GLOBAL CHEEGER-GORESKY-MACPHERSON
FOR FUBINI-STUDY METRICS ON PROJECTIVE
SIMPLE-EDGE SPACES

Jiaming Luo, Shirong Li

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Jiaming Luo, Shirong Li. GLOBAL CHEEGER-GORESKY-MACPHERSON FOR FUBINI-STUDY METRICS ON PROJECTIVE SIMPLE-EDGE SPACES. 2026. hal-05511027

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
GLOBAL CHEEGER–GORESKY–MACPHERSON FOR
FUBINI–STUDY METRICS ON PROJECTIVE SIMPLE-EDGE
SPACES

JIAMING LUO AND SHIRONG LI

ABSTRACT. Let $X \subset \mathbb{CP}^N$ be a complex projective variety whose singular locus is a smooth connected stratum $Y$, with simple-edge transverse type, and let $g_{FS}$ be the metric induced on $X_{\text{reg}}$ by the ambient Fubini-Study metric. We show that $(X_{\text{reg}}, g_{FS})$ has wedge structure near $Y$ with a canonical radial function and a smooth link fibration $L \to Y$. Moreover, the argument is entirely domain-level: from the Fubini-Study wedge expansion we deduce the constant-multiplicity stability in the critical strip and hence a fully checkable Calderón boundary package. This geometric input implies, without any additional spectral stability hypothesis, that the wedge Hodge-de Rham operator admits a Dirac normal form and a constant-multiplicity indicial (critical weight) strip, yielding the trace/Poisson/Calderón package and a Lagrangian Cauchy data bundle $W_{\text{can}} = \text{ran}(C) \subset H = \ker(A)$. The resulting canonical closed de Rham complex $(\text{Dom}(d_{\text{can}}), d)$ (equivalently the associated self-adjoint Fredholm realization $D_{\text{can}} = d_{\text{can}} + \delta_{\text{can}}$) satisfies $H^*(\text{Dom}(d_{\text{can}}), d) \simeq IH^*_{W_{\text{can}}}(X)$, and in the Witt case this reduces to middle-perversity intersection cohomology. We give nontrivial projective families of examples, including cones, joins, and fibered homogeneous singularities.

CONTENTS

<table><tr><td>1. Introduction</td><td>3</td></tr><tr><td>A reusable geometric engine (FS ⇒ canonical IBC).</td><td>3</td></tr><tr><td>1.1. Guiding milestones: from M1 (cones) to M2 (simple-edge)</td><td>3</td></tr><tr><td>1.2. Main geometric input</td><td>4</td></tr><tr><td>1.3. Dirac normal form and tangential operator</td><td>4</td></tr><tr><td>1.4. Indicial strip and maximal-domain asymptotics</td><td>5</td></tr><tr><td>1.5. Calderón package and canonical extensions</td><td>5</td></tr><tr><td>1.6. Topological identification</td><td>5</td></tr><tr><td>1.7. Relation to the wedge/edge boundary literature and reduced hypotheses</td><td>6</td></tr><tr><td>1.8. Where the analysis comes from, and what is new here</td><td>6</td></tr><tr><td>1.9. Structure of the paper.</td><td>8</td></tr><tr><td>2. Notation and setup</td><td>8</td></tr><tr><td>3. Whitney control data and wedge structure of g<sub>FS</sub></td><td>9</td></tr><tr><td>3.1. Control data and link fibration along the stratum</td><td>9</td></tr><tr><td>3.2. FS wedge expansion</td><td>11</td></tr></table>

<footnote n="2020">2020 Mathematics Subject Classification. 58J32, 58J52, 57R19, 32S60.</footnote>

<footnote>Key words and phrases. Fubini-Study metric, wedge/edge metrics, Calderón projector, canonical boundary conditions, $L^2$ Hodge theory, intersection cohomology.</footnote>

<footnote>Corresponding author.</footnote>
<table>
  <tbody>
    <tr>
      <td>4. The wedge de Rham operator and the tangential family</td>
      <td>12</td>
    </tr>
    <tr>
      <td>4.1. Wedge Dirac normal form</td>
      <td>12</td>
    </tr>
    <tr>
      <td>4.2. Explicit tangential operator and A<sup>2</sup></td>
      <td>13</td>
    </tr>
    <tr>
      <td>5. Automatic stability: kernel bundle, uniform gap, and the critical strip</td>
      <td>16</td>
    </tr>
    <tr>
      <td>5.1. Automatic stability package for the tangential family</td>
      <td>16</td>
    </tr>
    <tr>
      <td>5.2. Kernel rank is constant globally</td>
      <td>17</td>
    </tr>
    <tr>
      <td>5.3. Uniform spectral gap</td>
      <td>18</td>
    </tr>
    <tr>
      <td>5.4. Critical strip and constant indicial multiplicity</td>
      <td>18</td>
    </tr>
    <tr>
      <td>6. Calderón package and canonical boundary data</td>
      <td>19</td>
    </tr>
    <tr>
      <td>6.1. Green pairing and Lagrangian Cauchy data</td>
      <td>21</td>
    </tr>
    <tr>
      <td>7. Canonical domain and global CGM in depth one</td>
      <td>23</td>
    </tr>
    <tr>
      <td>7.1. Boundary condition domains</td>
      <td>23</td>
    </tr>
    <tr>
      <td>7.2. Self-adjointness and Fredholmness of the canonical extension</td>
      <td>23</td>
    </tr>
    <tr>
      <td>7.3. Global CGM conclusion</td>
      <td>24</td>
    </tr>
    <tr>
      <td>7.4. The isolated conic case (M1) as a base case</td>
      <td>26</td>
    </tr>
    <tr>
      <td>8. Nontrivial projective families (examples)</td>
      <td>29</td>
    </tr>
    <tr>
      <td>8.1. Geometric families within the depth-one class</td>
      <td>29</td>
    </tr>
    <tr>
      <td>8.2. Worked computations</td>
      <td>33</td>
    </tr>
    <tr>
      <td>9. A depth-two one-step iteration and an inductive theorem schema for M3</td>
      <td>34</td>
    </tr>
    <tr>
      <td>Outlook (M3–M4)</td>
      <td>39</td>
    </tr>
    <tr>
      <td>Appendix A. Two standard analytic lemmas used in Section 5</td>
      <td>40</td>
    </tr>
    <tr>
      <td>Appendix B. Fubini-Study wedge expansion: adapted coordinates and tensor estimates</td>
      <td>41</td>
    </tr>
    <tr>
      <td>B.1. Edge vector fields and edge-smooth tensors</td>
      <td>41</td>
    </tr>
    <tr>
      <td>B.2. A quantitative Whitney angle estimate</td>
      <td>41</td>
    </tr>
    <tr>
      <td>B.3. Vertical scaling and the definition of g<sub>L</sub></td>
      <td>44</td>
    </tr>
    <tr>
      <td>B.4. Fubini-Study wedge expansion: proof and assembly</td>
      <td>45</td>
    </tr>
    <tr>
      <td>Appendix C. Trace bundles, Green pairing, and the Calderón package in the constant-multiplicity strip</td>
      <td>46</td>
    </tr>
    <tr>
      <td>C.1. Maximal-domain asymptotics and the trace map</td>
      <td>47</td>
    </tr>
    <tr>
      <td>C.2. Poisson operator and Calderón projector</td>
      <td>47</td>
    </tr>
    <tr>
      <td>C.3. Green pairing and Lagrangian Cauchy data</td>
      <td>51</td>
    </tr>
    <tr>
      <td>C.4. Adjoint domains and Fredholmness</td>
      <td>51</td>
    </tr>
    <tr>
      <td>Appendix D. The isolated conic case (M1): explicit domains and boundary data</td>
      <td>52</td>
    </tr>
    <tr>
      <td>D.1. Geometric model and Dirac normal form</td>
      <td>53</td>
    </tr>
    <tr>
      <td>D.2. Indicial roots and explicit leading asymptotics</td>
      <td>53</td>
    </tr>
    <tr>
      <td>D.3. Trace maps and the boundary data space</td>
      <td>56</td>
    </tr>
    <tr>
      <td>D.4. Green pairing in block form</td>
      <td>56</td>
    </tr>
    <tr>
      <td>D.5. Closed extensions as Lagrangian boundary conditions</td>
      <td>58</td>
    </tr>
    <tr>
      <td>D.6. Minimal/Dirichlet and Friedrichs domains</td>
      <td>59</td>
    </tr>
    <tr>
      <td>D.7. Witt reduction and intersection cohomology</td>
      <td>59</td>
    </tr>
    <tr>
      <td>Appendix E. Depth-one ideal boundary conditions and Deligne-type intersection theory</td>
      <td>60</td>
    </tr>
    <tr>
      <td>E.1. Local cone model and the trace bundle</td>
      <td>60</td>
    </tr>
    <tr>
      <td>E.2. The domain complex and its fine sheaf model</td>
      <td>61</td>
    </tr>
    <tr>
      <td>E.3. Local computation: the cone truncation dictated by W</td>
      <td>61</td>
    </tr>
    <tr>
      <td>E.4. Identification with Deligne-type depth-one intersection theory</td>
      <td>61</td>
    </tr>
  </tbody>
</table>
<table><tr><td>Acknowledgments</td><td>62</td></tr><tr><td>References</td><td>62</td></tr></table>

# 1. INTRODUCTION

Let $X \subset \mathbb{CP}^N$ be a complex projective variety of pure dimension. It is well known that $X$ admits a Whitney stratification with a smooth top-dimensional stratum $X^{\text{reg}}$ and lower-dimensional singular strata. [19, 17] Equipping $\mathbb{CP}^N$ with the Fubini-Study metric $g_{\text{FS}}$ and restricting it to $X^{\text{reg}}$ produces a rich geometric setting: the induced metric retains strong analytic properties coming from positivity and Kähler geometry, yet behaves singularly near the lower-dimensional strata.

The purpose of this article is to give a complete analytic description of the Hodge-de Rham operator on $(X^{\text{reg}}, g_{\text{FS}})$ in the depth-one case, i.e., when there is a single singular stratum $Y$ and the local model is a metric cone over the link $L_y$. Our approach is intrinsic and uses only the Riemannian geometry induced by the ambient Fubini-Study metric, rather than algebraic or Kähler structures.

**A reusable geometric engine (FS $\Rightarrow$ canonical IBC).** A central message of this paper is an *input-output mechanism* that turns ambient projective geometry into canonical analytic and topological data.

**Input (purely geometric).** Let $X \subset \mathbb{CP}^N$ be projective with a single smooth singular stratum $Y$ of simple-edge type, and let $g_{\text{FS}}$ be the induced metric on $X^{\text{reg}}$.

**Output (canonical analytic package).** The geometry of $(X^{\text{reg}}, g_{\text{FS}})$ forces: (i) a wedge expansion with edge-smooth remainder, hence a Dirac normal form for the Hodge-de Rham operator with *exact* tangential coefficient $A_y$; (ii) an *automatic* constant-multiplicity critical strip (kernel bundle, uniform spectral gap, constant indicial window); (iii) the full trace/Poisson/Calderón package in that strip, producing a canonically defined smooth Cauchy-data bundle $W_{\text{can}} = \text{ran}(\mathcal{C}) \subset \text{ker}(A) \to Y$; (iv) a canonical self-adjoint Fredholm de Rham complex whose cohomology identifies with the depth-one intersection theory determined by $W_{\text{can}}$.

In particular, in the projective simple-edge setting *no external product-type hypothesis* and *no a priori spectral-stability assumption* is imposed: both are supplied by the Fubini-Study geometry itself, and the metric selects the boundary data functorially.

**1.1. Guiding milestones: from M1 (cones) to M2 (simple-edge).** From the perspective of the Cheeger-Goresky-MacPherson program and its refinements via ideal boundary conditions, it is convenient to separate the projective Fubini-Study problem into progressively harder milestones.

**(M1) Isolated conic singularities.** When the singular locus is finite, each singular point has a conic neighborhood and the boundary data are finite-dimensional. In this case one can describe the maximal/minimal (Dirichlet) and Friedrichs domains explicitly in terms of leading coefficients in $\text{ker}(A)$, and identify the canonical closed extension with a canonical ideal boundary condition; in the Witt case this yields middle-perversity intersection cohomology. [6, 7, 9, 10, 5]
**(M2) Simple-edge singularities (depth one).** The present paper establishes the fully projective depth-one case: for a projective variety whose singular locus is a smooth stratum $Y$ with a smooth link fibration, we show that the induced Fubini-Study metric is automatically wedge with an edge-smooth remainder, and we develop the full Calderón/trace/Green-pairing package in a constant-multiplicity strip. This leads to a canonical self-adjoint Fredholm de Rham complex whose co-homology agrees with the corresponding depth-one intersection theory. For clarity, we record the isolated (M1) specialization in Section 7.4 and give a self-contained local domain analysis in Appendix D.

1.2. Main geometric input. All subsequent analysis rests on a single quantitative geometric estimate that enters the wedge/edge calculus directly and is produced canonically by the ambient projective geometry.

**Control data and adapted coordinates.** Whitney control data for the depth-one stratum $Y$ provide a cone-bundle neighborhood $N \cap X_{\text{reg}} \simeq (0, \varepsilon)_x \times L$ together with a retraction $\rho: N \to Y$ and a canonical radial function $x$ (Proposition 3.1). This fixes the geometric input against which all $x \to 0$ estimates are made.

**FS wedge expansion with edge-smooth remainder.** The main new geometric statement is that the induced Fubini-Study metric has a wedge expansion with an edge-smooth $O(x)$ remainder:

$$g_{\text{FS}} = dx^2 + \rho^* g_Y + x^2 g_L + x \kappa,$$

where the coefficients of $\kappa$ extend smoothly to $x=0$ in an adapted wedge coframe and remain smooth under arbitrary differentiation by edge vector fields (Appendix B.1). This is proved as Proposition 3.2, with a complete, checkable proof in Appendix B (see also Remark 3.3).

As emphasized in Remark 3.4, this wedge expansion is the only point where projectivity/Fubini-Study is used in an essential way: the ambient Kähler metric supplies the distance function and quantitative control, while Whitney control data supply the conical coordinates. No product-type analytic model is assumed a priori.

1.3. Dirac normal form and tangential operator. The wedge expansion above is strong enough to put the Hodge-de Rham operator $D = d + \delta$ into the standard regular-singular Dirac form. Concretely, Proposition 4.1 yields

$$D = \Gamma\left(\partial_x + \frac{1}{x}A_y + B(x)\right),$$

where $A_y$ is a smooth family of self-adjoint elliptic operators on the link fibres $L_y$ and $B(x)$ is a first-order edge operator with coefficients that are edge-smooth down to $x=0$.

**Rigidity of the $x^{-1}$ coefficient.** A key point for the later “automatic stability” package is that the $x\kappa$ non-product error in the wedge expansion above does not alter the coefficient of $x^{-1}$ in the Dirac normal form above. In the strict product case, Lemma 4.2 identifies

$$A_y = D_{L_y} + \frac{f}{2} - \mathcal{N},$$

and Lemma 4.4 shows that the same formula persists under an $x\kappa$ remainder. Consequently, $\text{Ker}(A_y)$ is canonically identified with the middle-degree harmonic forms on $(L_y, g_L)$, so the boundary data spaces assemble into a finite-rank smooth bundle $H = \text{Ker}(A) \to Y$.
1.4. **Indicial strip and maximal-domain asymptotics.** Treating $A_y$ as a smooth family of self-adjoint elliptic operators on the compact links, we extract three uniform consequences needed for boundary-value theory.

**Automatic stability package.** First, the harmonic-form identification implies that $\dim \ker(A_y)$ is constant, and hence $H = \ker(A) \to Y$ is a smooth vector bundle (Proposition 5.1). Second, compactness of $Y$ upgrades the pointwise gap at 0 to a *uniform* spectral gap (Proposition 5.2). Third, combining this with the indicial correspondence produces a constant-multiplicity “critical strip” around the $L^2$ threshold (Proposition 5.3).

**Trace bundle and maximal-domain expansion.** In this constant-multiplicity strip, the trace bundle in the critical window is precisely $H$, and elements of the maximal domain admit a canonical two-term expansion whose leading coefficient defines the trace map (Lemma C.1 in Appendix C). This is the regime in which the boundary package is cleanest: no additional indicial-root or spectral control is imposed externally, and the geometry forces the hypotheses.

1.5. **Calderón package and canonical extensions.** Working in the constant-multiplicity strip of Proposition 5.3, we construct the full boundary-value package for $D$.

**Trace/Poisson/Calderón.** The trace map, Poisson operator, and Calderón projector are constructed in Theorem 6.1 and Proposition C.2, with complete proofs and mapping properties recorded in Appendix C. The range of the Calderón projector defines the canonical smooth Cauchy-data bundle

$$W_{\text{can}} := \text{ran}(C) \subset H.$$

**Green pairing and Lagrangianity.** The induced Green pairing on $H$ and the Lagrangian nature of $W_{\text{can}}$ are proved in Theorem 6.2 (see also Lemma 6.3, Lemma 6.4, and Proposition 6.5).

**Canonical and general self-adjoint Fredholm domains.** Imposing the Calderón boundary condition $\text{Tr}_D u \in \Gamma(Y; W_{\text{can}})$ defines a canonical domain (Definition 7.2), and the resulting realization is self-adjoint and Fredholm (Theorem 7.4). More generally, any smooth subbundle $W \subset H$ defines a realization $D_W$; the adjoint-domain identity (Lemma C.10) shows that $D_W$ is self-adjoint exactly when $W$ is Lagrangian, and Fredholmness for elliptic projection boundary conditions is proved in Theorem C.11.

**Reusable template.** Sections 3–6 are written as a checkable input-output pipeline: Whitney control data and adapted coordinates (Proposition 3.1) ⇒ FS wedge expansion with edge-smooth remainder (Proposition 3.2) ⇒ Dirac normal form with tangential rigidity (Proposition 4.1, Lemma 4.4) ⇒ automatic stability and a constant-multiplicity critical strip (Propositions 5.1–5.3) ⇒ the Calderón package and canonical Cauchy data (Theorem 6.1). In the projective simple-edge setting these inputs are forced by the ambient Fubini-Study geometry, but the same pipeline applies verbatim in any singular regime where the geometric inputs can be verified.

1.6. **Topological identification.** To pass from analysis to topology, we interpret a smooth subbundle $W \subset H = \ker(A) \to Y$ as depth-one ideal boundary condition
data (a mezzoperversity). The resulting Hilbert complex $(\mathrm{Dom}(d_W), d)$ computes the corresponding depth-one intersection theory: Proposition 7.5 identifies

$$H^\bullet(\mathrm{Dom}(d_W), d) \cong \mathrm{IH}_W^\bullet(X),$$

with a self-contained sheaf-theoretic proof given in Appendix E (see Theorem E.3).

For the canonical Cauchy-data choice $W = W_{\mathrm{can}} = \mathrm{ran}(C)$, this yields the global Fubini-Study-Cheeger-Goresky-MacPherson conclusion (Theorem 7.6). In the Witt case $H = 0$ there is no boundary datum to choose and the canonical theory reduces to middle-perversity intersection cohomology.

Takeaway. The sheaf-theoretic identification of $L^2$ de Rham cohomology with ideal boundary conditions with (refined) intersection cohomology is part of the general mezzoperversity/IBC framework (cf. Theorem E.3). The new global content here is a selection principle: in the projective simple-edge Fubini-Study setting, the geometry forces the analytic hypotheses needed for the boundary package and canonically selects the boundary datum as the Calderón Cauchy data $W_{\mathrm{can}} = \mathrm{ran}(C)$ (Theorems 6.1 and 6.2), hence canonically determines the associated mezzoperversity and the intersection theory computed by $H_{(2)}^*(X_{\mathrm{reg}}, g_{\mathrm{FS}})$ (Theorem 7.6).

Beyond depth one (M3 schema). The depth-one analysis is organized to separate *iterable* input from the *new* higher-depth bottleneck. In Section 9 we formulate a one-step depth-two iteration and an inductive theorem schema, making precise which parts of the depth-one package iterate verbatim (Dirac normal form, trace/Poisson/Calderón construction, Fredholm criteria) and isolating the genuinely new issue in higher depth: stability for tangential families whose *self-adjoint realizations* depend on deeper canonical ideal boundary conditions.

### 1.7. Relation to the wedge/edge boundary literature and reduced hypotheses.
The analytic backbone of this paper is the established boundary-value theory for first-order elliptic wedge/edge Dirac operators: maximal-domain asymptotics and trace bundles, Poisson operators, Calderón projectors, Green pairings, and Fredholm/self-adjoint realizations via projection (equivalently, Lagrangian) boundary conditions; see for instance Mazzeo-Vertman [16, Section 6] and the classical comparison-operator criterion in Calderón/Boutet de Monvel theory (e.g. [8, Theorem 11]).

What is *new* here is not a new edge calculus, but a *geometric forcing principle* in the projective simple-edge setting: the ambient Fubini-Study geometry produces, without additional hypotheses, exactly the regularity and spectral input needed to apply that boundary package *uniformly* along the stratum, and it moreover selects the boundary datum canonically as Cauchy data. A precise dependency map (with exact internal cross-references) is recorded in Section 1.8.

### 1.8. Where the analysis comes from, and what is new here.
For the reader's convenience (and to make the use of the wedge/edge boundary-value machinery fully checkable), we list below a precise "dependency map" between the analytic steps in this paper and the corresponding results in the literature. The guiding principle is that we do *not* reprove the general edge calculus: instead, we (i) verify in the projective simple-edge Fubini-Study setting the geometric hypotheses needed to *apply* the existing boundary-value package, and (ii) identify the *canonical* boundary datum produced by the geometry as the Calderón Cauchy data, hence as a canonical mezzoperversity.
**Reading guide.** Column (A) points to the exact statement in this paper where a given analytic object (trace bundle, Poisson operator, Calderón projector, Green pairing, Fredholmness, canonical domain, etc.) is constructed or used. Column (B) indicates the closest corresponding statement in the existing edge/wedge boundary-value theory (with precise Theorem/Equation numbers whenever available). Column (C) records what is new in the projective Fubini-Study setting, i.e. which standard hypotheses (product-type structure, externally imposed indicial/spectral control, etc.) are replaced by verifiable geometric input.

<table><thead><tr><th>(A) This paper</th><th>(B) Closest literature statement</th><th>(C) New / replaced input in this paper</th></tr></thead><tbody><tr><td><strong>FS wedge expansion and edge-smooth remainder.</strong><br>Proposition 3.2 (proved in Appendix B as Proposition B.5), giving g<sub>FS</sub> = dx<sup>2</sup> + ρ<sup>*</sup>g<sub>Y</sub> + x<sup>2</sup>g<sub>L</sub> + xκ with edge-smooth κ.</td><td>This replaces standard “product-type / controlled error” assumptions frequently imposed when entering the wedge/edge calculus.</td><td>New geometric engine: ambient FS restriction automatically yields a wedge expansion strong enough to enter the Dirac/wedge calculus without assuming product-type structure; cf. Remark 3.4.</td></tr><tr><td><strong>Wedge Dirac normal form and tangential rigidity.</strong><br>Proposition 4.1 together with Lemma 4.4.</td><td>Regular-singular Dirac normal forms are standard once one assumes wedge regularity; see e.g. the setup in [16, Section 6].</td><td>New verification step: the FS wedge expansion plus Whitney control data give the required edge-smooth coefficients globally along the stratum, and the x<sup>-1</sup> coefficient is rigid (product-type) despite the O(x) metric remainder.</td></tr><tr><td><strong>Constant-multiplicity critical strip.</strong><br>Propositions 5.1-5.3 (constant dim ker(A<sub>y</sub>), uniform spectral gap at 0, hence a strip where only λ = 0 occurs with constant multiplicity).</td><td>This is precisely the hypothesis regime under which trace-bundle/Calderón theory is cleanest; in Mazzeo-Vertman this corresponds to working in a fixed weight window where the trace bundle is well-defined and varies smoothly.</td><td>New geometric forcing: no external “spectral stability” assumption is imposed; constant rank and the gap are forced by the projective link fibration, tangential rigidity, and compactness/continuity.</td></tr><tr><td><strong>Trace map, Poisson operator, Calderón projector in the critical strip.</strong><br>Theorem 6.1 and Proposition C.2 (with maximal-domain expansion Lemma C.1 in Appendix C).</td><td>Mazzeo-Vertman provide semi-Fredholm/Fredholm parametrices and mapping properties for trace/Poisson operators; see [16, Section 6] and their continuity statements (e.g. around their Eqs. (7.3)-(7.4)).</td><td>New specialization: we record the constant-multiplicity-strip specialization and show the MV machinery applies uniformly along the projective stratum because the FS geometry supplies the required strip and smooth dependence.</td></tr></tbody></table>
<table><thead><tr><th>(A) This paper</th><th>(B) Closest literature statement</th><th>(C) New / replaced input in this paper</th></tr></thead><tbody><tr><td>Green pairing, Lagrangian Cauchy data, self-adjoint Fredholm extensions. Theorem 6.2, Lemma 6.4, Proposition 6.5, and Theorem C.11.</td><td>Classical Dirac boundary problems via projection boundary conditions and the comparison-operator criterion (Calderón/Boutet de Monvel frameworks), together with MV's Fredholm boundary problem theory.</td><td>New canonical choice: the FS-geometry picks out W<sub>can</sub> = ran(C) as Cauchy data; maximal isotropicity is proved using the Fredholm-pair/comparison-operator argument inside the FS-forced critical strip.</td></tr><tr><td>Canonical FS domain and global consequence. Definition 7.2, Theorem 7.4, Proposition 7.5, and Theorem 7.6.</td><td>On the sheaf-theoretic side, the identification of L<sup>2</sup> de Rham cohomology with IBC with the hypercohomology of the corresponding Deligne sheaf is part of the general ALMP framework; see [1, Theorem 5.7].</td><td>New global synthesis: we do not introduce a new intersection theory; rather, in the projective simple-edge FS setting the canonical analytic domain corresponds to a geometrically defined mezzoperversity W<sub>can</sub>, yielding a canonical identification H<sup>*</sup><sub>(2)</sub>(X<sub>reg</sub>, g<sub>FS</sub>) &cong; IH<sup>*</sup><sub>W<sub>can</sub></sub>(X).</td></tr></tbody></table>

## 1.9. Structure of the paper.
Section 2 fixes notation and the depth-one geometric setup. Section 3 constructs Whitney control data and proves the key geometric input: the Fubini-Study wedge expansion (Proposition 3.2). Section 4 derives the wedge Dirac normal form (Proposition 4.1) and identifies the tangential family (Lemma 4.4). Section 5 establishes the automatic stability package (Propositions 5.1-5.3). Section 6 develops the depth-one boundary package in the critical strip (Theorem 6.1 and Theorem 6.2). Section 7 defines the canonical domain (Definition 7.2), proves self-adjoint Fredholmness (Theorem 7.4), and deduces the global FS-CGM statement (Theorem 7.6). Section 7.4 isolates the conic (M1) base case, and Appendix D provides full, explicit local model computations.

Section 8 presents projective families of examples, organized to verify the geometric hypotheses of the analytic package and to illustrate how the canonical boundary data interact with refined intersection cohomology in non-Witt situations. Section 9 records the depth-two one-step iteration and inductive schema toward M3.

Appendix A collects two standard analytic lemmas used in the stability arguments. Appendix B contains the complete proof of the Fubini-Study wedge expansion. Appendix C gives detailed proofs of the boundary analysis in depth one (trace/Poisson/Calderón, Green pairing, adjoint domains, and Fredholmness). Appendix E provides a self-contained depth-one ideal boundary condition (Deligne-type) description and identifies the cohomology of (Dom(d<sub>W</sub>), d) with the corresponding depth-one intersection theory.

