A Residual-Excess Framework for Defect Resolutions: Finite-Node Saito Gluing and Integral Hodge Obstructions

1. Introduction: Residuals, Excess, and Defect Matrices

1.1. The Basic Problem: Cones Under-Resolve Defects

Let $\mathcal{T}$ be a triangulated category and let

$$R:A⟶A^{†}$$

be a morphism in $\mathcal{T}$. The cone of R is the third term in a distinguished triangle

$$A\stackrel{R}{\to }{A}^{†}⟶Cone\left(R\right)\stackrel{+1}{\to }.$$

Thus $Cone\left(R\right)$ records the aggregate defect of the morphism R.

The starting point of this paper is that, in geometric situations, $Cone\left(R\right)$ may under-resolve the defect in two different ways. First, it records that a defect exists, but it does not record how the morphism R is realized. Second, if one compares the same construction across coefficient systems, for example integral and rational coefficients, the rational cone may miss torsion information killed by rationalization.

The first problem is addressed by residual triples. A factorization

$$A\stackrel{u}{\to }B\stackrel{v}{\to }{A}^{†},vu=R,$$

contains more information than R alone. The object B is an interface through which the return map R is realized. We call such a factorization a defect resolution of R.

The residual triple associated to the defect resolution is

$$\mathsf{T}(u,v):=\left(Cone\left(u\right),Cone\left(v\right),Cone\left(R\right)\right).$$

The first cone records the outgoing defect of the map into the interface. The second records the incoming defect of the map out of the interface. The third records the aggregate return defect. The octahedral axiom relates the three cones:

$$Cone\left(u\right)⟶Cone\left(R\right)⟶Cone\left(v\right)\stackrel{+1}{\to }.$$

Thus the triple is not an arbitrary list of cones. It is a factorization-refined defect object.

The key point is that $\mathsf{T}(u,v)$ is generally finer than $Cone\left(R\right)$ alone. Distinct factorizations of the same return map R can have isomorphic aggregate cones but non-isomorphic residual triples. In a category such as $D^{b}MHM\left(X\right)$, this difference can be visible in support, Hodge filtration, weight filtration, Tate twist, realization, or extension class.

The second problem is addressed by excess. If $C^{\mathbb{Z}}$ is an integral object or residual package and $C^H$ is its rational or mixed-Hodge-module realization, then a comparison morphism

$$C^{\mathbb{Z}}\otimes \mathbb{Q}⟶C^H$$

may fail to be an isomorphism before or after passing through the relevant defect-resolution construction. We define the corresponding excess object by

$$Exc(C^{\mathbb{Z}},C^H):=Cone\left(C^{\mathbb{Z}}\otimes \mathbb{Q}⟶C^H\right)[-1],$$

whenever such a comparison morphism is specified. Excess measures coefficient-change defect. In particular, torsion objects killed by rationalization may appear as nonzero excess even when the rational shadow vanishes.

Thus residuals and excess measure different failures. Residual triples are horizontal invariants: they refine a defect at fixed coefficients by recording how a return map factors through an interface. Excess objects are vertical invariants: they measure the failure of a defect-resolution package to compare cleanly across coefficient systems.

1.2. The Residual–Excess Matrix

The two refinements introduced above can be organized in a single matrix. One axis records residual data at a fixed coefficient level. The other records what is lost, or fails to compare, when one passes between coefficient levels.

Let

$$A^{\mathrm{coeff}}\stackrel{u^{\mathrm{coeff}}}{\to }{B}^{\mathrm{coeff}}\stackrel{v^{\mathrm{coeff}}}{\to }{\left(A^{†}\right)}^{\mathrm{coeff}},v^{\mathrm{coeff}}{u}^{\mathrm{coeff}}=R^{\mathrm{coeff}},$$

be a defect resolution in a coefficient-refined theory. Here $\mathrm{coeff}$ denotes any coefficient-refined enhancement, such as an integral or finite-coefficient package. Suppose also that there is a corresponding rational or mixed-Hodge realization

$$A^H\stackrel{u^H}{\to }{B}^H\stackrel{v^H}{\to }{\left(A^{†}\right)}^H,v^H{u}^H=R^H.$$

The superscript H denotes the rational or mixed-Hodge level. We assume that comparison morphisms from the coefficient-refined data to the rational/MHM data have been specified and are compatible with the two factorizations.

At the coefficient-refined level, the residual triple is

$${\mathsf{T}}^{\mathrm{coeff}}(u,v)=\left(Cone\left(u^{\mathrm{coeff}}\right),Cone\left(v^{\mathrm{coeff}}\right),Cone\left(R^{\mathrm{coeff}}\right)\right).$$

At the rational/MHM level, the residual triple is

$${\mathsf{T}}^H(u,v)=\left(Cone\left(u^H\right),Cone\left(v^H\right),Cone\left(R^H\right)\right).$$

The comparison maps between corresponding residual cones have their own cones. These comparison cones are the excess objects.

We package these data in the residual–excess matrix

$$\mathfrak{D}(u,v)=\begin{array}{cccc}& Cone\left(u\right)& Cone\left(v\right)& Cone\left(R\right)\\ \mathrm{coeff}& C_u^{\mathrm{coeff}}& C_v^{\mathrm{coeff}}& C_R^{\mathrm{coeff}}\\ H/\mathbb{Q}& C_u^H& C_v^H& C_R^H\\ Exc& E_u& E_v& E_R\end{array}$$

where

$$C_u^{\mathrm{coeff}}=Cone\left(u^{\mathrm{coeff}}\right),C_u^H=Cone\left(u^H\right),$$

and similarly for v and R. If the coefficient-refined row is integral, then the comparison map for the u-entry has the form

$$C_u^{\mathbb{Z}}\otimes \mathbb{Q}⟶C_u^H,$$

and the corresponding excess object is

$$E_u:=Cone\left(C_u^{\mathbb{Z}}\otimes \mathbb{Q}⟶C_u^H\right)[-1].$$

The definitions of $E_v$ and $E_R$ are analogous. In finite-coefficient or Bockstein situations, the comparison morphism is the appropriate finite-coefficient or connecting comparison map, and the excess object is again the shifted cone of that specified comparison.

The rows of $\mathfrak{D}(u,v)$ are constrained by octahedral triangles. The columns are constrained by coefficient comparison. Thus the matrix separates two forms of hidden defect data: factorization-sensitive information in the horizontal direction, and coefficient-sensitive information in the vertical direction. Equivalently, residuals record how a defect factors through an interface at fixed coefficients, while excess records the obstruction to lifting, realizing, or comparing that residual package across coefficient change.

For the finite-node Saito situation, the same matrix takes the form

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ \mathrm{coeff}& Cone\left(u_{\Sigma }^{\mathrm{coeff}}\right)& Cone\left(v_{\Sigma }^{\mathrm{coeff}}\right)& Cone\left(N^{\mathrm{coeff}}\right)\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }^H\right)& Cone\left(v_{\Sigma }^H\right)& Cone\left(N^H\right)\\ Exc& E_{\mathrm{can},\Sigma }& E_{\mathrm{var},\Sigma }& E_{N,\Sigma }\end{array}$$

whenever coefficient-refined finite-node comparison data are available. The canonical and variation columns are especially important. They ask whether the extraction and reinsertion residuals seen rationally are genuine realizations of coefficient-refined residual data, or whether torsion excess intervenes. This matrix is therefore not merely a diagrammatic convenience. It is the main organizational device of the paper: horizontally it records what is controlled by the octahedral axiom, and vertically it records what is controlled by coefficient comparison.

Principle 1

(Residual–excess bridge). Let a rational or mixed-Hodge defect-resolution package be given by

$$A^H\stackrel{u^H}{\to }{B}^H\stackrel{v^H}{\to }{\left(A^{†}\right)}^H,v^H{u}^H=R^H.$$

Suppose one asks whether this package is the realization or rationalization of a coefficient-refined defect-resolution package

$$A^{\mathrm{coeff}}\stackrel{u^{\mathrm{coeff}}}{\to }{B}^{\mathrm{coeff}}\stackrel{v^{\mathrm{coeff}}}{\to }{\left(A^{†}\right)}^{\mathrm{coeff}}.$$

Once comparison morphisms are fixed, the excess column of the residual–excess matrix records the obstruction to this comparison. In particular, vanishing of the relevant excess objects means that the coefficient-refined residual package compares cleanly with the rational/MHM one.

Remark 1

(Equivalence versus inclusion). The paper does not claim that every possible coefficient-change defect arises from the lift obstruction of a residual triple. The framework uses the direction needed here: whenever coefficient-refined and rational residual packages are related by comparison morphisms, the vertical cones measure the failure of that comparison. Whether all natural excess classes arise from such defect-resolution lifts is a separate problem, left open here.

This principle is used as an organizing framework, not as a universal classification theorem. In the finite-ODP case, the local Milnor fiber is torsion-free, so the local excess entries vanish. In Diaz-type finite-coefficient Bockstein examples, a nonzero integral torsion class lies in the kernel of rationalization and calibrates the nonzero-excess direction.

1.3. Relation with Saito and Kerr–Laza

The nearby/vanishing-cycle case motivating this paper is classical. For a one-parameter degeneration, nearby and vanishing cycles carry canonical and variation morphisms

$$can:{\psi }_{\pi }⟶{\varphi }_{\pi },var:{\varphi }_{\pi }⟶{\psi }_{\pi },$$

whose composites are controlled by monodromy; see [1,2,3]. In Saito’s mixed-Hodge-module setting one has

$${can}^H:{\psi }_{\pi ,1}^H⟶{\varphi }_{\pi ,1}^H,{Var}^H:{\varphi }_{\pi ,1}^H⟶{\psi }_{\pi ,1}^H(-1),$$

and

$${Var}^H{can}^H=N,$$

where

$$N:{\psi }_{\pi ,1}^H⟶{\psi }_{\pi ,1}^H(-1)$$

is nilpotent monodromy [4,5].

The octahedral structure associated with the can/var/N diagram is not new. It is part of Saito’s mixed-Hodge-module formalism and appears explicitly in Saito’s work. Kerr–Laza use this structure, with an appendix by Saito, in their treatment of degenerations, Clemens–Schmid type sequences, local invariant cycles, phantom cohomology, and the decomposition theorem over a curve [6]. In particular, the can/var/monodromy octahedron is standard background for this paper, not its novelty.

Our use is different. We treat a factorization of a fixed return map as defect-resolution data, and then compare such data across coefficient levels. The paper asks two related questions. First, what Hodge-typed information is retained by a factorization

$$A\stackrel{u}{\to }B\stackrel{v}{\to }{A}^{†},vu=R,$$

beyond the aggregate cone $Cone\left(R\right)$? Second, when such a factorization is seen rationally, what obstruction prevents it from being the rationalization of an integral defect-resolution package?

In the finite ordinary-double-point case, the full vanishing-cycle object is replaced by an explicit node-supported Hodge interface

$$W_{\Sigma }^H=\underset{p_k\in \Sigma }{⨁}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

The main construction is the reduced finite-node factorization

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1),v_{\Sigma }{u}_{\Sigma }=N$$

on the finite-node sector. The goal is to extract the support, Hodge type, Tate twist, realization, Ext-class data, and coefficient-comparison behavior associated with this reduced interface.

1.4. Why Mixed Hodge Modules are Used

The formal operation of taking cones is not the main point. The main point is that, in $D^{b}MHM\left(X\right)$, a defect resolution carries Hodge-typed comparison data.

For a complex algebraic variety X, Saito constructed an abelian category

$$MHM\left(X\right)$$

and an exact faithful realization functor

$$rat:MHM\left(X\right)⟶Perv(X;\mathbb{Q})$$

[4,5]. Thus an object of $MHM\left(X\right)$ has an underlying rational perverse sheaf, but also carries Hodge and weight data.

This matters here for four reasons.

First, a mixed-Hodge-module residual records support. In the finite-node case, the interface object is supported on the finite ordinary-double-point set $\Sigma$.

Second, it records Hodge-theoretic normalization. A point-supported summand

$$i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

realizes to the rational skyscraper sheaf $i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}$, but the Tate twist $(-1)$ is invisible after applying rat.

Third, heart-level MHM residuals give exact sequences in $MHM\left(X\right)$, and exactness then realizes to exactness in $Perv(X;\mathbb{Q})$.

Fourth, applying hypercohomology to MHM objects gives mixed Hodge structures. Thus an MHM residual cone has a cohomological shadow in mixed Hodge structures and, in degeneration settings, may be compared with the limiting mixed Hodge structure.

For these reasons, this paper does not merely assign names to cones. In the main finite-node construction, we construct the interface $W_{\Sigma }^H$, the maps $u_{\Sigma }$ and $v_{\Sigma }$, the corrected extension produced by $Cone\left(v_{\Sigma }\right)[-1]$, and the nodewise Ext residual module. These data are not recoverable from $Cone\left(N\right)$ alone.

At the same time, mixed Hodge modules are rational objects. They are therefore not designed to see integral torsion killed by rationalization. This is why the excess axis is needed. Residual triples explain what the rational factorization remembers. Excess objects explain what the rational factorization cannot see when integral or finite-coefficient information is present.

1.5. The Finite–Node Model Theorem

The main geometric example is a one-parameter conifold degeneration

$$\pi :\mathcal{X}⟶\Delta$$

whose central fiber $X_0$ has finitely many ordinary double points

$$\Sigma =\{p_1,\dots ,p_r\}.$$

Let $i_k:\left\{p_k\right\}\hookrightarrow X_0$ be the inclusion. The node-supported Hodge interface is

$$W_{\Sigma }^H:={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

The finite-node mixed-Hodge-module construction produces maps

$$u_{\Sigma }:{\psi }_{\pi ,1}^H⟶W_{\Sigma }^H,v_{\Sigma }:W_{\Sigma }^H⟶{\psi }_{\pi ,1}^H(-1),$$

satisfying

$$v_{\Sigma }{u}_{\Sigma }=N$$

on the finite-node sector. Thus $W_{\Sigma }^H$ gives a defect resolution of finite-node monodromy.

The variation cone gives the corrected MHM extension:

$$P_{\mathrm{var},\Sigma }^H=Cone\left(v_{\Sigma }\right)[-1],$$

with

$$0⟶I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H⟶0.$$

Thus the incoming residual records the reinsertion of the node-supported Hodge interface into the central-fiber intersection-complex package.

The canonical cone records the extraction side. Under the explicit heart-level extraction hypothesis that

$$u_{\Sigma }:{\psi }_{\pi ,1}^H\to W_{\Sigma }^H$$

is an epimorphism in $MHM\left(X_0\right)$, the shifted cone

$$K_{\mathrm{can},\Sigma }^H:=Cone\left(u_{\Sigma }\right)[-1]$$

lies in the MHM heart and fits into

$$0⟶K_{\mathrm{can},\Sigma }^H⟶{\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H⟶0.$$

Without this epimorphism hypothesis, $Cone\left(u_{\Sigma }\right)$ remains a derived MHM residual. Thus the hypothesis is used only to identify the canonical-side residual with a heart-level kernel.

The return cone is

$$Cone\left(N:{\psi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1)\right).$$

The octahedral triangle relates the extraction residual, reinsertion residual, and monodromy residual.

The extension class of the variation residual lies in

$${Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H).$$

Since

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),$$

this Ext group decomposes nodewise:

$${Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{Ext}_{MHM\left(X_0\right)}^1(i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),I{C}_{X_0}^H).$$

This nodewise residual module is one of the concrete outputs of the paper.

1.6. The Zero–Excess Calibration: Ordinary Double Points

The finite ordinary-double-point model also gives the first calibration of the excess axis. Analytically near an ordinary double point of a complex threefold, the local model is

$$f=x_1^2+x_2^2+x_3^2+x_4^2.$$

Its Milnor fiber has the homotopy type of $S^3$. Hence the local integral vanishing cohomology is torsion-free:

$${\tilde{H}}^3(F;\mathbb{Z})\cong \mathbb{Z},{\tilde{H}}^m(F;\mathbb{Z})=0(m\ne 3).$$

Consequently, the local node interface has no hidden torsion killed by rationalization. In the residual–excess matrix, the ODP case has nontrivial horizontal residual data but vanishing local vertical excess:

$$\begin{array}{cccc}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right)\\ \mathbb{Z}& C_{u,\Sigma }^{\mathbb{Z}}& C_{v,\Sigma }^{\mathbb{Z}}& C_{N,\Sigma }^{\mathbb{Z}}\\ H/\mathbb{Q}& C_{u,\Sigma }^H& C_{v,\Sigma }^H& C_{N,\Sigma }^H\\ Exc& 0& 0& 0\end{array}\mathrm{locally}\mathrm{at}\mathrm{each}\mathrm{ordinary}\mathrm{double}\mathrm{point}.$$

This statement is local at the node. It does not assert that every possible global integral extension subtlety vanishes. Rather, it says that the ordinary double point itself is a torsion-free local model. Thus finite ODPs calibrate the horizontal residual axis of the theory: residual data can be meaningful even when local coefficient excess is zero.

1.7. The Nonzero–Excess Calibration: Diazś Bockstein Class

The contrasting calibration comes from Diaz’s Enriques-product construction. Let

$$V=S_1\times S_2$$

be the product of two Enriques surfaces. Diaz uses two finite-coefficient torsion inputs:

$${\alpha }_1\in H^1(S_1,\mathbb{Z}/2\left(1\right)),{\beta }_2\in H^2(S_2,\mathbb{Z}/2\left(1\right)).$$

Here ${\alpha }_1$ is the class of the K3 double cover of $S_1$, and ${\beta }_2$ is chosen to have nonzero Brauer image. Their external cup product is

$${\Theta }_2={\pi }_1^{\ast }{\alpha }_1\cup {\pi }_2^{\ast }{\beta }_2\in H^3(V,\mathbb{Z}/2\left(2\right)).$$

The coefficient sequence

$$0⟶\mathbb{Z}\left(2\right)\stackrel{\times 2}{\to }\mathbb{Z}\left(2\right)⟶\mathbb{Z}/2\left(2\right)⟶0$$

gives the Bockstein

$$\delta :H^3(V,\mathbb{Z}/2\left(2\right))⟶H^4(V,\mathbb{Z}\left(2\right)).$$

The resulting class

$${\Delta }_2:=\delta \left({\Theta }_2\right)\in H^4(V,\mathbb{Z}\left(2\right))$$

is integral 2-torsion. Hence

$${\Delta }_2\otimes \mathbb{Q}=0.$$

Under Diaz’s unramified-survival condition, ${\Delta }_2$ is non-algebraic.

This is the nonzero-excess calibration. The class ${\Delta }_2$ lies in the kernel of rationalization, so it is invisible to rational mixed Hodge modules, but it is not zero integrally. It is therefore a coefficient-change defect: a nonzero integral obstruction whose rational shadow vanishes.

In matrix form, Diaz’s mechanism may be summarized as

$$\begin{array}{cc}\mathrm{coefficient}\mathrm{level}& \mathrm{Diaz}--\mathrm{Bockstein}\mathrm{package}\\ \mathbb{Z}/2& {\Theta }_2\in H^3(V,\mathbb{Z}/2\left(2\right))\\ \mathbb{Z}& {\Delta }_2=\delta \left({\Theta }_2\right)\in H^4(V,\mathbb{Z}\left(2\right))\\ \mathbb{Q}& 0\\ Exc& {\Delta }_2\ne 0,{\Delta }_2\otimes \mathbb{Q}=0.\end{array}$$

Thus Diaz calibrates the vertical excess axis. It shows why a rational residual theory, even a refined one, must be supplemented by coefficient-change excess if the goal is to understand integral Hodge obstruction channels.

