A reversible five-dimensional centre-manifold reduction at the Anderson–Henningson fold: a three-sphere of non-isotropic Kapustin–Witten solutions below the fold

Abstract.

We study the Haydys–Witten (HW) equations for $SU(2)$ on $\mathbb R_s\times S^3_r\times(0,L)$ with Nahm poles at both ends, near the Anderson–Henningson fold $L=\pi r$, in the complete gauge-fixed deformation space. By [20], the static operator at the fold has kernel $\mathbb R\zeta\oplus\mathcal K$, with $\mathcal K\cong\mathbb R^4$ a spin-½ multiplet, and the AH instanton has index $-3$. (A) We construct a five-dimensional, parameter-dependent centre manifold that contains every small bounded solution. It is equivariant under $SO(4)=(SU(2)_L\times SU(2)_{\rm diag})/\pm1$, acting on $\mathcal K$ as on $\mathbb R^4$. It is also reversible under a new reversor $R$, built from $(s,y,q)\mapsto(-s,L-y,\bar q)$ and the gauge transformation $q\mapsto\bar q$. (B) Hence the reduced field is $\dot x=g(x^2,|w|^2,\bar\nu)$, $\dot w=x\,\tilde h(x^2,|w|^2,\bar\nu)\,w$, where $\bar\nu=L^2/r^2-\pi^2$. In the units of [17, 18, 20], $g=\tfrac56x^2-\tfrac43\bar\nu-\tfrac58|w|^2+O_4$ and $\tilde h=-\tfrac54+O_2$. An exact computation of the second-order centre-manifold terms shows that all weighted-cubic coefficients vanish: the coefficients of $|w|^4$, $\bar\nu|w|^2$ and $x^2|w|^2$ in the reduced potential are $0$. (C) For $-\delta<\bar\nu<0$ the KW solutions near the fold form exactly one orbit $\mathcal S_{\bar\nu}\cong S^3$: $x=0$ and $|w|^2=-\tfrac{32}{15}\bar\nu+O(\bar\nu^2)$. These solutions are not left-invariant. Their stabiliser is the $SO(3)$ of isometries fixing a point of $S^3$, and the family is Morse–Bott nondegenerate with small eigenvalues $\pm\sqrt{10|\bar\nu|/3}\,(1+O(\bar\nu))$ and $0^{(3)}$. For $\bar\nu>0$ the solutions are exactly the isotropic $k_\pm$, and $Rk_+=k_-$. (D) The reduced field has an explicit strict Lyapunov function $H=\int_0^x\dot x\,dx$, valid to all orders. Hence near the fold the only HW trajectories are constants and, for $\bar\nu>0$, the AH instanton; there is none from $k_-$ to $k_+$. A fixed ball is isolating uniformly through $\bar\nu=0$, and its Conley homology is $\mathbb Z$ in degrees $1,4$. So the index $-3=-\dim S^3$ of [20] is the shadow of $\mathcal S_{\bar\nu}$. verify.py: 65 checks, FALLOS = 0.

1. Introduction and results

Henningson [1] and Anderson–Henningson [2] studied the HW equations on $\mathbb R_s\times S^3_r\times I_y$ with Nahm poles at both ends of $I=(0,L)$, in the spherically symmetric class. Two static Kapustin–Witten (KW) solutions $k_\pm$ are born at $L=\pi r$ (period expansion [2, (3.8)]), and [2, (4.7), (4.10)–(4.13)] give a formal tunneling profile. Our accepted papers proved:

• existence, local uniqueness and transversality of the instanton $u:k_+\to k_-$ inside the isotropic class [17];
• uniqueness and transversality in the larger $V_4$ class [18];
• the structure of the complete gauge-fixed operator [20]. At the degenerate solution $k_*$ at the fold, $\ker\mathcal L_*=\mathbb R\zeta\oplus\mathcal K$, with $\zeta$ isotropic and $\mathcal K\cong\mathbb R^4$ a spin-½ multiplet. At $k_\pm$ the small eigenvalues are $\tfrac53x_\pm$ and $-\tfrac54x_\pm$ (the latter four times), and $D_u$ has index $-3$, with $\ker D_u=\mathbb R\partial_su$ and $\operatorname{coker}D_u\cong\mathbb R^4$.

[20, Rem. 7.3] left open the nonlinear question: what is the complete local picture near the fold, and do non-isotropic KW solutions exist? This paper answers it by a five-dimensional centre-manifold reduction. Every statement is proved unless it is labelled numerical. The imported ingredients are listed in §8.

Conventions. We work at the fold with radius $1/\rho$ and interval $(0,\pi)$, using scale invariance; then $\bar\nu:=L^2/r^2-\pi^2=\pi^2\mu$ with $\mu:=\rho^2-1$. Fields are deformations $\xi=(\alpha_i,\psi_i,a_s,a_y)$ of the background $A=\rho T_i\sigma^i$, $\varphi=\cot(y)T_i\sigma^i$. Here $T_a$ satisfies $[T_a,T_b]=\epsilon_{abc}T_c$, and $\sigma^i$ is the left-invariant coframe with $d\sigma^i=-2\rho\,\sigma^j\wedge\sigma^k$. Let $X=L^2(S^3\times(0,\pi);\mathbb R^{24})$, and let $Z=\operatorname{dom}\mathcal L$ with the graph norm. The gauge-fixed HW system ((E1)–(E3) and the 5D Coulomb condition, [20, §2]) is $$\partial_s\xi=F(\xi;\mu)=\mu\,e_{\rm iso}+\mathcal L_\rho\,\xi+Q(\xi,\xi),\qquad \mathcal L_\rho=J\partial_y+\rho\,\mathcal D+\cot(y)V_1, \tag{1.1}$$ where:

• $e_{\rm iso}$ is the constant field $\alpha_i=T_i$;
• $\mathcal D$ is the unit-sphere twisted Dirac operator of [20];
• $Q$ is the pointwise symmetric bilinear form collecting the commutator terms. It has no derivatives and does not depend on $\rho$.

That the forcing is exactly $\mu e_{\rm iso}$, and that $\mathcal L_\rho$ depends on $\rho$ only through $\rho\mathcal D$, is checked in verify.py §1 and §5.

Let $N:=x_2T_1-x_1T_2-x_0T_3$, where $q=(x_0,\dots,x_3)\in S^3\subset\mathbb R^4$ are quaternionic coordinates, $n=(x_2,-x_1,-x_0)$ and $|n|^2=1-x_3^2$. Write $\zeta=\sin^2y\,e_{\rm iso}$. Let $V_0$ be the field with $$a_y=\sin y\,N,\qquad \psi_i=\tfrac12\sin y\,[T_i,N],\qquad \alpha=a_s=0 .$$ $V_0$ spans $\mathcal K$ under $SO(4)$ [20, Thm. B]. One has $\|\zeta\|^2=\tfrac94\pi^3=2\|V_0\|^2$. Normalized coordinates on $\ker\mathcal L_*$ are $$\xi_c=-\tfrac{x}{\pi}\,\zeta+\tfrac{\sqrt2}{\pi}\sum_aw_aV_a ,$$ with $(V_a)$ an $SO(4)$-orthogonal basis of $\mathcal K$ with $\|V_a\|=\|V_0\|$, and with time $t=s/\pi$. These coincide with the coordinates and time of [17, 18, 20], where $\dot x=\tfrac56x^2-\tfrac43\bar\nu+\dots$, and they make the reduced metric conformally Euclidean.

