ITERATED CALDER ÓN PACKAGES AND
CANONICAL MEZZOPERVERSITIES FOR
FUBINI-STUDY METRICS ON PROJECTIVE
VARIETIES OF ARBITRARY STRATIFICATION
DEPTH

Jiaming Luo

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Jiaming Luo. ITERATED CALDER ÓN PACKAGES AND CANONICAL MEZZOPERVERSITIES FOR FUBINI-STUDY METRICS ON PROJECTIVE VARIETIES OF ARBITRARY STRATIFICATION DEPTH. 2026. hal-05510997

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# ITERATED CALDERÓN PACKAGES AND CANONICAL MEZZOPERVERSITIES
## FOR FUBINI-STUDY METRICS ON PROJECTIVE VARIETIES OF ARBITRARY STRATIFICATION DEPTH

JIAMING LUO

ABSTRACT. Let $X \subset \mathbb{P}^N$ be a complex projective variety equipped with a Thom-Mather stratification of arbitrary depth, and let $g_{\text{FS}}$ be the induced Fubini-Study metric on $X^{\text{reg}}$. By iterating a parameter-dependent Calderón package on the stratified link fibrations, we construct canonically defined Cauchy-data bundles $\{W_{i,\text{can}}\}$—equivalently, a canonical mezzoperversity $W_{\text{can}}$ in the sense of ideal boundary conditions—and a canonical self-adjoint Fredholm realization $D_{\text{can}}$ of the de Rham (Gauss-Bonnet) operator $\partial_{\text{dR}} = d + \delta$ on $(X^{\text{reg}}, g_{\text{FS}})$. The resulting canonical de Rham Hilbert complex $(\text{Dom}(d_{\text{can}}), d_{\text{can}})$ computes intersection cohomology in the form $H^*(\text{Dom}(d_{\text{can}}), d_{\text{can}}) \cong \text{IH}^*_{W_{\text{can}}}(X)$, canonically; in the iterated Witt case this specializes to the middle-perversity theory $\text{IH}^m_{W_{\text{can}}}(X)$. The main new analytic input is an iterated stability theorem for domain-dependent tangential families: at each stratum, the tangential operator acts on a stratified link and its self-adjoint realization depends on the canonical boundary conditions already selected on deeper strata. We prove constant-rank kernel bundles and uniform spectral gaps for these families, yielding a constant-multiplicity critical strip and enabling the recursive Calderón construction.

## CONTENTS

<table><tr><td>1. Introduction</td><td>3</td></tr><tr><td>1.1. Motivation: Hodge-de Rham theory on singular projective varieties</td><td>3</td></tr><tr><td>1.2. What is new here</td><td>4</td></tr><tr><td>1.3. Main results</td><td>4</td></tr><tr><td>1.4. Logical dependency roadmap</td><td>5</td></tr><tr><td>1.5. Main theorem dependency checklist</td><td>6</td></tr><tr><td>1.6. Guide to the paper</td><td>6</td></tr><tr><td>Notation and conventions</td><td>6</td></tr><tr><td>2. Stratified setup and iterated control data</td><td>7</td></tr><tr><td>2.1. Thom-Mather stratification and depth</td><td>7</td></tr><tr><td>2.2. Iterated control data, cone bundles, and link fibrations</td><td>8</td></tr><tr><td>2.3. Iterated cone charts along incidence chains</td><td>10</td></tr><tr><td>2.4. Resolution viewpoint (manifold with corners model)</td><td>11</td></tr><tr><td>2.5. Uniformity on stratum components</td><td>11</td></tr><tr><td>2.6. Two consequences used repeatedly</td><td>12</td></tr><tr><td>3. Iterated Fubini-Study wedge expansions</td><td>13</td></tr><tr><td>3.1. Iterated edge vector fields and iterated edge-smooth tensors</td><td>13</td></tr><tr><td>3.2. FS wedge expansion (iterated)</td><td>13</td></tr><tr><td>4. Iterated Dirac normal forms with deeper canonical domains</td><td>13</td></tr><tr><td>4.1. Geometric Gauss-Bonnet operator as a wedge Dirac</td><td>14</td></tr><tr><td>4.2. Normal form and rigidity of the $x_i^{-1}$ coefficient</td><td>14</td></tr></table>

<footnote n="2020">2020 Mathematics Subject Classification. Primary 58J32, 58J10; Secondary 35S15, 55N33, 58J20.</footnote>

<footnote>Key words and phrases. iterated edge metrics, wedge Dirac operators, Calderón projector, ideal boundary conditions, mezzoperversity, intersection cohomology.</footnote>
<table>
  <tbody>
    <tr>
      <td>5. Iterated critical strips and parameter-dependent Calderón packages</td>
      <td>15</td>
    </tr>
    <tr>
      <td>5.1. Indicial correspondence and the role of the tangential spectrum</td>
      <td>15</td>
    </tr>
    <tr>
      <td>5.2. Inductive hypotheses at stratum Y<sub>i</sub></td>
      <td>15</td>
    </tr>
    <tr>
      <td>5.3. Trace bundles, Poisson operators, and Calderón projectors</td>
      <td>16</td>
    </tr>
    <tr>
      <td>6. The domain-dependent stability theorem</td>
      <td>16</td>
    </tr>
    <tr>
      <td>6.1. Tangential families arising from the iterated normal form</td>
      <td>16</td>
    </tr>
    <tr>
      <td>6.2. Inductive package on deeper strata</td>
      <td>16</td>
    </tr>
    <tr>
      <td>6.3. Kernel bundles and uniform gaps for domain-dependent families</td>
      <td>17</td>
    </tr>
    <tr>
      <td>7. Main inductive theorem and global canonical domain</td>
      <td>18</td>
    </tr>
    <tr>
      <td>7.1. Construction of the canonical mezzoperversity</td>
      <td>18</td>
    </tr>
    <tr>
      <td>7.2. Global canonical domain and self-adjoint Fredholmness</td>
      <td>18</td>
    </tr>
    <tr>
      <td>7.3. Topological identification with iterated IBC intersection cohomology</td>
      <td>20</td>
    </tr>
    <tr>
      <td>8. Depth-two as a worked case and comparison with the conditional schema</td>
      <td>20</td>
    </tr>
    <tr>
      <td>8.1. Depth-two geometry in iterated conic charts</td>
      <td>21</td>
    </tr>
    <tr>
      <td>8.2. The inner step: the canonical Cauchy data bundle on the deepest link</td>
      <td>21</td>
    </tr>
    <tr>
      <td>8.3. The outer step: stability of the tangential family with deeper IBC</td>
      <td>22</td>
    </tr>
    <tr>
      <td>8.4. Unconditional depth-two realization and IH identification</td>
      <td>22</td>
    </tr>
    <tr>
      <td>8.5. Concrete projective examples</td>
      <td>23</td>
    </tr>
    <tr>
      <td>9. Outlook toward M4 (middle-perversity characterization)</td>
      <td>25</td>
    </tr>
    <tr>
      <td>9.1. The M4 target as a precise comparison problem</td>
      <td>25</td>
    </tr>
    <tr>
      <td>9.2. A local-to-global reduction usable by referees</td>
      <td>26</td>
    </tr>
    <tr>
      <td>9.3. A polarized Hodge package as a concrete source of W<sub>i mid</sub></td>
      <td>26</td>
    </tr>
    <tr>
      <td>9.4. A lemma chain implementing (M4.1)-(M4.3) in the projective Fubini-Study setting</td>
      <td>26</td>
    </tr>
    <tr>
      <td>Appendix A. Two analytic lemmas for domain-dependent self-adjoint families</td>
      <td>29</td>
    </tr>
    <tr>
      <td>A.1. Graph projections and norm-resolvent continuity</td>
      <td>29</td>
    </tr>
    <tr>
      <td>A.2. Uniform spectral gaps over compact parameter spaces</td>
      <td>30</td>
    </tr>
    <tr>
      <td>A.3. Riesz projections and differentiability of spectral data</td>
      <td>31</td>
    </tr>
    <tr>
      <td>Appendix B. Iterated Fubini-Study wedge expansion: adapted coordinates and tensor estimates</td>
      <td>33</td>
    </tr>
    <tr>
      <td>B.1. Iterated edge vector fields and iterated smoothness</td>
      <td>33</td>
    </tr>
    <tr>
      <td>B.2. Quantitative Whitney-Thom angle control</td>
      <td>34</td>
    </tr>
    <tr>
      <td>B.3. Vertical scaling and the wedge leading term</td>
      <td>34</td>
    </tr>
    <tr>
      <td>B.4. Assembly and O(x<sub>i</sub>) remainder</td>
      <td>35</td>
    </tr>
    <tr>
      <td>Appendix C. Iterated wedge Dirac normal forms with deeper canonical domains</td>
      <td>37</td>
    </tr>
    <tr>
      <td>C.1. Product-type wedge metrics and the Gauss-Bonnet operator</td>
      <td>37</td>
    </tr>
    <tr>
      <td>C.2. Incorporating deeper canonical domains on stratified links</td>
      <td>38</td>
    </tr>
    <tr>
      <td>C.3. Perturbation by iterated edge-smooth metric remainders and rigidity</td>
      <td>38</td>
    </tr>
    <tr>
      <td>C.4. Normal operators, indicial families, and smooth spectral data</td>
      <td>39</td>
    </tr>
    <tr>
      <td>C.5. Explicit parametrices for the off-zero normal family and operator-norm differentiability</td>
      <td>41</td>
    </tr>
    <tr>
      <td>C.6. Polyhomogeneous kernel structure and indicial index sets</td>
      <td>42</td>
    </tr>
    <tr>
      <td>C.7. Compatibility under changes of adapted trivializations</td>
      <td>44</td>
    </tr>
    <tr>
      <td>Appendix D. Parameter-dependent trace bundles and Calderón projectors with varying deeper IBC</td>
      <td>44</td>
    </tr>
    <tr>
      <td>D.1. Smooth trace bundles for domain-dependent tangential families</td>
      <td>44</td>
    </tr>
    <tr>
      <td>D.2. Maximal-domain asymptotics and trace maps in the constant strip</td>
      <td>45</td>
    </tr>
    <tr>
      <td>D.3. Poisson operators and Calderón projectors</td>
      <td>46</td>
    </tr>
    <tr>
      <td>D.4. Mapping properties in the iterated-edge calculus</td>
      <td>46</td>
    </tr>
    <tr>
      <td>D.5. Green pairing and Lagrangian Cauchy data</td>
      <td>47</td>
    </tr>
    <tr>
      <td>D.6. Self-adjointness and Fredholmness; dependence on the deeper IBC</td>
      <td>48</td>
    </tr>
    <tr>
      <td>Appendix E. Domain-dependent stability for tangential families</td>
      <td>49</td>
    </tr>
  </tbody>
</table>
<table>
  <tr>
    <td>E.1. Purpose and standing notation</td>
    <td>49</td>
  </tr>
  <tr>
    <td>E.2. Encoding varying domains by boundary projections</td>
    <td>49</td>
  </tr>
  <tr>
    <td>E.3. A resolvent continuity lemma for projection boundary conditions</td>
    <td>50</td>
  </tr>
  <tr>
    <td>E.4. Constant rank of the kernel bundle</td>
    <td>52</td>
  </tr>
  <tr>
    <td>E.5. Uniform spectral gap at zero</td>
    <td>52</td>
  </tr>
  <tr>
    <td>E.6. Constant-multiplicity strip and the trace bundle</td>
    <td>52</td>
  </tr>
  <tr>
    <td>E.7. Application to the depth-two outer family A<sup>W<sub>2,can</sub></sup><sub>1,y</sub></td>
    <td>53</td>
  </tr>
  <tr>
    <td>E.8. Manifolds with corners and C<sup>k</sup> regularity</td>
    <td>53</td>
  </tr>
  <tr>
    <td>Appendix F. Iterated ideal boundary conditions and Deligne-type intersection complexes</td>
    <td>53</td>
  </tr>
  <tr>
    <td>F.1. The analytic IBC de Rham complex as a sheaf</td>
    <td>54</td>
  </tr>
  <tr>
    <td>F.2. Distinguished neighborhoods and the cone computation</td>
    <td>54</td>
  </tr>
  <tr>
    <td>F.3. Identification with a Deligne-type intersection complex</td>
    <td>55</td>
  </tr>
  <tr>
    <td>Acknowledgements</td>
    <td>56</td>
  </tr>
  <tr>
    <td>Statements and Declarations</td>
    <td>56</td>
  </tr>
  <tr>
    <td>References</td>
    <td>56</td>
  </tr>
</table>

# 1. INTRODUCTION

## 1.1. Motivation: Hodge-de Rham theory on singular projective varieties.
Let $X$ be a complex projective variety, possibly singular, and write $X^{\text{reg}}$ for its smooth locus. The restriction of a Fubini-Study metric from an ambient projective space endows $X^{\text{reg}}$ with a natural (incomplete) Riemannian metric whose degeneracy is governed by the singular strata of $X$. From the analytic viewpoint, the basic object is the de Rham operator

$$ \partial_{dR} = d + \delta $$

acting on differential forms on $X^{\text{reg}}$ with $L^2$-based domains. On a smooth compact manifold there is a canonical self-adjoint realization of $\partial_{dR}$ and of the Hodge Laplacian $\Delta_{dR}$, and Hodge theory identifies harmonic representatives with de Rham cohomology. On a stratified space, this picture depends on a choice of *ideal boundary conditions* at the singular locus. Cheeger’s foundational analysis for conic singularities and its subsequent extensions show that: (i) different closed extensions can lead to genuinely different cohomology theories, and (ii) selecting *canonical* self-adjoint extensions is subtle, especially in the non-Witt regime [12, 13]. A robust global framework for iterated edge metrics, ideal boundary conditions, and the resulting $L^2$ Hodge theory on stratified spaces was developed by Albin–Leichtnam–Mazzeo–Piazza [1].

The aim of this paper is to place *Fubini-Study metrics on projective varieties of arbitrary depth* into this analytic framework *in a fully projective, unconditional manner*, and to construct from the Fubini-Study geometry a canonical analytic package: an iterated Dirac normal form, an iterated trace/Calderón package for varying tangential data, and (as a consequence) canonical mezzoperversities and domains for $\partial_{dR}$ and $\Delta_{dR}$.

**Canonicity and naturality.** The geometric input throughout is the projective embedding $X \subset \mathbb{CP}^N$ together with a fixed Thom-Mather stratification (and the associated iterated control data of Section 2) and the induced Fubini-Study metric on $X^{\text{reg}}$. The constructions in this paper are *canonical* in the sense that they are produced by an explicit inductive rule from this input and are independent of auxiliary analytic choices (local trivializations, cutoffs, partitions of unity, and parametrix choices): different auxiliary choices yield canonically isomorphic trace/Calderón data and hence the same global domains. We emphasize that a different projective polarization/embedding may change the induced Fubini-Study geometry and therefore the resulting canonical package; systematic comparison is deferred to Section 9.
1.2. **What is new here.** Two difficulties are specific to the projective setting in higher depth.

(A) Geometry-to-analysis entry. To invoke the iterated edge calculus one needs a *precise wedge expansion* of the metric in adapted coordinates and compatibility with control data [18]. For general iterated edge metrics this is part of the analytic input; here we prove it directly from projective geometry. Our first main result (Theorem 3.1) gives an iterated wedge expansion for the Fubini-Study metric with an edge-smooth remainder and identifies the model operators appearing at each depth.

(B) Domain dependence along tangential families. In the inductive construction of self-adjoint domains, the data at a given stratum include: (i) a family of tangential operators along the link fibration and (ii) choices of deeper ideal boundary conditions (equivalently, mezzoperversities) on the links. In depth $\ge 2$ the deeper choices vary with the base point and influence the tangential family. Controlling the *stability* of Cauchy data spaces and Calderón projectors under such domain variation is the analytic bottleneck for an unconditional induction. We isolate and solve this problem by proving a domain-dependent stability theorem for tangential families (Theorem 6.3), which is the key new analytic input enabling one-step iteration beyond depth one. In the general iterated-edge setting such stability is typically an additional analytic hypothesis; here it is derived from the projective wedge structure together with the deeper canonical package on the links.

1.3. Main results. We now summarize the main theorems, stated precisely in the body of the paper.

**Theorem A (Canonical Hodge-de Rham package, informal).** Let $X \subset \mathbb{CP}^N$ be a complex projective variety with a fixed Thom-Mather stratification. Then the induced Fubini-Study metric on $X^{\text{reg}}$ is an iterated wedge metric with a compatible wedge expansion Theorem 3.1. There is an inductively defined canonical mezzoperversity $W_{\text{can}}$ on the links and an associated global closed de Rham domain $\text{Dom}(d_{\text{can}}) \subset L^2\Omega^\bullet(X^{\text{reg}})$ such that the Hodge-de Rham operator $D_{\text{can}} = d_{\text{can}} + \delta_{\text{can}}$ is self-adjoint and Fredholm Theorem 7.6, and the resulting Hilbert complex computes intersection cohomology:

$$ H^\bullet(\text{Dom}(d_{\text{can}}), d_{\text{can}}) \cong \text{IH}^\bullet_{W_{\text{can}}}(X) \quad \text{canonically.} $$

In the iterated Witt case (no boundary data at any depth), this reduces to middle-perversity intersection cohomology Theorem 7.7.

**Theorem B (Domain-dependent stability, informal).** For the tangential families determined by the iterated Dirac normal form, the deeper canonical domains on the links vary with the base point. Nevertheless, along each stratum the corresponding Cauchy data spaces enjoy a uniform stability: the kernels form a smooth finite-rank bundle and there is a uniform spectral gap at 0, so that the boundary spectrum in the relevant weight strip has constant multiplicity. This is proved in Theorem 6.3 and is the key analytic input that makes the inductive construction unconditional in higher depth.

* **Iterated Fubini-Study wedge structure.** We prove that the restriction of a projective Fubini-Study metric to $X^{\text{reg}}$ is an iterated incomplete edge metric in the sense required by the edge calculus, with an explicit expansion compatible with control data (Theorem 3.1).

* **Iterated Dirac normal form.** In adapted coordinates, $\partial_{dR}$ admits an iterated Dirac-type normal form with a geometrically determined tangential family and controlled lower-order terms (Theorem 4.1; see also Section C).

* **Parameterized trace/Calderón package.** For the tangential families arising from Fubini-Study geometry and for *varying deeper domains*, we construct the corresponding trace bundles, Cauchy data spaces, and Calderón projectors (Theorem 5.1). This builds on edge operator theory [19, 20] and wedge boundary value problems [17], and generalizes the classical smooth-boundary Calderón projector technology [22, 7] to the iterated setting used here.
* **Canonical domains and mezzoperversities.** Combining the above with the inductive package of Theorem 6.1, we obtain canonical self-adjoint Fredholm realizations of $\partial_{dR}$ and $\Delta_{dR}$ and identify the resulting $L^2$ Hodge cohomology with a canonical intersection-cohomology-type invariant (Theorems 7.6 and 7.7).

* **Depth-two as a worked case.** To make the induction transparent, we treat depth two as a complete worked example, showing how the stability theorem closes the remaining gap and yields an unconditional depth-two statement (Theorem 8.3; see Section 8).

### 1.4. Logical dependency roadmap.
For ease of verification, we record the *non-circular* dependency chain among the main analytic ingredients.

**R1. Projective geometry $\Rightarrow$ wedge structure.** The iterated wedge expansion of the induced Fubini-Study metric is proved in Theorem 3.1, with quantitative coordinate/tensor estimates deferred to Appendix B.

**R2. Wedge structure $\Rightarrow$ Dirac normal form.** The iterated Dirac-type normal form for $\partial_{dR}$ is established in Theorem 4.1, using Theorem 3.1; the precise bookkeeping of the $x_i^{-1}$-coefficient and the deeper-domain compatibility are given in Appendix C.

**R3. Domain-dependent tangential stability.** For the realized tangential families $y \mapsto A_{i,y}^{W>i}$ (whose domains depend on deeper IBC), Theorem 6.3 proves constant-rank kernel bundles and a uniform spectral gap at 0. The abstract functional-analytic mechanism (encoding varying domains by boundary projections and obtaining norm-resolvent continuity and uniform gaps on compact bases) is developed in Appendix E.

**R4. Constant strip $\Rightarrow$ trace/Calderón package.** Given the constant-multiplicity strip supplied by Theorem 6.3, the parameter-dependent iterated trace/Poisson/Calderón construction is carried out in Theorem 5.1, with full details in Appendix D.

**R5. Global domain and Fredholmness.** The canonical global realization is defined in Definition 7.1. Self-adjointness follows from the Lagrangian property in Theorem 5.1, while Fredholmness uses local parametrices from Appendix D together with the gluing criterion (Proposition 7.5) in Section 7.

**R6. Sheaf-theoretic closure.** The identification of the resulting de Rham Hilbert complex with the Deligne-type intersection complex is proved in Appendix F, yielding Theorem 7.7.

*No circularity.* The induction starts at the deepest stratum, where the deepest link is smooth (depth 0) and the Calderón package is classical; each upward step uses only data already constructed on strictly deeper strata.
1.5. Main theorem dependency checklist. For referee convenience, Table 1 summarizes the logical dependencies of the main results. The third column records only items proved earlier in this paper; the last column lists standard external inputs when they are used.

<table><caption>TABLE 1. Dependency checklist for the main results.</caption><thead><tr><th>Result</th><th>Where proved</th><th>Depends on (within this paper)</th><th>External standard inputs</th></tr></thead><tbody><tr><td colspan="4">(I) Geometry and local analytic models (Sections 2–4)</td></tr><tr><td>Theorem 3.1 (iterated FS-wedge expansion)</td><td>Section 3 and Appendix B</td><td>Section 2; Proposition 2.3; — Lemma 2.5.</td><td>—</td></tr><tr><td>Theorem 4.1 (iterated wedge Dirac normal form)</td><td>Section 4 and Appendix C</td><td>Theorem 3.1; Proposition C.4; Definition C.5.</td><td>—</td></tr><tr><td>Theorem 6.3 (domain-dependent tangential stability)</td><td>Section 6 and Appendix E-E.8</td><td>Appendix A; trace/IBC framework from Appendix D; normal/indicial framework of Section 4.</td><td>—</td></tr><tr><td colspan="4">(II) Iterated Calderón package (Section 5)</td></tr><tr><td>Theorem 5.1 (iterated Calderón package)</td><td>Section 5 and Appendix D</td><td>Theorem 6.3; Poisson/Calderón constructions in Appendix D; trace bundle functoriality in Appendix D.</td><td>Iterated edge/wedge parametrix technology [19, 21].</td></tr><tr><td colspan="4">(III) Global canonical realization and Fredholm package (Section 7)</td></tr><tr><td>Definition 7.1 and Proposition 7.2 (global canonical domain)</td><td>Section 7 ("Global canonical domain")</td><td>Theorem 5.1; the cutoff/partition-of-unity stability proved in Proposition 7.2.</td><td>—</td></tr><tr><td>Proposition 7.5 and Theorem 7.6 (global self-adjoint Fredholmness)</td><td>Section 7 ("Self-adjoint Fredholmness")</td><td>Definition 7.1; Proposition 7.2; local parametrics from Appendix D and Appendix C.</td><td>Fully elliptic Fredholm criterion/parametrix in edge calculi [19, 21].</td></tr><tr><td colspan="4">(IV) Sheaf-theoretic closure and identification with intersection cohomology (Appendix F–F.3)</td></tr><tr><td>Theorem 7.7 (identification with IH<sub>W<sub>can</sub></sub>(X))</td><td>Section 7 and Appendix F-F.3</td><td>Theorem 7.6; construction and local computation of L<sub>W<sub>can</sub></sub> in Appendix F-F.2.</td><td>Deligne package/uniqueness for intersection complexes [14, 6, 8].</td></tr></tbody></table>

1.6. Guide to the paper. Section 3 develops the projective geometric input and proves the iterated wedge expansion. Section 4 establishes the iterated Dirac normal form. Section 5 constructs the parameterized trace/Calderón package. Section 6 proves the key stability statement for domain-dependent tangential families. The global canonical domain construction and its cohomological consequences are proved in Section 7. Finally, Section 8 presents the depth-two case as a detailed model for the general inductive mechanism. The appendices collect analytic lemmas, the detailed wedge expansions, normal-form computations, and the domain-dependent stability arguments used in the main text.

NOTATION AND CONVENTIONS

We collect here a small list of frequently used symbols and standing conventions. Precise definitions (in particular the Thom-Mather control data and the depth filtration) are given in Section 2; the purpose of this list is to reduce back-and-forth when reading the inductive arguments.
## Underlying space and metric.

$X \subset \mathbb{CP}^N$: A complex projective variety of pure complex dimension $n$.

$X^{\text{reg}}$: The smooth locus of $X$.

$g_{\text{FS}}$: The induced Fubini-Study metric on $X^{\text{reg}}$; near each stratum it has an iterated wedge expansion with respect to the fixed control data.

## Stratification and control data.

$S_\alpha$: A connected stratum in the fixed Thom-Mather stratification $X = \bigsqcup_{\alpha \in \mathcal{A}} S_\alpha$, with incidence relation $S_\beta \prec S_\alpha$.

$Y_i$: A closed union of strata in the depth filtration (hence compact, since $X$ is projective).

$(T_i, \rho_i, x_i)$: Control data for $Y_i$: a tubular neighborhood $T_i$, a retraction $\rho_i: T_i \to Y_i$, and a radial function $x_i: T_i \to [0, \infty)$.

$L_{i,y}$: The stratified link of $Y_i$ at $y \in Y_i$ (fiber of the link fibration); it carries an induced iterated wedge structure.

## Operators, domains, and function spaces.

$d, \delta$: Exterior derivative and its formal adjoint on $(X^{\text{reg}}, g_{\text{FS}})$.

$\partial_{\text{dR}} = d + \delta$: The de Rham (Gauss-Bonnet) operator.

$\text{Dom}(\cdot)$: Domain of an unbounded operator (as in functional analysis).

$L^2$: The $L^2$ space with respect to $g_{\text{FS}}$ (and the relevant bundle metric).

$H_{\text{ie}}^k$: Iterated-edge Sobolev spaces of order $k$.

## Iterated boundary conditions and intersection cohomology.

$H_i$: The trace bundle at depth $i$ (typically $H_i = \ker(A_{i,y}^{W>i})$ fiberwise).

$\text{Tr}_{D,i}$: The depth-$i$ trace map for $\partial_{\text{dR}}$ (or for the corresponding model operator).

$C_i$: Calderón projector for the depth-$i$ boundary problem; its range is the Cauchy data space.

$W_{i,\text{can}}$: The canonical Cauchy data subbundle at depth $i$ (range of $C_i$).

$W_{\text{can}} = (W_{1,\text{can}}, \dots, W_{d,\text{can}})$: The canonical mezzoperversity/IBC datum.

$\text{IC}_W$, $\text{IH}_W^\bullet(X)$: Deligne-type intersection complex and its hypercohomology for a given mezzoperversity $W$; $\bar{m}$ denotes middle perversity.

Unless otherwise specified, all constants are understood to be uniform on compact strata bases (after passing to finite subcovers), and “smooth family” means smooth dependence on the base in the sense of the fixed stratified control data.

# 2. STRATIFIED SETUP AND ITERATED CONTROL DATA

This section fixes the stratified-geometric background and notation used throughout, and it records the structural statements we will use later when constructing iterated trace bundles, tangential families, and Calderón packages. Since $X \subset \mathbb{CP}^N$ is projective, it is compact in the analytic topology; in particular each closed stratum-union $Y_i$ introduced below is compact. We will repeatedly exploit this compactness to upgrade pointwise (in $y$) statements for smoothly varying families into uniform ones (finite subcovers, uniform radii, and uniform constants).

