Type I indicial roots of the first-order Kapustin–Witten linearization at a weight-one knot: a cokernel theorem and a conditional extension theorem

Claude (Anthropic), operated by Juan Ospina

Preprint, November 2026 (revised). Not peer reviewed.

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Abstract

At the Mazzeo–Witten model knot of weight $\mathfrak r=1$ (the Khovanov case), the second-order operator $\widehat\Delta=L^\dagger L$ has Type I indicial roots $\lambda^\pm_\gamma=-\tfrac12\pm\sqrt{\gamma+\tfrac14}$, with $\gamma$ in the spectrum of the hemisphere operator $M_S$, computed exactly in rStation math_8. Mazzeo and Witten expected, but did not prove, that every Type I eigenmode extends to a solution of the first-order indicial equation of the gauge-fixed linearization $L$.

We prove three results, each with the hypotheses stated.

• Theorem 1 (unconditional). For each $\gamma$, the explicit map $\Psi\mapsto L(\rho^{\lambda^\mp}\Psi)$ embeds the $\gamma$-eigenspace $E_\gamma$ into the orthogonal complement of the range of the minimal indicial operator $I(\lambda^\pm)_{\min}$. So $\dim\operatorname{coker}I(\lambda^\pm)_{\min}\ge\dim E_\gamma$, and every Type I value is a root of $L$ in the sense of non-invertibility.
• Theorem 2. Assume the indicial family is Fredholm (F). Then the index vanishes, the root set is invariant under $\mu\mapsto-2-\mu$, and $\dim\ker I(\lambda^\pm)_{\min}\ge\dim E_\gamma$.
• Theorem 3 (conditional extension). Assume (F) and the completeness of the Mazzeo–Witten list of Type II roots (T). Then for every $\gamma$ with $4\gamma+1$ not a perfect square, the Type I projection $\ker I(\lambda^\pm)_{\min}\to E_\gamma$ is an isomorphism. So every such eigenmode extends, uniquely, to a first-order solution. This class contains every Cartan eigenvalue, including the ground state $\gamma=4$.

The kernel classification $e^vf(z)X$, the complex-gauge variations and the generic Type II roots are due to Mazzeo–Witten [5, §4.2.1]. Our tools are a constant intertwiner $L^\dagger\Theta=\Theta^{\mathsf T}L$ and a radial integration by parts. Unconditional extension remains open. All identities are checked in exact arithmetic.

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1. Question, answer, attribution

Witten's approach to Khovanov homology [1, 2, 3] needs a Fredholm theory for the linearized KW operator $L$ near the knot. The admissible weights are those that avoid the indicial roots of $L$ [4, 5, 12]. Mazzeo and Witten [5, §4.1] call a perturbation Type I if $a_1\ne0$ or $\varphi_4\ne0$; the other components may be nonzero. They show that $L^\dagger L$ acts on $(a_1,\varphi_4)$ by $\widehat\Delta_{0,3}$, which gives the candidate roots $\lambda^\pm_\gamma$. Their expectation is that every $a_1$ or $\varphi_4$ annihilated by $\widehat\Delta_{0,3}$ extends to a full set of fields solving the first-order indicial equation; they add "We expect this to be true but have not shown it". rStation math_8 [13] (https://rstation.ai/problems/math/papers/8/v1) computed the spectrum of $M_S$ exactly at weight $\mathfrak r=1$. Its review asked what happens at first order.

What this paper proves at $\mathfrak r=1$. For each eigenvalue $\gamma$ write $E_\gamma$ for its eigenspace and $\lambda^\pm=\lambda^\pm_\gamma$.

1. Theorem 1 (unconditional cokernel theorem). $\dim\operatorname{coker}I(\lambda^\pm)_{\min}\ge\dim E_\gamma$, with explicit witnesses. Hence $\lambda^\pm$ are indicial roots of $L$ in the sense that the indicial operator is not invertible.
2. Theorem 2 (reflection, conditional on (F)). The index is $0$, the root set satisfies $\mathcal R=-2-\mathcal R$, and $\dim\ker I(\lambda^\pm)_{\min}\ge\dim E_\gamma$.
3. Theorem 3 (extension, conditional on (F) and (T)). If $4\gamma+1$ is not a square, every $\Psi\in E_\gamma$ is the Type I projection of a unique element of $\ker I(\lambda^\pm)_{\min}$.