Theorem A (equivariant reversible centre manifold). Fix $k\ge4$. There exist $\delta>0$, neighbourhoods $\mathcal U\subset Z$ of $0$ and $(-\delta,\delta)\ni\bar\nu$, and a $C^k$ map $\Psi:\ker\mathcal L_*\times(-\delta,\delta)\to Z\cap(\ker\mathcal L_*)^\perp$ with $$\Psi(0,0)=0,\qquad D_{\xi_c}\Psi(0,0)=0,\qquad \partial_{\bar\nu}\Psi(0,0)=\pi^{-2}\Psi_\mu\neq0,$$ where $\Psi_\mu$ is the field $\psi_i=-\tfrac13\sin y\cos y\,T_i$. Tangency holds only in the centre variables. Equivalently, in the shifted coordinates $\tilde\xi=\xi-\mu\Psi_\mu$ of §2 the manifold is the graph of $\tilde\Psi=\Psi-\mu\Psi_\mu$, which is tangent jointly, $D\tilde\Psi(0,0)=0$. $\Psi$ has the following properties.

1. $\mathcal M_{\bar\nu}=\{\xi_c+\Psi(\xi_c,\bar\nu)\}$ is locally invariant under (1.1).
2. Every solution $\xi(s)$ of (1.1) with $\xi(s)\in\mathcal U$ for all $s\in\mathbb R$ lies on $\mathcal M_{\bar\nu}$. In particular every KW solution in $\mathcal U$ does.
3. $\Psi$ is $SO(4)$-equivariant, where $SO(4)$ acts on the fields by left translations and by the diagonal combination of right translations with rotations of the form and Lie indices.
4. $\Psi$ commutes with the reversor $R$ of Proposition 3.3. This reversor anticommutes with $F$ ($F(R\xi)=-RF(\xi)$) and is an involution. It acts on $\zeta$ by $-1$, and on $\mathcal K$ as a reflection $\sigma\in O(4)\setminus SO(4)$ (eigenvalues $-1,-1,-1,+1$).

Theorem B (the reduced field). In normalized coordinates $(x,w)\in\mathbb R\times\mathbb R^4$ the reduced field is $$\dot x=g(x^2,|w|^2,\bar\nu),\qquad \dot w=x\,\tilde h(x^2,|w|^2,\bar\nu)\,w, \tag{1.2}$$ with $g,\tilde h$ of class $C^{k-2}$ and $$g=\tfrac56x^2-\tfrac43\bar\nu-\tfrac58|w|^2+O_4,\qquad x\tilde h=-\tfrac54x+O_3 . \tag{1.3}$$ Here $O_m$ means weighted order $\ge m$, with $x,w$ of weight $1$ and $\bar\nu$ of weight $2$. Equivalently, all weighted-cubic Taylor coefficients of the reduced field vanish: $$[x^3],\ [x\bar\nu],\ [x|w|^2]\ \text{in }\dot x;\qquad [x^2w],\ [|w|^2w],\ [\bar\nu w]\ \text{in }\dot w .$$ These are coefficients of the reduced vector field. The quantities $\kappa,\beta$ of the original question are coefficients of a potential; they are the coefficients of $|w|^4$ and $\bar\nu|w|^2$ in the Lyapunov function $H$ of Theorem D, and they vanish identically (Remark 4.1). The truncation of (1.2) at weighted order $2$ is the gradient of the following function, for the Euclidean metric of the normalized coordinates. At the origin this metric is a constant multiple of the $L^2$ metric on $\ker\mathcal L_*$; it is not claimed to be the metric induced on the nonlinear manifold $\mathcal M_{\bar\nu}$: $$\mathsf W=\tfrac5{18}x^3-\tfrac43\bar\nu\,x-\tfrac58\,x|w|^2 .$$ The vanishing of the cubic coefficients is computed exactly (§4, verify.py §4–5); the parity structure is forced by Theorem A(3, 4).

Theorem C (KW solutions near the fold). There is $\delta>0$ such that, for $0<|\bar\nu|<\delta$, the KW solutions in $\mathcal U$ (in the Coulomb slice) are the following.

1. For $\bar\nu>0$: exactly the two isotropic solutions $k_\pm$, with $x_\pm=\pm\sqrt{X(\bar\nu)}$, $X=\tfrac85\bar\nu+O(\bar\nu^2)$ and $w=0$. The reversor exchanges them: $Rk_+=k_-$.

2. For $\bar\nu<0$: exactly one $SO(4)$-orbit $\mathcal S_{\bar\nu}=\{x=0,\ |w|^2=\Sigma(\bar\nu)\}\cong S^3$, with $\Sigma=-\tfrac{32}{15}\bar\nu+O(\bar\nu^2)$. These solutions:

   • are not invariant under $SU(2)_L$, so they are neither isotropic nor $V_4$-symmetric, and they are not gauge-equivalent to any solution of [17, 18];
   • have stabiliser $SO(3)$. This is the group of isometries of $S^3$ fixing a point $p_0$ (for $V_0$, $p_0=\pm(0,0,0,1)$), lifted by gauge rotations;
   • form an $R$-invariant set.

   Their linearization has exactly five eigenvalues near $0$: $\pm\sqrt{10|\bar\nu|/3}\,(1+O(|\bar\nu|))$ (in time $t$) and $0$ with multiplicity three, which is semisimple with eigenspace $T\mathcal S_{\bar\nu}$. So $\mathcal S_{\bar\nu}$ is a Morse–Bott nondegenerate critical manifold, with one reduced unstable normal direction: one unstable direction of the reduced field, i.e. one positive eigenvalue among the five small eigenvalues of the linearization.

3. For $\bar\nu=0$: only $k_*$.

To second order (radius $r=1/\rho>1$, interval $(0,\pi)$, $\mu=\rho^2-1<0$, $W^2=-\tfrac{64}{15}\mu$), the solution of $\mathcal S$ through the $V_0$-direction is $$\begin{aligned}A_y&=W\sin y\,N,\\ A&=\big(\rho T_i+\tfrac{5}{32}W^2x_3\sin^2y\,[T_i,N]\big)\sigma^i,\\ \varphi&=\big(\cot y\,T_i+\tfrac W2\sin y\,[T_i,N]-\tfrac{\mu}{3}\sin y\cos y\,T_i-\tfrac{3W^2}{32}\sin y\cos y\,(n_iN-\tfrac{|n|^2}{3}T_i)\big)\sigma^i,\end{aligned}$$ each up to $O(|\mu|^{3/2})$.