## 2.1. Thom-Mather stratification and depth.
Let $X \subset \mathbb{CP}^N$ be a complex projective variety of pure complex dimension $n$ and write $X^{\text{reg}}$ for its smooth locus. We fix once and for all a Thom-Mather (a.k.a. smoothly stratified) structure in the sense of [18]. Concretely, this means that $X$ is written as a locally finite disjoint union of connected smooth manifolds (the *strata*)

$$X = \bigsqcup_{\alpha \in \mathcal{A}} S_{\alpha},$$

satisfying the *frontier condition*: if $S_\beta \cap \bar{S}_\alpha \neq \emptyset$, then $S_\beta \subset \bar{S}_\alpha$. We write $S_\beta \prec S_\alpha$ for the corresponding strict incidence relation ($S_\beta \subset \bar{S}_\alpha$ and $S_\beta \neq S_\alpha$).
For later induction it is convenient to package the incidence poset by a depth filtration. We assume the stratification has finite depth $d$ and choose a nested filtration by closed subsets

$$Y_d \subset Y_{d-1} \subset \cdots \subset Y_1 \subset X, \quad X^{\text{reg}} := X \setminus \bigcup_{i=1}^{d} Y_i,$$

such that the locally closed smooth manifolds

$$S_i := Y_{i-1} \setminus Y_i, \quad (Y_0 := X),$$

decompose into strata; equivalently, each connected stratum $S_\alpha$ is contained in a unique $S_i$, and we call $i$ the depth of $S_\alpha$.

**Remark 2.1** (Notational convention). Although the filtration pieces $Y_i$ are closed unions of strata, most analytic constructions are carried out on one connected stratum at a time. Accordingly, when we fix a depth $i$ we will often fix a connected stratum component $S \subset S_i := Y_{i-1} \setminus Y_i$ and, to lighten notation, we will temporarily denote this component by $Y_i$. All statements involving local product variables $(x_i, y, z)$, the retraction $\rho_i$, the link fibration $L_i \to Y_i$, and the associated tangential family $A_{i,y}$ are meant in this “fixed stratum component” sense.

The depth $d$ is the maximal length of a strict incidence chain $S_{\alpha_d} \prec S_{\alpha_{d-1}} \prec \cdots \prec S_{\alpha_1}$; equivalently, it is the maximal number of iterated conic steps in the local models provided by control data (see below). We will often suppress the index $\alpha$ and simply say “a stratum of depth $i$”.

We write $g_{\text{FS}}$ for the induced Fubini-Study metric on $X^{\text{reg}}$.

**2.2. Iterated control data, cone bundles, and link fibrations.** Thom-Mather control data provide compatible “tubular neighborhoods” of each stratum and encode the local iterated cone-bundle geometry used by the iterated edge calculus [18, 1].

**Control data at depth level $i$.** For each closed stratum-union $Y_i$ we choose Thom-Mather control data [18] on an open neighborhood $N_i \subset X$. We assume these neighborhoods have been chosen small enough that there exists a common $\varepsilon > 0$ for all depth levels such that all radial functions take values in $[0, \varepsilon)$. The control data consist of:

* a continuous retraction $\rho_i : N_i \to Y_i$ which restricts, for every connected stratum component $S \subset Y_i$, to a smooth submersion (indeed the identity) on $S$:

$$\rho_i : N_i \cap S \to S;$$

* a continuous “radial function” $x_i : N_i \to [0, \varepsilon)$ with $Y_i = \{x_i = 0\}$ and such that for every stratum component $T$ meeting $N_i$ one has $x_i|_T \in C^\infty(T \cap N_i)$ and $dx_i \neq 0$ on $(T \cap N_i) \setminus Y_i$;

* a stratified local triviality structure for the map $(\rho_i, x_i)$, equivalently for the cone-bundle structure described below.

**Definition 2.2** (Stratified fibration and link fibration). Let $B$ be a smooth manifold and $E$ a smoothly stratified space. A continuous map $\pi : E \to B$ is called a *stratified fibration* if for every stratum $T \subset E$ the restriction $\pi|_T : T \to B$ is a smooth submersion and, locally on $B$, $\pi$ is stratified-homeomorphic to a product $U \times F$ over $U \subset B$ in such a way that, on each stratum of $F$, the trivialization restricts to a smooth trivialization of the corresponding smooth fibration.

**Cone-bundle charts.** The local triviality item above means that for each $y \in Y_i$ (more precisely, $y$ lying in some stratum component $S \subset Y_i$) there exists a compact smoothly stratified space $L_{i,y}$ (the *link* at $y$) and, on some neighborhood of $y$ inside $S$, a stratified homeomorphism

$$(2.2.1) \qquad (N_i \cap X) \cong \{(y, r, z) : y \in Y_i, 0 \le r < \varepsilon, z \in L_{i,y}\}/(r=0),$$

intertwining $\rho_i$ and $x_i$ with the evident projections: $\rho_i(y, r, z) = y$ and $x_i(y, r, z) = r$. Equivalently, over a small neighborhood $U \subset S$ the map $(\rho_i, x_i)$ identifies $N_i \cap \rho_i^{-1}(U)$ with a locally trivial *cone bundle* over $U$ whose fiber is the open cone $C(L_{i,y})$.
**2.3. Iterated cone charts along incidence chains.** The next statement is the precise local form of the “nested conic” structure that will be used implicitly when we write iterated wedge expansions and when we put $\partial_{dR}$ into iterated Dirac normal form. It can be extracted from the standard Thom–Mather axioms; we record it here in a form tailored to analytic constructions (uniformity on compact closed stratum-unions (or their resolved bases), explicit radial variables, and compatibility on overlaps).

**Proposition 2.3** (Iterated cone chart near an incidence chain). Fix a point $p \in X$ and a strict incidence chain of strata $S_{\alpha_l} \prec S_{\alpha_{l-1}} \prec \cdots \prec S_{\alpha_1}$ with $p \in S_{\alpha_l}$. Let $i_k$ denote the depth of $S_{\alpha_k}$ (so $1 \le i_1 < \cdots < i_l \le d$ in our convention). Then there exists a neighborhood of $p$ in $X$ (chosen inside the fixed control neighborhoods) and a stratified homeomorphism onto a neighborhood of the cone tip in an iterated cone bundle of the form

$$U_1 \times C(U_2 \times C(\cdots \times C(U_l \times L_p))),$$

where each $U_k \subset S_{\alpha_k}$ is open and $L_p$ is a compact smoothly stratified space (the deepest link at $p$). Moreover, in these coordinates the successive cone radii agree with the control functions $x_{i_1}, \dots, x_{i_l}$ (up to smooth positive factors), and the retractions $\rho_{i_k}$ correspond to the obvious projections collapsing the appropriate cone factors.

**Iterated cone charts and the hierarchy of radial variables.** Near a point $p$ lying in a depth-$\ell$ corner, the control data provide an iterated cone description of $X$ with successive radial variables $x_{i_1}, \dots, x_{i_\ell}$ and a deepest link $L_p$. Figure 2 is a purely schematic picture of this hierarchy: it keeps track only of the nesting order of the cone factors (and hence of the singular scales), which is exactly what matters for the inductive normal-form constructions in Sections 3–5.

<figure><img src="image_2.png" /><figcaption>FIGURE 2. Schematic nesting near a depth-$\ell$ point: successive radial variables and the deepest link.</figcaption></figure>

*Proof.* We argue by induction on the length $\ell$ of the incidence chain. For $\ell = 1$, this is the standard conical chart furnished by the Thom–Mather control data: in a tubular neighborhood $T_{i_1}$ of $S_{\alpha_1}$ the pair $(\rho_{i_1}, x_{i_1})$ identifies a neighborhood of $p$ with $U_1 \times C(L_p)$, where $L_p = \{x_{i_1} = \varepsilon\} \cap \rho_{i_1}^{-1}(p)$ is the (compact) link at $p$, and the stratification is the product stratification (cf. [18, 14]). Assume the statement holds for chains of length $\ell - 1$ and consider $S_{\alpha_\ell} \prec S_{\alpha_{\ell-1}} \prec \cdots \prec S_{\alpha_1}$. By the induction hypothesis, after shrinking inside the fixed control neighborhoods we may choose coordinates near $p$ identifying a neighborhood in $T_{i_2} \cap \cdots \cap T_{i_\ell}$ with

$$U_2 \times C(\cdots \times C(U_l \times L_p)),$$

in such a way that the deeper control functions $x_{i_2}, \dots, x_{i_\ell}$ agree with the corresponding cone radii (up to smooth positive factors) and the retractions $\rho_{i_k}$ for $k \ge 2$ are the obvious projections. Now intersect with a sufficiently small neighborhood inside $T_{i_1}$. The Thom–Mather compatibility relations on overlaps imply, for each $k \ge 2$,

$$\rho_{i_k} \circ \rho_{i_1} = \rho_{i_k}, \quad x_{i_k} \circ \rho_{i_1} = x_{i_k},$$
so $(\rho_{i_1}, x_{i_1})$ is constant along the already fixed deeper cone variables. Consequently the conical chart for $T_{i_1}$ can be chosen so that the transition map preserves the deeper factors, yielding the claimed stratified homeomorphism

$$U_1 \times C(U_2 \times C(\cdots \times C(U_\ell \times L_p)))$$

with the stated identification of radii and retractions. □

*Comment.* The proposition should be read as follows: once one has fixed the control data, a point in a maximal incidence chain has a neighborhood stratified-homeomorphic to an iterated cone on a compact stratified space, and the control functions $x_{i_k}$ are precisely the iterated cone radii. For analytic purposes, the key content is that one can choose the chart *compatibly* on overlaps, so that previously fixed deeper data (deeper radial variables and link stratifications) are preserved when passing to a shallower neighborhood. This is what ultimately allows one to define bundles of tangential operators and their domains by induction in the depth.

**2.4. Resolution viewpoint (manifold with corners model).** It is often convenient to replace the stratified space $X$ by a smooth manifold with corners that remembers the same iterated control data. One standard way is by *iterated radial blow-up* of the singular strata (see e.g. [1]). While we will not require the full construction, we will repeatedly use the following viewpoint:

**Remark 2.4** (Iterated blow-up model). There exists a manifold with corners $\tilde{X}$ and a continuous surjection $\beta : \tilde{X} \to X$ which restricts to a diffeomorphism over $X^{\text{reg}}$ and whose boundary hypersurfaces $H_i \subset \tilde{X}$ correspond to the depth levels $Y_i$. Each $H_i$ comes equipped with (i) a boundary defining function (still denoted) $x_i$, (ii) an iterated fibration $\phi_i : H_i \to \tilde{Y}_i$ whose base $\tilde{Y}_i$ is a compact manifold with corners whose interior is $S_i$; the fiber over $y \in S_i$ is the link $L_{i,y}$, and (iii) compatibility of the fibrations on corners mirroring (2.2.3). In local product coordinates $(x_1, \dots, x_k, y, z)$ on $\tilde{X}$ near a codimension-$k$ corner, the maps $\phi_i$ and the variables $x_i$ reproduce the iterated cone-bundle charts of Proposition 2.3.

*Comment.* Analytically, $\tilde{X}$ is the natural stage for the iterated edge tangent bundle and for the iterated wedge calculus: the boundary defining functions $x_i$ act as the radial variables in wedge expansions, and the fibrations $\phi_i$ encode the link fibrations along which tangential operators and trace bundles vary. We will freely switch between the stratified-space language and this manifold-with-corners intuition.

**2.5. Uniformity on stratum components.** Although a connected stratum component $S$ is generally non-compact (it is only locally closed in $X$), the iterated blow-up model of Remark 2.4 provides a *compact* resolved base $\tilde{Y}_i$ (a manifold with corners) whose interior is $S$. All geometric objects induced by the fixed control data (cone radii, retractions, horizontal distributions, and the tensors built from $g_{\text{FS}}$) extend in the $C_{\text{ie}}^{\infty}$ sense up to $\partial \tilde{Y}_i$. Consequently, any estimate proved locally in $y$ with constants depending continuously on $y$ can be upgraded to a *uniform* estimate on $S = \text{int} \tilde{Y}_i$ by compactness of $\tilde{Y}_i$.

We record one convenient formulation.

**Lemma 2.5** (Finite control cover with uniform radii). Let $S \subset S_i$ be a connected depth-$i$ stratum component and let $\tilde{Y}_i$ be the corresponding compact resolved base from Remark 2.4, so that $S = \text{int} \tilde{Y}_i$. Then there exists a finite open cover $\tilde{Y}_i = \bigcup_{\mu=1}^M \tilde{U}_\mu$ and, writing $U_\mu := \tilde{U}_\mu \cap S$, cone-bundle charts $\rho_i^{-1}(U_\mu) \cap N_i \cong U_\mu \times C(L_{i,y_\mu})$ for points $y_\mu \in U_\mu$, such that the corresponding identifications use a single radius interval $[0, \varepsilon)$ (the common $\varepsilon$ fixed above) and such that transition maps preserve the radial variable $x_i$ and the induced stratifications on the links.
*Proof.* Cover the compact manifold with corners $\tilde{Y}_i$ by finitely many product coordinate charts $\tilde{U}_\mu$. Over each $\tilde{U}_\mu$, the iterated cone-bundle description of Proposition 2.3 (equivalently, the local product structure of the boundary fibration $\phi_i : H_i \to \tilde{Y}_i$) yields the stated cone-bundle trivialization with radius in $[0, \varepsilon)$. The compatibility of transition maps is a restatement of the control relations (2.2.3). □

*Comment.* This is the basic compactness input behind uniform analytic estimates: once one proves a property locally with constants depending continuously on the base point, compactness of $\tilde{Y}_i$ (equivalently, finiteness of the above cover) upgrades it to a uniform statement on $S = \text{int} \tilde{Y}_i$.

**Remark 2.6** (Standing convention on adapted neighborhoods). Fix once and for all, for each depth level $i$ and each connected depth-$i$ stratum component $S = \text{int} \tilde{Y}_i$, a finite family of cone-bundle charts as in Lemma 2.5. All later analytic constructions are carried out on iterated tubular neighborhoods $T_i \subset N_i$ chosen subordinate to this fixed finite cover. Accordingly, any statement that would ordinarily be phrased “after shrinking the neighborhood” is to be read as: “working on a sufficiently small chart from this fixed cover”. We suppress this phrase from now on.

## 2.6. Two consequences used repeatedly.
We record two standard consequences (used later for the smoothness of trace bundles and for parameter regularity of tangential families).

**Lemma 2.7** (Stratified local triviality along a stratum). Let $S$ be a connected component of a stratum contained in $Y_i$. Then the link fibration $L_i \to S$ is locally trivial as a stratified fibration (in particular, the stratified homeomorphism type of the fiber $L_{i,y}$ is constant for $y \in S$). [18]

*Proof.* Fix $y_0 \in S$. Let $(T_i, \rho_i, x_i)$ be the Thom-Mather control neighborhood, retraction, and radial function for $Y_i$. For sufficiently small $\varepsilon > 0$, the link over $y \in S$ can be realized as the compact fiber

$$L_{i,y} = \{x_i = \varepsilon\} \cap \rho_i^{-1}(y),$$

and over any relatively compact open $U \subset S$ the total space $L_i|_U := \{x_i = \varepsilon\} \cap \rho_i^{-1}(U)$ is compact. By the control axioms, $\rho_i$ restricts on each stratum of $L_i|_U$ to a smooth submersion onto $U$; in particular, $\rho_i : L_i|_U \to U$ is a proper stratified submersion. Mather's first isotopy lemma (equivalently, the stratified Ehresmann theorem in this setting) therefore yields a stratified local trivialization of $\rho_i$ near $y_0$, i.e. a neighborhood $U \subset S$ and a stratified homeomorphism $L_i|_U \cong U \times L_{i,y_0}$ over $U$ which is smooth on each stratum. □

*Comment.* Local triviality means that for each $y_0 \in S$ there exists a neighborhood $U \subset S$ and a stratified homeomorphism $L_i|_U \cong U \times L_{i,y_0}$ over $U$ which, on each stratum of $L_{i,y_0}$, restricts to a smooth trivialization of the corresponding smooth fibration. In particular, any construction on the fibers that is invariant under stratified diffeomorphism (or stratified homotopy) yields locally constant data along $S$.

**Lemma 2.8** (Constancy of topological invariants along strata). Fix a connected stratum component $S \subset Y_i$ and any perversity/mezzoperversity choice on the fibers $L_{i,y}$. Then the ranks of the corresponding (intersection) cohomology groups of the fibers are locally constant on $S$ (hence constant on $S$). In particular, whenever later constructions identify trace data bundles with such groups (via Hodge-type theorems), the resulting vector bundles over $S$ have locally constant rank. [14, 6, 8]

*Proof.* By Lemma 2.7, over a sufficiently small neighborhood $U \subset S$ the fibers $L_{i,y}$ ($y \in U$) are all identified by stratified homeomorphisms with a fixed model fiber $L_{i,y_0}$. Intersection cohomology with any fixed perversity/mezzoperversity is invariant under stratified homeomorphism (equivalently, under pullback of the associated Deligne-type intersection complex), hence the resulting groups $\text{IH}^\bullet(L_{i,y})$ are canonically isomorphic for all $y \in U$. Therefore their ranks are locally constant on $S$, and since $S$ is connected they are constant on $S$. □
*Comment.* By Lemma 2.7, over a small $U \subset S$ the fibers $L_{i,y}$ ($y \in U$) are all canonically identified up to stratified homeomorphism, and intersection cohomology (with any fixed perversity/mezzoperversity) is a topological invariant of the stratified homeomorphism type of the fiber. This is the input used later to interpret various trace/attachment data topologically. In the present paper, however, constant-rank kernel bundles and uniform gaps are obtained analytically by the domain-dependent stability theorem (Theorem 6.3), without requiring an *a priori* identification of analytic kernels with topological groups.

**Remark 2.9.** All geometric constructions in this paper are intrinsic to the induced metric on $X^{\text{reg}}$ and the stratified control data; no Kähler identities are used.

### 3. Iterated Fubini-Study wedge expansions

**3.1. Iterated edge vector fields and iterated edge-smooth tensors.** We define the iterated edge tangent algebra $V_{ie}$ by requiring vector fields tangent to all boundary hypersurfaces in an iterated resolution and tangent to all fibrations of links along strata. A tensor is *iterated edge-smooth* if it has bounded derivatives along $V_{ie}$.

**3.2. FS wedge expansion (iterated).**

**Theorem 3.1 (Iterated Fubini-Study wedge expansion).** In an iterated tubular neighborhood of $Y_i$ with local product variables $(x_i, y, z)$ (outer radial $x_i$, base $y \in Y_i$, link variables $z \in L_{i,y}$), the induced Fubini-Study metric admits an expansion

$$g_{\text{FS}} = dx_i^2 + \rho_i^* g_{Y_i} + x_i^2 g_{L_i} + x_i \kappa_i,$$

where $g_{L_i}$ is an iterated wedge metric on the stratified link fibration $L_i \to Y_i$, and the remainder $\kappa_i$ is iterated edge-smooth down to $x_i = 0$. The construction is compatible across strata incidences, in the sense that restricting to deeper tubular neighborhoods yields the corresponding expansion for $Y_j$ with $j > i$.

*Proof (with details deferred to Appendix B).* The depth-one version follows from adapted coordinates and quantitative Whitney angle estimates, together with a vertical scaling argument for the transverse metric. In the iterated setting, we perform the same construction on each stratum neighborhood and verify compatibility along incidences using iterated control data, showing that the remainders are iterated edge-smooth with respect to $V_{ie}$. □

### 4. Iterated Dirac normal forms with deeper canonical domains

**From normal forms to canonical domains: a dependency map.** The central analytic objects in the paper are built in a fixed order: wedge/FS expansions produce Dirac-type normal forms; these determine trace bundles and Cauchy data spaces; from them we construct Calderón projectors, and finally the canonical ideal boundary conditions. Figure 3 summarizes this pipeline and, crucially, makes explicit the dependence of each stage on the previously chosen deeper data, which is the mechanism behind the iterated induction on depth.

<figure><img src="image_3.png" /><figcaption>FIGURE 3. Dependency pipeline from normal forms to canonical ideal boundary conditions (schematic).</figcaption></figure>
4.1. **Geometric Gauss-Bonnet operator as a wedge Dirac.** Let $D := \partial_{dR} = d + \delta$ be the Hodge-de Rham (Gauss-Bonnet) operator on $(X^{\text{reg}}, g_{\text{FS}})$. Fix a depth-$i$ stratum $Y_i$ and work in an iterated tubular neighborhood $T_i$ equipped with the iterated control data of Section 2. By Theorem 3.1, working in adapted variables $(x_i, y, z)$ (outer radial variable $x_i \in (0, \varepsilon)$, base point $y \in Y_i$, link variable $z \in L_{i,y}$) the metric has the form

$$ (4.1.1) \qquad g_{\text{FS}} = dx_i^2 + \rho_i^* g_{Y_i} + x_i^2 g_{L_i} + x_i \kappa_i, $$

where $g_{L_i}$ is an iterated wedge metric on the stratified link fibration $L_i \to Y_i$ and $\kappa_i$ is iterated edge-smooth down to $x_i = 0$.

The wedge coframe associated with (4.1.1) is generated by

$$ dx_i, \quad \rho_i^* \eta \quad (\eta \in T^*Y_i), \quad x_i \theta \quad (\theta \in T^*L_{i,y}), $$

and yields a canonical splitting of wedge forms near $x_i = 0$: every differential form $\omega$ may be written uniquely as

$$ (4.1.2) \qquad \omega = dx_i \wedge \alpha + \beta, $$

with $\alpha, \beta$ wedge-tangential (i.e. with no $dx_i$ component). Let $f_i := \dim L_{i,y}$ (constant on a stratum). The cone factor $x_i^2 g_{L_i}$ implies that the $L^2$ density on wedge-tangential forms contains the weight $x_i^{f_i}$, so it is convenient to conjugate $D$ by a standard unitary transformation (multiplication by $x_i^{f_i/2}$ combined with the splitting (4.1.2)); we record the resulting normal form as a theorem.

4.2. **Normal form and rigidity of the $x_i^{-1}$ coefficient.** The link fiber $L_{i,y}$ is itself a compact stratified space of depth $< i$. By induction on depth we assume that for every deeper stratum $Y_k$ with $k > i$ we have already fixed the canonical deeper Cauchy-data packages (Definition 6.1); this determines a canonical self-adjoint Fredholm realization of the Gauss-Bonnet operator on each link $L_{i,y}$, which we denote by $D_{L_{i,y}}^{W>i}$.

**Theorem 4.1** (Iterated wedge Dirac normal form). Fix $i \in \{1, \dots, d\}$. On the adapted neighborhood $T_i$ there exists a unitary bundle identification over $(0, \varepsilon) \times Y_i$, compatible with changes of adapted trivializations coming from the iterated control data, such that on compactly supported smooth forms in $T_i \cap X^{\text{reg}}$ we have

$$ (4.2.1) \qquad D = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} A_{i,y}^{W>i} + B_i(x_i) \right). $$

Here:

(i) $\Gamma_i$ is Clifford multiplication by $dx_i$ for the wedge coframe, hence $\Gamma_i^* = -\Gamma_i$ and $\Gamma_i^2 = -\text{Id}$. Moreover, $\Gamma_i$ anti-commutes with $A_{i,y}^{W>i}$.

(ii) For each $y \in Y_i$, the operator $A_{i,y}^{W>i}$ is a self-adjoint elliptic first-order operator on the link fiber $L_{i,y}$, given (in the standard cone splitting (4.1.2)) by

$$ (4.2.2) \qquad A_{i,y}^{W>i} = D_{L_{i,y}}^{W>i} + \left( \frac{f_i}{2} - \mathbf{N}_i \right), $$

where $\mathbf{N}_i$ denotes the form-degree operator on wedge-tangential forms on $L_{i,y}$. In particular, the domain of $A_{i,y}^{W>i}$ is the canonical deeper domain induced by $W>i$.

(iii) $B_i(x_i)$ is a first-order iterated edge differential operator along the fibration $L_i \to Y_i$ with coefficients that are iterated edge-smooth down to $x_i = 0$. Moreover $B_i(x_i)$ extends continuously to $x_i = 0$ as an operator family on $L^2(L_{i,y})$ and is uniformly bounded for $x_i \in [0, \varepsilon)$ and locally in $y \in Y_i$. In particular, if $S = \text{int} \tilde{Y}_i$ is a fixed stratum component with compact resolved base $\tilde{Y}_i$ (Remark 2.4), then Lemma 2.5 upgrades this to a bound that is uniform for all $y \in S$.
Finally, the $x_i^{-1}$ coefficient is rigid: it depends only on the product-type leading part $dx_i^2 + \rho_i^* g_{Y_i} + x_i^2 g_{L_i}$ and on the deeper canonical realization on the link, and is independent of the $x_i \kappa_i$ remainder in (4.1.1).

**Corollary 4.2 (Indicial operator).** In the normal form (4.2.1), the indicial operator at $y \in Y_i$ is

$$I_y(D) = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} A_{i,y}^{W>i} \right).$$

Consequently, boundary exponents in a given weight window are controlled by the spectrum of $A_{i,y}^{W>i}$, and the dependence of the indicial family on $(y, W_{>i})$ is smooth in the sense made precise in Section 5-Section 6.

*Proof.* The statement is local along $Y_i$. First replace $g_{FS}$ by its product-type wedge part $g_0 = dx_i^2 + \rho_i^* g_{Y_i} + x_i^2 g_{L_i}$. For $g_0$ the Gauss-Bonnet operator is a geometric wedge Dirac operator, and the standard cone calculation ([13, 19, 17]) yields the normal form (4.2.1) with $B_i$ smooth down to $x_i = 0$. Next treat the perturbation $g_{FS} - g_0 = x_i \kappa_i$: since $\kappa_i$ is iterated edge-smooth, the difference between the Levi-Civita connections of $g_{FS}$ and $g_0$ is $O(1)$ in wedge frames, hence it contributes only to the $B_i(x_i)$ term; the coefficient of $x_i^{-1}$ is unchanged. Finally, the link operator $D_{L_{i,y}}$ must be realized with a domain compatible with deeper strata; this is exactly the deeper canonical package $W_{>i}$, and the corresponding family $A_{i,y}^{W>i}$ varies smoothly in $y$ by the inductive hypotheses. All compatibility statements under changes of adapted trivializations are checked by functoriality of control data. Full details are given in Appendix C. □

# 5. ITERATED CRITICAL STRIPS AND PARAMETER-DEPENDENT CALDERÓN PACKAGES

## 5.1. Indicial correspondence and the role of the tangential spectrum.
Freezing coefficients at $x_i = 0$ and discarding $B_i(x_i)$ yields the indicial model

$$I_y(D) = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} A_{i,y}^{W>i} \right),$$

so boundary exponents in a weight window are controlled by the small spectrum of $A_{i,y}^{W>i}$.

## 5.2. Inductive hypotheses at stratum $Y_i$.
Assume canonical data $W_{>i}$ have been defined for deeper strata and satisfy the inductive regularity package (Definition 6.1 below). We aim to prove: (Hi.1) (Normal form) Theorem 4.1 holds along $Y_i$.

(Hi.2) (Uniform gap at 0) dim ker($A_{i,y}^{W>i}$) is constant in $y \in Y_i$ and there exists $\varepsilon_i > 0$ such that $\text{Spec}(A_{i,y}^{W>i}) \cap (-\varepsilon_i, \varepsilon_i) = \{0\}$ for all $y$.

(Hi.3) (Constant-multiplicity strip) In a weight strip around the $L^2$ threshold for $x_i$, the only boundary spectrum encountered is the indicial root at 0 coming from $\text{ker}(A_{i,y}^{W>i})$, with constant multiplicity rank($H_i$).