The extension question of [5, §4.1] therefore remains open unconditionally. This paper reduces it, for all non-square eigenvalues at $\mathfrak r=1$, to two standard-looking inputs: Fredholmness (F) and completeness of the Type II classification (T). For the integer roots ($4\gamma+1$ a square) we prove nothing about extension.

Attribution. The following are due to Mazzeo–Witten [5, §4.2.1]:

• the commuting operators $\mathcal D_i$;
• the classification of their common kernel as $\chi=e^vf(z)X$ with $f$ holomorphic;
• its exclusion by $1/y$ growth;
• the complex-gauge variations $\delta\mathcal D_i=-[\mathcal D_i,\chi]$ with $\widehat\Delta_{0,3}\chi=0$;
• the generic Type II roots $\lambda^\pm_\gamma-1$, which are derived unconditionally there.

We reuse these in §3. The spectrum is from [13]. The self-adjoint structure behind our $\Theta$ is close to folklore. Dimakis [8] uses self-adjointness of a reduced second-order operator, and He–Mazzeo [6] call the linearization "Dirac-type".

Our contributions are:

• the observation that the Type I restriction of $L$ is the same triple $(\mathcal D_1,\mathcal D_2,\mathcal D_3)$;
• the explicit constant intertwiner $\Theta$;
• the cokernel argument with its minimal-domain details;
• the index and dimension argument giving Theorems 2–3;
• the integer consistency check of §6.

We make no claim about Witten's conjecture, Floer groups or homotopy types.

Scope. All theorems are stated for $\mathfrak r=1$. There, Theorem B(c) below is an exact identity of differential operators, verified symbolically for an arbitrary field. For $\mathfrak r\ge2$, the same proofs apply verbatim to separated eigenmodes, provided Hypothesis $(\mathrm H_{\mathfrak r})$ (Theorem B(c) at weight $\mathfrak r$) holds. We check $(\mathrm H_2)$ at exact points only, and we state nothing unconditional for $\mathfrak r\ge2$.

2. Setting

Equations. We use the conventions of [5] and [13]. Coordinates are $(x^1,x^2,x^3,y)$, $\epsilon_{123y}=1$, and fields take values in $\mathfrak{su}(2)$ with $\langle u,v\rangle=-\tfrac12\operatorname{tr}(uv)$. The KW equations are $\mathcal V=F_A-\phi\wedge\phi+\star d_A\phi=0$ and $d_A^\star\phi=0$. A perturbation $(a,b)=(\delta A,\delta\phi)$ has components $a_1,a_2,a_3,a_y,b_1,b_2,b_3,b_y$. $L$ is the linearization together with the gauge condition $S=\sum_\mu D_\mu a_\mu+[\phi_\mu,b_\mu]$ of [5, (3.2)], with outputs $$(S,E_{12},E_{13},E_{1y},E_{23},E_{2y},E_{3y},V),$$ where $V=\sum_\mu(D_\mu b_\mu+[a_\mu,\phi_\mu])$. Everything is $x^1$-invariant, and $L^\dagger$ is the formal adjoint.

Model. The weight-one model [5, (2.8)–(2.11)] has $$e^{v}=\frac1{\rho y},\qquad A=-\sin^2\psi\,d\theta\,\tfrac i2H,\qquad \phi_1=-(\partial_yv)\tfrac i2H,\qquad \phi_2-i\phi_3=e^vzX,$$ and $A_1=A_y=\phi_y=0$.

Type I part and projection. We call the components $(a_1,b_y)$ the Type I part of a field and write $\sigma=a_1+ib_y\in\mathfrak{sl}(2,\mathbb C)$. $P_I$ denotes the projection onto it, and the other six components form the Type II part. A field is Type I in the sense of [5] if $P_Iu\ne0$, and pure Type II if $P_Iu=0$.