Theorem D (trajectories and the local Conley/Floer picture). Shrinking $\mathcal U$ and $\delta$ if necessary, every solution of the gauge-fixed HW system (1.1) with $\xi(s)\in\mathcal U$ for all $s$ is one of the following.

1. For $0<\bar\nu<\delta$: $k_+$, $k_-$, or a time-translate of the AH instanton $u$ of [17]. In particular:

   • $u$ is the unique trajectory from $k_+$ to $k_-$ in $\mathcal U$, in the complete space;
   • there is no trajectory from $k_-$ to $k_+$, although that pair has index $+3$.

2. For $-\delta<\bar\nu<0$: a constant trajectory at a point of $\mathcal S_{\bar\nu}$. There are no trajectories between distinct points of the family.

3. A neighbourhood $\bar B$ of $0$ in the centre coordinates is an isolating neighbourhood of the reduced flow, uniformly for $|\bar\nu|<\delta$. Grade critical elements by the reduced index, the number of positive eigenvalues among the five small ones: $k_+$ has $1$, $k_-$ has $4$, and $\mathcal S$ has reduced normal index $1$. The homology Conley index of $\operatorname{Inv}(\bar B)$ is $\mathbb Z$ in degrees $1$ and $4$ (the homology of $S^1\vee S^4$) for every $|\bar\nu|<\delta$, including $\bar\nu=0$, where $\operatorname{Inv}(\bar B)=\{k_*\}$.

   • For $\bar\nu>0$ it is computed from the Morse decomposition $(k_+,k_-)$ by the attractor–repeller exact sequence, whose connection map vanishes for degree reasons.
   • For $\bar\nu<0$ it is the Thom space of the reduced unstable bundle of $\mathcal S$.

   This is not an ordinary Morse–Smale complex: the connection $k_+\to k_-$ is non-transverse, and no perturbation is used. The Conley exact sequence needs no transversality, and the connection matrix is $0$ by degree.

What this means. The index $-3$ of the AH instanton [20] is $1-4=-\dim S^3$. On the other side of the fold the pair $(k_+,k_-)$ is replaced by a three-sphere of non-isotropic KW solutions. The topology of that sphere, $H_*(S^3)$ shifted by one, is what the obstructed pair carries homologically: the arrow $k_+\to k_-$ is not a differential (wrong degree), and it does not need to be. There is no "coupled concatenation" of $k_\pm$ with the family, because they never coexist: the family exists only for $L<\pi r$ and the pair only for $L>\pi r$. This is a saddle-type equivariant bifurcation in which the reversibility forces the family to lie exactly on $\{x=0\}$.

2. Hypotheses of the centre-manifold theorem

We apply Haragus–Iooss [5, Ch. 2, Thm. 2.20 (Hilbert-space version) and Thm. 3.3 (parameters and equivariance)], as in [17, §5], to $\partial_s\xi=\mathcal L\xi+\mathcal R(\xi,\mu)$. Here $\mathcal L=\mathcal L_1$ (the fold operator $\mathcal L_*$ of [20]) and $$\mathcal R(\xi,\mu)=\mu\,e_{\rm iso}+(\rho(\mu)-1)\,\mathcal D\xi+Q(\xi,\xi),\qquad \rho(\mu)=\sqrt{1+\mu}.$$

(H1) Spectrum. By [20, Thms. A–B and Prop. 4.2], $\mathcal L$ is self-adjoint on $Z$, with $$\sigma(\mathcal L)\cap(-1,1)=\{0\},$$ where $0$ is an eigenvalue of multiplicity $5$, with eigenspace $\mathbb R\zeta\oplus\mathcal K$. Hence the central spectrum is $\sigma(\mathcal L)\cap i\mathbb R=\{0\}$ (semisimple, finite multiplicity), and the rest of the spectrum is real with $|\lambda|\ge1$.

(H2) Resolvent. For self-adjoint $\mathcal L$ and $\omega\in\mathbb R$, $\|(i\omega-\mathcal L)^{-1}\|_{X\to X}\le|\omega|^{-1}$. This is the Hilbert-space hypothesis of [5, Thm. 2.20].

(H3) Nonlinearity. We need $\mathcal R\in C^k(Z\times(-\delta,\delta);X)$ with $\mathcal R(0,0)=0$ and $D_\xi\mathcal R(0,0)=0$. Since $\mathcal R$ is polynomial in $\xi$ and analytic in $\mu$, it suffices that $\mathcal D:Z\to X$ and $Q:Z\times Z\to X$ are bounded. This is Lemma 2.1.

Lemma 2.1. $Z\hookrightarrow H^1(S^3\times(0,\pi))$ continuously. Hence $Z\hookrightarrow L^4$, and $\|Q(\xi,\eta)\|_X\le C\|\xi\|_Z\|\eta\|_Z$.

Proof.

• Bound on $\mathcal D$. By [20, Thm. A], $\|\mathcal Lu\|^2=\|Ju'+\cot(y)V_1u\|^2+\|\mathcal Du\|^2$, so $\|\mathcal Du\|\le\|\mathcal Lu\|$.

• Tangential derivatives. Write $\mathcal D=\sum N_ie_i+V_0$ with $\|V_0\|=3$. On the sector $H_j$ one has $(\sum N_ie_i)^2\ge4j^2\ge\tfrac13\cdot4j(j+1)=\tfrac13\Delta_{S^3}$ for $j\ge\tfrac12$ [20, Lemma 4.4]. Hence $\|\nabla_{S^3}u\|\le\sqrt3(\|\mathcal Du\|+3\|u\|)$.

• Normal derivative. From (1.1), $u'=\cot(y)Bu+g$ with $B=JV_1$ and $g=J(\mathcal Du-\mathcal Lu)$, so $\|g\|\le2\|\mathcal Lu\|$. In an eigenbasis of the constant symmetric matrix $B$ (eigenvalues $\pm1,\pm2$), each component on $(0,\tfrac\pi2]$ is given by a Hardy-type integral:

  • for $\sigma\ge1$: $u_\sigma=\sin^\sigma y\,\big(c+\int_{\pi/2}^yg_\sigma\sin^{-\sigma}\big)$;
  • for $\sigma\le-1$: $u_\sigma=\sin^{-|\sigma|}y\int_0^yg_\sigma\sin^{|\sigma|}$, where the constant vanishes by $L^2$ [20, Prop. 4.2].

  The kernels $y^{\sigma-1}t^{-\sigma}\mathbf 1_{t>y}$ ($\sigma\ge1$) and $y^{-|\sigma|-1}t^{|\sigma|}\mathbf 1_{t<y}$ ($|\sigma|\ge1$) are bounded on $L^2(0,\tfrac\pi2)$ by Schur's test. So $\|u/\sin y\|+\|u'\|\le C(\|\mathcal Lu\|+\|u\|)$ on $(0,\tfrac\pi2]$. The end $y=\pi$ is symmetric. These estimates are pointwise in the $L^2(S^3)$-valued sense, so they hold on all of $Z$.