The point is not that Diaz’s Bockstein class is itself a finite-node Saito residual. It is not. Its role here is different: it supplies a clean nonzero-excess model. Together, finite ODPs and Diaz’s construction show that residuals and excess are independent but composable defect invariants.

1.8. Relation with Phantom and Closed–Stratum Contributions

Kerr–Laza’s decomposition theorem over a curve framework isolates point-supported terms $W_{\sigma }^j$ in perverse direct images and relates them to vanishing and phantom cohomology [6]. In that setting, a perverse sheaf over a disk has nearby/vanishing quiver data, and the decomposition separates the local-system part from point-supported closed-stratum contributions.

The finite-ODP object $W_{\Sigma }^H$ should be viewed as the MHM-level node-supported interface selecting the ordinary-double-point contribution of this type. More precisely, it is the finite-node, Hodge-normalized interface through which the local ODP vanishing sector factors the finite-node part of monodromy. Later sections compare this interface with extension-ladder and recollement residuals, and isolate the boundary/phantom contribution selected by the ordinary-double-point sector.

This comparison is not a formal consequence of the octahedral axiom alone. It depends on the finite-ODP geometry and on the point-supported MHM structure of $W_{\Sigma }^H$. In the residual–excess language, these are horizontal comparison results inside the rational or MHM row. They should be distinguished from vertical coefficient-change excess, which measures information lost under rationalization.

1.9. The Examples Treated in the Paper

The paper treats the finite-node can–var factorization as the main residual example, and Diaz’s Bockstein class as the main excess calibration. Several standard comparison formalisms are used as supporting defect-resolution families.

1.

Nearby/vanishing can–var residuals. The main defect resolution is

$${\psi }_{\pi ,1}^H\to W_{\Sigma }^H\to {\psi }_{\pi ,1}^H(-1),$$

with return operator N. Its residual triple is the rational/MHM row of the finite-node residual–excess matrix.

2.

Recollement open/closed residuals. For $j:U\hookrightarrow X_0$ and $i:\Sigma \hookrightarrow X_0$, the recollement triangles

$$j_{!}{j}^{\ast }M\to M\to i_{\ast }{i}^{\ast }M\stackrel{+1}{\to }$$

and

$$i_{\ast }{i}^{!}M\to M\to j_{\ast }{j}^{\ast }M\stackrel{+1}{\to }$$

give closed-stratum residuals. These are standard in perverse and MHM recollement [2,4,5,7].

3.

Minimal/middle/maximal extension residuals. For $L_U^H={\mathbb{Q}}_U^H\left[3\right]$, the comparison

$$j_{!}{L}_U^H\to j_{\ast }{L}_U^H$$

has image $j_{!\ast }{L}_U^H=I{C}_{X_0}^H$. The associated cones record boundary residuals between lower, middle, and upper extensions.

4.

Monodromy invariant/coinvariant residuals. The return operator $N:{\psi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1)$ has cone $Cone\left(N\right)$. Under heart-level hypotheses, its perverse cohomology identifies $kerN$ and $cokerN$.

5.

Specialization and limiting-mixed-Hodge residuals. Applying hypercohomology to MHM residual triangles gives exact sequences of mixed Hodge structures. In the finite-node case,

$${\mathbb{H}}^0(X_0,W_{\Sigma }^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1),$$

up to the fixed support and degree conventions.

6.

Coefficient-change excess. Whenever integral, finite-coefficient, and rational realizations of a defect-resolution package can be compared, the cone of the comparison map gives an excess object. In the ordinary-double-point case, the local excess vanishes because the Milnor fiber is torsion-free. In Diaz’s Enriques-product example, the Bockstein class gives a nonzero excess killed by rationalization.

7.

MacPherson–Vilonen zig-zag residuals. In an MV category $C(F,G;T)$, an object is a tuple

$$(A,B,u,v)$$

with

$$F\left(A\right)\stackrel{u}{\to }B\stackrel{v}{\to }G\left(A\right),vu=T_A.$$

Its residual triple is

$$\left(Cone\left(u\right),Cone\left(v\right),Cone\left(T_A\right)\right).$$

In finite-coefficient applications, the boundary shadow of this triple is related to Bockstein obstruction channels, as in [8].

The first five families are computed in finite-ODP, MHM, or cohomological form. The sixth introduces the coefficient-change axis. The final MV family supplies a formal categorical setting in which finite-coefficient Bockstein data can be organized by gluing and boundary maps.

1.10. Main Contributions

The central contribution of this paper is the residual–excess framework. A defect attached to a morphism admits two orthogonal refinements: a horizontal, factorization-sensitive refinement measured by residual triples, and a vertical, coefficient-sensitive refinement measured by excess objects. These two refinements assemble into a residual–excess matrix. The finite-ODP Saito interface and the Diaz Bockstein class serve as complementary calibration witnesses for the two axes.

More concretely, the paper contributes the following.

1.

We define structured defect resolutions of a morphism $R:A\to A^{†}$ and attach to each such resolution a residual triple constrained by the octahedral axiom.

2.

We show that residual triples are strictly finer than aggregate cones: distinct factorizations of the same return map can have isomorphic aggregate cones but non-isomorphic residual triples.

3.

We introduce excess objects as coefficient-change defects, defined as cones of comparison morphisms between integral or finite-coefficient residual data and rational or MHM residual data.

4.

We assemble residuals and excess into the residual–excess matrix, which records horizontal factorization data and vertical coefficient-comparison data in a single object.

5.

We construct the finite-node Saito interface $W_{\Sigma }^H={⨁}_{p_k\in \Sigma }{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$, prove that it factors finite-node monodromy, recover the corrected variation extension, and identify the nodewise Ext residual module.

6.

We identify two calibration regimes. Finite ordinary double points give the zero-excess local model: residual data are nontrivial, but the local Milnor fiber is torsion-free. Diaz’s Enriques-product Bockstein construction gives the nonzero-excess model: an integral torsion class lies in the kernel of rationalization and, under the survival criterion, gives a non-algebraic integral Hodge obstruction.

1.11. Scope

This paper does not claim that the can/var/monodromy octahedron is new. That structure belongs to Saito’s formalism and is used in the Kerr–Laza–Saito study of degenerations. The contribution here is different: we isolate factorization-refined defect data, apply it to a reduced finite-node Saito interface, compute the resulting corrected extension and nodewise Ext residual module, and place this residual package inside a broader residual–excess matrix.

The paper also does not construct a full integral mixed-Hodge-module theory. Integral and finite-coefficient objects are used here only where comparison data are available or where they serve as calibration examples. In particular, the ordinary-double-point discussion proves a local zero-excess statement because the Milnor fiber is torsion-free. The Diaz discussion is used as a nonzero-excess calibration: it shows how a finite-coefficient Bockstein class can produce integral torsion killed by rationalization.

Finally, this paper does not prove the rational Hodge conjecture, nor does it claim to solve the integral Hodge conjecture in general. Its purpose is to construct a defect language that separates two mechanisms: residual factorization data at fixed coefficients and excess obstruction data across coefficient change. The residual–excess matrix is proposed as the common framework in which finite-node Saito gluing and integral Hodge obstruction channels can be compared.

2. Saito–Kerr–Laza Background and MHM Defect Data

This section fixes the mixed-Hodge-module and degeneration-theoretic background used in the finite-node construction. The nearby/vanishing-cycle can–var–monodromy formalism is standard. It belongs to Saito’s mixed Hodge module theory and is used explicitly in the degeneration framework of Kerr–Laza, with Saito’s appendix, where the octahedral formalism is tied to Clemens–Schmid type sequences, local invariant cycles, phantom cohomology, and the decomposition theorem over a curve [4,5,6].

Our use is more specific. We isolate a finite ordinary-double-point interface

$$W_{\Sigma }^H=\underset{p_k\in \Sigma }{⨁}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

as a reduced, node-supported Hodge interface through which the finite-node part of monodromy factors. This interface supplies the rational or mixed-Hodge row of the residual–excess matrix introduced in the introduction.

Thus the goal of this section is not to reprove the Saito–Kerr–Laza formalism. Rather, it records the parts of that formalism needed for the paper’s defect language: nearby/vanishing cycles, can–var–monodromy, point-supported Hodge interfaces, heart-level cones, and realization. The excess direction will later compare this rational/MHM package with integral or finite-coefficient data. The present section therefore describes the rational Hodge-theoretic row of the matrix.

2.1. Mixed Hodge Modules and Realization

Let X be a complex algebraic variety. Saito constructs an abelian category

$$MHM\left(X\right)$$

of mixed Hodge modules on X, equipped with an exact faithful realization functor

$$rat:MHM\left(X\right)⟶Perv(X;\mathbb{Q})$$

[4,5]. An object $M\in MHM\left(X\right)$ has an underlying rational perverse sheaf

$$rat\left(M\right)\in Perv(X;\mathbb{Q}),$$

but also carries filtered D-module data, Hodge filtration, weight filtration, and Tate-twist information.

We will use the following consequences of Saito’s theory.

1.

$MHM\left(X\right)$ is abelian.

2.

The realization functor rat is exact.

3.

The realization functor rat is faithful.

4.

Nearby cycles, vanishing cycles, direct images, restrictions, and duality exist in the mixed-Hodge-module setting in the forms used below.

5.

If $i:\left\{p\right\}\hookrightarrow X$ is the inclusion of a point, then $i_{\ast }{\mathbb{Q}}_{\left\{p\right\}}^H$ is the point-supported pure Hodge module whose rational realization is $i_{\ast }{\mathbb{Q}}_{\left\{p\right\}}$.

Exactness implies that every short exact sequence in $MHM\left(X\right)$ realizes to a short exact sequence in $Perv(X;\mathbb{Q})$. Faithfulness implies that nonvanishing of a realized perverse morphism implies nonvanishing of the MHM morphism that realizes it.

The converse direction is the one used in this paper: an MHM object or morphism may carry Hodge filtration, weight filtration, support, and Tate-twist data that are invisible after applying rat. Thus a factorization in $D^{b}MHM\left(X\right)$ can carry information not recoverable from its rational perverse realization.

This is the first reason MHM is the correct setting for the finite-node residual package. The residual triple does not merely consist of three abstract cones. Its entries carry Hodge-typed data. In the finite-node setting, this includes the support on $\Sigma$, the Tate twist $(-1)$, the realization to rational skyscraper sheaves, and the extension class of the corrected object.

There is also a limitation. Mixed Hodge modules are rational. They are not designed to see integral torsion killed by rationalization. This is the reason for the excess axis of the residual–excess matrix. The MHM row records the rational Hodge-theoretic residual package; the excess row records the failure of integral or finite-coefficient data to be recovered from that rational row.

2.2. Realization Forgets the Tate Normalization

Let

$$i_k:\left\{p_k\right\}\hookrightarrow X_0$$

be a point inclusion. The point-supported Tate-twisted Hodge module

$$i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

has rational realization

$$rat\left(i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)\right)\cong i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}.$$

The Tate twist $(-1)$ is therefore part of the MHM structure but is invisible after rational realization.

This distinction is essential in the finite-ODP construction. Each node has a rank-one rational perverse contribution after realization, but its mixed-Hodge-module avatar is

$$i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

The Tate twist is compatible with the monodromy normalization

$$N:{\psi }_{\pi ,1}^H⟶{\psi }_{\pi ,1}^H(-1).$$

Thus the node interface is not only a rational point-supported sheaf. It is a Hodge-normalized point-supported interface.

This gives a first example of information lost under a forgetful functor. Rational realization forgets the Tate normalization, while rationalization of integral data forgets torsion. These are distinct losses. The former is still visible in $MHM\left(X\right)$, whereas the latter requires the excess axis of the residual–excess matrix. The paper keeps these two losses separate: Tate-normalized residual data live in the MHM row, while torsion killed by rationalization is measured by excess.

2.3. Nearby Cycles, Vanishing Cycles, and Monodromy

Let

$$\pi :\mathcal{X}⟶\Delta$$

be a one-parameter degeneration, and let

$$X_0:={\pi }^{-1}\left(0\right).$$

For $M\in MHM\left(\mathcal{X}\right)$, Saito’s theory gives unipotent nearby and vanishing cycles

$${\psi }_{\pi ,1}^H\left(M\right),{\varphi }_{\pi ,1}^H\left(M\right),$$

together with morphisms

$${can}^H:{\psi }_{\pi ,1}^H\left(M\right)⟶{\varphi }_{\pi ,1}^H\left(M\right),$$

$${Var}^H:{\varphi }_{\pi ,1}^H\left(M\right)⟶{\psi }_{\pi ,1}^H\left(M\right)(-1),$$

and nilpotent monodromy

$$N:{\psi }_{\pi ,1}^H\left(M\right)⟶{\psi }_{\pi ,1}^H\left(M\right)(-1).$$

With Saito’s normalization,

$${Var}^H\circ {can}^H=N.$$

After applying realization, this becomes the usual can–var–monodromy relation for rational nearby and vanishing cycles.

Thus the full nearby/vanishing interface gives a defect resolution of monodromy:

$${\psi }_{\pi ,1}^H\left(M\right)\stackrel{{can}^H}{\to }{\varphi }_{\pi ,1}^H\left(M\right)\stackrel{{Var}^H}{\to }{\psi }_{\pi ,1}^H\left(M\right)(-1),{Var}^H{can}^H=N.$$

The associated triple

$$\left(Cone\left({can}^H\right),Cone\left({Var}^H\right),Cone\left(N\right)\right)$$

is the classical nearby/vanishing monodromy triple. The use of the octahedral axiom for this can–var–N system is part of the Saito formalism and appears explicitly in Kerr–Laza–Saito’s treatment of degenerations [6].

The finite-node construction below is not a rediscovery of this octahedron. It is a reduced version of the interface in the ordinary-double-point case:

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1),v_{\Sigma }{u}_{\Sigma }=N$$

on the finite-node sector.

In the language of this paper, the full can–var interface and the reduced finite-node interface both give residual triples. The difference is that the reduced interface remembers the ordinary-double-point support and its Hodge-normalized point contributions. This is why the interface $W_{\Sigma }^H$ is treated as geometric data, not merely as a formal middle object in a factorization.

2.4. Point-Supported Terms and Phantom Cohomology in Kerr–Laza

We recall the part of Kerr–Laza’s framework that is relevant here. For a family over a curve and a semisimple complex $K^{•}$, the decomposition over the curve gives point-supported summands of the form

$${}^{p}R^j{f}_{\ast }{K}^{•}\cong j_{\ast }{V}^j\left(K^{•}\right)\left[1\right]\oplus \underset{\sigma \in \Sigma }{⨁}{i}_{\sigma \ast }{W}_{\sigma }^j\left(K^{•}\right),$$

where the $V^j\left(K^{•}\right)$ are local-system terms over the smooth locus and the $W_{\sigma }^j\left(K^{•}\right)$ are point-supported terms over the special points $\sigma$ [6]. Kerr–Laza also express vanishing cohomology in terms of a limiting part and a point-supported term; in their notation, one obtains a decomposition of the form

$$H_{\mathrm{van},\sigma }^{\ell }\left(K^{•}\right)\cong V_{lim}^{\ell }\left(K^{•}\right)/ker(T_{\sigma }-I)\oplus W_{\sigma }^{\ell }\left(K^{•}\right).$$

They define phantom cohomology by

$$H_{\mathrm{ph},\sigma }^{\ell }\left(K^{•}\right):=ker\left\{sp:H^{\ell }(X_{\sigma },K^{•})\to H_{lim,\sigma }^{\ell }\left(K^{•}\right)\right\},$$

and identify it with the relevant point-supported term $W_{\sigma }^{\ell -1}\left(K^{•}\right)$, equivalently with the image of the appropriate local cohomology composite [6].

This is the published degeneration-theoretic framework into which the present finite-node construction fits. The object

$$W_{\Sigma }^H=\underset{p_k\in \Sigma }{⨁}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

should be viewed as the finite ordinary-double-point MHM interface selecting the ODP contribution of this point-supported $W_{\sigma }$-type data. The comparison with Kerr–Laza’s point-supported and phantom terms is one of the main reasons to isolate $W_{\Sigma }^H$ as an interface object, rather than only working with the aggregate cone $Cone\left(N\right)$.

This comparison is horizontal in the residual–excess matrix. It takes place inside the rational/MHM row. It explains which part of the rational degeneration package is being isolated by the finite-node interface. It does not, by itself, detect integral torsion killed by rationalization. That coefficient-sensitive information belongs to the vertical excess direction.

2.5. Saito’s Divisor-Gluing Datum

Let

$$g:X⟶\mathbb{C}$$

be a regular function, set

$$Y:=g^{-1}\left(0\right),U:=X\setminus Y,$$

and let

$$j:U\hookrightarrow X,i:Y\hookrightarrow X$$

be the inclusions. Saito’s divisor-gluing formalism describes mixed Hodge modules along Y by open data, divisor data, and morphisms satisfying a monodromy relation [4,5,9].

The form used here is:

$$M^{\prime },M^{\prime \prime },u:{\psi }_{g,1}^H\left(M^{\prime }\right)⟶M^{\prime \prime },v:M^{\prime \prime }⟶{\psi }_{g,1}^H\left(M^{\prime }\right)(-1),$$

with

$$vu=N.$$

Thus Saito gluing supplies a defect resolution

$${\psi }_{g,1}^H\left(M^{\prime }\right)\stackrel{u}{\to }{M}^{\prime \prime }\stackrel{v}{\to }{\psi }_{g,1}^H\left(M^{\prime }\right)(-1)$$

of the monodromy return map N.

In the finite-ODP setting, the divisor is $X_0={\pi }^{-1}\left(0\right)$, and the interface object is

$$M^{\prime \prime }=W_{\Sigma }^H=\underset{p_k\in \Sigma }{⨁}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

The maps are the finite-node maps

$$u_{\Sigma }:{\psi }_{\pi ,1}^H\to W_{\Sigma }^H,v_{\Sigma }:W_{\Sigma }^H\to {\psi }_{\pi ,1}^H(-1).$$

The variation-side cone gives the corrected object

$$P_{\mathrm{var},\Sigma }^H\simeq Cone\left(v_{\Sigma }\right)[-1],$$

and the corresponding exact sequence is

$$0⟶I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H⟶0.$$

This exact sequence is the principal MHM output of the finite-node construction. It is a rational/Hodge-theoretic residual object. Later, when coefficient-comparison data are available, one may ask whether an integral variation residual rationalizes to $P_{\mathrm{var},\Sigma }^H$. The cone of that comparison is the variation excess

$$E_{\mathrm{var},\Sigma }.$$

Thus Saito gluing supplies the horizontal residual package, while excess records whether that package is the full rationalization of an integral one.

2.6. Heart-Level Cones in $D^{b}MHM$

Let

$$f:M⟶N$$

be a morphism in $MHM\left(X\right)$. Since $MHM\left(X\right)$ is the heart of the standard MHM t-structure on $D^{b}MHM\left(X\right)$, the cone $Cone\left(f\right)$ lies in $D^{b}MHM\left(X\right)$.

If f is an epimorphism in $MHM\left(X\right)$, then

$$Cone\left(f\right)[-1]\cong ker\left(f\right)$$

in the MHM heart. If f is a monomorphism, then

$$Cone\left(f\right)\cong coker\left(f\right)$$

in the MHM heart. For a general morphism, $Cone\left(f\right)$ may have more than one nonzero perverse cohomology object, and the residual remains a derived MHM residual.

