depending continuously (resp. $C^k$) on $y$ in operator norm, such that

$$ (E.3.1) \qquad (A_y^{\max} + i) Q_y = \mathrm{Id} + S_y, $$

$$ (E.3.2) \qquad T_y = \gamma_y + S'_y, $$

$$ (E.3.3) \qquad \gamma_y K_y = \mathrm{Id}_{\mathcal{H}_b} + R_y, $$

where $S_y, S'_y, R_y$ are compact and depend continuously (resp. $C^k$) in operator norm. Define the comparison operator

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

Then:

(1) $G_y$ is Fredholm of index 0 for $y$ near $y_0$. Moreover, ellipticity of the projection boundary problem implies that $G_{y_0}$ is invertible, hence $G_y$ is invertible for $y$ in a neighborhood of $y_0$, and $y \mapsto G_y^{-1}$ is continuous (resp. $C^k$) in operator norm.

(2) Let

$$ \mathcal{R}_y := Q_y - K_y G_y^{-1} P_y T_y Q_y. $$

Then $\mathcal{R}_y$ maps into $\mathrm{Dom}(A_y)$ and there exists a compact family $E_y \in \mathcal{K}(L^2(L; E))$, depending continuously (resp. $C^k$) in operator norm, such that

$$ (A_y + i) \mathcal{R}_y = \mathrm{Id} + E_y. $$

Consequently,

$$ (E.3.4) \qquad (A_y + i)^{-1} = \mathcal{R}_y (\mathrm{Id} + E_y)^{-1}, $$

and the right-hand side depends continuously (resp. $C^k$) on $y$ in operator norm.

**Proof.** The existence of $Q_y, K_y, T_y$ with (E.3.1)-(E.3.3) is part of the standard collar parametrix/Poisson construction for elliptic projection boundary problems; cf. Section D. Fredholmness of $G_y$ (indeed $G_y = \mathrm{Id} + \mathrm{compact}$) follows from (E.3.3) and (E.3.2). Since $G_{y_0}$ is invertible by ellipticity, the Banach inverse mapping theorem implies invertibility for $y$ near $y_0$ and continuity (resp. $C^k$) of $y \mapsto G_y^{-1}$ in operator norm.

For (2), set $\mathcal{R}_y$ as above. Using (E.1.1) and (E.3.3), one checks that $\mathcal{R}_y f$ satisfies the boundary condition $P_y \gamma_y (\mathcal{R}_y f) = 0$, hence $\mathcal{R}_y f \in \mathrm{Dom}(A_y)$. Next, apply $(A_y^{\max} + i)$ and use (E.3.1):

$$ (A_y + i) \mathcal{R}_y = (A_y^{\max} + i) Q_y - (A_y^{\max} + i) K_y G_y^{-1} P_y T_y Q_y = \mathrm{Id} + S_y - F_y, $$

where $F_y := (A_y^{\max} + i) K_y G_y^{-1} P_y T_y Q_y$. By construction of the Poisson operator, $(A_y^{\max} + i) K_y$ is compact (indeed smoothing) and varies continuously (resp. $C^k$) in operator norm; therefore $F_y$ is compact with the same regularity. Set $E_y := S_y - F_y$.

Finally, since $A_y$ is self-adjoint, $A_y + i$ is invertible, so $\mathrm{Id} + E_y = (A_y + i) \mathcal{R}_y$ is invertible on $L^2(L; E)$. The map $E \mapsto (\mathrm{Id} + E)^{-1}$ is smooth on the open set of invertible elements in the Banach algebra $\mathrm{Id} + \mathcal{K}(L^2)$, hence $y \mapsto (\mathrm{Id} + E_y)^{-1}$ is continuous (resp. $C^k$) in operator norm. Together with (E.3.4), this yields the claimed regularity of the resolvent. $\square$

**Remark E.3** (From $\Psi^0$-topology to operator-norm regularity). In applications (notably in Section D), the boundary projections $P_y$ arise as order-0 pseudodifferential projections on $\mathcal{H}_b$ varying $C^k$ in the natural $\Psi^0$-topology (uniform symbol bounds with $C^k$ dependence in the parameter). By the standard boundedness theorem for order-0 pseudodifferential operators, this implies $C^k$ dependence of $P_y$ in operator norm on $\mathcal{H}_b$, hence hypothesis (R2) in Lemma E.1 is automatic once the symbolic family is fixed.
**E.4. Constant rank of the kernel bundle.** Norm-resolvent continuity implies stability of spectral projections by the holomorphic functional calculus.