• Conclusion. This gives $Z\hookrightarrow H^1$. The Sobolev embedding $H^1\hookrightarrow L^4$ in dimension $4$, applied on the Lipschitz domain $S^3\times(0,\pi)$, and Hölder's inequality give the bound on $Q$. ∎

(H4) Symmetries.

• $SO(4)$ acts unitarily on $X$, preserves $Z$, commutes with $\mathcal L_\rho$ and $Q$, and fixes $e_{\rm iso}$. Isometries and constant gauge rotations preserve the equations and the background [20, Prop. 4.2(2)].
• The reversor $R$ (Proposition 3.3) is a unitary involution of $X$ preserving $Z$, with $R\mathcal L_\rho=-\mathcal L_\rho R$, $Q(R\cdot,R\cdot)=-RQ(\cdot,\cdot)$ and $Re_{\rm iso}=-e_{\rm iso}$.
• $\mathcal L$ commutes with $SO(4)$ and anticommutes with $R$, so the spectral projection $P$ onto $\ker\mathcal L$ commutes with both.

Lemma 2.2 (removing the hyperbolic forcing; cf. [17, Lemma 5.2 and the correction remark after it]). One has $$\Pi_he_{\rm iso}=\big(1-\tfrac43\sin^2y\big)e_{\rm iso}\neq0,\qquad \mathcal L\Psi_\mu=-\Pi_he_{\rm iso},\qquad \Psi_\mu\perp\ker\mathcal L_*,\qquad R\Psi_\mu=\Psi_\mu,$$ and $\Psi_\mu$ is $SO(4)$-invariant. The substitution $\xi=\mu\Psi_\mu+\tilde\xi$ transforms (1.1) into $$\partial_s\tilde\xi=\mathcal L\tilde\xi+\tilde{\mathcal R}(\tilde\xi,\mu),\qquad \tilde{\mathcal R}=\tfrac43\mu\,\zeta+(\rho(\mu)-1)\mathcal D(\tilde\xi+\mu\Psi_\mu)+Q(\tilde\xi+\mu\Psi_\mu,\tilde\xi+\mu\Psi_\mu),$$ with $\tilde{\mathcal R}(0,0)=0$, $D_{\tilde\xi}\tilde{\mathcal R}(0,0)=0$ and $\Pi_h\partial_\mu\tilde{\mathcal R}(0,0)=0$. The forcing is now $\tfrac43\mu\zeta=\mu Pe_{\rm iso}\in PX$. The shifted system is still $SO(4)$-equivariant, and $R$ is still a reversor, because $R\Psi_\mu=\Psi_\mu$ and $Re_{\rm iso}=-e_{\rm iso}$.

Proof. The identities are verified exactly in verify.py §4, §6 and §7b. Since $\rho(\mu)-1=O(\mu)$, we get $\partial_\mu\tilde{\mathcal R}(0,0)=Pe_{\rm iso}$. ∎

In the unshifted coordinates, differentiating the hyperbolic invariance equation in $\bar\nu$ at the origin gives $\mathcal L_h\partial_{\bar\nu}\Psi(0,0)=-\pi^{-2}\Pi_he_{\rm iso}\neq0$. This is why the joint tangency claimed in the first version was false, exactly as in [17]. Equivalently, appending $\mu'=0$ produces an extended linear part with a Jordan block on $(\zeta,0),(\Psi_\mu,1)$, and the shift straightens it.

Proof of Theorem A. We use the parameter extension of [5, Ch. 2, Thm. 3.3] together with its Hilbert-space criterion [5, Ch. 2, Thm. 2.20], in the form used in [17, Prop. 5.3]. The hypotheses for the shifted system are:

• $X=Y$ and $Z$ Hilbert spaces, and $\mathcal L:Z\to X$ bounded (Hypothesis 2.1);
• $\tilde{\mathcal R}\in C^k(Z\times(-\delta,\delta),X)$ with $\tilde{\mathcal R}(0,0)=0$ and $D_{\tilde\xi}\tilde{\mathcal R}(0,0)=0$, by (H3) and Lemma 2.1 (the parameter hypotheses of Thm. 3.3);
• the spectral decomposition with $\sigma_0=\{0\}$ semisimple of multiplicity $5$ and a gap, by (H1) (Hypothesis 2.4);
• the resolvent bound (H2), which by Thm. 2.20 implies the linear Hypothesis 2.7.

Thm. 3.3 then gives $\tilde\Psi\in C^k$ with $\tilde\Psi(0,0)=0$ and $D_{\xi_c}\tilde\Psi(0,0)=0$, together with items 1–2. Differentiating the invariance equation $D_{\xi_c}\tilde\Psi\cdot f=\mathcal L_h\tilde\Psi+\Pi_h\tilde{\mathcal R}$ in $\mu$ at $(0,0)$, and using $f(0,0)=0$ and Lemma 2.2, gives $\mathcal L_h\partial_\mu\tilde\Psi(0,0)=0$, so $\partial_\mu\tilde\Psi(0,0)=0$. Undoing the shift gives the tangency statements for $\Psi$. Items 3 and 4 follow from the equivariant and reversible versions [5, Ch. 2, §3]: the symmetries act linearly on $\ker\mathcal L_*$, commute with $P$ and preserve the shifted system. The action of $R$ on $\ker\mathcal L_*$ is Proposition 3.3(c). The shift does not change the centre coordinates, since $P\Psi_\mu=0$. So the reduced field is the same in both coordinate systems. ∎

3. Symmetries

Lemma 3.1 (the multiplet). The field $V_0$ of §1 lies in $\ker\mathcal L_*$, and so does $\zeta$. Among the fields $\sin y\cdot(\text{linear})$, the kernel is exactly $4$-dimensional and $L^2$-orthogonal with equal norms; among $\sin^2y\cdot(\text{constant})$ it is exactly $\mathbb R\zeta$ (verify.py §2, solved from the general ansatz). $SO(4)$ acts on $\mathcal K$ as on $\mathbb R^4\cong\mathbb H$ by $w\mapsto g\,w\,\bar h$. The stabiliser of $w\neq0$ is a conjugate of $SO(3)$. Under the identification of $\mathcal K$ with linear functions on $S^3$ (the profile of $\mathcal K$ is $\sin y\cdot v(q)$, with $v$ linear), it is the group of isometries fixing the corresponding point $p_0=w/|w|\in S^3$.

Proposition 3.2 (invariant forms). The $SO(4)$-invariant polynomials on $\mathbb R\times\mathbb R^4$ are the polynomials in $x$ and $|w|^2$. The equivariant maps are $(a(x,|w|^2),\,b(x,|w|^2)\,w)$. The same holds for smooth maps, by Schwarz's theorem; Whitney's theorem gives the smooth version of the parity statements below.