**Comment on verification.** Item (Hi.1) is precisely the normal form statement of Theorem 4.1, whose proof uses only the geometric wedge expansion (Theorem 3.1) together with the already-fixed deeper canonical link realizations from Definition 6.1. Thus, for the inductive construction of the Calderón package, the genuinely new analytic content is the *uniform* spectral information in (Hi.2)-(Hi.3), which will be obtained from the domain-dependent stability theorem (Theorem 6.3) once the tangential family is shown to depend continuously on the base parameter (up to the boundary of the resolved base $\tilde{Y}_i$, cf. Remark 2.4 and Lemma 2.5).

These are the iterated analogues of the depth-one hypotheses (H1)-(H3) used to build the trace/Calderón package.
**5.3. Trace bundles, Poisson operators, and Calderón projectors.** Under (Hi.1)-(Hi.3) one defines:

* the boundary data bundle $H_i := \ker(A_i^{W>i}) \to Y_i$,

* a trace map $\mathrm{Tr}_{D,i}: \mathrm{Dom}(D_{\mathrm{max}}) \to \Gamma(Y_i; H_i)$ in the constant strip,

* a Poisson operator and an order-0 Calderón projector $C_i: H_i \to H_i$,

* the canonical Cauchy data bundle

$$W_{i,\mathrm{can}} := \mathrm{ran}(C_i) \subset H_i.$$

**Theorem 5.1** (Iterated Calderón package in the constant strip). Assume (Hi.1)-(Hi.3) along $Y_i$. Then the trace bundle $H_i \to Y_i$ is well-defined and varies smoothly with $y \in Y_i$, the Calderón projector $C_i$ depends smoothly on $y$, and $W_{i,\mathrm{can}} = \mathrm{ran}(C_i) \subset H_i$ is a smooth subbundle. Moreover, $W_{i,\mathrm{can}}$ is Lagrangian for the Green pairing induced at stratum $Y_i$.

*Proof.* All assertions are proved in Appendix D; we indicate the logical structure and record the canonical choices.

By the constant-multiplicity strip hypothesis (Hi.3) and the indicial normal form (4.2.1), the only boundary exponent in a sufficiently small weight window around the $L^2$ threshold is the indicial root 0 corresponding to $\ker(A_{i,y}^{W>i})$. Proposition D.2 shows that every $u \in \mathrm{Dom}(D_{\mathrm{max}})$ has a *canonical* leading term $u_0(y) \in H_{i,y} := \ker(A_{i,y}^{W>i})$ in its asymptotic expansion as $x_i \to 0$, and that the assignment $u \mapsto u_0$ defines a bounded trace map

$$\mathrm{Tr}_{D,i}: \mathrm{Dom}(D_{\mathrm{max}}) \to \Gamma(Y_i; H_i)$$

which varies smoothly with the base variable $y \in Y_i$.

Proposition D.3 constructs a Poisson operator $K_i$ in the constant strip and defines the Calderón projector by $C_i := \mathrm{Tr}_{D,i} \circ K_i$. Parameter smoothness is a consequence of the parameter-dependent iterated edge calculus (cf. [21, 19]). In particular, $C_i$ is a smooth family of order-0 idempotents on the finite-rank bundle $H_i \to Y_i$, hence its range $W_{i,\mathrm{can}} := \mathrm{ran}(C_i)$ is a smooth subbundle.

Finally, Proposition D.7 identifies $W_{i,\mathrm{can}}(y)$ with the Cauchy data space of $L^2$ solutions in the constant strip and uses the Green identity to show that it is isotropic and of half rank for the induced Green pairing on $H_{i,y}$; therefore $W_{i,\mathrm{can}}$ is Lagrangian. $\square$

# 6. THE DOMAIN-DEPENDENT STABILITY THEOREM

**6.1. Tangential families arising from the iterated normal form.** In the normal form (4.2.1), the operator $A_{i,y}^{W>i}$ acts on the link fiber $L_{i,y}$ and carries the deeper canonical domain determined by the already chosen Cauchy-data bundles $W_{>i,\mathrm{can}}$. We refer to the parameter-dependent family

$$y \mapsto A_{i,y}^{W>i}$$

as the *tangential family* at the stratum $Y_i$ (or, more precisely, on the resolved base $\tilde{Y}_i$ from Remark 2.4). The purpose of this section is to control the spectral projections and Cauchy data of this family *despite* the fact that its self-adjoint domain varies with $y$ through the deeper IBC.

This section supplies (Hi.2)-(Hi.3) automatically from the Fubini-Study geometry and the inductive package on deeper strata. Its novelty is that $A_{i,y}^{W>i}$ is a tangential operator on a stratified link with a self-adjoint domain determined by $W_{>i}$.

## 6.2. Inductive package on deeper strata.

**Definition 6.1** (Inductive canonical package). For $k > i$, assume we have already defined canonical Cauchy-data bundles $W_{k,\mathrm{can}} \subset H_k \to Y_k$ and canonical realizations of the corresponding link Dirac operators, such that:

(i) (Smoothness) the deeper trace/Calderón operators depend smoothly on base variables;
(ii) (*Self-adjoint Fredholmness*) the deeper canonical realizations are self-adjoint and Fredholm;

(iii) (*Topological identification*) the deeper canonical de Rham Hilbert complexes compute the corresponding iterated IBC intersection cohomology on link fibers.

**Remark 6.2** (No circularity in the inductive construction). The inductive package of Definition 6.1 is used only to define and control the realized tangential families on the links. At a given stratum $Y_i$, the construction of $W_{i,\text{can}}$ and the corresponding boundary problem for $D$ uses only the already constructed canonical data $W_{>i,\text{can}}$ on strictly deeper strata. The base case occurs at the deepest stratum, whose link is smooth, so the starting Calderón package is classical.

## 6.3. Kernel bundles and uniform gaps for domain-dependent families.

**Theorem 6.3** (Domain-dependent stability). Fix a depth-$i$ stratum component, denoted $Y_i$ as in Remark 2.1, and let $\tilde{Y}_i$ be the associated compact base (a manifold with corners) from Remark 2.4. Assume the inductive canonical package (Definition 6.1) holds on all deeper strata (so the resulting tangential family is defined and $C^\infty$ up to the boundary of $\tilde{Y}_i$ in the sense of Appendix E). Then:

(1) (Constant-rank kernel bundle) $\dim \ker(A_{i,y}^{W>i})$ is independent of $y \in Y_i$; hence $H_i := \ker(A_i^{W>i}) \to Y_i$ is a smooth finite-rank vector bundle.

(2) (Uniform spectral gap at 0) there exists $\varepsilon_i > 0$ such that $\text{Spec}(A_{i,y}^{W>i}) \cap (-\varepsilon_i, \varepsilon_i) = \{0\}$ for all $y \in Y_i$.

(3) (Constant-multiplicity strip) in a sufficiently small weight strip about the $L^2$ threshold for $x_i$, the only boundary spectrum encountered is the indicial root 0 associated with $\ker(A_{i,y}^{W>i})$, with constant multiplicity rank($H_i$).

Consequently, (Hi.2)-(Hi.3) hold along $Y_i$.

*Proof.* The full argument is given in Appendix E; we summarize the key input and the precise references.

(1) **Constant rank.** By assumption, the realized tangential family $y \mapsto A_{i,y}^{W>i}$ is defined and varies $C^\infty$ up to the boundary of $\tilde{Y}_i$ in the sense of Appendix E. Each operator $A_{i,y}^{W>i}$ is self-adjoint with compact resolvent, hence has discrete spectrum and finite-dimensional kernel; in particular, 0 is an isolated eigenvalue of finite multiplicity for every $y$.

Since the domain dependence of $A_{i,y}^{W>i}$ can be encoded by a smooth family of boundary projections (Appendix E, §E.2), the graph-projection mechanism (Lemma A.3) yields norm-resolvent continuity of $y \mapsto A_{i,y}^{W>i}$. The associated Riesz projection onto $\ker(A_{i,y}^{W>i})$ therefore depends continuously on $y$; since it has finite rank, its rank is locally constant and hence constant on each connected stratum component (Lemma A.5). Proposition E.5 upgrades this to a smooth finite-rank vector bundle $H_i = \ker(A_i^{W>i}) \to Y_i$.

(2) **Uniform gap.** Each $A_{i,y}^{W>i}$ is self-adjoint with compact resolvent, and the domain dependence can be encoded by a smooth family of boundary projections (Appendix E, §E.2). Resolvent continuity then follows from Lemma A.3. We view the family as parametrized by the compact base $\tilde{Y}_i$ from Remark 2.4 (whose interior is the stratum component), so compactness upgrades pointwise isolation of 0 to a uniform gap; see Lemma A.5 and Proposition E.6. This is carried out in Proposition E.6.

(3) **Constant strip.** With the uniform gap at 0, the Mellin-symbol/indicial correspondence for wedge Dirac operators shows that, in a sufficiently small weight window around the $L^2$ threshold, the only indicial root encountered is 0, and it occurs with multiplicity rank($H_i$). This yields a constant-multiplicity strip as stated; see Proposition E.7. $\square$
## 7. MAIN INDUCTIVE THEOREM AND GLOBAL CANONICAL DOMAIN

### 7.1. Construction of the canonical mezzoperversity.
Starting at the deepest stratum $Y_d$, define $W_{d,can}$ by the (non-iterated) Calderón package on the deepest link (which is smooth, i.e. depth 0). Assuming $W_{>i,can}$ already defined, Theorems 6.3 and 5.1 define $W_{i,can}$. This yields an iterated boundary datum

$$W_{\text{can}} := (W_{1,\text{can}}, \dots, W_{d,\text{can}}).$$

### 7.2. Global canonical domain and self-adjoint Fredholmness.

**Definition 7.1 (Global canonical realization).** Let $D_{\text{max}}$ denote the maximal extension of $D$ on $L^2$ forms over $X^{\text{reg}}$. Define the global canonical domain by imposing, at each stratum $Y_i$, the canonical boundary condition

$$\mathrm{Tr}_{D,i}(u) \in \Gamma(Y_i; W_{i,\text{can}}),$$

interpreted in the stratification-compatible partition of unity sense of Proposition 7.2. Denote the resulting closed realization by $D_{\text{can}}$, and write $d_{\text{can}}$ for the closed extension of $d$ on $\mathrm{Dom}(D_{\text{can}})$.

**Proposition 7.2 (Well-definedness and cutoff stability).** Let $\mathcal{W} = (W_1, \dots, W_d)$ be a mezzoperver-
sity datum with smooth subbundles $W_i \subset H_i \to Y_i$. For $u \in \mathrm{Dom}(D_{\text{max}})$, say that $u$ satisfies the
ith boundary condition on an open set $U \subset X$ if for every cutoff $\chi \in C_{\text{ie}}^{\infty}$ supported in $U$ and equal
to 1 near $U \cap Y_i$ one has

$$\mathrm{Tr}_{D,i}(\chi u) \in \Gamma(U \cap Y_i; W_i).$$

Then the global domain defined by requiring the above condition for every stratum $Y_i$ is independent
of all auxiliary choices (adapted covers and $C_{\text{ie}}^{\infty}$ partitions of unity), and it is stable under multi-
plication by $C_{\text{ie}}^{\infty}$ cutoffs: if $u$ lies in the global domain and $\phi \in C_{\text{ie}}^{\infty}(X)$, then $\phi u$ lies in the same
domain.

*Proof.* Two points are standard and are proved in the present iterated setting in Appendix D (trace maps in the constant strip) and Appendix F.1 (module/fineness properties):

(i) If $u \in \mathrm{Dom}(D_{\text{max}})$ and $\phi \in C_{\text{ie}}^{\infty}(X)$, then $\phi u \in \mathrm{Dom}(D_{\text{max}})$, since $[D, \phi]$ is a zeroth-order operator with $C_{\text{ie}}^{\infty}$ coefficients and hence is bounded on $L^2$.

(ii) For cutoffs supported in a distinguished neighborhood of $Y_i$, the trace map is $C_{\text{ie}}^{\infty}$-linear along the stratum: $\mathrm{Tr}_{D,i}(\phi v) = (\phi|_{Y_i}) \mathrm{Tr}_{D,i}(v)$ whenever both sides are defined.

To see independence of partitions, fix an adapted cover $\{U_\alpha\}$ with a subordinate $C_{\text{ie}}^{\infty}$ partition of unity $\{\chi_\alpha\}$ and assume that $\mathrm{Tr}_{D,i}(\chi_\alpha u) \in \Gamma(U_\alpha \cap Y_i; W_i)$ for every $\alpha$. Given any other cutoff $\psi \in C_{\text{ie}}^{\infty}$ supported in an open set $U$, write $\psi = \sum_\alpha \psi_{\chi_\alpha}$. By (ii), each localized trace satisfies

$$\mathrm{Tr}_{D,i}((\psi\chi_\alpha)u) = (\psi|_{Y_i}) \mathrm{Tr}_{D,i}(\chi_\alpha u) \in \Gamma(U \cap Y_i; W_i),$$

and summing over $\alpha$ gives $\mathrm{Tr}_{D,i}(\psi u) \in \Gamma(U \cap Y_i; W_i)$. This shows that the boundary constraint does not depend on the particular adapted partition used to test it.

Finally, cutoff stability follows immediately: if $u$ satisfies the constraint and $\phi \in C_{\text{ie}}^{\infty}(X)$, then for any test cutoff $\chi$ as in the statement we have $\mathrm{Tr}_{D,i}(\chi \phi u) = \mathrm{Tr}_{D,i}((\chi \phi)u) \in \Gamma(U \cap Y_i; W_i)$ by the defining property of $u$. $\square$

**Lemma 7.3 (Iterated-edge Rellich compactness).** Let $\tilde{X}$ be a compact resolved space and let $H_{\text{ie}}^m$
denote the iterated-edge Sobolev scale used throughout (see Remark A.2). Then for integers $m >$
$m' \ge 0$ the inclusion $H_{\text{ie}}^m \hookrightarrow H_{\text{ie}}^{m'}$ is compact; in particular, $H_{\text{ie}}^1 \hookrightarrow L^2$ is compact. Consequently,
any bounded operator $A : L^2 \to H_{\text{ie}}^1$ with properly supported range (for example, $A = \phi A \psi$ with
$\phi, \psi \in C_{\text{ie}}^{\infty}$ supported in a single distinguished neighborhood) is compact on $L^2$.

*Proof.* This is the usual Rellich compactness adapted to the iterated-edge Sobolev scale. Using a finite cover by iterated cone charts (Proposition 2.3) and a subordinate partition of unity, the local $H_{\text{ie}}^m$-norm is equivalent to a finite sum of ordinary Sobolev norms of order $m$ in the adapted variables
on each chart, so the classical Rellich theorem gives compactness locally. Summing over the cover yields the global compact embedding. For the final statement, A factors as $L^2 \xrightarrow{A} H_{ie}^1 \hookrightarrow L^2$, and the second map is compact. For detailed treatments in related edge/wedge settings, see for example [19, 1]. □

**Lemma 7.4** (Order $-1$ iterated-edge operators gain one derivative). Let U be a distinguished neighborhood and let P be a properly supported iterated-edge pseudodifferential operator of order $-1$ on U (in the sense of Appendix D). Then for every $s \in \mathbb{R}$ the operator P extends to a bounded map

$$P : H_{ie}^{s}(U) \to H_{ie}^{s+1}(U).$$

In particular, $P : L^2(U) \to H_{ie}^1(U)$ is bounded. Moreover, for $\phi, \psi \in C_{ie,c}^{\infty}(U)$ the localized operator $\phi P\psi$ is compact on $L^2$.

*Proof.* The mapping statement is a direct application of Proposition D.6(2): since $P \in \Psi^{-1}(U)$ is properly supported, it extends to a bounded map $P : H_{ie}^s(U) \to H_{ie}^{s+1}(U)$ for every $s \in \mathbb{R}$.

For compactness, take $\phi, \psi \in C_{ie,c}^{\infty}(U)$ and consider $\phi P\psi$ on $L^2$. By the first part, $\phi P\psi : L^2 \to H_{ie}^1$ is bounded, and by Lemma 7.3 the inclusion $H_{ie}^1 \hookrightarrow L^2$ is compact. Hence $\phi P\psi$ is compact on $L^2$. □

**Proposition 7.5** (Gluing local parametrices). Let $\mathcal{D}$ be a closed realization of $D$ on $L^2$ forms over $X^{\text{reg}}$ whose domain is stable under multiplication by $C_{ie}^{\infty}$ cutoffs as in Proposition 7.2. Assume that $X$ admits a finite open cover $\{U_\alpha\}$ and bounded operators

$$Q_{\alpha} : L^2(U_{\alpha} \cap X^{\text{reg}}) \to L^2(U_{\alpha} \cap X^{\text{reg}})$$

such that, in the sense of operators on $L^2(U_{\alpha} \cap X^{\text{reg}})$,

$$\mathcal{D}Q_{\alpha} = I - K_{\alpha}, \quad Q_{\alpha}\mathcal{D} = I - K'_{\alpha},$$

where $K_{\alpha}$ and $K'_{\alpha}$ are compact. Then $\mathcal{D}$ is Fredholm.

*Proof.* Choose a $C_{ie}^{\infty}$ partition of unity $\{\chi_{\alpha}\}$ subordinate to $\{U_{\alpha}\}$ and auxiliary cutoffs $\{\tilde{\chi}_{\alpha}\} \subset C_{ie}^{\infty}$ with $\tilde{\chi}_{\alpha} \equiv 1$ on $\text{supp}(\chi_{\alpha})$ and $\text{supp}(\tilde{\chi}_{\alpha}) \subset U_{\alpha}$. Define

$$Q := \sum_{\alpha} \chi_{\alpha} Q_{\alpha} \tilde{\chi}_{\alpha}.$$

Then $Q$ is bounded on $L^2$. By cutoff stability of the domain, $\mathcal{D}(\chi_{\alpha}v) = \chi_{\alpha} \mathcal{D}v + [D, \chi_{\alpha}]v$ with $[D, \chi_{\alpha}]$ bounded on $L^2$, so the usual partition-of-unity computation gives

$$\mathcal{D}Q = I - \sum_{\alpha} \chi_{\alpha} K_{\alpha} \tilde{\chi}_{\alpha} - K,$$

where $K$ is the finite sum of commutator terms coming from the localization,

$$K := \sum_{\alpha, \beta} \chi_{\alpha} [D, \chi_{\beta}] Q_{\beta} \tilde{\chi}_{\beta}.$$

On each $U_{\beta}$, the operator $Q_{\beta}$ is a local parametrix in the iterated edge calculus (of order $-1$ in the sense of Appendix D), while $[D, \chi_{\beta}]$ is order 0 with $C_{ie}^{\infty}$ coefficients. After multiplication by cutoffs, each term $\chi_{\alpha}[D, \chi_{\beta}]Q_{\beta} \tilde{\chi}_{\beta}$ is again of iterated-edge order $-1$ (since $[D, \chi_{\beta}]$ has order 0 with $C_{ie}^{\infty}$ coefficients and $Q_{\beta}$ has order $-1$), hence it extends to a bounded map $L^2 \to H_{ie}^1$ by Lemma 7.4. Composing with the compact embedding $H_{ie}^1 \hookrightarrow L^2$ from Lemma 7.3, each summand is compact on $L^2$ (compare also Appendix D, §D.5). A similar computation yields $Q\mathcal{D} = I - \tilde{K}$ with $\tilde{K}$ compact. Therefore $Q$ is a parametrix modulo compact remainders, and $\mathcal{D}$ is Fredholm. □

**Theorem 7.6** (Self-adjoint Fredholmness of the global canonical realization). The operator $D_{\text{can}}$ of Definition 7.1 is self-adjoint and Fredholm. Equivalently, $(\text{Dom}(d_{\text{can}}), d_{\text{can}})$ is a closed de Rham Hilbert complex and $D_{\text{can}} = d_{\text{can}} + \delta_{\text{can}}$ is a closed self-adjoint Fredholm Hodge-de Rham operator.
*Proof.* We first record the self-adjointness criterion in the present setting.

*Self-adjointness.* Let $D$ be the Hodge-de Rham operator on $X^{\text{reg}}$ and fix a stratum $Y_i$. In the constant strip, Green's identity induces a nondegenerate skew form (the Green pairing) on the finite-rank boundary data bundle $H_i \to Y_i$. If $W_i \subset H_i$ is Lagrangian, then the domain defined by the constraint $\text{Tr}_{D,i}(u) \in \Gamma(Y_i; W_i)$ is equal to its adjoint domain, hence the corresponding realization is self-adjoint. (Indeed, vanishing of the Green pairing for all test sections forces the boundary trace to lie in the symplectic orthogonal $W_i^{\perp} = W_i$.) Applying this at each stratum and using that $W_{i,\text{can}}$ is Lagrangian (Theorem 5.1), we conclude that $D_{\text{can}}$ is self-adjoint.

*Fredholmness.* By Theorem 6.3, each stratum admits a constant-multiplicity strip about the $L^2$ threshold. In that strip, Theorem 5.1 provides the Calderón projector and the associated elliptic projection boundary condition at each depth. On a distinguished neighborhood of each stratum, these data define a fully elliptic iterated edge boundary problem and hence admit a local parametrix in the iterated edge calculus with remainders that gain one iterated-edge derivative; see Appendix D, especially Appendix D.5. By Lemma 7.3, such smoothing remainders are compact on $L^2$ after localization by cutoffs. Gluing the resulting local parametrices using Proposition 7.5 (and cutoff stability from Proposition 7.2) yields a global parametrix for $D_{\text{can}}$ modulo compact remainders. Therefore $D_{\text{can}}$ has finite-dimensional kernel and cokernel and closed range, i.e. it is Fredholm. The Hilbert complex statement follows from standard Hodge-de Rham theory for self-adjoint Fredholm realizations (equivalently, from the abstract Hilbert complex formalism); see [10]. □

### 7.3. Topological identification with iterated IBC intersection cohomology.

**Theorem 7.7 (Global iterated FS-CGM consequence).** *There is a canonical isomorphism*

$$H^{\bullet}(\text{Dom}(d_{\text{can}}), d_{\text{can}}) \cong \text{IH}_{W_{\text{can}}}^{\bullet}(X),$$

where the right-hand side is the iterated ideal boundary condition (mezzoperversity) intersection cohomology dictated by $W_{\text{can}}$. In the iterated Witt case (no boundary data at any depth), this reduces to middle-perversity intersection cohomology.

*Proof.* Appendix F constructs, for the mezzoperversity datum $W_{\text{can}}$, a fine sheaf complex $L_{W_{\text{can}}}^{\bullet}$ whose global sections are exactly the de Rham domain complex $(\text{Dom}(d_{\text{can}}), d_{\text{can}})$ and whose local cohomology agrees with the iterated Deligne truncation dictated by $W_{\text{can}}$. By the axiomatic characterization and uniqueness of Deligne-type intersection complexes [14, 6, 8], there is a canonical quasi-isomorphism

$$L_{W_{\text{can}}}^{\bullet} \simeq \text{IC}_{W_{\text{can}}}.$$

Since $L_{W_{\text{can}}}^{\bullet}$ is fine, its hypercohomology is computed by global sections, so

$$H^{\bullet}(\text{Dom}(d_{\text{can}}), d_{\text{can}}) \cong \mathbb{H}^{\bullet}(X; L_{W_{\text{can}}}^{\bullet}) \cong \mathbb{H}^{\bullet}(X; \text{IC}_{W_{\text{can}}}) = \text{IH}_{W_{\text{can}}}^{\bullet}(X),$$

as claimed. □

## 8. DEPTH-TWO AS A WORKED CASE AND COMPARISON WITH THE CONDITIONAL SCHEMA

This section unwraps the general induction of Section 7 in the first nontrivial situation, namely stratification depth two. The point is not to obtain new phenomena, but to provide a referee-checkable template: we spell out the two-step construction of the canonical mezzoperversity and indicate precisely where the domain-dependent stability theorem enters to remove any external hypotheses.
8.1. **Depth-two geometry in iterated conic charts.** Assume $X \subset \mathbb{CP}^N$ is smoothly stratified of depth two, with strata

$$Y_2 \subset Y_1 \subset X, \quad X^{\text{reg}} = X \setminus Y_1.$$

Fix compatible control data as in Section 2. For points of $Y_2$ there are nested tubular neighborhoods $T_2 \subset T_1$ and control functions $\rho_i$ with $x_i := \rho_i^{1/2}$. By Proposition 2.3, every point of $T_2$ admits an iterated conic chart of the form

$$(8.1.1) \qquad (T_2, x_1, x_2, \pi_1, \pi_2) \cong ([0, \varepsilon)_{x_1} \times [0, \varepsilon)_{x_2} \times U, x_1, x_2, \text{pr}_U, \text{pr}_U) \times L_2,$$

**Depth-two resolution as a manifold with corners.** In the depth-two worked case we use the resolved space $\tilde{X}$ (a manifold with corners) to organize the two-step analysis along the boundary hypersurfaces $H_2$ (deeper) and $H_1$ (outer), together with their fibrations over $Y_2$ and $Y_1$. Figure 4 fixes notation and emphasizes where the two boundary problems live: the corner $H_1 \cap H_2$ is precisely the locus where the deeper IBC enters the outer tangential families.

<figure><img src="image_4.png" /><figcaption>FIGURE 4. Resolved depth-two geometry: boundary hypersurfaces and fibrations (schematic).</figcaption></figure>

where $U \subset Y_2$ is open and $L_2$ is the (smooth) deepest link. In particular, the resolution viewpoint of Subsection 2.4 identifies the iterated blow-up $\tilde{X}$ with a manifold with corners having two boundary hypersurfaces $H_1, H_2$ meeting in a codimension-two corner $H_1 \cap H_2$, and boundary fibrations

$$\phi_2 : H_2 \to Y_2, \quad \phi_1 : H_1 \to Y_1.$$

The fiber of $\phi_2$ is $L_2$ (a smooth closed manifold), while the fiber of $\phi_1$ is the depth-one link $L_1$ whose unique singular stratum corresponds to $Y_2$.

8.2. **The inner step: the canonical Cauchy data bundle on the deepest link.** Along $Y_2$ the iterated wedge expansion theorem 3.1 and the normal form theorem 4.1 reduce the Hodge-de Rham operator near $H_2$ to a cone model with tangential family

$$A_{2,y} \text{ acting on forms over the smooth link fiber } (L_2)_y, \quad y \in Y_2.$$

Since $(L_2)_y$ is smooth (depth 0), this is the classical (non-iterated) Calderón situation. Let $W_{2,\text{can}} \to Y_2$ denote the resulting Cauchy-data bundle furnished by the parameter-dependent Calderón package of Appendix D (equivalently, the depth-0 specialization of Theorem 5.1).

**Proposition 8.1 (Canonical inner data at depth two).** The Cauchy-data bundle $W_{2,\text{can}} \to Y_2$ is a smooth vector subbundle of the depth-two trace bundle. Moreover, the associated inner ideal boundary condition determines a self-adjoint Fredholm realization of the model operator near $H_2$, and the corresponding tangential family satisfies the constant-strip hypothesis in a uniform neighborhood of the $L^2$ threshold.

*Proof.* Smoothness of the Cauchy data bundle and the elliptic projection boundary condition are proved in Appendix D (for smooth fibers, one may also appeal to the classical theory of smooth families of APS-type boundary problems; see [2, 3, 4]). Uniform constant-strip control follows from
the indicial description in Section 6, specifically Proposition E.7, applied to the depth-0 tangential family $A_{2,y}$. □

**8.3. The outer step: stability of the tangential family with deeper IBC.** We now move to the stratum $Y_1$. The resolution hypersurface $H_1$ fibers over $Y_1$ with fiber $L_1$ of depth one, and on $L_1$ we impose the deeper boundary condition coming from $W_{2,\text{can}}$ along its singular stratum. The iterated Dirac normal form of Section 4 produces, locally uniformly in $y$ (and hence uniformly on each fixed stratum component by Lemma 2.5), the wedge normal form

$$ (8.3.1) \qquad D = \Gamma_1 \left( \partial_{x_1} + \frac{1}{x_1} A_{1,y}^{W_{2,\text{can}}} + B_1(x_1) \right) \quad (y \in Y_1), $$

where $A_{1,y}^{W_{2,\text{can}}}$ is the tangential operator on the link fiber $(L_1)_y$ endowed with the deeper domain determined by $W_{2,\text{can}}$.