Whenever this paper identifies a shifted cone with a kernel, cokernel, or extension object in $MHM\left(X\right)$, the relevant monomorphism, epimorphism, or perverse-degree hypothesis is stated explicitly. In particular, the canonical-side object

$$K_{\mathrm{can},\Sigma }^H:=Cone\left(u_{\Sigma }\right)[-1]$$

is a heart-level kernel only under the stated heart-level extraction hypothesis that $u_{\Sigma }$ is an epimorphism in $MHM\left(X_0\right)$. Without that hypothesis, $Cone\left(u_{\Sigma }\right)$ remains a derived MHM residual.

This distinction matters for the residual–excess matrix. The entries of the MHM row may be heart-level objects or derived residuals, depending on the available exactness hypotheses. Excess should be formed after fixing the level at which the comparison is made. Thus one should compare heart-level objects with heart-level objects, or derived residuals with derived residuals, rather than silently mixing the two.

2.7. The Rational/MHM Row of the Residual–Excess Matrix

The preceding constructions supply the rational or MHM row of the residual–excess matrix. For the finite-node interface one obtains the factorization

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1),v_{\Sigma }{u}_{\Sigma }=N.$$

The associated MHM residual row is

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right).\end{array}$$

Under the heart-level extraction hypothesis, the canonical entry may be written as

$$K_{\mathrm{can},\Sigma }^H=Cone\left(u_{\Sigma }\right)[-1].$$

The variation entry is the corrected extension

$$P_{\mathrm{var},\Sigma }^H=Cone\left(v_{\Sigma }\right)[-1].$$

The return entry is the monodromy cone

$$Cone\left(N:{\psi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1)\right).$$

The octahedral axiom gives the horizontal constraint:

$$Cone\left(u_{\Sigma }\right)⟶Cone\left(N\right)⟶Cone\left(v_{\Sigma }\right)\stackrel{+1}{\to }.$$

This is the rational/MHM residual triangle. It is the row that later becomes part of the full residual–excess matrix.

If an integral or finite-coefficient finite-node package is available, one may compare its rationalization with this row. The vertical cones of those comparison morphisms are the canonical, variation, and return excess objects:

$$E_{\mathrm{can},\Sigma },E_{\mathrm{var},\Sigma },E_{N,\Sigma }.$$

The ordinary-double-point case will provide the local zero-excess calibration: the Milnor fiber is torsion-free, so the local coefficient-change excess at each node vanishes. Diaz’s Enriques–Brauer Bockstein construction provides the contrasting nonzero-excess calibration, where integral torsion lies in the kernel of rationalization.

2.8. Checklist for Hodge–Typed Residual Data

For every Hodge-typed defect resolution

$$A^H\stackrel{u}{\to }{B}^H\stackrel{v}{\to }{A}^{H,†},vu=R,$$

we verify or state hypotheses for the following items.

1.

The objects $A^H$, $B^H$, and $A^{H,†}$ lie in $MHM\left(X\right)$ or $D^{b}MHM\left(X\right)$.

2.

The maps u, v, and $R=vu$ are morphisms in the relevant category.

3.

The interface $B^H$ is geometrically identified. In the finite-ODP case,

$$B^H=W_{\Sigma }^H.$$

4.

When Saito gluing is used, the maps are strict filtered morphisms and are compatible with the relevant V-filtrations.

5.

The composite $vu$ is identified with a meaningful return operator, such as monodromy N.

6.

The residual triple

$$\left(Cone\left(u\right),Cone\left(v\right),Cone\left(R\right)\right)$$

is formed in the appropriate derived category.

7.

The octahedral triangle relating the three residual cones is recorded:

$$Cone\left(u\right)⟶Cone\left(R\right)⟶Cone\left(v\right)\stackrel{+1}{\to }.$$

8.

Heart-level interpretations of cones are made only under stated monomorphism, epimorphism, or perverse-degree hypotheses.

9.

Realization recovers the expected rational perverse shadow.

10.

Hypercohomology, when applied, gives mixed Hodge structures.

11.

The paper states what information is present in the factorized residual triple that is not recoverable from $Cone\left(R\right)$ alone.

The final item is essential. Residual language is used only when the factorization carries geometric, Hodge-theoretic, or interface information beyond the aggregate defect cone.

2.9. Checklist for Excess Data

The residual package above lives at fixed coefficients. To form the residual–excess matrix, one also needs coefficient-comparison data. Whenever the paper discusses an excess object, we verify or state hypotheses for the following items.

1.

There is an integral, finite-coefficient, or otherwise coefficient-refined object $C^{\mathrm{int}}$.

2.

There is a rational or MHM object $C^H$ representing the corresponding rational defect datum.

3.

A comparison morphism is specified:

$$C^{\mathrm{int}}\otimes \mathbb{Q}⟶C^H,$$

or the appropriate finite-coefficient-to-integral comparison is specified.

4.

The excess object is defined as the shifted cone of that comparison:

$$Exc(C^{\mathrm{int}},C^H):=Cone\left(C^{\mathrm{int}}\otimes \mathbb{Q}⟶C^H\right)[-1].$$

5.

The paper states whether the excess vanishes, is expected to vanish, or is known to be nonzero in the example under consideration.

6.

When the excess is torsion, the paper states explicitly that it lies in the kernel of rationalization.

7.

When nonzero excess is used as an integral Hodge obstruction channel, the survival or non-algebraicity criterion is stated separately.

This checklist prevents two common confusions. First, excess is not the same thing as a residual cone. A residual cone measures factorization data at fixed coefficients. An excess object measures failure of comparison across coefficient levels. Second, a nonzero torsion excess is not detected by the rational/MHM row. It must be recorded by the vertical axis of the residual–excess matrix.

3. Structured Defect–Resolution Categories

This section gives the formal framework for the residual side of the paper. The point is not merely that a morphism admits factorizations. The point is that, in a triangulated category, a factorization of a fixed return map carries a residual-triple structure constrained by the octahedral axiom. In an enriched setting such as $D^{b}MHM\left(X\right)$, the same package also carries support, Hodge filtration, weight filtration, Tate twist, realization, and Ext-class data.

Thus the object used in this paper is not just the bare category of factorizations of a morphism. It is the structured defect-resolution category: factorizations equipped with their residual triples and octahedral comparison data. These structured defect resolutions form the horizontal rows of the residual–excess matrix introduced in the introduction. The vertical direction of that matrix, namely excess, will later compare such residual rows across coefficient systems.

The finite-node Saito interface constructed later is a distinguished object of this structured category for the monodromy return map

$$N:{\psi }_{\pi ,1}^H⟶{\psi }_{\pi ,1}^H(-1).$$

Its residual triple records the canonical-side residual, the variation-side residual, and the aggregate monodromy residual. The ordinary-double-point case will then serve as the local zero-excess calibration for this residual package.

3.1. Bare Factorizations and Structured Defect Resolutions

Let $\mathcal{T}$ be a category and let

$$R:A⟶A^{†}$$

be a morphism in $\mathcal{T}$. The bare category of factorizations of R has objects

$$A\stackrel{u}{\to }B\stackrel{v}{\to }{A}^{†},vu=R,$$

and morphisms given by maps between the intermediate objects commuting with both legs.

In this paper we use a structured version of this construction. When $\mathcal{T}$ is triangulated, a factorization of R has three associated cones:

$$Cone\left(u\right),Cone\left(v\right),Cone\left(R\right).$$

These cones are constrained by the octahedral axiom. Therefore a factorization is not only an intermediate object B; it is an interface together with a constrained residual system.

Definition 1

(Structured defect-resolution object). Let $\mathcal{T}$ be a triangulated category and let

$$R:A\to A^{†}$$

be a morphism. A structured defect resolution of R is a datum

$$\mathfrak{R}=(B,u,v;{\mathsf{T}}_R,{\Delta }_{\mathrm{oct}})$$

where

$$A\stackrel{u}{\to }B\stackrel{v}{\to }{A}^{†},vu=R,$$

$${\mathsf{T}}_R(B,u,v)=\left(Cone\left(u\right),Cone\left(v\right),Cone\left(R\right)\right),$$

and ${\Delta }_{\mathrm{oct}}$ is the octahedral comparison triangle

$$Cone\left(u\right)⟶Cone\left(R\right)⟶Cone\left(v\right)⟶Cone\left(u\right)\left[1\right].$$

The object B is called the interface object.

Remark 2

(Notation). The symbol $\mathfrak{R}$ is used here for a structured residual object. Later, the symbol $\mathfrak{D}(u,v)$ will denote the larger residual–excess matrix obtained after comparing residual data across coefficient systems. Thus $\mathfrak{R}$ is horizontal residual data, while $\mathfrak{D}(u,v)$ packages both horizontal residuals and vertical excess.

Definition 2

(Structured defect-resolution category). The structured defect-resolution category

$${\mathbf{DefRes}}_{\mathcal{T}}\left(R\right)$$

has as objects the structured defect resolutions of R. A morphism

$${\mathfrak{R}}_1⟶{\mathfrak{R}}_2$$

from

$${\mathfrak{R}}_1=(B_1,u_1,v_1;{\mathsf{T}}_{R,1},{\Delta }_{\mathrm{oct},1})$$

to

$${\mathfrak{R}}_2=(B_2,u_2,v_2;{\mathsf{T}}_{R,2},{\Delta }_{\mathrm{oct},2})$$

is an interface morphism

$$\alpha :B_1\to B_2$$

such that

$$\alpha u_1=u_2,v_2\alpha =v_1,$$

and such that the induced morphisms of cone triangles are compatible with the octahedral comparison triangles ${\Delta }_{\mathrm{oct},1}$ and ${\Delta }_{\mathrm{oct},2}$, up to the fixed cone/sign convention.

There is a forgetful functor

$$U:{\mathbf{DefRes}}_{\mathcal{T}}\left(R\right)⟶{Fact}_{\mathcal{T}}\left(R\right),$$

where ${Fact}_{\mathcal{T}}\left(R\right)$ is the bare factorization category of R. The functor U forgets the residual-triple and octahedral comparison data. The paper uses ${\mathbf{DefRes}}_{\mathcal{T}}\left(R\right)$, not merely ${Fact}_{\mathcal{T}}\left(R\right)$, because the former is the structure that tracks defects at fixed coefficients.

Remark 3

(Relation with standard factorization categories). The bare factorization category ${Fact}_{\mathcal{T}}\left(R\right)$ is a standard construction. The structured category ${\mathbf{DefRes}}_{\mathcal{T}}\left(R\right)$ used here adds the residual triple and its octahedral constraint. In $D^{b}MHM\left(X\right)$, the interface object and the residual triple may also carry Hodge-theoretic data. This extra structure is what is used in the finite-node construction.

3.2. Residual Triples as Horizontal Defect Data

For a structured defect resolution

$$\mathfrak{R}=(B,u,v;{\mathsf{T}}_R,{\Delta }_{\mathrm{oct}})\in {\mathbf{DefRes}}_{\mathcal{T}}\left(R\right),$$

we call

$${\mathsf{T}}_R\left(\mathfrak{R}\right)=\left(Cone\left(u\right),Cone\left(v\right),Cone\left(R\right)\right)$$

the residual triple of $\mathfrak{R}$.

The first cone is the outgoing residual, the second cone is the incoming residual, and the third cone is the aggregate return residual. In the finite-node Saito application, these become the canonical-side residual, the variation-side residual, and the monodromy return residual.

The aggregate cone

$$Cone\left(R\right)$$

is fixed for all objects of ${\mathbf{DefRes}}_{\mathcal{T}}\left(R\right)$. The two factor cones

$$Cone\left(u\right),Cone\left(v\right)$$

depend on the interface B. Thus the residual triple records how the defect of R is realized, not only that R has a defect.

In the terminology of the introduction, residual triples are horizontal defect data. They live inside one coefficient theory or one triangulated category. They do not by themselves measure the failure of an integral object to rationalize to a given Hodge object. That latter failure is measured by excess. The residual triple supplies one row of the residual–excess matrix.

Remark 4

(Functoriality). In an ordinary triangulated category, cones are functorial only up to the standard choices. Thus the residual-triple assignment should be understood up to those choices. In an enhanced setting, such as a dg category or a stable ∞-category, the residual-triple assignment can be promoted to a genuine functor to a category of triples equipped with octahedral comparison data. The finite-node MHM arguments below only require the induced comparison maps and the octahedral triangles.

3.3. Octahedral Rigidity

The residual triple of a structured defect resolution is constrained by the octahedral axiom.

Proposition 1

(Octahedral residual comparison). Let

$$\mathfrak{R}=(B,u,v;{\mathsf{T}}_R,{\Delta }_{\mathrm{oct}})\in {\mathbf{DefRes}}_{\mathcal{T}}\left(R\right)$$

be a structured defect resolution of

$$R:A\to A^{†}.$$

Then there is a distinguished triangle

$$Cone\left(u\right)⟶Cone\left(R\right)⟶Cone\left(v\right)⟶Cone\left(u\right)\left[1\right],$$

up to the fixed cone/sign convention.

Proof. 

Apply the octahedral axiom to the composable pair

$$A\stackrel{u}{\to }B\stackrel{v}{\to }{A}^{†}.$$

The composite is $R=vu$. Choosing cone triangles for u, v, and R, the octahedral axiom supplies a distinguished triangle relating the three cones:

$$Cone\left(u\right)⟶Cone\left(R\right)⟶Cone\left(v\right)⟶Cone\left(u\right)\left[1\right],$$

up to the standard sign convention. □

This proposition is formal, but it is structural. It says that the residual triple is not a freely chosen triple of cones. It is an octahedrally constrained invariant attached to a specific interface realizing R. In the residual–excess matrix, this proposition gives the horizontal constraint on each residual row.

Remark 5

(Horizontal versus vertical constraints). The octahedral triangle is a horizontal constraint: it relates the three residual entries associated to a fixed coefficient theory. Excess will impose a different kind of constraint. It is vertical: it is obtained by taking cones of comparison morphisms between residual rows at different coefficient levels. Thus octahedral rigidity and coefficient-change excess are orthogonal, but they can be assembled into a single matrix.

3.4. Trivial Factorizations

There are two tautological factorizations of R:

$$A\stackrel{{id}_A}{\to }A\stackrel{R}{\to }{A}^{†}$$

and

$$A\stackrel{R}{\to }{A}^{†}\stackrel{{id}_{A^{†}}}{\to }{A}^{†}.$$

We call them the left-trivial and right-trivial factorizations.

Lemma 1

(Initial and terminal bare factorizations). In the bare factorization category ${Fact}_{\mathcal{T}}\left(R\right)$, the left-trivial factorization is initial and the right-trivial factorization is terminal.

Proof. 

Let

$$A\stackrel{u}{\to }B\stackrel{v}{\to }{A}^{†},vu=R,$$

be a factorization of R.

A morphism from the left-trivial factorization to $(B,u,v)$ is a map

$$\alpha :A\to B$$

such that

$$\alpha {id}_A=u\mathrm{and}v\alpha =R.$$

The first equation forces $\alpha =u$, and the second equation is exactly $vu=R$. Hence the morphism exists and is unique.

A morphism from $(B,u,v)$ to the right-trivial factorization is a map

$$\beta :B\to A^{†}$$

such that

$$\beta u=R\mathrm{and}{id}_{A^{†}}\beta =v.$$

The second equation forces $\beta =v$, and the first equation is exactly $vu=R$. Hence the morphism exists and is unique. □

Remark 6.

The trivial factorizations show that the bare factorization category is nonempty. They are usually not the geometrically interesting objects. In the finite-node setting, the geometrically meaningful interfaces are the full vanishing-cycle interface ${\varphi }_{\pi ,1}^H$ and the reduced node interface $W_{\Sigma }^H$.

3.5. Trivial-Return Rigidity

The simplest illustration of octahedral rigidity occurs when the return map is the identity.

Lemma 2

(Trivial-return rigidity). Let

$$A\stackrel{u}{\to }B\stackrel{v}{\to }A$$

be a structured defect resolution of

$${id}_A:A\to A.$$

Then

$$Cone\left(v\right)\cong Cone\left(u\right)\left[1\right],$$

up to the fixed cone/sign convention.

Proof. 

By Proposition 1, there is a distinguished triangle

$$Cone\left(u\right)⟶Cone\left({id}_A\right)⟶Cone\left(v\right)⟶Cone\left(u\right)\left[1\right].$$

Since the cone of an isomorphism is zero,

$$Cone\left({id}_A\right)\cong 0.$$

The triangle therefore becomes

$$Cone\left(u\right)⟶0⟶Cone\left(v\right)⟶Cone\left(u\right)\left[1\right].$$

Thus

$$Cone\left(v\right)\cong Cone\left(u\right)\left[1\right],$$

up to the fixed sign convention. □

Thus a trivial aggregate return can still have nontrivial factor-level residuals. The two factor cones are not arbitrary; they are forced to be shifts of one another. This illustrates why residual data should not be collapsed to the aggregate cone alone.

3.6. Strict Refinement over the Aggregate Cone

The next proposition records the basic reason that the structured defect-resolution package is not determined by the aggregate cone.

Proposition 2

(Residual triples are not determined by the aggregate cone). There exist two structured defect resolutions of the same morphism

$$R:A\to A^{†}$$

in $D^{b}MHM\left(\mathrm{pt}\right)$ whose aggregate cones

$$Cone\left(R\right)$$

are isomorphic, but whose residual triples are not isomorphic.

Proof. 

Since $MHM\left(\mathrm{pt}\right)$ is the category of graded-polarizable mixed Hodge structures, it is enough to work with Hodge structures on a point.

Let

$$A=A^{†}={\mathbb{Q}}^H,R=0:{\mathbb{Q}}^H\to {\mathbb{Q}}^H.$$

Then

$$Cone\left(R\right)=Cone(0:{\mathbb{Q}}^H\to {\mathbb{Q}}^H)\cong {\mathbb{Q}}^H\oplus {\mathbb{Q}}^H\left[1\right].$$

Consider the factorization

$${\mathbb{Q}}^H\stackrel{0}{\to }0\stackrel{0}{\to }{\mathbb{Q}}^H.$$

Its residual triple is

$${\mathsf{T}}_1=\left(Cone({\mathbb{Q}}^H\to 0),Cone(0\to {\mathbb{Q}}^H),Cone\left(R\right)\right).$$

Thus

$$Cone({\mathbb{Q}}^H\to 0)\cong {\mathbb{Q}}^H\left[1\right],Cone(0\to {\mathbb{Q}}^H)\cong {\mathbb{Q}}^H.$$

Now consider the factorization

$${\mathbb{Q}}^H\stackrel{0}{\to }{\mathbb{Q}}^H(-1)\stackrel{0}{\to }{\mathbb{Q}}^H.$$

Its residual triple is

$${\mathsf{T}}_2=\left(Cone({\mathbb{Q}}^H\to {\mathbb{Q}}^H(-1)),Cone({\mathbb{Q}}^H(-1)\to {\mathbb{Q}}^H),Cone\left(R\right)\right).$$

Since both maps are zero,

$$Cone({\mathbb{Q}}^H\to {\mathbb{Q}}^H(-1))\cong {\mathbb{Q}}^H(-1)\oplus {\mathbb{Q}}^H\left[1\right],$$

and

$$Cone({\mathbb{Q}}^H(-1)\to {\mathbb{Q}}^H)\cong {\mathbb{Q}}^H\oplus {\mathbb{Q}}^H(-1)\left[1\right].$$

Both factorizations have the same morphism $R=0$, hence the same aggregate cone

$$Cone\left(R\right)\cong {\mathbb{Q}}^H\oplus {\mathbb{Q}}^H\left[1\right].$$

However, the second residual triple contains copies of ${\mathbb{Q}}^H(-1)$ in its first two entries, while the first does not. Since

$${\mathbb{Q}}^H(-1)\neg \cong {\mathbb{Q}}^H$$

as mixed Hodge structures, the residual triples are not isomorphic. Therefore $Cone\left(R\right)$ does not determine the residual triple. □

Remark 7.