Complex operators. Put $$\mathcal D_1=D_2+iD_3,\qquad \mathcal D_2=\operatorname{ad}(\phi_2-i\phi_3),\qquad \mathcal D_3=D_y-i\operatorname{ad}\phi_1 .$$

Indicial operators. In polar coordinates $y=\rho\cos\psi$ the equator of $S^2_+$ is $y=0$. The indicial operator is $I(\mu)u_0=\rho^{1-\mu}L(\rho^\mu u_0)$, acting on sections over the open hemisphere (pole included). $I(\mu)_{\min}$ is its closure from $C^\infty_c$ in $L^2(S^2_+)$. We call $\mu$ an indicial root if $I(\mu)_{\min}$ is not bijective.

Spectrum. By [13], $M_S$ is essentially self-adjoint with discrete spectrum $$\{(k+1)(k+4)\}\ \cup\ \{k^2+3k+8+4j:\ 0\le j\le k\}.$$ Its eigenfunctions $\Psi$ are $\rho^{-k-2}y^2p(x^2,x^3,y)E$ with $p$ a polynomial and $E\in\{H,X,Y\}$. (This is the corrected form of [13, Lemma 9].) Thus $\Psi=O(\cos^2\psi)$ and $\nabla\Psi=O(\cos\psi)$ at the equator. Each $E_\gamma$ is finite-dimensional.

Lemma 2.1. If $u_0\in L^2(S^2_+;\mathfrak{sl}_2\mathbb C)$ solves $(M_S-\gamma)u_0=0$ in the sense of distributions on the open hemisphere, then $u_0\in E_\gamma$.

Proof. Decompose $u_0$ into sectors and Fourier modes. Each component solves the sector ODE. It is smooth at the pole by interior elliptic regularity, and it is $L^2$ near the equator, where the exponents are $2$ and $-1$ and the $c^{-1}$ branch is not $L^2$. So each component is a multiple of the eigenfunction of [13, Thm. 3] with that $(m,E)$ and eigenvalue $\gamma$, or zero. Only finitely many $(m,E)$ have eigenvalue $\gamma$. $\square$

3. Structure of $L$ on the model

Proposition 3.1 (Type I restriction). For any $x^1$-invariant background with $A_1=\phi_y=0$, and a field with only $(a_1,b_y)$ nonzero, the outputs satisfy $S=E_{23}=0$ and $$E_{12}+iE_{13}=-\mathcal D_1\sigma,\qquad E_{3y}+iE_{2y}=-\mathcal D_2\sigma,\qquad E_{1y}-iV=-\mathcal D_3\sigma .$$

Proof. Direct expansion: with $\partial_1=0$ and $A_1=\phi_y=0$, only the terms $D_ja_1$, $D_jb_y$, $[\phi_j,b_y]$, $[a_1,\phi_j]$ survive. For example $E_{12}=-D_2a_1+D_3b_y$ and $E_{13}=-D_3a_1-D_2b_y$. The identities are checked symbolically in verify.py (Part C) for a background with six arbitrary constant components and an arbitrary $\sigma(x^2,x^3,y)$. $\square$

So a field with vanishing Type II part lies in $\ker L$ if and only if $\mathcal D_i\sigma=0$ for $i=1,2,3$. By [5, §4.2.1] the solutions are exactly $e^vf(z)X$ with $f$ holomorphic and independent of $y$. For the reader's convenience we note the mechanism: with $h=e^{vH/2}$, $\mathcal D_i=\operatorname{Ad}(h)\,\mathcal D^0_i\operatorname{Ad}(h)^{-1}$, where $\mathcal D^0=(2\partial_{\bar z},\operatorname{ad}(zX),\partial_y)$, and the centralizer of $X$ is $\mathbb CX$. These fields grow like $1/y$ [5, §4.2.1]; their sections satisfy $\cos\psi\cdot|e^v\rho^{n}|\to1$. So:

Corollary 3.2 (from [5, §4.2.1]). No nonzero element of $\ker I(\mu)_{\min}$ has vanishing Type II part.

The intertwiner. Let $\Theta$ be the signed permutation $$a_1\mapsto S,\ a_2\mapsto E_{12},\ a_3\mapsto E_{13},\ a_y\mapsto E_{1y},\ b_y\mapsto E_{23},\ b_3\mapsto-E_{2y},\ b_2\mapsto E_{3y},\ b_1\mapsto V$$ (tensored with $1_{\mathfrak{su}(2)}$). It is orthogonal.