Proposition 3.3 (the reversor). Let $\iota(q)=\bar q$ be the inversion of $S^3$ (orientation reversing), and let $h(q)=\bar q$ as a gauge transformation. For a configuration $(A,\varphi)$ on $\mathbb R\times S^3\times(0,\pi)$ define $$(R(A,\varphi))(s,y,\cdot)=h\cdot\iota^*\big((A,\varphi)(-s,\pi-y,\cdot)\big)$$ (gauge action $A\mapsto hAh^{-1}+h\,dh^{-1}$, $\varphi\mapsto h\varphi h^{-1}$), with the induced signs $a_s\mapsto-a_s$ and $a_y\mapsto-a_y$ from $ds\mapsto-ds$ and $dy\mapsto-dy$. In left-frame components of the deformation: $$(R\xi)^b_j(q,y)=-\textstyle\sum_{a,i}(\mathrm{Ad}_q)_{ab}(\mathrm{Ad}_q)_{ij}\,\xi^a_i(\bar q,\pi-y)\quad(\xi=\alpha,\psi),\qquad (Ra_\bullet)^b=-\textstyle\sum_a(\mathrm{Ad}_q)_{ab}\,a_\bullet^a(\bar q,\pi-y).$$ Then:

• (a) $R$ fixes the background, for every $\rho$. Indeed $\iota^*\theta_L=-\theta_R$, and gauging by $h$ maps $\tfrac a2\theta_L\mapsto\tfrac{2\rho-a}{2}\theta_L$ and $\tfrac b2\theta_L\mapsto-\tfrac b2\theta_L$, while $-\cot(\pi-y)=\cot y$.
• (b) $R^2=1$, $R$ is unitary on $X$, and $F(R\xi;\mu)=-RF(\xi;\mu)$. Hence $s\mapsto R\xi(-s)$ is a solution whenever $\xi(s)$ is.
• (c) $R\zeta=-\zeta$, and $R|_{\mathcal K}$ is a reflection with eigenvalues $(-1,-1,-1,+1)$.

In isotropic variables, $R$ is the symmetry $u(s,y)\mapsto-u(-s,\pi-y)$ of the AH equation $\partial_su+i\partial_yu=\bar u^2-r^{-2}$ [17].

Proof. (a) is the computation indicated. For (b), the forcing satisfies $Re_{\rm iso}=-e_{\rm iso}$, and the quadratic part satisfies $Q(R\cdot,R\cdot)=-RQ$. The latter holds because $Q$ is built from brackets and the cross product in the form index, and is therefore equivariant under the same rotation $\mathrm{Ad}_q^{\sf T}\in SO(3)$ applied to both indices, while $Q(-z,-z)=Q(z,z)$. For the linear part, $\mathcal L_\rho R+R\mathcal L_\rho$ is a first-order differential operator, so it vanishes once it vanishes on $f(y)\,m\,e_k$ for all $24$ components $e_k$, all $m\in\{1,x_0,\dots,x_3\}$ and arbitrary $f$; this is checked exactly in verify.py §6, for symbolic $\rho$. verify.py §6 also checks $Q(Rz,Rz)=-RQ(z,z)$ for generic symbolic $z\in\mathbb R^{24}$, $R^2=1$, and the control that omitting $y\mapsto\pi-y$ destroys the property. (c) is computed exactly (verify.py §6). ∎

Corollary 3.4 (parity). In normalized coordinates the reduced field $f=(f_x,f_w)$ satisfies $f(gz)=gf(z)$ for $g\in SO(4)$, and $f(\hat Rz)=-\hat Rf(z)$ for $\hat R(x,w)=(-x,\sigma w)$. That is, $f_x(-x,\sigma w)=f_x(x,w)$ and $f_w(-x,\sigma w)=-\sigma f_w(x,w)$. With Proposition 3.2 this gives (1.2): $g$ is even in $x$ and $\tilde h:=f_w/(x\,w)$ is even in $x$.

4. Computation of the reduced field

The Taylor coefficients of the reduced field $f(\xi_c,\mu)=P\,F(\xi_c+\Psi(\xi_c,\mu);\mu)$ are unique, even though $\Psi$ need not be. With $\xi_c=xZ+WV_0$ in raw coordinates, write $\Psi=x^2\Psi_{xx}+xW\Psi_{xw}+W^2\Psi_{ww}+\mu\Psi_\mu+O_3$. Since $P\mathcal L=0$, the terms of weighted order $\le3$ are $$f=P\Big[\mu e_{\rm iso}+Q(\xi_c,\xi_c)+\mu\,\partial_\mu\mathcal L\,\xi_c+2Q\big(\xi_c,\Psi_2(\xi_c,\mu)\big)\Big]+O_4 ,\qquad \Psi_2=-\mathcal L_h^{-1}\Pi_h\big(Q(\xi_c,\xi_c)+\mu e_{\rm iso}\big). \tag{4.1}$$ The invariance equation $\mathcal L_h\Psi-D\Psi\cdot f=-\Pi_h\mathcal R$ differs from $\mathcal L_h\Psi_2=-\Pi_h\mathcal R_2$ only at weighted order $\ge3$ in $\Psi$, hence at order $\ge4$ in $f$, since $f=O_2$ [5, Ch. 2, §2].

Quadratic coefficients (exact, verify.py §3). In raw coordinates: $$\tfrac{\langle\zeta,Q(\zeta,\zeta)\rangle}{\|\zeta\|^2}=-\tfrac56,\qquad \tfrac{\langle V_0,2Q(\zeta,V_0)\rangle}{\|V_0\|^2}=\tfrac54,\qquad \tfrac{\langle\zeta,Q(V_0,V_0)\rangle}{\|\zeta\|^2}=\tfrac5{16},\qquad \tfrac{\langle\zeta,e_{\rm iso}\rangle}{\|\zeta\|^2}=\tfrac43 .$$ Moreover $\langle V_c,Q(V_a,V_b)\rangle=0$, $\langle V,Q(\zeta,\zeta)\rangle=\langle V,e_{\rm iso}\rangle=0$, and $\langle\zeta,Q(V_a,V_b)\rangle\propto\delta_{ab}$. The compatibility $\langle V,2Q(\zeta,V)\rangle=2\langle\zeta,Q(V,V)\rangle$ shows that the quadratic truncation is the gradient of a cubic. The change to normalized coordinates ($x\mapsto-\pi x$, $W\mapsto\pi w/\sqrt2$, $s=\pi t$, $\mu=\bar\nu/\pi^2$) gives (1.3), with the leading terms $\tfrac56x^2-\tfrac43\bar\nu-\tfrac58|w|^2$ and $-\tfrac54xw$. The ratio $\lambda_{1/2}/\lambda_{\rm iso}=-\tfrac54/\tfrac53=-\tfrac34$ reproduces [20, Thm. C].