The example is intentionally elementary. Its purpose is to prove factorization sensitivity. The finite-ODP construction below is the geometric case in which the interface is not an artificial Hodge structure ${\mathbb{Q}}^H(-1)$, but the node-supported Hodge module

$$W_{\Sigma }^H=\underset{p_k\in \Sigma }{⨁}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

Remark 8

(Residual refinement is not excess). Proposition 2 is a fixed-coefficient statement. It shows that residual triples refine aggregate cones even before one compares integral and rational theories. This should be distinguished from excess, which measures failure of comparison across coefficient systems. In the residual–excess matrix, the proposition says that a single horizontal row contains more information than its aggregate return entry.

3.7. Residual Rows and Coefficient Comparison

The formalism above produces residual rows. If the same return map is available in two coefficient theories, for example an integral theory and a rational or MHM realization, then one may compare the resulting residual rows.

Suppose, for instance, that we have an integral defect resolution

$$A^{\mathbb{Z}}\stackrel{u^{\mathbb{Z}}}{\to }{B}^{\mathbb{Z}}\stackrel{v^{\mathbb{Z}}}{\to }{\left(A^{†}\right)}^{\mathbb{Z}},v^{\mathbb{Z}}{u}^{\mathbb{Z}}=R^{\mathbb{Z}},$$

and a rational or Hodge realization

$$A^H\stackrel{u^H}{\to }{B}^H\stackrel{v^H}{\to }{\left(A^{†}\right)}^H,v^H{u}^H=R^H.$$

Then the two residual triples are

$${\mathsf{T}}^{\mathbb{Z}}(u,v)=\left(Cone\left(u^{\mathbb{Z}}\right),Cone\left(v^{\mathbb{Z}}\right),Cone\left(R^{\mathbb{Z}}\right)\right)$$

and

$${\mathsf{T}}^H(u,v)=\left(Cone\left(u^H\right),Cone\left(v^H\right),Cone\left(R^H\right)\right).$$

If rationalization maps the integral residual row to the Hodge residual row, then the cones of the vertical comparison maps are excess objects.

This paper does not assume that such integral data always exist. The point is rather to separate the two tasks. The present section constructs the horizontal residual formalism. Later sections specify when vertical coefficient-comparison data exist and what their excess objects measure.

3.8. The Finite-Node Object of $\mathbf{DefRes}\left(N\right)$

In the finite-ODP setting, the return map is

$$N:{\psi }_{\pi ,1}^H⟶{\psi }_{\pi ,1}^H(-1).$$

The full nearby/vanishing factorization

$${\psi }_{\pi ,1}^H\stackrel{{can}^H}{\to }{\varphi }_{\pi ,1}^H\stackrel{{Var}^H}{\to }{\psi }_{\pi ,1}^H(-1)$$

is a structured defect resolution of N. The reduced finite-node factorization

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1)$$

is another structured defect resolution of N, on the finite-node sector.

We denote this reduced finite-node residual object by

$${\mathcal{R}}_{\Sigma }^H:=\left(W_{\Sigma }^H,u_{\Sigma },v_{\Sigma };{\mathsf{T}}_{\Sigma }^H\left(N\right),{\Delta }_{\mathrm{oct},\Sigma }\right)\in {\mathbf{DefRes}}_{D^{b}MHM\left(X_0\right)}\left(N\right),$$

where

$${\mathsf{T}}_{\Sigma }^H\left(N\right)=\left(Cone\left(u_{\Sigma }\right),Cone\left(v_{\Sigma }\right),Cone\left(N\right)\right),$$

and ${\Delta }_{\mathrm{oct},\Sigma }$ denotes the octahedral comparison triangle

$$Cone\left(u_{\Sigma }\right)⟶Cone\left(N\right)⟶Cone\left(v_{\Sigma }\right)\stackrel{+1}{\to }.$$

The purpose of the finite-node construction is to study ${\mathcal{R}}_{\Sigma }^H$. It records the ordinary-double-point interface

$$W_{\Sigma }^H=\underset{p_k\in \Sigma }{⨁}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),$$

the corrected extension

$$0\to I{C}_{X_0}^H\to P_{\mathrm{var},\Sigma }^H\to W_{\Sigma }^H\to 0,$$

and the nodewise Ext residual classes

$${ϵ}_k^H\in {Ext}_{MHM\left(X_0\right)}^1(i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),I{C}_{X_0}^H).$$

These are data of the structured defect-resolution package. They are not determined by the aggregate cone $Cone\left(N\right)$ alone.

In the residual–excess matrix, ${\mathcal{R}}_{\Sigma }^H$ supplies the rational/MHM residual row:

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right).\end{array}$$

The next geometric task is to construct the interface $W_{\Sigma }^H$ and the maps $u_{\Sigma },v_{\Sigma }$. The later coefficient-comparison task is to ask whether ${\mathcal{R}}_{\Sigma }^H$ is the rational or MHM realization of a coefficient-refined residual object, and to identify the resulting excess objects when it is not.

4. The Finite–ODP Saito Interface

This section constructs the reduced finite-node interface used in the main theorem. The construction is local at each ordinary double point and then assembled over the finite set of nodes. The output is a defect resolution

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1)$$

of finite-node monodromy.

This section also records the local zero-excess property of ordinary double points. The Milnor fiber of a complex threefold ordinary double point has the homotopy type of $S^3$, hence its local integral vanishing cohomology is torsion-free. Thus the local node interface has no hidden finite-coefficient torsion killed by rationalization. In the residual–excess matrix, finite ODPs provide the basic zero-excess calibration: the residual row is nontrivial, but the local excess row vanishes.

4.1. Geometric Setup and Point-Support Convention

Let

$$\pi :\mathcal{X}⟶\Delta$$

be a one-parameter degeneration of complex threefolds. Assume that $\pi$ is smooth over

$${\Delta }^{\ast }=\Delta \setminus \left\{0\right\}.$$

Write

$$X_t:={\pi }^{-1}\left(t\right),X_0:={\pi }^{-1}\left(0\right).$$

Assume that $X_0$ has finitely many ordinary double points

$$\Sigma =\{p_1,\dots ,p_r\}.$$

Let

$$U:=X_0\setminus \Sigma ,j:U\hookrightarrow X_0,i_k:\left\{p_k\right\}\hookrightarrow X_0$$

be the natural inclusions.

We use the perverse normalization appropriate for threefold degenerations. Let

$${\mathbb{Q}}_{\mathcal{X}}^H\left[3\right]$$

denote the shifted constant Hodge module on the smooth total space, or its local replacement on a sufficiently small analytic neighborhood. Set

$${\psi }_{\pi ,1}^H:={\psi }_{\pi ,1}^H\left({\mathbb{Q}}_{\mathcal{X}}^H\left[3\right]\right),{\varphi }_{\pi ,1}^H:={\varphi }_{\pi ,1}^H\left({\mathbb{Q}}_{\mathcal{X}}^H\left[3\right]\right).$$

The maps

$${can}^H:{\psi }_{\pi ,1}^H\to {\varphi }_{\pi ,1}^H,{Var}^H:{\varphi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1)$$

satisfy

$${Var}^H{can}^H=N.$$

We also fix the point-support convention used throughout the paper.

Convention 2

(Point-support normalization). For a point inclusion $i_k:\left\{p_k\right\}\hookrightarrow X_0$, the object

$$i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

denotes the point-supported Hodge module placed in the $MHM\left(X_0\right)$ heart. With this convention,

$${\mathbb{H}}^0\left(X_0,i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)\right)\cong {\mathbb{Q}}^H(-1),$$

and

$${\mathbb{H}}^m\left(X_0,i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)\right)=0(m\ne 0).$$

Equivalently, the point object is placed in the perverse/MHM heart, not in the unshifted topological degree convention.

This convention is used later when interpreting the hypercohomology of $W_{\Sigma }^H$ as a direct sum of Tate-twisted point Hodge structures.

4.2. The Local Ordinary–Double-Point Block

Fix $p_k\in \Sigma$. Analytically near $p_k$, an ordinary double point is modeled by

$$f:{\mathbb{C}}^4\to \mathbb{C},f(x_1,x_2,x_3,x_4)=x_1^2+x_2^2+x_3^2+x_4^2.$$

The Milnor fiber of a complex threefold ordinary double point has the homotopy type of $S^3$. Hence

$${\tilde{H}}^m(F_{p_k};\mathbb{Q})\cong \left\{\begin{array}{cc}\mathbb{Q},\hfill & m=3,\hfill \\ 0,\hfill & m\ne 3,\hfill \end{array}\right.$$

and, integrally,

$${\tilde{H}}^m(F_{p_k};\mathbb{Z})\cong \left\{\begin{array}{cc}\mathbb{Z},\hfill & m=3,\hfill \\ 0,\hfill & m\ne 3.\hfill \end{array}\right.$$

This is the standard local topology of an isolated hypersurface ordinary double point [3,10].

Thus the local vanishing sector at $p_k$ is one-dimensional in the normalized perverse degree. The corresponding point-supported MHM block is

$$W_k^H:=i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

Its rational realization is

$$rat\left(W_k^H\right)\cong i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}.$$

Definition 3

(Local ODP Hodge block). The local ODP Hodge block at $p_k$ is

$$W_k^H:=i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)\in MHM\left(X_0\right).$$

Remark 9

(Local zero-excess input). The integral statement

$${\tilde{H}}^3(F_{p_k};\mathbb{Z})\cong \mathbb{Z}$$

has no torsion. Therefore the local ODP vanishing block has no hidden finite-coefficient torsion killed by rationalization. This is the local zero-excess property of ordinary double points. It does not say that the finite-node residual package is trivial. Rather, it says that the local coefficient-change excess attached to the ODP Milnor block vanishes.

4.3. The Node–Supported Interface

Define

$$W_{\Sigma }^H:={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{W}_k^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

Then $W_{\Sigma }^H$ is supported on $\Sigma$, and

$$rat\left(W_{\Sigma }^H\right)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}.$$

Lemma 3

(Finite support and realization). The object $W_{\Sigma }^H$ is a finite point-supported mixed Hodge module on $X_0$. Its rational realization is

$$rat\left(W_{\Sigma }^H\right)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}.$$

Proof. 

Each $W_k^H=i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$ is a point-supported mixed Hodge module. Finite direct sums exist in $MHM\left(X_0\right)$, so $W_{\Sigma }^H\in MHM\left(X_0\right)$. Its support is the union of the supports of its summands, namely $\Sigma$. Since rat is exact and commutes with finite direct sums,

$$rat\left(W_{\Sigma }^H\right)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}rat\left(W_k^H\right).$$

The Tate twist does not change the underlying rational perverse sheaf, so

$$rat\left(W_k^H\right)\cong i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}.$$

This gives the asserted realization. □

The object $W_{\Sigma }^H$ is the interface used in the reduced finite-node defect resolution. It retains the support and Tate normalization of the ODP vanishing sector, even though rational realization forgets the Tate twist.

4.4. Relation with Kerr–Laza Point-Supported Terms

Kerr–Laza’s decomposition over a curve separates local-system terms from point-supported terms $W_{\sigma }^j$, and their phantom cohomology is identified with such point-supported contributions [6]. In the present finite-ODP setting, $W_{\Sigma }^H$ is the MHM-level interface selecting the ordinary-double-point point-supported contribution.

We state this as a guiding comparison at this stage. The precise comparison with boundary and phantom residuals is treated later.

Principle 3

(Finite-ODP point-supported interface). For a finite ordinary-double-point degeneration, the object

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

is the Hodge-normalized node-supported interface corresponding to the ODP point-supported contribution in the Kerr–Laza $W_{\sigma }$-type decomposition.

The reason this principle is not stated as a theorem here is that the Kerr–Laza $W_{\sigma }$-terms are expressed after taking perverse direct images over a curve and passing to cohomological or MHS-level invariants, while $W_{\Sigma }^H$ is an MHM object on $X_0$. Later we isolate the precise comparison needed for the ODP boundary/phantom contribution.

In the residual–excess matrix, this comparison is horizontal. It identifies which rational/MHM point-supported contribution is selected by the finite-node interface. It does not measure coefficient-change excess. The local zero-excess statement for ODPs instead comes from the torsion-free integral cohomology of the Milnor fiber.

4.5. The Local Outgoing Map

The canonical morphism

$${can}^H:{\psi }_{\pi ,1}^H\to {\varphi }_{\pi ,1}^H$$

sends nearby-cycle data to vanishing-cycle data. At $p_k$, the ODP vanishing sector contains the rank-one Hodge line represented by $W_k^H$.

Let

$$A_k^H\subseteq {\varphi }_{\pi ,1}^H$$

denote the local ODP sector through which the finite-node contribution of ${can}^H$ factors. Equivalently, $A_k^H$ is the local rank-one vanishing-cycle contribution at $p_k$, with its induced Hodge normalization. The ODP calculation gives a quotient onto the rank-one Hodge block:

$$q_k:A_k^H\to W_k^H.$$

Define

$$u_k:=q_k\circ {can}^H:{\psi }_{\pi ,1}^H\to W_k^H.$$

Remark 10

(On the local sector $A_k^H$). The notation $A_k^H$ is used only to isolate the local ODP contribution. No global splitting of ${\varphi }_{\pi ,1}^H$ is asserted here. The construction requires the local rank-one quotient determined by the ODP vanishing cycle.

4.6. The Local Incoming Map

The variation morphism

$${Var}^H:{\varphi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1)$$

returns vanishing-cycle data to nearby-cycle data with Tate twist. Restricting ${Var}^H$ to the ODP Hodge block gives

$$v_k:W_k^H\to {\psi }_{\pi ,1}^H(-1).$$

Lemma 4

(Nonzero local maps). The morphisms $u_k$ and $v_k$ are nonzero on the local ODP sector.

Proof. 

It is enough to check nonvanishing after rational realization, since rat is faithful on morphisms in MHM.

The rational local vanishing cohomology of a threefold ordinary double point is one-dimensional. In the local model

$$f=x_1^2+x_2^2+x_3^2+x_4^2,$$

the Milnor fiber has the homotopy type of $S^3$, so the reduced cohomology is

$${\tilde{H}}^3(F_{p_k};\mathbb{Q})\cong \mathbb{Q}.$$

Under the perverse normalization ${\mathbb{Q}}_{\mathcal{X}}^H\left[3\right]$, this gives a rank-one local vanishing-cycle contribution supported at $p_k$.

The canonical morphism can identifies the local nearby-cycle vanishing direction with this rank-one vanishing-cycle line. If the component of can on the ODP line were zero, then the local monodromy operator

$$N_k=Var\circ can$$

would vanish on the ODP sector. This contradicts the Picard–Lefschetz local monodromy calculation for an ordinary double point. Hence the ODP component of can, and therefore $u_k=q_k{can}^H$, is nonzero.

Since $N_k={Var}^H{can}^H$ is nonzero on the same rank-one sector and $u_k$ is nonzero, the induced variation map

$$v_k:W_k^H\to {\psi }_{\pi ,1}^H(-1)$$

is also nonzero. □

4.7. Normalization by Local Monodromy

Let $N_k$ denote the local nilpotent monodromy operator on the contribution associated with $p_k$. Since

$${Var}^H{can}^H=N,$$

the composite $v_k{u}_k$ is a scalar multiple of $N_k$ on the rank-one ODP sector. We fix the scalar by a nodewise normalization.

Proposition 3

(Local normalization). After fixing a scalar normalization of $u_k$, one has

$$v_k{u}_k=N_k$$

on the local ODP sector.

Proof. 

By Lemma 4, both $u_k$ and $v_k$ are nonzero on the local ODP sector. This sector is rank one. Therefore the two nonzero endomorphisms $v_k{u}_k$ and $N_k$ differ by a unique nonzero scalar:

$$v_k{u}_k=c_k{N}_k,c_k\in {\mathbb{Q}}^{\times }.$$

Replacing $u_k$ by $c_k^{-1}{u}_k$, and leaving $v_k$ fixed, gives

$$v_k{u}_k=N_k.$$

This normalization is fixed once for the node $p_k$. □

Remark 11

(Dependence on normalization). The representative morphism $u_k$ depends on the chosen scalar normalization. The support, Tate twist, interface object $W_k^H$, and target Ext group

$${Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H)$$

do not. Thus the nodewise Hodge interface is intrinsic, while the equality $v_k{u}_k=N_k$ fixes a convenient representative of the local factorization.

4.8. Global Assembly

Define

$$u_{\Sigma }:{\psi }_{\pi ,1}^H\to W_{\Sigma }^H$$

by requiring

$${pr}_k{u}_{\Sigma }=u_k$$

for each projection ${pr}_k:W_{\Sigma }^H\to W_k^H$. Define

$$v_{\Sigma }:W_{\Sigma }^H\to {\psi }_{\pi ,1}^H(-1)$$

by requiring

$$v_{\Sigma }{\iota }_k=v_k$$

for each inclusion ${\iota }_k:W_k^H\to W_{\Sigma }^H$.

Proposition 4

(Global return relation). On the finite-node sector,

$$v_{\Sigma }{u}_{\Sigma }=N.$$

Proof. 

On the local sector associated with $p_k$, the composite $v_{\Sigma }{u}_{\Sigma }$ restricts to $v_k{u}_k$. By Proposition 3,

$$v_k{u}_k=N_k.$$

The finite-node part of N is the direct sum of the local operators $N_k$. Hence

$$v_{\Sigma }{u}_{\Sigma }=N$$

on the finite-node sector. □

4.9. Filtered Admissibility

The filtered admissibility check is local at the nodes.

Lemma 5

(Point-supported filtered structure). For each k, the object

$$W_k^H=i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

is a point-supported mixed Hodge module whose underlying filtered D-module is the point-supported delta module with Tate-twisted Hodge filtration. Its V-filtration along a local divisor through $p_k$ is the standard point-supported V-filtration.

Proof. 

The object ${\mathbb{Q}}_{\left\{p_k\right\}}^H$ is the pure Hodge module on a point. Its Tate twist remains a Hodge module on the point with shifted Hodge and weight data. Pushing forward by the closed immersion $i_k$ gives a point-supported Hodge module on $X_0$. Its underlying D-module is the delta module at $p_k$. Since the support is a point, the Kashiwara–Malgrange filtration along a local divisor through $p_k$ is the standard point-supported V-filtration. □

Proposition 5

(Local filtered admissibility). The local datum

$$\left({\psi }_{\pi ,1}^H,W_k^H,u_k,v_k\right)$$

is admissible for the point-supported ODP part of Saito’s divisor-gluing formalism. In particular, $u_k$ and $v_k$ are strict filtered morphisms compatible with the relevant V-filtrations, and

$$v_k{u}_k=N_k$$

holds in the filtered setting after normalization.