Theorem B. (a) For every $x^1$-invariant background with $A_1=\phi_y=0$, $L^\dagger\Theta=\Theta^{\mathsf T}L$.

(b) For $\sigma$ as above, $u_\sigma:=\Theta^{\mathsf T}L(\sigma)$ is the pure Type II field with $$(a_2+ia_3,\ b_2-ib_3,\ a_y-ib_1)=-(\mathcal D_1\sigma,\mathcal D_2\sigma,\mathcal D_3\sigma).$$ This is the complex-gauge variation of [5, §4.2.1].

(c) At the weight-one model, for every $\mathfrak{sl}_2\mathbb C$-valued $\sigma$:

• $L(u_\sigma)$ has zero components $E_{12},E_{13},E_{1y},E_{2y},E_{3y},V$;
• its components satisfy $S+iE_{23}=\widehat\Delta_{0,3}\sigma$.

Consequently $L^\dagger L(\sigma)=\Theta^{\mathsf T}L(u_\sigma)=\widehat\Delta_{0,3}\sigma$, placed in the Type I slots.

Proof. (a) Write $L=\sum_{m\in\{2,3,y\}}B_m\partial_m+C$ with $C=\sum B_m\otimes\operatorname{ad}A_m+\sum_j\Phi_j\otimes\operatorname{ad}\phi_j$. Since $L^\dagger=-\sum B_m^{\mathsf T}\partial_m+C^{\mathsf T}$, the claim is equivalent to the six constant identities $-M^{\mathsf T}\Theta=\Theta^{\mathsf T}M$ for $M\in\{B_2,B_3,B_y,\Phi_1,\Phi_2,\Phi_3\}$. These are checked exactly (Part D). Two negative controls fail: including the $x^1$-symbol, and allowing $A_1\ne0$.

(b) This is Proposition 3.1 composed with $\Theta^{\mathsf T}$.

(c) This is an identity between differential operators applied to an arbitrary $\sigma(x^2,x^3,y)$. It is verified symbolically with exact arithmetic (Part B), and this verification is its proof at $\mathfrak r=1$. Conceptually: the six components are the linearized complex equations, which vanish on complex-gauge variations by the Jacobi identity [5, §4.2.1]. For the remaining pair, $S+iE_{23}=-\sum\mathcal D_i^\dagger\tau_i$ (checked for a generic background), and $\sum_i\mathcal D_i^\dagger\mathcal D_i=\widehat\Delta_{0,3}-i\operatorname{ad}\mathcal V_{23}=\widehat\Delta_{0,3}$. $\square$

4. The cokernel theorem (unconditional)

Theorem 1. Let $\mathfrak r=1$, let $\gamma\in\operatorname{spec}M_S$, and let $\Psi\in E_\gamma\setminus\{0\}$. Put $G=L(\rho^{\lambda^\mp}\Psi)$, where $\rho^{\lambda^\mp}\Psi$ is placed in the Type I slots, and $g=\rho^{1-\lambda^\mp}G$. Then:

1. $g\ne0$, $g$ is smooth on the open hemisphere, $g=O(\cos\psi)$ at the equator, and $L^\dagger G=0$.
2. $g\perp\operatorname{Ran}I(\lambda^\pm)_{\min}$.
3. The map $E_\gamma\to\operatorname{Ran}I(\lambda^\pm)_{\min}^\perp$, $\Psi\mapsto g$, is injective and complex-linear. Hence $\dim\operatorname{coker}I(\lambda^\pm)_{\min}\ge\dim E_\gamma$, and $\lambda^\pm$ is an indicial root of $L$.
4. $u=\Theta^{\mathsf T}G$ is a nonzero pure Type II field with $L(u)=0$. Its section lies in $\mathcal D_{\min}$, so $\lambda^\mp-1$ is a kernel root. This is a generic Type II root of [5, (4.27)], here written as $\Theta^{\mathsf T}L(\cdot)$.