**Lemma E.4 (Spectral projections vary continuously).** Assume Lemma E.1. Let $\gamma$ be a small positively oriented circle in $\mathbb{C}$ enclosing 0 and no other points of $\text{Spec}(A_{y_0})$ for some $y_0 \in Y$. Then for $y$ sufficiently close to $y_0$ the circle $\gamma$ encloses the same number of eigenvalues (counted with multiplicity) of $A_y$, and the spectral projection

$$ \Pi_y := \frac{1}{2\pi i} \int_{\gamma} (A_y - \lambda)^{-1} d\lambda $$

depends continuously (resp. $C^k$) on $y$ in operator norm. In particular, $\text{rank}(\Pi_y)$ is locally constant.

*Proof.* By Lemma E.1, the resolvent depends continuously (resp. $C^k$) in operator norm for $\lambda \in \gamma$, hence the contour integral defining $\Pi_y$ depends continuously (resp. $C^k$). Finite rank follows from compact resolvent. Since $\Pi_y$ remains a projection and depends continuously in operator norm, its rank is locally constant. $\square$

**Proposition E.5 (Constant rank of $\ker(A_y)$).** Assume Lemma E.1. If 0 is isolated in $\text{Spec}(A_y)$ for every $y \in Y$, then $\text{dim} \ker(A_y)$ is locally constant on $Y$, hence constant on each connected component. Consequently, the kernels assemble into a $C^k$ finite-rank vector bundle

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

with $H_y = \ker(A_y)$.

*Proof.* Apply Lemma E.4 with $\gamma$ enclosing only 0. Then $\text{rank}(\Pi_y) = \text{dim} \ker(A_y)$ is locally constant. Moreover, $H$ is locally the range of $\Pi_y$, and $C^k$ dependence of $\Pi_y$ yields $C^k$ local frames obtained by applying $\Pi_y$ to a fixed basis at $y_0$. $\square$

**E.5. Uniform spectral gap at zero.** Compactness of the base promotes pointwise spectral separation into a uniform gap.

**Proposition E.6 (Uniform spectral gap from compactness).** Assume Lemma E.1 and that $Y$ is compact. If 0 is isolated in $\text{Spec}(A_y)$ for each $y \in Y$, then there exists $\varepsilon > 0$ such that

$$ \text{Spec}(A_y) \cap (-\varepsilon, \varepsilon) = \{0\} \quad \text{for all } y \in Y. $$

*Proof.* Argue by contradiction. If no such $\varepsilon$ exists, then there are $y_n \in Y$ and nonzero $\lambda_n \in \text{Spec}(A_{y_n})$ with $\lambda_n \to 0$. By compactness, after passing to a subsequence we have $y_n \to y_\infty \in Y$. Norm-resolvent continuity (Lemma E.1) implies convergence of spectral projections on small contours; in particular, eigenvalues cannot accumulate at a point that is isolated in the limiting spectrum. Hence 0 would fail to be isolated in $\text{Spec}(A_{y_\infty})$, contradicting the hypothesis. $\square$

**E.6. Constant-multiplicity strip and the trace bundle.** We now convert the uniform tangential gap into a uniform indicial strip for the outer wedge operator. This is the specific input needed for the trace-bundle construction in Section D.2.

**Proposition E.7 (Constant-multiplicity strip).** Assume the wedge normal form

$$ (E.6.1) \qquad D = \Gamma \left( \partial_x + \frac{1}{x} A_y + B(x) \right), $$

where $A_y$ is self-adjoint on $L^2(L; E)$ with compact resolvent and $B(x)$ is bounded and iterated-edge smooth up to $x=0$. Assume Proposition E.5 and Proposition E.6.

Then there exists $\delta \in (0, \varepsilon)$ such that, uniformly in $y \in Y$, the indicial family

$$ I_y(\lambda) := \Gamma(\lambda + A_y) $$
is invertible for all $\lambda \in \mathbb{C}$ with $0 < |\Re\lambda| < \delta$. Equivalently, in the weight strip $|\Re\lambda| < \delta$ about the $L^2$ threshold the only indicial root is $\lambda = 0$, with multiplicity rank($H$). Consequently, the trace bundle in this strip identifies canonically with $H = \ker(A) \to Y$ and has constant rank.