Proof. 

The maps ${can}^H$ and ${Var}^H$ are morphisms in Saito’s filtered nearby/vanishing-cycle formalism. The map $u_k$ is obtained from ${can}^H$ by projection onto the rank-one ODP Hodge block. By Lemma 5, this block has the standard point-supported V-filtration. Therefore the projection is compatible with the induced V-filtration. The same argument applies to $v_k$, which is induced by ${Var}^H$ on the same block.

Morphisms in MHM are strict for the Hodge filtration. The relation $v_k{u}_k=N_k$ holds in the filtered setting because it is obtained by restricting

$${Var}^H{can}^H=N$$

to the normalized local ODP sector. □

Proposition 6

(Global filtered admissibility). The global finite-node datum

$$\left({\psi }_{\pi ,1}^H,W_{\Sigma }^H,u_{\Sigma },v_{\Sigma }\right)$$

is admissible on the finite-node sector, and

$$v_{\Sigma }{u}_{\Sigma }=N$$

holds in the filtered setting on that sector.

Proof. 

The supports of the objects $W_k^H$ are disjoint points. Therefore the filtered D-module structures, Hodge filtrations, and V-filtrations on

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{W}_k^H$$

are computed componentwise. By Proposition 5, each local datum is filtered admissible. Taking the finite direct sum gives global filtered admissibility. The equality $v_{\Sigma }{u}_{\Sigma }=N$ follows componentwise from $v_k{u}_k=N_k$. □

4.10. The Finite–ODP Saito Interface

Definition 4

(Finite-ODP Saito interface). The finite-ODP Saito interface associated with the degeneration $\pi :\mathcal{X}\to \Delta$ is the defect resolution

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1),$$

where

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),$$

and

$$v_{\Sigma }{u}_{\Sigma }=N$$

on the finite-node sector.

This is the reduced finite-node factorization of monodromy used in the main theorem. It gives the residual triple

$$\left(Cone\left(u_{\Sigma }\right),Cone\left(v_{\Sigma }\right),Cone\left(N\right)\right),$$

whose entries are the extraction residual, reinsertion residual, and monodromy return residual.

4.11. The ODP Residual Row and Local Zero-Excess

The finite-ODP Saito interface supplies the rational/MHM residual row of the residual–excess matrix:

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right).\end{array}$$

The octahedral axiom supplies the horizontal constraint

$$Cone\left(u_{\Sigma }\right)⟶Cone\left(N\right)⟶Cone\left(v_{\Sigma }\right)\stackrel{+1}{\to }.$$

The same local ODP calculation also gives the zero-excess calibration. Since

$${\tilde{H}}^3(F_{p_k};\mathbb{Z})\cong \mathbb{Z}$$

and there is no torsion in the local Milnor cohomology, the integral local vanishing block rationalizes to the rational local vanishing block without torsion loss. Thus the local coefficient-change excess at each ODP node vanishes:

$$\begin{array}{cccc}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right)\\ \mathbb{Z}& C_{u,\Sigma }^{\mathbb{Z}}& C_{v,\Sigma }^{\mathbb{Z}}& C_{N,\Sigma }^{\mathbb{Z}}\\ H/\mathbb{Q}& C_{u,\Sigma }^H& C_{v,\Sigma }^H& C_{N,\Sigma }^H\\ Exc& 0& 0& 0\end{array}\mathrm{locally}\mathrm{at}\mathrm{each}\mathrm{ordinary}\mathrm{double}\mathrm{point}.$$

This statement is local at the ODP Milnor block. It does not assert that all global integral extension issues vanish. It asserts that ordinary double points are torsion-free local models. Hence finite ODPs calibrate the horizontal residual axis: the residual package is meaningful, but its local coefficient excess is zero.

5. The Finite–Node Defect Resolution of Monodromy

We now apply the defect-resolution formalism to the finite-ODP Saito interface constructed above. This is the main rational/MHM computation of the paper. The classical nearby/vanishing diagram

$${\psi }_{\pi ,1}^H\stackrel{{can}^H}{\to }{\varphi }_{\pi ,1}^H\stackrel{{Var}^H}{\to }{\psi }_{\pi ,1}^H(-1)$$

is the full can–var factorization of monodromy. The finite-node construction replaces the full vanishing-cycle interface by the reduced node-supported Hodge interface

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

Thus

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1),v_{\Sigma }{u}_{\Sigma }=N,$$

is a distinguished object of $\mathbf{DefRes}\left(N\right)$, the structured defect-resolution category of the finite-node monodromy map

$$N:{\psi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1).$$

This section constructs the horizontal residual row of the residual–excess matrix:

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right).\end{array}$$

The vertical excess direction is not yet used in the proof. It enters when one compares this rational/MHM residual row with integral or finite-coefficient data. For ordinary double points, the local Milnor fiber is torsion-free, so the local coefficient-change excess attached to the ODP node block vanishes. The residual row constructed here is therefore the zero-excess local calibration row for finite ODPs.

5.1. The Finite–Node Defect–Resolution Datum

The data are

$$A^H={\psi }_{\pi ,1}^H,B^H=W_{\Sigma }^H,A^{H,†}={\psi }_{\pi ,1}^H(-1),$$

and

$$R=N.$$

The residual triple is

$${\mathsf{T}}_{\Sigma }^H\left(N\right):=\left(Cone\left(u_{\Sigma }\right),Cone\left(v_{\Sigma }\right),Cone\left(N\right)\right).$$

The three entries are called the extraction residual, the reinsertion residual, and the monodromy return residual.

The main point is that ${\mathsf{T}}_{\Sigma }^H\left(N\right)$ contains information not recoverable from $Cone\left(N\right)$ alone. The interface $W_{\Sigma }^H$ records the support of the finite-node contribution, its Tate twist, and the point-supported Hodge type of the ordinary-double-point vanishing sector. These are horizontal residual data: they refine the monodromy cone at fixed rational/MHM coefficients.

5.2. The Variation–Side Corrected Extension

Define

$$P_{\mathrm{var},\Sigma }^H:=Cone\left(v_{\Sigma }\right)[-1].$$

This is the variation-side residual object. It is the incoming or reinsertion residual in the finite-node factorization.

Theorem 4

(Finite-node variation extension). The object

$$P_{\mathrm{var},\Sigma }^H=Cone\left(v_{\Sigma }\right)[-1]$$

lies in $MHM\left(X_0\right)$ and fits into a short exact sequence

$$0⟶I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H⟶0.$$

After applying rational realization, one obtains

$$0⟶I{C}_{X_0}⟶rat\left(P_{\mathrm{var},\Sigma }^H\right)⟶{\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}⟶0.$$

Proof. 

The finite-node datum

$$\left({\psi }_{\pi ,1}^H,W_{\Sigma }^H,u_{\Sigma },v_{\Sigma }\right)$$

is filtered-admissible by Proposition 6. Therefore Saito’s divisor-gluing formalism applies to this datum [4,5,9]. The gluing construction produces an object

$$P_{\mathrm{var},\Sigma }^H\in MHM\left(X_0\right)$$

whose open restriction is the middle extension of the smooth-locus Hodge module and whose divisor-side quotient is the interface object $W_{\Sigma }^H$.

Since the open part is $L_U^H={\mathbb{Q}}_U^H\left[3\right]$, its middle extension is

$$j_{!\ast }{L}_U^H=I{C}_{X_0}^H.$$

Hence the gluing object fits into

$$0⟶I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H⟶0.$$

It remains to identify this object with the shifted cone. The gluing datum contains

$$v_{\Sigma }:W_{\Sigma }^H\to {\psi }_{\pi ,1}^H(-1).$$

The associated gluing triangle has the form

$$W_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1)⟶P_{\mathrm{var},\Sigma }^H\left[1\right]\stackrel{+1}{\to }.$$

Thus

$$P_{\mathrm{var},\Sigma }^H\left[1\right]\cong Cone\left(v_{\Sigma }\right),$$

and so

$$P_{\mathrm{var},\Sigma }^H\cong Cone\left(v_{\Sigma }\right)[-1].$$

Finally, rat is exact and

$$rat\left(W_{\Sigma }^H\right)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}.$$

Applying rat to the MHM short exact sequence gives the realized exact sequence. □

Remark 12

(Cone convention and invariance). In an ordinary triangulated category, cones are defined up to the usual non-unique isomorphism. The theorem should be read with the finite-node gluing datum fixed. In an enhanced setting, such as a dg or stable ∞-categorical enhancement of $D^{b}MHM\left(X_0\right)$, the same construction may be represented by a functorial cofiber/fiber object. The heart-level invariant used below is the extension class of the short exact sequence

$$0\to I{C}_{X_0}^H\to P_{\mathrm{var},\Sigma }^H\to W_{\Sigma }^H\to 0.$$

Remark 13

(Variation residual versus variation excess). The theorem constructs the rational/MHM variation residual. If an integral variation object $P_{\mathrm{var},\Sigma }^{\mathbb{Z}}$ is later specified together with a comparison morphism

$$P_{\mathrm{var},\Sigma }^{\mathbb{Z}}\otimes \mathbb{Q}⟶P_{\mathrm{var},\Sigma }^H,$$

then the corresponding variation excess is

$$E_{\mathrm{var},\Sigma }:=Cone\left(P_{\mathrm{var},\Sigma }^{\mathbb{Z}}\otimes \mathbb{Q}⟶P_{\mathrm{var},\Sigma }^H\right)[-1].$$

Thus $P_{\mathrm{var},\Sigma }^H$ is a residual object; $E_{\mathrm{var},\Sigma }$ is a coefficient-comparison object.

5.3. The Canonical–Side Extraction Residual

The extraction side is controlled by

$$u_{\Sigma }:{\psi }_{\pi ,1}^H\to W_{\Sigma }^H.$$

The cone $Cone\left(u_{\Sigma }\right)$ exists in $D^{b}MHM\left(X_0\right)$ without additional assumptions. To identify it with a heart-level kernel object, we state the hypothesis explicitly.

Hypothesis 5

(Heart-level extraction hypothesis). The map

$$u_{\Sigma }:{\psi }_{\pi ,1}^H\to W_{\Sigma }^H$$

is an epimorphism in $MHM\left(X_0\right)$.

Definition 5

(Canonical extraction residual). Under Hypothesis 5, set

$$K_{\mathrm{can},\Sigma }^H:=Cone\left(u_{\Sigma }\right)[-1].$$

Proposition 7

(Canonical residual exact sequence). Assume Hypothesis 5. Then

$$K_{\mathrm{can},\Sigma }^H\cong ker\left(u_{\Sigma }\right)$$

in $MHM\left(X_0\right)$, and there is a short exact sequence

$$0⟶K_{\mathrm{can},\Sigma }^H⟶{\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H⟶0.$$

Proof. 

Since $u_{\Sigma }$ is an epimorphism in the abelian category $MHM\left(X_0\right)$, there is a short exact sequence

$$0⟶ker\left(u_{\Sigma }\right)⟶{\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H⟶0.$$

The corresponding distinguished triangle is

$$ker\left(u_{\Sigma }\right)⟶{\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H⟶ker\left(u_{\Sigma }\right)\left[1\right].$$

Comparing this with the cone triangle of $u_{\Sigma }$ gives

$$Cone\left(u_{\Sigma }\right)\cong ker\left(u_{\Sigma }\right)\left[1\right],$$

hence

$$Cone\left(u_{\Sigma }\right)[-1]\cong ker\left(u_{\Sigma }\right).$$

The stated exact sequence follows. □

Remark 14.

Without Hypothesis 5, the extraction residual remains the derived MHM object $Cone\left(u_{\Sigma }\right)$. The hypothesis is needed only for the heart-level kernel interpretation. The finite-node defect-resolution datum and its octahedral residual triangle exist independently of this heart-level strengthening.

Remark 15

(Canonical residual versus canonical excess). The object $K_{\mathrm{can},\Sigma }^H$ is the heart-level canonical residual, when Hypothesis 5 holds. If an integral extraction residual $K_{\mathrm{can},\Sigma }^{\mathbb{Z}}$ is later specified together with a comparison morphism

$$K_{\mathrm{can},\Sigma }^{\mathbb{Z}}\otimes \mathbb{Q}⟶K_{\mathrm{can},\Sigma }^H,$$

then the corresponding canonical excess is

$$E_{\mathrm{can},\Sigma }:=Cone\left(K_{\mathrm{can},\Sigma }^{\mathbb{Z}}\otimes \mathbb{Q}⟶K_{\mathrm{can},\Sigma }^H\right)[-1].$$

Thus the canonical residual and canonical excess occupy different entries in the residual–excess matrix.

5.4. The Monodromy Return Residual and Octahedral Constraint

The return residual is

$$R_{N,\Sigma }^H:=Cone\left(N:{\psi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1)\right).$$

Theorem 6

(Octahedral constraint for the finite-node defect resolution). The residual triple

$${\mathsf{T}}_{\Sigma }^H\left(N\right)=\left(Cone\left(u_{\Sigma }\right),Cone\left(v_{\Sigma }\right),Cone\left(N\right)\right)$$

is constrained by a distinguished triangle

$$Cone\left(u_{\Sigma }\right)⟶Cone\left(N\right)⟶Cone\left(v_{\Sigma }\right)⟶Cone\left(u_{\Sigma }\right)\left[1\right],$$

up to the fixed cone/sign convention.

Proof. 

Apply the octahedral axiom to the composable morphisms

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1).$$

Their composite is $v_{\Sigma }{u}_{\Sigma }=N$. The octahedral axiom gives the claimed distinguished triangle. □

Remark 16.

This is not presented as a new use of the octahedral axiom. The can–var–N octahedral structure is part of Saito’s formalism and appears in the Kerr–Laza–Saito degeneration framework. The point here is the reduced finite-node factorization through $W_{\Sigma }^H$, which retains node support, Tate twist, and Ext data not visible from $Cone\left(N\right)$ alone.

5.5. The Finite-Node Residual Row

Combining the preceding constructions, the finite-node Saito interface gives the rational/MHM residual row

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right).\end{array}$$

When Hypothesis 5 holds, the canonical entry may be represented in the MHM heart by

$$K_{\mathrm{can},\Sigma }^H=Cone\left(u_{\Sigma }\right)[-1].$$

The variation entry is represented in the MHM heart by

$$P_{\mathrm{var},\Sigma }^H=Cone\left(v_{\Sigma }\right)[-1],$$

and fits into

$$0⟶I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H⟶0.$$

The return entry is the monodromy cone

$$Cone\left(N:{\psi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1)\right).$$

The horizontal constraint on this row is the octahedral triangle

$$Cone\left(u_{\Sigma }\right)⟶Cone\left(N\right)⟶Cone\left(v_{\Sigma }\right)\stackrel{+1}{\to }.$$

This is the finite-node residual package. It is a fixed-coefficient, rational/MHM object. It should be distinguished from the excess package, which is obtained only after comparing this row with integral or finite-coefficient data.

For ordinary double points, the local Milnor fiber is torsion-free. Therefore, locally at each node, the coefficient-change excess associated with the ODP Milnor block vanishes. In this sense, the finite-node Saito interface is the zero-excess calibration of the residual–excess matrix:

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right)\\ {Exc}_{\mathrm{local}}& 0& 0& 0.\end{array}$$

This local zero-excess statement does not assert that every global integral extension issue vanishes. It asserts that the ordinary-double-point local vanishing block itself contributes no torsion excess killed by rationalization.

6. The Nodewise Ext Residual Module

The variation-side exact sequence of Theorem 4 defines an extension class. This class is the main heart-level invariant attached to the finite-node interface. It is the variation-side entry of the rational/MHM residual row, refined from a cone-level object to an explicit Ext class in $MHM\left(X_0\right)$.

The point of this section is to isolate the computable part of the finite-node residual package. The interface

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

is a finite direct sum of point-supported Hodge modules. Consequently, the variation residual class decomposes into nodewise Hodge residual classes. This nodewise Ext module records how each ordinary double point contributes to the corrected extension.

This is a horizontal residual invariant. It lives inside the rational/MHM row of the residual–excess matrix. It should be distinguished from an excess object, which arises only after comparing this MHM Ext class, or the corrected extension that represents it, with an integral or finite-coefficient lift.

6.1. The Finite-Node Variation Residual Class

The exact sequence

$$0⟶I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H⟶0$$

defines a class

$$\left[P_{\mathrm{var},\Sigma }^H\right]\in {Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H).$$

Definition 6

(Finite-node variation residual class). The class

$$\left[P_{\mathrm{var},\Sigma }^H\right]\in {Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H)$$

is called the finite-node variation residual class.

This class records how the node-supported Hodge interface is attached to the global intersection-complex Hodge module. It is not determined by the rational point support alone. It lives in the mixed-Hodge-module Ext group and retains the Hodge normalization of the node interface.

Equivalently, the class

$$\left[P_{\mathrm{var},\Sigma }^H\right]$$

is the heart-level form of the variation-side residual

$$Cone\left(v_{\Sigma }\right)[-1].$$

Thus the cone-level residual and the Ext-class residual encode the same variation-side attachment at different levels of precision. The cone gives the derived residual object. The Ext class records the corresponding short exact sequence in the MHM heart.

Remark 17

(Residual class versus aggregate monodromy cone). The class

$$\left[P_{\mathrm{var},\Sigma }^H\right]$$

is not determined by the aggregate monodromy cone

$$Cone\left(N\right).$$

It depends on the chosen finite-node interface $W_{\Sigma }^H$ and on the variation-side map

$$v_{\Sigma }:W_{\Sigma }^H\to {\psi }_{\pi ,1}^H(-1).$$

This is the sense in which the corrected extension is a genuinely factorization-sensitive residual invariant.

6.2. Nodewise Decomposition

Since

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{W}_k^H,W_k^H=i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),$$

finite additivity gives a direct-sum decomposition of the Ext group.

Proposition 8

(Nodewise residual module). There is a canonical isomorphism

$${Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H).$$

Consequently the residual class decomposes as

$$\left[P_{\mathrm{var},\Sigma }^H\right]=({ϵ}_1^H,\dots ,{ϵ}_r^H),$$

where

$${ϵ}_k^H\in {Ext}_{MHM\left(X_0\right)}^1(i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),I{C}_{X_0}^H).$$

Proof. 

In any abelian category, finite direct sums in the first argument give finite products on $Hom(-,I{C}_{X_0}^H)$. Since the index set is finite, finite products and finite direct sums agree. Passing to the first right-derived functor gives

$${Ext}^1\left({\displaystyle \underset{k=1}{\overset{r}{⨁}}}{W}_k^H,I{C}_{X_0}^H\right)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{Ext}^1(W_k^H,I{C}_{X_0}^H).$$

Applying this isomorphism to the extension class of $P_{\mathrm{var},\Sigma }^H$ gives the tuple

$$({ϵ}_1^H,\dots ,{ϵ}_r^H).$$

□

Definition 7

(Nodewise residual classes). The components

$${ϵ}_k^H\in {Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H)$$

are called the nodewise Hodge residual classes.

Thus the finite-node variation residual class decomposes as a finite list of local attachment channels. Each component records the contribution of one ordinary double point to the variation-side corrected extension.

6.3. Local Meaning of ${ϵ}_k^H$

Each ${ϵ}_k^H$ measures the attachment of the local ODP Hodge block

$$W_k^H=i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

to the global intersection-complex Hodge module. Equivalently, it measures the contribution of the k-th node to the corrected extension

$$0\to I{C}_{X_0}^H\to P_{\mathrm{var},\Sigma }^H\to W_{\Sigma }^H\to 0.$$

The local ODP calculation shows that $W_k^H$ is rank one and Tate-twisted. Therefore ${ϵ}_k^H$ is the local Hodge extension channel through which that rank-one vanishing sector is attached to $I{C}_{X_0}^H$.