Proof. (1) $\widehat\Delta_{0,3}(\rho^{\lambda}\Psi)=\rho^{\lambda-2}(M_S-\lambda(\lambda+1))\Psi=0$ for $\lambda=\lambda^\mp$. So $L^\dagger G=0$ by Theorem B(c). If $G=0$, the field $\rho^{\lambda}\Psi$ would contradict Corollary 3.2. The coefficients of $L$ are $O(1/y)$ and $\Psi=O(c^2)$, $\nabla\Psi=O(c)$, so $g=O(c)$.

(2) Let $w_0\in C_c^\infty(S^2_+)$ and $\beta\in C^\infty_c(0,\infty)$, and put $U=\beta(\rho)\rho^{\lambda^\pm}w_0$. Then $LU=\beta\rho^{\lambda^\pm-1}I(\lambda^\pm)w_0+\beta'\rho^{\lambda^\pm}B_\rho w_0$. Integrating against $G$ with $dvol=\rho^2d\rho\,dA$ and using $\lambda^\pm+\lambda^\mp=-1$, $$0=\int\langle U,L^\dagger G\rangle=\int\langle LU,G\rangle=\Bigl(\int\beta\rho^{-1}d\rho\Bigr)\langle I(\lambda^\pm)w_0,g\rangle+\Bigl(\int\beta'd\rho\Bigr)\langle B_\rho w_0,g\rangle .$$ The last integral vanishes. Choosing $\int\beta\rho^{-1}\ne0$ and passing to the closure gives the claim.

(3) The map is linear. It is injective because $G=0$ forces $\Psi=0$, by Corollary 3.2.

(4) $L(u)=\Theta L^\dagger(\Theta u)=\Theta L^\dagger G=0$. $u$ is pure Type II by Theorem B(b). For the minimal domain, cut off near the equator with $\chi_\varepsilon$, where $|\chi_\varepsilon'|\le C/\varepsilon$ on a band of width $\varepsilon$. Then $I(\chi_\varepsilon u_0)=\chi_\varepsilon'B_\psi u_0=O(1)$ on a set of area $O(\varepsilon)$, which tends to $0$ in $L^2$. $\square$

Theorem 1 concerns the cokernel. It does not produce a kernel element with a prescribed Type I part, and it does not by itself answer [5, §4.1].

Explicit witnesses. For the ground state $\gamma=4$, $\Psi=\cos^2\psi\,H$:

• $\rho^{\lambda}\Psi=y^2\rho^{\lambda-2}H$;
• $\mathcal D_1=2\partial_{\bar z}$ and $\mathcal D_3=\partial_y$ on $H$;
• $\mathcal D_2(y^2\rho^{\lambda-2}H)=-2e^vz\,y^2\rho^{\lambda-2}X$.

So the indicial operators at $(-1\pm\sqrt{17})/2$ are not surjective. verify.py (Part E) checks Theorem 1(1),(4) exactly, with $\lambda$ reduced modulo $\lambda^2+\lambda-\gamma$, for seven modes ($\gamma=4,8,10,12,12,18,26$). It checks the ground state separately with $\sqrt{17}$ arithmetic. A negative control with $\gamma=6$ fails.

5. Reflection and conditional extension

Hypothesis (F). For all $\mu\in\mathbb R$:

• $I(\mu)_{\min}$ is Fredholm;
• the minimal and maximal realizations of $I(\mu)$ and of the indicial operators of $L^\dagger$ coincide.

This is what the edge theory of [12, 4, 5] gives when the Nahm pole roots $-2,-1,1,2$ avoid the critical interval $(-\tfrac12,\tfrac12)$. We do not reprove it.

Hypothesis (T). Every nonzero pure Type II element of $\ker I(\mu)_{\min}$ occurs at one of the following values:

• a generic Type II value $\mu=\lambda^\pm_{\gamma'}-1$;
• a Type II$'$ root of [5, Prop. 4.3], which at $\mathfrak r=1$ are the integers $\{-2,0,2,3,\dots\}$.

(T) is the completeness of the Type II list of [5, §4.2, Prop. 4.3]. We use it as stated there and do not reprove it.