**Proof.** For the model part $D_0 = \Gamma(\partial_x + \frac{1}{x}A_y)$, a formal solution $u = x^\lambda v$ gives

$$D_0(x^\lambda v) = x^{\lambda-1}\Gamma(\lambda + A_y)v,$$

so the indicial family is $I_y(\lambda) = \Gamma(\lambda + A_y)$ and the indicial roots are exactly $\lambda = -\mu$ with $\mu \in \text{Spec}(A_y)$. Since $B(x)$ contributes only order $x^\lambda$ terms, it does not change the indicial family. By Proposition E.6, there is $\varepsilon > 0$ such that $\text{Spec}(A_y) \cap (-\varepsilon, \varepsilon) = \{0\}$ for all $y$. Choose any $\delta \in (0, \varepsilon)$; then for $0 < |\Re\lambda| < \delta$ we have $-\Re\lambda \notin \text{Spec}(A_y)$, hence $\lambda + A_y$ is invertible and so is $I_y(\lambda)$. The root at $\lambda = 0$ corresponds to the eigenvalue 0 of $A_y$, with multiplicity dim $\ker(A_y) = \text{rank}(H)$ (constant by Proposition E.5). The trace-bundle identification follows by the standard leading-term map that sends $u \in \text{Dom}_{\max}(D)$ to the $x^0$ coefficient in its asymptotic expansion in the strip. $\square$

**E.7. Application to the depth-two outer family $A_{1,y}^{W_{2,\text{can}}}$**. We explain how the abstract criteria are met in the depth-two one-step iteration.

**Theorem E.8 (Outer stability without extra rigidity).** In the depth-two setup of Section 8, assume:

(S1) The inner canonical Calderón projector $C_2(y)$ (hence $W_{2,\text{can}}(y) = \text{ran}(C_2(y))$) depends $C^\infty$ on $y \in Y_1$ in the wedge/edge pseudodifferential topology.

(S2) The outer tangential operator $A_{1,y}$ is constructed functorially from the link geometry and depends $C^\infty$ on $y$ at the level of coefficients/trivializations.

Then the realized outer family $A_{1,y}^{W_{2,\text{can}}}$ satisfies the outer hypotheses (Hi.2)-(Hi.3) of Subsection 5.2: $\ker(A_{1,y}^{W_{2,\text{can}}})$ has constant rank and forms a smooth bundle $H_1 \to Y_1$, there is a uniform spectral gap at 0, and there exists a constant-multiplicity outer strip in which the trace bundle is $H_1$.

**Proof.** By (S1) and the parameter-dependent Calderón package of Appendix D, the family $y \mapsto C_2(y)$ is smooth in $\Psi^0$, hence the induced family of boundary projections $P_y$ encoding the deeper IBC is smooth in operator norm (cf. (P2) above). By (S2), the coefficients of the tangential operator $A_{1,y}$ vary smoothly. Therefore Lemma E.1 applies to $A_y := A_{1,y}^{W_{2,\text{can}}}$ on the compact resolved base $\tilde{Y}_1$ (a compact manifold with corners with interior $Y_1$). Proposition E.5 yields constant rank of the kernel bundle, Proposition E.6 yields a uniform gap, and Proposition E.7 produces the constant-multiplicity strip and identifies the trace bundle with $H_1$. $\square$

**Remark E.9.** Theorem E.8 is the analytic substitute for the “iterated rigidity” template recorded in Section 8: smoothness of the deeper canonical Calderón projector forces smoothness of the realized tangential family, and compactness of the (resolved) base forces a uniform gap.

**E.8. Manifolds with corners and $C^k$ regularity.** In the iterated setting the parameter space can naturally be a manifold with corners (for example, a boundary face in a blow-up). The arguments above remain valid verbatim if $Y$ is a compact manifold with corners and all parameter dependences are understood in the sense of Theorem A.8 (“$C^k$ up to the boundary”). Indeed, the proofs only use operator-norm continuity (or $C^k$) on $Y$ and compactness of $Y$. In particular, the conclusions of Lemma E.1 and Propositions E.5–E.7 hold with “$C^\infty$” replaced by “$C^k$” throughout.