The ordinary-double-point hypothesis is important here. It implies that the local Milnor fiber has torsion-free integral vanishing cohomology:

$${\tilde{H}}^3(F_{p_k};\mathbb{Z})\cong \mathbb{Z}.$$

Thus the local ODP block contributes a nontrivial residual channel but no local torsion excess. In the residual–excess matrix, the class

$${ϵ}_k^H$$

belongs to the horizontal MHM residual row, while the local vertical excess attached to the ODP Milnor block vanishes.

Remark 18.

A full numerical computation of

$${Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H)$$

depends on the local and global geometry of $X_0$. The present paper uses the nodewise decomposition as the finite-node residual module. In examples where the local model and global incidence data are explicit, the next step is to compute the dimensions and generators of the individual summands.

6.4. The Nodewise Residual Module in the Residual–Excess Matrix

The nodewise Ext module refines the variation-side entry of the residual–excess matrix. The finite-node residual row contains

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right).\end{array}$$

After shifting the variation cone to the MHM heart, the variation entry becomes the corrected extension

$$P_{\mathrm{var},\Sigma }^H=Cone\left(v_{\Sigma }\right)[-1].$$

This extension is classified by

$$\left[P_{\mathrm{var},\Sigma }^H\right]\in {Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H).$$

The nodewise decomposition gives

$$\left[P_{\mathrm{var},\Sigma }^H\right]=({ϵ}_1^H,\dots ,{ϵ}_r^H).$$

Thus the variation column of the matrix contains increasingly refined horizontal data:

$$Cone\left(v_{\Sigma }\right),P_{\mathrm{var},\Sigma }^H=Cone\left(v_{\Sigma }\right)[-1],({ϵ}_1^H,\dots ,{ϵ}_r^H).$$

The first is the derived residual cone. The second is its heart-level corrected extension. The third is the nodewise Ext-class decomposition.

If an integral variation residual

$$P_{\mathrm{var},\Sigma }^{\mathbb{Z}}$$

is specified in an integral or finite-coefficient enhancement, and if a comparison morphism

$$P_{\mathrm{var},\Sigma }^{\mathbb{Z}}\otimes \mathbb{Q}⟶P_{\mathrm{var},\Sigma }^H$$

is given, then the variation excess is

$$E_{\mathrm{var},\Sigma }:=Cone\left(P_{\mathrm{var},\Sigma }^{\mathbb{Z}}\otimes \mathbb{Q}⟶P_{\mathrm{var},\Sigma }^H\right)[-1].$$

This excess object is not the same as the nodewise residual class. The nodewise classes ${ϵ}_k^H$ measure how the rational/MHM node interface is attached to $I{C}_{X_0}^H$. The excess object measures whether this rational attachment is the rationalization of an integral attachment.

6.5. Zero-Excess Interpretation for ODP Nodes

For ordinary double points, the local Milnor fiber calculation gives

$${\tilde{H}}^3(F_{p_k};\mathbb{Z})\cong \mathbb{Z}$$

and no local torsion. Therefore the local node interface itself has no coefficient-change excess. In matrix form, the local ODP contribution has the shape

$$\begin{array}{cc}\mathrm{entry}& \mathrm{local}\mathrm{ODP}\mathrm{contribution}\\ H/\mathbb{Q}\mathrm{residual}& {ϵ}_k^H\in {Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H)\\ {Exc}_{\mathrm{local}}& 0.\end{array}$$

This is the precise sense in which ordinary double points provide the zero-excess calibration for the nodewise residual module. The residual class ${ϵ}_k^H$ may be nonzero and geometrically meaningful, but it is not explained by hidden local torsion. It is a Hodge-theoretic attachment channel inside the rational/MHM row.

By contrast, in a torsion-bearing situation such as the Enriques–Brauer Bockstein mechanism used by Diaz, the nonzero information lies in the vertical coefficient-change direction: a finite-coefficient class has a Bockstein image which is integral torsion and is killed by rationalization. This is why Diaz’s construction calibrates the nonzero-excess axis, while ordinary double points calibrate the zero-excess residual axis.

6.6. Summary of the Nodewise Residual Invariant

The finite-node construction produces three levels of variation-side information:

$$Cone\left(v_{\Sigma }\right),P_{\mathrm{var},\Sigma }^H,({ϵ}_1^H,\dots ,{ϵ}_r^H).$$

The first is the derived variation residual. The second is the heart-level corrected extension. The third is the nodewise Ext residual module.

These objects refine the aggregate monodromy cone by recording how the node-supported Hodge interface is reinserted into the global intersection-complex package. They are rational/MHM residual data. In the ordinary-double-point case, they are accompanied by zero local coefficient excess because the local Milnor fiber is torsion-free. This separation is the main lesson of the finite-node model: residual data may be nontrivial even when local excess vanishes.

7. Boundary, Phantom, and Extension–Ladder Comparisons

We now compare the finite-node defect resolution with two standard sources of closed-stratum data: recollement and the extension ladder

$$j_{!}{L}_U^H\to j_{!\ast }{L}_U^H\to j_{\ast }{L}_U^H.$$

This section is the place where the factorization viewpoint earns its keep. The finite-node can–var factorization and the extension/recollement factorizations are not identical, but they meet at a point-supported ordinary-double-point contribution.

The comparison made in this section is horizontal in the residual–excess matrix. All objects live in the rational/MHM row. Thus this section compares different rational residual packages:

$$\mathrm{can}--\mathrm{var}\mathrm{residuals},\mathrm{recollement}\mathrm{residuals},\mathrm{extension}-\mathrm{ladder}\mathrm{residuals}.$$

It does not yet introduce coefficient-change excess. Excess appears only when one compares such rational/MHM data with integral or finite-coefficient data. The point here is to identify which closed-stratum rational residual is selected by the finite-ODP interface $W_{\Sigma }^H$.

7.1. Recollement Residuals

Let

$$j:U\hookrightarrow X_0,i:\Sigma \hookrightarrow X_0$$

be the open and closed inclusions. For

$$M^H\in D^{b}MHM\left(X_0\right),$$

recollement gives distinguished triangles

$$j_{!}{j}^{\ast }{M}^H⟶M^H⟶i_{\ast }{i}^{\ast }{M}^H\stackrel{+1}{\to },$$

and

$$i_{\ast }{i}^{!}{M}^H⟶M^H⟶j_{\ast }{j}^{\ast }{M}^H\stackrel{+1}{\to }.$$

Thus the lower and upper recollement residuals are

$$R_{j!,\Sigma }^H\left(M^H\right):=Cone(j_{!}{j}^{\ast }{M}^H\to M^H)\cong i_{\ast }{i}^{\ast }{M}^H,$$

and

$$R_{j\ast ,\Sigma }^H\left(M^H\right):=Cone(M^H\to j_{\ast }{j}^{\ast }{M}^H)\cong i_{\ast }{i}^{!}{M}^H\left[1\right].$$

For

$$M^H=P_{\mathrm{var},\Sigma }^H,$$

one has

$$j^{\ast }{P}_{\mathrm{var},\Sigma }^H\cong {\mathbb{Q}}_U^H\left[3\right],$$

because $W_{\Sigma }^H$ is supported on $\Sigma$. Therefore

$$R_{j!,\Sigma }^H\left(P_{\mathrm{var},\Sigma }^H\right)\mathrm{and}{R}_{j\ast ,\Sigma }^H\left(P_{\mathrm{var},\Sigma }^H\right)$$

are supported on $\Sigma$.

These recollement residuals record closed-stratum data of $P_{\mathrm{var},\Sigma }^H$. They are not, by definition, the same as the finite-node can–var residuals

$$Cone\left(u_{\Sigma }\right),Cone\left(v_{\Sigma }\right),Cone\left(N\right).$$

Rather, they give another horizontal residual package in the same rational/MHM row. The comparison problem is to identify the ordinary-double-point part of these recollement residuals and compare it with $W_{\Sigma }^H$.

7.2. The Extension Ladder

Let

$$L_U^H:={\mathbb{Q}}_U^H\left[3\right].$$

The standard extension ladder is

$$j_{!}{L}_U^H⟶j_{!\ast }{L}_U^H⟶j_{\ast }{L}_U^H.$$

In the finite-ODP setting,

$$j_{!\ast }{L}_U^H=I{C}_{X_0}^H.$$

The total lower-to-upper comparison

$$j_{!}{L}_U^H\to j_{\ast }{L}_U^H$$

has image $I{C}_{X_0}^H$. Define

$$R_{!,\ast }^H:=Cone(j_{!}{L}_U^H\to j_{\ast }{L}_U^H),$$

$$R_{!,!\ast }^H:=Cone(j_{!}{L}_U^H\to I{C}_{X_0}^H),$$

and

$$R_{!\ast ,\ast }^H:=Cone(I{C}_{X_0}^H\to j_{\ast }{L}_U^H).$$

All three restrict to zero on U. Hence they are supported on $\Sigma$.

The octahedral axiom applied to

$$j_{!}{L}_U^H⟶I{C}_{X_0}^H⟶j_{\ast }{L}_U^H$$

gives

$$R_{!,!\ast }^H⟶R_{!,\ast }^H⟶R_{!\ast ,\ast }^H\stackrel{+1}{\to }.$$

This is another residual triangle. It is horizontal in the same sense as the finite-node residual triple: it compares three rational/MHM objects at fixed coefficients. It should not be confused with a vertical excess triangle.

Remark 19

(Extension residuals versus can–var residuals). The extension-ladder residuals and the can–var residuals answer different questions. The extension ladder compares lower, middle, and upper extensions of the smooth-locus Hodge module. The can–var factorization compares nearby, vanishing, and monodromy data. In a finite-ODP degeneration these two packages meet along a point-supported ODP contribution, but they are not formally identical.

7.3. The Corrected Object Between the Extension Ladder and the Node Interface

The corrected object

$$P_{\mathrm{var},\Sigma }^H$$

sits beyond the middle extension by the node-supported quotient:

$$0⟶I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H⟶0.$$

Thus the four relevant objects are

$$j_{!}{L}_U^H,I{C}_{X_0}^H,P_{\mathrm{var},\Sigma }^H,j_{\ast }{L}_U^H.$$

They all restrict to $L_U^H$ on U, but they differ by closed-stratum data on $\Sigma$.

The can–var factorization and the extension-ladder factorization therefore organize related but different boundary information. The can–var factorization identifies the ODP vanishing interface $W_{\Sigma }^H$. The extension ladder records the full lower-to-upper boundary room. Their comparison requires selecting the ordinary-double-point quotient from the boundary residual.

In the residual–excess matrix, the situation may be read as follows:

$$\begin{array}{cccc}& \mathrm{lower}/\mathrm{middle}\mathrm{side}& \mathrm{corrected}\mathrm{finite}-\mathrm{node}\mathrm{side}& \mathrm{upper}/\mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& j_{!}{L}_U^H,I{C}_{X_0}^H& P_{\mathrm{var},\Sigma }^H,W_{\Sigma }^H& j_{\ast }{L}_U^H,Cone\left(N\right).\end{array}$$

This is schematic, not a claim of equality among all entries. Its purpose is to emphasize that the comparison is being made inside the rational/MHM row: one is comparing which closed-stratum residuals are selected by the finite-node interface.

7.4. ODP-Selected Boundary Quotient

We isolate the precise comparison object.

Definition 8

(ODP-selected boundary quotient). Let

$$R_{!,\ast }^H=Cone(j_{!}{L}_U^H\to j_{\ast }{L}_U^H)$$

be the total boundary residual. The ODP-selected boundary quotient, denoted

$$R_{!,\ast }^{H,\mathrm{ODP}},$$

is the maximal point-supported quotient of the perverse cohomology of $R_{!,\ast }^H$ generated by the local ordinary-double-point vanishing sectors.

This definition separates the total boundary residual from the ordinary-double point part of that residual. We do not claim

$$R_{!,\ast }^H\cong W_{\Sigma }^H.$$

The correct comparison is with the ODP-selected quotient.

Conjecture 7

(Boundary-interface comparison). For a finite ordinary-double-point degeneration, the ODP-selected boundary quotient is naturally isomorphic to the node-supported Hodge interface:

$$R_{!,\ast }^{H,\mathrm{ODP}}\cong W_{\Sigma }^H.$$

Equivalently, the ODP-selected quotient of the extension-ladder boundary residual agrees with the reduced finite-node can–var interface.

Remark 20

(Expected proof strategy and prior work). The local input for Conjecture 7 is standard in the theory of isolated hypersurface singularities. For a threefold ordinary double point, the local vanishing contribution is rank one in the normalized perverse degree and has Tate type ${\mathbb{Q}}^H(-1)$. This is the node case of the odd-dimensional hypersurface singularity calculations in Kerr–Laza’s study of vanishing cohomology [11]. The single-node corrected perverse extension is also treated in the conifold setting in earlier work of the author [12]. Conjecture 7 globalizes this local statement over a finite set of disjoint nodes. The remaining issue is to verify that the local identifications assemble compatibly and that no non-ODP closed-stratum contribution survives in the ODP-selected quotient.

Remark 21

(Why this remains horizontal). Conjecture 7 is a rational/MHM comparison. It compares two point-supported residual packages inside the $H/\mathbb{Q}$ row of the residual–excess matrix. It is therefore not an excess statement. If one later constructs integral versions of $R_{!,\ast }^{H,\mathrm{ODP}}$ and $W_{\Sigma }^H$, then the failure of their rationalizations to agree would define a vertical excess object. For ordinary double points, the local Milnor cohomology is torsion-free, so no local torsion excess is expected from the ODP node block itself.

Remark 22

(Integral prediction). The residual–excess framework predicts that the integral analogue of the ODP-selected comparison should remain zero-excess locally for ordinary double points, because the ODP Milnor fiber is torsion-free. By contrast, torsion-bearing boundary packages, such as Coble–Enriques or ${\displaystyle \frac{1}{4}}(1,1)$-type local models, are expected to produce nonzero excess. This prediction is not used in the proof of the finite-node MHM statements; it records how the present comparison should behave under future integral refinements.

7.5. Phantom Cohomology Shadow

Kerr–Laza identify point-supported terms $W_{\sigma }^j$ in the decomposition over a curve and relate them to phantom cohomology

$$H_{\mathrm{ph},\sigma }^{\ell }=ker\left(sp:H^{\ell }\left(X_{\sigma }\right)\to H_{lim,\sigma }^{\ell }\right)$$

[6]. In the finite-ODP setting, the hypercohomology of $W_{\Sigma }^H$ gives

$${\mathbb{H}}^0(X_0,W_{\Sigma }^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1),$$

with the fixed point-support convention.

Thus the cohomological shadow of the finite-node interface is a point-supported Tate contribution. This is the MHM-level source of the ordinary-double-point phantom/closed-stratum contribution. The MHM object $W_{\Sigma }^H$ retains the individual node supports and Tate twists before passing to cohomology.

In the residual–excess framework, this phantom shadow is again horizontal. It is a cohomological shadow of the rational/MHM finite-node interface. It should be separated from the vertical excess phenomenon represented by torsion-bearing Bockstein examples such as Diaz’s Enriques-product construction, where the nonzero data lie in the kernel of rationalization.

7.6. Summary of the Boundary Comparison

The comparison in this section has three outputs.

First, recollement gives closed-stratum residuals supported on $\Sigma$:

$$R_{j!,\Sigma }^H\left(M^H\right),R_{j\ast ,\Sigma }^H\left(M^H\right).$$

Second, the extension ladder gives a lower-to-upper boundary residual

$$R_{!,\ast }^H=Cone(j_{!}{L}_U^H\to j_{\ast }{L}_U^H)$$

and its octahedral decomposition through $I{C}_{X_0}^H$. Third, the finite-node can–var construction selects the ODP interface

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

The proposed bridge between the extension-ladder residual and the finite-node can–var residual is the ODP-selected boundary quotient

$$R_{!,\ast }^{H,\mathrm{ODP}}.$$

The conjectural comparison

$$R_{!,\ast }^{H,\mathrm{ODP}}\cong W_{\Sigma }^H$$

is the precise finite-ODP statement. It is a comparison of rational/MHM closed-stratum residuals. It is not a coefficient-change excess statement.

This distinction is important. The boundary comparison explains how the finite-node residual package sits inside the rational degeneration formalism. The excess formalism explains something different: what is lost when one passes from integral or finite-coefficient data to rational/MHM data. Ordinary double points provide a zero-excess local model, while Diaz-type Bockstein classes provide nonzero-excess calibration examples.

8. Cohomological and Limiting–Mixed–Hodge Shadows

The previous sections constructed MHM-level residuals. Applying hypercohomology gives mixed-Hodge-structure shadows. This section records the part needed for the finite-node construction. The distinction between levels is important. The MHM object remembers support on $X_0$, individual node labels, Tate twists, and extension data in $MHM\left(X_0\right)$. Its hypercohomology remembers the resulting mixed Hodge structures after taking global sections. Thus passing from MHM t MHS is another horizontal passage inside the rational/Hodge row of the residual–excess matrix. It is not, by itself, a coefficient-change excess operation.

8.1. Hypercohomology of the Node Interface

Since

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),$$

the fixed point-support convention gives

$${\mathbb{H}}^0(X_0,W_{\Sigma }^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1),$$

and

$${\mathbb{H}}^m(X_0,W_{\Sigma }^H)=0\mathrm{for}m\ne 0.$$

Lemma 6

(Hypercohomology of the node interface). With the point-support convention of Convention 2, one has

$${\mathbb{H}}^0(X_0,W_{\Sigma }^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1),$$

and

$${\mathbb{H}}^m(X_0,W_{\Sigma }^H)=0(m\ne 0).$$

Proof. 

Hypercohomology commutes with finite direct sums. For each point inclusion $i_k:\left\{p_k\right\}\hookrightarrow X_0$, the object

$$i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

is placed in the MHM heart with only degree-zero hypercohomology, equal to

$${\mathbb{Q}}^H(-1).$$

Summing over k gives the result. □

Thus the cohomological shadow of the node-supported interface is a direct sum of Tate-twisted point Hodge structures. This is the MHS-level shadow of the MHM object $W_{\Sigma }^H$. The MHM object remembers the individual supports $p_k$, while the hypercohomology remembers the resulting direct sum.

8.2. The Cohomological Variation Residual

The extension

$$0\to I{C}_{X_0}^H\to P_{\mathrm{var},\Sigma }^H\to W_{\Sigma }^H\to 0$$

gives a distinguished triangle

$$I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H\stackrel{+1}{\to }.$$

Applying hypercohomology gives a long exact sequence of mixed Hodge structures. Using Lemma 6, the point-supported term contributes the connecting morphism

$$\partial :{\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1)⟶{\mathbb{H}}^1(X_0,I{C}_{X_0}^H).$$

This morphism is the cohomological shadow of the MHM extension class

$$\left[P_{\mathrm{var},\Sigma }^H\right]\in {Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H).$$

Equivalently, the nodewise residual class

$$\left[P_{\mathrm{var},\Sigma }^H\right]=({ϵ}_1^H,\dots ,{ϵ}_r^H)$$

has a global cohomological shadow

$$\partial =\underset{k}{⨁}{\partial }_k,$$

where each ${\partial }_k$ is the hypercohomological contribution of the nodewise attachment channel

$${ϵ}_k^H\in {Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H).$$

This passage from ${ϵ}_k^H$ to ${\partial }_k$ loses some local geometric information: support and object-level extension data are compressed into a mixed-Hodge-structure morphism.