Second-order terms (explicit, verified exactly, verify.py §4). Each solves $\mathcal L\Psi=-\Pi_h(\cdot)$ and is $L^2$-orthogonal to $\ker\mathcal L_*$:

• $\Psi_{ww}$: $\alpha_i=\tfrac5{32}x_3\sin^2y\,[T_i,N]$ and $\psi_i=-\tfrac3{32}\sin y\cos y\,(n_iN-\tfrac{|n|^2}3T_i)$;
• $\Psi_\mu$: $\psi_i=-\tfrac13\sin y\cos y\,T_i$, the term $\bar\nu\psi_1$ of [17];
• $\Psi_{xx}$: $\psi_i=-\tfrac16\sin^3y\cos y\,T_i$;
• $\Psi_{xw}$: $\alpha_i=-\tfrac58\sin^2y\cos y\,[T_i,N]$ and $\psi_i=\tfrac14x_3\sin^3y\,T_i$.

They were found by solving the linear ODE systems in each harmonic sector ($j=0,1$ for $\Psi_{ww}$, $j=\tfrac12$ for $\Psi_{xw}$) with the ansatz $\sin y\,P(\cos y)+\sin^2y\,R(\cos y)$, $P,R$ polynomial. A solution of degree $\le2$ exists and is unique modulo the kernel, by [20, Thm. B]. It is then checked directly against the HW equations in quaternionic coordinates.

Cubic coefficients. Inserting into (4.1), verify.py §5 finds exactly: $$\kappa=[|w|^2w]_{\dot w}=0,\quad \beta=[\bar\nu w]_{\dot w}=0,\quad \gamma=[x^2w]_{\dot w}=0,\quad [x^3]_{\dot x}=[x\bar\nu]_{\dot x}=[x|w|^2]_{\dot x}=[w^3]_{\dot x}=0 .$$ Two of these vanish structurally in a way the computation exhibits:

• $\langle\Psi_{ww},\mathcal L\Psi_{ww}\rangle=0$, which is the quartic term of the Lyapunov–Schmidt potential;
• $\partial_\mu\mathcal L\,V_0=0$, because $\mathcal K\subset\ker\mathcal D$.

As a control, adding $\zeta$ to $\Psi_{ww}$ would make $\kappa\neq0$. All these vanishings are predicted by Corollary 3.4, which forces $[x^3],[x\bar\nu],[x|w|^2]$ (odd in $x$ in $\dot x$) and $[x^2w],[|w|^2w],[\bar\nu w]$ (even in $x$ in $\dot w/w$) to vanish. The computation is thus an independent check of Proposition 3.3.

Proof of Theorem B. (1.2) is Corollary 3.4. The coefficients in (1.3) are the quadratic coefficients above in normalized form, and the vanishing of the cubic ones gives the remainders $O_4$ and $O_3$. ∎

Remark 4.1 (vector-field versus potential coefficients). The coefficients computed above are those of the reduced vector field. The question posed after [20] was phrased for a potential $\mathsf W=\tfrac5{18}x^3-\tfrac43\nu x+\alpha x|w|^2+\kappa|w|^4+\beta\nu|w|^2$. We do not claim that the reduced field is a gradient for the induced metric. The natural function is the strict Lyapunov function $H=\int_0^xf_x$ of §6, which satisfies $H=\mathsf W+O_5$ with $\alpha=-\tfrac58$. $H$ is odd in $x$ to all orders, so every even-in-$x$ coefficient vanishes identically: $\kappa=\beta=0$, and so are the coefficients of $\bar\nu^2$, $x^2|w|^2$, and so on (verify.py §7b). The vanishing vector-field coefficients $[|w|^2w]_{\dot w}$ and $[\bar\nu w]_{\dot w}$ are the corresponding statements for the field.

5. Critical points (proof of Theorem C)

Equilibria of (1.2) are of two kinds.

• Isotropic ($w=0$). $g(x^2,0,\bar\nu)=0$, and $\partial_{x^2}g=\tfrac56$, $\partial_{\bar\nu}g=-\tfrac43$. By the implicit function theorem, $x^2=X(\bar\nu)=\tfrac85\bar\nu+O(\bar\nu^2)$, so $x=\pm\sqrt{X}$ exists iff $\bar\nu>0$, and $x_-=-x_+$ exactly. These are the objects of [17], since equilibria with $w=0$ are $SO(4)$-fixed and hence isotropic. $R$ maps $(x,0)\mapsto(-x,0)$, so $Rk_+=k_-$.
• Non-isotropic ($w\ne0$). Then $x\tilde h=0$ with $\tilde h=-\tfrac54+O_2\ne0$, so $x=0$ exactly. Then $g(0,|w|^2,\bar\nu)=0$ with $\partial_{|w|^2}g=-\tfrac58$ gives $|w|^2=\Sigma(\bar\nu)=-\tfrac{32}{15}\bar\nu+O(\bar\nu^2)$, which exists iff $\bar\nu<0$.

By Theorem A(2) these are all the KW solutions in $\mathcal U$.

Spectra.

• At $k_\pm$ the linearization of (1.2) is $\mathrm{diag}\big(\tfrac53x_\pm,-\tfrac54x_\pm\cdot1_4\big)(1+O(\bar\nu))$, as in [20].
• At $p=(0,w_0)\in\mathcal S$, using $x=0$, $\partial_xg|_{x=0}=0$ and $\partial_w(x\tilde hw)|_{x=0}=0$, it is $$\begin{pmatrix}0&2g_{S}\,w_0^{\sf T}\\ \tilde h\,w_0&0\end{pmatrix},\qquad g_S=-\tfrac58+O_2,\quad \tilde h=-\tfrac54+O_2,$$ with eigenvalues $\pm\sqrt{2g_S\tilde h}\,|w_0|=\pm\tfrac54|w_0|(1+O(\bar\nu))=\pm\sqrt{10|\bar\nu|/3}\,(1+O(\bar\nu))$, and $0$ on $w_0^\perp=T_p\mathcal S$ (semisimple).

Since $\mathcal M_{\bar\nu}$ is invariant and $p$ is an equilibrium, $T_p\mathcal M$ is invariant under $DF(p)$. By Kato's stability of isolated spectral groups [8, IV.§3.5], applied to $DF(p)=\mathcal L+O(|\bar\nu|^{1/2})$ as an operator $Z\to X$, exactly five eigenvalues lie near $0$, and they are those above. Hence $\ker DF(p)=T_p\mathcal S$, and $\mathcal S$ is Morse–Bott with normal index $1$.

Non-isotropy and stabiliser.

• $SU(2)_L$ acts on $\mathcal K\cong\mathbb H$ by left multiplication, which has no fixed vector $\ne0$, so no point of $\mathcal S$ is left-invariant.
• If such a solution were gauge-equivalent to a left-invariant one, both would be represented in the Coulomb slice near $k_*$ (slice theorem; the Nahm-pole configuration is irreducible [20, Rem. 6.1]), contradicting the classification above.
• The stabiliser is Lemma 3.1.
• The second-order formula of Theorem C is $\xi=WV_0+W^2\Psi_{ww}+\mu\Psi_\mu+O(|\mu|^{3/2})$ at $x=0$, with $W^2=-\tfrac{64}{15}\mu$ (raw form of $\Sigma$). ∎

6. Trajectories (proof of Theorem D)

By Theorem A(2), the HW trajectories in $\mathcal U$ are exactly the reduced trajectories that stay in a closed ball $\bar B=\bar B_{\delta_1}\subset\mathbb R^5$ of the centre coordinates. Write the reduced field as $f=(f_x,f_w)$, of class $C^{k}$ with $k\ge4$.