# APPENDIX F. ITERATED IDEAL BOUNDARY CONDITIONS AND DELIGNE-TYPE INTERSECTION COMPLEXES

We view this appendix as an iterated version of the Deligne-sheaf description of intersection cohomology [14, 6] adapted to ideal boundary conditions/mezzoperversities in the sense of [1, 5]. This appendix records the sheaf-theoretic package behind the global identification (Theorem 7.7):
the de Rham complex endowed with an iterated ideal boundary condition (mezzoperversity) datum $W$ represents a Deligne-type intersection complex. We keep the discussion self-contained and only use the stratified local models from Section 2 together with the constant-strip trace theory and normal forms established in Section 5-Section 6.

**F.1. The analytic IBC de Rham complex as a sheaf.** Let $X$ be a compact smoothly stratified space with strata $\{Y_i\}$ ordered by depth. Write $j: X^{\text{reg}} \hookrightarrow X$ for the inclusion of the top stratum. Fix a mezzoperversity datum

$$W = (W_1, \dots, W_d),$$

where for each $i$ we have a smooth subbundle $W_i \subset H_i \to Y_i$ of the trace bundle

$$H_i = \ker(A_i^{W_{>i}}) \to Y_i,$$

with $A_i^{W_{>i}}$ the tangential operator at $Y_i$ endowed with the deeper boundary data $W_{>i} = (W_{i+1}, \dots, W_d)$ (see Section 4 and Appendix D).

For an open set $U \subset X$, define $L_W^k(U)$ to be the space of $L^2$ $k$-forms on $U \cap X^{\text{reg}}$ which lie in the domain of the closed extension $d_W$ and satisfy the boundary conditions prescribed by $W$ at every stratum meeting $U$ (equivalently: for each $i$, the canonical trace along $Y_i \cap U$ takes values in $W_i$ whenever that stratum is encountered in $U$). Exterior differentiation defines maps $d: L_W^k(U) \to L_W^{k+1}(U)$, and these assignments form a complex of sheaves $L_W^\bullet$ on $X$.

**Lemma F.1 (Fine resolution).** The complex $L_W^\bullet$ is a fine complex of sheaves. In particular, its hypercohomology is computed by global sections:

$$\mathbb{H}^\bullet(X; L_W^\bullet) \cong \mathbb{H}^\bullet(\Gamma(X; L_W^\bullet)).$$

*Proof.* Let $\tilde{X}$ be the iterated resolution (a compact manifold with corners) associated with the Thom-Mather control data (Section 2). Denote by $C_{ie}^\infty(X)$ the algebra of functions on $X^{\text{reg}}$ which lift to smooth functions on $\tilde{X}$; it admits partitions of unity subordinate to arbitrary open covers of $X$, since $\tilde{X}$ does. Moreover, if $\phi \in C_{ie}^\infty(X)$ and $u \in L_W^k(U)$, then $\phi u \in L_W^k(U)$: $\phi$ is bounded with all $\mathcal{V}_{ie}$-derivatives bounded, hence multiplication preserves the graph norms defining $\text{Dom}(d_W)$; and on each stratum the trace map is local and satisfies $\gamma_i(\phi u) = (\phi|_{Y_i}) \gamma_i(u)$, so the condition $\gamma_i(u) \in W_i$ is stable under multiplication. Thus one can form stratification-compatible partitions of unity and use them to construct the usual sheaf homotopies, proving that $L_W^\bullet$ is fine. $\square$

**F.2. Distinguished neighborhoods and the cone computation.** Let $p \in Y_i$ and let $U_p \cong B \times C(L)$ be a distinguished neighborhood as in Section 2, where $B$ is a ball in $Y_i$ and $L$ is the (compact) link at $p$, endowed with its induced stratification of strictly smaller depth. Write $f = \text{dim } L$ and set

$$\ell = \lfloor \frac{f}{2} \rfloor.$$

Over $U_p$ the constant-strip expansion (Proposition D.2) and the wedge Dirac normal form (Section 4, Appendix C) identify the maximal domain near the cone tip with iterated conormal expansions whose leading coefficient lies in $\ker(A_{i,p}^{W_{>i}}) = H_{i,p}$; the boundary condition $\gamma_i(\cdot) \in W_{i,p}$ is precisely the selection of an allowed subspace at the borderline exponent.

