makes that amplitude constant, and in the first-order kernel for every \(b\). The drift also shears the shell labels. In shell coordinates, with fixed labels on each shell, its variation is that direction plus an off-diagonal \(\ell = 1\) term \(\varepsilon\beta\sin\theta\,(dr\,d\theta + d\theta\,dr)\), which is itself in the first-order kernel. On the round base, for the radial modulation \(2\varepsilon\cos\theta\), \(\beta = -\tan r\).} \end{enumerate} \emph{Proof.} (1) is Proposition 7.4 with those choices; flatness is the case \(k = 0\). (2) and (3) are Proposition 7.4 and a first-order expansion. For the drift in shell coordinates, \(x = x_0(r) + \Phi\hat n\) gives \(g_{rr} = \lvert x_0' + \Phi'\hat n\rvert^2 + \ldots\) and \(g_{r\theta} = \Phi\,x_0'\cdot\partial_\theta\hat n\). The drift's variation and the \(\ell = 1\) direction both lie in the first-order kernel, so their difference, the off-diagonal term, does too. \(\square\) The shell-coordinate formulas, with the round-base value of \(\beta\), are checked symbolically (\texttt{test\_338} P8). The first-order kernel statements, for an \(r\)-dependent \(\ell = 1\) amplitude and for the off-diagonal term alone, are checked with the full operator on three non-round bases (\texttt{test\_337} P7). Part (2) is checked with the full operator, exactly, on a non-round base: at two points, with \(\mathbb M\) positive definite, and also for an axisymmetric tilt (\texttt{test\_338} P2, P5, P6). \emph{In plain terms.} In three dimensions an \(\ell = 1\) change of the radial coefficient of nested spheres can only come from translating their centres, and the translation also shears the labels on the shells. In \(E^4\) it can come instead from tilting the spheres' 3-planes, and then nothing else changes. An earlier version asserted that the \(\ell = 1\) amplitude must equal \(\Phi\,\vec W\cdot\hat n\). Part (2) refutes this whenever \(\alpha \not\equiv 0\) (Appendix C). \textbf{Proposition 7.6 (the Lorentz model of a shell; nesting).} \emph{Put} \[V = \frac1{2u_3}\big(1,\ -u_1,\ -u_2,\ u_1^2 + u_2^2 + u_3^2\big) \in \mathbb R^4, \qquad \mathcal Q(V) = 4(V_0V_3 - V_1^2 - V_2^2),\] \emph{a quadratic form of signature \((1,3)\), with polarisation \(\mathcal B(X, Y) = \tfrac12\big(\mathcal Q(X + Y) - \mathcal Q(X) - \mathcal Q(Y)\big)\), so that \(\mathcal B(X, X) = \mathcal Q(X)\). Then:} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{\(\mathcal Q(V) = 1\);} \item \emph{\(-\mathcal Q(dV) = (du_1^2 + du_2^2 + du_3^2)/u_3^2\), so \((u_1, u_2, u_3)\) is \(V\) on the hyperboloid model of \(H^3\) and \(\lvert\vec W\rvert = \lvert V'\rvert\);} \item \emph{\(E = \tfrac12\mathcal B(V, N)\), with \(N = (1, -p, -q, p^2 + q^2)\) null;} \item \emph{with \(\mathcal Y = \Phi V\), \(\mathcal Q(\mathcal Y') = \Phi'^2 - \Phi^2\lvert\vec W\rvert^2\).} \end{enumerate} \emph{So \(H > 0\) in every direction iff \(\Phi' > \Phi\lvert\vec W\rvert\), iff \(r \mapsto \Phi V(r)\) is a future-directed timelike curve, \(\mathcal B(\mathcal Y', V) = \Phi' > 0\). For the flat member of Corollary 7.5(1) this is the condition that the shells are nested round spheres. In Szekeres's setting it is the no-shell-crossing condition of Hellaby and Krasiński (2002, their eq. 92), restated as a causal condition. The Möbius freedom of the chart is \(SO^+(1,3)\) acting on \(V\).} \emph{Proof.} (1)--(4) are direct computation, and the last statement is Lemma 7.3 with \(\min_{\hat n}\vec W\cdot\hat n = -\lvert\vec W\rvert\). \(\square\) Items (1)--(4) are checked symbolically in \texttt{test\_330} P6, and Lemma 7.3 in \texttt{test\_330} P5. Each shell is intrinsically round, with Gauss curvature \(1/\Phi^2\) for every chart (\texttt{test\_338} P7). \hypertarget{second-order-at-ell-1}{% \subsection{\texorpdfstring{7.4 Second order at \(\ell = 1\)}{7.4 Second order at \textbackslash ell = 1}}\label{second-order-at-ell-1}} The \(\ell = 1\) direction of Theorem F is a first-order kernel, not a symmetry. \textbf{Theorem G (second order, round base).} \emph{On the round base, on interior shells \(0 < r < \pi\), hold the deformation \(A \mapsto 1 + \varepsilon\cos\theta\) pure to second order. Then the residual first appears at \(O(\varepsilon^2)\):} \[C^{(2)}_{r\theta r} = -\frac{2\sin\theta\cos\theta\cos^2r}{\sin^4r}, \quad C^{(2)}_{r\theta\theta} = -\frac{\sin^2\theta\cos r}{\sin^3r}, \quad C^{(2)}_{r\varphi\varphi} = \frac{\sin^4\theta\cos r}{\sin^3r}, \quad C^{(2)}_{\theta\varphi\varphi} = -\frac{2\sin^3\theta\cos\theta}{\tan^2r}.\] \emph{Its angular content is pure \(\ell = 2\). The mechanism is \((\ell{=}1)\times(\ell{=}1) \to \ell{=}0 \oplus \ell{=}2\), with the \(\ell = 0\) part killed by the gradient prefactor. The obstruction is removed in two ways:} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{(Tilt.) A second-order correction \(\delta A/A = \varepsilon\cos\theta + \varepsilon^2\cot^2r\,[\tfrac16 + \tfrac13P_2(\cos\theta)]\), with no off-diagonal term. This is the round sphere with shells in tilting 3-planes, \(b = (\varepsilon/\sin^2r)\hat e_3\) (Corollary 7.5(2)), and the \(r\)-only \(\ell = 0\) part never obstructs.} \item \emph{(Drift.) Write the metric as \(g_{rr} = 1 + 2\varepsilon\cos\theta + \varepsilon^2\mu(r)\) and \(g_{r\theta} = \varepsilon\beta(r)\sin\theta\), with the shells unchanged. The off-diagonal term is the one a drift of the centres carries in shell coordinates (Corollary 7.5(3)). The \(O(\varepsilon^2)\) residual is quadratic in \(\beta\). Its \(r\theta r\) component is \(-(\beta + \tan r)^2/\tan^2r\) times \(2\sin\theta\cos\theta/\sin^4r\): a perfect square, with its double root at \(\beta = -\tan r\), the value the drifting hypersurface carries. With \(\beta = -\tan r\) and \(\mu = \sec^2r\) every component vanishes.} \end{enumerate} \emph{Proof and scope.} \begin{itemize} \tightlist \item \emph{Components.} They are computed with the full nonlinear chain in exact jets, at two radii and two angles, with the other five components zero (\texttt{test\_339} P1). The audit had recomputed all four (P1-X03). \item \emph{Route 1.} It removes all components with one function, and its identification with the tilt is the exact expansion of Corollary 7.5(2) (\texttt{test\_338} P4). That it removes every component through \(O(\varepsilon^2)\) is \texttt{test\_339} P2. \item \emph{Route 2.} It is computed exactly, for \(s = 0, 1, 2, 3\) in \(\beta = -s\tan r\) (\texttt{test\_339} P3). The radial companion \(\mu\) does not enter the residual. \item \emph{Interior shells.} The statement is on \(0 < r < \pi\). Both \(\cos\theta\) and \(\cot^2r\) are singular at the poles, and nothing here is a smooth global deformation of \(S^3\). \(\square\) \end{itemize} The two routes are the two rigid motions of Corollary 7.5. Second order selects neither: it rigidifies the extension within each class. \textbf{Observation 7.7 (general bases: exact checks, not a theorem).