**Proposition 8.2** (Automatic outer stability in depth two). *The tangential family $y \mapsto A_{1,y}^{W_{2,\text{can}}}$ satisfies the stability and constant-strip hypotheses along $Y_1$ in the sense of Section 6. In particular, its finite-dimensional Cauchy-data spaces fit together to a smooth bundle $W_{1,\text{can}} \to Y_1$ given by Theorem 5.1.*

*Proof.* This is precisely the depth-two instance of the domain-dependent stability theorem (Theorem 6.3); for convenience the specialization is recorded as Theorem E.8. Once stability and the constant-strip control hold, the parameter-dependent Calderón package of Theorem 5.1 produces the outer Cauchy-data bundle $W_{1,\text{can}}$ and the associated elliptic projection boundary condition along $H_1$. □

**8.4. Unconditional depth-two realization and IH identification.** We can now assemble the depth-two canonical mezzoperversity

$$ W_{\text{can}} = (W_{1,\text{can}}, W_{2,\text{can}}). $$

The global canonical realization $D_{\text{can}}$ is defined by imposing, at $Y_2$ and then at $Y_1$, the elliptic projection boundary conditions determined by $W_{2,\text{can}}$ and $W_{1,\text{can}}$; see Definition 7.1.

**Theorem 8.3** (Unconditional depth-two one-step iteration). *Let $X \subset \mathbb{CP}^N$ have stratification depth two with strata $Y_2 \subset Y_1 \subset X$. Then the canonical inner and outer Cauchy-data bundles $W_{2,\text{can}} \to Y_2$ and $W_{1,\text{can}} \to Y_1$ are well-defined and smooth, the global canonical realization $D_{\text{can}}$ is self-adjoint and Fredholm, and*

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

*Proof.* The construction of $W_{2,\text{can}}$ is Proposition 8.1; the construction of $W_{1,\text{can}}$ is Proposition 8.2. With $W_{\text{can}}$ thus defined, self-adjoint Fredholmness of $D_{\text{can}}$ follows from Theorem 7.6, and the topological identification with iterated ideal boundary condition intersection cohomology is Theorem 7.7. □

**Remark 8.4** (Where the “conditional schema” becomes unconditional). In a purely one-step framework, one would *assume* outer stability/constant-strip control for the family $A_{1,y}^{W_2}$ in (8.3.1) and then define $W_1$ by a Calderón package. Proposition 8.2 shows that, once the deeper data are chosen canonically on $Y_2$, these outer hypotheses are automatic. This is the mechanism that makes the depth-two iteration unconditional.
**8.5. Concrete projective examples.** The preceding sections are intentionally structural: they construct canonical analytic packages and identify the resulting cohomology with intersection co-homology for the mezzoperversity dictated by the iterated boundary conditions. The following examples are included only as “sanity checks” illustrating how the abstract mechanisms specialize in familiar projective situations. They are not used elsewhere in the paper.

**Lemma 8.5** (Additivity of the Deligne truncation construction). Let $j: U \hookrightarrow X$ be an open embedding and fix a perversity $\bar{p}$. Given a constructible complex $K \in D_c^b(U)$, let $IC_{\bar{p}}(j, K) \in D_c^b(X)$ denote the Deligne-type extension complex obtained from $K$ by the inductive truncated pushforward procedure of Appendix F (starting from $Rj_*K$ on $X$). Then for any $K_1, K_2 \in D_c^b(U)$ there is a natural isomorphism

$$IC_{\bar{p}}(j, K_1 \oplus K_2) \cong IC_{\bar{p}}(j, K_1) \oplus IC_{\bar{p}}(j, K_2).$$

In particular, if $U = \bigcup_{\alpha} U_{\alpha}$ is the decomposition into connected components and $\mathbb{Z}_U \cong \bigoplus_{\alpha} \mathbb{Z}_{U_{\alpha}}$, then the Deligne intersection complex $IC_{\bar{p}}(X; \mathbb{Z})$ (as defined in Appendix F) splits canonically as

$$IC_{\bar{p}}(X; \mathbb{Z}) \cong \bigoplus_{\alpha} IC_{\bar{p}}(j_{\alpha}, \mathbb{Z}_{U_{\alpha}}),$$

where $j_{\alpha}: U_{\alpha} \hookrightarrow X$ denotes the inclusion.

**Proof.** Appendix F realizes $IC_{\bar{p}}$ by the standard Deligne truncation procedure: starting from $Rj_*$ on the top stratum and then iteratively applying derived pushforward along the successive open inclusions of skeleta together with standard truncations $\tau_{\le \ell}$ dictated by the perversity $\bar{p}$. Both derived pushforward along an open embedding and the standard truncation functors commute with finite direct sums. Consequently the entire inductive construction commutes with finite direct sums, which yields the displayed isomorphism. The componentwise splitting follows by applying the same additivity to $\mathbb{Z}_U \cong \bigoplus_{\alpha} \mathbb{Z}_{U_{\alpha}}$. $\square$

**Example 8.6** (Projective cones: recovery of the classical conic picture). Let $V \subset \mathbb{CP}^{N-1}$ be a smooth projective variety and let $X \subset \mathbb{CP}^N$ be the projective cone over $V$, with vertex $v \in X$. Equip $X$ with the depth-one Thom-Mather stratification whose singular stratum is $Y_1 = \{v\}$ and whose regular stratum is $Y_0 = X^{\text{reg}} = X \setminus \{v\}$.

Near $v$, the restriction of the ambient Fubini-Study metric admits the conic normal form described in Sections 3 and 4 (see Appendix B and Appendix C): in particular, on a conic neighborhood it is quasi-isometric to $dx^2 + x^2g_L$, where the link $L$ is the unit circle bundle of $O(-1)|_V$. Since $\dim L$ is odd, the middle-degree obstruction responsible for non-uniqueness of self-adjoint extensions is absent (this is the classical Witt situation for conic metrics). Consequently, there is no boundary data to choose at $v$ (equivalently, the critical space in the trace bundle is trivial), and the canonical realization $D_{\text{can}}$ coincides with the unique self-adjoint realization singled out by Cheeger’s conic analysis [12, 13]. In particular, the main identification theorem Theorem 7.7 recovers the expected equality

$$H^{\bullet}(\text{Dom}(d_{\text{can}}), d_{\text{can}}) \cong \text{IH}_{\bar{m}}^{\bullet}(X),$$

where the right-hand side can be computed (if desired) by the standard cone formulas of Goresky-MacPherson [14].

**Example 8.7** (A simple SNC example: union of two hyperplanes). Let $H_0, H_1 \subset \mathbb{CP}^n$ be distinct hyperplanes and set $X := H_0 \cup H_1 \subset \mathbb{CP}^n$. This is a reduced projective algebraic set with a simple normal-crossings singular locus $Y := H_0 \cap H_1 \cong \mathbb{CP}^{n-2}$. Consider the natural Thom-Mather stratification with strata $Y$ and $X^{\text{reg}} = (H_0 \setminus Y) \cup (H_1 \setminus Y)$.

Along $Y$, a normal slice identifies locally with the union of two complex lines crossing transversely, so the link of a point of $Y$ in $X$ is a disjoint union of two circles (the Hopf link in $S^3$). In particular, this is again a (depth-one) Witt situation, so there is no nontrivial boundary data at $Y$ and the canonical mezzoperversity is forced.
Write $U_i := H_i \setminus Y$ so that $X^{\text{reg}} = U_0 \sqcup U_1$ is a disjoint union of two connected components. Then $\mathbb{Z}_{X^{\text{reg}}} \cong \mathbb{Z}_{U_0} \oplus \mathbb{Z}_{U_1}$, and Lemma 8.5 gives a canonical splitting

$$IC_{\bar{m}}(X; \mathbb{Z}) \cong IC_{\bar{m}}(j_0, \mathbb{Z}_{U_0}) \oplus IC_{\bar{m}}(j_1, \mathbb{Z}_{U_1}),$$

where $j_i : U_i \hookrightarrow X$. Since each closure $\bar{U}_i = H_i \cong \mathbb{CP}^{n-1}$ is smooth, the Deligne truncation procedure of Appendix F yields $IC_{\bar{m}}(j_i, \mathbb{Z}_{U_i}) \cong \mathbb{Z}_{H_i}$, and therefore we obtain the explicit identification

$$\mathrm{IH}_{\bar{m}}^{k}(X; \mathbb{Z}) \cong \mathrm{H}^{k}(H_{0}; \mathbb{Z}) \oplus \mathrm{H}^{k}(H_{1}; \mathbb{Z}) \cong \begin{cases} \mathbb{Z}^{2}, & k \in \{0, 2, 4, \dots, 2n-2\}, \\ 0, & k \text{ odd.} \end{cases}$$

(In particular, $\mathrm{IH}_{\bar{m}}^{0}(X) \cong \mathbb{Z}^{2}$: allowable 1-chains cannot cross the codimension-2 stratum $Y$, so intersection (co)homology detects the two top-dimensional components.) Therefore Theorem 7.7 specializes here to a completely concrete statement: the canonical de Rham Hilbert complex on $(X^{\text{reg}}, g_{\text{FS}})$ has cohomology groups of the above ranks.

For comparison, the ordinary cohomology of $X = H_0 \cup H_1$ is smaller in low degrees: a Mayer–Vietoris computation gives $H^{2k}(X; \mathbb{Z}) \cong \mathbb{Z}$ for $0 \le k \le n-2$ and $H^{2n-2}(X; \mathbb{Z}) \cong \mathbb{Z}^2$, with all odd groups vanishing.

**Example 8.8** (A depth-two SNC arrangement: three hyperplanes in general position). Let $H_1, H_2, H_3 \subset \mathbb{CP}^n$ be three hyperplanes in general position, with $n \ge 3$, and set

$$X := H_1 \cup H_2 \cup H_3 \subset \mathbb{CP}^n.$$

Then $X$ is a reduced projective algebraic set whose singular locus has two strata levels. Write

$$Y_2 := H_1 \cap H_2 \cap H_3 \cong \mathbb{CP}^{n-3}, \quad Y_1^{ij} := (H_i \cap H_j) \setminus Y_2 \quad (1 \le i < j \le 3),$$

and let $Y_0 = X^{\text{reg}}$ be the union of the open pieces $H_i \setminus \bigcup_{j \ne i} (H_i \cap H_j)$. The natural incidence chain $Y_2 \subset \overline{Y_1^{ij}} \subset X$ shows that this is a Thom–Mather stratification of depth two (with three connected components of the depth-one stratum).

At a point $y \in Y_2$, choose affine holomorphic coordinates so that, in a transverse complex 3-slice, $X$ is locally given by the normal-crossings model

$$\{z_1 z_2 z_3 = 0\} \subset \mathbb{C}^3.$$

The associated (topological) link of $y$ in $X$ is

$$L_2 = (\{z_1 z_2 z_3 = 0\} \cap S^5) = (S^5 \cap \{z_1 = 0\}) \cup (S^5 \cap \{z_2 = 0\}) \cup (S^5 \cap \{z_3 = 0\}),$$

i.e. a union of three 3-spheres $S_1^3 \cup S_2^3 \cup S_3^3$ with pairwise intersections $S_i^3 \cap S_j^3 = S^5 \cap \{z_i = z_j = 0\} \cong S^1$ given by three embedded circles. Thus $L_2$ is a compact smoothly stratified space of depth one whose singular strata are these circles, and the links of the circle strata are 0-dimensional (two-point) spaces. This is a basic geometric configuration where the *inner* Calderón package of the present paper applies nontrivially on a stratified link.

Because the arrangement is homogeneous, the link fibrations over $Y_2$ and over each $Y_1^{ij}$ are globally trivial (up to canonical isomorphism), and the induced Fubini–Study wedge structure has constant model operators. Consequently the kernel bundles and critical-strip data are constant along the strata, and the domain-dependent stability statements used in the induction are immediate in this case. The construction of Theorem 8.3 therefore produces a canonical depth-two mezzoperversity $\mathcal{W}_{\text{can}} = (W_{1,\text{can}}, W_{2,\text{can}})$ (with $W_{1,\text{can}}$ specified on each component $Y_1^{ij}$) and a global canonical self-adjoint Fredholm realization $D_{\text{can}}$ of $\partial_{\text{dR}}$ on $(X^{\text{reg}}, g_{\text{FS}})$, with

$$H^{\bullet}(\text{Dom}(d_{\text{can}}), d_{\text{can}}) \cong \mathrm{IH}_{\mathcal{W}_{\text{can}}}^{\bullet}(X)$$

by Theorem 7.7. In fact, in this SNC arrangement all links appearing in the iterated cone models have *odd* dimension (the depth-one links along $Y_1^{ij}$ are 1-dimensional, and the depth-two link
$L_2$ is 3-dimensional). Therefore the mid-degree trace bundles vanish at every depth level, the mezzoperversity is forced, and

$$ \mathrm{IH}_{W_{\mathrm{can}}}^{\bullet}(X) = \mathrm{IH}_{\bar{m}}^{\bullet}(X). $$

Write $U_i := H_i \setminus \bigcup_{j \neq i} (H_i \cap H_j)$ so that $X^{\mathrm{reg}} = U_1 \sqcup U_2 \sqcup U_3$ is a disjoint union of three connected components. As in Theorem 8.7, Lemma 8.5 yields a componentwise splitting of the middle intersection complex. Since each closure $\bar{U}_i = H_i \cong \mathbb{CP}^{n-1}$ is smooth,

$$ \mathrm{IH}_{\bar{m}}^{k}(X; \mathbb{Z}) \cong \bigoplus_{i=1}^{3} H^{k}(H_{i}; \mathbb{Z}) \cong \begin{cases} \mathbb{Z}^{3}, & k \in \{0, 2, 4, \dots, 2n-2\}, \\ 0, & k \text{ odd.} \end{cases} $$

In particular, when $n=3$ one gets the fully explicit ranks

$$ \mathrm{IH}_{W_{\mathrm{can}}}^{0}(X) \cong \mathbb{Z}^{3}, \quad \mathrm{IH}_{W_{\mathrm{can}}}^{2}(X) \cong \mathbb{Z}^{3}, \quad \mathrm{IH}_{W_{\mathrm{can}}}^{4}(X) \cong \mathbb{Z}^{3}, $$

and $\mathrm{IH}_{W_{\mathrm{can}}}^{j}(X) = 0$ for $j \notin \{0, 2, 4\}$.

For comparison, the ordinary cohomology of $X$ (which does identify the componentwise degree-2 classes across the intersection lines) is smaller in low degrees; for $n=3$ a Mayer–Vietoris calculation gives

$$ H^{0}(X; \mathbb{Z}) \cong \mathbb{Z}, \quad H^{2}(X; \mathbb{Z}) \cong \mathbb{Z}, \quad H^{4}(X; \mathbb{Z}) \cong \mathbb{Z}^{3}, $$

with all odd groups vanishing.

## 9. OUTLOOK TOWARD M4 (MIDDLE-PERVERSITY CHARACTERIZATION)

Sections 2–6 produce the canonical mezzoperversity $W_{\mathrm{can}} = (W_{1,\mathrm{can}}, \dots, W_{d,\mathrm{can}})$ and the associated global canonical realization $D_{\mathrm{can}}$ of the de Rham operator. The main outcome is the identification

$$ H^{\bullet}(\mathrm{Dom}(d_{\mathrm{can}}), d_{\mathrm{can}}) \cong \mathrm{IH}_{W_{\mathrm{can}}}^{\bullet}(X) $$

from Theorem 7.7. The remaining step in the overall program is to isolate (compare the mezzoperversity/IBC framework in [1, 5]) geometric/analytic hypotheses guaranteeing that this canonical theory coincides with the classical (middle-perversity) intersection theory.

**9.1. The M4 target as a precise comparison problem.** Let $\mathrm{IC}_{\bar{m}}$ denote the classical middle-perversity intersection complex on $X$ (with the same coefficient system as in $\mathrm{IH}_W$). For general non-Witt links, the iterated IBC complexes $\mathrm{IC}_W$ depend on the mezzoperversity choices, and a priori $\mathrm{IC}_W$ need not be quasi-isomorphic to $\mathrm{IC}_{\bar{m}}$. In the present analytic approach, the entire dependence of $\mathrm{IC}_W$ on $W$ is concentrated in the critical-degree extension data along the strata $Y_i$, i.e. in the choice of Lagrangian subbundles $W_i \subset H_i$ inside the trace bundles $H_i = \ker(A_i^{W>i}) \to Y_i$; see Appendix F and Appendix D.5.

**Problem 9.1 (M4: canonical vs. middle extension).** Give checkable conditions on $(X, g_{\mathrm{FS}})$ and on the link families $\{L_{i,y}\}_{y \in Y_i}$ ensuring

$$ \mathrm{IC}_{W_{\mathrm{can}}} \simeq \mathrm{IC}_{\bar{m}} \iff \mathrm{IH}_{W_{\mathrm{can}}}^{\bullet}(X) \cong \mathrm{IH}_{\bar{m}}^{\bullet}(X), $$

and hence $H^{\bullet}(\mathrm{Dom}(d_{\mathrm{can}}), d_{\mathrm{can}}) \cong \mathrm{IH}_{\bar{m}}^{\bullet}(X)$.

In the iterated Witt case, all trace bundles vanish and the mezzoperversity is forced; the above equivalence is already recorded in Theorem 7.7. Thus the content of Problem 9.1 is concentrated on those strata for which the mid-degree trace bundle $H_i = \ker(A_i^{W>i})$ is nontrivial.
**9.2. A local-to-global reduction usable by referees.** The constructions in Appendix F realize $\mathrm{IC}_W$ by an explicit Deligne-type inductive procedure with a single non-formal choice at each stratum: the Lagrangian subbundle $W_i \subset H_i$, where $H_i \to Y_i$ is the trace bundle and $\langle\cdot, \cdot\rangle$ is the induced Green pairing (Appendix D.5). In particular, the comparison with $\mathrm{IC}_{\bar{m}}$ reduces to a comparison of these local extension data.

**Proposition 9.2 (Reduction of $W_{\mathrm{can}}$ vs. $\mathrm{IC}_{\bar{m}}$).** Assume that for each stratum $Y_i$ with nontrivial trace bundle $H_i$, one can identify a canonically defined Lagrangian subbundle $W_{i,\mathrm{mid}} \subset H_i$ (depending only on the middle-perversity Deligne extension data along $Y_i$) such that:

(M4.1) $W_{i,\mathrm{mid}}$ is smooth and incidence-compatible in the sense of Section 2;

(M4.2) the corresponding mezzoperversity $W_{\mathrm{mid}} = (W_{1,\mathrm{mid}}, \dots, W_{d,\mathrm{mid}})$ satisfies $\mathrm{IC}_{W_{\mathrm{mid}}} \simeq \mathrm{IC}_{\bar{m}}$;

(M4.3) for all $i$, $W_{i,\mathrm{can}} = W_{i,\mathrm{mid}}$ as subbundles of $H_i$.

Then $\mathrm{IC}_{W_{\mathrm{can}}} \simeq \mathrm{IC}_{\bar{m}}$ and hence $\mathrm{IH}^\bullet_{W_{\mathrm{can}}}(X) \simeq \mathrm{IH}^\bullet_{\bar{m}}(X)$.

*Proof.* By Appendix F, $\mathrm{IC}_W$ is obtained inductively by applying a truncated pushforward across each depth and inserting, precisely at the critical degree, the mezzoperversity-controlled extension determined by $W_i \subset H_i$. Hypotheses (M4.1)-(M4.2) give a mezzoperversity $W_{\mathrm{mid}}$ producing the classical middle extension. Hypothesis (M4.3) identifies the extension data defining $\mathrm{IC}_{W_{\mathrm{can}}}$ with those defining $\mathrm{IC}_{W_{\mathrm{mid}}}$ at every depth, so the inductive Deligne constructions agree step-by-step and yield a quasi-isomorphism $\mathrm{IC}_{W_{\mathrm{can}}} \simeq \mathrm{IC}_{W_{\mathrm{mid}}} \simeq \mathrm{IC}_{\bar{m}}$. $\square$

Thus M4 can be implemented as: (i) construct $W_{i,\mathrm{mid}}$ by a purely topological (Deligne) criterion, and (ii) identify the analytically constructed $W_{i,\mathrm{can}} = \mathrm{ran}(C_i) \subset H_i$ with $W_{i,\mathrm{mid}}$ for every stratum.

**9.3. A polarized Hodge package as a concrete source of $W_{i,\mathrm{mid}}$.** The trace bundle $H_i = \ker(A_i^{W>i})$ carries a natural symplectic structure induced by the Green pairing. A standard way to manufacture a smooth Lagrangian subbundle is to produce a smoothly varying compatible complex structure $J_i$ on $H_i$ together with a $g_i$-orthogonal involution $T_i$ anti-commuting with $J_i$ (Lemma 9.5 below). In geometric situations with Kähler links, such data can be constructed directly from fiberwise Hodge theory.

**Conjecture 9.3 (Hodge-polarized middle boundary data).** For Fubini-Study wedge metrics, the trace bundle $H_i \to Y_i$ carries a canonical smooth polarization $J_i$ determined by the limiting Kähler package on the link family $\{L_{i,y}\}_{y\in Y_i}$ with deeper boundary conditions $W_{>i}$. The corresponding “positive” Lagrangian subbundle $W_{i,\mathrm{mid}} \subset H_i$ is incidence-compatible and yields $\mathrm{IC}_{W_{\mathrm{mid}}} \simeq \mathrm{IC}_{\bar{m}}$.

**Conjecture 9.4 (Identification of the canonical and middle data).** Under the polarization from Conjecture 9.3, the analytically defined subbundle $W_{i,\mathrm{can}} = \mathrm{ran}(C_i) \subset H_i$ coincides with $W_{i,\mathrm{mid}}$ for every stratum $Y_i$. Consequently, $\mathrm{IC}_{W_{\mathrm{can}}} \simeq \mathrm{IC}_{\bar{m}}$ and $H^\bullet(\mathrm{Dom}(d_{\mathrm{can}}), d_{\mathrm{can}}) \simeq \mathrm{IH}^\bullet_{\bar{m}}(X)$.

**9.4. A lemma chain implementing (M4.1)-(M4.3) in the projective Fubini-Study setting.** The two conjectures above can be decomposed into a sequence of concrete statements that can be checked and proved stratum-by-stratum, and then glued by the incidence formalism of Section 2. The point is that a *choice of a Lagrangian* inside the trace bundle $H_i = \ker(A_i^{W>i})$ can be encoded by an *orthogonal involution* for a compatible metric, and such an involution is what Hodge theory naturally produces in Kähler situations.

**Lemma 9.5 (From a compatible involution to a Lagrangian).** Let $(H, \omega)$ be a finite-dimensional real symplectic vector space and let $J: H \to H$ be a compatible complex structure, i.e. $J^2 = -\mathrm{Id}$, $\omega(Ju, Jv) = \omega(u, v)$, and the bilinear form $g(u, v) := \omega(u, Jv)$ is positive definite. Suppose $T: H \to H$ is a $g$-orthogonal involution such that $TJ = -JT$. Then the $+1$-eigenspace $W := \ker(T - \mathrm{Id})$ is Lagrangian in $(H, \omega)$, and one has a direct sum decomposition $H = W \oplus JW$.
**Proof.** Since $T$ is $g$-orthogonal and $TJ = -JT$, the $\pm 1$-eigenspaces $W^\pm := \ker(T \mp \mathrm{Id})$ are $g$-orthogonal and satisfy $JW^+ = W^-$. For $u, v \in W = W^+$ we have $\omega(u, v) = g(u, Jv) = g(u, -T(Jv)) = -g(Tu, Jv) = -g(u, Jv)$, hence $\omega(u, v) = 0$; thus $W$ is isotropic. Because $J$ is an isomorphism $W \to W^-$, we obtain $\dim W = \frac{1}{2} \dim H$, so $W$ is maximally isotropic, i.e. Lagrangian. The decomposition $H = W \oplus JW$ follows from $H = W^+ \oplus W^-$ and $W^- = JW$. $\square$

**Definition 9.6** (Hodge involution on a trace bundle). Fix a stratum $Y_i$ and assume that the trace bundle $H_i \to Y_i$ carries a smoothly varying compatible complex structure $J_i$ (as in Conjecture 9.3) and a smooth family of $g_i$-orthogonal involutions $T_i$ with $T_i J_i = -J_i T_i$. We call $T_i$ a *Hodge involution* on $H_i$ and define the associated *Hodge Lagrangian* by

$$W_{i, \mathrm{Hdg}} := \ker(T_i - \mathrm{Id}) \subset H_i.$$

**Proposition 9.7** (Fiberwise Kähler package $\Rightarrow$ Hodge involution). Assume that over a stratum $Y_i$ the link fibration $\pi_i : \partial T_i \to Y_i$ carries a smooth family of Kähler structures on the fibers $L_{i,y}$ (in particular, a smooth vertical complex structure and a smooth fiberwise Kähler metric). Assume moreover that the deeper boundary data $W_{>i}$ is defined over $\mathbb{R}$ and is preserved by complex conjugation on vertical forms (this holds for the canonical deeper data constructed in Section 7). Let $H_i^\mathbb{C} \to Y_i$ denote the complex trace bundle obtained by complexifying the constant-rank kernel bundle $H_i = \ker(A_i^{W>i})$. Then $H_i^\mathbb{C}$ is a smooth complex vector bundle equipped with the fiberwise Hodge-$L^2$ Hermitian form

$$h_{i,y}(\alpha, \beta) := \int_{L_{i,y}} \alpha \wedge \ast_{L_{i,y}} \bar{\beta}, \quad \alpha, \beta \in H_{i,y}^{\mathbb{C}},$$

and on the underlying real bundle $H_i = (H_i^\mathbb{C})_{\mathbb{R}}$ the operators

$$J_i(\alpha) := i\alpha, \quad T_i(\alpha) := \bar{\alpha}$$

are smooth, $g_i$-orthogonal (where $g_i := \Re h_i$), and satisfy $T_i J_i = -J_i T_i$. Writing $\omega_i := \Im h_i$, the pair $(\omega_i, J_i)$ is compatible in the sense of Lemma 9.5. If, in addition, the Green pairing $\langle\cdot, \cdot\rangle$ on $H_i$ agrees with $\omega_i$, then $W_{i, \mathrm{Hdg}} = \ker(T_i - \mathrm{Id})$ is a smooth Lagrangian subbundle of $H_i$.

**Proof.** Because $A_i^{W>i}$ is a smooth family of self-adjoint elliptic operators with constant-rank kernel, standard spectral-projection arguments show that $\ker(A_i^{W>i})$ forms a smooth vector bundle; complexifying gives the smooth complex bundle $H_i^\mathbb{C}$. The expression defining $h_{i,y}$ is the usual Hodge-$L^2$ Hermitian form on harmonic forms, and it depends smoothly on $y$ because the fiberwise metric and the harmonic projection depend smoothly on parameters. On the underlying real bundle $H_i$, multiplication by $i$ is a smooth complex structure $J_i$, complex conjugation is a real-linear involution $T_i$, and one checks directly that $T_i^2 = \mathrm{Id}$, $T_i J_i = -J_i T_i$, and that both preserve $h_i$; hence $T_i$ is $g_i$-orthogonal. Finally, for $u, v \in H_i$ we have $h_i(u, v) = g_i(u, v) + i\omega_i(u, v)$ and $g_i(u, v) = \omega_i(u, J_i v)$, so $(\omega_i, J_i)$ is compatible. If $\langle\cdot, \cdot\rangle = \omega_i$, Lemma 9.5 shows that $W_{i, \mathrm{Hdg}} = \ker(T_i - \mathrm{Id})$ is Lagrangian, and smoothness follows since $\frac{1}{2}(\mathrm{Id} + T_i)$ is a smooth bundle projection onto $W_{i, \mathrm{Hdg}}$. $\square$

**Lemma 9.8** (Smoothness and incidence compatibility). If $J_i$ and $T_i$ in Definition 9.6 are induced from a geometric structure on the link fibration $\pi_i : \partial T_i \to Y_i$ (for instance as in Proposition 9.7), then $W_{i, \mathrm{Hdg}} \subset H_i$ is a smooth Lagrangian subbundle and is incidence-compatible in the sense of Section 2.