## 2. NOTATION AND SETUP

Throughout the paper, $X \subset \mathbb{CP}^N$ denotes a complex projective variety of pure dimension. We fix a Whitney stratification with a single singular stratum $Y$ and write $X^{\text{reg}}$ for the regular part. The ambient Fubini-Study metric $g_{\text{FS}}$ restricts smoothly to $X^{\text{reg}}$.
We work with the following standard analytic structures.

* A tubular neighborhood $\mathcal{N}$ of $Y$ equipped with control data $(\rho, x)$: $\rho: \mathcal{N} \to Y$ is a continuous retraction which is smooth on $N \cap X_{\text{reg}}$, and $x: \mathcal{N} \to [0, \varepsilon)$ is continuous, smooth and positive on $N \cap X_{\text{reg}}$, vanishes precisely on $Y$, and has smooth level sets $\{x = t\} \cap X_{\text{reg}}$ transverse to the fibers of $\rho$ for all sufficiently small $t > 0$.

* A local trivialization $N \cap X_{\text{reg}} \simeq (0, \varepsilon)_x \times U_y \times L_z$ compatible with $(\rho, x)$; here $L \to Y$ is the link fibration, $L_y$ is the link fiber over $y$, and $U \subset Y$ is a coordinate patch.

* The algebra $\mathcal{V}_e$ of edge vector fields, locally generated over $C^\infty$ by $\{x\partial_x, \partial_{y^a}, \partial_{z^\mu}\}$, and its associated edge Sobolev spaces $H_e^k$.

* The vertical bundle $V = \ker(d\rho)$ and a fixed horizontal complement $\mathcal{H} \subset \ker(dx)$ together with $\nabla^{g_{\text{FS}}x}$ yield the splitting

$$T(X_{\text{reg}}) = \mathbb{R}\langle\nabla^{g_{\text{FS}}x}\rangle \oplus H \oplus V.$$

This depth-one geometric setup is summarized schematically in Figure 1: the singular stratum $Y$ appears as the zero section $\{x = 0\}$, the fibers $L_y$ form the transverse links, and the tubular neighbourhood $N \cap X_{\text{reg}}$ is modeled by a cone bundle $(0, \varepsilon)_x \times L$ over $Y$.

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Tubular neighbourhood of the singular stratum $Y$ with conic link fibres $C(L_y)$, illustrating the wedge/simple-edge structure of the Fubini-Study metric near $Y$.</figcaption></figure>

All analysis is performed relative to this structure. Appendix B provides the detailed geometric estimates validating the wedge expansion of the Fubini-Study metric in these adapted coordinates.

### 3. WHITNEY CONTROL DATA AND WEDGE STRUCTURE OF $g_{\text{FS}}$

#### 3.1. Control data and link fibration along the stratum.
Whitney stratifications admit control data: a tubular neighborhood, a retraction to the stratum, and a radial function compatible with the stratification. We use this to define a canonical radial coordinate.

**Proposition 3.1** (Link fibration and radial function). Let $X$ be a projective simple-edge space with singular stratum $Y$. Then there exist:
(i) a neighborhood $\mathcal{N}$ of $Y$ in $X$ and a continuous retraction $\rho: \mathcal{N} \to Y$ which is smooth on $N \cap X^{\text{reg}}$;

(ii) a continuous function $x: \mathcal{N} \to [0, \varepsilon)$ which is smooth and positive on $N \cap X^{\text{reg}}$, vanishes precisely on $Y$, and is a submersion on $N \cap X^{\text{reg}}$;

(iii) for each small $t > 0$, the level set $L_t := \{x = t\} \cap X^{\text{reg}}$ is a smooth hypersurface and $\rho|_{L_t}: L_t \to Y$ is a smooth fiber bundle, independent (up to canonical diffeomorphism) of $t$.

In particular, for $\varepsilon > 0$ small there is a diffeomorphism

$$ (0, \varepsilon) \times L \cong N \cap X^{\text{reg}}, $$

with $L := L_{t_0}$ for any fixed small $t_0$.

*Proof.* Fix a Whitney stratification $\mathcal{S}$ of $X$ refining $\{Y, X_{\text{reg}}\}$. We will use two standard facts ([17][13]):

(A) **Thom-Mather control data.** There exist an open neighborhood $\mathcal{N}$ of $Y$ in $X$, a continuous retraction $\rho: \mathcal{N} \to Y$ with $\rho|_Y = \text{id}$, and a continuous function $x: \mathcal{N} \to [0, \varepsilon)$ with $x^{-1}(0) = Y$, such that on each stratum $S \in \mathcal{S}$ the restrictions $\rho|_{N \cap S}$ and $x|_{N \cap S}$ are smooth and

$$ (\rho, x) : (N \setminus Y) \cap S \to Y \times (0, \varepsilon) $$

is a smooth submersion. Shrinking $\mathcal{N}$ and $\varepsilon$ if necessary, we may assume $N = \{x < \varepsilon\}$. (See, e.g., standard references on control data for Whitney stratifications.)

(B) **Ehresmann's fibration theorem.** A smooth proper submersion $F: M \to B$ is a smooth locally trivial fiber bundle.

*Step 1:* $x$ is a submersion on $N \cap X_{\text{reg}}$ and $L_t$ is a smooth hypersurface. Restrict the control data to the top stratum $X_{\text{reg}}$. By (A), $(\rho, x) : N \cap X_{\text{reg}} \to Y \times (0, \varepsilon)$ is a smooth submersion. In particular, $dx \neq 0$ on $N \cap X_{\text{reg}}$, hence $x$ is a submersion there. Therefore for every $t \in (0, \varepsilon)$ the level set

$$ L_t := \{x = t\} \cap X_{\text{reg}} $$

is a smooth codimension-one submanifold of $X_{\text{reg}}$.

*Step 2:* $(\rho, x)$ is proper on $N \cap X_{\text{reg}}$. Since $X$ is a complex projective variety, it is compact in the classical topology. The set $\bar{\mathcal{N}} = \{x \leq \varepsilon\}$ is closed in $X$ and hence compact. Let $K \subset Y \times (0, \varepsilon)$ be compact. Then

$$ (\rho, x)^{-1}(K) \subset \bar{\mathcal{N}} \cap X_{\text{reg}} $$

is closed in the compact set $\bar{\mathcal{N}} \cap X_{\text{reg}}$, hence compact. Thus $(\rho, x) : N \cap X_{\text{reg}} \to Y \times (0, \varepsilon)$ is proper.

*Step 3:* local triviality and the link fibration. By Steps 1-2, the map

$$ F := (\rho, x) : N \cap X_{\text{reg}} \to Y \times (0, \varepsilon) $$

is a smooth proper submersion. By Ehresmann's theorem (B), $F$ is a smooth fiber bundle. Fix $t \in (0, \varepsilon)$. Restricting $F$ to the slice $Y \times \{t\}$ yields

$$ \rho|_{L_t} : L_t = F^{-1}(Y \times \{t\}) \to Y, $$

which is therefore a smooth locally trivial fiber bundle with compact fibers. This proves the claimed link fibration.
Step 4: independence of t and a product decomposition in the x-direction. Be-
cause F is a submersion, at each $p \in N \cap X_{\text{reg}}$ the differential $dF_p : T_p(N \cap X_{\text{reg}}) \to T_{\rho(p)}Y \oplus \mathbb{R}$ is surjective. In particular, the affine subspace

$$
\{V_p \in T_p(N \cap X_{\text{reg}}) : d\rho_p(V_p) = 0, \, dx_p(V_p) = 1\}
$$

is nonempty and varies smoothly with $p$. Choose a smooth global section $V$ satisfying

$$
d\rho(V) = 0, \quad dx(V) = 1
$$

(using a partition of unity on $N \cap X_{\text{reg}}$). Let $\Phi_s$ denote its (maximal) flow.

Along any integral curve $\gamma(s) = \Phi_s(p)$ one has

$$
\frac{d}{ds} x(\gamma(s)) = dx(V) = 1 \Rightarrow x(\Phi_s(p)) = x(p) + s,
$$

and also $\rho(\Phi_s(p)) = \rho(p)$ since $d\rho(V) = 0$. Hence for each $t \in (0, \varepsilon)$ and $p \in L_{t_0}$ the time $s = t - t_0$ is admissible (because $x(p) + s = t \in (0, \varepsilon)$), and $\Phi_{t-t_0}$ defines a diffeomorphism

$$
\Psi_{t,t_0} : L_{t_0} \xrightarrow{\approx} L_t, \quad \rho \circ \Psi_{t,t_0} = \rho.
$$

In particular, all $\rho|_{L_t}$ are canonically isomorphic as fibrations over $Y$.

Fix $t_0 \in (0, \varepsilon)$ and set $L := L_{t_0}$. Define

$$
\Theta : (0, \varepsilon) \times L \to N \cap X_{\text{reg}}, \quad \Theta(t, p) := \Phi_{t-t_0}(p).
$$

Then $x(\Theta(t, p)) = t$ and $\Theta(t, \cdot) = \Psi_{t,t_0}$ is a diffeomorphism onto $L_t$. Thus $\Theta$ is a
diffeomorphism, giving the desired identification $N \cap X_{\text{reg}} \cong (0, \varepsilon) \times L$ (compatible
with $(\rho, x))$. $\square$

**3.2. FS wedge expansion.** We next record the metric statement needed for the analysis.

**Proposition 3.2** (FS wedge form with edge-smooth remainder). Let $X$ be a projective simple-edge space and let $g_{\text{FS}}$ be the induced FS metric on $X^{\text{reg}}$. On $N \cap X^{\text{reg}} \cong (0, \varepsilon) \times L$ from Proposition 3.1, there exist smooth metrics $g_Y$ on $Y$ and $g_L$ along link fibers, and an edge-smooth symmetric 2-tensor $\kappa$ such that

$$
g_{\text{FS}} = dx^2 + x^2 g_L + \rho^* g_Y + x \kappa.
$$

Here “edge-smooth” means the coefficients of $\kappa$ are smooth and bounded together
with all derivatives with respect to the edge vector fields

$$
x \partial_x, \quad \partial_{y^a} \text{ (horizontal lifts),} \quad \partial_{z^\mu} \text{ (vertical).}
$$

Moreover, the remainder $\kappa$ is uniformly bounded together with its iterated edge
derivatives on $N \cap X_{\text{reg}}$ (see Appendix B).

Proof. This is exactly the content of the Fubini–Study wedge expansion proved in Appendix B.4, Proposition B.5, specialized to the ambient metric $g = g_{\text{FS}}$ and the control-data identification $N \cap X^{\text{reg}} \cong (0, \varepsilon) \times L$ from Proposition 3.1.

For the reader’s convenience we indicate how the objects are chosen and how
the error term is shown to be edge-smooth. Following **Step 1** of Proposition B.5,
replace the control function $x$ by $x = \psi(r)$ with $r = \text{dist}_{\text{FS}}(\cdot, Y)$ and $\psi'(0) = 1$, so
that $|dx|_{g_{\text{FS}}} = 1 + O(x)$ near $Y$. Let $V := \ker(d\rho) \cap \ker(dx)$ be the vertical bundle
(tangent to the link fibers at fixed $x$) and choose a smooth horizontal complement
$H \subset \ker(dx)$ with $d\rho|_H : H \to TY$ an isomorphism. Then Lemma B.3 (Angle control $\Rightarrow$ mixed terms are $O(x)$) yields uniform estimates

$$g_{\text{FS}}(H, V) = O(x), \quad g_{\text{FS}}(\nabla x, H) = O(x), \quad g_{\text{FS}}(\nabla x, V) = O(x),$$

and the same $O(x)$ order is stable under iterated differentiation by edge vector fields in an adapted chart. Moreover, Lemma B.4 (Vertical scaling) identifies the leading vertical block:

$$g_{\text{FS}}|_{V \times V} = x^2 g_L + O(x^3),$$

again with stability under edge differentiation. Define $g_Y$ by restricting $g_{\text{FS}}$ to the horizontal subspace at $x=0$ (transported by $\rho$ via the chosen horizontal lift), and set

$$g_0 := dx^2 + \rho^* g_Y + x^2 g_L, \quad \kappa := x^{-1}(g_{\text{FS}} - g_0).$$

By the preceding mixed-term and vertical-scaling estimates, every component of $g_{\text{FS}} - g_0$ vanishes to order $x$ in an adapted wedge coframe, and this vanishing order is preserved under edge derivatives. Using the definition of edge-smooth tensors from Appendix B.1, it follows that $\kappa$ is edge-smooth on $N \cap X^{\text{reg}}$, and hence

$$g_{\text{FS}} = dx^2 + x^2 g_L + \rho^* g_Y + x\kappa,$$

as claimed.

<figure><img src="image_3.png" /></figure>

*Remark 3.3* (Where the missing estimates are proved). The geometric input Proposition 3.2 is proved in Appendix B.4, Proposition B.5. In particular, the choice of $x = \psi(\text{dist}_{\text{FS}}(\cdot, Y))$ and the estimate $|dx|_{g_{\text{FS}}} = 1+O(x)$ are in **Step 1** of Proposition B.5, the $O(x)$ control of all mixed components is Lemma B.3, and the identification of the leading vertical block $x^2g_L$ is Lemma B.4. The notion of edge-smooth tensors used here is recalled in Appendix B.1.

*Remark 3.4.* Proposition 3.2 is the only place where projectivity/FS is used in an essential way: it supplies an ambient smooth Kähler metric to measure distances and control expansions, while Whitney control data provides the conical (wedge) coordinates. No “product analytic type” assumption is required.

## 4. THE WEDGE DE RHAM OPERATOR AND THE TANGENTIAL FAMILY

### 4.1. Wedge Dirac normal form.
Let $D = \partial_{dR} = d + \delta$ on $(X^{\text{reg}}, g_{\text{FS}})$. In wedge-adapted orthonormal frames, wedge Dirac operators admit the standard normal form.

**Proposition 4.1** (Dirac normal form). Under the hypotheses of Proposition 3.2, in an edge-adapted orthonormal coframe one has

$$(4.1.1) \qquad D = \Gamma\left(\partial_x + \frac{1}{x}A + B(x)\right),$$

where $\Gamma = c(dx)$ is Clifford multiplication by $dx$, $A = A_y$ is a self-adjoint elliptic first-order operator on each compact link fiber $L_y$ varying smoothly with $y$, and $B(x)$ is a first-order edge operator with edge-smooth coefficients down to $x=0$.

*Proof.* Work in a tubular neighborhood $N \cap X_{\text{reg}} \cong (0, \varepsilon)_x \times L$ over the singular stratum $Y$ and use the FS wedge expansion established in Proposition 3.2 (proved in Appendix B):

$$g = dx^2 + \rho^* g_Y + x^2 g_L + x \kappa,$$

with $\kappa$ edge-smooth down to $x=0$.
**Step 1: wedge-orthonormal frame.** Let {$e_i$} be an orthonormal frame for $(Y, g_Y)$ and {$f_\alpha$} an orthonormal frame for the vertical metric $(L_y, g_L)$. Define a wedge frame on $X_{\text{reg}}$ by

$$E_0 := \partial_x, \quad E_i := \tilde{e}_i, \quad E_\alpha := x^{-1} \tilde{f}_\alpha,$$

where tildes denote horizontal/vertical lifts. The $x\kappa$ term produces only edge-smooth corrections to the connection one-forms in this wedge frame (i.e. coefficients that remain bounded under edge derivatives).

**Step 2: local formula for $\partial_{dR}$.** In a wedge-orthonormal frame, the Hodge-de Rham operator $D = \partial_{dR}$ can be written in the standard Dirac form

$$D = \sum_j c(\chi^j) \nabla_{E_j} + \mathcal{R},$$

where {$\chi^j$} is the dual coframe, $c(\cdot)$ is Clifford multiplication, and $\mathcal{R}$ is a zeroth-order endomorphism built from curvature terms. Because the only singular scaling in the frame is $E_\alpha = x^{-1}\tilde{f}_\alpha$, the only potential $x^{-1}$ singularity comes from the vertical part of the connection along the link fibers.

**Step 3: extraction of the $x^{-1}$ tangential operator.** A direct computation shows that the coefficient of $x^{-1}$ in $D$ is exactly the fiberwise Hodge-de Rham operator on $(L_y, g_L)$ (together with the standard degree-shift term coming from the form bundle grading). In particular, the $O(x)$ perturbation $x\kappa$ contributes only to the *bounded* operator $B(x)$ below and does not affect the $x^{-1}$ coefficient.

**Step 4: normal form.** Collecting the $\partial_x$ term and the $x^{-1}$ tangential term yields

$$D = \Gamma\left(\partial_x + \frac{1}{x}A_y + B(x)\right),$$

where $\Gamma = c(dx)$ up to a bounded bundle automorphism, $A_y$ is the fiberwise self-adjoint elliptic tangential operator on $L_y$, and $B(x)$ is a first-order edge operator with edge-smooth coefficients down to $x = 0$. $\square$

4.2. **Explicit tangential operator and $A^2$.** In the product-type case, the tangential operator can be written explicitly and its square is Laplace type, which connects $\text{Ker}(A_y)$ to link cohomology.

**Lemma 4.2 (Tangential formula and Laplace-type square).** Assume a strict product-type wedge metric near $Y$,

$$g = dx^2 + x^2 g_L + \rho^* g_Y \quad \text{on } (0, \varepsilon) \times L.$$

Then the normal form (4.1.1) holds with

$$(4.2.1) \qquad A = D_L + \frac{f}{2} - \mathbf{N},$$

where $f = \dim_{\mathbb{R}} L_y$, $D_L = d_L + \delta_L$ is the Hodge-de Rham Dirac along the link fibers $(L_y, g_L)$ and $\mathbf{N}$ is the vertical form-degree operator. Moreover,

$$(4.2.2) \qquad A^2 = \Delta_L + \left(\mathbf{N} - \frac{f}{2}\right)^2,$$

with $\Delta_L = D_L^2$ the Hodge Laplacian on $(L_y, g_L)$. In particular,

$$\text{Ker}(A_y) = \{u : \Delta_{L_y} u = 0, \mathbf{N}u = \frac{f}{2}u\}, \quad \text{and hence} \quad \dim \text{Ker}(A_y) = b_{f/2}(L_y).$$
*Proof.* Fix $y \in Y$ and work fiberwise on the link $(L_y, g_L)$. Write forms as $\chi = \alpha(x)+dx\wedge\beta(x)$, use the wedge Hodge star on $dx^2+x^2g_L+\rho^*g_Y$, and note the vertical Hodge star scales as $\ast_L^{(x)} = x^{f-2N}\ast_L$ on vertical forms. Differentiating $x^{f-2N}$ yields the $(\frac{f}{2}-N)x^{-1}$ term in the normal form, while vertical derivatives scale by $x^{-1}$ in an orthonormal wedge frame, giving (4.2.1). Since $D_L$ changes vertical degree by $\pm 1$, it anticommutes with $N - \frac{f}{2}$, so $(D_L - (N - \frac{f}{2})^2) = D_L^2 + (N - \frac{f}{2})^2$, giving (4.2.2). $\square$

*Remark 4.3* (FS remainder does not change the $x^{-1}$–tangential term). For the Fubini–Study metric, Proposition B.5 shows that

$$g_{\text{FS}} = dx^2 + x^2 g_L + \rho^* g_Y + x\kappa,$$

with $\kappa$ edge-smooth. Although this is only product-type to leading order, the $x\kappa$ remainder cannot create any new $x^{-1}$ singularities in the wedge Dirac expression (4.1.1); it only modifies the $O(1)$ term $B(x)$. Lemma 4.4 below makes this precise, upgrading Lemma 4.2 from a product-type computation to an identity for the actual tangential family $A_y$ arising from $g_{\text{FS}}$.

**Lemma 4.4** (Tangential operator formula persists under an $x\kappa$ remainder). Assume a wedge metric of the form

$$g = dx^2 + x^2 g_L + \rho^* g_Y + x\kappa$$

with $\kappa$ edge-smooth (in the sense stated in Proposition 3.2). Let $D = \partial_{dR} = d + \delta$ be the de Rham Dirac operator. Then in the normal form (4.1.1) one has the same tangential operator as in the strict product-type case:

$$A_y = D_{L_y} + \frac{f}{2} - N,$$

where $f = \dim_{\mathbb{R}} L_y$, $D_{L_y} = d_{L_y} + \delta_{L_y}$ is the fiberwise Hodge–de Rham operator for $(L_y, g_L)$, and $N$ is the vertical form-degree operator. Consequently, the square identity (4.2.2) holds fiberwise:

$$A_y^2 = \Delta_{L_y} + \left(N - \frac{f}{2}\right)^2,$$

and in particular

$$\dim \ker(A_y) = b_{f/2}(L_y)$$

*Proof.* Write the Fubini–Study wedge expansion (Proposition 3.2) as

$$g = g_0 + x\kappa, \quad g_0 := dx^2 + \rho^* g_Y + x^2 g_L,$$

where $\kappa$ is edge-smooth and hence has coefficients bounded together with all iterated derivatives along $V_e$ in any adapted wedge coframe.

Fix an adapted product chart $N \cap X_{\text{reg}} \cong (0, \varepsilon)_x \times U_y \times L_z$ and choose local orthonormal coframes $(\phi^i)$ on $(U, g_Y)$ and $(\theta^\alpha)$ along the fibers $(L_y, g_L)$. Set the product-type wedge coframe for $g_0$

$$\chi_0 := (dx, \phi^i, x\theta^\alpha).$$

In this coframe the metric matrix of $g_0$ is exactly I.
**Step 1: metric matrix and an elementary orthonormalization.** Let $G(x)$ denote the matrix of $g$ in the coframe $\chi_0$. Since $g = g_0 + x\kappa$ and $\kappa$ is edge-smooth, we can write

$$G(x) = I + xK(x),$$

where the entries of $K(x)$ are uniformly bounded and remain bounded under all iterated edge derivatives. For $x$ sufficiently small, $G(x)$ is uniformly positive definite. Define

$$M(x) := I - \frac{1}{2}xK(x).$$

A direct expansion gives

$$M(x)^T G(x) M(x) = \left(I - \frac{1}{2}xK^T\right) \left(I + xK\right) \left(I - \frac{1}{2}xK\right) = I + x\left(K - \frac{1}{2}K - \frac{1}{2}K^T\right) + O(x^2) = I + x \text{Skew}(K) + O(x^2),$$

where $\text{Skew}(K) = (K - K^T)/2$ and the $O(x^2)$ term has uniformly bounded coefficients (with bounded edge derivatives). Since $\text{Skew}(K)$ is skew-symmetric, there exists a smooth matrix $Q(x) = I + O(x)$ with bounded edge derivatives such that

$$Q(x)^T (I + x \text{Skew}(K) + O(x^2)) Q(x) = I + O(x^2)$$

(e.g. take $Q(x) = \exp(-\frac{1}{2}x \text{Skew}(K))$ and expand; only the fact $Q(x) = I + O(x)$ is used). Setting $\tilde{M}(x) := M(x)Q(x)$ we obtain

$$\tilde{M}(x)^T G(x) \tilde{M}(x) = I + O(x^2).$$

Thus the coframe

$$\chi := \tilde{M}(x) \chi_0$$

is wedge-orthonormal for $g$ up to an $O(x^2)$ error, and in particular satisfies

$$\chi = \chi_0 + O(x),$$

with all coefficients and iterated edge derivatives bounded.

**Step 2: where an $x^{-1}$ singularity can occur.** Let $(E_0, E_i, E_\alpha)$ be the dual frame to $\chi$. Because $\chi = \chi_0 + O(x)$, the dual frame satisfies

$$E_0 = \partial_x + O(1), \quad E_i = \tilde{e}_i + O(1), \quad E_\alpha = x^{-1} \tilde{f}_\alpha + O(1),$$

where $\tilde{e}_i$ are horizontal lifts and $\tilde{f}_\alpha$ are vertical lifts. In particular, the only source of an $x^{-1}$ factor is the vertical scaling $E_\alpha = x^{-1}\tilde{f}_\alpha + O(1)$; the $O(1)$ corrections produced by the $x\kappa$ remainder cannot generate additional $x^{-1}$ singularities.

**Step 3: identification of the coefficient of $x^{-1}$.** The de Rham Dirac operator can be written in a wedge-orthonormal frame as

$$D = \sum_j c(\chi^j) \nabla_{E_j} + \mathcal{R},$$

where $\mathcal{R}$ is zeroth order. By **Step 2**, the $x^{-1}$ coefficient in this expression comes exactly from the vertical part and coincides with the product-type computation for $g_0$. Hence the coefficient of $x^{-1}$ in the wedge normal form

$$D = \Gamma\left(\partial_x + \frac{1}{x}A_y + B(x)\right)$$

is the same as in Lemma 4.2, namely

$$A_y = D_{L_y} + \frac{f}{2} - N,$$
where $\mathcal{N}$ is vertical form degree and $D_{L_y}$ is the fiberwise de Rham Dirac operator. The square identity

$$A_y^2 = \Delta_{L_y} + \left(N - \frac{f}{2}\right)^2$$

follows fiberwise exactly as in Lemma 4.2. □

## 5. AUTOMATIC STABILITY: KERNEL BUNDLE, UNIFORM GAP, AND THE CRITICAL STRIP

### 5.1. Automatic stability package for the tangential family. We keep the wedge Dirac normal form

$$D = \Gamma\left(\partial_x + \frac{1}{x}A_y + B(x)\right)$$

from Proposition 4.1 and the tangential identification of Lemma 4.4. In this section we treat $A_y$ as a smooth family of self-adjoint elliptic operators on the link fibers and extract three uniform consequences: constant nullity, a uniform spectral gap at 0, and a constant-multiplicity indicial strip.

Family trivialization. Fix $y_0 \in Y$ and choose a coordinate neighborhood $U \ni y_0$ over which the link fibration trivializes:

$$L|_U \cong U \times L_0.$$

Using this identification (and, if desired, a smooth family of unitary maps to fix the $L^2$ density), we may view the tangential family as a smooth map

$$y \mapsto A_y \in \mathcal{L}(H^1\Omega^\bullet(L_0), L^2\Omega^\bullet(L_0)),$$

i.e. in local coordinates $A_y$ is a first-order elliptic differential operator on the fixed closed manifold $L_0$ whose coefficients depend smoothly on $y$. Consequently, for each fixed $\lambda \notin \text{Spec}(A_{y_0})$ the resolvent $(\lambda - A_y)^{-1} : L^2 \to H^1$ exists for all $y$ near $y_0$ and depends smoothly on $y$; differentiating the identity $(\lambda - A_y)(\lambda - A_y)^{-1} = I$ yields

$$\partial_{y_j}(\lambda - A_y)^{-1} = (\lambda - A_y)^{-1}(\partial_{y_j} A_y)(\lambda - A_y)^{-1},$$

and similarly for higher derivatives.

Summary of the stability package. The goal of this section is to extract three uniform consequences from the tangential family $A_y$ appearing in the wedge Dirac normal form

$$D = \Gamma\left(\partial_x + \frac{1}{x}A_y + B(x)\right).$$

First, Proposition 5.1 identifies $\ker(A_y)$ with the middle-degree harmonic forms on the link fiber $L_y$, hence $\dim \ker(A_y)$ is constant along $Y$ and the kernel spaces assemble into a smooth vector bundle $H = \ker(A) \to Y$ (via a Riesz projection). Second, Proposition 5.2 uses min-max continuity together with the compactness of $Y$ to upgrade the pointwise spectral gap at 0 to a uniform gap $\text{Spec}(A_y) \cap (-\varepsilon_0, \varepsilon_0) = \{0\}$ for all $y$. Finally, Proposition 5.3 combines the uniform gap with the Dirac indicial correspondence to produce a constant-multiplicity indicial strip, so that the trace bundle in the critical window is precisely $H$ and the boundary analysis in Sections 6-7 can be carried out uniformly in $y$.
## 5.2. Kernel rank is constant globally.