} On a general base the drift of Corollary 7.5 with radial modulation \(2\varepsilon A^2\cos\theta\) has, in shell coordinates, \(g_{rr} = A^2(1 + 2\varepsilon\cos\theta) + \varepsilon^2\mu\) and \(g_{r\theta} = \varepsilon\beta\sin\theta\), with \(\beta_{\rm pred} = -\Phi A^2/\Phi'\) and \(\mu_{\rm pred} = A^4/\Phi'^2\). Put \(\beta = s\,\beta_{\rm pred}(r)\) and \(\mu = \mu_{\rm pred}(r)\). The following hold in exact arithmetic on three disjoint non-round bases, at eight base--radius pairs and three angular points, and on two bases with \(\mathfrak D \equiv 0\): \begin{itemize} \tightlist \item each \(O(\varepsilon^2)\) component is quadratic in \(s\) and vanishes at \(s = 1\), where the metric is that of the drifting hypersurface; \item every component has a double root there, the \(\mathfrak D\) family included, as on the round base; \item terms depending on \(r\) alone never obstruct: changing the radial coefficient by a function of \(r\) leaves a shell metric, which is realizable, so such a term has zero first variation. The residual is checked to be unchanged when \(\mu\) is rescaled or removed, with \(\beta\) frozen. \end{itemize} If instead \(\beta\) and \(\mu\) are frozen at their values at the point, the \((\Lambda - 2)\) families are unchanged. The \(\mathfrak D\) family, however, acquires a simple root at \(s = 1\), with a second root that varies from base to base. The freezing drops \(\beta'\), and only the \(\mathfrak D\) family sees it. An earlier version reported that frozen picture, a transversal zero on a general base, as a property of the drift (Appendix C). The checks are \texttt{test\_340}, and they are not proved for every base (§13, problem 3). On the round base \(\beta_{\rm pred} = -\tan r\), which is Theorem G(2). \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{the-degenerate-locus}{% \section{8. The degenerate locus}\label{the-degenerate-locus}} \(F\) is built from \(\tilde K = \sqrt{\det\mathbb M}\,\mathbb M^{-1}\) and is undefined where \(\det\mathbb M = 0\). This section asks what can be said there. Two polynomial forms of the equation remove the root and the inverse. They show exactly what survives at \(\det\mathbb M = 0\), and on realizable metrics the answer is: nothing that \(\mathbb M\) can see. \hypertarget{two-polynomial-forms}{% \subsection{8.1 Two polynomial forms}\label{two-polynomial-forms}} Put \(\mathbb P = \mathrm{adj}\,\mathbb M\), raised and lowered with \(g\). By (2.3), on a realizable metric \[\mathbb P = \mathrm{adj}\,\mathrm{adj}\,K = (\det K)\,K,\] Thomas's identity (9.3). \(\mathbb P\) is polynomial in the curvature and even under \(K \mapsto -K\). On \(\{\det\mathbb M > 0\}\), \(\mathbb P = \sqrt{\det\mathbb M}\,\tilde K\), and \(\mathbb P\) is positive definite iff \(\tilde K\) is definite (\texttt{test\_341} P1). \textbf{Proposition 8.1 (two polynomial forms).} \emph{For every metric \(g\) put} \[\mathcal T(g) = 2(\det\mathbb M)\,\mathrm{Cod}(\mathbb P) - d(\det\mathbb M)\wedge\mathbb P, \qquad (d\varphi\wedge T)_{abc} = \partial_a\varphi\,T_{bc} - \partial_b\varphi\,T_{ac},\] \[\mathcal R(g)_{ab}{}^d = \tfrac12\big(\partial_a\det\mathbb M\;\delta_b{}^d - \partial_b\det\mathbb M\;\delta_a{}^d\big) - \big((\nabla_a\mathbb M)\,\mathbb P\big)^d{}_b + \big((\nabla_b\mathbb M)\,\mathbb P\big)^d{}_a .\] \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{\(\mathcal T\) is polynomial in the 3-jet of \(g\) and in \(g^{-1}\): of degree four in the curvature and one in its derivative. On \(\{\det\mathbb M > 0\}\), \(\mathcal T = 2(\det\mathbb M)^{3/2}F\).} \item \emph{\(\mathcal R\) is polynomial of degree two in the curvature and one in its derivative. On \(\{\det\mathbb M > 0\}\), \(\mathcal R_{ab}{}^d = \sqrt{\det\mathbb M}\;F_{ab}{}^c\,\mathbb M_c{}^d\).} \item \emph{So on \(\{\det\mathbb M > 0\}\), \(\mathcal T = 0\) iff \(\mathcal R = 0\) iff \(F = 0\), on both branches. Both third-order symbols are \(\sigma(F)\) composed with invertible algebraic maps, so Theorems B, C and D carry over to both on \(\mathcal U\).} \item \emph{On a realizable metric both vanish everywhere, the degenerate locus included.} \item \emph{For every metric, the nine components of \(\mathcal R\) satisfy the three linear identities \(\mathcal R_{ab}{}^b = 0\), which are the contracted Bianchi identity, and the cyclic identity \(\sum_{\mathrm{cyc}(a,b,c)}\mathcal R_{ab}{}^d\,\mathbb P_{dc} = 0\).} \end{enumerate} \emph{Proof.} Write \(\varphi = \sqrt{\det\mathbb M}\), so \(\mathbb P = \varphi\tilde K\) and \(2\varphi\,d\varphi = d\det\mathbb M\). \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \item \(\mathrm{Cod}(\varphi T) = d\varphi\wedge T + \varphi\,\mathrm{Cod}\,T\) gives \(2\varphi^2\mathrm{Cod}(\varphi\tilde K) - 2\varphi\,d\varphi\wedge\varphi\tilde K = 2\varphi^3\mathrm{Cod}\,\tilde K\). \item Differentiate \(\mathbb M\tilde K = \varphi\,\mathrm{Id}\): \(\mathbb M\,\nabla_a\tilde K = \partial_a\varphi\,\mathrm{Id} - (\nabla_a\mathbb M)\tilde K\). Antisymmetrise in \(a, b\) and multiply by \(\varphi\). Jacobi's formula \(\partial_a\det\mathbb M = \mathrm{tr}\big(\mathbb P\,\nabla_a\mathbb M\big)\) shows that \(\mathcal R\) is polynomial. \item \(\mathbb M\) is invertible and \(\varphi > 0\) there. Derivatives of the factors are of lower order. \item Where \(\det\mathbb M > 0\) this is (3). At a point of a realizable metric where \(\det\mathbb M = 0\), \(\det K = 0\), so \(\mathrm{rank}\,\mathbb M \leq 1\) by Proposition 8.2(1), and both forms vanish there by Proposition 8.2(2). No derivative of \(K\) is used. \item Tracing over \(b = d\) leaves \(\big((\nabla_b\mathbb M)\,\mathbb P\big)^b{}_a = (\mathrm{div}\,\mathbb M)_c\,\mathbb P^c{}_a\), and \(\mathrm{div}\,\mathbb M = -\mathrm{div}\,G = 0\). Where \(\det\mathbb M > 0\), \(F_{abc} = \mathcal R_{ab}{}^d\,\mathbb P_{dc}/\varphi^3\), so the cyclic identity of \(\mathrm{Cod}\,\tilde K\) (Lemma 4.6(1)) gives the second. It is a polynomial identity in the jet that holds on an open set of jets, so it holds for every metric. \(\square\) \end{enumerate} \(\mathcal R\) has the bidegree of Rivertz's six equalities (§1). After its three trace identities it has six components, and the cyclic identity leaves the five of Theorem C. Whether Rivertz's equalities are the trace-free part of \(\mathcal R\) is not checked here (§13). Thomas clears the root by the same factor \((\det\mathbb M)^{3/2}\) as \(\mathcal T\) (his §10). He then eliminates the unknown and arrives at a resultant system, which has no principal symbol to compute. \(\mathcal T\) and \(\mathcal R\) keep a tensor unknown. \emph{Checks:} \texttt{test\_341} P2 for \(\mathcal T\), against \(F\) computed independently on both branches. For \(\mathcal R\), \texttt{test\_331}: - on a generic metric with \(\det\mathbb M > 0\), \(\mathcal R = \sqrt{\det\mathbb M}\,F\mathbb M\) exactly, with \(F\) computed independently and nonzero, together with the identities (5); - \(\mathcal R = 0\) on two realizable metrics, one of them non-umbilic; - \(\mathcal R\) at the rank-one and rank-two points of §8.2. \hypertarget{at-detmathbb-m-0}{% \subsection{\texorpdfstring{8.2 At \(\det\mathbb M = 0\)}{8.2 At \textbackslash det\textbackslash mathbb M = 0}}\label{at-detmathbb-m-0}} \textbf{Proposition 8.2.} \emph{Let \(\det\mathbb M(p) = 0\).} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{(Rank of the adjugate.) \(\mathrm{rank}\,\mathrm{adj}\,K\) is 3, 1 or 0 as \(\mathrm{rank}\,K\) is 3, 2 or at most 1. So a rank-2 \(\mathbb M\) is never \(\mathrm{adj}\,K\), and on a realizable metric \(\det\mathbb M(p) = 0\) forces \(\mathrm{rank}\,\mathbb M(p) \leq 1\):} \[\mathrm{rank}\,K = 2 \iff \mathrm{rank}\,\mathbb M = 1, \qquad \mathrm{rank}\,K \leq 1 \iff \mathbb M = 0.\] \item \emph{(Rank at most one.) If \(\mathrm{rank}\,\mathbb M(p) \leq 1\), then \(\mathbb P(p) = 0\), \(d(\det\mathbb M)(p) = 0\) and \(\mathcal T(p) = \mathcal R(p) = 0\). Both forms are silent at every such point, whatever the rank nearby.} \item \emph{(Non-realizable points, where the forms are not vacuous.) If \(\mathrm{rank}\,\mathbb M(p) = 2\), with kernel spanned by a unit vector \(v\) and nonzero eigenvalues \(m_1, m_2\), then \(\mathbb P(p) = m_1m_2\,v\otimes v \neq 0\) and} \[\mathcal T(p) = -\,m_1m_2\,d(\det\mathbb M)\wedge v^\flat\otimes v^\flat .\] \emph{So \(\mathcal T(p) = 0\) iff \(d(\det\mathbb M)(p)\) is parallel to \(v^\flat\). This is a genuine first-order condition, which a rank-2 point of a metric can fail.} \end{enumerate} \emph{Proof.} (1) In an eigenframe, \(\mathrm{adj}\,\mathrm{diag}(k_1, k_2, k_3) = \mathrm{diag}(k_2k_3, k_1k_3, k_1k_2)\). (2) \(\mathbb P(p) = \mathrm{adj}\,\mathbb M(p) = 0\), and by Jacobi's formula \(d(\det\mathbb M)(p) = \mathrm{tr}\big(\mathbb P(p)\,\nabla\mathbb M(p)\big) = 0\). Every term of \(\mathcal T\) and \(\mathcal R\) carries an undifferentiated factor \(\det\mathbb M\), \(d\det\mathbb M\) or \(\mathbb P\) (\texttt{test\_344} P3, at points of rank one and zero with rank two or more nearby). (3) At \(p\) the first term of \(\mathcal T\) vanishes, and the second is \(d(\det\mathbb M)\wedge\mathbb P\). \(\square\) \emph{Example.} \(g = dr^2 + (1 + r + r^3)^2dx^2 + (1 + r + r^2)^2dy^2\) at \(r = 0\) has: \begin{itemize} \tightlist \item \(\mathbb M = \mathrm{diag}(-1, -2, 0)\), so \(\det\mathbb M = 0\); \item \(\partial_r\det\mathbb M = -12\); \item \(\mathbb P = \mathrm{diag}(0, 0, 2)\); \item \(\mathcal T_{ryy} = 24\), the only nonzero component up to antisymmetry in the first two indices; \item \(\mathcal R_{rx}{}^x = -6\) and \(\mathcal R_{ry}{}^y = 6\), likewise. \end{itemize} \(v = \partial_y\) is not parallel to \(d(\det\mathbb M) = -12\,dr\). Both forms are genuine conditions at a rank-two point, and they are different polynomial extensions of \(F\) there. The example is the audit's, rebuilt here (\texttt{test\_329} P1--P2; \texttt{test\_331} P3). \emph{The symbol near the locus.} On \(\mathcal U\) the symbol of \(F\) uses \((D\,\mathrm{adj}_K)^{-1}\). By Lemma 4.1(2) its kernel condition can be written without the inverse, as \(\sigma_2(D\mathbb M)(\xi)h \in \mathrm{span}\{D\,\mathrm{adj}_K(\xi\otimes\xi)\} = \mathrm{span}\{\mathrm{adj}_2Q_\xi\}\). That condition is polynomial in \(K\) and defined at \(\det K = 0\). It describes \(F\)'s kernel, not the symbols of \(\mathcal T\) and \(\mathcal R\), which vanish identically where \(\mathrm{rank}\,\mathbb M \leq 1\). Approaching the locus, the rank of \(\sigma(\xi)\) stays 2 (checked down to \(k_3 = 10^{-6}\)). What degenerates is the constant. On TT the largest singular value grows like \(1/k_3\) at every covector. The smallest stays bounded at covectors orthogonal to the direction whose principal curvature vanishes, and grows like \(1/k_3\) at the others. So their ratio grows like \(1/k_3\) exactly at those orthogonal covectors (\texttt{test\_341} P3). An earlier claim that the rank rises to 3 at the locus was a regularisation artefact and is withdrawn. \hypertarget{the-rank-one-branch}{% \subsection{8.3 The rank-one branch}\label{the-rank-one-branch}} On an open set where \(\mathrm{rank}\,K = 2\), Proposition 8.2(1) gives \(\mathbb M = \mu\,v\otimes v\) with \(\mu = k_1k_2 = R/2\) and \(v\) spanning \(\ker K\), the direction of relative nullity. \textbf{Proposition 8.3 (the rank-one Einstein tensor).} \emph{Let \((\Sigma^3, g)\) be any Riemannian three-manifold with \(-G = \mu\,v\otimes v\) on an open set, \(\lvert v\rvert = 1\), \(\mu \neq 0\). Then the contracted Bianchi identity splits as} \[\nabla_a(\mu v^a) = 0, \qquad \nabla_vv = 0.\] \emph{So the integral curves of \(v\) are geodesics, and \(\mu\,dV\) is conserved along them.} \emph{Proof.} \(0 = \nabla^a(\mu v_av_b) = \nabla^a(\mu v_a)\,v_b + \mu\,\nabla_vv_b\). Contract with \(v^b\) and use \(v^b\nabla v_b = 0\), which gives the first equation. The second follows because \(\mu \neq 0\). \(\square\) For a hypersurface this recovers, from the intrinsic geometry alone, the classical fact that the leaves of relative nullity are totally geodesic. On the cone over a Clifford torus, \(\mathbb M = \mathrm{diag}(-1/t^2, 0, 0)\): \(\partial_t\) is geodesic, \(\mathrm{div}(\mu\,\partial_t) = 0\), and \(\mathcal T\) and \(\mathcal R\) vanish identically (\texttt{test\_329} P3--P4; \texttt{test\_331} P4). The split itself is checked at a point for general \(\mu\) and \(v\) (\texttt{test\_342} Q1--Q2). \textbf{Theorem 8.4 (the Codazzi block system).} \emph{On a rank-one region, let \((e_1, e_2)\) be an orthonormal frame of \(v^\perp\), parallel along \(v\) (possible because \(v\) is geodesic, so parallel transport along \(v\) preserves \(v^\perp\)), and \(B_{ab} = \langle e_b, \nabla_{e_a}v\rangle\). \(B\) is intrinsic, since \(v\) is an eigenvector of the Einstein tensor. For a symmetric \(K\) with \(Kv = 0\) and block \(K_\perp\) on \(v^\perp\), the Gauss and Codazzi equations are equivalent to} \[\det K_\perp = \mu, \qquad BK_\perp = K_\perp B^{\mathsf T}, \qquad \nabla_vK_\perp = -BK_\perp,\] \emph{together with the two Codazzi equations along \(v^\perp\). At each point the first two leave one free parameter of the two that \(\mathbb M\) leaves: the ratio \(k_1/k_2\) and the principal angle. The trace of the third is, by Jacobi's formula, \(v(\ln\lvert\mu\rvert) = -\mathrm{tr}\,B\), which is Proposition 8.3's conservation law again.} \emph{Proof.} Contract Codazzi with \(v\) and use \(Kv = 0\): \((\nabla_aK)(e_b, v) = -(BK)_{ab}\). Symmetry in \(a, b\) gives the second equation, and the transport along \(v\) gives the third. \(\square\) Two consequences: \begin{itemize} \tightlist \item Equal principal curvatures force \(B\) to be symmetric, so the leaves of \(v^\perp\) have no twist. \item A nonzero pure rotation of the rulings (\(B\) skew, \(B \neq 0\)) forces \(\mathrm{tr}\,K_\perp = 0\), because then \(BK_\perp + K_\perp B = (\mathrm{tr}\,K_\perp)B\) must vanish. So \(\mu = \det K_\perp < 0\) there. \end{itemize} On the cone over a Clifford torus all three equations hold exactly, with \(B = I/t\) and \(K_\perp = \mathrm{diag}(\tan\alpha, -\cot\alpha)/t\) (\texttt{test\_342} P1--P5). \textbf{Corollary 8.5 (what \(F\), \(\mathcal T\) and \(\mathcal R\) see on the degenerate locus).