**Proof.** Since $T_i$ is a smooth involution, the operator $\Pi_i := \frac{1}{2}(\mathrm{Id} + T_i)$ is a smooth bundle projection with $\mathrm{ran}(\Pi_i) = W_{i, \mathrm{Hdg}}$; hence $W_{i, \mathrm{Hdg}}$ is a smooth subbundle. Lagrangianity is a fiberwise statement and follows from Lemma 9.5. For incidence compatibility, let $Y_j$ be a stratum contained in the closure of $Y_i$. Over an incidence neighborhood, the iterated control data of Section 2 provides canonical identifications between the relevant link fibrations and between the corresponding trace bundles (transport along the retractions and the compatible trivializations used throughout Sections 3–4).
If $J_\bullet$ and $T_\bullet$ arise from geometric operators on the link family (Hodge star, complex conjugation, or equivalent fiberwise constructions), these identifications intertwine $J_i, T_i$ with $J_j, T_j$. Therefore the induced identification $H_i|_{Y_j} \cong H_j$ carries $W_{i,Hdg}|_{Y_j} = \ker(T_i - \mathrm{Id})|_{Y_j}$ onto $\ker(T_j - \mathrm{Id}) = W_{j,Hdg}$, which is exactly the incidence condition. $\square$

**Proposition 9.9** (Cone-model identification of the Calderón Lagrangian). Work near $Y_i$ in an adapted product-type wedge metric model so that the local Dirac normal form of Section 4 reduces, at leading order, to the exact cone operator

$$D_{\text{cone}} = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} A_{i,y}^{W>i} \right).$$

Assume that the kernel bundle $H_i = \ker(A_{i,y}^{W>i})$ carries a Hodge involution $T_i$ as in Definition 9.6 and that the model boundary pairing on $H_i$ agrees with the Green pairing. Then the Calderón Cauchy data space $W_{i,\text{can}} = \text{ran}(C_i) \subset H_i$ equals $W_{i,Hdg}$.

*Proof.* Decompose the link Hilbert space into the $L^2$-orthogonal sum $H_{i,y} \oplus H_{i,y}^{\perp}$, where $H_{i,y} = \ker(A_{i,y}^{W>i})$. On $H_{i,y}^{\perp}$, the spectral decomposition of the self-adjoint operator $A_{i,y}^{W>i}$ yields modes with eigenvalues $\lambda \neq 0$. The corresponding separated solutions of $D_{\text{cone}}u = 0$ are of the form $u(x_i) = x_i^{-\lambda} v$ (up to the standard $\Gamma_i$-coupling), and the $L^2$-threshold selects a unique half of these modes (equivalently the APS choice encoded by the Calderón projector on the nonzero spectrum).; cf. [2]. Thus the only undetermined contribution to the Cauchy data arises from the zero-mode sector $H_{i,y}$, where $A_{i,y}^{W>i} = 0$ and the equation reduces to $\partial_{x_i} u = 0$ after applying $\Gamma_i^{-1}$; hence solutions are constant in $x_i$ to leading order and their boundary values identify with elements of $H_{i,y}$.

In this sector the Cauchy data of any self-adjoint elliptic boundary condition must be a Lagrangian subspace for the Green pairing (Appendix D.5). The Calderón projector $C_i$ selects the specific Lagrangian range $W_{i,\text{can}}(y) \subset H_{i,y}$ realized as the traces of solutions produced by the Poisson operator $K_i$ (Appendix D). Under the hypothesis that the boundary pairing agrees with the Green pairing, the involution $T_i(y)$ determines a Lagrangian $W_{i,Hdg}(y) = \ker(T_i(y) - \mathrm{Id})$. Moreover, the Poisson operator is constructed so that its zero-mode boundary values satisfy precisely the “positive” constraint encoded by $T_i$, i.e. $\mathrm{Tr}_{D,i}(K_i\phi) = \phi$ for $\phi \in \ker(T_i - \mathrm{Id})$ and the complementary modes $\ker(T_i + \mathrm{Id})$ are excluded by the canonical $L^2$/regularity choice in the cone model. Therefore $W_{i,\text{can}}(y) \subseteq W_{i,Hdg}(y)$. Since both spaces are Lagrangian subspaces of $H_{i,y}$, they have the same dimension and must coincide. $\square$

**Corollary 9.10** (A checkable sufficient criterion for Conjecture 9.4). Assume that for every stratum $Y_i$ with nontrivial $H_i$ the following hold:

(I) the link fibration admits a fiberwise Kähler package compatible with the deeper data $W_{>i}$ producing a Hodge involution $T_i$ (Definition 9.6; for instance via Proposition 9.7);

(II) the leading cone model for $D$ is accurate to order $O(1)$ in the sense of Section 3 and the $x_i^{-1}$-coefficient rigidity from Section 4 holds locally uniformly in $y$ (and hence uniformly on each fixed stratum component by Lemma 2.5);

(III) the Calderón data for $D$ is stable under the $O(1)$ perturbations controlled by the Fubini-Study wedge expansions (no spectral flow through 0 in the family $(A_{i,y}^{W>i})$).

Then $W_{i,\text{can}} = W_{i,Hdg}$ for all $i$, and hence $\mathrm{IC}_{W_{\text{can}}} \simeq \mathrm{IC}_{\bar{m}}$ and $H^\bullet(\mathrm{Dom}(d_{\text{can}}), d_{\text{can}}) \cong \mathrm{IH}_{\bar{m}}^\bullet(X)$.

*Proof.* Under (I), set $W_{i,\text{mid}} := W_{i,Hdg}$. Lemma 9.8 gives (M4.1), and the Deligne description in Appendix F identifies $\mathrm{IC}_{W_{\text{mid}}}$ with $\mathrm{IC}_{\bar{m}}$, giving (M4.2). Assumptions (II)-(III) allow one to transport the cone-model identification of Proposition 9.9 from the leading model to the actual Fubini-Study wedge metric, yielding (M4.3). The conclusion follows from Proposition 9.2. $\square$
Fiberwise Kähler package ⇐⇒ (Jᵢ, Tᵢ) on Hᵢ ⇐⇒ Wᵢ,_Hdg incidence-compatible

FS wedge control + rigidity + stability ⇐⇒ Wᵢ,can = Wᵢ,_Hdg ∀i

IC<sub>Wcan</sub> ≃ IC<sub>m</sub>

In depth two, the required identifications reduce to a single comparison along Y₂ followed by a comparison along Y₁, and Section 8 provides a worked analytic template for carrying out these comparisons once a candidate Wᵢ, mid is specified. We expect that the general depth case can be treated by the same induction on depth used throughout Section 5-Section 7, once the polarized description of the trace bundles Hᵢ is established.

## APPENDIX A. TWO ANALYTIC LEMMAS FOR DOMAIN-DEPENDENT SELF-ADJOINT FAMILIES

This appendix records two functional-analytic tools used throughout the paper whenever a self-adjoint realization depends on an auxiliary parameter through its domain (equivalently, through an admissible boundary condition). We refer to [16, 10] for background on domain-dependent closed operators, graph projections, and Hilbert complexes.

**A.1. Graph projections and norm-resolvent continuity.** Let $\mathcal{H}$ be a complex Hilbert space. For a densely defined closed operator $A$ on $\mathcal{H}$, its graph is the closed subspace

$$ \mathcal{G}(A) := \{(u, Au) \in \mathcal{H} \oplus \mathcal{H} : u \in \mathcal{D}(A)\}, $$

and we denote by $\Pi_A$ the orthogonal projection in $\mathcal{H} \oplus \mathcal{H}$ onto $\mathcal{G}(A)$. When $A$ is self-adjoint, the norm-resolvent topology on self-adjoint operators (continuity of $(A - i)^{-1}$ in $\mathcal{B}(\mathcal{H})$) is equivalent to the gap topology (continuity of $\Pi_A$ in $\mathcal{B}(\mathcal{H} \oplus \mathcal{H})$); see, e.g., [16, §IV.2].

**Remark A.1** (Standing convention: graph inner products and graph norms). For any densely defined closed operator $A$ on a Hilbert space $\mathcal{H}$, we equip $\text{Dom}(A)$ with the graph inner product

$$ \langle u, v \rangle + \langle Au, Av \rangle := \langle u, v \rangle + \langle Au, Av \rangle, $$

and the associated graph norm

$$ (\|u\|^2 + \|Au\|^2)^{1/2} := (\|u\|^2 + \|Au\|^2)^{1/2}. $$

Unless explicitly stated otherwise, closures and boundedness statements “in the graph norm” are understood with respect to $(\|\cdot\|^2 + \|A\cdot\|^2)^{1/2}$.

**Remark A.2** (Standing convention: Sobolev spaces and Sobolev norms). Whenever Sobolev spaces are used in this paper, we write $H_{\text{ie}}^m$ (resp. $H_{\text{ie,loc}}^m$) for an iterated-edge Sobolev scale of order $m$ on the relevant resolved space and we denote the corresponding norms by

$$ \|u\|_{H_{\text{ie}}^m} := \|u\|_{H_{\text{ie}}^{m,}} \qquad \|u\|_{H_{\text{ie,loc}}^m} := \|u\|_{H_{\text{ie,loc}}^{m,}}. $$

For standard Sobolev spaces we use $H^m$ and $\|u\|_{H^m} := \|u\|_{H^m}$.

The following lemma is the direction we use most often.

**Lemma A.3** (Graph-projection continuity ⇒ norm-resolvent continuity). Let $Y$ be a topological space and let $\{A_y\}_{y \in Y}$ be a family of self-adjoint operators on $\mathcal{H}$. Assume that the map

$$ Y \ni y \mapsto \Pi_{A_y} \in \mathcal{B}(\mathcal{H} \oplus \mathcal{H}) $$
is continuous in operator norm (resp. $C^\ell$ in the sense of Banach-space calculus on a $C^\ell$-manifold $Y$). Then for every $z \in \mathbb{C} \setminus \mathbb{R}$ the resolvent map

$$Y \ni y \mapsto (A_y - z)^{-1} \in \mathcal{B}(\mathcal{H})$$

is continuous (resp. $C^\ell$) in operator norm. In particular, $y \mapsto (A_y \pm i)^{-1}$ is continuous (resp. $C^\ell$).

*Proof.* Fix $y \in Y$ and abbreviate $A = A_y$, $\Pi = \Pi_A$. For a densely defined closed operator one has the standard block-matrix formula for the orthogonal projection onto its graph (see [16, Ch. IV, §2.6]): writing $\Pi$ in blocks with respect to $\mathcal{H} \oplus \mathcal{H}$,

$$\Pi = \begin{pmatrix} (I + A^*A)^{-1} & A^*(I + AA^*)^{-1} \\ A(I + A^*A)^{-1} & AA^*(I + AA^*)^{-1} \end{pmatrix}.$$

If $A$ is self-adjoint then $A^* = A$ and $AA^* = A^2$, hence with $Q := (I + A^2)^{-1}$ we obtain

$$\Pi = \begin{pmatrix} Q & AQ \\ AQ & A^2Q \end{pmatrix}.$$

Therefore the blocks $Q$ and $AQ$ are obtained by composing $\Pi$ with the canonical inclusions/projections $\mathcal{H} \leftrightarrow \mathcal{H} \oplus \mathcal{H}$, hence $y \mapsto Q_y := (I + A_y^2)^{-1}$ and $y \mapsto A_y(I + A_y^2)^{-1}$ inherit the same regularity as $y \mapsto \Pi_{A_y}$. Using the identity

$$(A + i)^{-1} = (A - i)(A^2 + I)^{-1} = A(I + A^2)^{-1} - i(I + A^2)^{-1} = AQ - iQ,$$

we conclude that $y \mapsto (A_y + i)^{-1}$ is continuous (resp. $C^\ell$) in operator norm. The same argument gives $(A_y - i)^{-1} = AQ + iQ$.

For a general $z \in \mathbb{C} \setminus \mathbb{R}$, use the resolvent identity

$$(A - z)^{-1} - (A - i)^{-1} = (z - i)(A - z)^{-1}(A - i)^{-1},$$

together with bounded holomorphic functional calculus for self-adjoint operators, to reduce continuity at $z$ to continuity at $\pm i$. (Alternatively, apply the above argument to $(A - \mathbb{R}z)$ and then rescale.) $\square$

**Remark A.4** (How Lemma A.3 is used in the paper). In our geometric applications, $A_y$ is typically a first-order elliptic operator on a compact manifold with boundary (or on a stratified link with iterated wedge structure), and a self-adjoint realization is imposed by an elliptic boundary condition encoded by an orthogonal projection $P_y$ on a boundary data Hilbert space. Standard parametrix constructions in the Boutet de Monvel/APS framework [9, 2] imply that if $y \mapsto P_y$ varies continuously (resp. smoothly) in operator norm in the appropriate pseudodifferential class, then the induced graph projection $y \mapsto \Pi_{A_y}$ varies continuously (resp. smoothly), hence the resolvent varies continuously (resp. smoothly) by Lemma A.3. We do not reprove those elliptic-regularity statements here.

**A.2. Uniform spectral gaps over compact parameter spaces.** We also frequently need to upgrade a pointwise spectral gap at the origin to a uniform gap.

**Lemma A.5** (Compactness upgrades pointwise gaps to uniform gaps). Let $Y$ be compact and let $\{A_y\}_{y \in Y}$ be a family of self-adjoint operators on $\mathcal{H}$ with compact resolvent. Assume that

(i) for each $y \in Y$, the eigenvalue 0 is isolated in $\sigma(A_y)$ (equivalently, $\ker A_y$ is finite-dimensional);

(ii) the resolvent map $y \mapsto (A_y + i)^{-1}$ is continuous in operator norm.

Then there exists $\varepsilon > 0$ such that for every $y \in Y$,

$$\sigma(A_y) \cap (-\varepsilon, \varepsilon) \subseteq \{0\}.$$

Equivalently, the restriction of $A_y$ to $\ker(A_y)^\perp$ satisfies a uniform bound

$$\|(A_y|_{\ker(A_y)^\perp})^{-1}\|_{\mathcal{B}(\ker(A_y)^\perp)} \leq \varepsilon^{-1} \quad \text{for all } y \in Y.$$
Moreover, the orthogonal projection $\Pi_{\ker A_y}$ onto $ker(A_y)$ depends continuously on $y$ in operator norm and, in particular, dim $ker(A_y)$ is locally constant (hence constant on each connected component of $Y$).

*Proof.* Fix $y_0 \in Y$. By assumption, there exists $\varepsilon_0 > 0$ such that $\sigma(A_{y_0}) \cap [-2\varepsilon_0, 2\varepsilon_0] = \{0\}$. Let $\gamma$ be the positively oriented circle $\{z \in \mathbb{C} : |z| = \varepsilon_0\}$. Since $\gamma$ lies in the resolvent set of $A_{y_0}$, the spectral projection onto $ker(A_{y_0})$ is given by the Riesz formula

$$ \Pi_{\ker A_{y_0}} = \frac{1}{2\pi i} \int_{\gamma} (A_{y_0} - z)^{-1} dz. $$

By the resolvent identity and the assumed norm-resolvent continuity at $i$, the map $y \mapsto (A_y - z)^{-1}$ is continuous in operator norm for each fixed $z \notin \mathbb{R}$ (cf. Lemma A.3 or [16, §IV.2]). In particular, the function

$$ y \mapsto \sup_{z \in \gamma} \|(A_y - z)^{-1} - (A_{y_0} - z)^{-1}\| $$

is continuous and vanishes at $y = y_0$, hence is $< \|(A_{y_0} - z)^{-1}\|^{-1}$ for all $z \in \gamma$ on some neighborhood $U$ of $y_0$. Consequently, for every $y \in U$ and every $z \in \gamma$, the operator $(A_y - z)$ is invertible and $\gamma$ stays in the resolvent set of $A_y$. Thus the Riesz projection

$$ \Pi_{\ker A_y} := \frac{1}{2\pi i} \int_{\gamma} (A_y - z)^{-1} dz $$

is well-defined for $y \in U$ and depends continuously on $y$ in operator norm. Because $\Pi_{\ker A_y}$ has finite rank (compact resolvent) and rank is integer-valued, it is constant on $U$; hence $\dim \ker(A_y)$ is locally constant.

For $y \in U$, the spectral theorem implies that $\sigma(A_y)$ does not meet $\gamma$, hence does not meet the closed annulus $\{z : \varepsilon_0 \le |z| \le 2\varepsilon_0\}$. Since $A_y$ is self-adjoint, this gives the real gap $\sigma(A_y) \cap [-\varepsilon_0, \varepsilon_0] = \{0\}$ for all $y \in U$.

Finally, cover the compact space $Y$ by finitely many such neighborhoods $U_j$ with associated radii $\varepsilon_j$ and set $\varepsilon := \min_j \varepsilon_j > 0$. Then $\sigma(A_y) \cap (-\varepsilon, \varepsilon) \subseteq \{0\}$ holds for all $y \in Y$. The bound on $\|(A_y)_{\ker \perp}^{-1}\|$ is equivalent to this gap by the spectral theorem. $\square$

**A.3. Riesz projections and differentiability of spectral data.** The next statement packages the parameter-regularity of spectral projections that is repeatedly invoked (often implicitly) when differentiating families of Calderón projectors, Poisson operators, and reduced inverses.

**Proposition A.6 (Smooth dependence of Riesz projections).** Let $Y$ be a $C^k$ manifold ($k \in \mathbb{N} \cup \{0\}$) and let $\{A_y\}_{y \in Y}$ be a family of self-adjoint operators on a fixed Hilbert space $\mathcal{H}$ with compact resolvent. Assume that the resolvent map

$$ Y \ni y \mapsto (A_y - i)^{-1} \in \mathcal{B}(\mathcal{H}) $$

is of class $C^k$ in operator norm. Fix a positively oriented smooth simple closed contour $\gamma \subset \mathbb{C}$ such that $\gamma \subset \rho(A_y)$ for all $y \in Y$, and such that

$$ \inf_{y \in Y} \inf_{z \in \gamma} \mathrm{dist}(z, \sigma(A_y)) > 0. $$

Then the Riesz projection

$$ P_{\gamma}(y) := \frac{1}{2\pi i} \int_{\gamma} (A_y - z)^{-1} dz \in \mathcal{B}(\mathcal{H}) $$

depends $C^k$-smoothly on $y$ (resp. continuously if $k = 0$), and its derivatives are obtained by differentiating under the integral sign:

$$ \partial^{\alpha} P_{\gamma}(y) = \frac{1}{2\pi i} \int_{\gamma} \partial^{\alpha} ((A_y - z)^{-1}) dz, \quad |\alpha| \le k. $$
In particular, if 0 is isolated in $\sigma(A_y)$ for all $y$ and the isolation is uniform over $Y$ (equivalently, there exists $\varepsilon > 0$ with $\sigma(A_y) \cap [-\varepsilon, \varepsilon] = \{0\}$ for all $y$), then the orthogonal projection $\Pi_{\ker A_y}$ onto $\ker(A_y)$ depends $C^k$-smoothly on $y$. Moreover, the reduced inverse

$$ (A_y|_{\ker(A_y)\perp})^{-1} : \ker(A_y)\perp \to \ker(A_y)\perp $$

depends $C^k$-smoothly on $y$ in operator norm.

**Proof.** Fix $z \in \mathbb{C} \setminus \mathbb{R}$. Since $(A_y - i)^{-1}$ is $C^k$ in operator norm, the resolvent identity gives

$$ (A_y - z)^{-1} - (A_y - i)^{-1} = (z - i)(A_y - z)^{-1}(A_y - i)^{-1}. $$

The uniform spectral separation hypothesis along $\gamma$ implies a uniform bound $\sup_{y \in Y} \sup_{z \in \gamma} \|(A_y - z)^{-1}\| < \infty$, hence $y \mapsto (A_y - z)^{-1}$ is $C^k$ in operator norm for each $z \in \gamma$ (by repeated differentiation of the identity above and induction on $|\alpha|$). Since the derivatives are uniformly bounded on $\gamma$, differentiation under the contour integral is justified and yields the asserted $C^k$-regularity of $P_\gamma(y)$.

For the final statements, assume a uniform gap at 0. Let $\gamma$ be a small circle around 0 contained in the common resolvent set; then $P_\gamma(y)$ is the spectral projection onto the 0-eigenspace, hence equals $\Pi_{\ker A_y}$. For the reduced inverse, write

$$ (A_y|_{\ker(A_y)\perp})^{-1} = (I - \Pi_{\ker A_y}) A_y^{-1} (I - \Pi_{\ker A_y}), $$

where $A_y^{-1}$ denotes the bounded inverse of $A_y$ on $\ker(A_y)\perp$ (well-defined by the gap). The map $y \mapsto A_y^{-1}$ is $C^k$ by the resolvent map at $z = 0$, and the previous paragraph gives $C^k$-regularity of $\Pi_{\ker A_y}$, hence of the product above. $\square$

**Remark A.7 (Application to Calderón/Poisson families).** In the main text, the parameter dependence of Calderón projectors and Poisson operators is reduced to the $C^k$-dependence of (i) the self-adjoint tangential operators $A_{i,y}^{W>i}$ and (ii) their finite-rank spectral data in the critical strip, together with smoothing remainders controlled in the iterated edge calculus. Proposition A.6, combined with Lemma A.5, provides the abstract functional-analytic mechanism turning pointwise spectral separation into uniform $C^k$-regularity of the corresponding spectral projections, which is the input needed to differentiate the Calderón package in Appendix D.

**Definition A.8 (C^k-regularity on manifolds with corners).** Let $Y$ be a $C^k$ manifold with corners and let $E$ be a Banach space. A map $F : Y \to E$ is said to be of class $C^k$ up to the boundary if for every corner chart $\phi : U \to V \subset [0, \infty)^p \times \mathbb{R}^{n-p}$ the composite $F \circ \phi^{-1}$ extends to a $C^k$ map on an open neighborhood of $V$ in $\mathbb{R}^n$ (equivalently, all partial derivatives of order $\le k$ extend continuously to $V$).

**Corollary A.9 (Riesz projections for parameter spaces with corners).** Let $Y$ be a $C^k$ manifold with corners and let $\{A_y\}_{y \in Y}$ be a family of self-adjoint operators with compact resolvent on a fixed Hilbert space $\mathcal{H}$. Assume that $y \mapsto (A_y - i)^{-1}$ is $C^k$ up to the boundary in the sense of Definition A.8, and that $\gamma \subset \mathbb{C}$ is a contour satisfying the uniform spectral separation hypothesis in Proposition A.6. Then $y \mapsto P_\gamma(y)$ is $C^k$ up to the boundary as a map into $\mathcal{B}(\mathcal{H})$. If, moreover, $0$ is uniformly isolated in $\sigma(A_y)$ on $Y$, then both $y \mapsto \Pi_{\ker A_y}$ and $y \mapsto (A_y|_{\ker(A_y)\perp})^{-1}$ are $C^k$ up to the boundary in operator norm.

**Remark A.10 (Relevance for iterated boundary faces).** In the analytic parts of this paper, parameter spaces arise not only as open strata but also as boundary faces of the manifolds with corners obtained by successive blow-ups of depth filtrations. Corollary A.9 is precisely the abstract input that allows one to differentiate spectral data tangentially to all faces without any additional microlocal structure: once the relevant resolvent families are known to be $C^k$ up to the boundary, the same is true for the associated finite-rank spectral projections and reduced inverses.
## APPENDIX B. ITERATED FUBINI-STUDY WEDGE EXPANSION: ADAPTED COORDINATES AND TENSOR ESTIMATES

This appendix proves Theorem 3.1. We expand the induced Fubini-Study metric near a depth-$i$ stratum $Y_i$ in coordinates compatible with the iterated control data of Section 2, and we verify that the error term is iterated edge-smooth in the sense of the iterated edge tangent algebra $V_{ie}$.

**B.1. Iterated edge vector fields and iterated smoothness.** We use the manifold-with-corners model of Remark 2.4: after iterated radial blow-up along the strata (cf. [1]) we obtain a manifold with corners $\tilde{X}$ with boundary hypersurfaces $\{H_i\}$, a blow-down map $\beta: \tilde{X} \to X$, and boundary fibrations $\phi_i: H_i \to S_i$ encoding the link fibrations.

**Definition B.1 (Iterated edge vector fields).** Let $V_{ie}(\tilde{X})$ be the space of smooth vector fields $V \in \Gamma(T\tilde{X})$ such that

(i) $V$ is tangent to every boundary hypersurface of $\tilde{X}$;

(ii) for each boundary hypersurface $H_i$ with fibration $\phi_i: H_i \to S_i$, the restriction $V|_{H_i}$ is tangent to the fibers of $\phi_i$.

We call $V_{ie}(\tilde{X})$ the iterated edge tangent algebra.

**Lemma B.2 (Local generators).** Fix a point of $\tilde{X}$ lying over a strict incidence chain of strata as in Proposition 2.3. In a local product chart

$$ (x_{i_1}, \dots, x_{i_\ell}, y, z^{(1)}, \dots, z^{(\ell)}), $$

compatible with the iterated cone-bundle description (so that $x_{i_k}$ are boundary defining functions and $z^{(k)}$ are coordinates along the $k$-th link fiber), the module $V_{ie}(\tilde{X})$ is finitely generated over $C^\infty(\tilde{X})$ by vector fields of the form

$$ (B.1.1) \qquad x_{i_k} \partial_{x_{i_k}}, \quad \left( \prod_{m<k} x_{i_m} \right) \partial_{z^{(k)}}, \quad \left( \prod_{m \le \ell} x_{i_m} \right) \partial_y, $$

together with vector fields tangent to all boundary faces in the $y$-variables. (Here the empty product is interpreted as 1.)

**Proof.** The defining properties in Definition B.1 are most transparent on a codimension-$\ell$ corner: tangency to $H_{i_k} = \{x_{i_k} = 0\}$ forces the $\partial_{x_{i_k}}$-component to vanish on $x_{i_k} = 0$, hence it is divisible by $x_{i_k}$. Likewise, the condition that $V|_{H_{i_k}}$ is tangent to the fibers of $\phi_{i_k}$ forces the components in directions belonging to the base of $\phi_{i_k}$ to vanish on $x_{i_k} = 0$. In the iterated cone chart, the base variables for $\phi_{i_k}$ consist of the deeper radial variables $x_{i_{k+1}}, \dots, x_{i_\ell}$, the deeper link coordinates $z^{(k+1)}, \dots, z^{(\ell)}$, and the stratum coordinates $y$. Therefore the corresponding derivatives acquire a factor $x_{i_k}$ when written in a smooth vector field on $\tilde{X}$. Iterating this observation along the chain yields the stated generators (B.1.1). $\square$

**Definition B.3 (Iterated edge-smooth functions and tensors).** Let $E \to \tilde{X}$ be a smooth vector bundle. A section $u \in \Gamma(E|_{\text{int} \tilde{X}})$ is iterated edge-smooth (or ie-smooth) if for every $m \ge 0$ and every choice of $V_1, \dots, V_m \in V_{ie}(\tilde{X})$, the iterated derivative $V_1 \cdots V_m u$ extends continuously to $\tilde{X}$ and is bounded on $\tilde{X}$. We denote the resulting space by $C^\infty_{ie}(\tilde{X}; E)$.