**Proposition 5.1** (Constant rank of $\ker(A)$). Let $X$ be projective simple-edge. Then $\dim \ker(A_y)$ is constant on $Y$. Equivalently, the boundary data space

$$ \mathcal{H} := \ker(A) \to Y $$

is a smooth finite-rank vector bundle.

*Proof. Step 1:* $\dim \ker(A_y)$ is constant. By Proposition 3.1, the link fibration $L \to Y$ is a smooth fiber bundle with compact fibers, hence all fibers $L_y$ are diffeomorphic and the Betti numbers $b_k(L_y)$ are independent of $y$.

By Lemma 4.4 we have fiberwise

$$ A_y^2 = \Delta_{L_y} + \left(N - \frac{f}{2}\right)^2. $$

If $u \in \ker(A_y)$ then $\langle A_y^2 u, u \rangle$ implies simultaneously $\Delta_{L_y} u = 0$ and $(N - \frac{f}{2})u = 0$. Thus $\ker(A_y)$ consists precisely of harmonic forms of vertical degree $f/2$ on $L_y$, and hence

$$ \dim \ker(A_y) = b_{f/2}(L_y), $$

which is constant in $y$.

**Step 2:** local smoothness of the kernel via a Riesz projection. Fix $y_0 \in Y$ and work over a trivializing neighborhood $U \ni y_0$ as in the family setup above, so $A_y$ acts on the fixed Hilbert space $L^2 \Omega^\bullet(L_0)$.

Let $m := \dim \ker(A_{y_0})$. Since $A_{y_0}$ is self-adjoint elliptic on a closed manifold, its spectrum is discrete and 0 is an isolated eigenvalue of multiplicity $m$. Let

$$ \mu_0 := \min\{|\lambda| : \lambda \in \operatorname{Spec}(A_{y_0}) \setminus \{0\}\} > 0, $$

and choose $r \in (0, \mu_0)$, for instance $r = \mu_0/2$. Let $\gamma = \{|\lambda| = r\}$.

We claim that after shrinking $U$ if necessary,

$$ \gamma \cap \operatorname{Spec}(A_y) = \emptyset, \quad \forall y \in U. $$

Indeed, by the min-max continuity of eigenvalues for smooth self-adjoint elliptic families (see Appendix A (spectral stability / Riesz projections)), the nonzero eigenvalues of $A_y$ vary continuously in $y$. Hence no eigenvalue can cross the circle $\gamma$ for $y$ sufficiently close to $y_0$.

Define the (finite-rank) Riesz projection

$$ \Pi(y) := \frac{1}{2\pi i} \int_{\gamma} (\lambda - A_y)^{-1} d\lambda \in \mathcal{L}(L^2, L^2). $$

By the family setup discussion, $(\lambda - A_y)^{-1}$ depends smoothly on $y$ for $\lambda \in \gamma$, hence so does $\Pi(y)$. Standard spectral theory gives

$$ \mathrm{ran}\Pi(y) = \bigoplus_{\lambda \in \mathrm{Spec}(A_y), |\lambda| < r} \ker(A_y - \lambda) = \ker(A_y), $$

where the last equality uses that 0 is the only eigenvalue inside $\gamma$.

Thus $\ker(A_y) = \mathrm{ran}\Pi(y)$ is the range of a smooth finite-rank projection on the trivial bundle $U \times L^2 \to U$, and therefore forms a smooth vector subbundle over $U$. Covering $Y$ by finitely many such charts and using that the bundle transition maps are induced by the trivializations of $L \to Y$ yields a smooth finite-rank bundle $H = \ker(A) \to Y$. $\square$
### 5.3. Uniform spectral gap.

**Proposition 5.2** (Uniform spectral gap at 0). Let X be projective simple-edge. Then there exists $\epsilon_0 > 0$ such that for all $y \in Y$,

$$ \mathrm{Spec}(A_y) \cap (-\epsilon_0, \epsilon_0) = \{0\}. $$

*Proof.* Let $m := \dim \ker(A_y)$, which is constant by Proposition 5.1. For each $y \in Y$, since $A_y$ is self-adjoint elliptic on the closed manifold $L_y$, the spectrum is discrete, real, and accumulates only at $\pm\infty$. Define the spectral gap function

$$ \lambda_*(y) := \min\{|\lambda| : \lambda \in \mathrm{Spec}(A_y) \setminus \{0\}\} = \text{the smallest positive eigenvalue of } |A_y|. $$
Then $\lambda_*(y) > 0$ for every $y$.

**Step 1: continuity of $\lambda_*(y)$ (min-max).** Work over a trivializing neighborhood $U \subset Y$ and view $A_y$ on a fixed space. Consider the quadratic form

$$ q_y(u) := \|A_y u\|_{L^2}^2, \quad u \in H^1\Omega^\bullet(L_0), $$

with the Rayleigh quotient $R_y(u) = q_y(u)/\|u\|_{L^2}^2$. Since coefficients of $A_y$ depend smoothly on $y$, the map $y \mapsto q_y(u)$ is continuous for each fixed $u$, and in fact locally uniform on the unit sphere of $H^1$ modulo compactness arguments (this is exactly the hypothesis behind Appendix A, Lemma A.1). The min-max principle then implies that the ordered eigenvalues of $A_y^2$ vary continuously in $y$, hence so does $\lambda_*(y)$.

**Step 2: compactness gives a uniform lower bound.** Since $Y$ is compact and $\lambda_*(y)$ is continuous and strictly positive, it attains a minimum:

$$ \epsilon_0 := \min_{y \in Y} \lambda_*(y) > 0. $$

Therefore for every $y \in Y$,

$$ \mathrm{Spec}(A_y) \cap (-\epsilon_0, \epsilon_0) = \{0\}, $$

which is the desired uniform spectral gap at 0. □

### 5.4. Critical strip and constant indicial multiplicity.

**Proposition 5.3** (A critical strip with constant multiplicity). There exists a weight strip $(\delta, \bar{\delta})$ around the $L^2$ threshold such that the only indicial roots of $D = \partial_{dR}$ in this strip correspond to the zero mode $\lambda = 0$ of $A_y$ (for all $y \in Y$). In particular, the trace bundle in this strip identifies with $\mathcal{H} = \mathrm{Ker}(A) \to Y$ and has constant rank.

*Proof.* Recall the wedge normal form

$$ D = \Gamma\left(\partial_x + \frac{1}{x}A_y + B(x)\right). $$

The corresponding indicial operator at fixed $y$ is obtained by freezing coefficients at $x = 0$ and discarding the $O(1)$ term $B(x)$, namely

$$ I_y(D) = \Gamma\left(\partial_x + \frac{1}{x}A_y\right). $$

Diagonalize $A_y$ on the fiber: if $A_y v = \lambda v$, then the model equation $I_y(D)u = 0$ admits separated solutions of the form

$$ u(x) \sim x^{-\lambda} v \quad (\text{and similarly } x^{+\lambda} v), $$
so the indicial exponents in any fixed weight window are controlled by the small spectrum of $A_y$ (this is the standard Dirac/wedge indicial correspondence).

By Proposition 5.2 there is a uniform gap $\epsilon_0 > 0$ such that $\text{Spec}(A_y) \cap (-\epsilon_0, \epsilon_0) = \{0\}$ for all $y$. Choose any $\delta \in (0, \epsilon_0)$; equivalently, choose the weight strip $(-\delta, \delta)$ around the $L^2$ threshold so narrow that it only sees indicial exponents coming from eigenvalues $\lambda$ with $|\lambda| < \delta$. Then the only contribution comes from $\lambda = 0$, uniformly in $y$.

Finally, the multiplicity of the $\lambda = 0$ mode is exactly dim $\ker(A_y)$, which is constant by Proposition 5.1. Hence the indicial data in this strip has constant multiplicity, and the trace bundle in this strip identifies with

$$H = \ker(A) \to Y$$

with constant rank.

□

Figure 2 provides a schematic visualization of this situation. In a narrow weight strip $\{|\Re\lambda| < \epsilon\}$ around the $L^2$-threshold, the only indicial root that occurs is $\lambda = 0$, and its multiplicity is the constant rank of the boundary data bundle $H = \ker(A) \to Y$. All other boundary spectrum lies outside this strip. This constant-multiplicity picture is exactly the input needed for the Calderón and trace-bundle constructions that follow in Section 6.

<figure><img src="image_4.png" /><figcaption>FIGURE 2. Schematic picture of the boundary spectrum of $A_y$ in the constant-multiplicity strip of Proposition 5.3. The only indicial root in the strip is $\lambda = 0$, with multiplicity equal to $\text{rank}(H)$, yielding the trace bundle $H \to Y$.</figcaption></figure>

## 6. CALDERÓN PACKAGE AND CANONICAL BOUNDARY DATA

We now appeal to the standard edge/wedge calculus for Dirac-type operators with a constant-multiplicity strip. The purpose of Sections 3–5 is exactly to verify these hypotheses *globally* for FS metrics.

**Analytic input checklist.** The standard edge/wedge Calderón package for Dirac-type operators applies once: (i) a wedge Dirac normal form with edge-smooth coefficients is available; (ii) there is a critical weight strip with constant indicial multiplicity around the $L^2$ threshold; (iii) the maximal domain admits a canonical leading expansion in that strip, defining a finite-rank trace bundle. In our setting, (i) is Proposition 4.1, whose hypotheses are ensured by the FS wedge expansion (Proposition 3.2, proved in Appendix B as Proposition B.5) together with the link fibration (Proposition 3.1); (ii) is Proposition 5.3 (using Lemma A.1 and Lemma A.2); and (iii) is recorded as Lemma C.1 in Appendix C.
**Theorem 6.1** (Trace and Calderón package in the critical strip). Let $D = \partial_{dR}$ on $(X_{\text{reg}}, g_{\text{FS}})$ for a projective simple-edge space $X$. In the weight strip of Proposition 5.3 there exist:

(1) a continuous trace map

$$ \mathrm{Tr}_D : \mathrm{Dom}(D_{\max}) \to \Gamma(Y; H), $$

extracting the leading coefficient in the maximal-domain expansion;

(2) a Calderón projector

$$ C : \Gamma(Y; H) \to \Gamma(Y; H), \quad C^2 = C, $$

which is an order-0 pseudodifferential projection (in the appropriate wedge/edge calculus along $Y$) and whose range equals the Cauchy data space

$$ W_{\text{can}} := \mathrm{Tr}_D(\ker(D) \cap L^2(X_{\text{reg}})) = \mathrm{ran}(C) \subset H; $$

(3) a Poisson operator

$$ P : \Gamma(Y; H) \to \ker(D) \cap L^2(X_{\text{reg}}) $$

satisfying

$$ \mathrm{Tr}_D(P(v)) = Cv, \quad \forall v \in \Gamma(Y; H). $$

In particular, $P$ is a right inverse to $\mathrm{Tr}_D$ on $\Gamma(Y; W_{\text{can}})$: if $w \in \Gamma(Y; W_{\text{can}})$ then $\mathrm{Tr}_D(P(w)) = w$.

*Proof.* We give a complete reference-level argument, indicating precisely where the edge/wedge boundary calculus is used.

**Step 1: reduction to a wedge Dirac operator with a constant-multiplicity strip.** By the FS wedge expansion (Proposition 3.2, proved in Appendix B as Proposition B.5) and the Dirac normal form (Proposition 4.1), the de Rham Dirac operator $D = \partial_{dR}$ is a first-order wedge Dirac-type operator with edge-smooth coefficients. By the uniform gap statement for the tangential family (Proposition 5.2) and the indicial-root description (Proposition 5.3), there exists a weight strip around the $L^2$ threshold in which the only indicial contribution is the $\lambda = 0$ mode, with constant multiplicity rk($H$).

**Step 2: maximal domain asymptotics and trace.** In the constant-multiplicity strip, standard Mellin/edge arguments imply that every $u \in \mathrm{Dom}(D_{\max})$ has a two-term leading asymptotic expansion

$$ u \sim x^{-f/2}(a(y) + x b(y)) + u', \quad u' \in x^{-f/2+\varepsilon} H_e^1, $$

where $a(y)$ takes values in the trace bundle $H = \ker(A) \to Y$. We define $\mathrm{Tr}_D(u) := a$. The existence/uniqueness/continuity of this expansion and the trace map are recorded in Lemma C.1.

**Step 3: Poisson operator and Calderón projector.** In the constant-multiplicity strip, the standard Calderón construction for (formally self-adjoint) Dirac-type operators produces an order-0 pseudodifferential idempotent $C$ on $\Gamma(Y; H)$ whose range equals the Cauchy data space $W_{\text{can}} = \mathrm{Tr}_D(\ker(D) \cap L^2)$. Once $C$ is fixed, define the Poisson operator by prescribing trace only on $W_{\text{can}}$ and extending by projection: choose a right inverse

$$ P_{\text{can}} : \Gamma(Y; W_{\text{can}}) \to \ker(D) \cap L^2(X_{\text{reg}}) \quad \text{with} \quad \mathrm{Tr}_D \circ P_{\text{can}} = \mathrm{Id}, $$
and set $P := P_{\text{can}} \circ C$ on $\Gamma(Y; H)$. Then $\text{Tr}_D(P(v)) = Cv$ for all $v$, and in particular $\text{Tr}_D(P(w)) = w$ for $w \in \Gamma(Y; W_{\text{can}})$. The mapping properties and the wedge/edge pseudodifferential nature of $C$ are recorded in Appendix C.

**Step 4: smooth dependence in y.** All constructions above depend smoothly on the base point $y \in Y$ because the normal operator family and the tangential operator family vary smoothly in $y$, and the constant-multiplicity strip prevents eigenvalue crossings inside the critical window. Hence $\text{ran}(C) \subset H$ is a smooth subbundle. $\square$

## 6.1. Green pairing and Lagrangian Cauchy data.

**Theorem 6.2 (Green pairing and Lagrangianity).** There is a nondegenerate fiber-wise bilinear pairing

$$B: \mathcal{H} \times \mathcal{H} \to \mathbb{R}$$

such that for $u, v \in \text{Dom}(D_{\text{max}})$ one has Green's formula

$$\langle Du, v \rangle_{L^2} - \langle u, Dv \rangle_{L^2} = B(\text{Tr}_D u, \text{Tr}_D v).$$

Moreover, the Cauchy data bundle $W_{\text{can}} = \text{ran}(C) \subset \mathcal{H}$ is Lagrangian for $B$, i.e.

$$B|_{W_{\text{can}} \times W_{\text{can}}} = 0, \quad W_{\text{can}} = W_{\text{can}}^{\perp B}.$$

**Proof. Step 1: the boundary pairing.** In a wedge-orthonormal frame, let $c(dx)$ denote Clifford multiplication by $dx$. Define the fiberwise pairing on $H = \ker(A) \to Y$ by

$$B_y(a, b) := \int_{L_y} \langle c(dx)a(y, \cdot), b(y, \cdot) \rangle \, \text{dvol}_{g_{L_y}}, \quad a, b \in H_y,$$

and set $B(a, b) := \int_Y B_y(a(y), b(y)) \, \text{dvol}_{g_Y}$. Nondegeneracy follows from $c(dx)^2 = -\text{Id}$ on boundary data and the $L^2$-inner product on the link.

**Step 2: Green's formula.** For $u, v \in \text{Dom}(D_{\text{max}})$ supported in the tubular neighborhood, integrate by parts on $(\varepsilon', \varepsilon) \times L_y$ using formal self-adjointness of $D$ in the wedge frame. The outer boundary term vanishes by support. Inserting the two-term maximal-domain expansions from Lemma C.1 and using that the remainder gains positive weight, the inner boundary flux converges as $\varepsilon' \downarrow 0$ and depends only on the leading coefficients $\text{Tr}_D u, \text{Tr}_D v \in \Gamma(Y; H)$. Hence

$$\langle Du, v \rangle_{L^2} - \langle u, Dv \rangle_{L^2} = B(\text{Tr}_D u, \text{Tr}_D v),$$

as proved in Lemma C.8.

**Step 3: Lagrangianity of Cauchy data.** Let $W_{\text{can}} = \text{ran}(C) \subset H$. If $a, b \in \Gamma(Y; W_{\text{can}})$, choose Poisson solutions $u = P(a)$ and $v = P(b)$, so $Du = Dv = 0$ and, since $Ca = a$ and $Cb = b$, we have $\text{Tr}_D u = \text{Tr}_D(P(a)) = Ca = a$ and similarly $\text{Tr}_D v = b$. Applying Green's formula gives $B(a, b) = 0$, so $W_{\text{can}}$ is isotropic. Maximal isotropicity is proved in Proposition 6.5, using the comparison operator from Appendix C.2.1. $\square$

### 6.1.1. Maximal isotropicity via Fredholm pairs.
Let $B$ be the Green boundary pairing on $H$ (see Appendix C.3). Write $W_{\text{can}} = \text{ran}(C) \subset H$.

**Lemma 6.3 (Isotropicity).** $W_{\text{can}}$ is isotropic with respect to $B$.
**Proof.** If $a, b \in W_{\text{can}}$, then $a = Ca$ and $b = Cb$. Let $u = P(a)$ and $v = P(b)$ be the Poisson solutions provided by Theorem 6.1 (Poisson operator), so $Du = Dv = 0$ and $\text{Tr}_{D}u = \text{Tr}_{D}(P(a)) = Ca = a$ and similarly $\text{Tr}_{D}v = b$. Green's formula (Lemma C.8) gives

$$0 = \langle Du, v \rangle - \langle u, Dv \rangle = B(\text{Tr}_{D}u, \text{Tr}_{D}v) = B(a, b). \quad \square$$

**Lemma 6.4** (A reference Lagrangian boundary projector). There exists an order-0 pseudodifferential projection $R$ on $H$ such that $\text{ran}(R)$ is Lagrangian for $B$ and the boundary value problem $(D, R)$ is Fredholm.

**Proof.** One may take $R$ to be an APS-type spectral projection of the model tangential operator in the strip, [2] or any fixed Lagrangian graph boundary condition in the finite-dimensional trace bundle $H$. Fredholmness in the wedge setting follows from the standard theory of projection boundary conditions for Dirac-type wedge/edge operators in the constant-multiplicity strip; see Appendix C.4 (in particular Lemma C.10 and Theorem C.11), and the classical expositions such as [8, Theorem 11] or [4, §§12–13] together with the wedge/edge adaptations in [11, 12]. $\square$

**Proposition 6.5** (Maximal isotropicity from Fredholm pair). $W_{\text{can}}$ is maximal isotropic, hence Lagrangian: $W_{\text{can}} = W_{\text{can}}^{\perp B}$.

**Proof.** Let $R$ be as in Lemma 6.4 and set $P := C$. Consider the comparison operator

$$T_{R,P} := RP + (I - R)(I - P)$$

on the boundary Hilbert space. Since the boundary value problem $(D, R)$ is Fredholm, Lemma C.12 (Appendix C.4) yields that $T_{R,P}$ is Fredholm. By Lemma C.6 (comparison operator criterion; Appendix C.2.1), this is equivalent to the fact that $(\text{ran}(P), \text{ran}(R))$ is a Fredholm pair, and it provides the kernel/cokernel description used below.

Since $\text{ran}(R)$ is Lagrangian and $\text{ran}(P) = W_{\text{can}}$ is isotropic (Lemma 6.3), the only possible obstruction to $W_{\text{can}}$ being Lagrangian is a nontrivial quotient $W_{\text{can}}^{\perp B}/W_{\text{can}}$. Moreover, because $\text{ran}(R)$ is Lagrangian, Lemma C.10 implies that $D_R$ is self-adjoint, hence $\text{ind}(D_R) = 0$; therefore Lemma C.12 gives $\text{ind}(T_{R,P}) = 0$. In particular,

$$\dim(\text{ran}(P) \cap \text{ran}(I - R)) = \dim(\text{ran}(R) \cap \text{ran}(I - P)),$$

by the index formula in Lemma C.6.

Since $\text{ran}(R)$ is Lagrangian and $\text{ran}(P) = W_{\text{can}}$ is isotropic (Lemma 6.3), the only possible obstruction to $W_{\text{can}}$ being Lagrangian is a nontrivial quotient $W_{\text{can}}^{\perp B}/W_{\text{can}}$. But the Fredholm-pair condition forces $\text{ran}(P)$ to be of the same (finite) codimension as a Lagrangian subspace in the trace bundle: concretely, the kernel/cokernel description in Lemma C.6 implies that the defect indices measuring $\text{ran}(P) \cap \text{ran}(I - R)$ and $\text{ran}(R) \cap \text{ran}(I - P)$ are finite and balance in the self-adjoint case. In particular, because $D$ is formally self-adjoint, one has $\text{ind}(T_{R,P}) = 0$ and therefore

$$\dim(\text{ran}(P) \cap \text{ran}(I - R)) = \dim(\text{ran}(R) \cap \text{ran}(I - P)).$$

Choosing $R$ Lagrangian and using isotropicity of $\text{ran}(P)$, these defect spaces identify with the symplectic complements, forcing $\text{ran}(P)$ to be maximal isotropic. Hence $W_{\text{can}}$ is Lagrangian. $\square$
## 7. CANONICAL DOMAIN AND GLOBAL CGM IN DEPTH ONE

In this section we assemble the boundary package developed in Section 6 and Appendix C into a canonical closed realization of the wedge de Rham operator. We first define closed boundary-condition domains in terms of the trace bundle, then show that the Calderón-induced choice is self-adjoint and Fredholm. Finally, we interpret the cohomology of the resulting closed de Rham complex as a depth-one ideal boundary condition theory (equivalently, a Deligne-type intersection theory), obtaining the global FS-CGM consequence.

### 7.1. Boundary condition domains.

**Definition 7.1** (Closed domains from boundary subbundles). Let $W \subset \mathcal{H}$ be a smooth subbundle. Define

$$ \mathrm{Dom}(D_W) := \{u \in \mathrm{Dom}(D_{\mathrm{max}}) : \mathrm{Tr}_D u \in \Gamma(Y; W)\}. $$

Similarly define the first-order domain for $d$ by

$$ \mathrm{Dom}(d_W) := \{u \in L^2 \Omega^*(X^{\mathrm{reg}}) : u \in \mathrm{Dom}(d_{\mathrm{max}}), \mathrm{Tr}_D u \in \Gamma(Y; W)\}, $$

and analogously for $\delta$ (equivalently, define $\mathrm{Dom}(D_W)$ and then intersect with the graded components).

**Definition 7.2** (Calderón-induced canonical domain). Let $W_{\mathrm{can}} \subset H = \ker(A) \to Y$ be the Cauchy data bundle produced by the Calderón projector $C$ in Theorem 6.1 and Appendix C. We define the canonical (Calderón) domain for the wedge Dirac operator $D$ by

$$ \mathrm{Dom}(D_{\mathrm{can}}) := \{u \in \mathrm{Dom}(D_{\mathrm{max}}) : \mathrm{Tr}_D(u) \in \Gamma(Y; W_{\mathrm{can}})\}, $$

and set $D_{\mathrm{can}} := D|_{\mathrm{Dom}(D_{\mathrm{can}})}$.

Similarly, we define the canonical closed extension of the de Rham complex by

$$ \mathrm{Dom}(d_{\mathrm{can}}) := \{\chi \in \mathrm{Dom}(d_{\mathrm{max}}) : \mathrm{Tr}_D(\chi) \in \Gamma(Y; W_{\mathrm{can}})\}, \quad d_{\mathrm{can}} := d|_{\mathrm{Dom}(d_{\mathrm{can}})}. $$

**Remark 7.3** (Closedness). The trace map $\mathrm{Tr}_D : \mathrm{Dom}(D_{\mathrm{max}}) \to \Gamma(Y; H)$ constructed in Appendix C.1 is continuous with respect to the graph norm on $\mathrm{Dom}(D_{\mathrm{max}})$. Hence for any smooth subbundle $W \subset H$, the set $\{u \in \mathrm{Dom}(D_{\mathrm{max}}) : \mathrm{Tr}_D u \in \Gamma(Y; W)\}$ is a closed subspace of $\mathrm{Dom}(D_{\mathrm{max}})$, and therefore $D_W$ is a closed operator on $L^2$. The same statement holds for $d_W$ and $\delta_W$ (equivalently, for the graded components of $D_W = d_W + \delta_W$).

### 7.2. Self-adjointness and Fredholmness of the canonical extension.

**Theorem 7.4** (Self-adjointness and Fredholmness of the canonical extension). The canonical realization $D_{\mathrm{can}}$ of Definition 7.2 is self-adjoint and Fredholm on $L^2$. In particular, the canonical complex $(\mathrm{Dom}(d_{\mathrm{can}}), d)$ is closed and yields a Fredholm Hodge-de Rham operator $D_{\mathrm{can}} = d_{\mathrm{can}} + \delta_{\mathrm{can}}$.

*Proof. Step 1: Green pairing.* By the Green formula in the constant-multiplicity strip (Lemma C.8), there is a nondegenerate boundary pairing $B$ on the trace bundle $H \to Y$ such that, for all $u, v \in \mathrm{Dom}(D_{\mathrm{max}})$ supported in a tubular neighborhood of $Y$,

$$ \langle Du, v \rangle_{L^2} - \langle u, Dv \rangle_{L^2} = B(\mathrm{Tr}_D u, \mathrm{Tr}_D v). $$

By a cutoff argument (localizing near $Y$) this identity controls adjointness for boundary-condition domains.
**Step 2:** Adjoint domains from boundary orthogonals. Let $W \subset H$ be a smooth subbundle and let $W^{\perp_B} \subset H$ denote the fiberwise B-orthogonal complement. Then the adjoint domain identity holds (Lemma C.10):

$$ \mathrm{Dom}(D_W)^* = \mathrm{Dom}(D_{W^{\perp_B}}). $$

**Step 3:** The Calderón Cauchy data bundle is Lagrangian. Let $C$ be the Calderón projector (Theorem 6.1 / Proposition C.2) and set $W_{\mathrm{can}} := \mathrm{ran}(C) \subset H$. The Cauchy data space is Lagrangian for the Green pairing, i.e. $W_{\mathrm{can}} = W_{\mathrm{can}}^{\perp_B}$ (Proposition C.9, equivalently the maximal-isotropicity argument of Section 6.1.1).

**Step 4:** Self-adjointness. Applying **Step 2** with $W = W_{\mathrm{can}}$ and using **Step 3** gives

$$ \mathrm{Dom}(D_{\mathrm{can}})^* = \mathrm{Dom}(D_{W_{\mathrm{can}}})^* = \mathrm{Dom}(D_{W_{\mathrm{can}}^{\perp_B}}) = \mathrm{Dom}(D_{W_{\mathrm{can}}}) = \mathrm{Dom}(D_{\mathrm{can}}), $$
so $D_{\mathrm{can}}$ is self-adjoint.

**Step 5:** Fredholmness. Fredholmness for elliptic projection boundary conditions in the constant-multiplicity strip is proved in Appendix C.4: by Theorem C.11, the boundary condition defined by the Calderón projector yields a Fredholm realization. In particular, $D_{\mathrm{can}}$ is Fredholm; since it is self-adjoint, its Fredholm index is 0.

**Step 6:** Closedness of the canonical de Rham complex. By Remark 7.3, $\mathrm{Dom}(d_{\mathrm{can}})$ is closed in $\mathrm{Dom}(d_{\mathrm{max}})$ for the graph norm, hence $d_{\mathrm{can}}$ is closed; similarly for $\delta_{\mathrm{can}}$. Therefore $D_{\mathrm{can}} = d_{\mathrm{can}} + \delta_{\mathrm{can}}$ is a closed self-adjoint Fredholm Hodge-de Rham operator. $\square$

**7.3. Global CGM conclusion.** To state the topological consequence, we interpret boundary subbundles $W \subset H$ as depth-one ideal boundary condition data (mezzoperversities).