} \begin{itemize} \tightlist \item On a rank-one region of a realizable metric, \(F\) is \(0/0\), and \(\mathcal T\) and \(\mathcal R\) vanish identically. Everything \(\mathbb M\) determines there is \((\mu, v)\). Realizability passes to the block system of Theorem 8.4, which is the Sbrana--Cartan system. \item Generic hypersurfaces with an open rank-two region are rigid there. The isometrically deformable ones form the four classes of Sbrana (1909) and Cartan (1916): surface-like, ruled, continuous and discrete. For the modern treatment see Dajczer, Florit and Tojeiro (1998) and Dajczer and Tojeiro (2019). The classification describes the \textbf{deformable} subfamilies, not every rank-two hypersurface. These classes are invisible to \(F\), \(\mathcal T\) and \(\mathcal R\), and visible to the block system. \item Across a \emph{seam}, where one principal curvature passes through zero, \(\tilde K\) flips sign while \(K = \mathrm{sign}(\det K)\,\tilde K\) stays continuous. So \(K\) is fixed on the seam from either side, and the block system is needed only for an open rank-two region (\texttt{test\_342} P6). \end{itemize} \emph{Proof.} The first item is Proposition 8.2(2) with Theorem 8.4. The second is the cited classification. The third is Theorem A(3) on each side. \(\square\) \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{the-chiral-form}{% \section{9. The chiral form}\label{the-chiral-form}} Let \((e_1, e_2, e_3, N)\) be an adapted frame along a hypersurface, with coframe \(\theta^i\). The flat connection of \(E^4\) restricts to an \(\mathfrak{so}(4)\)-valued 1-form \(\Omega\). Its tangential block is the Levi-Civita connection \(\omega_{ij}\) of \(g\), and its normal block is the shape operator, \(\Omega_{Ni} = K_{ij}\theta^j\). In Euclidean signature \(\mathfrak{so}(4) = \mathfrak{su}(2)_+\oplus\mathfrak{su}(2)_-\) splits into two commuting \textbf{real} halves. Write \(A^\pm\) for the two parts of \(\Omega\). In components, \(A^{\pm\,i} = \Gamma^i \pm K^i\), with \(\Gamma^i = \tfrac12\epsilon^{ijk}\omega_{jk}\) and \(K^i = K_{ij}\theta^j\). For any metric \(g\) and any symmetric \(K\), define \(\Omega\) by the same formulas. It is then an \(\mathfrak{so}(4)\)-connection on \(T\Sigma\oplus\mathbb R\), and need not be flat. We write \(F_A\) for the curvature of a connection \(A\); the subscript distinguishes it from the operator \(F\). \textbf{Theorem 9.1 (Gauss and Codazzi as chiral flatness).} \emph{\([A^+, A^-] = 0\), and the curvatures of the two halves are} \[F_{A^\pm} = \mathcal G \pm \mathcal C,\] \emph{where \(\mathcal G\) is the \(\mathfrak{su}(2)\)-dual of the Gauss residual \(R_{ij} - K^i\wedge K^j\), and \(\mathcal C = d_\Gamma K\), the dual of the Codazzi residual. So \((g, K)\) satisfies the Gauss and Codazzi equations iff both chiral connections are flat.} \emph{Proof.} The curvature of \(\Omega\) is \(d\Omega + \Omega\wedge\Omega\). Its tangential block is the Gauss residual and its mixed block the Codazzi residual. Projecting onto the two \(\mathfrak{su}(2)\) ideals gives the even and odd parts. \(\square\) Theorem 9.1 is the connection form of Lawn and Roth's spinorial characterization of hypersurfaces of \(\mathbb R^4\) (Lawn and Roth 2010), as Proposition 9.4 makes explicit. \textbf{Corollary 9.2.} \emph{On \(\mathcal U\), take \(K = \tilde K\). Then \(\mathcal G = 0\) identically (Theorem A(2)), so} \[F_{A^\pm} = \pm\,F(g), \qquad F(g) = \tfrac12\big(F_{A^+} - F_{A^-}\big).\] \emph{The realizability operator is half the difference of the two chiral curvatures, which are equal and opposite.} \textbf{Proposition 9.3 (reversing the normal swaps the halves).} \emph{Under \(N \mapsto -N\), \(K \mapsto -K\) and \(\Omega \mapsto R\,\Omega R\) with \(R = \mathrm{diag}(1,1,1,-1)\), which exchanges \(A^+\) and \(A^-\). On \(\mathcal U\) this is the branch involution \(\tilde K \mapsto -\tilde K\) of Theorem A(3).} \textbf{Proposition 9.4 (spinors).} \emph{Let \(\Sigma\) be spin, with Clifford multiplication normalised by \(X\cdot X\cdot = -\lvert X\rvert^2\), and let \(K\) be symmetric. For one choice of orientation of \(T\Sigma\oplus\mathbb R\), the connections that \(A^+\) and \(A^-\) induce on the spinor bundle are} \[\nabla^\pm_X = \nabla_X \mp \tfrac12K(X)\cdot ,\] \emph{(the other orientation exchanges them), and the curvature of \(\nabla^\pm\) is the image of \(F_{A^\pm}\). Hence:} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \emph{A nonzero \(\nabla^\pm\)-parallel spinor exists near a point iff \(F_{A^\pm} = 0\) there.} \item \emph{(Lawn and Roth 2010.) \((g, K)\) satisfies the Gauss and Codazzi equations iff it carries nonzero parallel spinors for both \(\nabla^+\) and \(\nabla^-\). If \(K\) is Codazzi, one of them suffices: then \(F_{A^+} = F_{A^-} = \mathcal G\).} \item \emph{For \(g \in \mathcal U\) and \(K = \tilde K(g)\), \(F_{A^\pm} = \pm F(g)\). So, on a simply connected \(\Sigma\), \(g\) is realizable iff it carries one nonzero spinor with \(\nabla_X\psi = \tfrac12\tilde K(X)\cdot\psi\), iff it carries one with the opposite sign. No Codazzi hypothesis is needed, because \(\tilde K\) solves the Gauss equation by construction.} \end{enumerate} \emph{Proof.} Under \(\mathfrak{so}(4) = \mathfrak{su}(2)_+\oplus\mathfrak{su}(2)_-\) each half-spin representation is trivial on one ideal. Restricted to \(\mathfrak{so}(3)\) both are the spin representation, and the mixed generators act as \(\pm\tfrac12e_i\cdot\), with opposite signs on the two halves. So the connection that \(\Omega\) induces on a half-spin bundle is \(\nabla \mp \tfrac12K\cdot\), and it depends only on \(A^\pm\). Its curvature is the image of \(F_{A^\pm}\), because the representation is a Lie algebra homomorphism. \begin{enumerate} \def\labelenumi{(\arabic{enumi})} \item The image of a nonzero element of \(\mathfrak{su}(2)\) is \(v\cdot\) with \(v \neq 0\), and \((v\cdot)^2 = -\lvert v\rvert^2\). So it is invertible, and annihilates no nonzero spinor. A parallel spinor forces the curvature to annihilate it. Conversely, a flat \(\nabla^\pm\) has a two-dimensional space of local parallel spinors. \item is Theorem 9.1 with (1). When \(\mathcal C = 0\), both curvatures equal \(\mathcal G\). \item is Corollary 9.2 with (1). Simple connectivity makes local parallel spinors global. \(\square\) \end{enumerate} These spinors are the generalized Killing spinors of Bär, Gauduchon and Moroianu (2005). Their theorem embeds a metric that carries one, for a Codazzi tensor, in a Ricci-flat manifold with a parallel spinor. By (2), in dimension three that ambient manifold is flat near \(\Sigma\), as Lawn and Roth also note. The Dirac operator sees less. With the convention above, \(\sum_ie_i\cdot K(e_i)\cdot = -\mathrm{tr}\,K\), so \(D\psi = \mp\tfrac12(\mathrm{tr}\,K)\psi\) for a \(\nabla^\pm\)-parallel \(\psi\). The sign depends on the Clifford convention; only \(\mathrm{tr}\,K\) appears. \textbf{Remark 9.5 (why the split is real).} In Euclidean signature \(\mathrm{Spin}(4) = SU(2)\times SU(2)\) has two real factors, so \(A^\pm\) are real connections. For spacelike hypersurfaces of \(\mathbb R^{3,1}\) the structure algebra is \(\mathfrak{so}(3,1) \cong \mathfrak{sl}(2,\mathbb C)\), which does not split over \(\mathbb R\): its two halves are complex conjugates, and the analogous connection is complex. That complex connection is the self-dual connection of Ashtekar (1986). \emph{Checks.