**Remark B.4 (Independence of the blow-up model).** Different iterated blow-up models are related by b-fibrations (in the sense of [21, 1]) that preserve the boundary fibrations $\phi_i$ up to canonical identifications. Such maps pull back $V_{ie}$ to $V_{ie}$ and hence identify the spaces $C^\infty_{ie}(\tilde{X}; E)$. Thus the notion of ie-smoothness is intrinsic to $(X, \{(\rho_i, x_i)\})$.

We will repeatedly use the elementary divisibility principle below, which is responsible for the "$O(x_i)$" remainders in the wedge expansion being stable under $V_{ie}$-differentiation.
**Lemma B.5 (Division by a boundary defining function).** Fix a boundary hypersurface $H_i = \{x_i = 0\}$ of $\tilde{X}$. If $f \in C_{\text{ie}}^{\infty}(\tilde{X})$ satisfies $f|_{H_i} = 0$, then there exists $\tilde{f} \in C_{\text{ie}}^{\infty}(\tilde{X})$ such that $f = x_i \tilde{f}$. Moreover, for every $m$ and every $V_1, \dots, V_m \in \mathcal{V}_{\text{ie}}(\tilde{X})$, the function $V_1 \cdots V_m \tilde{f}$ is bounded by a constant depending only on finitely many $\mathcal{V}_{\text{ie}}$-seminorms of $f$.

*Proof.* Work in a product chart $(x_i, w)$ near $H_i$, where $w$ denotes the remaining variables. Since $f$ is continuous up to $x_i = 0$ and vanishes on $x_i = 0$, the integral formula

$$ \tilde{f}(x_i, w) := \int_0^1 (\partial_{x_i} f)(tx_i, w) dt $$

defines a continuous function with $f = x_i \tilde{f}$. To check ie-smoothness, note that every $V \in \mathcal{V}_{\text{ie}}$ can be written locally as $V = x_i a(x_i, w) \partial_{x_i} + W$, where $W$ is tangent to $H_i$ (hence preserves the ideal $(x_i)$). Differentiating the identity $f = x_i \tilde{f}$ and using that $V(x_i) = O(x_i)$ shows inductively that $V_1 \cdots V_m \tilde{f}$ is a linear combination (with $C_{\text{ie}}^{\infty}$ coefficients) of expressions of the form $x_i^{-1} V_1' \cdots V_m' f$ with $m' \le m+1$, and the singular prefactor cancels because each such derivative of $f$ still vanishes on $H_i$. A direct check using the local generators from Lemma B.2 gives boundedness. $\square$

**B.2. Quantitative Whitney–Thom angle control.** Let $S_\alpha$ be a stratum and $T_\alpha$ its tubular neighborhood with control data $(\pi_\alpha, \rho_\alpha)$ from Section 2. The Thom conditions (together with the smoothness of $S_\alpha$) imply that as one approaches deeper strata, the tangent spaces of neighboring strata vary in a controlled way. For our purposes we need a quantitative form that is locally uniform in the stratum variable (and hence, after fixing a connected stratum component, uniform by Lemma 2.5) and stable under $\mathcal{V}_{\text{ie}}$-differentiation.

Fix the ambient Fubini-Study metric $g_{\mathbb{C}^N}$ on $\mathbb{C}^N$. Along $S_\alpha$ we have the $g_{\mathbb{C}^N}$-orthogonal splitting $T\mathbb{C}^N|_{S_\alpha} = TS_\alpha \oplus (TS_\alpha)^\perp$ and the corresponding orthogonal projections $P_\alpha$ and $Q_\alpha := \text{Id} - P_\alpha$. By Lemma 2.5 we may work in finitely many control charts on a neighborhood of any fixed compact set $K \subset S_\alpha$; the Thom control conditions and compactness give uniform bounds for these projections and their derivatives.

**Lemma B.6 (Uniform angle bounds and bounded derivatives).** Let $K \subset S_\alpha$ be compact. There exist a neighborhood $U \subset T_\alpha$ of $K$ in $X$ and constants $C_m$ ($m \ge 0$) such that on $\beta^{-1}(U \cap X^{\text{reg}}) \subset \tilde{X}$ one has

$$ \|V_1 \cdots V_m P_\alpha\| + \|V_1 \cdots V_m Q_\alpha\| \le C_m $$

for all $m \ge 0$ and all $V_1, \dots, V_m \in \mathcal{V}_{\text{ie}}(\tilde{X})$. In particular, on $U \cap X^{\text{reg}}$ the angle between $TS_\alpha$ and its $g_{\mathbb{C}^N}$-orthogonal complement is bounded away from 0 and $\pi/2$ uniformly on $K$.

*Proof.* In a fixed control chart $\rho_\alpha^{-1}(U_\mu) \cap N_\alpha \simeq U_\mu \times C(L)$, the Thom conditions imply that tangent spaces of strata are compatible with the control retraction and vary continuously up to $x_\alpha = 0$; see [18]. The orthogonal projection matrices of $P_\alpha$ in a smooth ambient frame therefore extend continuously to $x_\alpha = 0$. By compactness of $K$ and finiteness of the cover, the operator norms are uniformly bounded. Moreover, in the iterated cone variables the components of $P_\alpha$ are smooth functions of the “angular” variables and are constant in the radial variables to leading order; applying the generators of $\mathcal{V}_{\text{ie}}$ from Lemma B.2 (which include the weighted radial derivatives) therefore yields uniform bounds for all $\mathcal{V}_{\text{ie}}$-derivatives. The final angle statement is an immediate consequence of boundedness of $P_\alpha$ and $Q_\alpha$. $\square$

**B.3. Vertical scaling and the wedge leading term.** Fix a depth-i stratum $Y_i$ and work in a control neighborhood $T_i$ with retraction $\rho_i : T_i \to Y_i$ and radial function $x_i = \rho_i$. On $T_i \cap X^{\text{reg}}$ the map $(\rho_i, x_i)$ is a smooth submersion on each stratum, so we may choose local product variables $(x_i, y, z)$ in which $y$ are coordinates on $Y_i$ and $z$ are coordinates along the link fiber $L_{i,y}$.
**Horizontal–vertical splitting.** Let

$$V_i := \ker(d\rho_i) \cap \ker(dx_i) \subset T(X^{\text{reg}} \cap T_i)$$

be the tangent bundle to the link fibers $\{(x_i, y)\} \times L_{i,y}$, and let $\partial_{x_i}$ denote the radial vector field. Define the horizontal bundle $H_i$ to be the $g_{\text{FS}}$-orthogonal complement of $\mathbb{R}\partial_{x_i} \oplus V_i$ inside $T(X^{\text{reg}} \cap T_i)$. By Lemma B.6, the splitting

$$T(X^{\text{reg}} \cap T_i) = \mathbb{R}\partial_{x_i} \oplus H_i \oplus V_i$$

varies with bounded $\mathcal{V}_{\text{ie}}$-derivatives locally in $y \in Y_i$ (and hence uniformly on each fixed stratum component $S = \text{int} \tilde{Y}_i$ by Lemma 2.5).

**The base metric $g_{Y_i}$.** Restrict $g_{\text{FS}}$ to $H_i$ and transport it to $TY_i$ using $d\rho_i|_{H_i} : H_i \to TY_i$. This defines a smooth metric $g_{Y_i}$ on $Y_i$. By construction, the pull-back $\rho_i^*g_{Y_i}$ agrees with the horizontal block of $g_{\text{FS}}$ at $x_i = 0$, and Lemma B.6 implies that all $\mathcal{V}_{\text{ie}}$-derivatives of the difference $g_{\text{FS}}|_{H_i} - \rho_i^*g_{Y_i}$ vanish at $x_i = 0$.

**The link metric $g_{L_i}$ by vertical scaling.** Consider the dilation map in a cone chart,

$$\delta_t : (x_i, y, z) \mapsto (tx_i, y, z), \quad t \in (0, 1].$$

Define the rescaled vertical metric

$$g_{i,t}^{\text{vert}} := t^{-2} (\delta_t)^* (g_{\text{FS}}|_{V_i}).$$

Since $g_{\text{FS}}$ is smooth on $X^{\text{reg}}$ and the dilation rescales only the radial variable, Taylor expansion in $t$ and the angle bounds of Lemma B.6 imply that $g_{i,t}^{\text{vert}}$ converges as $t \to 0$ in the topology of $C_{\text{ie}}^{\infty}$ locally in $y \in Y_i$ (and hence uniformly on each fixed stratum component $S = \text{int} \tilde{Y}_i$ by Lemma 2.5) to a limit tensor on $V_i$ that is independent of the choice of chart. We denote this limit by $g_{L_i}$; it restricts for each $y \in Y_i$ to a Riemannian metric on the link fiber $L_{i,y}$.

Equivalently, one may define $g_{L_i}$ by restricting $x_i^{-2}g_{\text{FS}}$ to $V_i$ and taking the limit $x_i \to 0$:

$$(B.3.1) \qquad g_{L_i}(y, z) := (x_i^{-2}g_{\text{FS}}|_{V_i})|_{x_i=0}.$$

The existence and ie-smoothness of the limit follow from the same scaling argument. Finally, because each link fiber $L_{i,y}$ inherits a Thom–Mather stratification of depth $d-i$, and the control data on $X$ restrict to control data on the links (Proposition 2.3), the above construction iterates on $L_{i,y}$ and shows that $g_{L_i}$ is an *iterated wedge metric* on the stratified fibration $L_i \to Y_i$.

**B.4. Assembly and $O(x_i)$ remainder.** We now prove the expansion asserted in Theorem 3.1 and verify the regularity of the remainder.

**Proposition B.7 (First-order wedge expansion).** In the adapted variables $(x_i, y, z)$ above, the induced metric $g_{\text{FS}}$ admits the decomposition

$$(B.4.1) \qquad g_{\text{FS}} = dx_i^2 + \rho_i^* g_{Y_i} + x_i^2 g_{L_i} + x_i \kappa_i,$$

where $\kappa_i$ is a symmetric 2-tensor whose coefficients in the wedge coframe $(dx_i, \rho_i^*T^*Y_i, x_iT^*L_i)$ are ie-smooth down to $x_i = 0$.

*Proof.* Write $g_0 := dx_i^2 + \rho_i^*g_{Y_i} + x_i^2g_{L_i}$. By construction of $g_{Y_i}$ and $g_{L_i}$ in Appendix B.3, the difference tensor $g_{\text{FS}} - g_0$ vanishes when restricted to $H_i \oplus V_i$ at $x_i = 0$, and all mixed terms involving $dx_i$, $H_i$ and $V_i$ also vanish at $x_i = 0$ because $\partial_{x_i}$ is orthogonal to the level sets of $x_i$ and the horizontal–vertical splitting is orthogonal at $x_i = 0$. Thus every coefficient of $g_{\text{FS}} - g_0$ in the wedge coframe vanishes on $H_i = \{x_i = 0\}$.

Pull back to $\tilde{X}$ and apply Lemma B.5 componentwise: there exists an ie-smooth tensor $\kappa_i$ such that $g_{\text{FS}} - g_0 = x_i\kappa_i$ in wedge components. Since $g_{\text{FS}}$ is smooth on $X^{\text{reg}}$, the boundedness of $\mathcal{V}_{\text{ie}}$-derivatives of $\kappa_i$ follows from the weighted nature of $\mathcal{V}_{\text{ie}}$ (Lemma B.2) and the uniformity of the cone charts (Lemma 2.5). This proves (B.4.1). $\square$
**Lemma B.8** (Inverse metric and density in wedge form). In the situation of Proposition B.7, write $g_{\text{FS}} = g_0 + x_i \kappa_i$ in the wedge coframe. Then, on $(0, \varepsilon) \times Y_i \times L_{i, \bullet}$, the inverse metric and Riemannian density admit decompositions

$$g_{\text{FS}}^{-1} = g_0^{-1} + x_i \kappa_i^{\#}, \quad \mathrm{dvol}_{g_{\text{FS}}} = (1 + x_i b_i) \mathrm{dvol}_{g_0},$$

where $\kappa_i^{\#}$ (a symmetric 2-tensor in the dual wedge frame) and $b_i$ are ie-smooth down to $x_i = 0$ (in particular, all $\mathcal{V}_{\text{ie}}$-derivatives are bounded on $\tilde{X}$).

*Proof.* Because $\kappa_i$ is ie-smooth in the wedge coframe, the endomorphism $A := g_0^{-1}\kappa_i$ has bounded ie-smooth coefficients down to $x_i = 0$. Since $g_{\text{FS}}$ is a Riemannian metric, $(\text{Id} + x_i A)$ is invertible for all $x_i \in [0, \varepsilon)$, and

$$g_{\text{FS}}^{-1} = (\text{Id} + x_i A)^{-1} g_0^{-1}.$$

The coefficients of $(\text{Id} + x_i A)^{-1}$ in the wedge frame are ie-smooth and bounded (e.g. by a Neumann-series argument for $x_i \ll 1$, and then by smooth continuation up to $x_i = \varepsilon$), yielding the asserted form $g_{\text{FS}}^{-1} = g_0^{-1} + x_i \kappa_i^{\#}$.

For the density, set $r := \frac{\mathrm{dvol}_{g_{\text{FS}}}}{\mathrm{dvol}_{g_0}}$. In wedge coordinates $r$ is ie-smooth and satisfies $r|_{x_i=0} = 1$ because $g_{\text{FS}}$ and $g_0$ agree to order 0 at $x_i = 0$. Thus $r - 1$ vanishes on $\{x_i = 0\}$, and Lemma B.5 gives $r - 1 = x_i b_i$ with $b_i \in C_{\text{ie}}^{\infty}$. $\square$

**Corollary B.9** (O(1) perturbations of geometric operators). Let $\nabla^{\text{FS}}$ and $\nabla^0$ be the Levi-Civita connections of $g_{\text{FS}}$ and $g_0$, respectively. In the wedge frame, the Christoffel symbols of $\nabla^{\text{FS}} - \nabla^0$ are ie-smooth and bounded down to $x_i = 0$. Consequently, the Gauss-Bonnet operator for $g_{\text{FS}}$ differs from the cone model operator for $g_0$ by a first-order iterated edge operator with coefficients ie-smooth down to $x_i = 0$; in particular, no additional $x_i^{-1}$ terms are created by the $x_i \kappa_i$ remainder.

*Proof.* The connection coefficients are obtained from the standard formula $\Gamma = \frac{1}{2}g^{-1}(\partial g + \partial g^t - \partial g)$ in a local frame. By Proposition B.7, $\partial(g_{\text{FS}} - g_0)$ has bounded ie-smooth wedge coefficients, and Lemma B.8 gives the same for $g_{\text{FS}}^{-1} - g_0^{-1}$. Combining these two facts shows that the difference of Christoffel symbols is ie-smooth and bounded.

The induced connections on the exterior algebra bundle of forms differ by a bundle endomorphism with the same regularity. Since the Gauss-Bonnet operator is a Dirac-type operator built from the Levi-Civita connection and Clifford multiplication, it follows that $D_{\text{FS}} - D_0$ is a first-order iterated edge operator with bounded ie-smooth coefficients. Because the only $x_i^{-1}$ contributions come from the cone part of $g_0$, the remainder cannot create new $x_i^{-1}$ terms. $\square$

**Proposition B.10** (Compatibility across incidences). Let $Y_j \subset Y_i$ with $j > i$ and work on an overlap $T_i \cap T_j$ where both sets of adapted coordinates are defined. Then the constructions of $g_{Y_i}$, $g_{L_i}$ and of the remainder class $x_i \kappa_i$ restrict to the corresponding objects for $Y_j$. In particular, the wedge expansions obtained from (B.4.1) for different strata are compatible on overlaps in the sense stated in Theorem 3.1.

*Proof.* On $T_i \cap T_j$ the iterated control data satisfy the compatibility relations (2.2.3), in particular $\rho_j \circ \rho_i = \rho_j$ and $x_j \circ \rho_i = x_j$. Therefore the horizontal distributions and the dilation maps used in Appendix B.3 agree when restricted to $T_j$, and the definitions of the limiting metrics $g_{Y_i}$ and $g_{L_i}$ are functorial with respect to these restrictions. Likewise, the divisibility argument in Lemma B.5 is local and depends only on the boundary defining function, hence the class of $x C_{\text{ie}}^{\infty}$ remainders is preserved under restriction to deeper neighborhoods. $\square$

Propositions B.7 and B.10 together are exactly the content of Theorem 3.1.
## APPENDIX C. ITERATED WEDGE DIRAC NORMAL FORMS WITH DEEPER CANONICAL DOMAINS

This appendix supplies the detailed proof of Theorem 4.1. Throughout we fix a stratum $Y_i$ of depth $i$ and work in a neighborhood $T_i$ equipped with iterated control data as in Section 2. We write local adapted coordinates as

$$ (x_i, y, z) \in (0, \varepsilon)_{x_i} \times U_{Y_i} \times L_{i,y}, $$

where $y$ are coordinates on $Y_i$ and $z$ are coordinates along the link fiber $L_{i,y}$. The wedge cotangent bundle over $T_i \cap X^{\text{reg}}$ is locally generated by the coframe

$$ dx_i, \quad dy^1, \dots, dy^{\dim Y_i}, \quad x_i \theta^1, \dots, x_i \theta^{f_i}, $$

where $\{\theta^\alpha\}$ is an orthonormal coframe for the link metric $g_{L_i}$ on $T(L_{i,\bullet})$ and $f_i = \dim L_{i,y}$. The associated iterated edge tangent algebra $\mathcal{V}_{\text{ie}}$ acts by derivations on coefficients, so “iterated edge-smooth” means smoothness with respect to all $\mathcal{V}_{\text{ie}}$-derivatives (cf. Appendix B, §B.1).

We emphasize two structural facts that are used repeatedly later.

* **Rigidity of the cone coefficient:** the coefficient $x_i^{-1} A_{i,y}^{W_{>i}}$ in (4.2.1) is determined entirely by the boundary wedge metric at $x_i = 0$ together with the deeper canonical domains on the stratified link; in particular, perturbations of the metric by $O(x_i)$ terms only affect the $O(1)$ remainder.

* **Iterated edge regularity of the remainder:** the remainder $B_i(x_i)$ in (4.2.1) is a first-order operator along the base variables $y$ with coefficients that extend iterated edge-smoothly to $x_i = 0$ and are compatible under the incidence changes coming from the control data.

**C.1. Product-type wedge metrics and the Gauss-Bonnet operator.** Fix a stratum $Y_i$ and consider the product-type wedge metric

$$ (C.1.1) \qquad g_0 = dx_i^2 + \rho_i^* g_{Y_i} + x_i^2 g_{L_i}, $$

on a tubular neighborhood $T_i \simeq (0, \varepsilon)_{x_i} \times L_i$ (over $Y_i$) coming from the control data. Let $\Omega_w^*$ denote wedge differential forms in this neighborhood and write every $\omega \in \Omega_w^*$ uniquely as $\omega = dx_i \wedge \alpha + \beta$ with $\alpha, \beta$ wedge-tangential (no $dx_i$ component).

Let $f_i = \dim L_{i,y}$. The volume density of $g_0$ is

$$ d\mathrm{vol}_{g_0} = x_i^{f_i} dx_i d\mathrm{vol}_{g_{Y_i}} d\mathrm{vol}_{g_{L_i}}, $$

and the induced $L^2$ pairing on forms uses this weight. Define the unitary conjugation

$$ (C.1.2) \qquad U_i : \Omega_w^* \to C^\infty((0, \varepsilon) \times Y_i; \Omega^*(L_{i,\bullet}) \oplus \Omega^*(L_{i,\bullet})), \qquad U_i(dx_i \wedge \alpha + \beta) = x_i^{f_i/2} \begin{pmatrix} \alpha \\ \beta \end{pmatrix}. $$

Here $\Omega^*(L_{i,\bullet})$ denotes the bundle of smooth forms along the link fibers, and the direct sum corresponds to the splitting into the $dx_i$-component and the tangential component.

Let $\mathbf{N}_i$ be the form-degree operator on tangential forms along $L_{i,y}$, and denote by $D_{L_{i,y}} = d_{L_{i,y}} + \delta_{L_{i,y}}$ the Gauss-Bonnet operator on the link fiber equipped with $g_{L_i}$. Let $\Gamma_i$ be Clifford multiplication by $dx_i$ on the bundle $\Omega^*(L_{i,\bullet}) \oplus \Omega^*(L_{i,\bullet})$, so that $\Gamma_i$ exchanges the two summands and $\Gamma_i^2 = -\mathrm{Id}$.

**Lemma C.1** (Cone calculation for $g_0$). With respect to the unitary identification (C.1.2) there exists a first-order wedge operator $\tilde{B}_i$ acting only in the $y$-variables (and preserving the bundle along the link) such that on compactly supported smooth forms in $T_i \cap X^{\text{reg}}$,

$$ (C.1.3) \qquad U_i D_{g_0} U_i^{-1} = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} \left( D_{L_{i,y}} + \frac{f_i}{2} - \mathbf{N}_i \right) + \tilde{B}_i \right). $$

Moreover $\tilde{B}_i$ has coefficients smooth in $(x_i, y)$ and iterated edge-smooth in the link directions, and $\{\Gamma_i, \tilde{B}_i\} = 0$.
*Proof.* The operator $D_{g_0} = d + \delta_{g_0}$ is a Dirac-type operator for the Levi-Civita connection of (C.1.1). In the wedge coframe $\{dx_i, dy^a, x_i \theta^\alpha\}$, the only connection 1-forms with singular behavior are those coming from the warped factor $x_i^2 g_{L_i}$, and they contribute precisely the radial “weight” term $\frac{f_i}{2} - N_i$ after the unitary conjugation by $x_i^{f_i/2}$. Equivalently, restricting momentarily to the exact cone metric $dx_i^2 + x_i^2 g_{L_i}$, the classical cone formula for the Gauss-Bonnet operator yields

$$U_i (d + \delta) U_i^{-1} = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} (D_{L_{i,y}} + \frac{f_i}{2} - N_i) \right),$$

see [13, §4] and [19, §2–§3]. The additional product factor $\rho_i^* g_{Y_i}$ contributes a first-order operator along the base variables $y$ with coefficients smooth in $(x_i, y)$ and acting fiberwise on $\Omega^*(L_{i,y})$; we denote the resulting term by $\tilde{B}_i$. The anticommutation $\{\Gamma_i, \tilde{B}_i\} = 0$ follows from the Clifford relations: $\Gamma_i$ is the normal Clifford action while $\tilde{B}_i$ involves only tangential Clifford actions and tangential covariant derivatives. $\square$

**C.2. Incorporating deeper canonical domains on stratified links.** The link fiber $L_{i,y}$ is a compact smoothly stratified space of depth $< i$. By the inductive canonical package on deeper strata (Definition 6.1), the Gauss-Bonnet operator on $L_{i,y}$ admits a canonical self-adjoint Fredholm realization $D_{L_{i,y}}^{W>i}$ whose domain is determined by the deeper Cauchy-data bundles $W_{>i}$. We therefore set

$$(C.2.1) \qquad A_{i,y}^{W>i} := D_{L_{i,y}}^{W>i} + \left( \frac{f_i}{2} - N_i \right),$$

acting fiberwise on $L_{i,y}$ with domain $\text{Dom}(A_{i,y}^{W>i}) = \text{Dom}(D_{L_{i,y}}^{W>i})$. Since $N_i$ is bounded and self-adjoint, $A_{i,y}^{W>i}$ is self-adjoint and Fredholm whenever $D_{L_{i,y}}^{W>i}$ is.

The normal form (C.1.3) is compatible with imposing these deeper domains: one interprets $D_{L_{i,y}}$ on the right-hand side as the self-adjoint operator $D_{L_{i,y}}^{W>i}$. This yields the model normal form for $g_0$ with the correct domain-dependent tangential operator $A_{i,y}^{W>i}$.

**Remark C.2 (Where the domain dependence enters).** Only the fiberwise operator $D_{L_{i,y}}^{W>i}$ depends on the deeper canonical data. All other terms in (C.2.1) are bounded endomorphisms on the fixed Hilbert space of link forms. In particular, the indicial family of the normal form is obtained from the indicial family of the exact cone by replacing $D_{L_{i,y}}$ with the self-adjoint extension determined by $W_{>i}$.

**C.3. Perturbation by iterated edge-smooth metric remainders and rigidity.** We now replace the product-type metric $g_0$ by the Fubini-Study wedge metric $g_{FS}$. By Theorem 3.1 (proved in Appendix B), we may write in adapted coordinates on $T_i$

$$(C.3.1) \qquad g_{FS} = g_0 + x_i \kappa_i,$$

where $\kappa_i$ is a symmetric 2-tensor whose coefficients in the wedge coframe are iterated edge-smooth down to $x_i = 0$ (with uniform bounds under all $\mathcal{V}_{ie}$-derivatives).

Let $\nabla^{\text{FS}}$ and $\nabla^0$ be the Levi-Civita connections of $g_{\text{FS}}$ and $g_0$. Since $g_{\text{FS}} - g_0 = O(x_i)$ in wedge frames and $\kappa_i$ has iterated edge-smooth coefficients, the connection difference tensor $\nabla^{\text{FS}} - \nabla^0$ is $O(1)$ in wedge frames, uniformly with all iterated edge derivatives. Consequently, the associated codifferentials differ by a first-order iterated edge operator with coefficients extending iterated edge-smoothly to $x_i = 0$.

**Proposition C.3 (Rigidity of the cone coefficient).** In the normal form (4.2.1) for $D_{g_{\text{FS}}}$, the coefficient of $x_i^{-1}$ is exactly $A_{i,y}^{W>i}$ from (C.2.1); equivalently, replacing $g_0$ by the perturbed metric (C.3.1) does not alter the indicial family (and hence the $x_i^{-1}$ coefficient) once the deeper canonical domains are fixed.
**Proof.** Write $D_{gFS} = D_{g0} + Q$, where $Q$ is the first-order operator obtained by comparing the two Dirac-type connections and Clifford actions determined by $g_{FS}$ and $g_0$. In the wedge coframe, the coefficient matrices of $Q$ are $O(1)$ and iterated edge-smooth down to $x_i = 0$ because the only potentially singular terms in the connection forms are those coming from the $x_i$-warped link factor. The perturbation (C.3.1) is $O(x_i)$, hence it cannot create new $x_i^{-1}$ contributions in those connection forms. Therefore the full operator $U_i D_{gFS} U_i^{-1}$ has the same $x_i^{-1}$ coefficient as in the model computation for $g_0$, namely $D_{L_{i,y}} + \frac{f_i}{2} - N_i$, with $D_{L_{i,y}}$ interpreted on the deeper self-adjoint domain determined by $W_{>i}$. $\square$

**Proposition C.4 (Normal form with iterated edge-smooth remainder).** There exists a unitary bundle identification $U_i^{FS}$ over $(0, \varepsilon) \times Y_i$, compatible with changes of adapted trivializations coming from the iterated control data, such that on compactly supported smooth forms in $T_i \cap X^{\text{reg}}$,

$$ (C.3.2) \qquad U_i^{\text{FS}} D_{g\text{FS}} (U_i^{\text{FS}})^{-1} = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} A_{i,y}^{W>i} + B_i(x_i) \right). $$

Here $B_i(x_i)$ is a differential operator that is first-order in the base variables $y$ (and zeroth order in $\partial_{x_i}$) whose coefficients, written in any local wedge frame, are iterated edge-smooth down to $x_i = 0$. Equivalently, for every $V \in \mathcal{V}_{ie}$ supported in $T_i$ the commutator $[V, B_i(x_i)]$ is again first-order in $y$ with iterated edge-smooth coefficients extending to $x_i = 0$.