**Proposition 7.5** (Depth-one IBC and cohomology of $(\mathrm{Dom}(d_W), d)$). In depth one (simple-edge), a choice of a smooth subbundle $W \subset H = \ker(A) \to Y$ determines a canonical ideal boundary condition (mezzoperversity) datum. The corresponding Hilbert complex $(\mathrm{Dom}(d_W), d)$ computes the associated depth-one intersection theory:

$$ H^\bullet(\mathrm{Dom}(d_W), d) \cong IH_W^\bullet(X). $$

In particular, in the Witt case $H = 0$ there is no datum to choose and the resulting theory reduces to middle-perversity intersection cohomology.

**Proof.** A self-contained sheaf-theoretic proof is given in Appendix E. More precisely, Appendix E constructs the fine sheaf complex $\mathcal{L}_W^\bullet$ associated to the domain condition and shows that it is quasi-isomorphic to the depth-one Deligne-type intersection complex $\mathrm{IC}_W$; hence

$$ H^\bullet(\mathrm{Dom}(d_W), d) \cong \mathbb{H}^\bullet(X; \mathcal{L}_W^\bullet) \cong \mathbb{H}^\bullet(X; \mathrm{IC}_W) = IH_W^\bullet(X), $$

see Theorem E.3. (Conceptually, this is exactly the depth-one specialization of the general ALMP IBC/mezzoperversity framework.) $\square$

**Theorem 7.6** (Global FS-CGM consequence). Let $X$ be projective simple-edge and let $D_{\mathrm{can}}(d)$ be the canonical FS domain from Definition 7.2. Then

$$ H^\bullet(D_{\mathrm{can}}(d), d) \cong IH_{W_{\mathrm{can}}}^\bullet(X), $$
where $W_{\text{can}} = \text{ran}(C) \subset H$ is the Calderón Cauchy data bundle. In particular, in the Witt case $H = 0$ (equivalently $b_{f/2}(L_y) = 0$) one has

$$H^{\bullet}(D_{\text{can}}(d), d) \cong IH_{m}^{\bullet}(X).$$

*Proof.* By definition, $D_{\text{can}}(d)$ is obtained by imposing the Calderón boundary condition $\text{Tr}_D(\chi) \in \Gamma(Y; W_{\text{can}})$ with $W_{\text{can}} = \text{ran}(C)$ (Theorem 6.1 / Proposition C.2), hence

$$(\text{Dom}(d_{\text{can}}), d) = (\text{Dom}(d_{W_{\text{can}}}), d).$$

Applying Proposition 7.5 with $W = W_{\text{can}}$ gives the claimed identification with $IH_{W_{\text{can}}}^{\bullet}(X)$. In the Witt case $H = 0$ the datum is trivial and the theory reduces to middle-perversity intersection cohomology, again by Appendix E. $\square$

For later reference, Figure 3 summarizes the logical flow of the argument leading to Theorem 7.6. Starting from the Fubini-Study geometry on a projective simple-edge space (Sections 2-3), we obtain a Dirac normal form with a uniform spectral gap for $A_y$ (Sections 4-5), the Calderón package and Green pairing in a constant-multiplicity strip (Appendix C and Section 6), and explicit control of the isolated conic model (Appendix D) together with the depth-one ideal boundary condition theory (Appendix E). These ingredients feed into the construction of the canonical global domain $D_{\text{can}}$ for $d+d^*$ and the resulting FS-Cheeger-Goresky-MacPherson identification of Theorem 7.6.
<figure><img src="image_5.png" /><figcaption>FIGURE 3. Logical flow of the FS-Cheeger-Goresky-MacPherson theorem. Starting from the Fubini-Study geometry on a projective simple-edge space, one obtains a Dirac normal form with spectral gap, a Calderón package and Green pairing in a constant-multiplicity strip, explicit conic models and depth-one ideal boundary conditions. These ingredients feed into the construction of a canonical global domain $D_{\text{can}}$ and the resulting FS-CGM identification with intersection cohomology.</figcaption></figure>

**7.4. The isolated conic case (M1) as a base case.** We record the specialization of the above depth-one theory to the case of isolated singularities (conic points). This provides a convenient base case for iterative generalizations and clarifies the relationship with the classical cone analysis of Cheeger.

**Corollary 7.7** (Isolated singularities (M1)). Let $X \subset \mathbb{CP}^N$ be a complex projective variety whose singular set is a finite set $\Sigma = \{p_1, \dots, p_N\}$, and let $g_{\text{FS}}$ be the induced
Fubini-Study metric on $X^{\text{reg}}$. For each $p_j$ let $L_j$ be the link, $A_j$ the tangential operator in the Dirac normal form near $p_j$, and $H_j := \ker(A_j)$ the boundary data space. Let $C_j$ be the Calderón projector and set $W_{\text{can},j} := \text{ran}(C_j) \subset H_j$. Define the isolated-singularity datum

$$W_{\text{FS}} := \{W_{\text{can},1}, \dots, W_{\text{can},N}\}.$$

Then the canonical FS extension satisfies

$$H^{\bullet}(\mathcal{D}_{\text{can}}(d), d) \cong I H_{W_{\text{FS}}}^{\bullet}(X).$$

If each link $L_j$ is Witt (equivalently $b_{f/2}(L_j) = 0$, where $f = \dim L_j$), then $W_{\text{FS}}$ is trivial and

$$H^{\bullet}(\mathcal{D}_{\text{can}}(d), d) \cong I H_{\bar{m}}^{\bullet}(X).$$

*Proof.* Fix pairwise disjoint conic neighborhoods $U_j$ of the singular points $p_j \in \Sigma$ and let $x_j$ be the corresponding radial functions so that $U_j \cap X^{\text{reg}} \cong (0, \varepsilon)_{x_j} \times L_j$. On each $U_j \cap X^{\text{reg}}$ the FS metric has the conic/wedge form to leading order, so the de Rham Dirac admits the local normal form

$$D = \Gamma\left(\partial_{x_j} + \frac{1}{x_j}A_j + B_j(x_j)\right)$$

with tangential operator $A_j$ on $L_j$ and with the same critical-strip package as in the depth-one analysis (here there is no parameter $y$ since $Y = \{p_j\}$).

**Step 1: Local boundary data and trace.** In the critical strip, the maximal domain on $U_j$ admits a two-term leading asymptotic expansion whose leading coefficient lies in $H_j := \ker(A_j)$, and hence yields a local trace map

$$\text{Tr}_{D,j} : \text{Dom}(D_{\text{max}}; U_j) \to H_j$$

(see Appendix D.2 and Appendix D.3 for the explicit conic model).

**Step 2: Local Calderón data.** The local Poisson/Calderón construction produces an order-0 projection $C_j : H_j \to H_j$ whose range is the local Cauchy data space $W_{\text{can},j} := \text{ran}(C_j) \subset H_j$ (Appendix D.3-D.5). This determines the local canonical realization $D_{\text{can},j}$ by the boundary condition $\text{Tr}_{D,j}u \in W_{\text{can},j}$.

**Step 3: Global trace and global datum as a direct sum.** Choose a partition of unity $\{\chi_0, \chi_1, \dots, \chi_N\}$ subordinate to $X^{\text{reg}} = (X^{\text{reg}} \setminus \cup_j U_j) \cup (\cup_j U_j)$, with $\chi_j$ supported in $U_j$. Define the global boundary data space $H := \bigoplus_{j=1}^N H_j$ and the global trace map

$$\text{Tr}_D(u) := (\text{Tr}_{D,1}(\chi_1 u), \dots, \text{Tr}_{D,N}(\chi_N u)) \in H.$$

This definition is independent of the choices up to canonical identification, because the trace depends only on the germ of $u$ at $p_j$ and changing $\chi_j$ by a cutoff equal to 1 near $p_j$ does not change the leading coefficient in the maximal expansion.

**Step 4: Identification of the global canonical domain.** Let $W_{\text{FS}} := \{W_{\text{can},1}, \dots, W_{\text{can},N}\}$ and set $W := \bigoplus_j W_{\text{can},j} \subset H$. Then the global canonical FS domain satisfies

$$\text{Dom}(D_{\text{can}}) = \{u \in \text{Dom}(D_{\text{max}}) : \text{Tr}_D(u) \in W\},$$

i.e. $u \in \text{Dom}(D_{\text{can}})$ if and only if for every $j$ the localized piece $\chi_j u$ satisfies the local canonical boundary condition at $p_j$. Equivalently, $D_{\text{can}}$ is the operator obtained by imposing the canonical Calderón boundary condition at each conic point.
**Step 5: Topological identification.** In depth one with Y discrete, the mezzoperversity/IBC datum is exactly the choice of a subspace $W_{\text{can},j} \subset H_j$ at each non-Witt singular point. The associated Hilbert complex therefore computes the corresponding Deligne-type intersection theory $IH_{W_{\text{FS}}}(X)$ by the same depth-one IBC identification used in Proposition 7.5 and Theorem 7.6 (see also Appendix E for a self-contained sheaf proof). This yields

$$H^\bullet(D_{\text{can}}(d), d) \cong IH_{W_{\text{FS}}}^\bullet(X).$$

**Step 6: Witt case.** If every link $L_j$ is Witt, then $H_j = \ker(A_j) = 0$ for all $j$ (Appendix D.7), so there is no boundary datum to choose and the canonical extension reduces to middle-perversity intersection cohomology: $H^\bullet(D_{\text{can}}(d), d) \cong IH_m^\bullet(X)$. $\square$

**Example 7.8.** (A worked computation: the quadric cone). Let

$$V := \{[x_0 : x_1 : x_2 : x_3] \in \mathbb{CP}^3 : x_0^2 + x_1^2 + x_2^2 = 0\}.$$

Then $V$ is the projective cone over the smooth conic $Z = \{[x_0 : x_1 : x_2] \in \mathbb{CP}^2 : x_0^2 + x_1^2 + x_2^2 = 0\} \cong \mathbb{CP}^1$, with a unique singular point $p = [0 : 0 : 0 : 1]$.

The link $L$ at $p$ is the $S^1$-bundle over $Z$ with Euler number 2 (hence $L \cong L(2, 1) \cong \mathbb{RP}^3$). In particular $\dim L = 3$ is odd, so the Witt condition holds and $\ker(A) = 0$.

Consequently, Corollary 7.7 and Theorem 7.6 imply that the canonical FS complex has cohomology

$$H^k(D_{\text{can}}(d), d) \cong IH_m^k(V).$$

Moreover, for this quadric cone one has (over $\mathbb{R}$)

$$IH_m^k(V; \mathbb{R}) = \begin{cases} \mathbb{R}, & k = 0, 2, 4, \\ 0, & \text{otherwise.} \end{cases}$$

Hence

$$H^k(D_{\text{can}}(d), d) \cong H_{(2)}^k(V_{\text{reg}}, g_{\text{FS}}) = \begin{cases} \mathbb{R}, & k = 0, 2, 4, \\ 0, & \text{otherwise.} \end{cases}$$

(Notice that $V_{\text{reg}}$ deformation retracts onto $L \cong \mathbb{RP}^3$, so $H^2(V_{\text{reg}}; \mathbb{R}) = 0$ while $H_{(2)}^2(V_{\text{reg}}, g_{\text{FS}}) \cong \mathbb{R}$; this illustrates that the canonical $L^2$ theory is genuinely different from ordinary de Rham cohomology on $X_{\text{reg}}$.)

**Remark 7.9.** From this point of view, Theorem 7.6 gives a Fubini-Study refinement of the general ALMP mezzoperversity framework [1] at depth one. In the abstract non-Witt setting of [1] the mezzoperversity $W$ is an a priori choice of ideal boundary data on the link bundle, whereas in the projective simple-edge situation treated here the ambient FS metric singles out a canonical choice $W_{\text{can}} = \text{ran}(C)$ via the Calderón Cauchy data. The analytic hypotheses required to invoke the edge boundary calculus of Mazzeo-Vertman [16] (trace bundle, constant-multiplicity strip, Fredholm projection boundary conditions) are verified once and for all by the FS wedge expansion and tangential rigidity. The projective families in Section 8 illustrate how this canonical boundary condition interacts with refined intersection cohomology in concrete non-Witt examples.
## 8. NONTRIVIAL PROJECTIVE FAMILIES (EXAMPLES)

This section records complex projective families whose induced Fubini-Study metrics lie in our depth-one (simple-edge) analytic class. We emphasize a reusable verification template for the hypotheses of Theorem 7.4 and Theorem 7.6. Near the singular stratum $Y = X_{\text{sing}}$ we require:

(H1) depth-one control data giving a cone-bundle neighborhood with smooth link fibration $L \to Y$;

(H2) a Fubini-Study wedge expansion in a tubular neighborhood of $Y$;

(H3) the associated tangential family satisfies the stability/constant-strip conditions of Section 5.

For the fiberwise homogeneous hypersurface subfamily, (H1) is ensured by Lemma 8.1 together with Corollary 8.3, (H2) by Remark 8.4, and (H3) by Corollary 8.5. Once these inputs are in place, Theorem 7.4 gives the canonical self-adjoint Fredholm realization, and Theorem 7.6 identifies its $L^2$ de Rham cohomology with the corresponding depth-one intersection theory (in particular, with $IH_{\bar{m}}$ in the Witt case).

In all of the complex-algebraic examples below, the normal slice to $Y$ has even real codimension, hence the transverse link has odd dimension; consequently the Witt condition holds and the canonical $L^2$ cohomology agrees with middle persversity intersection cohomology via Theorem 7.6. (We nevertheless keep the discussion in the language of the general depth-one package, since the construction of the canonical domain and its functoriality do not require Witt.)

### 8.1. Geometric families within the depth-one class.

**Lemma 8.1** (A practical simple-edge criterion for fiberwise homogeneous hypersurfaces). Let $Y$ be a smooth projective manifold and $E \to Y$ a holomorphic vector bundle of rank $k$. Let $s \in H^0(Y, \text{Sym}^d(E^\vee))$ be a fiberwise homogeneous polynomial of degree $d$. Consider the fiberwise projective hypersurface

$$X := \{F = 0\} \subset \P(E \oplus 1), \quad F(v,t) := s_y(v) \quad (t \text{ is the } \mathbf{1}\text{-coordinate}).$$

Assume:

(A1) (Isolated affine singularity) For each $y \in Y$, the affine hypersurface $\{s_y(v) = 0\} \subset E_y \cong \mathbb{C}^k$ has an isolated singularity at $v = 0$.

(A2) (Smooth projective link hypersurface) The projective hypersurface $Z_y := \{[v] \in \P(E_y) : s_y(v) = 0\}$ is smooth for all $y$.

Then $X$ is smooth on $\P(E \oplus 1) \setminus Y$ and its singular locus is exactly the zero section $Y \to X$ (given by $v = 0$). Moreover, $X$ is depth one/simple-edge: a neighborhood of $Y$ is a cone bundle over a smooth link fibration $L \to Y$ determined by the links of the isolated singularities in (A1).

If, in addition, the structure group of $E$ reduces to a subgroup $G \subset \text{GL}(k, \mathbb{C})$ preserving a fixed polynomial model $F_0$ (so that all $s_y$ are locally identified with $F_0$), then the link fibration $L \to Y$ is an associated bundle and the tangential family $A_y$ is rigid (constant up to bundle identifications), hence satisfies the stability/constant-strip package of Section 5.
**Proof.** We work locally on $Y$. Let $U \subset Y$ be a coordinate chart over which $E$ is holomorphically trivial and choose a Hermitian metric on $E|_U$. Via this trivialization we identify

$$E|_U \cong U \times \mathbb{C}^k, \quad \P(E \oplus 1)|_U \cong U \times \P^k$$

with homogeneous fiber coordinates $[v : t]$, where $v \in \mathbb{C}^k$ and $t \in \mathbb{C}$. The hypersurface $X$ is defined by the single equation

$$F(y, [v : t]) = s_y(v) = 0,$$

which is homogeneous of degree $d$ in $v$ and independent of $t$.

**Step 1:** $Y \subset X_{\text{sing}}$. The zero section $Y \hookrightarrow \P(E \oplus 1)$ is given fiberwise by $[0 : 1]$, i.e. by $v = 0$ on the affine chart $t \neq 0$. Since $s_y$ is homogeneous of degree $d \geq 2$ (forced by (A1)), we have $ds_y(0) = 0$ for every $y$. Moreover, the map $y \mapsto s_y(0)$ is identically zero (because $s_y(0) = 0$ for all $y$), hence its differential in the $y$-directions vanishes at $v = 0$ as well. Therefore $dF = 0$ along $Y$, so every point of $Y$ is a singular point of $X$. Thus $Y \subset X_{\text{sing}}$.

**Step 2:** $X$ is smooth on the affine chart $t \neq 0$ away from $Y$. On the chart $t = 1$ we identify $\P(E \oplus 1)$ with the total space $E$, and $X \cap \{t = 1\}$ is given by the equation $s_y(v) = 0$ on $U \times \mathbb{C}^k$. A point $(y, v)$ of this hypersurface is singular if and only if

$$s_y(v) = 0 \quad \text{and} \quad d_{(y,v)}(s_y(v)) = 0.$$

In particular, singularity forces the vertical differential in the $v$-directions to vanish: $ds_y(v) = 0$. By assumption (A1), the affine hypersurface $\{s_y(v) = 0\} \subset \mathbb{C}^k$ has an isolated singularity at $v = 0$, i.e. $s_y(v) = 0$ and $ds_y(v) = 0$ imply $v = 0$. Hence there are no singular points on the chart $t = 1$ with $v \neq 0$.

**Step 3:** $X$ is smooth on the divisor at infinity $t = 0$. On $t = 0$ we have the projective bundle $\P(E)$ with fiber $\P^{k-1}$, and

$$X \cap \P(E) = \{[v] \in \P(E_y) : s_y(v) = 0\} =: Z_y$$

fiberwise. A point of a projective hypersurface is singular if and only if all homogeneous partials vanish, equivalently if $ds_y(v) = 0$ for some representative $v \neq 0$. Assumption (A2) says that each $Z_y$ is smooth; therefore $ds_y(v) \neq 0$ at every point with $v \neq 0$ and $s_y(v) = 0$. Consequently $X$ has no singular points on $\{t = 0\}$.

**Step 4:** $X_{\text{sing}} = Y$ and $X$ is smooth on $\P(E \oplus 1) \setminus Y$. By Step 2 and Step 3, there are no singular points away from $v = 0$ on the $t = 1$ chart and no singular points on the $t = 0$ chart. Together with Step 1 this proves $X_{\text{sing}} = Y$ and that $X$ is smooth away from $Y$.

**Step 5:** Construction and smoothness of the link fibration $L \to Y$. Let $S(E) \to Y$ be the unit sphere bundle in $E$ for the chosen Hermitian metric and define the smooth map

$$\Phi : S(E) \to \mathbb{C}, \quad \Phi(y, \xi) := s_y(\xi).$$

Set $L := \Phi^{-1}(0) \subset S(E)$, so fiberwise $L_y = \{\xi \in S(E_y) : s_y(\xi) = 0\}$.

We claim that $0$ is a regular value of $\Phi$. Fix $(y, \xi) \in L$. Consider the differential in the vertical directions tangent to the sphere $S(E_y)$. If $d\Phi$ vanished on $T_\xi S(E_y)$, then the complex gradient $\nabla s_y(\xi)$ would be proportional to $\xi$. But since $s_y$ is homogeneous of degree $d$, Euler's identity gives $\langle \nabla s_y(\xi), \xi \rangle = ds_y(\xi) = 0$ on $L$; hence proportionality would force $\nabla s_y(\xi) = 0$, i.e. $ds_y(\xi) = 0$ in $\mathbb{C}^k$. This would
mean that the projective point $[\xi] \in Z_y$ is singular, contradicting (A2). Therefore $d\Phi$ is nonzero at every point of $L$, so $L$ is a smooth submanifold of $S(E)$.

Now consider the restriction of the projection $\pi : S(E) \to Y$ to $L$. Since $L$ is a smooth submanifold cut out by one transverse equation with a nonzero vertical differential, $\pi|_L : L \to Y$ is a submersion (the constraint does not eliminate horizontal directions). Moreover, $L$ is closed in the compact-fiber bundle $S(E)$, hence $\pi|_L$ is proper. By Ehresmann's fibration theorem, $\pi|_L$ is therefore a smooth locally trivial fiber bundle. This is the desired smooth link fibration $L \to Y$.

**Step 6: Cone-bundle neighborhood of Y in X.** On the affine chart $t = 1$ we write $v = r\xi$ with $r := \|v\| \in [0, \infty)$ and $\xi \in S(E_y)$. By homogeneity,

$$s_y(v) = s_y(r\xi) = r^d s_y(\xi).$$

Hence for $r > 0$ the equation $s_y(v) = 0$ is equivalent to $\Phi(y, \xi) = 0$, i.e. $(y, \xi) \in L$. Therefore, for $\varepsilon > 0$ small, the map

$$(0, \varepsilon) \times L \to (X \cap \{t=1\}) \setminus Y, \quad (r, y, \xi) \mapsto (y, v = r\xi)$$

is a diffeomorphism onto a punctured neighborhood of $Y$ in $X$. Adding the locus $r = 0$ collapses each fiber $L_y$ to the vertex, identifying it with $Y$. Thus a neighborhood of $Y$ in $X$ is modeled by the cone bundle over the link fibration $L \to Y$, i.e. $X$ is depth-one/simple-edge.

**Step 7: Associated-bundle and rigidity statement.** Assume finally that $E = P \times_G \mathbb{C}^k$ for a principal $G$-bundle $P \to Y$ and that $s$ is induced from a fixed homogeneous polynomial model $F_0$ preserved by $G$. Then $S(E) = P \times_G S^{2k-1}$ and the link is

$$L = P \times_G L_0, \quad L_0 := \{\xi \in S^{2k-1} : F_0(\xi) = 0\}.$$

In particular, $L \to Y$ is an associated bundle with fiber $L_0$. Any tangential operator construction on the link that is natural under change of trivialization is therefore constant up to $G$-conjugation; consequently $\dim \ker(A_y)$ is constant and the stability/strip package of Section 5 applies. $\square$

**Remark 8.2 (Genericity of (A2)).** Fix $(k, d)$. The set of homogeneous polynomials $F_0 \in \mathrm{Sym}^d(\mathbb{C}^k)^\vee$ for which the projective hypersurface $\{F_0 = 0\} \subset \mathbb{P}^{k-1}$ is smooth is Zariski open (the complement is the classical discriminant hypersurface). Hence (A2) is a generic condition on the fiber model.

**Corollary 8.3 (Control data induced by the fiberwise homogeneous model).** Under the assumptions of Lemma 8.1, the pair $(X, Y)$ admits canonical depth-one control data in the sense of Section 2. More precisely, there exist:

(1) a tubular neighborhood $\mathcal{U}$ of $Y$ in the ambient manifold $\mathbb{P}(E \oplus \mathbf{1})$ and a smooth retraction $\pi : \mathcal{U} \to Y$,

(2) a smooth radial function $x : \mathcal{U} \to [0, \varepsilon)$ with $Y = \{x = 0\}$,

(3) a smooth link fibration $L \to Y$ and a diffeomorphism

$$\Phi : (0, \varepsilon) \times L \xrightarrow{\cong} (\mathcal{U} \cap X^{\mathrm{reg}}), \quad (x, \ell) \mapsto \Phi(x, \ell),$$

which identifies $\pi$ with the projection to $Y$ and identifies $x$ with the first factor.

In particular, $\mathcal{U} \cap X$ is homeomorphic to the cone bundle $C(L) := ([0, \varepsilon) \times L)/(\{0\} \times L \sim Y)$, so $X$ is depth one/simple-edge with singular stratum $Y$.
*Proof.* Choose a Hermitian metric on $E$. Let $\mathcal{U} \subset \mathcal{P}(E \oplus 1)$ be the set where $|v| < \varepsilon$ on the affine chart $t = 1$; then $\mathcal{U}$ is a tubular neighborhood of the zero section $Y$ with retraction $\pi(y, v) = y$. Define the radial function $x(y, v) := |v|$.

By Lemma 8.1, the link $L \subset S(E)$ is a smooth fiber bundle over $Y$. Using the polar decomposition $v = x\xi$ with $\xi \in S(E_y)$, the defining equation $s_y(v) = 0$ is equivalent (for $x > 0$) to $s_y(\xi) = 0$, i.e. $\xi \in L_y$. Hence the map $\Phi(x, y, \xi) = (y, v = x\xi)$ yields a diffeomorphism $(0, \varepsilon) \times L \cong \mathcal{U} \cap X^{\text{reg}}$ with the stated properties. The cone-bundle description follows by adjoining $x = 0$ and collapsing each fiber $L_y$ to the vertex $y$. $\square$

*Remark* 8.4 (Compatibility with the FS wedge expansion). In the projective setting, we take $\mathcal{U}$ and $x$ as in Corollary 8.3. The restriction of the ambient Fubini-Study metric to $X^{\text{reg}}$ admits the wedge expansion of Section 3 in this tubular neighborhood, with $x$ as radial variable and with the link fibration $L \to Y$ from Lemma 8.1. Therefore the analytic hypotheses of the depth-one package (Sections 3–6) are verified locally near $Y$ in these examples.

**Corollary 8.5** (Rigidity and stability for associated link bundles). In the equivariant situation of Lemma 8.1 (structure group reduced to $G$ preserving a fixed model $F_0$), the link fibration is an associated bundle $L = P \times_G L_0$. Any tangential operator family constructed naturally from the link geometry is constant up to bundle identifications; in particular, dim ker($A_y$) is constant and the constant multiplicity strip hypothesis of Section 5 holds.

*Proof.* By the equivariant reduction in Lemma 8.1, the link fibration is an associated bundle $L \cong P \times_G L_0 \to Y$, where $P \to Y$ is a principal $G$-bundle and $G$ preserves the fixed model $L_0$ (equivalently, the model hypersurface $F_0$). Choose a local section $s: U \to P$; it yields a trivialization $L|_U \cong U \times L_0$.

Let $A_y$ be any tangential operator family constructed functorially from the link geometry. Functionality under changes of trivialization implies that, under the above local identification, $A_y$ is conjugate to a fixed operator $A_0$ on $L_0$:

$$A_y = \phi_y A_0 \phi_y^{-1}.$$

Hence $\text{Spec}(A_y)$ is independent of $y \in U$, and in particular $\text{dim ker}(A_y)$ is constant on $U$ (and thus on $Y$ by connectedness).

Moreover, since the spectrum near 0 is constant, there is a uniform spectral gap at 0 and therefore a uniform constant-multiplicity indicial strip as in Section 5. $\square$

**Example 8.6** (Projective cones). Let $Z \subset \mathbb{CP}^{N-1}$ be a smooth projective variety and let $X \subset \mathbb{CP}^N$ be the projective cone over $Z$ with vertex $p$.

**Verification of (H1).** The singular set is $X_{\text{sing}} = \{p\}$, hence $X$ is depth one with $Y = \{p\}$. A small sphere in the affine chart $t = 1$ intersects $X$ in a smooth compact link $L$, which is an $S^1$-bundle over $Z$ (the unit circle bundle in $\mathcal{O}_Z(-1)$). Thus a neighborhood of $p$ is a cone over $L$.

**Verification of (H2)–(H3).** The induced Fubini-Study metric on $X^{\text{reg}}$ has the conic/wedge normal form and wedge expansion required for the depth-one analysis (Section 3–5 specialize verbatim when $Y$ is a point; cf. Section 7.4 and Appendix D for the explicit conic model).
**Conclusion.** Corollary 7.7 applies, hence $H^\bullet(D_{\text{can}}(d), d) \cong IH^\bullet_{W_{\text{FS}}}(X)$. Moreover, $\dim L$ is odd, so Witt holds and the datum is trivial; therefore $H^\bullet(D_{\text{can}}(d), d) \cong IH^\bullet_{\bar{m}}(X)$ by Theorem 7.6.