} All exact. Theorem 9.1 is checked on a generic metric with a generic \(K\), with the Gauss and Codazzi blocks computed separately. On the paraboloid \(z = \lvert x\rvert^2/2\) in \(\mathbb R^4\) both halves are flat, and \(1.1K\) breaks Gauss alone. Corollary 9.2 is checked on a non-realizable metric with \(K = \tilde K\), and Proposition 9.3 directly (\texttt{test\_343}). For Proposition 9.4 (\texttt{test\_334}): \begin{itemize} \tightlist \item the two half-spin representations; \item the spinor curvature as the image of \(F_{A^\pm}\), with Gauss and Codazzi computed separately; \item parallel spinors of both signs on the paraboloid; \item Killing spinors on the round \(S^3\), and none for \(K = 2\,\mathrm{Id}\); \item the Dirac identity. \end{itemize} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{where-the-problem-sits-not-load-bearing}{% \section{10. Where the problem sits (not load-bearing)}\label{where-the-problem-sits-not-load-bearing}} The classical programmes that settle local realizability in codimension one do not reach \(n = 3\): \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3333}} >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3333}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright programme \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright requires \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright at \(n = 3\) \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot Thomas (1936, §6): Codazzi follows from Gauss & second fundamental form of rank \(\geq 4\) & the rank is at most 3 \\ Vilms (1977), via Chow (1949) & \(n \geq 5\) & not reached \\ conformal deformation of hypersurfaces (Cartan 1917), a related rigidity theory & \(n \geq 5\) & not reached \\ \end{longtable} This is not an accident. The Gauss equation has \(n^2(n^2-1)/12\) components and \(K\) has \(n(n+1)/2\). The difference is \[\frac{n^2(n^2 - 1)}{12} - \frac{n(n+1)}2 = \frac{n(n-3)(n+1)(n+2)}{12},\] which vanishes only at \(n = 3\). Equivalently, \(\dim\Lambda^2 = n\) only at \(n = 3\), and Hodge duality identifies the curvature operator on \(\Lambda^2\) with an endomorphism of \(T\Sigma\). At \(n = 3\) that endomorphism is \(\mathbb M\): in an eigenframe of \(K\) the curvature operator is \(e_i\wedge e_j \mapsto k_ik_j\,e_i\wedge e_j\), and its eigenvalues are the diagonal of \(\mathrm{adj}\,K\). Vilms's determinant condition (his Theorem 3(ii)) is then the gate \(\det\mathbb M > 0\). Fodor (2019) reaches the same sign structure through the logarithm of the curvature operator. Thomas's rank condition is what makes Codazzi \textbf{redundant} in higher dimensions, not what makes the criterion \textbf{statable}. For \(n \geq 4\) with rank at least 4, Codazzi follows from Gauss. At \(n = 3\) it is independent, and that independence is the subject of this paper. \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{regularity-not-load-bearing}{% \section{11. Regularity (not load-bearing)}\label{regularity-not-load-bearing}} \(F\) needs three derivatives of the metric: two for \(\mathbb M\) and one for \(\mathrm{Cod}\). This is a different question from the thresholds of isometric embedding, which concern immersions of low regularity: \begin{itemize} \tightlist \item In codimension one, \(C^{1,\alpha}\) isometric immersions are flexible, in the sense of the \(h\)-principle (Gromov 1986), for \(\alpha < 1/(1 + n^2 - n)\), which is \(1/7\) at \(n = 3\) (Cao, Hirsch and Inauen 2025). This improves Borisov's \(1/(1 + n(n+1))\) (Borisov 1965, 2004; Conti, De Lellis and Székelyhidi 2012). \item They are rigid for \(\alpha > 2/3\) (Borisov 1959; Conti, De Lellis and Székelyhidi 2012). \item For surfaces in \(\mathbb R^3\) the critical exponent is conjectured to be \(1/2\). \end{itemize} The results of this paper are statements of the rigid regime. \textbf{Observation 11.1 (the chiral form's one nonlinearity).} In Theorem 9.1, \(\mathcal C = d_\Gamma K\) is linear in \(K\) and passes to weak limits. The only nonlinearity is \(K\wedge K\) in \(\mathcal G\), the \(2\times2\) minors of \(K = d\nu\): the Jacobian of the Gauss map \(\nu\). For an immersion of class \(C^{1,\alpha}\), \(\nu \in C^{0,\alpha}\). Its \(2\times2\) minors \(d\nu^a\wedge d\nu^b = d(\nu^a\,d\nu^b)\) are defined as distributions, and depend continuously on \(\nu\) in \(C^{0,\alpha}\), exactly when \(\alpha > 1/2\) (Brezis and Nguyen 2011, sharp). So in chiral form the realizability equations make weak sense exactly above \(\alpha = 1/2\), the exponent conjectured for surfaces. This is an observation about where the equations can be read, not a rigidity or flexibility theorem. The estimate that would make it one is not attempted here. \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{higher-dimensions-not-load-bearing}{% \section{12. Higher dimensions (not load-bearing)}\label{higher-dimensions-not-load-bearing}} For a hypersurface \(\Sigma^n \subset E^{n+1}\), write \(T_m\) for the Newton transformations of \(K\). Then \[\mathrm{adj}\,K = T_{n-1}, \qquad -G = T_2,\] in every dimension, and the even-order \(T_{2k}\) are intrinsic, being polynomial in the curvature (Reilly 1973; Labbi 2007). For odd \(n\), \(T_{n-1}\) is therefore intrinsic. On \(\{\det T_{n-1} > 0\}\) the tensor \[\tilde K = (\det T_{n-1})^{1/(n-1)}\,T_{n-1}^{-1}\] satisfies \(\mathrm{adj}\,\tilde K = T_{n-1}\). For \(n \geq 5\) this is a \textbf{contracted} identity, not the Gauss equation. The Riemann tensor has more components than \(T_{n-1}\) carries, and equality of the contracted quantities does not give \(R = \tilde K\wedge\tilde K\). Full Gauss is a further algebraic condition on the metric. At \(n = 3\) the two coincide, because the Riemann tensor is determined by the Einstein tensor and \(T_2 = T_{n-1}\). For even \(n\), \(\mathrm{adj}\,K\) changes sign with \(K\) and is not intrinsic. So the reduction of §3 is special to \(n = 3\), and the higher analogue of \(F\) is an operator on the metrics satisfying the extra algebraic conditions. It is not developed here. \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{open-problems}{% \section{13. Open problems}\label{open-problems}} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item \textbf{The symbol module at every shape operator.} Theorem C's identification \(g_q = M_K^q\), and with it involutivity, is proved in exact arithmetic at sixteen shape operators, the umbilic one included, and so holds off a proper real-algebraic subset \(\mathcal X\). Is \(\mathcal X\) empty? The count of five conditions is already proved at every \(K\) by a single monomial minor (Theorem C(2)). The same computation for \(\tau_4\) and the flag restrictions, over \(\mathbb Q(k_1, k_2, k_3)\), would settle it. \item \textbf{Local structure near \(\gamma\).} The equivariant implicit-function construction that Theorem E(5) makes available: a manifold chart for the realizable set modulo diffeomorphisms (the Ebin--Marsden half). \item \textbf{Theorem G on a general base,} as a theorem rather than as the exact checks of Observation 7.7. Is the drift's second-order zero a double root on every base? \item \textbf{Round shells.