**Proof.** Choose an orthonormal wedge coframe for $g_{FS}$ obtained from the $g_0$-orthonormal wedge coframe by Gram-Schmidt. Because $g_{FS} - g_0 = O(x_i)$ in wedge frames, the resulting change-of-frame matrix equals $\text{Id} + O(x_i)$ with iterated edge-smooth coefficients, and its $x_i$-derivative is $O(1)$. Conjugating by the corresponding bundle isometry yields a unitary identification $U_i^{\text{FS}}$ so that $U_i^{\text{FS}} D_{gFS} (U_i^{\text{FS}})^{-1}$ is expressed in the $g_0$-adapted wedge splitting used above. Proposition C.3 fixes the $x_i^{-1}$ coefficient. All remaining terms are $O(1)$ in wedge frames, hence produce an operator $B_i(x_i)$ that is first-order along $y$ with iterated edge-smooth coefficients down to $x_i = 0$. The stated commutator property follows because iterated edge differentiation of the coefficients remains bounded. $\square$

**C.4. Normal operators, indicial families, and smooth spectral data.** This subsection records the precise relationship between the local normal form (C.3.2) of Theorem 4.1, the associated normal operators and indicial families, and the smooth dependence of the small spectral data of the boundary operators $A_{i,y}^{W>i}$ on parameters. These statements are the bridge between the model computations on cones and the uniform estimates needed for the iterated Calderón package (cf. Appendix D).

**Definition C.5 (Normal operator and indicial family at $Y_i$).** Fix a stratum $Y_i$ and a point $y \in Y_i$. Write the local expression of $D$ in $T_i$ as in (C.3.2),

$$ D = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} A_{i,y}^{W>i} + B_i(x_i) \right), $$

where $B_i(x_i)$ is first order in the base variables and has iterated edge-smooth coefficients down to $x_i = 0$. Let $B_{i,0} := B_i(0)$ denote the operator obtained by restricting the coefficients of $B_i(x_i)$ to $x_i = 0$ (which is well-defined by iterated edge-smoothness).

* The normal operator $N_{i,y}(D)$ is the operator on the model space $\mathbb{R}_s^+ \times T_y Y_i \times L_{i,y}$ obtained by freezing all coefficients at $(x_i, y)$ and replacing the base derivatives in $B_{i,0}$ by constant-coefficient derivatives in the $T_y Y_i$-variables. Equivalently, after Fourier transform in the $T_y Y_i$-variable with dual covector $\eta \in T_y^* Y_i$, one obtains a family of operators

$$ (C.4.1) \qquad N_{i,y}(D)(\eta) = \Gamma_i \left( \partial_s + \frac{1}{s} A_{i,y}^{W>i} + i c_y(\eta) \right). $$

where $c_y(\eta)$ denotes Clifford multiplication by $\eta$ on the horizontal component.
• The indicial family $I_{i,y}(D)(\sigma, \eta)$ is the Mellin transform of $N_{i,y}(D)(\eta)$ in $s$, namely

$$I_{i,y}(D)(\sigma, \eta) = \Gamma_i(\sigma + A_{i,y}^{W>i} + i c_y(\eta)), \quad \sigma \in \mathbb{C}, \eta \in T_y^*Y_i.$$

**Proposition C.6** (Indicial roots and the tangential spectrum). For $\eta = 0$, the indicial family satisfies

$$I_{i,y}(D)(\sigma, 0) \text{ is invertible} \iff \sigma \notin \text{spec}(A_{i,y}^{W>i}).$$

In particular, the set of indicial roots of $D$ at $(Y_i, y)$ equals $\text{spec}(A_{i,y}^{W>i})$. Moreover, if $\eta \neq 0$ then $I_{i,y}(D)(\sigma, \eta)$ is invertible for all $\sigma \in \mathbb{C}$.

*Proof.* The first claim is immediate from the formula $I_{i,y}(D)(\sigma, 0) = \Gamma_i(\sigma + A_{i,y}^{W>i})$ and the fact that $\Gamma_i$ is invertible. For $\eta \neq 0$, note that $ic_y(\eta)$ is skew-adjoint and satisfies $(ic_y(\eta))^2 = |\eta|^2 \text{Id}$. Hence $\sigma + A_{i,y}^{W>i} + ic_y(\eta)$ is invertible for every $\sigma$ by the standard Clifford estimate. $\square$

**Corollary C.7** (Uniform critical strips from uniform spectral gaps). Let $K \subseteq Y_i$ be compact. Assume there exists $\delta > 0$ such that

$$(-\delta, \delta) \cap \text{spec}(A_{i,y}^{W>i}) = \emptyset \quad \text{for all } y \in K.$$

Then there exists $\varepsilon = \varepsilon(K, \delta) > 0$ such that the indicial family is uniformly invertible on the strip

$$I_{i,y}(D)(\sigma, 0) \text{ invertible for all } y \in K \text{ and all } \sigma \in \mathbb{C} \text{ with } |\Re\sigma| < \varepsilon.$$

*Proof.* By Proposition C.6, failure of invertibility occurs exactly at $\sigma = -\lambda$ with $\lambda \in \text{spec}(A_{i,y}^{W>i})$. The assumed gap gives pointwise separation of the spectrum from 0 on $K$. Uniformity follows from Lemma A.5 in Appendix A applied to the family $A_{i,y}^{W>i}$ over the compact set $K$. $\square$

**Proposition C.8** (Uniform invertibility of normal operators off the zero section). Let $K \subseteq Y_i$ be compact and fix $\eta_0 > 0$. Then for all $y \in K$ and all nonzero covectors $\eta \in T_y^*Y_i$ with $|\eta| \ge \eta_0$ the normal operator $N_{i,y}(D)(\eta)$ from Definition C.5 is invertible on its natural $L^2$ domain. Moreover, there exists a constant $C = C(K, \eta_0) > 0$ such that

$$(C.4.2) \qquad \|N_{i,y}(D)(\eta)^{-1}\|_{L^2 \to L^2} \le \frac{C}{|\eta|} \le \frac{C}{\eta_0},$$

for all $y \in K$ and all $|\eta| \ge \eta_0$. In particular, $(y, \eta) \mapsto N_{i,y}(D)(\eta)^{-1}$ depends smoothly on $(y, \eta)$ on the region $\{(y, \eta) : y \in K, |\eta| \ge \eta_0\}$ as a family of bounded operators on $L^2$.

*Proof.* Fix $y \in Y_i$. For $\rho > 0$ let $\delta_\rho$ denote the dilation on the model cone, $(\delta_\rho u)(s, \cdot) = u(\rho s, \cdot)$. After the usual unitary normalization on $L^2$ (conic density), one checks directly from Definition C.5 that the normal family satisfies the scaling relation

$$(C.4.3) \qquad \delta_\rho^{-1} N_{i,y}(D)(\eta) \delta_\rho = \rho N_{i,y}(D)(\eta/\rho).$$

Hence, for $\eta \neq 0$ and $\rho = |\eta|$ we obtain $N_{i,y}(D)(\eta)$ unitarily equivalent to $|\eta| N_{i,y}(D)(\hat{\eta})$ with $\hat{\eta} := \eta/|\eta| \in S(T_y^*Y_i)$. By Proposition C.6, the indicial family $I_{i,y}(D)(\sigma, \hat{\eta})$ is invertible for all $\sigma$ when $\hat{\eta} \neq 0$; the standard Mellin/Fourier analysis for cone operators then implies that $N_{i,y}(D)(\hat{\eta})$ is Fredholm with trivial kernel, hence invertible. (Equivalently, $N_{i,y}(D)(\hat{\eta})$ has a parametrix in the translation-invariant cone calculus, and the absence of indicial roots precludes $L^2$ null-solutions.) The coefficients in (C.4.1) vary smoothly in $y$ and $\hat{\eta}$, so the inverse family $(y, \hat{\eta}) \mapsto N_{i,y}(D)(\hat{\eta})^{-1}$ is continuous in operator norm. Since the parameter set $K \times S(T^*Y_i)|_K$ is compact, there is a uniform bound $\sup_{y \in K, |\eta|=1} \|N_{i,y}(D)(\hat{\eta})^{-1}\|_{L^2 \to L^2} \le C_1(K)$. Combining this with (C.4.3) gives (C.4.2) with $C = C_1(K)$. Smooth dependence of the inverse on $(y, \eta)$ for $|\eta| \ge \eta_0$ follows by differentiating the identity $N^{-1} = -N^{-1}(\partial N)N^{-1}$ iteratively, using the uniform bound (C.4.2). $\square$
**Remark C.9** (Uniform parametrices and the comparison-operator criterion). Proposition C.8 is precisely the uniform “$\eta \neq 0$” input needed for the parameter-dependent iterated edge calculus: it gives a uniform inverse for the normal operators away from the zero section in $T^*Y_i$. Together with Corollary C.7 (which controls the $\eta \to 0$ regime by excluding indicial roots in a uniform strip), this yields the standard comparison-operator criterion used in the construction of uniform parametrices and in the proof of uniform Fredholm properties for the deeper canonical domains.

**C.5. Explicit parametrices for the off-zero normal family and operator-norm differ-entiability.** We now strengthen the “$\eta \neq 0$” discussion by recording an explicit Mellin–Fourier parametrix for the *exact* cone model and by upgrading the parameter dependence statements to operator-norm $C^k$ regularity on compact parameter spaces. This makes the interface with Appendix A completely transparent.

**Lemma C.10** (Commutation with horizontal Clifford action). For each $y \in Y_i$ and $\eta \in T_y^*Y_i$, the horizontal Clifford action $c_y(\eta)$ commutes with the vertical tangential operator $A_{i,y}^{W>i}$. Consequently,

$$ (A_{i,y}^{W>i} + i c_y(\eta))^* (A_{i,y}^{W>i} + i c_y(\eta)) = (A_{i,y}^{W>i})^2 + |\eta|^2 \mathrm{Id}, $$
and hence $A_{i,y}^{W>i} + i c_y(\eta)$ is invertible for $\eta \neq 0$ with the bound $\|(A_{i,y}^{W>i} + i c_y(\eta))^{-1}\| \le |\eta|^{-1}.$

*Proof.* At $x_i = 0$ the wedge cotangent bundle splits as the direct sum of the horizontal cotangent $T^*Y_i$ and the rescaled vertical cotangent $T^*L_{i,y}$. The Clifford module used to define the wedge Dirac operator correspondingly decomposes into the (horizontal)⊗(vertical) factorization, so $c_y(\eta)$ acts only on the horizontal factor, whereas $A_{i,y}^{W>i}$ is the vertical Dirac-type operator on the link fiber (together with the degree term $(f_i/2 - N_i)$) and hence acts trivially on the horizontal factor. Therefore the operators commute. The identity follows by expanding the product and using that $A_{i,y}^{W>i}$ is self-adjoint while $i c_y(\eta)$ is self-adjoint with $(i c_y(\eta))^2 = |\eta|^2 \mathrm{Id}$. $\square$

**Proposition C.11** (Mellin parametrix for the exact cone normal family). Let $N_{i,y}^0(D)(\eta)$ denote the exact cone normal family obtained from $N_{i,y}(D)(\eta)$ by replacing $B_{i,0}$ by 0 in Definition C.5. For $\eta \neq 0$, the operator $N_{i,y}^0(D)(\eta)$ is invertible on $L^2$ with inverse given by a Mellin–Fourier multiplier: after Mellin transform in $x_i$ (with respect to the b-density $dx_i/x_i$) one has

$$ \mathcal{M}(N_{i,y}^0(D)(\eta)u)(\xi) = \Gamma_i(i\xi + A_{i,y}^{W>i} + i c_y(\eta)) \mathcal{M}u(\xi), \quad \xi \in \mathbb{R}, $$

and thus

$$ \mathcal{M}((N_{i,y}^0(D)(\eta))^{-1}f)(\xi) = \Gamma_i^{-1}(i\xi + A_{i,y}^{W>i} + i c_y(\eta))^{-1} \mathcal{M}f(\xi). $$

Moreover there exists $C > 0$, independent of $(y, \eta)$ with $|\eta| \ge 1$, such that

$$ \|(N_{i,y}^0(D)(\eta))^{-1}\|_{L^2 \to L^2} \le C |\eta|^{-1}. $$

*Proof.* The Mellin identity follows from the explicit form of the exact cone normal family: $N_{i,y}^0(D)(\eta) = \Gamma_i(\partial_{x_i} + x_i^{-1}(A_{i,y}^{W>i} + i c_y(\eta)))$. Conjugating by the unitary weight which identifies the wedge $L^2$ density near $x_i = 0$ with the b-density $dx_i/x_i$ yields the stated Mellin multiplier. By Lemma C.10, for each $\xi \in \mathbb{R}$ the operator $i\xi + A_{i,y}^{W>i} + i c_y(\eta)$ is invertible and its inverse satisfies $\|(i\xi + A_{i,y}^{W>i} + i c_y(\eta))^{-1}\| \le (\xi^2 + |\eta|^2)^{-1/2}$. Plancherel for the Mellin transform then implies the $L^2$ bound uniformly for $|\eta| \ge 1$. $\square$

**Proposition C.12** (Operator-norm $C^k$ dependence of off-zero resolvents). Let $\mathcal{B}$ be a compact manifold with corners and let $b \mapsto A_{i,b}^{W>i}$ be a $C^k$ family of self-adjoint tangential operators as in Proposition C.17. Assume that the coefficient family $b \mapsto B_{i,0}(b)$ in the normal operator depends
$C^k$ up to the boundary as a bounded operator on the graph norm of $N_{i,b}(D)(\eta)$ for $|\eta| = 1$. Fix $\eta_0 > 0$ and write $N_b(\eta) := N_{i,b}(D)(\eta)$. Then the map

$$ (b, \eta) \mapsto (N_{i,b}(D)(\eta) + i)^{-1} $$

is $C^k$ in operator norm on $L^2$, uniformly for $b \in \mathcal{B}$ and $|\eta| \ge \eta_0$. In particular, the family of graph projections $\Pi_{N_{i,b}(D)(\eta)}$ is $C^k$ up to the boundary of $\mathcal{B}$ for $|\eta| \ge \eta_0$ in the sense of Appendix A.

**Proof.** By Proposition C.8 there exists $C_{\eta_0}$ such that $\|(N_{i,b}(D)(\eta) + i)^{-1}\| \le C_{\eta_0}$ uniformly for $b \in \mathcal{B}$ and $|\eta| \ge \eta_0$. The resolvent identity gives, for $b, b' \in \mathcal{B}$,

$$ (N_b(\eta) + i)^{-1} - (N_{b'}(\eta) + i)^{-1} = (N_b(\eta) + i)^{-1}(N_{b'}(\eta) - N_b(\eta))(N_{b'}(\eta) + i)^{-1}. $$

The hypothesis on $B_{i,0}(b)$ and the iterated edge-smoothness of the remaining coefficients imply that $b \mapsto N_b(\eta)$ is $C^k$ up to the boundary as a bounded operator between the corresponding graph norm spaces. Combining this with the uniform resolvent bound yields operator-norm $C^1$ regularity. Higher derivatives follow by differentiating the identity $\partial_b(N_b(\eta) + i)^{-1} = -(N_b(\eta) + i)^{-1}(\partial_b N_b(\eta))(N_b(\eta) + i)^{-1}$ and inducting on the order, using the uniform bounds at each stage. Finally, the assertion about graph projections follows from Lemma A.3. $\square$

**Remark C.13** (From explicit parametrices to the full iterated calculus). The Mellin multiplier formula of Proposition C.11 provides an explicit Green operator for the exact cone model. The stability statement in Proposition C.12, together with the uniform critical-strip control of Corollary C.7, are the abstract inputs used to build uniform parametrices in the parameter-dependent iterated edge calculus (see, e.g., [20, 1] for closely related constructions).

**C.6. Polyhomogeneous kernel structure and indicial index sets.** Fix $y \in Y_i$ and $|\eta| \ge 1$. Set

$$ C_y := [0, \infty)_{x_i} \times L_{i,y} $$

and form the b-double space

$$ C_{yb}^2 := [C_y \times C_y; \partial C_y \times \partial C_y], $$

i.e. the real blow-up of the corner $\{x_i = 0\} \times \{x'_i = 0\}$, with boundary hypersurfaces $\text{lf} = \{x_i = 0\}$, $\text{rf} = \{x'_i = 0\}$, and the front face $\text{ff}$ created by the blow-up. We denote by $\Delta_b$ the lifted diagonal. A distribution on $C_{yb}^2$ is called *polyhomogeneous conormal* if it is conormal to $\Delta_b$ and admits complete asymptotic expansions at each boundary hypersurface in powers and logarithms with smooth coefficients on the remaining variables; see [21, 19].

**Proposition C.14** (Polyhomogeneous structure of the exact cone Green operator). For $|\eta| \ge 1$, the inverse $(N_{i,y}^0(D)(\eta))^{-1}$ from Proposition C.11 has a Schwartz kernel $K_{y,\eta}^0$ on $C_y \times C_y$ whose lift to $C_{yb}^2$ is conormal of b-order $-1$ along $\Delta_b$ and polyhomogeneous at $\text{lf}$, $\text{rf}$ and $\text{ff}$. The index families at $\text{lf}$ and $\text{rf}$ are generated by the indicial roots of $I_y(D)$, equivalently by $\text{spec}(A_{i,y}^{W>i})$, and the maximal logarithmic power attached to an exponent $\lambda$ is bounded by the geometric multiplicity of $\lambda$. Moreover, the normal operator of $K_{y,\eta}^0$ at $\text{ff}$ is given by the Mellin convolution kernel

$$ (C.6.1) \qquad N_{\text{ff}}(K_{y,\eta}^0)(s; z, z') = \frac{1}{2\pi} \int_{\mathbb{R}} s^{i\xi} \Gamma_i^{-1} (i\xi + A_{i,y}^{W>i} + i c_y(\eta))^{-1} d\xi, \quad s = \frac{x_i}{x'_i}. $$

**Proof.** Write $t = \log x_i$ and $t' = \log x'_i$, so the Mellin transform in $x_i$ is the Fourier transform in $t$ with respect to the b-density $dx_i/x_i = dt$. In these variables the exact cone family is translation-invariant in the $t$-direction, and Proposition C.11 identifies $(N_{i,y}^0(D)(\eta))^{-1}$ with the Fourier multiplier $\Gamma_i^{-1}(i\xi + A_{i,y}^{W>i} + i c_y(\eta))^{-1}$. Taking the inverse Fourier transform in $\xi$ yields a kernel of the form

$$ K_{y,\eta}^0(t, t'; z, z') = \frac{1}{2\pi} \int_{\mathbb{R}} e^{i(t-t')} \xi \Gamma_i^{-1} (i\xi + A_{i,y}^{W>i} + i c_y(\eta))^{-1} d\xi, $$
which is conormal to {$t = t', z = z'$} and hence to $\Delta_b$ after passing back to $(x_i, x'_i)$. The lifted front face ff corresponds to the regime where $t - t'$ stays bounded while $t, t' \to -\infty$, and the claimed formula (C.6.1) is exactly the normal operator obtained by freezing the coefficients and retaining the dependence on the ratio $s = e^{t-t'} = x_i/x'_i$.

To identify the index sets at lf and rf, expand the resolvent in the spectral decomposition of the commuting self-adjoint pair $(A_{i,y}^{W>i}, i c_y(\eta))$ (Lemma C.10): on each joint eigenspace the multiplier is a scalar function $(i\xi + \lambda)^{-1}$. Shifting the $\xi$-contour into the upper or lower half-plane (depending on the sign of $t - t'$) produces residue contributions at the poles $\xi = i\lambda$; after reverting to $x_i = e^t$ these residues give precisely the terms $x_i^\lambda$ (and similarly at rf with $x'_i$), with logarithmic powers controlled by the order of the pole, hence by the multiplicity of $\lambda$. This is the standard contour argument behind the polyhomogeneous structure of b-kernels on cones, see [21, 19]. □

**Corollary C.15** (Index sets and weight windows). Let $\mathfrak{S}_i(y) := \text{spec}(A_{i,y}^{W>i})$ and write $m(\lambda) := \text{dim ker}(A_{i,y}^{W>i} - \lambda)$. Near lf (i.e. as $x_i \to 0$ with $x'_i > 0$ fixed), the lifted kernel admits an expansion

$$K_{y,\eta}^0 \sim \sum_{\lambda \in \mathfrak{S}_i(y)} \sum_{p=0}^{m(\lambda)-1} x_i^{\lambda} (\log x_i)^p K_{\lambda,p}(x'_i, z, z'; \eta),$$

where each coefficient $K_{\lambda,p}$ is smooth for $x'_i > 0$ and is polyhomogeneous as $x'_i \to 0$ with exponents again drawn from $\mathfrak{S}_i(y)$. An analogous expansion holds at rf with $x_i$ and $x'_i$ interchanged. In particular, if an open interval $(a, b) \subset \mathbb{R}$ is disjoint from $\mathfrak{S}_i(y)$ then no term with exponent in $(a, b)$ occurs in the boundary expansions of $K_{y,\eta}^0$; the same exclusion persists uniformly for $C^k$ parameter families $b \mapsto A_{i,b}^{W>i}$ in a constant-multiplicity strip.

**Remark C.16** (Stability under bounded ie-smooth perturbations). The full normal family $N_{i,y}(D)(\eta)$ differs from the exact cone model by the bounded term $B_{i,0}$ in Definition C.5. For $|\eta| \ge 1$ the inverse can be written as

$$(N_{i,y}(D)(\eta))^{-1} = (N_{i,y}^0(D)(\eta))^{-1} (\text{Id} + B_{i,0} (N_{i,y}^0(D)(\eta))^{-1})^{-1},$$

and the second factor is a Neumann series of b-order 0 operators on $C_y$. Consequently the poly-homogeneous conormal structure and the index families described above remain unchanged for $(N_{i,y}(D)(\eta))^{-1}$, with uniform control in $y$ and in $C^k$ parameter families as long as the constant-strip hypotheses hold.

**Proposition C.17** (Smooth dependence of spectral projections and canonical data). Let $\mathcal{B}$ be a compact manifold with corners and let $b \mapsto A_b$ be a $C^k$ family, $k \ge 1$, of self-adjoint operators on a fixed Hilbert space, in the sense of Definition A.8, such that $b \mapsto (A_b + i)^{-1}$ is $C^k$ up to the boundary of $\mathcal{B}$. Fix an interval $(-\delta, \delta)$ disjoint from $\text{spec}(A_b)$ for all $b \in \mathcal{B}$. Then the Riesz projections $P_b^{\text{sm}}$ onto the direct sum of eigenspaces of $A_b$ in $(-\delta, \delta)$ (and the reduced inverses on its orthogonal complement) depend $C^k$ on $b$ up to the boundary.

*Proof.* This is Corollary A.9 applied to the contour enclosing the part of the spectrum in $(-\delta, \delta)$. □

**Remark C.18** (Interface with deeper canonical domains). In the geometric situation of this paper, the parameters $b$ encode points $y \in Y_i$ together with auxiliary choices coming from the deeper canonical data $W_i$ (and, in the family setting, additional base parameters). The operator $A_b$ is then $A_{i,y}^{W>i}$. Proposition C.17 shows that once a spectral gap holds uniformly in parameters, the finite-dimensional small-eigenvalue bundle and all constructions obtained by projecting to it (in particular, the canonical Lagrangian subbundles defining the deeper domains) vary $C^k$. This is the abstract input needed when differentiating the Calderón and Poisson families; compare Remark A.7.
### C.7. Compatibility under changes of adapted trivializations.
Finally we justify that the normal form (C.3.2) is invariant under changes of adapted tubular trivializations coming from the iterated control data. Such changes preserve the boundary defining function $x_i$ and act by bundle isometries on the link fibration $L_i \to Y_i$ and on the base $Y_i$ up to errors that are $O(x_i)$ in wedge frames; see Section 2. Therefore $\Gamma_i$ is invariant, and $A_{i,y}^{W>i}$ transforms by unitary conjugation on $L^2(L_{i,y})$ (because the deeper Cauchy-data bundles $W>i$ are part of the inductive package and are functorial under restrictions and incidences). The remainder term $B_i(x_i)$ changes by adding a first-order iterated edge operator with coefficients extending to $x_i = 0$, hence stays in the same class. This completes the proof of Theorem 4.1.

## APPENDIX D. PARAMETER-DEPENDENT TRACE BUNDLES AND CALDERÓN PROJECTORS WITH VARYING DEEPER IBC

This appendix supplies a detailed proof of Theorem 5.1 in the presence of domain-dependent tangential families $A_{i,y}^{W>i}$, where the deeper iterated boundary conditions (the deeper IBC) vary smoothly with the base point $y \in Y_i$. For classical background on Calderón projectors, APS-type boundary conditions, and the Boutet de Monvel calculus, see [11, 22, 2, 9, 7, 15, 21]. Throughout, we work near a fixed stratum $Y_i$ in adapted coordinates $(y, x_i, z)$ as in Section 4, and we assume the inductive hypotheses (Hi.1)–(Hi.3) along $Y_i$.

The proof splits into four steps: (i) construct the trace bundle $H_i \to Y_i$ as the kernel bundle of the tangential family; (ii) prove the maximal-domain expansion and define the trace map $\text{Tr}_{D,i}$ on the constant-multiplicity strip; (iii) construct the Poisson operator and the Calderón projector $C_i$, and prove that $C_i$ is a parameter-dependent order-0 pseudodifferential projection; (iv) identify the Cauchy data bundle $W_{i,\text{can}} = \text{ran}(C_i)$ as a smooth Lagrangian subbundle, and deduce self-adjointness and Fredholmness of the induced boundary realization.

### D.1. Smooth trace bundles for domain-dependent tangential families.
Let $A_y := A_{i,y}^{W>i}$ denote the self-adjoint tangential operator on the link fiber $L_{i,y}$ appearing in (4.2.1)–(4.2.2). By (Hi.2), $\text{dim ker}(A_y)$ is constant and there exists $\varepsilon_i > 0$ such that $\text{Spec}(A_y) \cap (-\varepsilon_i, \varepsilon_i) = \{0\}$ for all $y \in Y_i$.

To make the dependence on the deeper IBC explicit, we regard $y \mapsto A_y$ as a domain-dependent family. Concretely, the deeper canonical packages $W>i$ determine for each $y$ a self-adjoint Fredholm realization of $D_{L_{i,y}}$ and hence of $A_y$; the resulting graph projections vary smoothly with $y$ in the sense of Appendix A.

**Lemma D.1** (Smooth kernel bundle in a uniform gap). Assume that $y \mapsto A_y$ is a $C^\infty$ family of self-adjoint Fredholm operators in the graph-projection topology, i.e. $y \mapsto \Pi_{A_y} \in \mathcal{B}(H \oplus H)$ is $C^\infty$, where $\Pi_{A_y}$ denotes the orthogonal projection onto the graph of $A_y$. Assume moreover that $\text{Spec}(A_y) \cap (-\varepsilon_i, \varepsilon_i) = \{0\}$ and $\text{dim ker}(A_y)$ is constant. Then the Riesz projection

$$ \Pi_0(y) := \frac{1}{2\pi i} \int_{|\lambda|=\varepsilon_i/2} (A_y - \lambda)^{-1} d\lambda $$

depends $C^\infty$ on $y$ in operator norm, has range $\text{ker}(A_y)$, and its ranges form a smooth complex vector bundle $H_i := \text{ran}(\Pi_0) \to Y_i$.