8.3. Comparison of MHM and MHS Residuals

The MHM residual and its MHS shadow are not the same object.

At the MHM level, the residual is the extension

$$0\to I{C}_{X_0}^H\to P_{\mathrm{var},\Sigma }^H\to W_{\Sigma }^H\to 0.$$

At the MHS level, one sees the long exact hypercohomology sequence and the connecting morphism

$${\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1)⟶{\mathbb{H}}^1(X_0,I{C}_{X_0}^H).$$

The MHM object remembers that the contribution is supported at the individual nodes $p_k$. The cohomological shadow remembers only the resulting mixed Hodge structure after global hypercohomology.

This is why the finite-node interface is constructed in $MHM\left(X_0\right)$ before passing to cohomology. At the MHM level, one can still distinguish the node-supported interface

$$W_{\Sigma }^H=\underset{k}{⨁}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),$$

the corrected object

$$P_{\mathrm{var},\Sigma }^H,$$

and the nodewise Ext classes

$${ϵ}_k^H.$$

After hypercohomology, these data contribute to mixed Hodge structures, but the object-level support package is no longer visible in the same form.

8.4. Limiting–Mixed–Hodge Interpretation

The cohomological shadow of the finite-node interface also has a limiting-mixed-Hodge interpretation. The nearby-cycle object carries the limiting mixed Hodge structure of the degeneration, and the nilpotent monodromy operator

$$N:{\psi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1)$$

induces the usual monodromy operator on the cohomological limiting mixed Hodge structure.

The node-supported interface

$$W_{\Sigma }^H$$

selects the finite ordinary-double-point contribution to this picture. Its hypercohomology

$${\mathbb{H}}^0(X_0,W_{\Sigma }^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1)$$

is the MHS-level trace of the node interface. The connecting morphism

$$\partial :{\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1)⟶{\mathbb{H}}^1(X_0,I{C}_{X_0}^H)$$

records how the finite-node contribution enters the global cohomological package.

This should be read as a shadow, not as a replacement for the MHM statement. The MHM statement controls the interface object and its extension class before global cohomology is taken. The limiting-MHS statement records the induced cohomological contribution after applying hypercohomology.

8.5. MHS Shadows and Excess

The MHS shadow remains part of the rational/Hodge row of the residual–excess matrix. It is obtained by applying hypercohomology to MHM residual data. It does not detect integral torsion killed by rationalization unless integral or finite-coefficient data are added.

For ordinary double points, this distinction is harmless locally. The Milnor fiber has

$${\tilde{H}}^3(F_{p_k};\mathbb{Z})\cong \mathbb{Z}$$

and no local torsion. Thus the MHS shadow of the ODP node interface has no hidden local torsion excess. The residual data may be nontrivial, but the local coefficient-change excess is zero.

By contrast, in torsion-bearing examples such as Diaz’s Enriques–Brauer Bockstein construction, the key class is integral torsion and which lies in the kernel of rationalization. Such a class is not detected by the rational MHS shadow alone. It requires the vertical excess axis of the residual–excess matrix. Thus there are three distinct levels: the MHM residual object, its MHS cohomological shadow, and the separate coefficient-change excess. The first two are rational/Hodge-level data. The third records integral or finite-coefficient information lost under rationalization.

8.6. Summary

The finite-node interface has the cohomological shadow

$${\mathbb{H}}^0(X_0,W_{\Sigma }^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1).$$

The variation residual extension gives a connecting morphism

$${\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1)⟶{\mathbb{H}}^1(X_0,I{C}_{X_0}^H),$$

which is the MHS-level shadow of the nodewise Ext residual class.

These cohomological shadows are useful, but they are weaker than the MHM objects from which they come. They forget individual object-level support and extension data. They also remain rational/Hodge-level objects; they do not detect torsion killed by rationalization. That torsion belongs to the excess axis, not to the MHS shadow alone.

9. Supporting Factorization Families

The finite-node Saito interface is the main geometric example of the paper. Two standard categorical sources of similar factorization patterns are adjunctions and MacPherson–Vilonen zig-zags. We record them only as comparison families.

For an adjunction $F:\mathcal{C}\rightleftarrows \mathcal{D}:G$, the unit and counit maps ${id}_{\mathcal{C}}\to GF$ and $FG\to {id}_{\mathcal{D}}$ give recovery maps whose cones measure failure of recovery across the adjunction. In this paper the relevant examples are recollement adjunctions $j_{!}⊣j^{\ast }⊣j_{\ast }$ and $i^{\ast }⊣i_{\ast }⊣i^{!}$, which underlie the closed-stratum residuals used above. These are horizontal residual examples, not excess objects.

MacPherson–Vilonen zig-zags give a second comparison family. An object of an MV category $C(F,G;T)$ is a tuple $(A,B,u,v)$ with $F\left(A\right)\stackrel{u}{\to }B\stackrel{v}{\to }G\left(A\right)$ and $vu=T_A$, hence it has a residual triple $(Cone\left(u\right),Cone\left(v\right),Cone\left(T_A\right))$. In finite-coefficient Bockstein applications, the MV boundary is not identical to this cone triple; it is a cohomological or obstruction-channel shadow of the zig-zag datum. Thus MV supplies useful language for comparing residual triples with Bockstein-type excess, but the present paper does not reprove the Bockstein obstruction results or construct a full integral motivic MV theory.

10. Comparative Structure and Finite–Node Output

The preceding sections no longer treat the residual families as equal examples. The central object is the finite-node defect resolution

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1),v_{\Sigma }{u}_{\Sigma }=N.$$

This is a distinguished object of

$$\mathbf{DefRes}\left(N\right),$$

where

$$N:{\psi }_{\pi ,1}^H\to {\psi }_{\pi ,1}^H(-1)$$

is the finite-node monodromy return map. The comparison with recollement, extension ladders, cohomological shadows, adjunctions, and MacPherson–Vilonen zig-zags is useful only insofar as it clarifies what this finite-node object records.

This section summarizes the finite-node output in the language of the residual–excess matrix. The finite-node construction supplies the rational/MHM residual row. Ordinary double points give the local zero-excess calibration because their Milnor fibers are integrally torsion-free. Thus the finite-node model shows that residual data can be nontrivial even when local coefficient-change excess vanishes.

10.1. The Main Finite–Node Package

The finite-node package consists of the following data.

$$\begin{array}{cc}\mathrm{datum}& \mathrm{object}\mathrm{or}\mathrm{map}\\ \mathrm{source}& {\psi }_{\pi ,1}^H\\ \mathrm{interface}& W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)\\ \mathrm{return}\mathrm{object}& {\psi }_{\pi ,1}^H(-1)\\ \mathrm{outgoing}\mathrm{map}& u_{\Sigma }:{\psi }_{\pi ,1}^H\to W_{\Sigma }^H\\ \mathrm{incoming}\mathrm{map}& v_{\Sigma }:W_{\Sigma }^H\to {\psi }_{\pi ,1}^H(-1)\\ \mathrm{return}\mathrm{map}& N=v_{\Sigma }{u}_{\Sigma }\end{array}$$

The associated residual triple is

$${\mathsf{T}}_{\Sigma }^H\left(N\right)=\left(Cone\left(u_{\Sigma }\right),Cone\left(v_{\Sigma }\right),Cone\left(N\right)\right).$$

The entries have the following interpretations.

$$\begin{array}{ccc}\mathrm{cone}& \mathrm{name}& \mathrm{interpretation}\\ Cone\left(u_{\Sigma }\right)& \mathrm{extraction}\mathrm{residual}& \mathrm{nearby}-\mathrm{cycle}\mathrm{data}\mathrm{not}\mathrm{captured}\mathrm{by}{W}_{\Sigma }^H\\ Cone\left(v_{\Sigma }\right)& \mathrm{reinsertion}\mathrm{residual}& \mathrm{insertion}\mathrm{of}{W}_{\Sigma }^H\mathrm{into}\mathrm{the}\mathrm{central}-\mathrm{fiber}\mathrm{package}\\ Cone\left(N\right)& \mathrm{return}\mathrm{residual}& \mathrm{aggregate}\mathrm{finite}-\mathrm{node}\mathrm{monodromy}\mathrm{defect}\end{array}$$

The key point is that the interface

$$W_{\Sigma }^H$$

is not recoverable from $Cone\left(N\right)$ alone. It records the point support, Tate twist, Hodge normalization, and nodewise decomposition of the ordinary-double-point contribution.

In residual–excess language, the preceding table is horizontal data. It belongs to the rational/MHM residual row:

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right).\end{array}$$

The vertical excess entries are not part of this construction alone. They would arise only after comparing this row with integral or finite-coefficient data.

10.2. Exact and Ext Structures

The main exact sequence is

$$0⟶I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H⟶0,$$

where

$$P_{\mathrm{var},\Sigma }^H=Cone\left(v_{\Sigma }\right)[-1].$$

This is the heart-level form of the reinsertion residual.

Under the heart-level extraction hypothesis, the extraction residual gives

$$0⟶K_{\mathrm{can},\Sigma }^H⟶{\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H⟶0,$$

where

$$K_{\mathrm{can},\Sigma }^H=Cone\left(u_{\Sigma }\right)[-1].$$

The extension class of the reinsertion residual lies in

$${Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H).$$

Since

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{W}_k^H,W_k^H=i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),$$

this Ext group decomposes nodewise:

$${Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H).$$

Thus the residual class decomposes as

$$\left[P_{\mathrm{var},\Sigma }^H\right]=({ϵ}_1^H,\dots ,{ϵ}_r^H),$$

with

$${ϵ}_k^H\in {Ext}_{MHM\left(X_0\right)}^1(i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),I{C}_{X_0}^H).$$

This nodewise Ext module is one of the concrete outputs of the paper. It is the finite-node residual module attached to the object

$$\left({\psi }_{\pi ,1}^H\to W_{\Sigma }^H\to {\psi }_{\pi ,1}^H(-1)\right)\in \mathbf{DefRes}\left(N\right).$$

The variation column of the residual row therefore admits three levels of description:

$$Cone\left(v_{\Sigma }\right),P_{\mathrm{var},\Sigma }^H,({ϵ}_1^H,\dots ,{ϵ}_r^H).$$

The first is the derived residual cone. The second is the heart-level corrected extension. The third is the nodewise Ext decomposition.

If an integral variation residual

$$P_{\mathrm{var},\Sigma }^{\mathbb{Z}}$$

and a comparison map

$$P_{\mathrm{var},\Sigma }^{\mathbb{Z}}\otimes \mathbb{Q}⟶P_{\mathrm{var},\Sigma }^H$$

are specified, then the cone of this comparison defines the variation excess. That object is not the same as the nodewise residual class. The nodewise classes measure rational/MHM attachment. Excess measures coefficient-change failure.

10.3. Triangulated Constraint

The residual triple is constrained by the octahedral triangle

$$Cone\left(u_{\Sigma }\right)⟶Cone\left(N\right)⟶Cone\left(v_{\Sigma }\right)⟶Cone\left(u_{\Sigma }\right)\left[1\right].$$

This triangle is not the novelty of the paper. The can–var–monodromy octahedron belongs to Saito’s formalism and appears in the Kerr–Laza–Saito treatment of degenerations. The novelty here is the reduced finite-node interface and the Hodge-typed residual data it retains.

The triangle says that the extraction residual, reinsertion residual, and return residual are not independent. The return cone $Cone\left(N\right)$ is the aggregate residual, while the two factor cones record how the aggregate defect is realized through $W_{\Sigma }^H$.

In the residual–excess matrix, this octahedral triangle is the horizontal constraint on the MHM row. Vertical excess constraints are different: they arise from cones of comparison morphisms between coefficient levels.

10.4. Relation with Recollement and Extension Residuals

Recollement and the extension ladder organize the same singular set from a different direction. Recollement gives

$$Cone(j_{!}{j}^{\ast }{M}^H\to M^H)\cong i_{\ast }{i}^{\ast }{M}^H$$

and

$$Cone(M^H\to j_{\ast }{j}^{\ast }{M}^H)\cong i_{\ast }{i}^{!}{M}^H\left[1\right].$$

For

$$M^H=P_{\mathrm{var},\Sigma }^H,$$

these residuals are supported on $\Sigma$.

The extension ladder

$$j_{!}{L}_U^H⟶I{C}_{X_0}^H⟶j_{\ast }{L}_U^H$$

produces the boundary residuals

$$R_{!,!\ast }^H,R_{!\ast ,\ast }^H,R_{!,\ast }^H.$$

These are also supported on $\Sigma$.

The finite-node can–var residual and the extension/recollement residuals should not be identified wholesale. The total boundary residual

$$R_{!,\ast }^H$$

contains all lower-to-upper extension-room data. The object

$$W_{\Sigma }^H$$

is the ordinary-double-point selected, Tate-twisted node interface. The comparison therefore proceeds through the ODP-selected quotient

$$R_{!,\ast }^{H,\mathrm{ODP}},$$

introduced above. The expected comparison is

$$R_{!,\ast }^{H,\mathrm{ODP}}\cong W_{\Sigma }^H.$$

This is the precise finite-node boundary-interface comparison problem.

This comparison remains horizontal. It compares rational/MHM residual packages inside the $H/\mathbb{Q}$ row. It does not assert anything about integral lifts or torsion excess.

10.5. Phantom and Cohomological Shadows

Kerr–Laza isolate point-supported terms $W_{\sigma }^j$ in the decomposition over a curve and identify phantom cohomology with such point-supported contributions. In the finite-ODP setting, the MHM interface $W_{\Sigma }^H$ has hypercohomology

$${\mathbb{H}}^0(X_0,W_{\Sigma }^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1),$$

and no other hypercohomology, with the point-support convention.

Thus the cohomological shadow of $W_{\Sigma }^H$ is a finite direct sum of Tate-twisted Hodge structures. This is the MHM-level source of the ordinary-double-point point-supported or phantom contribution. The MHM object retains the individual node supports before this data is collapsed by hypercohomology.

The phantom/cohomological shadow is still a rational/Hodge-level object. It is weaker than the MHM interface and it is different from coefficient-change excess. In ODP examples this distinction is harmless locally because the Milnor fiber is torsion-free. In torsion-bearing examples, however, the rational cohomological shadow may miss integral torsion lying in the kernel of rationalization.

10.6. Realization Comparison

Applying rational realization gives

$$rat\left(W_{\Sigma }^H\right)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}.$$

The realized variation extension is

$$0⟶I{C}_{X_0}⟶rat\left(P_{\mathrm{var},\Sigma }^H\right)⟶{\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}⟶0.$$

The realization forgets the Tate twist, Hodge filtration, and weight filtration. Therefore the rational perverse residual is only the shadow of the MHM residual. The Hodge-typed finite-node interface is

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1),$$

not merely its rational realization.

This realization comparison should also be separated from coefficient-change excess. Forgetting Hodge filtration and Tate normalization under rat is not the same as killing integral torsion under $-\otimes \mathbb{Q}$. The first loss is controlled by staying in $MHM\left(X_0\right)$. The second requires the vertical excess axis.

10.7. ODP as the Zero–Excess Calibration

The finite-node construction supplies a nontrivial rational/MHM residual package. At the same time, the ordinary-double-point local model has torsion-free integral Milnor cohomology:

$${\tilde{H}}^3(F_{p_k};\mathbb{Z})\cong \mathbb{Z},{\tilde{H}}^m(F_{p_k};\mathbb{Z})=0(m\ne 3).$$

Therefore the local ODP node block has no hidden torsion killed by rationalization. In the residual–excess matrix, the finite-ODP local model has the form

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right)\\ {Exc}_{\mathrm{local}}& 0& 0& 0.\end{array}$$

This is a local statement about the ODP Milnor block. It does not assert that every global integral extension issue vanishes. It asserts that finite ODPs are the zero-excess calibration of the theory: residual data may be geometrically meaningful even when local coefficient excess is zero.

10.8. Contrast with Diaz–Type Nonzero Excess

The zero-excess ODP situation should be contrasted with torsion-bearing finite-coefficient constructions such as Diaz’s Enriques–Brauer Bockstein mechanism. There one starts with a finite-coefficient class

$${\Theta }_2\in H^3(S_1\times S_2,\mathbb{Z}/2\left(2\right))$$

and takes its Bockstein image

$${\Delta }_2=\delta \left({\Theta }_2\right)\in H^4(S_1\times S_2,\mathbb{Z}\left(2\right)).$$

The class ${\Delta }_2$ is integral 2-torsion, hence

$${\Delta }_2\otimes \mathbb{Q}=0.$$

Under Diaz’s survival criterion, it is non-algebraic. Thus Diaz’s construction is a nonzero-excess calibration: the key data lie in the kernel of rationalization.

The contrast is the central reason for separating residuals from excess. Finite ODPs show that residual data can be nontrivial even with zero local excess. Diaz-type Bockstein classes show that nonzero excess can carry integral Hodge obstruction data invisible to rational/MHM residuals.

10.9. What the Comparison Shows

The comparison shows that the formal operation

$$(u,v)\mapsto \left(Cone\left(u\right),Cone\left(v\right),Cone\left(vu\right)\right)$$

is useful only after the factorization and the interface have been identified. In the finite-node case, the interface is

$$W_{\Sigma }^H.$$

The main data retained by the finite-node defect resolution are:

1.

the support of the ordinary-double-point contribution;

2.

the Tate-twisted Hodge type $i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$;

3.

the corrected MHM extension

$$0\to I{C}_{X_0}^H\to P_{\mathrm{var},\Sigma }^H\to W_{\Sigma }^H\to 0;$$

4.

the nodewise residual classes

$${ϵ}_k^H\in {Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H);$$

5.

the cohomological Tate shadow

$${\mathbb{H}}^0(X_0,W_{\Sigma }^H)\cong {\displaystyle \underset{k=1}{\overset{r}{⨁}}}{\mathbb{Q}}^H(-1);$$

6.

the boundary/phantom comparison problem

$$R_{!,\ast }^{H,\mathrm{ODP}}\stackrel{?}{\cong }{W}_{\Sigma }^H.$$

These are the finite-node outputs of the paper. They are rational/MHM residual outputs. The accompanying local excess statement is different: because the ODP Milnor fiber is torsion-free, the local coefficient-change excess at the nodes vanishes. Thus the finite-node model calibrates the residual axis of the residual–excess matrix, while Diaz-type Bockstein examples calibrate the nonzero-excess axis.

11. Toward Higher–Order Strata

The finite-ODP case is deliberately rigid. The singular set is finite, each ordinary double point contributes a rank-one local vanishing sector, and the interface object is a finite direct sum of point-supported Hodge modules:

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

This rigidity is why the reduced defect resolution can be written explicitly and why the local excess calculation is clean: the Milnor fiber of a threefold ordinary double point is $S^3$, so the local integral vanishing cohomology is torsion-free.

For higher-dimensional singular strata, the same formal shape may persist, but both axes of the residual–excess matrix become more complicated. Horizontally, the finite direct sum of point Hodge lines must be replaced by stratum-supported mixed Hodge modules. Vertically, the coefficient-comparison problem may become nontrivial: higher strata, quotient strata, or strata with monodromy may carry integral or finite-coefficient information killed by rationalization.