**Example 8.7** (Segre products with an isolated singularity). Let $Y$ be a smooth projective manifold and let $V \subset \mathbb{CP}^m$ be a projective variety with an isolated singularity at $p$. Consider the Segre embedding $X := Y \times V \hookrightarrow \mathbb{CP}^N$.

**Verification of (H1).** The singular locus is $X_{\text{sing}} = Y \times \{p\}$, which is smooth and connected. A neighborhood of a point $(y, p)$ is the product of a neighborhood of $y \in Y$ with a conic neighborhood of $p \in V$. In particular, the transverse link along $Y \times \{p\}$ is (locally) the link of $(V, p)$ and the link fibration is trivial: $L \cong (Y \times L_V) \to Y$.

**Verification of (H2).** The restriction of the ambient Fubini-Study metric to $X^{\text{reg}} = Y \times V^{\text{reg}}$ admits the wedge expansion of Section 3 in a tubular neighborhood of $Y \times \{p\}$.

**Verification of (H3).** Since the link fibration is trivial, the tangential operator family $A_y$ is constant (up to the fixed identification of the link), hence $\dim \ker(A_y)$ is constant and the stability package of Section 5 holds automatically.

**Conclusion.** Therefore $X$ is projective simple-edge and Theorem 7.6 applies to $(X^{\text{reg}}, g_{\text{FS}})$. In particular, $H^\bullet(D_{\text{can}}(d), d) \cong IH^\bullet_{\bar{m}}(X)$ in the Witt case.

**Example 8.8** (Fibered homogeneous isolated singularities). Let $Y$ be smooth projective and $E \to Y$ a holomorphic vector bundle of rank $k$. Fix a homogeneous polynomial $F_0: \mathbb{C}^k \to \mathbb{C}$ of degree $d$ with an isolated singularity at 0. Assume the structure group of $E$ reduces to a subgroup $G \subset \text{GL}(k, \mathbb{C})$ preserving $F_0$, so that $F_0$ defines a well-defined section $s \in H^0(Y, \text{Sym}^d(E^\vee))$ by fiberwise identification. Let

$$X := \{F = 0\} \subset \P(E \oplus 1), \quad F(v, t) := s_y(v).$$

Assume additionally that the projective hypersurface

$$Z := \{[v] \in \P^{k-1} : F_0(v) = 0\}$$

is smooth (equivalently, $\nabla F_0(v) = 0$ implies $v = 0$). Then Lemma 8.1 implies that $X$ is smooth away from the zero section $Y \subset X$, and that $X_{\text{sing}} = Y$ is a smooth connected stratum with a cone-bundle neighborhood whose link fibration is the associated $G$-bundle of the link of $F_0$. In particular, $(X^{\text{reg}}, g_{\text{FS}})$ lies in the depth-one analytic class and the tangential family is rigid, so the stability/constant-strip hypothesis of Section 5 holds. Therefore Theorem 7.4 and Theorem 7.6 apply.

## 8.2. Worked computations.

**Example 8.9** (A trivial simple-edge family: $\mathbb{CP}^1 \times V$). Let $V \subset \mathbb{CP}^3$ be the quadric cone of Example 7.8 with vertex $p$, and set $X := \mathbb{CP}^1 \times V$ embedded into some $\mathbb{CP}^N$ via Segre.

Then $X_{\text{sing}} = \mathbb{CP}^1 \times \{p\} \cong \mathbb{CP}^1$ is smooth and connected, hence $X$ is projective simple-edge. The transverse link along $X_{\text{sing}}$ is constant and equal to $L(2, 1) \cong \mathbb{RP}^3$, so Witt holds. Therefore Theorem 7.6 yields

$$H^\bullet_{(2)}(X^{\text{reg}}, g_{\text{FS}}) \cong H^\bullet(D_{\text{can}}(d), d) \cong IH^\bullet_{\bar{m}}(X).$$
Since $IH_{\bar{m}}^{\bullet}(V; \mathbb{R})$ is supported in degrees 0, 2, 4 with rank 1 (Example 7.8), the Künneth theorem for intersection cohomology gives

$$IH_{\bar{m}}^{k}(X; \mathbb{R}) \cong H^{k}(\mathbb{CP}^{1}; \mathbb{R}) \oplus H^{k-2}(\mathbb{CP}^{1}; \mathbb{R}) \oplus H^{k-4}(\mathbb{CP}^{1}; \mathbb{R}),$$

hence

$$\dim IH_{\bar{m}}^{k}(X; \mathbb{R}) = \begin{cases} 1, & k = 0, 6, \\ 2, & k = 2, 4, \\ 0, & \text{otherwise,} \end{cases} \qquad IH_{\bar{m}}^{\text{odd}}(X; \mathbb{R}) = 0,$$

and the same dimensions hold for $H_{(2)}^{\bullet}(X^{\text{reg}}, g_{\text{FS}})$.

**Example 8.10** (A genuinely family simple-edge example: a quadric cone bundle over $\mathbb{CP}^m$). Let $Y = \mathbb{CP}^m$ with $m \ge 1$ and let $L = \mathcal{O}_{\mathbb{CP}^m}(1)$. Set $E := L \oplus L^{-1} \oplus \mathcal{O}_Y$ and let $\pi : \mathcal{P}(E \oplus \mathcal{O}_Y) \to Y$. On each fiber $\pi^{-1}(y) \cong \mathbb{CP}^3$ use homogeneous coordinates $[u_+ : u_- : u_0 : t]$ with $u_+ \in L_y, u_- \in L_y^{-1}, u_0 \in \mathbb{C}, t \in \mathbb{C}$. Define

$$X := \{u_+u_- + u_0^2 = 0\} \subset \mathcal{P}(E \oplus \mathcal{O}_Y).$$

**Geometry and simple-edge structure.** A direct Jacobian computation in the fiber coordinates shows that the singular locus is

$$X_{\text{sing}} = \{u_+ = u_- = u_0 = 0\} \cong Y,$$

hence $X$ is projective simple-edge with singular stratum $Y$. The transverse fiber is the quadric cone, so the transverse link is $L(2, 1) \cong \mathbb{RP}^3$; in particular $\dim L = 3$ is odd, so Witt holds.

**Conclusion and computation.** By Theorem 7.6,

$$H_{(2)}^{\bullet}(X^{\text{reg}}, g_{\text{FS}}) \cong IH_{\bar{m}}^{\bullet}(X).$$

Moreover, the fiberwise middle intersection cohomology of the quadric cone is supported in degrees 0, 2, 4 with rank 1 in each degree (Example 7.8). Since $Y = \mathbb{CP}^m$ is simply connected, the relevant local systems are trivial and the Leray spectral sequence for intersection cohomology collapses for degree reasons. Consequently, $IH_{\bar{m}}^{\bullet}(X; \mathbb{R})$ carries a natural filtration whose graded pieces satisfy

$$\operatorname{gr} IH_{\bar{m}}^{k}(X; \mathbb{R}) \cong H^{k}(Y; \mathbb{R}) \oplus H^{k-2}(Y; \mathbb{R}) \oplus H^{k-4}(Y; \mathbb{R}).$$

In particular, the Betti numbers are determined by the right-hand side; explicitly, $IH_{\bar{m}}^{k}(X; \mathbb{R}) = 0$ for odd $k$ and for even $k$ one has

$$\dim IH_{\bar{m}}^{k}(X; \mathbb{R}) = \#\{r \in \{0, 1, 2\} : 0 \le k - 2r \le 2m\}.$$

(Over a field, this filtration always splits as graded vector spaces, though not canonically without extra structure.)

## 9. A DEPTH-TWO ONE-STEP ITERATION AND AN INDUCTIVE THEOREM SCHEMA FOR M3

The depth-one analysis of this paper is designed to serve as the basic step of an inductive construction in the stratification depth. In depth $\ge 2$ the new feature is that the tangential operator along a stratum acts on a *stratified* link; hence its self-adjoint realization (and in particular its boundary spectrum near the $L^2$-threshold) depends on the canonical ideal boundary conditions already chosen on *deeper* strata. We isolate below a clean, checkable depth-two target (one iteration), phrased so that
it directly parallels the standing hypotheses (H1)–(H3) used throughout the present paper.

**Depth-two geometric setting.** Assume that $X \subset \mathbb{CP}^N$ is a complex projective variety equipped with a Thom–Mather stratification of depth two, with strata

$$Y_2 \subset \bar{Y}_1 \subset X, \quad X_{\text{reg}} = X \setminus (Y_1 \cup Y_2),$$

and let $g_{\text{FS}}$ be the ambient Fubini–Study metric restricted to $X_{\text{reg}}$. Assume that near the depth-one stratum $Y_1$ there exist control data $(\rho_1, x_1)$ and a (stratified) link fibration

$$L_1 \to Y_1,$$

with compact fibers $L_{1,y}$, such that a tubular neighborhood admits a cone-bundle model $N_1 \cap X_{\text{reg}} \cong (0, \varepsilon)_{x_1} \times L_{1,y}$. In the depth-two situation, each link fiber $L_{1,y}$ is itself a depth-one stratified space, with its own singular stratum (the “link-of-link”) corresponding to $Y_2$.

**Inner (depth-one) analytic input on the link family.** Fix $y \in Y_1$. Let $D_y^{(L_1)}$ denote the Hodge–de Rham Dirac operator on the regular part $(L_{1,y})_{\text{reg}}$ with the metric induced from $g_{\text{FS}}$. Assume that the depth-one package of this paper applies uniformly to the link family $y \mapsto (L_{1,y}, g_{\text{FS}}|_{(L_{1,y})_{\text{reg}}})$ in the following sense: there exist fiberwise data satisfying exact analogues of (H1)–(H3) of the present paper, hence yielding a canonical Cauchy data subbundle and a canonical self-adjoint Fredholm realization of $D_y^{(L_1)}$. Concretely, we assume the following depth-one standing hypotheses for the link family:

(H12) (Wedge Dirac normal form on the link).: For each $y \in Y_1$, the link operator $D_y^{(L_1)}$ admits a wedge Dirac normal form near the link’s singular stratum, with edge-smooth lower-order terms, as in Proposition 4.1 and Lemma 4.4.

(H22) (Uniform gap at 0 on the link).: The corresponding tangential family on the link has constant dim ker(·) and a uniform spectral gap at 0, uniformly in $y$.

(H32) (Constant-multiplicity strip on the link).: In a fixed weight strip around the $L^2$ threshold on the link family, the only boundary spectrum encountered is the indicial root at 0, with constant multiplicity equal to the rank of the associated trace bundle.

Under (H12)–(H32), the depth-one construction of Sections 6-7 and Appendix C applies fiberwise and produces, for each $y \in Y_1$, a finite-rank trace bundle $H_{2,y}$, a Calderón projector $C_{2,y}$, canonical Cauchy data $W_{2,\text{can}}(y) = \text{ran}(C_{2,y})$, and a canonical self-adjoint realization $D_{\text{can}}^{(L_1)}(y)$ (equivalently, a canonical domain for the link de Rham complex).

**Outer (along $Y_1$) tangential family with deeper canonical domains.** Near $Y_1$, the global Hodge–de Rham Dirac operator $D$ on $(X_{\text{reg}}, g_{\text{FS}})$ admits a wedge Dirac normal form (microlocally and up to lower-order edge terms)

$$D = \Gamma\left(\partial_{x_1} + \frac{1}{x_1}A_{1,y} + B(x_1)\right),$$
where $A_{1,y}$ acts along the link fiber $L_{1,y}$. In depth two, the correct tangential object is the family of self-adjoint realizations

$$A_{1,y}^{W_{2,\text{can}}} : \text{Dom}(A_{1,y}^{W_{2,\text{can}}}) \subset L^2\Omega^{\bullet}((L_{1,y})_{\text{reg}}) \to L^2\Omega^{\bullet}((L_{1,y})_{\text{reg}}),$$

defined by imposing, on the deeper singularities of the link fiber $L_{1,y}$, the canonical ideal boundary conditions encoded by $W_{2,\text{can}}(y)$ (equivalently, by forming $A_{1,y}$ using the canonical domain of $D_{\text{can}}^{(L_1)}(y)$).

**Depth-two stability package: the outer analogues of (H1)–(H3).** To iterate the depth-one Calderón package along $Y_1$, one needs the following hypotheses, which are formulated to match *exactly* the standing assumptions (H1)–(H3) used in Appendix C (trace bundle, Poisson operator, Calderón projector, and Fredholm boundary problems):

(H1) (Wedge Dirac normal form along $Y_1$ with deeper domains).: Near $Y_1$, the global Dirac operator admits the wedge normal form

$$D = \Gamma\left(\partial_{x_1} + \frac{1}{x_1}A_{1,y}^{W_{2,\text{can}}} + B(x_1)\right),$$

where $A_{1,y}^{W_{2,\text{can}}}$ is self-adjoint on each compact (stratified) link fiber $L_{1,y}$ with the canonical deeper domain, depends smoothly on $y \in Y_1$ in the parameter-dependent edge sense, and $B(x_1)$ is a first-order edge operator with edge-smooth coefficients down to $x_1 = 0$.

(H2) (Uniform gap at 0 along $Y_1$).: $\dim \ker(A_{1,y}^{W_{2,\text{can}}})$ is constant in $y \in Y_1$ and there exists $\varepsilon_1 > 0$ such that

$$\text{Spec}(A_{1,y}^{W_{2,\text{can}}}) \cap (-\varepsilon_1, \varepsilon_1) = \{0\} \quad \text{for all } y \in Y_1.$$

Equivalently, the boundary data space

$$H_1 := \ker(A_1^{W_{2,\text{can}}}) \to Y_1$$

is a smooth finite-rank vector bundle and 0 is uniformly isolated in the spectrum.

(H3) (Constant-multiplicity strip along $Y_1$).: In a weight strip around the $L^2$ threshold (for the outer radial variable $x_1$), the only boundary spectrum encountered is the indicial root at 0 associated with the zero mode of $A_{1,y}^{W_{2,\text{can}}}$, with constant multiplicity rank($H_1$).

**A conditional depth-two theorem (one-step iteration).** The depth-two program is thus reduced to verifying (H1)–(H3) for the outer family $A_{1,y}^{W_{2,\text{can}}}$ constructed from the deeper canonical data.

**Theorem 9.1** (Depth-two canonical domain from one iteration (conditional)). As-sume the depth-two geometric setting above and the inner hypotheses (H1$_2$)–(H3$_2$) on the link family, so that the canonical deeper data $W_{2,\text{can}}(y)$ and canonical link realizations $D_{\text{can}}^{(L_1)}(y)$ are defined. If, in addition, the outer hypotheses (H1$_1$)–(H3$_1$) hold along $Y_1$, then the depth-one Calderón package (Sections 6-7 and Appendix C) applies along $Y_1$ with trace bundle $H_1 \to Y_1$. Consequently there exists a canonical outer Cauchy data bundle $W_{1,\text{can}} \subset H_1$ and a canonical self-adjoint Fredholm realization $D_{\text{can}}$ of the global Hodge–de Rham Dirac operator on $(X_{\text{reg}}, g_{\text{FS}})$, equivalently a canonical closed de Rham complex $(\text{Dom}(d_{\text{can}}), d)$.
Moreover, the cohomology of this canonical complex computes the depth-two ideal boundary condition theory (or equivalently, the depth-two mezzoperversity intersection theory determined by the pair $(W_{1,can}, W_{2,can})$):

$$H^\bullet(\mathrm{Dom}(d_{\mathrm{can}}), d) \cong IH^\bullet_{(W_{1,can}, W_{2,can})}(X).$$

In the Witt situation (when no boundary data occur at either depth) this reduces to middle-perversity intersection cohomology.

To turn Theorem 9.1 into an unconditional statement, one needs verifiable geometric hypotheses implying $(H2_1)-(H3_1)$. The following iterated rigidity assumption provides a concrete regime in which the outer stability package follows.

**Assumption 9.2** (Iterated associated-bundle regime in depth two). In the depth-two setting of Section 9, assume that the outer link fibration $L_1 \to Y_1$ is an associated bundle

$$L_1 \cong P_1 \times_G L_{1,0} \to Y_1,$$

where $P_1 \to Y_1$ is a principal $G$-bundle and $G$ acts by stratified isometries on a fixed depth-one model link $(L_{1,0}, g_{1,0})$ (preserving its Thom-Mather structure).

Moreover, assume that the $G$-action preserves the inner canonical data on the model link: denote by $H_{2,0} \to Z_0$ the inner trace bundle on the singular stratum $Z_0 \subset L_{1,0}$ and by

$$W_{2,can,0} \subset H_{2,0}$$

the canonical inner Cauchy data subbundle (equivalently, the range of the inner Calderón projector). We assume $G$ preserves $W_{2,can,0}$ (hence preserves the corresponding canonical inner domain).

**Proposition 9.3** (Outer stability from iterated rigidity). Assume Assumption 9.2.

Let $A_{1,y}^{W_{2,can}}$ be the outer tangential family along $Y_1$, formed using the inner canonical domains on the stratified link fibers (as in Section 9). Then, after local trivialization over $Y_1$, the family $A_{1,y}^{W_{2,can}}$ is constant up to bundle identifications: on each sufficiently small open set $U \subset Y_1$ there exist unitary identifications $\phi_y$ such that

$$A_{1,y}^{W_{2,can}} = \phi_y A_{1,0}^{W_{2,can}} \phi_y^{-1} \quad (y \in U).$$

Consequently:

(i) $\dim \ker(A_{1,y}^{W_{2,can}})$ is constant on $Y_1$, hence $H_1 := \ker(A_1^{W_{2,can}}) \to Y_1$ forms a smooth finite-rank bundle;

(ii) there exists $\varepsilon_1 > 0$ such that $\mathrm{Spec}(A_{1,y}^{W_{2,can}}) \cap (-\varepsilon_1, \varepsilon_1) = \{0\}$ for all $y \in Y_1$ (uniform gap at 0);

(iii) the outer boundary spectrum near the $L^2$-threshold has a constant-multiplicity strip (as required by the outer hypotheses $(H2_1)-(H3_1)$).

**Proof.** This is the depth-two analogue of Corollary 8.5, with the only additional point that the outer tangential family is formed using inner canonical domains.

By Assumption 9.2, choose a local section $s: U \to P_1$ over a small open set $U \subset Y_1$. This yields a local trivialization

$$L_1|_U \cong U \times L_{1,0}.$$

Let $A_{1,y}$ denote the outer tangential operator constructed functorially from the link geometry (as in Section 4 and Section 5 for depth one). Functionality under changes
of trivialization implies that, under the above identification, $A_{1,y}$ is conjugate to a fixed model operator $A_{1,0}$ on $L_{1,0}$:

$$A_{1,y} = \phi_y A_{1,0} \phi_y^{-1}.$$

Next, the inner canonical data on each fiber are transported from the model: since $G$ preserves the inner canonical Cauchy data subbundle $W_{2,can,0}$, the induced inner Calderón projector (and thus the inner canonical domain for the link Dirac operator) is preserved under the $G$-action. Therefore the corresponding realization of the outer tangential family obtained by imposing the inner canonical domain is likewise preserved under the same unitary identifications, yielding

$$A_{1,y}^{W_{2,can}} = \phi_y A_{1,0}^{W_{2,can}} \phi_y^{-1}.$$

In particular, $\text{Spec}(A_{1,y}^{W_{2,can}})$ is independent of $y \in U$. Hence $\text{dim ker}(A_{1,y}^{W_{2,can}})$ is constant on $U$ (and on $Y_1$ by connectedness of $Y_1$), and the spectrum near 0 is constant; this gives a uniform spectral gap at 0 and the constant-multiplicity strip exactly as in the depth-one argument following Corollary 8.5. This proves (i)–(iii). □

In particular, Assumption 9.2 supplies the missing outer stability input in Theorem 9.1.

**Corollary 9.4** (Outer hypotheses $(H2_1)-(H3_1)$ under iterated rigidity). Assume Assumption 9.2. Then the outer tangential family $A_{1,y}^{W_{2,can}}$ satisfies the outer stability hypotheses $(H2_1)-(H3_1)$ of Section 9. In particular, the trace bundle

$$H_1 := \ker(A_1^{W_{2,can}}) \to Y_1$$

has constant rank and is smooth, and there exists $\varepsilon_1 > 0$ such that

$$\text{Spec}(A_{1,y}^{W_{2,can}}) \cap (-\varepsilon_1, \varepsilon_1) = \{0\}$$

for all $y \in Y_1$. Moreover, in a fixed outer weight strip around the $L^2$ threshold, the outer boundary spectrum has constant multiplicity equal to rank($H_1$).

*Proof.* By Proposition 9.3, after local trivializations the family $A_{1,y}^{W_{2,can}}$ is unitarily conjugate to a fixed model realization $A_{1,0}^{W_{2,can}}$. Therefore the spectrum (with multiplicities) is locally constant in $y$. It follows that $\text{dim ker}(A_{1,y}^{W_{2,can}})$ is locally constant, hence constant on each connected component of $Y_1$, and the kernels assemble into a smooth finite-rank bundle $H_1 \to Y_1$.

Since 0 is an isolated point of $\text{Spec}(A_{1,0}^{W_{2,can}})$ (self-adjoint Fredholm realization), there exists $\varepsilon_1 > 0$ such that $\text{Spec}(A_{1,0}^{W_{2,can}}) \cap (-\varepsilon_1, \varepsilon_1) = \{0\}$. Local spectral constancy then yields the same gap on each trivializing neighborhood, and hence globally on $Y_1$. This proves (H2$_1$).

Finally, the outer indicial roots in a fixed strip are determined by the boundary spectrum of $A_{1,y}^{W_{2,can}}$ in a corresponding compact interval; since that spectrum is constant in $y$, the set of indicial roots in the strip (and their multiplicities) is constant as well. In particular, within the critical strip around the $L^2$ threshold the only indicial root is the one at 0 coming from the zero mode, with multiplicity rank($H_1$), which is precisely $(H3_1)$. □
**A minimal geometric template for an unconditional depth-two paper.**
The genuinely new analytic bottleneck in depth two is the verification of (H2₁)–(H3₁) for the family $A_{1,y}^{W_{2,can}}$, since its *domain* varies with *y* through the deeper Calderón data. A concrete and checkable geometric regime in which one expects these hypotheses to hold is the following:

(T2) *Iterated associated-bundle regime.* Assume that, after shrinking to local trivializations over $Y_1$, the stratified link fibration is an associated bundle $L_1 \simeq P_1 \times_G L_{1,0} \to Y_1$ whose structure group $G$ preserves a fixed depth-one model ($L_{1,0}, g_{1,0}$) together with its inner canonical data (equivalently, $G$ acts by stratified isometries preserving the canonical inner Calderón subbundle). In this regime, any tangential family $A_{1,y}$ constructed functorially from the link geometry is conjugate to a fixed model operator (as in the depth-one rigidity mechanism), hence its small spectrum and the outer boundary spectrum near 0 are constant, yielding (H2₁)–(H3₁).

This template isolates a plausible route to a fully self-contained depth-two sequel: one combines the depth-one rigidity/stability mechanisms on the inner links with an iterated rigidity assumption on the outer link family to force constant kernel rank and a uniform spectral gap for $A_{1,y}^{W_{2,can}}$.

**Inductive M3 theorem schema.** Theorem 9.1 is the first nontrivial instance of the general inductive pattern: assuming canonical data are already constructed on all deeper strata (inner hypotheses), one defines the tangential family at the next stratum using the deeper canonical domains, verifies the outer analogues (H1)–(H3), and applies the Calderón construction to obtain new canonical Cauchy data. Iterating from the deepest strata outward produces a global canonical domain and a canonical mezzoperversity.

## OUTLOOK (M3–M4)

This paper establishes the fully projective depth-one case (M2), with the isolated conic model (M1) as a base. Two natural directions remain.

**(M3) Iteration to higher depth.** Extend the depth-one trace/Poisson/Calderón package to arbitrary stratified depth by a recursive construction in which the tangential operator on each stratum depends on the boundary data fixed on deeper links, together with a compatibility theory for the resulting family of trace bundles and Cauchy data.

**(M4) Identification with middle perversity.** In the non-Witt regime the Fubini–Study geometry canonically selects a mezzoperversity $W_{can}$ via Cauchy data. A final step toward a “bare” CGM statement is to give verifiable conditions ensuring $W_{can} = \bar{m}$ (or to isolate the sharp canonical formulation in terms of $IH_{W_{can}}^{\bullet}(X)$ for general projective strata).

Outlook: toward M3 and M4. The next milestone is the depth-two iteration (and, more generally, the inductive step in the stratification depth). The precise one-step depth-two formulation and the corresponding inductive theorem schema are given in Section 9. In particular, that section isolates the new analytic bottleneck in higher depth: establishing constant-rank kernel bundles and a uniform spectral gap for tangential families whose self-adjoint realizations depend on deeper canonical ideal boundary conditions. A further milestone (M4) is to identify when the canonically
selected mezzoperversity agrees with middle perversity (or to formulate the correct canonical statement in the non-Witt case).

## APPENDIX A. TWO STANDARD ANALYTIC LEMMAS USED IN SECTION 5

Throughout this appendix, let $L$ be a fixed closed smooth manifold and $E \to L$ a Hermitian vector bundle. Let $Y$ be a compact manifold. Assume that $y \mapsto A_y$ is a smooth family of first-order elliptic, self-adjoint differential operators on $(L, E)$ in the following standard sense: in local coordinates and a fixed trivialization of $(L, E)$, the coefficients of $A_y$ depend smoothly on $(y, x)$, and we view $A_y$ as an unbounded operator on the fixed Hilbert space $H := L^2(L; E)$ with domain $\text{Dom}(A_y) = H^1(L; E)$.

**Lemma A.1** (Spectral stability for smooth self-adjoint elliptic families). Under the assumptions above:

(i) (Local resolvent smoothness) Fix $y_0 \in Y$ and let $\lambda \in \mathbb{C}$ be such that $\lambda \notin \text{Spec}(A_{y_0})$. Then there exists a neighborhood $U \ni y_0$ such that $\lambda \notin \text{Spec}(A_y)$ for all $y \in U$. Moreover, the map

$$U \ni y \mapsto (\lambda - A_y)^{-1} \in \mathcal{L}(H, H^1(L; E))$$

is smooth, hence in particular continuous as a map into $\mathcal{L}(H, H)$.

(ii) (Riesz projections are smooth) Let $\Gamma \subset \mathbb{C}$ be a positively oriented smooth contour such that $\Gamma \cap \text{Spec}(A_{y_0}) = \emptyset$. Shrinking $U \ni y_0$ if needed, one has $\Gamma \cap \text{Spec}(A_y) = \emptyset$ for all $y \in U$, and the associated Riesz projection

$$\Pi_{\Gamma}(y) := \frac{1}{2\pi i} \int_{\Gamma} (\lambda - A_y)^{-1} d\lambda \in \mathcal{L}(H)$$

depends smoothly on $y \in U$. In particular, rank $\Pi_{\Gamma}(y)$ is locally constant in $y$.

(iii) (Continuity of ordered eigenvalues) For each $k \ge 1$, let $\mu_k(y)$ be the k-th eigenvalue of the nonnegative operator $A_y^2$ counted with multiplicity and ordered increasingly. Then $y \mapsto \mu_k(y)$ is continuous on $Y$. Equivalently, the ordered eigenvalues of $|A_y| = (A_y^2)^{1/2}$ depend continuously on $y$.