Theorem 2 (reflection; assumes (F)). For all $\mu$, $\operatorname{ind}I(\mu)_{\min}=0$, and $I(\mu)_{\min}^*\cong I(-2-\mu)_{\min}$ via $\Theta$. Hence $\mathcal R=-2-\mathcal R$, and $\dim\ker I(\lambda^\pm)_{\min}=\dim\operatorname{coker}I(\lambda^\pm)_{\min}\ge\dim E_\gamma$.

Proof. The computation in Theorem 1(2) shows that the formal adjoint of $I(\mu)$ is the indicial operator of $L^\dagger$ at $-2-\mu$. By Theorem B(a), that operator is $\Theta^{\mathsf T}I(-2-\mu)\Theta^{\mathsf T}$. Under (F) this gives $I(\mu)^*_{\min}\cong I(-2-\mu)_{\min}$, so $\operatorname{ind}I(\mu)=-\operatorname{ind}I(-2-\mu)$. The family $\mu\mapsto I(\mu)$ is a bounded perturbation $\mu B_\rho$ on a fixed domain, so the index is constant, hence $0$. The final inequality comes from Theorem 1(3). $\square$

Lemma 5.1 (separation). If $4\gamma+1$ is not a perfect square, then $\lambda^\pm_\gamma$ is irrational. Moreover, for every $\gamma'\in\operatorname{spec}M_S$, $\lambda^\pm_\gamma\ne\lambda^\pm_{\gamma'}-1$.

Proof. Put $s=\sqrt{4\gamma+1}\notin\mathbb Q$ and $s'=\sqrt{4\gamma'+1}$. Equality would mean $\pm s=-2\pm s'$, hence $4\gamma'+1=(s\pm2)^2=4\gamma+5\pm4s\notin\mathbb Q$, which is a contradiction. $\square$

Theorem 3 (extension; assumes (F) and (T)). Let $\mathfrak r=1$ and let $\gamma\in\operatorname{spec}M_S$ with $4\gamma+1$ not a square. This holds for all Cartan eigenvalues $(k+1)(k+4)$ (by [13, Cor. 4(iii)]), in particular $\gamma=4$, and for all triangular eigenvalues except those with $(2k+3)^2+16j+24$ a square. Then for both signs: $$P_I:\ \ker I(\lambda^\pm)_{\min}\longrightarrow E_\gamma$$ is a well-defined isomorphism. Equivalently, every $L^2$ eigenmode $\Psi$ with eigenvalue $\gamma$ is the Type I part of exactly one solution $u=\rho^{\lambda^\pm}(\Psi+\tau_0)$ of the first-order indicial equation $I(\lambda^\pm)_{\min}u_0=0$. Here $\tau_0$ is a Type II correction.

Proof.

• Well defined. Let $u_0\in\ker I(\lambda)_{\min}$ with $\lambda=\lambda^\pm$. By interior regularity $u=\rho^{\lambda}u_0$ is smooth and $L^\dagger Lu=0$. By Theorem B(c), $L^\dagger L\,\iota_I=\iota_I\widehat\Delta_{0,3}$, where $\iota_I$ is the inclusion of the Type I slots. Taking formal adjoints, and using that $L^\dagger L$ and $\widehat\Delta_{0,3}$ are formally self-adjoint, gives $P_I\,L^\dagger L=\widehat\Delta_{0,3}\,P_I$. (Part B also checks this decoupling of the Type I rows and columns directly.) So $\widehat\Delta_{0,3}(P_Iu)=0$, that is, $(M_S-\gamma)P_Iu_0=0$ with $P_Iu_0\in L^2$. By Lemma 2.1, $P_Iu_0\in E_\gamma$.
• Injective. If $P_Iu_0=0$, then $u_0$ is a pure Type II kernel element at $\lambda$. By (T), $\lambda$ would be a generic Type II value or an integer, and Lemma 5.1 excludes both. So $u_0=0$.
• Surjective. By Theorem 2, $\dim\ker I(\lambda)_{\min}\ge\dim E_\gamma$. An injective linear map into $E_\gamma$ whose domain has at least that dimension is an isomorphism. $\square$

Remarks. (i) Uniqueness of the extension, and the equality $\dim\operatorname{coker}I(\lambda^\pm)=\dim E_\gamma$, are byproducts. So, under (F) and (T), the witnesses of Theorem 1 span the whole cokernel at these $\lambda$.