Lemma 6.1 (exact structure). For $|\bar\nu|<\delta$ and $z=(x,w)\in\bar B$:

1. $f_w(x,w)\in\mathbb Rw$. Consequently $w(t)/|w(t)|$ is constant along every orbit with $w\neq0$, and $\{w=0\}$ is invariant.
2. Fix a unit vector $e$ with $\sigma e=e$ (Proposition 3.3(c)), and put $\phi(x,\rho):=\langle f_w(x,\rho e),e\rangle$ and $F(x,\rho):=f_x(x,\rho e)$. Then:
   • on each half-plane $\Pi_{e'}=\{(x,\rho e'):\rho\ge0\}$ the flow is the planar system $\dot x=F(x,\rho)$, $\dot\rho=\phi(x,\rho)$, independently of the unit vector $e'$;
   • $F$ is even in $x$ and in $\rho$, and $\phi$ is odd in $x$ and in $\rho$;
   • by Hadamard's lemma, $\partial_\rho F=\rho\,a(x,\rho)$ and $\phi=x\rho\,d(x,\rho)$, with $a,d$ of class $C^{k-2}$ and $a(0,0)=d(0,0)=-\tfrac54$ at $\bar\nu=0$ (by (1.3)).

Proof.

1. The stabiliser $SO(3)$ of $w\neq0$ fixes $f_w(x,w)$ by equivariance, and its fixed set in $\mathbb R^4$ is $\mathbb Rw$. At $w=0$, $f_w(x,0)$ is fixed by all of $SO(4)$, hence $0$. Write $\dot w=b(t)w$ with $b$ continuous. Then $w(t)=\exp(\int b)\,w(t_0)$, and by uniqueness $w\equiv0$ if $w(t_0)=0$.
2. Independence of $e'$ follows from transitivity of $SO(4)$ on $S^3$ and equivariance. Parity in $\rho$ comes from $-1\in SO(4)$. Parity in $x$ comes from Corollary 3.4 with $\sigma e=e$: $F(-x,\rho)=F(x,\rho)$ and $\phi(-x,\rho)=\langle-\sigma f_w(x,\rho e),\sigma e\rangle=-\phi(x,\rho)$. The values at the origin come from (1.3). ∎

Proposition 6.2 (strict Lyapunov function). Define $$H(x,w;\bar\nu):=\int_0^xf_x(x',w;\bar\nu)\,dx' .$$ $H$ is $C^k$ on $\bar B$, $SO(4)$-invariant, odd in $x$, and $H=\mathsf W+O_5$. Shrinking $\delta_1,\delta$, along every reduced orbit in $\bar B$, $$\frac{dH}{dt}=f_x^2+x^2|w|^2\,m(x,|w|,\bar\nu),\qquad m:=\bar a\,d\ \ge\ 1,\qquad \bar a(x,\rho):=\frac1x\int_0^xa(x',\rho)\,dx'. \tag{6.1}$$ In particular $\dot H\ge0$, with equality exactly at equilibria. So $H$ is a strict Lyapunov function of the actual reduced flow, including all higher-order terms.

Proof. On $\Pi_e$ we have $H=\int_0^xF(x',\rho)\,dx'$, so $H_x=F$ and $H_\rho=\int_0^x\rho\,a(x',\rho)\,dx'=x\rho\,\bar a$. Hence $$\dot H=F^2+x\rho\,\bar a\cdot x\rho\,d .$$ At $\bar\nu=0$, $\bar a(0,0)d(0,0)=\tfrac{25}{16}$; by continuity in $(x,\rho,\bar\nu)$, $m\ge1$ on a small $\bar B\times(-\delta,\delta)$. Now suppose $\dot H=0$. Then $F=0$ and $x\rho=0$, so $\dot x=0$, $\dot\rho=x\rho d=0$ and (by Lemma 6.1) $\dot w=0$: the point is an equilibrium. Conversely $\dot H=0$ at equilibria. $H$ is odd in $x$ because $f_x$ is even in $x$. verify.py §7b checks (6.1) exactly for the general reversible equivariant field truncated at weighted order 5, with symbolic higher coefficients, and gives $m(0)=\tfrac{25}{16}$. ∎

Remark 6.3 (the HW energy). The HW equations are the gradient flow in $s$ of a gauge-invariant functional whose critical points are KW solutions (Haydys [14]; Witten [3, §5]). Restricted to $\mathcal M_{\bar\nu}$ this would give a Lyapunov function as well. That route requires showing that the Nahm-pole boundary terms vanish for the renormalised functional. We do not use it: $H$ is constructed from the reduced field alone, and Proposition 6.2 needs only Theorems A–B.

Proof of Theorem D.

• Limit sets. Let $z(t)\in\bar B$ for all $t$. $H$ is bounded on $\bar B$ and nondecreasing along the orbit. By LaSalle's invariance principle [15], the $\alpha$- and $\omega$-limit sets are nonempty, compact, connected, invariant, and contained in $\{\dot H=0\}$, which is the set of equilibria. By Lemma 6.1(1) the orbit lies in one closed half-plane $\Pi_{e'}$ (or in $\{w=0\}$), which meets the equilibrium set in finitely many points. Hence $z(t)$ converges to equilibria $z_\mp$ as $t\to\mp\infty$. If the orbit is nonconstant, the strictness in (6.1) gives $H(z_-)<H(z_+)$: there are no returns and no homoclinic orbits.

• $\bar\nu>0$. The equilibria are $k_\pm$ (Theorem C). Since $H$ is odd in $x$ and $x_-=-x_+$, $H(k_-)=-H(k_+)$. Moreover $H(k_+)=\int_0^{x_+}f_x(x',0)\,dx'<0$, because $f_x(x,0)=G(x^2)$ with $G'=\tfrac56+O(|\bar\nu|^{1/2})>0$ and $G(x_+^2)=0$, so $f_x(x,0)<0$ for $|x|<x_+$. Hence the only possible nonconstant orbits run from $k_+$ to $k_-$. Such an orbit lies in $W^u(k_+)$, which is one-dimensional and $SO(4)$-invariant. An orbit through a point with $w\neq0$ is three-dimensional, so $W^u(k_+)\subset\{w=0\}$. On that invariant line $\dot x=f_x(x,0)<0$ on $(x_-,x_+)$, so the connection exists and is unique up to translation; it is the isotropic instanton $u$ of [17].

• $\bar\nu<0$. The equilibria are $\mathcal S$, where $H=0$ because $x=0$. A nonconstant orbit would need $0=H(z_-)<H(z_+)=0$, which is impossible.