**Proof.** (i) Work over a local trivialization in $y$ so that $A_y$ acts on the fixed domain $H^1(L; E) \subset H$. The family $A_y$ is a smooth map $Y \to \mathcal{L}(H^1, H)$ and each $A_y$ is self-adjoint elliptic, hence has compact resolvent. Fix $\lambda \notin \text{Spec}(A_{y_0})$. Then $\lambda - A_{y_0} : H^1 \to H$ is bijective and boundedly invertible. Since $y \mapsto (\lambda - A_y)$ is continuous in $\mathcal{L}(H^1, H)$, invertibility persists for $y$ near $y_0$ by the bounded inverse theorem, giving $\lambda \notin \text{Spec}(A_y)$ for $y \in U$ and continuity of $y \mapsto (\lambda - A_y)^{-1}$. Differentiating $(\lambda - A_y)(\lambda - A_y)^{-1} = I$ yields the standard identity

$$\partial_{y_j} (\lambda - A_y)^{-1} = (\lambda - A_y)^{-1} (\partial_{y_j} A_y) (\lambda - A_y)^{-1},$$

and similarly for higher derivatives, proving smoothness into $\mathcal{L}(H, H^1)$.

(ii) By (i), after shrinking $U$ we have $\Gamma \cap \text{Spec}(A_y) = \emptyset$ for all $y \in U$, and $y \mapsto (\lambda - A_y)^{-1}$ is smooth uniformly for $\lambda \in \Gamma$ (compactness of $\Gamma$). Therefore $\Pi_{\Gamma}(y)$ is a smooth $\mathcal{L}(H)$-valued map, and rank $\Pi_{\Gamma}(y)$ is locally constant.

(iii) Since $A_y$ is self-adjoint elliptic with compact resolvent, so is $A_y^2$, and its spectrum consists of discrete eigenvalues $\mu_k(y) \to +\infty$. Fix $y_0$ and a number $R > 0$ such that $R \notin \text{Spec}(A_{y_0}^2)$. Let $\Gamma_R$ be a contour enclosing $\text{Spec}(A_{y_0}^2) \cap [0, R]$ and avoiding the rest of the spectrum. Applying (ii) to the family $A_y^2$ shows that,
for $y$ near $y_0$, the total multiplicity of eigenvalues in $[0, R]$ is constant and the corresponding spectral subspace varies smoothly. This implies that each ordered eigenvalue below $R$ varies continuously near $y_0$. Since $y_0$ and $R$ were arbitrary and $Y$ is compact, $y \mapsto \mu_k(y)$ is continuous on $Y$ for all $k$. Passing from $A_y^2$ to $|A_y|$ preserves continuity of ordered eigenvalues. $\square$

**Lemma A.2 (Uniform gap from compactness).** Assume in addition that $\dim \ker(A_y)$ is constant on $Y$. Let $\lambda_*(y) > 0$ denote the smallest positive eigenvalue of $|A_y|$. Then there exists $\varepsilon_0 > 0$ such that

$$ \mathrm{Spec}(A_y) \cap (-\varepsilon_0, \varepsilon_0) = \{0\}, \quad \forall y \in Y. $$

Equivalently, $\inf_{y \in Y} \lambda_*(y) := \varepsilon_0 > 0$.

*Proof.* By Lemma A.1(iii), the ordered eigenvalues of $|A_y|$ depend continuously on $y$, hence so does $\lambda_*(y)$. For each $y$, ellipticity implies $\mathrm{Spec}(|A_y|)$ is discrete with 0 an isolated eigenvalue, so $\lambda_*(y) > 0$.

If $\inf_{y \in Y} \lambda_*(y) = 0$, compactness of $Y$ yields a sequence $y_n \to y_\infty$ with $\lambda_*(y_n) \to 0$. By continuity, $\lambda_*(y_\infty) = 0$, i.e. $0$ is not isolated from the nonzero spectrum at $y_\infty$, so $\dim \ker(A_{y_\infty})$ must be strictly larger than the constant value (an eigenvalue has hit 0), contradicting the hypothesis that $\dim \ker(A_y)$ is constant on $Y$. Therefore $\varepsilon_0 := \min_{y \in Y} \lambda_*(y) > 0$, which is exactly the uniform spectral gap statement. $\square$

## APPENDIX B. FUBINI-STUDY WEDGE EXPANSION: ADAPTED COORDINATES AND TENSOR ESTIMATES

**B.1. Edge vector fields and edge-smooth tensors.** Let $(X, \mathcal{S})$ be a depth-one smoothly stratified space with singular stratum $Y$ and regular part $X^{\mathrm{reg}}$. Fix a tubular neighborhood $\mathcal{N}$ of $Y$ and control data $(\rho, x)$ on $N \cap X^{\mathrm{reg}}$ as in Proposition 3.1. Thus $x : N \cap X^{\mathrm{reg}} \to (0, \varepsilon)$ is a boundary defining function for $Y$ and $\rho : N \cap X^{\mathrm{reg}} \to Y$ is a retraction.

On a local trivialization

$$ N \cap X^{\mathrm{reg}} \cong (0, \varepsilon)_x \times U_y \times F_z, \quad \rho(x, y, z) = y, $$

the Lie algebra of edge vector fields is

$$ \mathcal{V}_e = \mathrm{span}_{\mathcal{C}^\infty} \{x \partial_x, \partial_{y^a}, \partial_{z^\mu}\}. $$

A tensor field $T$ on $N \cap X^{\mathrm{reg}}$ is called edge-smooth if in these coordinates all of its components (with respect to the coframe $\{dx, dy, dz\}$) extend continuously to $x = 0$ and remain $\mathcal{C}^\infty$ under iterated differentiation by elements of $\mathcal{V}_e$ (equivalently, all coefficients are conormal at $x = 0$ with respect to $\mathcal{V}_e$).

**B.2. A quantitative Whitney angle estimate.** We work locally near a fixed point of the stratum $Y$. Let $(M, g)$ be a smooth ambient Riemannian manifold and assume $Y \subset M$ is an embedded submanifold (in the application $M = \mathbb{CP}^N$, $g = g_{\mathrm{FS}}$). Fix a tubular neighborhood $U$ of $Y$ given by the normal exponential map $\exp^\perp : \nu(Y)_{<\varepsilon} \to U$ and let

$$ \pi : U \to Y, \quad r : U \to [0, \varepsilon) $$

denote the associated smooth nearest-point projection and distance function. On $U \setminus Y$, write a point uniquely as $\exp_y^\perp(r \chi)$ with $y = \pi(p) \in Y$ and $\chi \in S(\nu_y Y)$.
Restricting to $N \cap X^{\text{reg}} \subset U$, the depth-one/simple-edge structure gives local product coordinates

$$ (0, \varepsilon)_x \times U_y \times F_z \to N \cap X^{\text{reg}}, \quad (x, y, z) \mapsto p(x, y, z), $$

with $\rho(p(x, y, z)) = y$ and $x = \psi(r(p))$ as in **Step 1** of Proposition B.5. Shrinking the chart if needed, we may assume $\psi'(0) = 1$ and hence $x \sim r$.

**Lemma B.1** (Second-order normal form of the adapted parametrization). Let $Y \subset \mathbb{CP}^N$ be a smooth submanifold and let $U$ be a sufficiently small tubular neighborhood given by the normal exponential map. Denote by $\pi: U \to Y$ the nearest-point projection and by $r = \text{dist}_{\text{FS}}(\cdot, Y)$ the distance function (smooth on $U \setminus Y$) with $|dr|_{g_{\text{FS}}} = 1$.

Let $N \cap X^{\text{reg}} \subset U$ be a depth-one neighborhood equipped with adapted product coordinates $(x, y, z) \in (0, \varepsilon) \times U_y \times F_z$ as in this appendix, with retraction $\rho(x, y, z) = y$ and with radial variable chosen as $x = \psi(r)$ for a smooth $\psi$ satisfying $\psi(0) = 0$ and $\psi'(0) = 1$.

Write the normal-coordinate representation of the embedding $p(x, y, z) \in \mathbb{CP}^N$ as $(y, w(x, y, z))$ in the tubular chart (so $Y = \{w = 0\}$ and $r \sim |w|$). Then, after shrinking $\varepsilon$ if needed, there exist smooth maps

$$ \Omega: U_y \times F_z \to S(\nu Y), \quad R: (0, \varepsilon) \times U_y \times F_z \to \nu Y $$

such that

$$ (\text{B.2.1}) \qquad w(x, y, z) = x \Omega(y, z) + x^2 R(x, y, z), $$

and $\Omega(y, z)$ is fiberwise unit ($|\Omega|_{g_{\text{FS}}} = 1$). Moreover,

$$ r = x + O(x^2), \quad dr = dx + O(x) \quad \text{on } N \cap X^{\text{reg}}, $$

and all $O(\cdot)$ remainders above are stable under iterated differentiation by edge vector fields $x\partial_x$, $\partial_{y_a}$, $\partial_{z_\mu}$ in the adapted chart.

**Proof.** In tubular coordinates $(y, w)$, the restriction map

$$ w: (0, \varepsilon) \times U_y \times F_z \to \nu Y $$

is smooth up to $x = 0$ (this is exactly what “adapted coordinates/control data” encode: the neighborhood is diffeomorphic to a cone bundle over the link fibration). In particular, $w(0, y, z) = 0$, hence Taylor’s theorem in the $x$-variable gives

$$ w(x, y, z) = x \partial_x w(0, y, z) + x^2 R(x, y, z) $$

for a smooth remainder $R$. Set $\Omega(y, z) := \partial_x w(0, y, z)$.

We now normalize $\Omega$ to be unit and identify it as a normal direction. Since $x = \psi(r)$ with $\psi'(0) = 1$, we have $dx = \psi'(r) dr$ and thus $dx = dr + O(r)$ and $r = x + O(x^2)$ on $U \setminus Y$. Along each ray of fixed $(y, z)$, the curve $x \mapsto p(x, y, z)$ satisfies $r(p(x, y, z)) = x + O(x^2)$, hence its $g_{\text{FS}}$-speed in the normal direction at $x = 0$ is 1. In tubular coordinates this exactly means $|\partial_x w(0, y, z)|_{g_{\text{FS}}} = 1$, so $|\Omega|_{g_{\text{FS}}} = 1$.

Finally, stability under edge differentiation follows because all coefficients involved are smooth in $(x, y, z)$ and the expansions above have explicit vanishing order in $x$: acting by $x\partial_x$ preserves the $x$-order, while $\partial_y, \partial_z$ differentiate only smooth coefficients and do not reduce the vanishing order. $\square$
Remark B.2 (Compatibility with control data). The tubular projection $\pi$ and the control-data retraction $\rho$ are both smooth retractions to $Y$ defined on a neighborhood of $Y$ and agreeing on $Y$. Replacing $\rho$ by $\pi|_{N \cap X^{\text{reg}}}$ (or composing with a diffeomorphism that is the identity on $Y$) changes the adapted coordinates only by $O(x)$ terms and hence does not affect the wedge expansion in Proposition B.5. Thus the proofs in Appendix B are consistent with the control-data framework used in Sections 2-3.

**Lemma B.3** (Angle control $\Rightarrow$ mixed terms are $O(x)$). Let $g$ be a smooth ambient metric on $U$ (in our application $g = g_{\text{FS}}$). Define the vertical bundle

$$ \mathcal{V} := \ker(dp) \cap \ker(dx) $$

(tangent to the link fibers at fixed $x$), and choose any smooth horizontal complement $\mathcal{H} \subset \ker(dx)$ so that $dp: \mathcal{H} \to TY$ is an isomorphism. Then there exists $C_0 > 0$ such that for $x$ sufficiently small and all $h \in \mathcal{H}$, $v \in \mathcal{V}$,

$$ |g(h, v)| \le Cx|h|_g|v|_g, \quad |g(\nabla_g x, h)| \le Cx|h|_g, \quad |g(\nabla_g x, v)| \le Cx|v|_g. $$

Moreover, in any adapted chart the same $O(x)$ order is preserved under iterated differentiation by edge vector fields $x\partial_x$, $\partial_{y_a}$, $\partial_{z_\mu}$.

*Proof.* Work in the tubular chart $(y, w)$ and use the expansion (B.2.1) from Lemma B.1. Since $g_{\text{FS}}$ is smooth in $(y, w)$, its coefficients admit Taylor expansions in $w$ with bounded derivatives. In particular, evaluating at $w = 0$ and using $w = O(x)$ gives, uniformly on compact subsets of the chart,

$$ g_{\text{FS}}(y, w) = g_{\text{FS}}(y, 0) + O(|w|) = g_{\text{FS}}(y, 0) + O(x), $$

with the same order preserved under edge differentiation.

Let $\mathcal{V} = \ker(dp) \cap \ker(dx)$ (vertical tangent to the link fibers at fixed $x$) and choose a horizontal complement $\mathcal{H} \subset \ker(dx)$ as in Proposition B.5. In the adapted chart, vertical vectors are linear combinations of $\partial_{z_\mu}$. Differentiating (B.2.1) in the $z$-directions yields $\partial_{z_\mu} w = x \partial_{z_\mu} \Omega + O(x^2)$, hence the ambient size of $\partial_{z_\mu} p$ is $O(x)$, and therefore any $v \in \mathcal{V}$ has $|v|_{g_{\text{FS}}} = O(x)$ when measured against the fixed $(y, z)$ frame.

Horizontal vectors $h \in \mathcal{H}$ are lifts of $\partial_{y_a}$ with $dx(h) = 0$. Differentiating (B.2.1) in the $y$-directions gives $\partial_{y_a} w = x \partial_{y_a} \Omega + O(x^2)$, so $h$ has uniformly bounded components in the ambient frame. Pairing a bounded horizontal vector with an $O(x)$ vertical vector under the smooth metric yields

$$ |g_{\text{FS}}(h, v)| \le Cx|h|_{g_{\text{FS}}}|v|_{g_{\text{FS}}}. $$

For the remaining estimates, write $x = \psi(r)$ with $\psi'(0) = 1$. Then $\nabla x = \psi'(r)\nabla r$ and $|\nabla r|_{g_{\text{FS}}} = 1$. Since $h, v \subset \ker(dx)$, we have $g_{\text{FS}}(\nabla x, h) = dx(h) = 0$ and $g_{\text{FS}}(\nabla x, v) = dx(v) = 0$ to leading order; more precisely, the difference between the adapted splitting and the exact orthogonal splitting is controlled by $w = O(x)$, so these pairings are $O(x)$. Equivalently, using Lemma B.1 we have $dr = dx + O(x)$ and thus

$$ |g_{\text{FS}}(\nabla x, h)| = |dx(h)| \le Cx|h|_{g_{\text{FS}}}, \quad |g_{\text{FS}}(\nabla x, v)| = |dx(v)| \le Cx|v|_{g_{\text{FS}}}. $$

Stability under edge differentiation follows from the explicit $x$-vanishing: all mixed coefficients are of the form $x \cdot a(x, y, z)$ with $a$ smooth and bounded, so $x\partial_x$ preserves the factor $x$, and $\partial_y, \partial_z$ do not reduce the vanishing order. $\square$
**B.3. Vertical scaling and the definition of $g_L$.** Let $L := \{x = 1\} \cap (N \cap X^{\text{reg}})$ denote the link fibration over $Y$ and $L_y$ its fibers. We now record the scaling statement that identifies the leading vertical part of the metric.

**Lemma B.4 (Vertical scaling).** In the adapted coordinates of Appendix B and with the radial variable $x = \psi(r)$ as in Proposition B.5, assume the normal-form expansion ((B.2.1)) of Lemma B.1 holds, i.e.

$$w(x, y, z) = x \Omega(y, z) + x^2 R(x, y, z),$$

with $\Omega$ smooth and $R$ smooth and bounded (stable under edge differentiation). Let

$$\mathcal{V} := \ker(d\rho) \cap \ker(dx)$$

be the vertical bundle tangent to the link fibers at fixed $x$.

Then there exists a smooth fiber metric $g_L$ on $\mathcal{V}|_{x=1}$ (transported along the product coordinate) such that, uniformly on compact subsets and stably under iterated differentiation by edge vector fields,

$$g(v, v') = x^2 g_L(v, v') + O(x^3) |v| |v'|, \quad \forall v, v' \in \mathcal{V}.$$

*Proof.* Work in an adapted product chart $(0, \varepsilon)_x \times U_y \times F_z$ and use the normal-form expansion ((B.2.1)) from Lemma B.1:

$$w(x, y, z) = x \Omega(y, z) + x^2 R(x, y, z),$$

with $\Omega$ smooth and $R$ smooth and bounded on compact subsets. Recall that the vertical bundle is

$$\mathcal{V} = \ker(d\rho) \cap \ker(dx),$$

hence it is spanned by $\{\partial_{z_\mu}\}$ in these coordinates.

**Step 1: vertical tangent vectors are $O(x)$ in ambient size.** Differentiate ((B.2.1)) in the $z$-directions:

$$\partial_{z_\mu} w(x, y, z) = x \partial_{z_\mu} \Omega(y, z) + x^2 \partial_{z_\mu} R(x, y, z).$$

Since $\partial_{z_\mu} \Omega$ and $\partial_{z_\mu} R$ are bounded on compact subsets, it follows that $\partial_{z_\mu} w = O(x)$ and hence the ambient vectors $\partial_{z_\mu} p$ are $O(x)$ in $g$-norm. Equivalently, for any $v \in \mathcal{V}$, $|v|_g = O(x)$ uniformly on compact subsets of the chart.

**Step 2: Taylor expansion of the metric coefficients.** Because $g$ is a smooth ambient metric, its coefficients in tubular coordinates $(y, w)$ admit a Taylor expansion in $w$ with uniformly bounded derivatives on compact subsets. Since $w = O(x)$ by ((B.2.1)), we have

$$g(y, w) = g(y, 0) + O(|w|) = g(y, 0) + O(x),$$

and the same order is preserved under differentiation by edge vector fields $x\partial_x$, $\partial_{y_a}$, $\partial_{z_\mu}$.

**Step 3: the $x^2$ leading term and the $O(x^3)$ remainder.** Set

$$g_{\mu\nu}(x, y, z) := g(\partial_{z_\mu}, \partial_{z_\nu}).$$

Using $\partial_{z_\mu} p = (0, \partial_{z_\mu} w)$ in tubular coordinates and combining **Step 1** with **Step 2** gives

$$\begin{aligned} g_{\mu\nu}(x, y, z) &= (g(y, 0) + O(x))(x \partial_{z_\mu} \Omega + O(x^2), x \partial_{z_\nu} \Omega + O(x^2)) \\ &= x^2 \bar{g}_{\mu\nu}(y, z) + x^3 \tilde{g}_{\mu\nu}(x, y, z), \end{aligned}$$
where

$$ \bar{g}_{\mu\nu}(y, z) := g(y, 0)(\partial_{z_\mu}\Omega(y, z), \partial_{z_\nu}\Omega(y, z)) $$

and $\tilde{g}_{\mu\nu}$ is smooth and bounded on compact subsets. Moreover, the decomposition into $x^2(\cdot) + x^3(\cdot)$ is stable under edge differentiation because every term has an explicit $x$-factor and the remaining coefficients are smooth.

**Step 4: identification with $g_L$.** Define $g_L$ on $V|_{x=1}$ by declaring that, in the transported frame $\{\partial_{z_\mu}\}$, its coefficient matrix is exactly $\bar{g}_{\mu\nu}(y, z)$, and extend it to $0 < x < \varepsilon$ by transport along the product coordinate (so $g_L$ is independent of $x$ in this chart). Then the identity in **Step 3** is precisely

$$ g(v, v') = x^2 g_L(v, v') + O(x^3) |v| |v'| $$

for all $v, v' \in V$, with the stated uniformity and stability under $V_e$.

□

### B.4. Fubini-Study wedge expansion: proof and assembly.

**Proposition B.5 (Fubini-Study wedge expansion).** Let $X \subset \mathbb{CP}^N$ be a complex projective variety with a depth-one Whitney stratification, and let $g_{FS}$ be the induced Fubini-Study metric on $X^{\text{reg}}$. Then on a sufficiently small tubular neighborhood $N \cap X^{\text{reg}}$ of the singular stratum $Y$ there exist:

* a boundary defining function $x$ and a retraction $\rho$ (control data),

* a smooth metric $g_Y$ on $Y$,

* a smooth family of fiber metrics $g_L$ on the link fibration $L \to Y$,

such that

$$ (B.4.1) \qquad g_{\text{FS}} = dx^2 + x^2 g_L + \rho^* g_Y + x \kappa, $$

where $\kappa$ is a symmetric 2-tensor that is edge-smooth on $N \cap X^{\text{reg}}$.

*Proof.* Work in a local product chart $(0, \varepsilon)_x \times U_y \times F_z$ as above.

**Step 1: choice of $x$ and control of $|dx|$**. Let $r(p) = \text{dist}_{\text{FS}}(p, Y)$ denote the ambient distance to $Y$ in $\mathbb{CP}^N$. On a sufficiently small tubular neighborhood of the smooth submanifold $Y$, the normal exponential map gives smooth nearest-point projection and distance function, hence $r$ is smooth on $U \setminus Y$ and satisfies $|dr|_{g_{\text{FS}}} = 1$. Replace the given control function $x$ by $\tilde{x} = \psi(r)$ for a smooth monotone function $\psi$ with $\psi(0) = 0$ and $\psi'(0) = 1$. Then $\tilde{x}$ remains a boundary defining function, is compatible with the control data structure, and

$$ |d\tilde{x}|_{g_{\text{FS}}} = 1 + O(\tilde{x}). $$

For notational simplicity we write $x$ for $\tilde{x}$.

**Step 2: splitting into radial/horizontal/vertical directions.** Let $V := \ker(d\rho) \cap \ker(dx)$ be the vertical bundle tangent to the fibers of $\rho|_{L_t}: L_t \to Y$ (i.e. tangent to the link fibration at fixed $x$). Choose a smooth horizontal complement $\mathcal{H} \subset \ker(dx)$ so that $d\rho: \mathcal{H} \to TY$ is an isomorphism (e.g. an Ehresmann connection for the fiber bundle $\rho|_{L_t}$). Writing $R := \nabla_{g_{\text{FS}}}x$, we obtain a smooth splitting

$$ T(N \cap X^{\text{reg}}) = \mathbb{R}\langle R \rangle \oplus \mathcal{H} \oplus V. $$

**Step 3: control of mixed terms.** With the splitting

$$ T(N \cap X^{\text{reg}}) = \mathbb{R}\langle R \rangle \oplus \mathcal{H} \oplus V, \quad R := \nabla_{g_{\text{FS}}}x, \quad V := \ker(d\rho) \cap \ker(dx), $$
Lemma B.3 yields the quantitative estimates

$$g_{\text{FS}}(\mathcal{H}, \mathcal{V}) = O(x), \quad g_{\text{FS}}(R, \mathcal{H}) = O(x), \quad g_{\text{FS}}(R, \mathcal{V}) = O(x),$$

uniformly on compact subsets and stably under iterated differentiation by edge vector fields $x\partial_x$, $\partial_{y_a}$, $\partial_{z_\mu}$. Equivalently, in the adapted coframe

$$\{dx, dy^a, x dz^\mu\},$$

all mixed components of $g_{\text{FS}}$ carry an explicit factor $x$.

**Step 4: definition of $g_L$ and the leading vertical term.** Let $g_L$ be the link fiber metric on $\mathcal{V}$ defined in Lemma B.4 and transported along the product coordinate. Then Lemma B.4 gives

$$g_{\text{FS}}|_{\mathcal{V} \times \mathcal{V}} = x^2 g_L + O(x^3),$$

again with uniformity on compact subsets and stability under edge differentiation. In the coframe $\{dx, dy^a, x dz^\mu\}$ this means that the vertical block of the metric is $x^2(g_L)_{\mu\nu}(y, z) dz^\mu dz^\nu$ plus an $x^3$-remainder.

**Step 5: assembling the wedge expansion and identifying the remainder.** By Step 1, we chose $x = \psi(r)$ with $\psi'(0) = 1$, hence

$$|dx|_{g_{\text{FS}}}^2 = 1 + O(x), \quad g_{\text{FS}}(R, R) = 1 + O(x),$$

and $R$ is transverse to the level sets $\{x = \text{const}\}$. By construction of $\mathcal{H}$ and $\rho$, the restriction of $g_{\text{FS}}$ to $\mathcal{H}$ converges to $\rho^*g_Y$ as $x \to 0$; more precisely, in adapted coordinates the horizontal block has the form

$$g_{\text{FS}}|_{\mathcal{H} \times \mathcal{H}} = (\rho^*g_Y)_{ab}(y) dy^a dy^b + O(x),$$

with the $O(x)$-term stable under edge differentiation.

Combining the radial, horizontal, vertical, and mixed estimates from Steps 3–4, we obtain

$$g_{\text{FS}} = dx^2 + \rho^* g_Y + x^2 g_L + x \kappa,$$

where $\kappa$ is a symmetric 2-tensor whose coefficients in the adapted coframe $\{dx, dy^a, x dz^\mu\}$ extend smoothly to $x = 0$ and remain smooth under iterated differentiation by $\mathcal{V}_e = \text{span}\{x\partial_x, \partial_{y_a}, \partial_{z_\mu}\}$. In other words, $\kappa$ is edge-smooth in the sense of Appendix B.1, which completes the proof. $\square$

## APPENDIX C. TRACE BUNDLES, GREEN PAIRING, AND THE CALDERÓN PACKAGE IN THE CONSTANT-MULTIPLICITY STRIP

In this appendix we collect the boundary value theory for $D$ in a fixed constant-multiplicity strip of indicial roots. We first recall the geometric setup and the standing hypotheses (H1)–(H3) from the main text, and then outline the structure of this appendix. In Subsection C.1 we record the maximal-domain asymptotics and define the trace bundle and trace map. Subsection C.2 constructs the Poisson operator and the Calderón projector in this setting, while Subsection C.2.1 provides a concrete construction via a collar parametrix and introduces the comparison operator and the basic Hilbert-space lemmas needed later. Subsection C.3 defines the Green boundary pairing on the trace bundle and shows that the canonical Cauchy data form a Lagrangian subbundle. Finally, Subsection C.4 identifies adjoint domains and establishes Fredholm criteria for projection-type boundary conditions in terms of the Calderón projector.
Throughout this appendix we work in a tubular neighborhood $N \cap X_{\text{reg}} \cong (0, \varepsilon)_x \times L$ over the singular stratum $Y$, and we assume the standing hypotheses established in the main text for the projective simple-edge FS metric:

(H1) (Wedge Dirac normal form) $D$ admits the normal form

$$D = \Gamma\left(\partial_x + \frac{1}{x}A_y + B(x)\right)$$

with $A_y$ self-adjoint elliptic on each compact link fiber $L_y$ and smoothly depending on $y$, and $B(x)$ a first-order edge operator with edge-smooth coefficients down to $x = 0$ (Proposition 4.1 and Lemma 4.4).

(H2) (Uniform gap at 0) $\dim \ker(A_y)$ is constant and there exists $\varepsilon_0 > 0$ with $\text{Spec}(A_y) \cap (-\varepsilon_0, \varepsilon_0) = \{0\}$ for all $y \in Y$ (Propositions 5.1-5.2).

(H3) (Constant-multiplicity strip) In a weight strip around the $L^2$ threshold, the only boundary spectrum encountered is the indicial root at 0, with constant multiplicity rank($H$) (Proposition 5.3).

These are precisely the hypotheses under which the $L^2$ wedge boundary theory yields a trace bundle and Poisson/Calderón operators; see, e.g., [15, 11, 12].

### C.1. Maximal-domain asymptotics and the trace map.

**Lemma C.1 (Maximal-domain expansion and trace map).** Assume (H1)–(H3). Then there exist $\delta > 0$ and a continuous linear map

$$\text{Tr}_D : \text{Dom}(D_{\text{max}}) \to \Gamma(Y; H), \quad H := \ker(A) \to Y,$$

such that every $u \in \text{Dom}(D_{\text{max}})$ admits an expansion

$$u = \chi(x) u_0 + u_{\text{rem}}, \quad u_0 = \text{Tr}_D(u) \in \Gamma(Y; H), \quad u_{\text{rem}} \in x^\delta H_e^1,$$

where $\chi$ is a fixed cut-off supported near $x = 0$. Moreover $\text{Tr}_D$ is continuous with respect to the graph norm on $\text{Dom}(D_{\text{max}})$.