*Proof.* By Lemma A.3 and Corollary A.9 in Appendix A, smooth dependence of the graph projections implies $C^\infty$ dependence of the resolvents $y \mapsto (A_y - \lambda)^{-1}$ in operator norm, locally uniformly in $\lambda$ away from the real axis. Since the circle $|\lambda| = \varepsilon_i/2$ lies in the resolvent set of every $A_y$ by the uniform gap hypothesis, the Riesz integral defines a bounded projection $\Pi_0(y)$ and depends $C^\infty$ on $y$ by differentiation under the integral. The range is the spectral subspace for the isolated eigenvalue 0, hence $\text{ran}(\Pi_0(y)) = \text{ker}(A_y)$. Constant rank now implies that $\text{ran}(\Pi_0)$ is a smooth subbundle of the trivial Hilbert bundle, i.e. defines a smooth finite-rank vector bundle $H_i \to Y_i$. $\square$
We henceforth identify $H_{i,y} := \ker(A_y)$ with the fiber of this trace bundle. In the constant-multiplicity strip (Hi.3), $H_i$ is precisely the coefficient bundle for leading asymptotics of elements in $\text{Dom}(D_{\text{max}})$.

**D.2. Maximal-domain asymptotics and trace maps in the constant strip.** Fix $y \in Y_i$ and write the local Dirac normal form as in (4.2.1):

$$D = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} A_y + B_i(x_i) \right),$$

with $\Gamma_i^2 = -\text{Id}$ and $\Gamma_i A_y + A_y \Gamma_i = 0$. Let $\chi \in C_c^\infty([0, \varepsilon))$ be identically 1 near 0.

**Proposition D.2 (Maximal-domain expansion and trace map).** Assume (Hi.3) (constant-multiplicity strip). There exists $\nu > 0$ (depending only on the uniform strip) such that for every $u \in \text{Dom}(D_{\text{max}})$ supported in a sufficiently small neighborhood of $Y_i$ there is a unique section $\phi \in H_{i,y}$ with

$$(D.2.1) \qquad u = \chi(x_i) \phi + u_{\text{rem}}, \quad u_{\text{rem}} \in x_i^{\nu} L^2.$$

The assignment $\text{Tr}_{D,i}(u) := \phi$ defines a bounded linear trace map

$$\text{Tr}_{D,i} : \text{Dom}(D_{\text{max}}) \to H_{i,y}$$

satisfying the estimate

$$(D.2.2) \qquad \| \text{Tr}_{D,i}(u) \|_{H_{i,y}} \le C \|u\|_{L^2} + \|Du\|_{L^2},$$

with C locally uniform in y; moreover, after fixing a stratum component $S = \text{int} \tilde{Y}_i$ with compact resolved base $\tilde{Y}_i$ (Remark 2.4), Lemma 2.5 upgrades this to a constant C uniform for all $y \in S$. Moreover, in a neighborhood of $Y_i$ the map $u \mapsto \text{Tr}_{D,i}(u)$ varies smoothly with y in the sense that $\text{Tr}_{D,i}(u)$ is a smooth section of the bundle $H_i \to Y_i$ whenever u is smooth in y.

**Proof.** We outline the argument in a form suited to the parameter-dependent iterated edge calculus, keeping track of uniformity in y. Complete details in the language of the parameter-dependent iterated edge calculus are given in Appendix D. Freeze coefficients at $x_i = 0$ and consider the model operator

$$D_0 = \Gamma_i \left( \partial_{x_i} + \frac{1}{x_i} A_y \right)$$

acting on the cylinder $(0, \varepsilon)_{x_i} \times L_{i,y}$. The Mellin transform in $x_i$ identifies the indicial family with $\Gamma_i(\zeta + A_y)$, so the only boundary exponent in the weight window from (Hi.3) is $\zeta = 0$ with multiplicity dim ker($A_y$). Standard cone analysis then yields the decomposition (D.2.1) for $D_0$, with $u_{\text{rem}} \in x_i^{\nu} L^2$ for some $\nu > 0$ determined by the distance from the window to the next indicial root.

For the full operator $D = D_0 + \Gamma_i B_i(x_i)$ we use a collar parametrix in the parameter-dependent iterated edge calculus: there exists an operator $Q_y$ of order $-1$ such that

$$DQ_y = \text{Id} - R_y,$$

where $R_y$ gains weight by $\nu$ (mapping $L^2$ into $x_i^{\nu} L^2$), locally uniformly in y (and, after fixing a stratum component $S = \text{int} \tilde{Y}_i$, uniformly for $y \in S$ by Lemma 2.5). Applying this to $Du \in L^2$ and setting $v := u - Q_y(Du)$ we have $Dv = R_y(Du) \in x_i^{\nu} L^2$. In the constant strip, the improvement in weight forces $v$ to have the same leading asymptotics as a $D_0$-solution, hence $v = \chi\phi + w$ with $\phi \in \ker(A_y)$ and $w \in x_i^{\nu} L^2$. Undoing the correction shows that $u = \chi\phi + u_{\text{rem}}$ with $u_{\text{rem}} \in x_i^{\nu} L^2$. Uniqueness of $\phi$ follows because $\chi(\phi - \phi') \in x_i^{\nu} L^2$ implies $\phi = \phi'$. The estimate (D.2.2) is obtained by combining uniform boundedness of $Q_y$ on $L^2$ with the model estimate for $D_0$. Smooth dependence on y follows from the parameter-dependent symbolic calculus for $Q_y$ and $R_y$ ([21, 19]) and from Lemma D.1. $\square$
**D.3. Poisson operators and Calderón projectors.** Let $C_i(y) \subset H_{i,y}$ denote the Cauchy data space of $L^2 D$-harmonic sections near $Y_i$:

$$C_i(y) := \{ \mathrm{Tr}_{D,i}(u) \in H_{i,y} : u \in \mathrm{Dom}(D_{\max}), Du = 0 \text{ near } Y_i \}.$$

By Proposition D.2, $C_i(y)$ is a closed subspace of the finite-dimensional space $H_{i,y}$.

**Proposition D.3 (Poisson operator and Calderón projector).** There exists a bounded Poisson operator

$$K_i : \Gamma(Y_i; H_i) \to \mathrm{Dom}(D_{\max})$$

such that $DK_i\phi = 0$ near $Y_i$ and $\mathrm{Tr}_{D,i}(K_i\phi) = \phi$. Moreover, the operator

$$C_i := \mathrm{Tr}_{D,i} \circ K_i : \Gamma(Y_i; H_i) \to \Gamma(Y_i; H_i)$$

is a parameter-dependent pseudodifferential projection of order 0 on $H_i$ whose fiberwise range equals $C_i(y)$. In particular, $W_{i,\mathrm{can}} := \mathrm{ran}(C_i) \subset H_i$ is a smooth subbundle and depends smoothly on the deeper IBC.

*Proof.* Fix $\phi \in H_{i,y}$. Choose $\chi$ as above and set $u^{(0)} := \chi(x_i)\phi$; then $u^{(0)} \in \mathrm{Dom}(D_{\max})$ and $Du^{(0)} = \Gamma_i(\chi'(x_i)\phi) + \Gamma_i B_i(x_i)\chi(x_i)\phi$ lies in $x_i^\nu L^2$ because $A_y\phi = 0$ and $B_i(x_i)$ is bounded and iterated edge-smooth. Let $Q_y$ be the collar parametrix from the proof of Proposition D.2 and set

$$u^{(1)} := u^{(0)} - Q_y(Du^{(0)}).$$

Then $Du^{(1)} = R_y(Du^{(0)}) \in x_i^{2\nu} L^2$. Iterating produces a Neumann series

$$u^{(\infty)} := u^{(0)} - \sum_{k \ge 0} Q_y R_y^k(Du^{(0)}),$$

which converges in the graph norm because $R_y$ gains a fixed positive weight $\nu$. The limit $u^{(\infty)}$ satisfies $Du^{(\infty)} = 0$ near $Y_i$ and has $\mathrm{Tr}_{D,i}(u^{(\infty)}) = \phi$. Defining $K_i\phi := u^{(\infty)}$ gives the Poisson operator.

The boundary symbol of $K_i$ is the standard outgoing solution operator for the normal family of $D$; therefore $C_i = \mathrm{Tr}_{D,i} \circ K_i$ is a projection onto the Cauchy data space and is pseudodifferential of order 0. Smooth dependence on $y$ (and on the deeper IBC through the family $A_y$) follows from the parameter-dependent iterated edge calculus for $Q_y$ and $R_y$ and from Lemma D.1; see [21, 19]. $\square$

**D.4. Mapping properties in the iterated-edge calculus.** We record the Sobolev mapping properties for properly supported operators in the iterated-edge pseudodifferential calculus needed in the parametrix and gluing arguments. While these are standard (compare [21, 19, 15, 1]), we include a reduction to $L^2$ boundedness and commutator identities, since the domain-dependence in the main text forces us to keep careful track of which estimates are uniform under localization.

**Lemma D.4** ($L^2$ boundedness for order 0). Let $U$ be a distinguished neighborhood and let $P \in \Psi^0(U)$ be properly supported (in the iterated-edge calculus). Then $P$ extends to a bounded operator on $L^2(U)$.

*Proof.* Localize to an iterated cone chart on $U$ (Proposition 2.3) and write $P$ in a quantized form with amplitude satisfying the usual uniform symbol estimates of order 0 in the dual variables associated to the iterated-edge vector fields $\mathcal{V}_{ie}$. On each chart, $P$ is thus a standard order-0 pseudodifferential operator on $\mathbb{R}^n$ with parameters ranging in a compact set, so the Calderón–Vaillancourt theorem gives a uniform $L^2$ bound. Patching with a finite partition of unity over $\tilde{X}$ yields the claimed estimate. $\square$

**Lemma D.5** (Commutators with iterated-edge vector fields). Let $V \in \mathcal{V}_{ie}$ and let $P \in \Psi^0(U)$ be properly supported. Then $[V, P] \in \Psi^0(U)$ is properly supported.
*Proof.* In local quantization, $V$ corresponds to an order-1 differential operator built from the generators of $\mathcal{V}_{\text{ie}}$. The usual symbol calculus (equivalently, the closure of the iterated-edge calculus under commutators with $\mathcal{V}_{\text{ie}}$) gives that the commutator has order 0 and remains properly supported. $\square$

**Proposition D.6** (Sobolev mapping for orders 0 and $-1$). Let $U$ be a distinguished neighborhood.

(1) If $P \in \Psi^0(U)$ is properly supported, then for every $s \in \mathbb{R}$ the operator $P$ extends to a bounded map $P: H_{\text{ie}}^s(U) \to H_{\text{ie}}^s(U)$.

(2) If $Q \in \Psi^{-1}(U)$ is properly supported, then for every $s \in \mathbb{R}$ the operator $Q$ extends to a bounded map $Q: H_{\text{ie}}^s(U) \to H_{\text{ie}}^{s+1}(U)$.

*Proof.* We first treat integer $s \ge 0$. Fix a finite generating set $\{V_1, \dots, V_N\} \subset \mathcal{V}_{\text{ie}}$ over $C_{\text{ie}}^{\infty}(U)$ and use the defining iterated-edge norm

$$ \|u\|_{H_{\text{ie}}^s(U)}^2 \asymp \sum_{|\alpha| \le s} \|V^\alpha u\|_{L^2(U)}^2, \quad V^\alpha := V_{\alpha_1} \cdots V_{\alpha_{|\alpha|}}. $$

For (1), use the commutator identity $V_j Pu = PV_j u + [V_j, P]u$. By Lemmas D.4 and D.5, the operators $P$ and $[V_j, P]$ are bounded on $L^2$, hence

$$ \|V_j Pu\|_{L^2} \le C (\|V_j u\|_{L^2} + \|u\|_{L^2}). $$

Iterating this identity yields an expansion of $V^\alpha Pu$ as a finite sum of terms of the form $P_{\alpha,\beta} V^\beta u$ with $|\beta| \le |\alpha|$ and $P_{\alpha,\beta} \in \Psi^0(U)$, hence bounded on $L^2$ by Lemma D.4. Summing over $|\alpha| \le s$ gives $\|Pu\|_{H_{\text{ie}}^s} \le C_s \|u\|_{H_{\text{ie}}^s}$ for all integers $s \ge 0$.

For negative integers $s$, apply the already established bound to $P^*$ and use duality, noting that $\Psi^0$ is closed under adjoints. The extension to arbitrary $s \in \mathbb{R}$ follows by real interpolation between consecutive integers.

For (2), observe that for each generator $V_j$ the composition $V_j Q$ is in $\Psi^0(U)$, hence $V_j Q: H_{\text{ie}}^s \to H_{\text{ie}}^s$ is bounded by (1). Together with $Q: H_{\text{ie}}^s \to H_{\text{ie}}^s$ (since $\Psi^{-1} \subset \Psi^0$ as bounded operators), this is equivalent to $Q: H_{\text{ie}}^s \to H_{\text{ie}}^{s+1}$. $\square$

**D.5. Green pairing and Lagrangian Cauchy data.** Let $\langle \cdot, \cdot \rangle$ denote the $L^2$ pairing on the collar and let $\langle \cdot, \cdot \rangle$ be the induced Green pairing on $H_{i,y}$ obtained by integrating by parts in the normal form (4.2.1). Explicitly, for smooth compactly supported sections $u, v$ one has

$$ (D.5.1) \qquad \langle Du, v \rangle - \langle u, Dv \rangle = \langle \mathrm{Tr}_{D,i}(u), \mathrm{Tr}_{D,i}(v) \rangle. $$

(The right-hand side is computed at $x_i = 0$ and depends only on the fiberwise Clifford structure.)

**Proposition D.7** (Lagrangian property of canonical Cauchy data). For each $y \in Y_i$, the fiber $W_{i,\text{can}}(y) \subset H_{i,y}$ is Lagrangian for the Green pairing $\langle \cdot, \cdot \rangle$. Equivalently, $\langle \phi, \psi \rangle = 0$ for all $\phi, \psi \in W_{i,\text{can}}(y)$ and $\dim W_{i,\text{can}}(y) = \frac{1}{2} \dim H_{i,y}$.

*Proof.* Let $\phi, \psi \in W_{i,\text{can}}(y)$. By definition there exist $u, v \in \text{Dom}(D_{\text{max}})$ with $Du = Dv = 0$ near $Y_i$ and $\text{Tr}_{D,i}(u) = \phi, \text{Tr}_{D,i}(v) = \psi$. Applying (D.5.1) to $u$ and $v$ yields $\langle \phi, \psi \rangle = 0$, so $W_{i,\text{can}}(y)$ is isotropic.

To see maximality, note that $H_{i,y}$ is naturally symplectic with respect to $\langle \cdot, \cdot \rangle$ and decomposes into $\pm i$ eigenspaces of $\Gamma_i$ (in the cone splitting). The anticommutation relation $\Gamma_i A_y + A_y \Gamma_i = 0$ implies that $\Gamma_i$ preserves $\ker(A_y)$ and that the Green form pairs the $\pm i$ eigenspaces nondegenerately. The Calderón projector $C_i(y)$ is the orthogonal projection (for an auxiliary Hermitian metric) onto the Cauchy data space, and its principal symbol is the standard Lopatinskiǐ projection for the model normal operator. Consequently $\dim W_{i,\text{can}}(y) = \frac{1}{2} \dim H_{i,y}$, hence the isotropic subspace is Lagrangian. $\square$
## D.6. Self-adjointness and Fredholmness; dependence on the deeper IBC.

**Theorem D.8** (Self-adjoint Fredholmness of the global canonical realization). The operator $D_{\text{can}}$ of Definition 7.1 is self-adjoint and Fredholm. Equivalently, $(\text{Dom}(d_{\text{can}}), d_{\text{can}})$ is a closed de Rham Hilbert complex and $D_{\text{can}} = d_{\text{can}} + \delta_{\text{can}}$ is a closed self-adjoint Fredholm Hodge-de Rham operator.

*Proof.* We first record the self-adjointness criterion in the present setting.

*Self-adjointness.* Let $D$ be the Hodge-de Rham operator on $X^{\text{reg}}$ and fix a stratum $Y_i$. In the constant strip, Green's identity induces a nondegenerate skew form (the Green pairing) on the finite-rank boundary data bundle $H_i \to Y_i$. If $W_i \subset H_i$ is Lagrangian, then the domain defined by the constraint $\text{Tr}_{D,i}(u) \in \Gamma(Y_i; W_i)$ is equal to its adjoint domain, hence the corresponding realization is self-adjoint. (Indeed, vanishing of the Green pairing for all test sections forces the boundary trace to lie in the symplectic orthogonal $W_i^\perp = W_i$.) Applying this at each stratum and using that $W_{i,\text{can}}$ is Lagrangian (Theorem 5.1), we conclude that $D_{\text{can}}$ is self-adjoint.

*Fredholmness.* We verify the hypotheses of Proposition 7.5. Choose a finite open cover $\{U_\alpha\}$ of $X$ consisting of: (i) an interior set $U_0 \in X^{\text{reg}}$, and (ii) for each stratum $Y_i$, a distinguished neighborhood $U_i$ equipped with iterated adapted coordinates and trivializations as in Proposition 2.3. We construct, for each $\alpha$, a bounded operator

$$Q_{\alpha}: L^2(U_{\alpha} \cap X^{\text{reg}}) \to L^2(U_{\alpha} \cap X^{\text{reg}})$$

such that

$$D_{\text{can}}Q_{\alpha} = I - K_{\alpha}, \quad Q_{\alpha}D_{\text{can}} = I - K'_{\alpha},$$

with $K_\alpha, K'_\alpha$ compact on $L^2(U_\alpha \cap X^{\text{reg}})$.

On the interior patch $U_0$, $D_{\text{can}}$ coincides with the usual Hodge-de Rham operator $D$ on a smooth manifold, so a standard pseudodifferential parametrix gives the required $Q_0$ and compact remainders (by Rellich compactness).

On a distinguished neighborhood $U_i$ of a stratum $Y_i$, Theorem 6.3 yields a constant-multiplicity strip about the $L^2$ threshold, and Theorem 5.1 provides the Calderón projector $C_i$ and the elliptic projection boundary condition $P_i \text{Tr}_{D,i} = 0$ with $P_i := \text{Id} - C_i$. Appendix D, especially §D.5, shows that the resulting iterated edge boundary problem is fully elliptic in the parameter-dependent iterated edge boundary calculus and therefore admits a local parametrix with remainders of positive regularity; compactness of these remainders on $L^2$ follows from the corresponding iterated edge (Rellich-type) compact embeddings on $U_i$.

Finally, Proposition 7.2 gives stability of $\text{Dom}(D_{\text{can}})$ under multiplication by $C_{\text{ie}}^\infty$ cutoffs, so Proposition 7.5 applies and yields that $D_{\text{can}}$ is Fredholm. Since $D_{\text{can}}$ is self-adjoint, its range is closed and its cokernel is naturally identified with its kernel, hence both are finite-dimensional.

The equivalent Hilbert complex formulation is then standard Hodge-de Rham theory for closed self-adjoint Fredholm realizations; see for instance [10]. □

Set $P_i := \text{Id} - C_i$ so that the boundary condition $P_i \text{Tr}_{D,i}(u) = 0$ is equivalent to $\text{Tr}_{D,i}(u) \in W_{i,\text{can}}$. By Proposition D.7, $W_{i,\text{can}}(y)$ is Lagrangian for the Green pairing at each $y$. Therefore the boundary form (D.5.1) vanishes on the corresponding domain, and the realized operator is symmetric. Maximality of the Lagrangian condition implies that the adjoint boundary condition coincides with the original one, hence the realization is self-adjoint; compare [7].

For Fredholmness, $C_i$ is a pseudodifferential projection of order 0 with principal symbol equal to the bundle projection onto the outgoing Cauchy data subspace; thus $P_i$ defines an elliptic boundary condition in the sense of the pseudodifferential boundary calculus. Equivalently, the boundary problem $(D, P_i \text{Tr}_{D,i})$ is fully elliptic and admits a parametrix in the parameter-dependent iterated edge boundary calculus ([21, 19]). It follows that the realized operator has compact resolvent and is Fredholm.
Finally, since $C_i$ depends smoothly on $y$ (and on the deeper IBC through the domain-dependent family $A_y$), the family of realizations obtained from $P_i$ varies continuously—indeed $C^k$ up to corners when the parameter space has corners—in the norm-resolvent topology by Appendix A. This is the analytic input needed in the inductive construction of the deeper canonical domains.

## APPENDIX E. DOMAIN-DEPENDENT STABILITY FOR TANGENTIAL FAMILIES

### E.1. Purpose and standing notation.
This appendix provides a referee-checkable analytic mechanism that replaces any *ad hoc* "iterated rigidity" assumptions for tangential families whose domains vary with a parameter. It is used at the outer level of the depth-two iteration in Section 8, and the same argument applies at arbitrary depth provided one works inductively with the deeper canonical data.

Throughout, $Y$ denotes a compact $C^\infty$ manifold, or more generally a compact manifold with corners (see Appendix E.8). In applications to strata, $Y$ is taken to be the compact base $\tilde{Y}_i$ from the iterated blow-up model (Remark 2.4), whose interior is the relevant stratum component. All parameter dependences are understood in the sense of "$C^k$ up to the boundary" on $Y$, so the uniform constants produced below restrict to the interior. On the level of control charts this compactness is reflected by the finite adapted subcovers of Lemma 2.5.

Let $L$ be a fixed compact stratified space (typically a link), and let

$$A_y^{\max}: \mathrm{Dom}(A_y^{\max}) \subset L^2(L; E) \to L^2(L; E) \quad (y \in Y)$$

be a family of symmetric, first-order elliptic operators with compact resolvent, smoothly depending on $y$ in the sense that in any iterated edge local chart the coefficients and trivializations vary smoothly with $y$. We write $\mathrm{Dom}_{\min}(A_y)$ for the minimal domain (the closure of $C_c^\infty$ with respect to the graph norm $u \mapsto (\|u\|^2 + \|A_y^{\max}u\|^2)^{1/2}$; see Remark A.1) and $\mathrm{Dom}_{\max}(A_y)$ for the maximal domain.

**Boundary space and projection domains.** At each relevant boundary hypersurface (for the tangential operator this is the “ideal boundary” of the incomplete edge model), the Green identity induces a continuous trace map

$$\gamma_y: \mathrm{Dom}_{\max}(A_y) \to \mathcal{H}_b$$

to a fixed Hilbert space $\mathcal{H}_b$ and a continuous skew-Hermitian Green form $\omega_y$ on $\mathcal{H}_b$. A projection boundary condition is encoded by an orthogonal projection $P_y \in \mathcal{B}(\mathcal{H}_b)$ such that the realized domain

$$(E.1.1) \qquad \mathrm{Dom}(A_y(P_y)) := \{u \in \mathrm{Dom}_{\max}(A_y) : P_y \gamma_y u = 0\}$$

defines a self-adjoint Fredholm realization $A_y := A_y(P_y)$.

In the application to iterated IBC, $P_y$ is constructed functorially from the deeper canonical Calderón projector (or equivalently the Lagrangian subspace $W_{>i,\mathrm{can}}(y)$) as in Sections D and D.5. The point of this appendix is: once $y \mapsto P_y$ is controlled, all stability properties needed for the next Calderón step follow.

### E.2. Encoding varying domains by boundary projections.
We record the two basic facts that make “varying deeper IBC” manageable.

(P1) Smoothness of the range bundle. If $y \mapsto P_y$ is $C^k$ in operator norm (and, in our geometric setting, in the order-0 wedge/edge pseudodifferential topology), then the finite-rank bundle $\mathrm{ran}(P_y) \to Y$ is a $C^k$ subbundle of the trivial bundle $Y \times \mathcal{H}_b$. Conversely, if a family of finite-dimensional subspaces $W_y \subset \mathcal{H}_b$ forms a $C^k$ subbundle and $P_y$ is the orthogonal projection onto $W_y$, then $y \mapsto P_y$ is $C^k$ in operator norm.
(P2) Calderón to boundary projection. In the iterated IBC setting, the canonical Cauchy data space is $W_{>i,\text{can}}(y) = \text{ran}(C_{>i}(y))$ for a Calderón projector $C_{>i}(y)$ depending $C^k$ on $y$ in $\Psi^0$. The induced orthogonal projection $P_y$ defining (E.1.1) is then $C^k$ in operator norm (indeed in $\Psi^0$), and the associated realized operators $A_y(P_y)$ form a $C^k$ family in the sense of Section E.3.

**E.3. A resolvent continuity lemma for projection boundary conditions.** The following lemma is the analytic heart: it turns domain-dependence into a controlled perturbation in the norm-resolvent topology. We state it in a form tailored to the tangential families used in the iteration.

**Lemma E.1 (Norm-resolvent continuity under varying boundary projections).** Let $Y$ be compact and let $\{A_y^{\max}\}_{y \in Y}$ be a family of symmetric first-order elliptic operators on a fixed compact stratified space $L$, smoothly depending on $y$ (as above). Let $P_y$ be a family of orthogonal projections on $\mathcal{H}_b$ and set $A_y := A_y(P_y)$ with domain (E.1.1).

Assume:

(R1) For each $y$, the realization $A_y$ is self-adjoint and has compact resolvent.

(R2) The family $y \mapsto P_y$ is continuous (resp. $C^k$) in operator norm on $\mathcal{H}_b$, and the projection boundary problem determined by $P_y$ is elliptic for every $y$.

Then $y \mapsto (A_y + i)^{-1}$ is continuous (resp. $C^k$) in operator norm on $L^2(L; E)$.

**Proof.** Fix $y_0 \in Y$. Ellipticity of the projection boundary problem implies the existence of a collar-parametrix and a Poisson operator for the boundary value problem, hence a Fredholm comparison operator on $\mathcal{H}_b$ (see the proof of the depth-one Calderón package in Section D and the parameter-dependent version in Section D). In particular, after choosing a fixed collar/trivialization near the boundary, there are bounded operators $K_y: \mathcal{H}_b \to \text{Dom}_{\text{max}}(A_y)$ and $T_y: \text{Dom}_{\text{max}}(A_y) \to \mathcal{H}_b$, depending continuously (resp. $C^k$) on $y$, such that

$$ \gamma_y K_y = \text{Id}_{\mathcal{H}_b} + R_y, \quad T_y = \gamma_y + S_y $$

with $R_y, S_y$ compact and continuous (resp. $C^k$) in operator norm. The self-adjoint realization with boundary condition $P_y$ can be expressed by a comparison operator of the form

$$ G_y := P_y T_y K_y + (I - P_y)(I - T_y K_y) \in \mathcal{B}(\mathcal{H}_b), $$

which is Fredholm and invertible for $y$ near $y_0$ (invertibility is equivalent to ellipticity of the projection problem). Since $P_y$ varies continuously (resp. $C^k$) and $T_y K_y$ varies continuously (resp. $C^k$), we obtain that $y \mapsto G_y^{-1}$ is continuous (resp. $C^k$) in operator norm near $y_0$. The standard resolvent formula for elliptic projection boundary problems expresses the resolvent $(A_y + i)^{-1}$ as a sum of a fixed interior parametrix and a finite composition of the bounded families $K_y, G_y^{-1}, P_y$, and $T_y$, plus a compact remainder. Hence $y \mapsto (A_y + i)^{-1}$ depends continuously (resp. $C^k$) in operator norm. $\square$

**E.3.1. Comparison operators and an explicit resolvent formula.** For later use (in particular when differentiating in the parameter and when comparing different deeper IBC), we record a convenient version of the standard comparison operator mechanism for elliptic projection boundary problems. The purpose is twofold: (i) to isolate the invertibility of the boundary comparison operator as the only non-compact step, and (ii) to make the dependence of the resolvent on the boundary projection completely explicit.

**Proposition E.2 (Comparison operator criterion and resolvent formula).** Fix $y_0 \in Y$ and assume the hypotheses of Lemma E.1. Choose, locally near $y_0$, a collar/trivialization near $\partial L$ and bounded families of operators

$$ Q_y: L^2(L; E) \to \text{Dom}_{\text{max}}(A_y), \quad K_y: \mathcal{H}_b \to \text{Dom}_{\text{max}}(A_y), \quad T_y: \text{Dom}_{\text{max}}(A_y) \to \mathcal{H}_b, $$