The purpose of this section is therefore programmatic. We explain what the finite-ODP model suggests, what must change for higher strata, and where nonzero excess may enter. No general theorem for arbitrary higher-order strata is claimed here.

11.1. Replacing Point Hodge Lines

In the finite-ODP case, the interface is

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

Each summand is a point-supported Hodge line. The support is zero-dimensional, the local vanishing sector is rank one, and the point-support V-filtration is standard.

Let $S\subset X_0$ be a positive-dimensional singular stratum, with inclusion

$$i_S:S\hookrightarrow X_0.$$

The expected analogue is not a finite direct sum of point modules. It should be a stratum-supported mixed Hodge module

$$W_S^H\in MHM\left(S\right),$$

or its pushforward

$$i_{S\ast }{W}_S^H\in MHM\left(X_0\right).$$

Depending on the singularity type along S, the object $W_S^H$ may be a variation of mixed Hodge structure, a local system with nontrivial monodromy, or a more general mixed Hodge module on the stratum.

Thus the first higher-strata replacement is

$${\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)⇝i_{S\ast }{W}_S^H.$$

The second replacement is conceptual: nodewise residual classes

$${ϵ}_k^H\in {Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H)$$

should be replaced by stratum-supported residual classes in groups such as

$${Ext}_{MHM\left(X_0\right)}^1(i_{S\ast }{W}_S^H,I{C}_{X_0}^H).$$

11.2. Expected Stratified Defect Resolution

The finite-node defect resolution

$${\psi }_{\pi ,1}^H\to W_{\Sigma }^H\to {\psi }_{\pi ,1}^H(-1)$$

suggests the stratum-supported form

$${\psi }_{\pi ,1}^H\stackrel{u_S}{\to }{i}_{S\ast }{W}_S^H\stackrel{v_S}{\to }{\psi }_{\pi ,1}^H(-1),$$

with return relation

$$v_S{u}_S=N_S,$$

where $N_S$ is the part of monodromy mediated by the stratum S.

If such a datum is constructed, it defines an object of

$$\mathbf{DefRes}\left(N_S\right),$$

and hence a residual triple

$$\left(Cone\left(u_S\right),Cone\left(v_S\right),Cone\left(N_S\right)\right).$$

This would be the horizontal residual row associated with the stratum S.

At this stage, however, this is a program rather than a theorem. The formal octahedral constraint applies once the maps $u_S$ and $v_S$ and the return relation $v_S{u}_S=N_S$ have been established. The difficult part is the construction of the interface $W_S^H$, the filtered admissibility of the maps, and the identification of the stratum-supported part of monodromy.

11.3. Stratified Residual Modules

In the finite-node case, the variation residual class lies in

$${Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H)$$

and decomposes as

$${Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H)\cong \underset{k}{⨁}{Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H).$$

This direct-sum decomposition is a special feature of a finite set of point-supported summands.

For a positive-dimensional stratum, the analogous group

$${Ext}_{MHM\left(X_0\right)}^1(i_{S\ast }{W}_S^H,I{C}_{X_0}^H)$$

is controlled by the geometry of the stratum, the monodromy of $W_S^H$, and the incidence relations with other strata. It need not decompose into a finite list of independent nodewise summands.

If the singular locus is stratified by strata $S_{\alpha }$, one expects a stratified residual package of the form

$$W_{\mathrm{strat}}^H=\underset{\alpha }{⨁}{i}_{\alpha \ast }{W}_{S_{\alpha }}^H,$$

when such a direct-sum description is available. More generally, closure relations among strata may force extension data among the $i_{\alpha \ast }{W}_{S_{\alpha }}^H$. In that case the residual object should be organized by a stratified Ext category rather than by a direct sum of nodewise Ext groups.

Thus the replacement is

$$\underset{k}{⨁}{Ext}^1(W_k^H,I{C}_{X_0}^H)⇝\mathrm{stratified}\mathrm{Ext}\mathrm{data}\mathrm{involving}{i}_{\alpha \ast }{W}_{S_{\alpha }}^H.$$

11.4. New Difficulties

The passage from finite ODPs to higher strata introduces several difficulties.

1.

The interface need not be rank one.

2.

The support need not be zero-dimensional.

3.

The V-filtration is no longer the standard point-supported filtration.

4.

Local systems along strata may have nontrivial monodromy.

5.

Ext groups need not decompose into finite nodewise summands.

6.

Quiver-like shadows become stratified diagrams, involving incidence and closure relations among strata.

7.

The local integral cohomology of the link or Milnor fiber may contain torsion. In that case, the vertical excess row of the residual–excess matrix may be nonzero.

8.

Integral or finite-coefficient comparison data may be required in addition to the rational/MHM residual data.

The last two points are the main new feature from the residual–excess perspective. In the finite-ODP model, local integral Milnor cohomology is torsion-free, so the local excess vanishes. For quotient-type singularities, Enriques/Coble-type boundary packages, or other torsion-bearing strata, the local or global coefficient-comparison problem may have nonzero excess. This is the geometric regime in which the vertical axis of the matrix becomes essential.

11.5. Higher Strata and Excess

The residual–excess matrix suggests two separate questions for a singular stratum S.

First, there is the rational/MHM residual question:

$$\mathrm{Can}\mathrm{one}\mathrm{construct}{\psi }_{\pi ,1}^H\stackrel{u_S}{\to }{i}_{S\ast }{W}_S^H\stackrel{v_S}{\to }{\psi }_{\pi ,1}^H(-1)\mathrm{with}{v}_S{u}_S=N_S?$$

This is the horizontal question. It asks for a stratum-supported defect-resolution object.

Second, there is the coefficient-change question:

$$\mathrm{Does}\mathrm{an}\mathrm{integral}\mathrm{or}\mathrm{finite}-\mathrm{coefficient}\mathrm{lift}\mathrm{of}\mathrm{this}\mathrm{residual}\mathrm{package}\mathrm{rationalize}\mathrm{to}\mathrm{the}\mathrm{MHM}\mathrm{one}\text{?}$$

This is the vertical question. Its failure is measured by excess.

Thus the expected higher-strata matrix has the schematic form

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ \mathbb{Z}\mathrm{or}\mathrm{finite}\mathrm{coeff}.& C_{u,S}^{\mathrm{int}}& C_{v,S}^{\mathrm{int}}& C_{N,S}^{\mathrm{int}}\\ H/\mathbb{Q}& C_{u,S}^H& C_{v,S}^H& C_{N,S}^H\\ Exc& E_{\mathrm{can},S}& E_{\mathrm{var},S}& E_{N,S}.\end{array}$$

For finite ODPs, the local excess row is zero. For torsion-bearing strata, one expects some of the excess entries to be nonzero.

This is where Diaz-type Bockstein mechanisms become relevant as calibration examples. They show that finite-coefficient classes can have integral Bockstein images killed by rationalization. Such phenomena are invisible to the rational/MHM row alone and must be recorded by the vertical excess row.

11.6. What Can Be Claimed Here

The paper makes no general theorem for arbitrary higher-order strata. The claims in this section are limited to the following.

1.

The finite-ODP construction gives the model form

$${\psi }^H\to i_{S\ast }{W}_S^H\to {\psi }^H(-1)$$

for a stratum-supported defect resolution.

2.

If such a datum is constructed and satisfies $v_S{u}_S=N_S$, then the octahedral constraint applies formally.

3.

The main rational/MHM technical difficulty is the construction of $W_S^H$ and the filtered admissibility of $u_S$ and $v_S$.

4.

The nodewise Ext residual module should be replaced by a stratified Ext or residual category.

5.

If the relevant local or global integral data contain torsion, then the vertical excess row may be nonzero.

6.

The present paper does not construct a full integral or motivic enhancement for arbitrary strata. It only identifies the residual and excess objects that such a theory would need to compare.

Thus the finite-ODP case is the first fully computable model. The higher strata problem is to replace the finite direct sum of point Hodge lines by stratum-supported mixed Hodge modules and to track the corresponding factorization-refined residual triples. The residual–excess framework adds one more requirement: whenever integral or finite-coefficient torsion is present, one must also track the coefficient-change excess invisible to the rational/MHM row.

12. Conclusions

12.1. What Has Been Constructed

This paper studies defect resolutions of a fixed return map

$$R:A\to A^{†}.$$

A defect resolution is a factorization

$$A\stackrel{u}{\to }B\stackrel{v}{\to }{A}^{†},vu=R.$$

Its residual triple is

$$\left(Cone\left(u\right),Cone\left(v\right),Cone\left(R\right)\right).$$

The cone $Cone\left(R\right)$ records the aggregate defect. The factorization records how that defect is realized through an interface B. The residual triple is therefore a factorization-refined object.

The paper then adds a second axis: excess. Residuals measure hidden structure at fixed coefficients. Excess measures coefficient-change defect, namely the failure of an integral or finite-coefficient residual package to compare cleanly with its rational or mixed-Hodge realization. These two directions assemble into the residual–excess matrix:

$$\begin{array}{cccc}& Cone\left(u\right)& Cone\left(v\right)& Cone\left(R\right)\\ \mathbb{Z}\mathrm{or}\mathrm{finite}\mathrm{coeff}.& C_u^{\mathrm{int}}& C_v^{\mathrm{int}}& C_R^{\mathrm{int}}\\ H/\mathbb{Q}& C_u^H& C_v^H& C_R^H\\ Exc& E_u& E_v& E_R.\end{array}$$

The horizontal rows are residual triples, constrained by octahedral triangles. The vertical entries are excess objects, defined as cones of coefficient comparison morphisms. Thus the matrix separates two kinds of hidden defect data: factorization-sensitive data and coefficient-sensitive data.

The main geometric example is the finite-ODP Saito interface

$${\psi }_{\pi ,1}^H\stackrel{u_{\Sigma }}{\to }{W}_{\Sigma }^H\stackrel{v_{\Sigma }}{\to }{\psi }_{\pi ,1}^H(-1),v_{\Sigma }{u}_{\Sigma }=N.$$

This is a distinguished object of

$$\mathbf{DefRes}\left(N\right).$$

Here

$$W_{\Sigma }^H={\displaystyle \underset{k=1}{\overset{r}{⨁}}}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

is the reduced node-supported Hodge interface.

The variation residual is

$$P_{\mathrm{var},\Sigma }^H=Cone\left(v_{\Sigma }\right)[-1],$$

and it fits into

$$0⟶I{C}_{X_0}^H⟶P_{\mathrm{var},\Sigma }^H⟶W_{\Sigma }^H⟶0.$$

Under the heart-level extraction hypothesis, the extraction residual is

$$K_{\mathrm{can},\Sigma }^H=Cone\left(u_{\Sigma }\right)[-1].$$

The return residual is $Cone\left(N\right)$, and the octahedral axiom relates the three cones:

$$Cone\left(u_{\Sigma }\right)⟶Cone\left(N\right)⟶Cone\left(v_{\Sigma }\right)\stackrel{+1}{\to }.$$

Thus the finite-node construction supplies the rational/MHM residual row of the residual–excess matrix.

12.2. What the Finite–Node Interface Records

The finite-node interface records data not recoverable from $Cone\left(N\right)$ alone.

1.

It records support:

$$supp\left(W_{\Sigma }^H\right)=\Sigma .$$

2.

It records Hodge type and Tate normalization:

$$W_k^H=i_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1).$$

3.

It records a heart-level corrected extension:

$$0\to I{C}_{X_0}^H\to P_{\mathrm{var},\Sigma }^H\to W_{\Sigma }^H\to 0.$$

4.

It records a nodewise Ext residual module:

$${Ext}_{MHM\left(X_0\right)}^1(W_{\Sigma }^H,I{C}_{X_0}^H)\cong \underset{k}{⨁}{Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H).$$

5.

It records nodewise residual classes:

$$\left[P_{\mathrm{var},\Sigma }^H\right]=({ϵ}_1^H,\dots ,{ϵ}_r^H).$$

6.

It has a cohomological Tate shadow:

$${\mathbb{H}}^0(X_0,W_{\Sigma }^H)\cong \underset{k}{⨁}{\mathbb{Q}}^H(-1).$$

7.

It isolates the finite-node boundary-interface comparison problem:

$$R_{!,\ast }^{H,\mathrm{ODP}}\stackrel{?}{\cong }{W}_{\Sigma }^H.$$

This is the precise sense in which the finite-node defect resolution is finer than the aggregate monodromy cone. The aggregate cone remembers the return defect. The finite-node interface remembers how that defect is mediated through the ordinary-double-point Hodge blocks.

12.3. Zero-Excess Calibration

The finite-ODP model also supplies the zero-excess calibration for the theory. At each ordinary double point, the local Milnor fiber has the homotopy type of $S^3$. Hence

$${\tilde{H}}^3(F_{p_k};\mathbb{Z})\cong \mathbb{Z},{\tilde{H}}^m(F_{p_k};\mathbb{Z})=0(m\ne 3).$$

There is no local integral torsion in the ODP vanishing block. Therefore the local coefficient-change excess at the ODP node vanishes.

In matrix form, the finite-ODP local model has the shape

$$\begin{array}{cccc}& \mathrm{canonical}\mathrm{side}& \mathrm{variation}\mathrm{side}& \mathrm{return}\mathrm{side}\\ H/\mathbb{Q}& Cone\left(u_{\Sigma }\right)& Cone\left(v_{\Sigma }\right)& Cone\left(N\right)\\ {Exc}_{\mathrm{local}}& 0& 0& 0.\end{array}$$

This does not mean that the residual data are trivial. On the contrary, the finite-node residual package contains the interface $W_{\Sigma }^H$, the corrected extension $P_{\mathrm{var},\Sigma }^H$, and the nodewise classes ${ϵ}_k^H$. The point is that these are rational/MHM residual data, not hidden local torsion data.

Thus ordinary double points show that residuals and excess are independent axes. Residuals may be nontrivial even when local excess vanishes.

12.4. Relation to Saito–Kerr–Laza

The can–var–monodromy octahedron is not new. It belongs to Saito’s mixed-Hodge-module formalism and is used in the Kerr–Laza–Saito study of degenerations, Clemens–Schmid sequences, local invariant cycles, and phantom cohomology. This paper uses that framework as background.

The contribution here is the reduced finite-node interface

$$W_{\Sigma }^H$$

and the associated defect-resolution package. Kerr–Laza’s point-supported $W_{\sigma }$-terms and phantom cohomology motivate the comparison with the ODP-selected boundary quotient

$$R_{!,\ast }^{H,\mathrm{ODP}}.$$

The expected comparison

$$R_{!,\ast }^{H,\mathrm{ODP}}\cong W_{\Sigma }^H$$

is the finite-node boundary-interface comparison problem isolated in this paper.

This comparison is horizontal. It takes place inside the rational/MHM row of the residual–excess matrix. It should be distinguished from vertical excess, which concerns comparison across coefficient systems.

12.5. Relation to Diaz–Type Nonzero Excess

The zero-excess ODP model should be contrasted with torsion-bearing finite-coefficient constructions such as Diaz’s Enriques-product Bockstein mechanism. In that setting one has a finite-coefficient class

$${\Theta }_2={\pi }_1^{\ast }{\alpha }_1\cup {\pi }_2^{\ast }{\beta }_2\in H^3(S_1\times S_2,\mathbb{Z}/2\left(2\right)),$$

and its Bockstein image

$${\Delta }_2=\delta \left({\Theta }_2\right)\in H^4(S_1\times S_2,\mathbb{Z}\left(2\right)).$$

The class ${\Delta }_2$ is integral 2-torsion, so

$${\Delta }_2\otimes \mathbb{Q}=0.$$

Under Diaz’s survival criterion, ${\Delta }_2$ is non-algebraic. This is the nonzero-excess calibration. The key data lie in the kernel of rationalization. They are invisible to the rational/MHM row alone and must be recorded by the vertical excess axis.

The framework predicts that ordinary double point comparisons should remain zero-excess at the local integral level, whereas Coble–Enriques or ${\displaystyle \frac{1}{4}}(1,1)$-type torsion-bearing boundary packages should produce nonzero excess. This prediction is not needed for the finite-ODP theorem, but it explains why the ODP and Diaz/Coble mechanisms occupy complementary calibration regimes.

Thus the two examples play complementary roles:

$$\mathrm{ODP}\text{:}\mathrm{residual}\mathrm{nontrivial},\mathrm{local}\mathrm{excess}\mathrm{zero},$$

while

$$\mathrm{Diaz}\text{:}\mathrm{excess}\mathrm{nontrivial},\mathrm{rational}\mathrm{shadow}\mathrm{zero}.$$

Together they justify the residual–excess matrix as the correct organizing object for comparing rational defect resolutions with integral Hodge obstruction channels.

12.6. Next Directions

There are several natural continuations.

First, one should compute the local summands

$${Ext}_{MHM\left(X_0\right)}^1(W_k^H,I{C}_{X_0}^H)$$

in explicit geometric examples. This would turn the nodewise residual module from a structural invariant into a computable invariant.

Second, one should prove the boundary-interface comparison

$$R_{!,\ast }^{H,\mathrm{ODP}}\cong W_{\Sigma }^H$$

under precise finite-ODP hypotheses, or determine the correction terms when it fails.

Third, one should develop the integral or finite-coefficient version of the finite-node residual row whenever such data are available. Given an integral variation residual

$$P_{\mathrm{var},\Sigma }^{\mathbb{Z}}$$

and a comparison morphism

$$P_{\mathrm{var},\Sigma }^{\mathbb{Z}}\otimes \mathbb{Q}⟶P_{\mathrm{var},\Sigma }^H,$$

one obtains the variation excess

$$E_{\mathrm{var},\Sigma }=Cone\left(P_{\mathrm{var},\Sigma }^{\mathbb{Z}}\otimes \mathbb{Q}⟶P_{\mathrm{var},\Sigma }^H\right)[-1].$$

Analogous canonical and return excess objects should be studied.

Fourth, higher-order strata should be treated by replacing the finite direct sum of point Hodge lines

$$\underset{k}{⨁}{i}_{k\ast }{\mathbb{Q}}_{\left\{p_k\right\}}^H(-1)$$

with stratum-supported mixed Hodge modules

$$i_{S\ast }{W}_S^H.$$

In such settings, local systems, monodromy along strata, and incidence relations among strata may produce stratified residual categories rather than nodewise Ext sums.

Finally, torsion-bearing examples should be studied through the vertical excess axis. Diaz-type Bockstein constructions show that finite-coefficient classes can produce integral torsion classes killed by rationalization. These examples calibrate the nonzero-excess regime and point toward the integral Hodge applications of the residual–excess framework.

12.7. Final Summary

The main message of the paper is that defect data have two independent refinements.

First, a defect can be refined horizontally by choosing a factorization through an interface:

$$A\stackrel{u}{\to }B\stackrel{v}{\to }{A}^{†}.$$

The resulting residual triple records how the defect is realized.

Second, a defect can be refined vertically by comparing coefficient systems. The resulting excess object records what is lost under rationalization.

The finite-ODP Saito interface supplies the basic rational/MHM residual model. Diaz-type Bockstein classes supply the contrasting nonzero-excess model. The residual–excess matrix packages both phenomena:

$$\mathrm{horizontal}\mathrm{residuals}+\mathrm{vertical}\mathrm{excess}=\mathrm{structured}\mathrm{defect}\mathrm{data}.$$

This is the framework proposed here for connecting finite-node Saito gluing, boundary / phantom residuals, and integral Hodge obstruction channels.