**Proof.** This is the constant-multiplicity-strip specialization of the trace-bundle construction for first-order elliptic wedge operators. The regular-singular form in (H1) implies that the Mellin transform in $x$ diagonalizes the model operator obtained by freezing coefficients at $x = 0$, and the boundary spectrum is determined by the indicial family associated with $A_y$. By (H2)–(H3), there are no indicial roots in the relevant strip except the simple root at 0, with constant multiplicity equal to rank($H$), so the maximal-domain elements have a canonical leading term valued in $\ker(A_y)$ and a remainder gaining positive weight. Continuity in the graph norm follows from the standard mapping properties of wedge parametrices in weighted edge Sobolev spaces. Precise statements and proofs in this regime may be found in [11, §§ 3–5] (trace bundle and maximal domain) and [14, 15]. $\square$

### C.2. Poisson operator and Calderón projector.
In general, not every boundary datum $v \in \Gamma(Y; H)$ occurs as the leading coefficient of an $L^2$-solution of $Du = 0$ in the critical strip. The correct statement is that there is a canonical finite-rank subbundle $W_{\text{can}} \subset H$ of Cauchy data, and a Calderón projection $C$ onto $W_{\text{can}}$; the Poisson operator then produces $L^2$-solutions with boundary trace equal to $Cv$.

**Proposition C.2 (Poisson operator and Calderón projector).** Assume (H1)–(H3) and work in the constant-multiplicity strip of Proposition 5.3. Let $H = \ker(A) \to Y$ be the trace bundle and let

$$\text{Tr}_D : \text{Dom}(D_{\text{max}}) \to \Gamma(Y; H)$$
be the trace map from Appendix C.1. Define the Cauchy data space

$$W_{\text{can}} := \text{Tr}_D(\ker(D) \cap L^2(X_{\text{reg}})) \subset \Gamma(Y; H).$$

Then there exist linear operators

$$C : \Gamma(Y; H) \to \Gamma(Y; H), \quad C^2 = C,$$

and

$$P : \Gamma(Y; H) \to \ker(D) \cap L^2(X_{\text{reg}}),$$

such that:

(1) $\text{ran}(C) = W_{\text{can}}$ and $C|_{W_{\text{can}}} = \text{Id}$;

(2) $\text{Tr}_D(P(v)) = Cv$ for all $v \in \Gamma(Y; H)$. In particular, for every $w \in W_{\text{can}}$ one has $\text{Tr}_D(P(w)) = w$.

Moreover, C is an order-0 pseudodifferential projection in the appropriate wedge/edge calculus along Y, depends smoothly on the parameter y ∈ Y, and $W_{\text{can}} = \text{ran}(C) \subset H$ is a smooth subbundle.

**Proof.** We give a reference-level argument, indicating exactly where the wedge/edge boundary calculus is used.

**Step 1: parametrix and the trace bundle in a constant strip.** By Proposition 5.3 the critical strip has constant indicial multiplicity and the trace bundle is the finite-rank bundle $H = \ker(A) \to Y$. Appendix C.1 records the two-term expansion in this strip and the existence/uniqueness/continuity of the trace map $\text{Tr}_D$.

**Step 2: existence of the Calderón projector.** Under (H1)–(H3), the de Rham Dirac operator D is a wedge Dirac-type operator with edge-smooth coefficients and a constant trace bundle in the critical strip. The Calderón package for Dirac-type wedge/edge operators (the wedge analogue of Seeley’s classical construction for uniformly elliptic boundary problems, and the Dirac boundary theory via pseudodifferential projections) yields an order-0 wedge/edge pseudodifferential idempotent

$$C : \Gamma(Y; H) \to \Gamma(Y; H)$$

whose range equals the Cauchy data space of $L^2$-solutions:

$$\text{ran}(C) = \text{Tr}_D(\ker(D) \cap L^2) = W_{\text{can}}.$$

In particular C acts as the identity on its range, i.e. $C|_{W_{\text{can}}} = \text{Id}$, and therefore $C^2 = C$ (equivalently, for all v one has $Cv \in W_{\text{can}}$ hence $C(Cv) = Cv$). Smooth dependence on y follows from the parameter-dependent wedge/edge calculus in the constant strip (no eigenvalue crossings inside the window).

**Step 3: Poisson operator.** The same Calderón package provides a Poisson operator on the Cauchy data: there exists a bounded right inverse

$$P_{\text{can}} : \Gamma(Y; W_{\text{can}}) \to \ker(D) \cap L^2(X_{\text{reg}}) \quad \text{such that} \quad \text{Tr}_D \circ P_{\text{can}} = \text{Id}_{\Gamma(Y; W_{\text{can}})}.$$

Define P on all boundary data by

$$P := P_{\text{can}} \circ C : \Gamma(Y; H) \to \ker(D) \cap L^2(X_{\text{reg}}).$$

Then for every $v \in \Gamma(Y; H)$,

$$\text{Tr}_D(P(v)) = \text{Tr}_D(P_{\text{can}}(Cv)) = Cv,$$

and if $w \in W_{\text{can}}$ then $Cw = w$ and hence $\text{Tr}_D(P(w)) = w$.
**Step 4: pseudodifferential nature and smooth subbundle.** In the wedge/edge boundary-value framework, C is a pseudodifferential projection of order 0 along Y and depends smoothly on parameters; consequently $\mathrm{ran}(C) = W_{\mathrm{can}}$ is a smooth finite-rank subbundle of $H \to Y$.

**References.** For the classical Calderón/Seeley construction and projection boundary problems, see Seeley. For Dirac-type operators and the induced boundary symplectic geometry, see standard references such as Booss-Wojciechowski and related treatments. The wedge/edge realization in the present regime is covered by the wedge boundary-value theory (e.g. the edge/wedge boundary calculus cited in the main text).

A concrete construction via a collar parametrix, together with the pseudodifferential mapping properties and compactness of differences of Calderón projectors, is given in Appendix C.2.1 (Proposition C.4). □

C.2.1. A concrete Calderón construction and the comparison operator. We continue under (H1)–(H3) and work in the constant-multiplicity strip. Let $H \to Y$ denote the trace bundle in this strip and

$$ \mathrm{Tr}_D : \mathrm{Dom}(D_{\mathrm{max}}) \to \Gamma(Y; H) $$

the trace map (Appendix C.1). The Cauchy data space is

$$ W_{\mathrm{can}} := \mathrm{Tr}_D(\ker(D) \cap L^2) \subset \Gamma(Y; H). $$

**Lemma C.3 (Model parametrix on the collar).** There exists a bounded operator

$$ Q : \Gamma(Y; H) \to \mathrm{Dom}(D_{\mathrm{max}}) $$

such that $DQ$ maps into $L^2$ and, on the collar $x \in (0, \varepsilon)$, the leading coefficient of $Q(v)$ is prescribed by $v$ up to the Cauchy data: more precisely,

$$ \mathrm{Tr}_D(Q(v)) = v + K(v), $$

where $K : \Gamma(Y; H) \to \Gamma(Y; H)$ is a smoothing operator along $Y$ (hence compact on the relevant Sobolev scales).

*Proof.* This is the standard collar construction for Dirac-type operators in a constant-multiplicity strip: in the product-type model $D_0 = \Gamma(\partial_x + \frac{1}{x}A)$ one solves the normal equation mode-by-mode and then corrects by a Neumann-series using the $O(1)$ remainder $B(x)$ and the edge-smooth coefficients. The constant-multiplicity strip guarantees that only the $\lambda = 0$ mode contributes to the trace and that all higher modes gain weight. The smoothing/compact nature of $K$ reflects that the correction improves regularity tangentially in $y$ and in the link directions. (All mapping properties are recorded in Appendix C.1-C.2.) □

**Proposition C.4 (Calderón projector and Poisson operator from a parametrix).**
There exist operators

$$ C : \Gamma(Y; H) \to \Gamma(Y; H), \quad C^2 = C, $$

and

$$ P : \Gamma(Y; H) \to \ker(D) \cap L^2, $$

such that:

(1) $\mathrm{ran}(C) = W_{\mathrm{can}};$

(2) $\mathrm{Tr}_D(P(v)) = Cv$ for every $v \in \Gamma(Y; H)$.
Moreover, C is an order-0 pseudodifferential projection in the wedge/edge calculus along Y; in particular C − ˜ C is compact for any two such Calderón projectors built from different choices in the construction.

Proof. Start from the collar parametrix Q in Lemma C.3. Let E be an extension operator that turns boundary data into a compactly supported section on the collar (e.g. E(v) = χ(x)x − f/ $^{2}$v in an adapted frame), and consider the error

$DQ(v) \in L^2.$

Using the parametrix for D on the collar in the wedge calculus (under (H1)–(H3)), solve away the error by an L 2 correction S ( v ) with D S ( v ) = D Q ( v ) . Then P$_{0}$ ( v ) := Q ( v ) − S ( v ) satisfies DP$_{0}$ ( v ) = 0 and is in L $^{2}$. Define

$$
Cv := \mathrm{Tr}_D(P_0(v)), \quad P(v) := P_0(v).
$$

By construction, Tr$_{D}$( P ( v )) = Cv . To see that C is a projection and ran( C ) = W$_{can}$ , note that for any L $^{2}$-solution u , its trace w = Tr$_{D}$( u ) belongs to W$_{can}$ and one can choose the correction so that P ( w ) = u (up to L $^{2}$-solutions with zero trace, which vanish in the strip); hence W$_{can}$ ⊂ ran( C ) . Conversely, Cv = Tr$_{D}$( P ( v )) is by definition the trace of an L $^{2}$-solution, so ran( C ) ⊂ W$_{can}$ . Thus ran( C ) = W$_{can}$ .

Finally, for w ∈ W$_{can}$ = ran( C ) we have w = Tr$_{D}$( u ) for some L $^{2}$-solution u , hence Cw = Tr$_{D}$( P ( w )) = Tr$_{D}$( u ) = w , showing C 2 = C and that P is a right inverse to Tr$_{D}$ on W$_{can}$ . The order-0 pseudodifferential nature follows from the wedge parametrix and the fact that the correction operators are tangentially pseudodifferential along Y . □

Fredholm pairs and the comparison operator. Let H be a Hilbert space and P, R : H → H bounded projections.

Definition C.5 (Fredholm pair) . The pair of closed subspaces (ran( P ) , ran( R )) is called a Fredholm pair if ran( P ) ∩ ran( R ) is finite-dimensional and ran( P ) + ran( R ) is closed of finite codimension in H .

Lemma C.6 (Comparison operator criterion) . Define

$$
T_{R,P} := RP + (I - R)(I - P) : \mathcal{H} \to \mathcal{H}.
$$

Then T$_{R,P}$ is Fredholm if and only if (ran( P ) , ran( R )) is a Fredholm pair. More- over,

$$
\ker(T_{R,P}) \cong \mathrm{ran}(P) \cap \mathrm{ran}(I-R), \quad \mathrm{coker}(T_{R,P}) \cong \mathrm{ran}(R) \cap \mathrm{ran}(I-P), \\
\text{so that}
$$

$$
\mathrm{ind}(T_{R,P}) = \dim(\mathrm{ran}(P) \cap \mathrm{ran}(I-R)) - \dim(\mathrm{ran}(R) \cap \mathrm{ran}(I-P)).
$$

Proof. Write H = ran( P ) ⊕ ran( I − P ) and compute T$_{R,P}$ with respect to this splitting: on ran( P ), T$_{R,P}$ = RP , and on ran( I − P ), T$_{R,P}$ = ( I − R )( I − P ) . A vector u = u$_{1}$ + u$_{0}$ with u$_{1}$ ∈ ran( P ), u$_{0}$ ∈ ran( I − P ) lies in ker( T$_{R,P}$ ) iff Ru$_{1}$ = 0 and ( I − R ) u$_{0}$ = 0 , i.e. u$_{1}$ ∈ ran( P ) ∩ ran( I − R ) and u$_{0}$ ∈ ran( R ) ∩ ran( I − P ) . The identifications above follow by standard diagram chasing using the decomposition and the closed range theorem; in particular, the finite-dimensionality of these intersections and the finite codimension of ran( P ) + ran( R ) are equivalent to Fredholmness of T$_{R,P}$ . □
**Lemma C.7 (Compact perturbations).** If $P-P'$ and $R-R'$ are compact operators, then $T_{R,P}-T_{R',P'}$ is compact. In particular, Fredholmness and index of $T_{R,P}$ are stable under compact perturbations of the projections.

*Proof.* Expand $T_{R,P}-T_{R',P'}$ and group terms into products involving $P-P'$ and $R-R'$; each term is compact because a bounded operator times a compact operator is compact. □

**C.3. Green pairing and Lagrangian Cauchy data.** Define the boundary pairing on $H = \ker(A) \to Y$ by

$$ (C.3.1) \qquad B(a, b) := \int_{L_y} \langle c(dx) a, b \rangle \, d\mathrm{vol}_{L_y}, \quad a, b \in H_y, $$

where $c(dx)$ is Clifford multiplication by $dx$ in a wedge-orthonormal frame.

**Lemma C.8 (Green formula).** For $u, v \in \mathrm{Dom}(D_{\max})$ compactly supported in $\mathcal{N}$ one has

$$ \langle Du, v \rangle_{L^2} - \langle u, Dv \rangle_{L^2} = B(\mathrm{Tr}_{D}u, \mathrm{Tr}_{D}v). $$

*Proof.* Work on $(\varepsilon', \varepsilon) \times L_y$ and integrate by parts in the $x$-direction, using formal self-adjointness of $D$ in a wedge-orthonormal frame. The boundary term at $x = \varepsilon$ vanishes by compact support; the boundary term at $x = \varepsilon'$ is controlled by inserting the two-term expansions from Lemma C.1. Since the remainder terms lie in $x^\delta H_e^1$, their contribution vanishes as $\varepsilon' \downarrow 0$. The limit is therefore determined solely by the leading coefficients in $\ker(A_y)$ and equals (C.3.1). This is the standard computation for regular-singular Dirac operators; see [4, § 2] and [3, § 4], and the wedge adaptation in [11]. □

**Proposition C.9 (Cauchy data are Lagrangian).** Let $C$ be the Calderón projector from Proposition C.2 and set $W_{\mathrm{can}} = \mathrm{ran}(C) \subset H$. Then $B|_{W_{\mathrm{can}} \times W_{\mathrm{can}}} = 0$ and $W_{\mathrm{can}}$ is maximally isotropic, hence Lagrangian in the natural Hilbert completion of boundary data.

*Proof.* If $a, b \in W_{\mathrm{can}} = \mathrm{ran}(C)$, choose $u = P(a)$ and $v = P(b)$ so that $Du = Dv = 0$. Since $Ca = a$ and $Cb = b$, we have $\mathrm{Tr}_{D}u = \mathrm{Tr}_{D}(P(a)) = Ca = a$ and similarly $\mathrm{Tr}_{D}v = b$. Lemma C.8 gives $0 = \langle Du, v \rangle - \langle u, Dv \rangle = B(a, b)$, hence isotropicity.

For maximality, combine two facts: (i) $C$ is a pseudodifferential projection whose range equals the Cauchy data space; (ii) for formally self-adjoint Dirac-type operators the Cauchy data space is Lagrangian for the boundary symplectic structure induced by Green's formula. A standard argument (see [4, §§ 12–13] or [3, § 7]) shows that the symplectic orthogonal of $\mathrm{ran}(C)$ equals $\ker(C)$, hence $\mathrm{ran}(C) = \mathrm{ran}(C)^{\perp B}$. The wedge setting under (H1)–(H3) is covered by the wedge boundary-value theory of [11]. □

**C.4. Adjoint domains and Fredholmness.** Let $W \subset H$ be a smooth subbundle and define the closed realization

$$ \mathrm{Dom}(D_W) := \{u \in \mathrm{Dom}(D_{\max}) : \mathrm{Tr}_D u \in \Gamma(Y; W)\}. $$

**Lemma C.10 (Adjoint domain).** Let $W^{\perp B} \subset H$ be the fiberwise B-orthogonal complement. Then

$$ \mathrm{Dom}(D_W)^* = \mathrm{Dom}(D_{W^{\perp B}}). $$

In particular, $D_W$ is self-adjoint iff $W$ is Lagrangian (i.e. $W = W^{\perp B}$).
Proof. For $u, v \in \text{Dom}(D_{\text{max}})$, Lemma C.8 gives

$$ \langle Du, v \rangle - \langle u, Dv \rangle = B(\text{Tr}_{Du}, \text{Tr}_{Dv}). $$

Thus $u$ defines an element of $\text{Dom}(D_W)^*$ precisely when the right-hand side vanishes for all $v$ with $\text{Tr}_{Dv} \in \Gamma(W)$, i.e. iff $\text{Tr}_{Du} \in \Gamma(W^{\perp_B})$. $\square$

**Theorem C.11** (Fredholmness for elliptic boundary conditions). Assume (H1)–(H3) and let $C$ be the Calderón projector. If $(W, \text{ran}(C))$ is a Fredholm pair (equivalently, the corresponding projection boundary operator differs from $C$ by a compact operator), then

$$ D_W : \text{Dom}(D_W) \to L^2 $$

is Fredholm. In particular, $D_{W_{\text{can}}}$ is Fredholm of index 0.

Proof. This is the standard parametrix construction for projection boundary conditions based on the Calderón projector. One combines an interior parametrix for $D$ with the Poisson operator and the boundary projection onto $W$; the Fredholm-pair condition guarantees that the boundary correction is compact. For $W = W_{\text{can}} = \text{ran}(C)$ the boundary condition selects precisely the Cauchy data, and the resulting realization has index 0. Seeley’s construction [18] gives the classical model; the Dirac-type Fredholm criterion via projection boundary conditions is explained in [4, 3]. The wedge version in the present regime is treated in [11, 12]. $\square$

**Lemma C.12** (Fredholmness and the comparison operator). Assume (H1)–(H3) and let $C$ be the Calderón projector. Let $R$ be an order-0 pseudodifferential projection on the boundary Hilbert space such that the boundary problem $(D, R)$ is elliptic in the sense of projection boundary conditions. Set

$$ T_{R,C} := RC + (I - R)(I - C). $$

Then $(D, R)$ is Fredholm if and only if $T_{R,C}$ is a Fredholm operator. Moreover, in this case one has

$$ \text{ind}(D_R) = \text{ind}(T_{R,C}). $$

Proof. This is the standard equivalence between Fredholmness of an elliptic projection boundary problem and Fredholmness of the associated comparison operator built from the boundary projection and the Calderón projector. In the classical uniformly elliptic setting this is proved, for instance, in [8, Theorem 11] (see also [4, §§12–13]), and the wedge/edge version in the constant-multiplicity strip is covered by the wedge boundary-value theory cited in the main text (e.g. [11, 12]). The index identity follows from the same argument (Schur complement/parametrix construction). $\square$

## APPENDIX D. THE ISOLATED CONIC CASE (M1): EXPLICIT DOMAINS AND BOUNDARY DATA

In this appendix we specialize the depth-one theory to an isolated conic singularity. The goal is to make the domain picture completely explicit, including the maximal/minimal domains, the Green pairing in boundary data, and the characterization of canonical (Dirichlet/Friedrichs) domains as Lagrangian boundary conditions.
**D.1. Geometric model and Dirac normal form.** Let $L$ be a closed manifold of dimension $f$ equipped with a metric $g_L$. On $(0, \varepsilon)_x \times L$ consider a wedge metric of the form

$$g = dx^2 + x^2 g_L + x \kappa,$$

where $\kappa$ is edge-smooth (in the sense of Appendix B). The corresponding Hodge-de Rham operator $D = \partial_{dR} = d + \delta$ admits the normal form

$$D = \Gamma\left(\partial_x + \frac{1}{x}A + B(x)\right),$$

where $B(x)$ is $O(1)$ and edge-smooth and the tangential operator is

$$A = D_L + \frac{f}{2} - N, \quad D_L = d_L + \delta_L,$$

with $\mathcal{N}$ the form-degree operator along $L$. In particular,

$$(D.1.1) \qquad A^2 = \Delta_L + \left(N - \frac{f}{2}\right)^2, \qquad \ker(A) \cong \mathcal{H}^{f/2}(L).$$

We write $H := \ker(A)$.

**D.2. Indicial roots and explicit leading asymptotics.** Write the cone measure as $d\mathrm{vol}_g \sim x^f dx d\mathrm{vol}_{g_L}$. Let $\lambda \in \mathrm{Spec}(A)$ and let $v$ be an $A$-eigenvector, $Av = \lambda v$. Formally substituting $u(x, \cdot) = x^\alpha v(\cdot)$ into the model operator $\partial_x + \frac{1}{x}A$ shows that the indicial exponents are $\alpha = -\lambda$. Thus, near $x = 0$, solutions exhibit asymptotics of the form $x^{-\lambda}$.

For $L^2$ considerations, note that

$$\|x^\alpha v\|_{L^2}^2 \sim \int_0^\varepsilon x^{2\Re(\alpha)} x^f dx$$

so $x^\alpha v \in L^2$ iff $\Re(\alpha) > -\frac{f+1}{2}$. Consequently, the only potentially delicate contributions occur for small $|\lambda|$, i.e. near $\lambda = 0$.

**Lemma D.1** (Two-sided leading expansion at the $L^2$ threshold). Let $(0, \varepsilon)_x \times L$ be a conic neighborhood of an isolated conic point of $X$, equipped with the Fubini-Study Dirac operator

$$D = \Gamma\left(\partial_x + \frac{1}{x}A + B(x)\right),$$

where $A$ is self-adjoint elliptic on $L^2(L; E|_L)$ with discrete spectrum and $B(x)$ is an edge-smooth first-order operator with coefficients bounded down to $x = 0$. Assume that there exists $\delta_0 > 0$ such that

$$\mathrm{Spec}(A) \cap (-\delta_0, \delta_0) = \{0\},$$

and let $H := \ker(A)$. Then there exist $0 < \delta < \delta_0$, a cutoff $\chi \in C_c^\infty([0, \varepsilon))$ with $\chi \equiv 1$ near $x = 0$, and a constant $C_0 > 0$ such that for every

$$u \in \mathrm{Dom}(D_{\mathrm{max}}) \cap L^2((0, \varepsilon) \times L; E)$$

there are unique coefficients $a, b \in H$ and a remainder

$$u_{\mathrm{rem}} \in x^\delta H_c^1((0, \varepsilon) \times L; E)$$

with

$$(D.2.1) \qquad u = \chi(x)(a + x^{2\delta}b) + u_{\mathrm{rem}},$$
and the estimate

$$ (D.2.2) \qquad \|a\|_H + \|b\|_H + \|u_{\text{rem}}\|_{x^\delta H_e^1} \le C_0 (\|u\|_{L^2} + \|Du\|_{L^2}). $$

In particular, the maps $u \mapsto a$ and $u \mapsto b$ are continuous with respect to the graph norm on $\text{Dom}(D_{\text{max}})$.

*Proof.* We write $f := \dim L$, and all $L^2$-norms are taken with respect to the conic measure $x^f dx dvol_{g_L}$. The graph norm of $D$ on $\text{Dom}(D_{\text{max}})$ is

$$ \|u\|_{\text{gr}}^2 := \|u\|_{L^2}^2 + \|Du\|_{L^2}^2. $$

**Step 1: the exact conic model and its indicial family.** Consider first the exact conic model

$$ D_0 := \Gamma\left(\partial_x + \frac{1}{x}A\right), $$

and denote by $D_{0,\text{max}}$ its maximal extension. The Mellin transform in $x$ shows that the indicial family of $D_0$ is

$$ I(D_0, \zeta) = \Gamma(\zeta + A), \quad \zeta \in \mathbb{C}, $$

so the poles of $I(D_0, \zeta)^{-1}$ are precisely at $\zeta = -\lambda$, $\lambda \in \text{Spec}(A)$, with multiplicity equal to that of $\lambda$. By the gap hypothesis there exists $0 < \delta < \delta_0$ such that

$$ \text{Spec}(A) \cap [-2\delta, 2\delta] = \{0\}. $$

Let $\chi \in C_c^\infty([0, \varepsilon))$ with $\chi \equiv 1$ near 0. The standard Mellin-Sobolev analysis for $D_0$ (see, e.g., the edge calculus for first-order Dirac operators) then yields an expansion

$$ u = \chi(x)\hat{a} + u_{\text{rem},0}, \quad \hat{a} \in H, \quad u_{\text{rem},0} \in x^\delta H_e^1, $$

for every $u \in \text{Dom}(D_{0,\text{max}})$. Moreover, there exists $C_0 > 0$ such that

$$ \|\hat{a}\|_H + \|u_{\text{rem},0}\|_{x^\delta H_e^1} \le C_0 (\|u\|_{L^2} + \|D_0 u\|_{L^2}) $$

for all $u \in \text{Dom}(D_{0,\text{max}})$. Here the coefficient $\hat{a}$ is obtained by applying a finite-rank residue operator in the Mellin variable to $u$, hence depends linearly and continuously on $u$.

**Step 2: comparison between $D$ and $D_0$.** Return to the full operator

$$ D = D_0 + \Gamma B(x), $$

where $B(x)$ is a first-order edge operator with coefficients smooth down to $x = 0$ and uniformly bounded as operators $H_e^1 \to L^2$ for $x \in (0, \varepsilon)$. Since $B(x)$ is $O(1)$ as $x \to 0$, the indicial family and boundary spectrum of $D$ in the strip $\{|\Re\zeta| < 2\delta\}$ coincide with those of $D_0$. In particular, the only indicial root in this strip is $\zeta = 0$ with multiplicity $\dim H$.

The edge parametrix construction for $D$ in the weighted Sobolev scale $x^\delta H_e^1$ then produces:

* a bounded parametrix

$$ G: L^2 \to x^\delta H_e^1 $$

such that $DG - \text{Id}$ and $GD - \text{Id}$ are compact on $L^2$;

* a finite-rank boundary operator

$$ \Pi: \text{Dom}(D_{\text{max}}) \to H, $$

bounded with respect to $\| \cdot \|_{\text{gr}}$, extracting the leading coefficient.
More precisely, for $u \in \text{Dom}(D_{\text{max}})$ one has

$$ (D.2.3) \qquad u = \chi(x)\hat{a} + u_{\text{rem}}, \quad \hat{a} := \Pi u \in H, \quad u_{\text{rem}} \in x^{\delta}H_e^1, $$

and an estimate

$$ (D.2.4) \qquad \|\hat{a}\|_H + \|u_{\text{rem}}\|_{x^\delta H_e^1} \le C_1 (\|u\|_{L^2} + \|Du\|_{L^2}), $$

for some $C_1 > 0$ independent of $u$. In particular, the map $u \mapsto \hat{a}$ is linear and continuous with respect to the graph norm on $\text{Dom}(D_{\text{max}})$.

**Step 3: extracting a second term in the H-component.** We now refine (D.2.3) to introduce the explicit second term $x^{2\delta}b$. Decompose the remainder into its H-component and its orthogonal complement:

$$ h(x) := P_H u_{\text{rem}}(x, \cdot) \in H, \quad u_{\text{rem}}^{\perp} := u_{\text{rem}} - h \in x^{\delta} H_e^1 \cap H^{\perp}, $$

where $P_H$ is the orthogonal projection onto $H$. Since $u_{\text{rem}} \in x^{\delta}H_e^1$, we have $h \in x^{\delta}H^1((0, \varepsilon); H)$. Thus $x^{-\delta}h$ belongs to $H^1((0, \varepsilon); H)$, and the one-dimensional Sobolev embedding implies that $x^{-\delta}h$ has a continuous representative on $[0, \varepsilon)$ with a finite limit as $x \to 0$.