**Proposition F.2 (Local cohomology on an iterated cone).** Let $U_p \cong B \times C(L)$ be as above, with $B$ contractible. Then the cohomology of the complex of sections over $U_p$ is given by

$$H^k(\Gamma(U_p; L_W^\bullet)) \cong \begin{cases} \mathrm{IH}_{W_{>i}}^k(L), & k < \ell, \\ \mathrm{IH}_{W_{>i}}^\ell(L), & k = \ell \text{ and } f \text{ odd,} \\ W_{i,p}, & k = \ell \text{ and } f \text{ even,} \\ 0, & k > \ell, \end{cases}$$
where $W_{>i}$ denotes the deeper datum restricted to the link and $\mathrm{IH}_{W_{>i}}^{\bullet}(L)$ is the corresponding iterated IBC intersection cohomology of $L$ (depth strictly smaller than that of $X$). Moreover, these identifications are natural with respect to restriction to smaller distinguished neighborhoods.

*Proof.* Since $B$ is contractible and the control data provide a product decomposition, all local systems in the statement are constant over $B$, and it suffices to compute at a fixed point of $B$. Write $x$ for the cone variable and decompose forms in wedge coframe as

$$u(x) = \alpha(x) + \frac{dx}{x} \wedge \beta(x),$$

with $\alpha(x), \beta(x)$ families of forms on $L$ taking values in the deeper domain determined by $W_{>i}$. The constant-strip trace theory and the normal form for the Hodge-de Rham operator show that closed $L^2$ forms admit asymptotic expansions whose leading term is $x^{-\ell}\alpha_0$ with $\alpha_0 \in H_{i,p} = \ker(A_{i,p}^{W_{>i}})$; when $f$ is odd there is no borderline ambiguity, while for $f$ even the coefficient $\alpha_0$ is exactly the boundary trace. The ideal boundary condition enforces $\alpha_0 \in W_{i,p}$.

For $k > \ell$, the usual cone homotopy operator in the radial variable (cf. [12, 13]) is $L^2$-bounded in the present wedge setting, and the constant-strip hypothesis excludes indicial roots in the critical strip; hence every closed $k$-form is locally exact in $\Gamma(U_p; L_W^\bullet)$, yielding vanishing. For $k < \ell$, the same homotopy shows that the cohomology is represented by forms independent of $x$ (pulled back from the link), and the deeper boundary conditions ensure that these representatives lie in the $W_{>i}$-IBC domain on $L$; thus $H^k(\Gamma(U_p; L_W^\bullet)) \cong \mathrm{IH}_{W_{>i}}^k(L)$. Finally, at $k = \ell$ the only potentially nontrivial classes are those with borderline exponent; by the normal form, their class is determined by the boundary coefficient in $H_{i,p}$, and the boundary condition selects exactly the subspace $W_{i,p}$ when $f$ is even (when $f$ is odd, the borderline degree does not occur and one obtains $\mathrm{IH}_{W_{>i}}^\ell(L)$). Naturality under restriction follows because the construction uses only local traces and the canonical cone homotopies. $\square$

**Lemma F.3 (Local Deligne-type conditions).** The cohomology sheaves $\mathcal{H}k(L_W^\bullet)$ satisfy the Deligne-type vanishing and attachment conditions dictated by $W$ in the sense of [14, 6, 8]: on a distinguished neighborhood $U_p \cong B \times C(L)$ as above, $\mathcal{H}k(L_W^\bullet)$ vanishes for $k > \ell$, agrees with the (constant) local system with fiber $\mathrm{IH}_{W_{>i}}^k(L)$ for $k < \ell$, and in the borderline degree $k = \ell$ (when $f$ is even) the attachment local system is precisely the prescribed subbundle $W_i \subset H_i$.