} Is every realizable metric whose \(r\)-shells are intrinsically round spheres locally the metric of a hypersurface swept by round 2-spheres (Proposition 7.4)? An intrinsically round shell need not be extrinsically round. \item \textbf{Quantitative realizability.} If \(F(g)\) is small, is \(g\) close to a realizable metric? The relevant apparatus is the rigidity theory of non-Euclidean elasticity, for instance Kupferman, Maor and Shachar (2019). Theorem E(4) gives the linear answer at \(\gamma\), on TT. \item \textbf{Coercive estimates off umbilics.} An elliptic estimate for \(F\) in the gauge \(\mathrm{div}\,h = 0\), \(\langle h, \tilde K\rangle = 0\) of Corollary 4.2(2), on both branches. TT-characteristic covectors are no obstruction to it. \item \textbf{Higher dimensions.} The operator built on \(T_{n-1}\) for odd \(n \geq 5\), on the metrics where full Gauss holds (§12), and whether its linearisation at the round \(S^n\) factorises as in Theorem E. \item \textbf{Attribution.} Rosenson (1940, 1941, 1943), cited by Vilms as a further solution of Gauss and Codazzi, has not been read here. If it contains the polynomial form of §8.1, the attribution there changes. \item \textbf{Rivertz's equalities.} Are Rivertz's six equalities the trace-free part of \(\mathcal R\) (Proposition 8.1(5)), up to an invertible linear change? The bidegrees and the counts agree. \end{enumerate} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{appendix-v.-verification-index-and-disclosure}{% \section{Appendix V. Verification index and disclosure}\label{appendix-v.-verification-index-and-disclosure}} Every displayed result is reproduced by at least one of the scripts deposited with this paper. These are tests 325--327 and 329--344, with their library \texttt{ratjets.py} and a run log, at DOI 10.5281/zenodo.23765515. Each script states its expectations before computing and exits non-zero on any failure. Where a check had to be re-posed after its first run, the script records the first run, what failed and why. Arithmetic is exact (rational, symbolic, or exact algebraic) except where the index says floating point. A few zero tests on exact algebraic expressions fall back to evaluation at 60 digits, as the scripts state. The numbering continues that of the author's archive (concept DOI 10.5281/zenodo.21264118). Earlier archive scripts that covered the same results are superseded for this paper; the bundle's README lists them. \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.3902}} >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.0610}} >{\raggedright\arraybackslash}p{(\columnwidth - 4\tabcolsep) * \real{0.5488}}@{}} \toprule\noalign{} \begin{minipage}[b]{\linewidth}\raggedright result \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright where \end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright verification \end{minipage} \\ \midrule\noalign{} \endhead \bottomrule\noalign{} \endlastfoot \(\det D\,\mathrm{adj}_K = -2(\det K)^2\) & §2.4 & \texttt{test\_330} P1 (symbolic) \\ Theorem A & §3 & \texttt{test\_335} P1--P4 (symbolic); \texttt{test\_330} P2 \\ Example 3.3 (Flamm) & §3 & \texttt{test\_330} P3 \\ Lemma 4.1 & §4 & \texttt{test\_326} P1 (symbolic) \\ The symbol (4.1) against the full nonlinear operator & §4 & \texttt{test\_335} P5 (both branches) \\ Theorem B(1): rank, kernel & §4 & \texttt{test\_325} P1; \texttt{test\_332} P3; with Lemma 4.1, \texttt{test\_326} P1 \\ Theorem B(2): the TT criterion and the counterexample & §4 & \texttt{test\_325} P2 (floating point, with the exact counterexample) \\ Corollary 4.2 & §4 & \texttt{test\_332} P3 \\ Lemma 4.6 & §4 & \texttt{test\_325} P6 \\ Theorem C(2): five conditions at every nondegenerate \(K\) & §4 & \texttt{test\_332} P1 (symbolic); \texttt{test\_325} P3 (floating point) \\ Theorem C(3): Cartan's test, \(\dim g_q\), \(g_q = M_K^q\) & §4 & \texttt{test\_332} P2 (exact, sixteen \(K\)); \texttt{test\_325} P4--P5, \texttt{test\_326} P2--P3 (floating point, \(q \leq 9\) at two \(K\)); flag bounds \texttt{test\_344} P4 \\ The Kulkarni--Nomizu determinant & §4 & \texttt{test\_326} P4 (symbolic) \\ Theorem D & §5 & proof; \texttt{test\_335} P5 \\ Theorem E(1) & §6 & \texttt{test\_327} P1--P2 \\ Theorem E(2)--(3) & §6 & \texttt{test\_336} P1 (operator identity on TT jets), P2 \\ Theorem E(4) & §6 & \texttt{test\_336} P2 (level 3) \\ Theorem E(5) & §6 & proof; \texttt{test\_336} P3 (the augmented symbol) \\ Remark 6.1: \(\zeta(0) = 10\), its scope, the determinant & §6 & \texttt{test\_336} P4; \texttt{test\_333} \\ Proposition 6.2 & §6 & \texttt{test\_327} P4; \texttt{test\_336} P5 \\ Proposition 6.3 & §6 & \texttt{test\_327} P3 \\ Theorem F & §7 & \texttt{test\_337} P1--P7; the components of \(\mathbb M_1\), \texttt{test\_344} P2 \\ Proposition 7.1 & §7 & \texttt{test\_337} P4, P8, P9 \\ Lemma 7.3; Proposition 7.6 & §7 & \texttt{test\_330} P5--P6; \texttt{test\_338} P7 \\ Proposition 7.4 & §7 & \texttt{test\_338} P1, P3; the chart's rotation, \texttt{test\_344} P1 \\ Corollary 7.5 & §7 & \texttt{test\_338} P2--P6, P8; \texttt{test\_337} P7 \\ Theorem G & §7 & \texttt{test\_339}; \texttt{test\_338} P4 \\ Observation 7.7 & §7 & \texttt{test\_340} \\ Proposition 8.1 & §8 & \texttt{test\_341} P1--P2 (\(\mathcal T\)); \texttt{test\_331} (\(\mathcal R\)) \\ Proposition 8.2 and its example & §8 & \texttt{test\_329} P1--P2; \texttt{test\_331} P3; (2) pointwise, \texttt{test\_344} P3 \\ The symbol near the degenerate locus & §8 & \texttt{test\_341} P3 (rank exact; singular values in floating point) \\ Proposition 8.3; Theorem 8.4; Corollary 8.5 & §8 & \texttt{test\_342}; \texttt{test\_329} P3--P4; \texttt{test\_331} P4 \\ Theorem 9.1; Corollary 9.2; Proposition 9.3 & §9 & \texttt{test\_343} \\ Proposition 9.4 & §9 & \texttt{test\_334} \\ The count of §10 & §10 & \texttt{test\_330} P4 \\ \end{longtable} \textbf{Disclosure.} Large-language-model systems (Anthropic's Claude) were used throughout, under the author's direction, for: \begin{itemize} \tightlist \item derivation and derivation-checking; \item drafting; \item writing and running the verification scripts; \item an independent referee read of this draft, by a separate instance that had not seen it written, and a second independent read by Google Deep Review. \end{itemize} Every load-bearing claim is gated by the scripts above. An independent machine-assisted audit (October 2026) of the earlier mathematics papers preceded this draft. Results credited to it in the text are its own, and each is also checked by a script here. All errors remain the author's. \hypertarget{appendix-c.-changes-from-the-deposited-versions}{% \section{Appendix C. Changes from the deposited versions}\label{appendix-c.