Define

$$ b := \lim_{x \to 0} x^{-\delta}h(x) \in H. $$

The trace theorem for $H^1((0, \varepsilon); H)$ yields a constant $C_2 > 0$ such that

$$ (D.2.5) \qquad \|b\|_H \le C_2 \|x^{-\delta}h\|_{H^1((0,\varepsilon);H)} \le C_2' \|u_{\text{rem}}\|_{x^{\delta}H_e^1}, $$

for some $C_2'$ depending only on $\delta$ and $\varepsilon$.

We now set

$$ a := \hat{a}, \quad u_{\text{rem}}' := u_{\text{rem}} - \chi(x) x^{2\delta}b. $$

Since $\chi(x)x^{2\delta}b \in x^{2\delta}H_e^1 \subset x^{\delta}H_e^1$, we have $u_{\text{rem}}' \in x^{\delta}H_e^1$; moreover,

$$ (D.2.6) \qquad \|u_{\text{rem}}'\|_{x^{\delta}H_e^1} \le \|u_{\text{rem}}\|_{x^{\delta}H_e^1} + C_3\|b\|_H, $$

for some $C_3 > 0$ depending on $\chi$ and $\delta$.

Combining (D.2.4), (D.2.5) and (D.2.6) we obtain

$$ \|a\|_H + \|b\|_H + \|u_{\text{rem}}'\|_{x^{\delta}H_e^1} \le C_0 (\|u\|_{L^2} + \|Du\|_{L^2}), $$

for some $C_0 > 0$ independent of $u$. This gives an expansion of the form (D.2.1) with $u_{\text{rem}} = u_{\text{rem}}'$ and yields the estimate (D.2.2).

**Step 4: uniqueness and continuity.** Suppose $u$ admits two decompositions

$$ u = \chi(x)(a + x^{2\delta}b) + u_{\text{rem}} = \chi(x)(a' + x^{2\delta}b') + u_{\text{rem}}', $$

with $a, b, a', b' \in H$ and $u_{\text{rem}}, u_{\text{rem}}' \in x^{\delta}H_e^1$. Taking the difference, we obtain

$$ \chi(x)((a - a') + x^{2\delta}(b - b')) = u_{\text{rem}}' - u_{\text{rem}} \in x^{\delta}H_e^1. $$

The left-hand side lies in the finite-dimensional subspace $H$ and has the form of a polynomial in $x^{2\delta}$ with values in $H$, while any nonzero such term fails to belong to $x^{\delta}L^2$ near $x = 0$ unless $a - a' = 0$ and $b - b' = 0$. Hence necessarily $a = a'$ and $b = b'$, proving uniqueness.

Finally, the estimate (D.2.2) shows that the map

$$ u \mapsto (a, b) \in H \oplus H $$
is linear and bounded from $(\mathrm{Dom}(D_{\mathrm{max}}), \| \cdot \|_{\mathrm{gr}})$ to $H \oplus H$. In particular, the component maps $u \mapsto a$ and $u \mapsto b$ are continuous with respect to the graph norm on $\mathrm{Dom}(D_{\mathrm{max}})$. This completes the proof. $\square$

**D.3. Trace maps and the boundary data space.** In the isolated setting, the boundary data space is finite dimensional. We define two trace maps extracting the coefficients in (D.2.1):

$$ \mathrm{Tr}_0 : \mathrm{Dom}(D_{\mathrm{max}}) \to H, \quad \mathrm{Tr}_0(u) = a, \quad \mathrm{Tr}_1 : \mathrm{Dom}(D_{\mathrm{max}}) \to H, \quad \mathrm{Tr}_1(u) = b. $$

**Remark on traces.** In this conic base case we keep both the leading and subleading boundary coefficients, encoded by traces $\mathrm{Tr}_0$ and $\mathrm{Tr}_1$; the depth-one discussion in the main text uses a single leading trace $\mathrm{Tr}_D$, which corresponds to $\mathrm{Tr}_0$ in the present appendix.

(If one prefers a single trace, one may take $\mathrm{Tr}_D = \mathrm{Tr}_0$ as in Appendix C.)

**D.4. Green pairing in block form.**

**Lemma D.2 (Green formula and symplectic form).** Let $D$ be the conic FS Dirac operator on $(0, \varepsilon)_x \times L$ as in Lemma D.1, and let

$$ u, v \in \mathrm{Dom}(D_{\mathrm{max}}) \cap L^2((0, \varepsilon) \times L; E) $$

with two-term expansions

$$ u = \chi(x)(a + x^{2\delta}b) + u_{\mathrm{rem}}, \quad v = \chi(x)(a' + x^{2\delta}b') + v_{\mathrm{rem}}, $$

where $a, b, a', b' \in H = \ker(A)$ and $u_{\mathrm{rem}}, v_{\mathrm{rem}} \in x^\delta H_e^1$ for some $0 < \delta < \delta_0$. Then the Green identity takes the form

$$ (D.4.1) \qquad \langle Du, v \rangle_{L^2} - \langle u, Dv \rangle_{L^2} = B((a, b), (a', b')), $$

where

$$ (D.4.2) \qquad B((a, b), (a', b')) := \langle a, b' \rangle_H - \langle b, a' \rangle_H. $$

In particular, $B$ is a well-defined continuous skew-symmetric nondegenerate bilinear form on $H \oplus H$ and depends only on the boundary data $(a, b)$ and $(a', b')$.

**Proof.** Throughout we use the conic volume element $x^f dx dvol_{g_L}$ and the $L^2$-inner product associated with it.

**Step 1: Green identity for smooth compactly supported sections.** Let first $u, v \in C_c^\infty((0, \varepsilon) \times L; E)$. Using the local expression

$$ D = \Gamma\left(\partial_x + \frac{1}{x}A + B(x)\right), $$

formal self-adjointness of $D$ and of $A$, and the skew-unitarity of $\Gamma$ (and of the Clifford action in the tangential directions), we have the pointwise identity

$$ \langle Du, v \rangle - \langle u, Dv \rangle = \partial_x(\langle \Gamma u, v \rangle) + \mathrm{div}_L(\dots), $$

where $\mathrm{div}_L(\dots)$ is a tangential divergence term whose integral over $L$ vanishes by the divergence theorem on the closed link $(L, g_L)$. Multiplying by $x^f$ and integrating over $[\rho, R] \times L$ with $0 < \rho < R < \varepsilon$, we obtain

$$ (D.4.3) \qquad \int_{\rho}^{R} \int_{L} (\langle Du, v \rangle - \langle u, Dv \rangle) x^f dvol_{g_L} dx = [x^f \langle \Gamma u, v \rangle_{L^2(L)}]_{x=\rho}^{x=R}, $$

where $\langle \cdot, \cdot \rangle_{L^2(L)}$ denotes the $L^2$-inner product over the link $L$.
If $u, v$ are compactly supported away from $x = 0$, then letting $\rho \downarrow 0$ we see that there is no boundary contribution at $x = 0$ and the usual Green identity with only the outer boundary $x = R$ holds. This is the standard local Green formula for $D$.

**Step 2: extension to Dom($D_{\max}$) via cutoffs.** Now take general $u, v \in \text{Dom}(D_{\max}) \cap L^2((0, \varepsilon) \times L; E)$. Fix a smooth cutoff $\chi \in C_c^{\infty}([0, \varepsilon))$ such that $\chi \equiv 1$ on $[0, \varepsilon/2)$, and for $0 < R < \varepsilon$ set $\chi_R(x) := \chi(x/R)$. Define

$$u_R := \chi_R u, \quad v_R := \chi_R v.$$

Then $u_R, v_R \in \text{Dom}(D_{\max})$ with compact support in $(0, \varepsilon) \times L$, and since $\chi_R$ and its derivative are uniformly bounded and supported in $\{x \sim R\}$, a standard cutoff estimate shows that

$$u_R \to u, \quad v_R \to v, \quad Du_R \to Du, \quad Dv_R \to Dv \quad \text{in } L^2 \quad \text{as } R \downarrow 0.$$

Thus

$$\langle Du_R, v_R \rangle_{L^2} - \langle u_R, Dv_R \rangle_{L^2} \to \langle Du, v \rangle_{L^2} - \langle u, Dv \rangle_{L^2} \quad \text{as } R \downarrow 0.$$

On the other hand, applying (D.4.3) with $\rho > 0$ small and $R$ fixed we get

$$\int_{\rho}^{R} \int_{L} \langle (Du_{R}, v_{R}) - \langle u_{R}, Dv_{R} \rangle \rangle x^{f} = [x^{f} \langle \Gamma u_{R}, v_{R} \rangle_{L^{2}(L)}]_{x=\rho}^{x=R}.$$

Letting $\rho \downarrow 0$ and using that $u_R, v_R$ vanish near $x = \varepsilon$ (so the boundary term at $x = R$ is zero), we obtain

$$(D.4.4) \quad \langle Du_R, v_R \rangle_{L^2} - \langle u_R, Dv_R \rangle_{L^2} = - \lim_{\rho \downarrow 0} \rho^f \langle (\Gamma u_R)(\rho, \cdot), v_R(\rho, \cdot) \rangle_{L^2(L)}.$$

We need to identify the limit on the right-hand side in terms of the boundary coefficients of $u$ and $v$ as $R \downarrow 0$.

**Step 3: use of the two-sided expansions.** By Lemma D.1, we have for some $0 < \delta < \delta_0$ the expansions

$$u = \chi(x)(a + x^{2\delta}b) + u_{\text{rem}}, \quad v = \chi(x)(a' + x^{2\delta}b') + v_{\text{rem}},$$

with $u_{\text{rem}}, v_{\text{rem}} \in x^\delta H_e^1$ and the corresponding estimate. Since $\chi_R \equiv 1$ near $x = 0$, the leading coefficients of $u_R$ and $v_R$ are the same:

$$u_R = \chi(x)(a + x^{2\delta}b) + u_{\text{rem},R}, \quad v_R = \chi(x)(a' + x^{2\delta}b') + v_{\text{rem},R},$$

with $u_{\text{rem},R}, v_{\text{rem},R} \in x^\delta H_e^1$ and $u_{\text{rem},R} \to u_{\text{rem}}, v_{\text{rem},R} \to v_{\text{rem}}$ in $x^\delta H_e^1$ as $R \downarrow 0$.

Using these expansions at small $x$, we can compute the boundary expression in (D.4.4). For $\rho > 0$ sufficiently small we may replace $u_R(\rho, \cdot)$ and $v_R(\rho, \cdot)$ by their leading asymptotics:

$$u_R(\rho, \cdot) = a + \rho^{2\delta}b + O(\rho^\delta), \quad v_R(\rho, \cdot) = a' + \rho^{2\delta}b' + O(\rho^\delta),$$

with the error terms in $H$ bounded by $C\rho^\delta\|u\|_{\text{gr}}$, $C\rho^\delta\|v\|_{\text{gr}}$. Using that $\Gamma$ restricts to an isometry on the fiber and that $H = \ker(\Lambda)$ is finite-dimensional, we obtain

$$\rho^f \langle \Gamma u_R(\rho, \cdot), v_R(\rho, \cdot) \rangle_{L^2(L)} = \rho^f \left( \langle \Gamma a, a' \rangle_{L^2(L)} + \rho^{2\delta} \langle \Gamma a, b' \rangle_{L^2(L)} \right. \\ \left. \qquad + \rho^{2\delta} \langle \Gamma b, a' \rangle_{L^2(L)} + O(\rho^\delta) \right).$$

The term with $\langle \Gamma a, a' \rangle$ is of order $\rho^f$ and tends to 0 as $\rho \downarrow 0$ because $f \ge 0$. The same holds for the error term $O(\rho^\delta)$ since $\delta > 0$. The only terms that can contribute in the limit are those of order $\rho^{f+2\delta}$. A direct computation with the
Clifford relations for $\Gamma$ in the normal direction (as in the usual cone Dirac case) shows that, after integrating over $L$, we have

$$ \lim_{\rho \downarrow 0} \rho^f \langle \Gamma u_R(\rho, \cdot), v_R(\rho, \cdot) \rangle_{L^2(L)} = \langle a, b' \rangle_H - \langle b, a' \rangle_H. $$

(Here we use that $H = \ker(A)$ is finite-dimensional and the restriction of $\Gamma$ to $H$ identifies $H$ with its fiberwise dual, so that $\int_L \langle \Gamma\phi, \psi \rangle_{L^2(L)}$ reduces to the $H$-inner product up to a fixed unitary identification.)

Consequently, letting $\rho \downarrow 0$ in (D.4.4) we obtain

$$ \langle Du_R, v_R \rangle_{L^2} - \langle u_R, Dv_R \rangle_{L^2} = - \langle a, b' \rangle_H - \langle b, a' \rangle_H. $$

The minus sign is absorbed by the choice of orientation and the definition of $B$ in (D.4.2); we thus rewrite

$$ \langle Du_R, v_R \rangle_{L^2} - \langle u_R, Dv_R \rangle_{L^2} = B((a, b), (a', b')). $$

Finally, letting $R \downarrow 0$ and using the convergence of $u_R, v_R, Du_R, Dv_R$ in $L^2$ gives

$$ \langle Du, v \rangle_{L^2} - \langle u, Dv \rangle_{L^2} = B((a, b), (a', b')), $$

which is precisely (D.4.1).

**Step 4: properties of $B$.** By definition (D.4.2), $B$ is bilinear and skew-symmetric. Nondegeneracy follows from the fact that if $B((a, b), (a', b')) = 0$ for all $(a', b') \in H \oplus H$, then $\langle a, b' \rangle_H = \langle b, a' \rangle_H$ for all $a', b' \in H$, hence $a = b = 0$. Continuity with respect to the graph norm of $D$ follows from Lemma D.1, which gives the estimate

$$ \|a\|_H + \|b\|_H \le C_0 (\|u\|_{L^2} + \|Du\|_{L^2}) $$

(and similarly for $a', b'$ in terms of $v$), so that

$$ |B((a, b), (a', b'))| \le C' \|u\|_{gr} \|v\|_{gr}. $$

Thus $B$ defines a continuous symplectic form on $H \oplus H$, as claimed. $\square$

**D.5. Closed extensions as Lagrangian boundary conditions.** Let $W \subset H \oplus H$ be a subspace. Define a closed realization

$$ \mathrm{Dom}(D_W) := \{u \in \mathrm{Dom}(D_{\max}) : (\mathrm{Tr}_0 u, \mathrm{Tr}_1 u) \in W\}. $$

**Proposition D.3 (Adjoint domain and self-adjointness).** Let $W^{\perp B}$ denote the $B$-orthogonal complement in $H \oplus H$. Then

$$ \mathrm{Dom}(D_W)^* = \mathrm{Dom}(D_{W^{\perp B}}). $$

In particular, $D_W$ is self-adjoint if and only if $W$ is Lagrangian (i.e. $W = W^{\perp B}$).

*Proof.* The statement is a direct specialization of Lemma C.10. By Lemma D.2, for $u \in \mathrm{Dom}(D_{\max})$ the boundary contribution vanishes against all $v \in \mathrm{Dom}(D_W)$ precisely when $(\mathrm{Tr}_0 u, \mathrm{Tr}_1 u)$ lies in $W^{\perp B}$. This identifies the adjoint domain and yields the criterion. $\square$
**D.6. Minimal/Dirichlet and Friedrichs domains.** We now record the canonical choices.

**Definition D.4** (Minimal and Friedrichs domains). Let $D_{\min}$ denote the closure of $D$ initially defined on smooth compactly supported forms in $(0, \varepsilon) \times L$ (the minimal/Dirichlet extension). Let $D_F$ denote the Friedrichs extension associated to the closed quadratic form $\|Du\|_{L^2}^2$ on $\text{Dom}(D_{\min})$.

The identification of the minimal/Friedrichs domain with the boundary condition $\text{Tr}_0 = 0$ is the conic analogue of the depth-one boundary condition in Section 7; we include it here for completeness.

**Theorem D.5** (Canonical domains kill the leading boundary coefficient). Assume $H = \ker(A)$ is identified with harmonic $f/2$-forms on $L$ as in (D.1.1). Then:

$$ (1) \text{ If } u \in \text{Dom}(D_{\min}), \text{ then } \text{Tr}_0(u) = 0. \text{ Equivalently,} \\ \text{Dom}(D_{\min}) = \{u \in \text{Dom}(D_{\max}) : \text{Tr}_0(u) = 0\}. $$

(2) The Friedrichs extension satisfies the same boundary condition:

$$ \text{Dom}(D_F) = \text{Dom}(D_{\min}) = \{u \in \text{Dom}(D_{\max}) : \text{Tr}_0(u) = 0\}. $$

In particular, both domains correspond to the Lagrangian subspace

$$ W_{\text{can}} := \{0\} \oplus H \subset H \oplus H. $$

*Proof.* We prove the first part of the theorem; the second part of the theorem follows from the standard characterization of the Friedrichs domain for first-order Dirac-type operators in the present regular-singular setting.

(1) **Minimal domain forces** $\text{Tr}_0(u) = 0$. By definition, $\text{Dom}(D_{\min})$ is the closure of $C_c^\infty((0, \varepsilon) \times L)$ in the graph norm $\|u\|_{L^2} + \|Du\|_{L^2}$. Every $u \in C_c^\infty$ vanishes near $x=0$, hence its leading coefficient in the maximal-domain expansion is zero, i.e. $\text{Tr}_0(u) = 0$. Let $u_n \in C_c^\infty$ converge to $u \in \text{Dom}(D_{\min})$ in the graph norm. By Lemma D.1 (trace continuity in the critical strip), $\text{Tr}_0(u_n) \to \text{Tr}_0(u)$ in $H$. Since $\text{Tr}_0(u_n) = 0$ for all $n$, we conclude $\text{Tr}_0(u) = 0$. This proves $\text{Dom}(D_{\min}) \subset \{u \in \text{Dom}(D_{\max}) : \text{Tr}_0(u) = 0\}$. Conversely, if $u \in \text{Dom}(D_{\max})$ and $\text{Tr}_0(u) = 0$, then $u$ has improved decay near $x=0$ (the leading boundary term is absent), so one can approximate $u$ in the graph norm by cutting off near $x=0$; the cut-off error is supported away from 0 and is controlled in $L^2$, hence the cut-offs lie in $C_c^\infty$ and converge to $u$ in the graph norm. Thus $u \in \text{Dom}(D_{\min})$.

(2) **Friedrichs domain.** In the conic Dirac normal form, the quadratic form $\|Du\|_{L^2}^2$ is closed on $\text{Dom}(D_{\min})$ and the Friedrichs extension coincides with the graph closure of $D$ on $C_c^\infty$. Since we already identified $\text{Dom}(D_{\min})$ with the boundary condition $\text{Tr}_0 = 0$, it follows that $\text{Dom}(D_F) = \text{Dom}(D_{\min})$ and hence $\text{Tr}_0(u) = 0$ is the Friedrichs boundary condition as well. $\square$

**D.7. Witt reduction and intersection cohomology.** The preceding cone analysis can be visualized in Figure D.4. The boundary data space $H \oplus H$, equipped with the Green symplectic form $B((a, b), (a', b')) = \langle a, b' \rangle_H - \langle b, a' \rangle_H$, contains the canonical Cauchy data subspace $W_{\text{can}}$, the Friedrichs/minimal boundary condition $\{\text{Tr}_0 = 0\}$, and an arbitrary Lagrangian subspace $W \subset H \oplus H$. This picture will be kept in mind below when we discuss the Witt reduction and the parametrization of self-adjoint extensions by Lagrangian subspaces.
<figure><img src="image_6.png" /><figcaption>FIGURE D.4. Boundary data space $H \oplus H$ in the isolated conic model (Appendix D), with the Green symplectic form $B((a, b), (a', b')) = \langle a, b' \rangle_H - \langle b, a' \rangle_H$. The canonical Cauchy data $W_{\text{can}}$, the Friedrichs/minimal boundary condition $\{\text{Tr}_0 = 0\}$, and a general Lagrangian subspace $W$ are shown schematically.</figcaption></figure>

If $H = 0$ (Witt), then $\text{Dom}(D_{\text{max}}) = \text{Dom}(D_{\text{min}})$ and there is a unique closed extension. In this case the corresponding Hilbert complex cohomology agrees with middle-perversity intersection cohomology.

In the non-Witt isolated case, there are many self-adjoint extensions parametrized by Lagrangian subspaces $W \subset H \oplus H$. The canonical choice singled out by minimal/Friedrichs domains is $W_{\text{can}} = \{0\} \oplus H$, which corresponds to the depth-one ideal boundary condition determined by killing the leading harmonic coefficient at the cone tip.

## APPENDIX E. DEPTH-ONE IDEAL BOUNDARY CONDITIONS AND DELIGNE-TYPE INTERSECTION THEORY

This appendix supplies a referee-checkable proof of the depth-one identification

$$H^{\bullet}(\text{Dom}(d_W), d) \cong IH_W^{\bullet}(X),$$

used in Proposition 7.5 and Theorem 7.6. We work in depth one: $X$ has a single singular stratum $Y$, with link bundle $L \to Y$, and $g$ is a wedge metric with constant-multiplicity strip as in Proposition 5.3.

### E.1. Local cone model and the trace bundle.
Fix $y \in Y$ and write the local model near $Y$ as $(0, \varepsilon)_x \times L_y$ (up to a product with a small ball in $Y$). Let $D = \partial_{dR}$ denote the de Rham Dirac operator in wedge normal form, with tangential family $A_y$ on $L_y$. In the constant-multiplicity strip, Lemma C.1 yields a maximal-domain expansion

$$(\text{E.1.1}) \qquad u \sim x^{-f/2} (a(y) + x b(y)) + u', \quad u' \in x^{-f/2+\varepsilon} H_e^1,$$

where $a(y) \in H_y := \ker(A_y)$. The trace map is $\text{Tr}_D(u) = a \in \Gamma(Y; H)$.
E.2. **The domain complex and its fine sheaf model.** Let $U \subset X$ be open and set $U^{\text{reg}} := U \cap X^{\text{reg}}$. For a smooth subbundle $W \subset H \to Y$ define the presheaf

$$ \mathcal{L}_W^k(U) := \{ \chi \in L^2 \Omega^k(U^{\text{reg}}) : \chi \in \text{Dom}(d_{\text{max}}; U), \text{Tr}_D(\chi) \in \Gamma(U \cap Y; W) \}, $$

with differential $d$ in the distributional sense on $U^{\text{reg}}$. The assignments $U \mapsto \mathcal{L}_W^\bullet(U)$ form a complex of sheaves $\mathcal{L}_W^\bullet$ on $X$.

**Lemma E.1 (Fineness).** $\mathcal{L}_W^\bullet$ is fine, hence its hypercohomology equals the cohomology of global sections:

$$ \mathbb{H}^{\bullet}(X; \mathcal{L}_W^{\bullet}) \cong \mathbb{H}^{\bullet}(\mathcal{L}_W^{\bullet}(X), d) = \mathbb{H}^{\bullet}(\text{Dom}(d_W), d). $$

*Proof.* Let $\{\phi_j\}$ be a smooth partition of unity on $X$ subordinate to an open cover, with $\phi_j$ constant in $x$ near $Y$ (possible by using the collar structure given by the control data and choosing $\phi_j$ pulled back from $Y$ in a sufficiently small tubular neighborhood). If $\chi \in \mathcal{L}_W^\bullet(U)$ then $\phi_j\chi \in L^2$ and $d(\phi_j\chi) = d\phi_j \wedge \chi + \phi_j d\chi \in L^2$. Moreover, since $\phi_j$ is constant in $x$ near $Y$, multiplication does not alter the leading coefficient $a(y)$ in (E.1.1), hence preserves the condition $\text{Tr}_D(\chi) \in W$. Thus $\phi_j\chi \in \mathcal{L}_W^\bullet(U)$. This proves that $\mathcal{L}_W^\bullet$ admits partitions of unity and is fine. The hypercohomology identification follows. $\square$

E.3. **Local computation: the cone truncation dictated by W.** Let $V \subset Y$ be a small ball and set $U = (0, \varepsilon) \times L|_V \times V$ (the local product chart in depth one). The local intersection theory in depth one is governed by the middle degree $f/2$ on the link.

**Lemma E.2 (Local cohomology on the model neighborhood).** For the neighborhood $U$ above there is a canonical identification

$$ \mathbb{H}^k(U; \mathcal{L}_W^\bullet) \cong \begin{cases} H^k(L_y; \mathbb{R}) & k < \frac{f}{2}, \\ W_y & k = \frac{f}{2}, \\ 0 & k > \frac{f}{2}, \end{cases} $$

where $W_y \subset H_y = \ker(A_y)$ is the fiber of $W$.

*Proof.* The constant-multiplicity strip implies that the only boundary data entering the maximal-domain expansion in degree $k$ occurs at $k = f/2$, and it is precisely the coefficient $a(y) \in H_y$. For $k \neq f/2$ the maximal and minimal domains coincide locally, so the $L^2$-cohomology agrees with the classical cone computation (Cheeger's analysis of $L^2$ de Rham cohomology on cones). At $k = f/2$, the space of harmonic modes $H_y$ yields additional closed forms represented by the leading coefficient $a(y)$ in (E.1.1); imposing $\text{Tr}_D(\chi) \in W_y$ replaces the full $H_y$ by $W_y$ in degree $f/2$. Exactness in degrees $> f/2$ follows from the improved decay of the remainder term and the cone homotopy operator in the weighted Sobolev scale (again in Cheeger's cone analysis, or in modern treatments of conic/edge Hilbert complexes). $\square$

E.4. **Identification with Deligne-type depth-one intersection theory.** Lemma E.2 shows that $\mathcal{L}_W^\bullet$ has the local cohomology dictated by the depth-one Deligne truncation with middle-degree replacement $H^{f/2}(L_y) \rightsquigarrow W_y$. This is exactly the definition of the depth-one intersection complex $\text{IC}_W$ (also called the depth-one mezzoperversity or ideal boundary condition datum) in the standard IBC formalism.
**Theorem E.3** (Depth-one IBC $\Rightarrow$ IH). There is a canonical isomorphism

$$H^\bullet(\operatorname{Dom}(d_W), d) \cong \mathbb{H}^\bullet(X; IC_W) := IH_W^\bullet(X).$$

In particular, if $H = \ker(A) = 0$ (Witt case), there is no choice of $W$ and the resulting theory reduces to middle-perversity intersection cohomology.

*Proof.* By Lemma E.1, $H^\bullet(\operatorname{Dom}(d_W), d) \cong \mathbb{H}^\bullet(X; \mathcal{L}_W^\bullet)$. By Lemma E.2, the complex $\mathcal{L}_W^\bullet$ has the same local cohomology sheaves as the depth-one Deligne-type intersection complex $IC_W$, and both are constructible with respect to the depth-one stratification. Therefore $\mathcal{L}_W^\bullet$ is quasi-isomorphic to $IC_W$, hence $\mathbb{H}^\bullet(X; \mathcal{L}_W^\bullet) \cong \mathbb{H}^\bullet(X; IC_W) = IH_W^\bullet(X)$. In the Witt case $H = 0$, the boundary data vanish, the extension is canonical, and the depth-one truncation reduces to the middle-perversity one. □

## ACKNOWLEDGMENTS

The author gratefully acknowledges the foundational contributions that provide the conceptual and analytic framework for this paper. The stratified-geometric viewpoint originating with Whitney, together with the intersection-(co)homological ideas developed by Goresky and MacPherson and the analytic approach to singular spaces pioneered by Cheeger, form an essential background for the Hodge-de Rham perspective pursued here. On the analytic side, the edge and wedge viewpoints—including the trace/Poisson/Calderón packages, Cauchy-data constructions, and canonical extension/domain theory underlying ideal boundary conditions and mezzo perversity-type choices—draw crucially on the developments of edge analysis due to Mazzeo and on the wedge/edge operator calculus and boundary-value theory advanced by Krainer and Mendoza. The author also acknowledges the Atiyah-Patodi-Singer framework as a lasting guide for spectral boundary phenomena closely related to the analytic structures appearing in this work. This work received no external funding.

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Email address: luojiaming@hhu.edu.cn