• $\bar\nu=0$. The only equilibrium is $0$, so every orbit in $\bar B$ is constant.

• Uniform isolation. For all $|\bar\nu|<\delta$, $\operatorname{Inv}(\bar B)$ is the union of the equilibria and (for $\bar\nu>0$) the segment $\{w=0,\ x_-\le x\le x_+\}$. This set lies in $B_{C|\bar\nu|^{1/2}}\subset\operatorname{int}\bar B$. So $\bar B$ is an isolating neighbourhood for the whole family, and the Conley index of $\operatorname{Inv}(\bar B)$ is independent of $\bar\nu$ by continuation [7, Ch. IV].

• Conley index.

  • For $\bar\nu<0$, $\mathcal S\cong S^3$ is a normally hyperbolic manifold of equilibria. Its reduced unstable bundle is a trivial line bundle: the unstable eigenvector $(-1,\,w_0/|w_0|)$ of §5 is a global nowhere-vanishing section. So the index is the Thom space $\Sigma(S^3_+)\simeq S^1\vee S^4$, with homology $\mathbb Z[1]\oplus\mathbb Z[4]$.
  • For $\bar\nu>0$, $(k_-,k_+)$ is an attractor–repeller decomposition of $\operatorname{Inv}(\bar B)$, with $CH(k_+)=\mathbb Z[1]$ and $CH(k_-)=\mathbb Z[4]$ (hyperbolic, with unstable dimensions $1$ and $4$). In the exact sequence $\cdots\to CH_n(k_-)\to CH_n\to CH_n(k_+)\xrightarrow{\partial}CH_{n-1}(k_-)\to\cdots$ the map $\partial$ is zero for degree reasons, so $CH_*=\mathbb Z[1]\oplus\mathbb Z[4]$. This agrees with continuation.

  The cascade description for $\bar\nu<0$ (a perfect Morse function on $S^3$, generators in degrees $1,4$, no cascades since there are no orbits between points of $\mathcal S$) gives the same groups with $d=0$. ∎

Remark 6.1 (relation to [20]). The trajectory $u$ has virtual dimension $-3$ but moduli of dimension $1$. Its obstruction bundle, $\operatorname{coker}D_u\cong\mathcal K$, is exactly the four spin-½ directions in which $W^s(k_-)$ and $W^u(k_+)$ fail to be transverse in $\mathbb R^5$ ($1+1-5=-3$). Conversely, $\mathcal M(k_-,k_+)$ has virtual dimension $+3$ and is empty. Theorem D(1) upgrades the local uniqueness of [17] from the isotropic class to the complete gauge-fixed space.

7. Discussion and open problems

1. The answer to the question of [20, Rem. 7.3]. Non-isotropic KW solutions exist, but on the other side of the fold ($L<\pi r$), and they form exactly one $S^3$. The quartic coefficient is not needed for their existence, and it is in fact zero. What matters is the sign of the $x|w|^2$ coupling, $-\tfrac58$, which is fixed by the $-\tfrac54$ of [20] through the gradient compatibility of §4.
2. Floer-theoretic meaning (interpretation). In a putative Floer theory built from the complete operator, the local contribution near the fold is $\mathbb Z[1]\oplus\mathbb Z[4]$ on both sides. The AH pair is not a cancelling pair. Generic holonomy-type perturbations preserving no symmetry would, by the stability of the Conley index, leave a local homology of rank $2$; we do not construct such perturbations.
3. Open problems.
   • The global continuation of $\mathcal S_{\bar\nu}$ as $\bar\nu$ decreases.
   • Triaxial metrics, which break $SO(4)$ to a finite group, so that $S^3$ should break into finitely many critical points with the same Conley index.
   • An explicit $\delta$.
   • A numerical solution of the static PDE for $\mathcal S_{\bar\nu}$ at finite $\bar\nu$; we do not provide one.
4. Literature. We found no prior construction of non-spherically-symmetric Nahm-pole solutions on $S^3\times I$. [2] works in the spherically symmetric ansatz. He [9] constructs rotationally invariant singular KW solutions on other geometries, Leung–Takahashi [10] prove energy bounds on $S^3\times\mathbb R^+$, and He [11] proves a gluing theorem with Nahm pole; none of these concerns the AH fold.

8. Changes from the first submission

1. §6 has been rewritten. The previous argument (an increase of the cubic $\mathsf W$ outside neighbourhoods, plus last-exit inequalities) did not control returns, as the referee pointed out. It is replaced by an explicit strict Lyapunov function $H=\int_0^xf_x$ for the full reduced field (Proposition 6.2), the exact planar reduction (Lemma 6.1), LaSalle's principle, and a uniform isolating neighbourhood through $\bar\nu=0$. The Conley conclusions are derived from that isolated invariant set.
2. Theorem A no longer asserts joint tangency. $\partial_{\bar\nu}\Psi(0,0)=\Psi_\mu/\pi^2\neq0$. Lemma 2.2 introduces the affine shift of [17, v2, Lemma 5.2], and the hypotheses of [5, Thms. 2.20, 3.3] are listed for the shifted system.
3. Clarifications:
   • Euclidean metric at the origin versus the induced metric;
   • vector-field coefficients versus potential coefficients (Remark 4.1);
   • the reduced unstable normal direction;
   • Conley homology versus a Morse–Smale complex (Theorem D(3)).

9. Status of claims

Proved here.

• Theorems A–D, Lemmas 2.1, 2.2, 3.1, 6.1, Propositions 3.2–3.3, 6.2, Corollary 3.4.
• The trajectory classification of Theorem D rests on the strict Lyapunov function of Proposition 6.2, valid for the full reduced field; no leading-order or numerical argument is used.
• The exact coefficients of §4 are computer-verified in exact arithmetic (sympy). The method is the following: explicit fields are checked against the HW equations in quaternionic coordinates, and $S^3$-integrals are evaluated with the monomial formula.

Imported.

• From [20]: self-adjointness, domain, gap and kernel of $\mathcal L_*$ (Thms. A–B, Prop. 4.2, Lemma 4.4).
• From [17, 18]: the isotropic centre manifold, $k_\pm$ and the instanton.
• From [5, Ch. 2]: the centre-manifold theorems, including their parameter-dependent, equivariant and reversible versions.
• Conley index theory (continuation, attractor–repeller sequence, Thom-space index of a normally hyperbolic manifold) [7], LaSalle's principle [15], Kato [8], Hadamard's lemma and Schwarz's theorem [12].

Numerical only. verify.py §8 (escape of invariant manifolds in the leading planar model) and one random-sample positivity check in §7b; neither is used in the proofs.

Not claimed.

• An explicit $\delta$.
• Global statements.
• Floer homology beyond the reduced Conley index.
• Perturbation theory.
• The triaxial case.

Verifier. verify.py runs 65 checks, exact except §8, and reports FALLOS = 0; it includes negative controls. verify.mpl mirrors the isotropic block and the reduced normal-form classification. We did not run it, because Maple is not available to us.

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