-changes-from-the-deposited-versions}} This paper supersedes, for local questions, mathematics paper 1 (Thrussell 2026a, deposited as version 3), mathematics paper 2 (Thrussell 2026b, version 2) and mathematics paper 4 (Thrussell 2026d, draft 1). The statements below change. Labels such as ``Theorem E'' refer to the paper named, and the corresponding result here is given in brackets. The global results of papers 2, 3 and 5 (Thrussell 2026b, c, e) are treated in the companion paper. \textbf{Mathematics paper 1.} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item Theorem A(2)'s proof assumed every eigenvalue of \(\mathbb M\) positive. It needs only nonzero eigenvalues with positive product (§3). \item The example of the Riemannian Schwarzschild metric was called complete. The exterior sheet is not. The complete hypersurface is the two-ended Flamm surface (Example 3.3). \item Global sufficiency needs a simply connected domain, or trivial monodromy. The round \(\mathbb{RP}^3\) satisfies \(F = 0\) and is not realizable (Remark 3.2). \item Theorem D(2) {[}here Theorem F(2){]} excludes \(\ell \geq 2\) only within its own family. This was corrected in the repository's version 4, which is not deposited. \item Proposition F's ``invariant splitting'' into translations, moduli and Lie directions overlaps, and the shell dipole is a local deformation on punctured shells, not a global smooth modulus (Proposition 6.2). \item Theorem E {[}here Theorem G{]} is stated on interior shells. Its general-base statements are exact checks on three bases, not a theorem for every base. The transversal zero they report comes from freezing the drift's coefficients at a point; with the drift itself the zero is a double root on every base checked (Observation 7.7). Its route 1 is now identified as the tilt (Corollary 7.5). \item Appendix N's description of \(\mathfrak D\)'s zero set as open is withdrawn (Proposition 7.1). \(F\)'s need for three derivatives is separated from the flexibility thresholds (§11). \item §10 on higher dimensions: the reduction gives a contracted identity, not the Gauss equation (§12). \end{enumerate} \textbf{Mathematics paper 2.} The meagreness proof local to the round sphere is replaced by one on the whole nondegenerate domain, on both branches (Theorem D). The TT levels are relabelled to the single convention of §2.6. \textbf{Mathematics paper 4.} \begin{enumerate} \def\labelenumi{\arabic{enumi}.} \tightlist \item ``Nine Codazzi components against six'': the count is nine components, four identities, five conditions, at every nondegenerate shape operator (Theorem C(2)). \item The rank and kernel theorems were established at sample shape operators. They are now proved for every nondegenerate \(K\) (Theorem B). \item TT-injectivity was claimed wherever \(\det\mathbb M \neq 0\). It holds exactly where \(\sigma_1g - K\) is definite (Theorem B(2)). Overdetermined ellipticity was claimed on the same set. Modulo diffeomorphisms it holds nowhere, because the symbol also kills the normal direction \(K\); modulo the linearised immersions it holds everywhere on \(\mathcal U\) (Corollary 4.2). \item ``The fourth kernel direction is conformal iff \(K \propto g\)'': at a fixed covector, iff \(K\vert_{\xi^\perp}\) is scalar (Theorem B(3)). \item Meagreness was argued from a cokernel. That argument is not a proof, and is replaced (Theorem D). \item ``\(P\) {[}here \(\mathbb P\){]} vanishes at \(\det\mathbb M = 0\) and the equation is vacuous there'': true only where \(\mathrm{rank}\,\mathbb M \leq 1\). ``The same symbol'': the same up to the factor \(2(\det\mathbb M)^{3/2}\) (Propositions 8.1--8.2). \item Sbrana and Cartan classify the deformable rank-two hypersurfaces, not all of them (Corollary 8.5). \item The Cartan characters \((2,2,1)\) and \(\mathrm{rank}\,\tau_4 = 15\) are wrong. The correct values are \(\mathrm{rank}\,\tau_4 = 12\) and characters \((36, 15, 4)\), from \(\dim g_{3,k} = 55, 19, 4\). Cartan's test then holds with equality at the sixteen shape operators of Theorem C(3), and so off a proper real-algebraic set. There the paper's conclusion, that the system is involutive, stands. \item The claimed rise of the inverse-free symbol's rank to 3 at the degenerate locus is withdrawn. The degeneration of the injectivity constant on TT stands (§8.2). \end{enumerate} \textbf{Elsewhere in the author's archive.} The claim that an \(\ell = 1\) radial coefficient forces a chart drift of matching amplitude (the ``amplitude half'' of what the archive calls the canal tie) is false. When \(\alpha \not\equiv 0\), which holds on the positive branch, an \(\ell = 1\) radial coefficient is realized by a tilt with no chart drift (Corollary 7.5(2)). \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} \hypertarget{references}{% \section{References}\label{references}} Agaoka, Y. and Hashinaga, T. (2022). Intrinsic characterization of 3-dimensional Riemannian submanifolds of \(\mathbb R^4\). arXiv:2206.03634. Allendoerfer, C. B. (1939). Rigidity for spaces of class greater than one. \emph{Amer. J. Math.} \textbf{61}, 633--644. Ashtekar, A. (1986). New variables for classical and quantum gravity. \emph{Phys. Rev.~Lett.} \textbf{57}, 2244--2247. Bär, C., Gauduchon, P. and Moroianu, A. (2005). Generalized cylinders in semi-Riemannian and spin geometry. \emph{Math. Z.} \textbf{249}, 545--580. arXiv:math/0303095. Beckner, W. (1993). Sharp Sobolev inequalities on the sphere and the Moser--Trudinger inequality. \emph{Ann. of Math.} \textbf{138}, 213--242. Beez, R. (1876). Zur Theorie des Krümmungsmaasses von Mannigfaltigkeiten höherer Ordnung. \emph{Z. Math. Phys.} \textbf{21}, 373--401. Berger, E., Bryant, R. and Griffiths, P. (1983). The Gauss equations and rigidity of isometric embeddings. \emph{Duke Math. J.} \textbf{50}, 803--892. Besse, A. L. (1987). \emph{Einstein Manifolds}. Springer, Berlin. {[}Chapter 16 appendix on Codazzi tensors, by A. Derdziński.{]} Borisov, Yu. F. (1959, 1960). On parallel displacement on a smooth surface. \emph{Vestnik Leningrad. Univ.} \textbf{14} (13), 20--26; \textbf{15} (19), 127--129. Borisov, Yu. F. (1965). \(C^{1,\alpha}\)-isometric immersions of Riemannian spaces. \emph{Dokl. Akad. Nauk SSSR} \textbf{163}, 11--13. Borisov, Yu. F. (2004). Irregular surfaces of the class \(C^{1,\beta}\) with an analytic metric. \emph{Sibirsk. Mat. Zh.} \textbf{45}, 25--61. Branson, T. (1995). Sharp inequalities, the functional determinant, and the complementary series. \emph{Trans. Amer. Math. Soc.} \textbf{347}, 3671--3742. Brezis, H. and Nguyen, H.-M. (2011). The Jacobian determinant revisited. \emph{Invent. Math.} \textbf{185}, 17--54. Bryan, P., Ivaki, M. N. and Scheuer, J. (2019). Negatively curved three-manifolds, hyperbolic metrics, isometric embeddings in Minkowski space and the cross curvature flow. arXiv:1906.02381. Bryant, R. L., Chern, S.-S., Gardner, R. B., Goldschmidt, H. L. and Griffiths, P. A. (1991). \emph{Exterior Differential Systems}. MSRI Publications \textbf{18}, Springer, New York. Buckley, R. G. and Schlegel, E. M. (2020). Physical geometry of the quasispherical Szekeres models. arXiv:1908.02697. Cao, W., Hirsch, J. and Inauen, D. (2025). A Nash--Kuiper theorem for isometric immersions beyond Borisov's exponent. arXiv:2503.13867. Cartan, É. (1916). La déformation des hypersurfaces dans l'espace euclidien réel à \(n\) dimensions. \emph{Bull. Soc. Math. France} \textbf{44}, 65--99. Cartan, É. (1917). La déformation des hypersurfaces dans l'espace conforme réel à \(n \geq 5\) dimensions. \emph{Bull. Soc. Math. France} \textbf{45}, 57--121. Cartan, É. (1927). Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien. \emph{Ann. Soc. Polon. Math.} \textbf{6}, 1--7. Chow, B. and Hamilton, R. S. (2004). The cross curvature flow of 3-manifolds with negative sectional curvature. \emph{Turkish J. Math.} \textbf{28}, 1--10. Chow, W.-L. (1949). On the geometry of algebraic homogeneous spaces. \emph{Ann. of Math.} \textbf{50}, 32--67. Conti, S., De Lellis, C. and Székelyhidi, L., Jr.~(2012). \(h\)-principle and rigidity for \(C^{1,\alpha}\) isometric embeddings. In \emph{Nonlinear Partial Differential Equations}, Abel Symposia \textbf{7}, Springer, 83--116. Dajczer, M. and Tojeiro, R. (2019). \emph{Submanifold Theory: Beyond an Introduction}. Springer, New York. Dajczer, M., Florit, L. and Tojeiro, R. (1998). On deformable hypersurfaces in space forms. \emph{Ann. Mat. Pura Appl.} \textbf{174}, 361--390. Derdziński, A. and Shen, C.-L. (1983). Codazzi tensor fields, curvature and Pontryagin forms. \emph{Proc. London Math. Soc.} (3) \textbf{47}, 15--26. Eiesland, J. (1925). The group of motions of an Einstein space. \emph{Trans. Amer. Math. Soc.} \textbf{27}, 213--245. Ferus, D. (1981). A remark on Codazzi tensors in constant curvature spaces. In \emph{Global Differential Geometry and Global Analysis}, Lecture Notes in Math. \textbf{838}, Springer, 257. Fodor, D. G. (2019). Isometric immersions of Riemannian manifolds in \(k\)-codimensional Euclidean space. arXiv:1908.11616. Goldschmidt, H. (1967). Integrability criteria for systems of nonlinear partial differential equations. \emph{J. Differential Geom.} \textbf{1}, 269--307. Gromov, M. (1986). \emph{Partial Differential Relations}. Springer, Berlin. Hartman, P. and Nirenberg, L. (1959). On spherical image maps whose Jacobians do not change sign. \emph{Amer. J. Math.} \textbf{81}, 901--920. Hellaby, C. and Buckley, R. G. (2020). Rotation, embedding and topology for the Szekeres geometry. arXiv:2012.03192. Hellaby, C. and Krasiński, A. (2002). You can't get through Szekeres wormholes, or regularity, topology and causality in quasispherical Szekeres models. \emph{Phys. Rev.~D} \textbf{66}, 084011. Huisken, G. and Sinestrari, C. (2009). Mean curvature flow with surgeries of two-convex hypersurfaces. \emph{Invent. Math.} \textbf{175}, 137--221. Janet, M. (1926). Sur la possibilité de plonger un espace riemannien donné dans un espace euclidien. \emph{Ann. Soc. Polon. Math.} \textbf{5}, 38--43. Karmarkar, K. R. (1948). Gravitational metrics of spherical symmetry and class one. \emph{Proc. Indian Acad. Sci.} \textbf{A27}, 56--60. Killing, W. (1885). \emph{Die nicht-euklidischen Raumformen in analytischer Behandlung}. Teubner, Leipzig. Kuiper, N. H. (1955). On \(C^1\)-isometric imbeddings, I, II. \emph{Nederl. Akad. Wetensch. Proc.} A \textbf{58}, 545--556, 683--689. Kupferman, R., Maor, C. and Shachar, A. (2019). Reshetnyak rigidity for Riemannian manifolds. \emph{Arch. Ration. Mech. Anal.} \textbf{231}. arXiv:1701.08892. Labbi, M.-L. (2007). On \((2k)\)-minimal submanifolds. arXiv:0706.3092. Lawn, M.-A. and Roth, J. (2010). Isometric immersions of hypersurfaces in 4-dimensional manifolds via spinors. \emph{Differential Geom. Appl.} \textbf{28}, 205--219. arXiv:0803.2621. Lin, C.-S. (1985). The local isometric embedding in \(\mathbb R^3\) of 2-dimensional Riemannian manifolds with nonnegative curvature. \emph{J. Differential Geom.} \textbf{21}, 213--230. Lin, C.-S. (1986). The local isometric embedding in \(\mathbb R^3\) of two-dimensional Riemannian manifolds with Gaussian curvature changing sign cleanly. \emph{Comm. Pure Appl. Math.} \textbf{39}, 867--887. Mardare, S. (2003). On isometric immersions of a Riemannian space with little regularity. \emph{Anal. Appl.} \textbf{1}, 433--453. Mardare, S. (2005). On Pfaff systems with \(L^p\) coefficients and their applications in differential geometry. \emph{J. Math. Pures Appl.} \textbf{84}, 1659--1692. Merton, G. (2013). Codazzi tensors with two eigenvalue functions. \emph{Proc. Amer. Math. Soc.} \textbf{141}, 3265--3273. Nadirashvili, N. and Yuan, Y. (2002). Counterexamples for local isometric embedding. arXiv:math/0208127. Nash, J. (1954). \(C^1\) isometric imbeddings. \emph{Ann. of Math.} \textbf{60}, 383--396. Obata, M. (1962). Certain conditions for a Riemannian manifold to be isometric with a sphere. \emph{J. Math. Soc. Japan} \textbf{14}, 333--340. Pandey, S. N. and Sharma, S. P. (1982). Insufficiency of Karmarkar's condition. \emph{Gen.~Relativity Gravitation} \textbf{14}, 113--115. Pogorelov, A. V. (1971). An example of a two-dimensional Riemannian metric admitting no local realization in \(E_3\). \emph{Dokl. Akad. Nauk SSSR} \textbf{198}, 42--43. Reilly, R. C. (1973). Variational properties of functions of the mean curvatures for hypersurfaces in space forms. \emph{J. Differential Geom.} \textbf{8}, 465--477. Rivertz, H. J. (1999). \emph{On Isometric and Conformal Immersions into Riemannian Manifolds}. Doctoral thesis, University of Oslo. Rosenson, N. (1940, 1941, 1943). Sur les espaces riemanniens de classe I; Seconde partie; Troisième partie. \emph{Izv. Akad. Nauk SSSR Ser. Mat.} \textbf{4}, 181--192; \textbf{5}, 325--351; \textbf{7}, 253--284. Sbrana, U. (1909). Sulla varietà ad \(n-1\) dimensioni deformabili nello spazio euclideo ad \(n\) dimensioni. \emph{Rend. Circ. Mat. Palermo} \textbf{27}, 1--45. Seiler, W. M. (2010). \emph{Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra}. Springer, Berlin. Szekeres, P. (1975). A class of inhomogeneous cosmological models. \emph{Commun. Math. Phys.} \textbf{41}, 55--64. Thomas, T. Y. (1936). Riemann spaces of class one and their characterization. \emph{Acta Math.} \textbf{67}, 169--211. Thrussell, J. (2026a). The Einstein adjugate and the Codazzi obstruction: realizability of Riemannian three-manifolds as hypersurfaces of Euclidean four-space (mathematics paper 1), version 3. Zenodo. doi:10.5281/zenodo.21978286. Thrussell, J. (2026b). Meagerness of the realizable set for hypersurfaces of Euclidean four-space (mathematics paper 2). Zenodo. doi:10.5281/zenodo.22171949. Thrussell, J. (2026c). An exclusion atlas for the Thurston geometries as hypersurfaces of Euclidean four-space (mathematics paper 3). Zenodo. doi:10.5281/zenodo.22172083. Thrussell, J. (2026d). The realizability operator: symbol, ellipticity, and meagreness (mathematics paper 4), draft 1. Zenodo. doi:10.5281/zenodo.22844150. Thrussell, J. (2026e). The atlas completed: exclusions, the ladder, and three routes to the round sphere (mathematics paper 5). Zenodo. doi:10.5281/zenodo.22844201. Vilms, J. (1977). Local isometric imbedding of Riemannian \(n\)-manifolds into Euclidean \((n+1)\)-space. \emph{J. Differential Geom.} \textbf{12}, 197--202. York, J. W., Jr.~(1973). Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity. \emph{J. Math. Phys.} \textbf{14}, 456--464. \end{document}