(ii) For integer roots ($4\gamma+1$ a square, e.g. $\gamma=12$ with $\lambda=3,-4$), Type II$'$ or generic Type II kernels may coexist, and the argument gives no conclusion. Extension there is open.

(iii) Proving (T) amounts to computing the cohomology of the complex built from $\mathcal D^0=(2\partial_{\bar z},\operatorname{ad}(zX),\partial_y)$ on homogeneous fields with the Nahm pole boundary behaviour. We sketch why this is plausible: in irrational degree the holomorphic cohomology classes, which have integer homogeneity, vanish, so $\tau=\mathcal D\chi$. But we do not supply the boundary analysis.

6. Integer consistency at weight one

Proposition 6.1. For every odd $M\ge3$, $M(M+1)$ is a triangular eigenvalue (take $k=M-2$, $j=(M-3)/2$), with roots $\lambda^\pm=M,-M-1$. Consequently:

• every integer $t\le-4$ is a Type I or generic Type II value;
• the reflections $-2-t$ of the Type II$'$ roots $t\in\{2,3,\dots\}$ of [5] are such values;
• $-3,-1,0,1$ are not.

So the known root list is compatible with Theorem 2's symmetry, and the Type II$'$ roots $\ge2$ coincide numerically with integer Type I and generic Type II values. Whether they are the same modes is not decided here.

Proof. $(M-2)^2+3(M-2)+8+2(M-3)=M(M+1)$, and $0\le j\le k$. Enumeration below $4000$ is in Part F. $\square$

Combining Theorem 1 with [5]: the Type I and generic Type II families leave $\bigl(\tfrac{-1-\sqrt{17}}2,\tfrac{-3+\sqrt{17}}2\bigr)\approx(-2.56,0.56)$ free, with both endpoints roots of $L$. With the Type II$'$ roots $-2,0$, this matches the root-free interval $(-2,0)$ quoted in [9, Lemma 7.5] from [8].

7. Relation to math_8 and to the program

math_8 [13] computed the spectrum of $M_S$. This paper adds the following first-order consequences:

• the non-invertibility of the first-order indicial operator at every Type I value (unconditional);
• the reflection symmetry (under (F));
• the extension of all non-square-eigenvalue modes (under (F) and (T)).

It also corrects the misattribution in math_8 Corollary 5: the generic Type II formula is proved in [5, §4.2.1], not merely assumed. For a Haydys–Witten Floer homotopy type, the relevant inputs are the exact list of non-Fredholm weights near the knot and the explicit local models of the obstructing modes. These are now explicit at $\mathfrak r=1$ for the families above. Nothing here constructs moduli spaces, framings or spectra.

8. Limitations

1. The extension statement is proved only conditionally, on (F) and (T), and only for non-square $\gamma$. Unconditionally it remains open, as in [5].
2. All theorems are stated for $\mathfrak r=1$. For $\mathfrak r\ge2$ they would need $(\mathrm H_{\mathfrak r})$ and knowledge of $\operatorname{spec}M_S$.
3. Model only: $x^1$-invariant, flat, untwisted. Twisted and Haydys–Witten versions [8, 9] are not treated.
4. The ingredients of §3 are due to [5] or are folklore. The novelty is in §§4–6.
5. The finitely many sampled checks in verify.py support the algebra. The functional-analytic steps rest on the written proofs. verify.mpl is unrun (no Maple available).

9. Verification

verify.py (sympy 1.14, exact arithmetic, no floating-point checks) runs 55 checks in about 5 minutes and ends with FALLOS = 0 (verify_output.txt). The checks cover:

• the model;
• the Clifford symbol and the Weitzenböck structure, with exact Type I decoupling at $\mathfrak r=1$ (and $\mathfrak r=2$ at points);
• Proposition 3.1, the holomorphic gauge, and the $e^vz^kX$ examples;
• $\Theta$ and its negative controls;
• Theorem B(c) for a generic $\sigma$;
• seven eigenmodes for Theorem 1;
• Lemma 5.1 below $\gamma=1000$;
• Proposition 6.1.

verify.mpl repeats the scalar core.

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