*Proof.* By Proposition F.2, the cohomology of sections over distinguished neighborhoods is constant on $B$ and vanishes above degree $\ell$. This is exactly the local vanishing condition. The description for $k < \ell$ identifies the restriction maps on overlaps with the standard restriction maps on the link and hence implements the usual attachment isomorphisms. In the borderline even-link case, the cohomology in degree $\ell$ is identified with $W_{i,p}$, and the restriction maps are fiberwise inclusions induced by the trace map and orthogonal projection onto $W_i$, which is precisely the Deligne-type attachment prescription encoded by the datum $W$. $\square$

**F.3. Identification with a Deligne-type intersection complex.** Let $\mathrm{IC}_W$ denote the Deligne-type intersection complex associated with the datum $W$. For the purposes of this paper, we take $\mathrm{IC}_W$ to be the (unique up to unique isomorphism) object of $\mathrm{D}^b(\mathrm{Sh}(X))$ characterized by the following two properties:

(1) $j^* \mathrm{IC}_W \simeq \mathbb{R}_{X^{\mathrm{reg}}}$ (the constant sheaf on the regular stratum);

(2) for each $p \in Y_i$ with distinguished neighborhood $U_p \cong B \times C(L)$ and $\ell = \lfloor \dim(L)/2 \rfloor$, the stalk cohomology satisfies the local conditions of Lemma F.3.

Equivalently, one may realize $\mathrm{IC}_W$ by the standard Deligne iterative extension-and-truncation procedure, with the borderline attachment in even-link degree prescribed by the subbundle $W_i$ (cf. [14, 6, 8]).
**Theorem F.4 (Sheaf identification).** There is a canonical quasi-isomorphism of complexes of sheaves

$$L_{\mathcal{W}}^{\bullet} \simeq \mathrm{IC}_{\mathcal{W}}.$$

Consequently,

$$H^{\bullet}(\Gamma(X; L_{\mathcal{W}}^{\bullet})) \cong \mathrm{IH}_{\mathcal{W}}^{\bullet}(X).$$

*Proof.* By Lemma F.1, it suffices to identify $L_{\mathcal{W}}^{\bullet}$ in the derived category of sheaves. On $X^{\mathrm{reg}}$ the complex $L_{\mathcal{W}}^{\bullet}$ is the usual de Rham complex, hence resolves the constant sheaf $\mathbb{R}_{X^{\mathrm{reg}}}$; this gives the first defining property of $\mathrm{IC}_{\mathcal{W}}$. Lemma F.3 gives the Deligne-type local vanishing and attachment conditions at each stratum. By the axiomatic characterization and uniqueness of Deligne-type intersection complexes (cf. [14, 6, 8]), these properties determine $\mathrm{IC}_{\mathcal{W}}$ uniquely up to canonical isomorphism, and therefore $L_{\mathcal{W}}^{\bullet}$ is canonically quasi-isomorphic to $\mathrm{IC}_{\mathcal{W}}$. Taking hypercohomology and using fineness yields the asserted identification with $\mathrm{IH}_{\mathcal{W}}^{\bullet}(X)$. $\square$

## ACKNOWLEDGEMENTS

This work is indebted to the foundational analysis of Cheeger on $L^2$ Hodge theory for conic singularities, and to the subsequent development of the global analytic framework for iterated edge metrics and ideal boundary conditions by Albin-Leichtnam-Mazzeo-Piazza. The wedge/edge operator technology used throughout draws on the analytic and microlocal foundations of Mazzeo, Vertman, and Krainer-Mendoza, and on the classical boundary-value and Calderón projector theory of Calderón, Seeley, Boutet de Monvel, Grubb, and Booß-Bavnbek-Wojciechowski, as well as the APS viewpoint of Atiyah-Patodi-Singer and its reformulation by Melrose. On the topological and sheaf-theoretic side, the perspective is guided by the intersection homology theory of Goresky-MacPherson, the theory of perverse sheaves of Beilinson-Bernstein-Deligne, the Borel volume on intersection cohomology, and Banagl's mezzoperversity viewpoint.

## STATEMENTS AND DECLARATIONS

**Competing Interests.** The author declares that there are no competing interests.

**Funding.** No external funding was received for this work.

**Author Contributions.** The author solely contributed to the conception, analysis, and writing of this manuscript.

**Data Availability.** Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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SCHOOL OF MATHEMATICS AND STATISTICS, KASHI UNIVERSITY, KASHI, CHINA

Email address: luojiaming@hhu.edu.cn