Solver-Agnostic Convergence Certificates for DSGE Computation in Economics

1. Introduction

Economic models are typically defined by systems of equilibrium restrictions, laws of motion, and expectation operators. Solving such models—whether as decentralised equilibria or as planner problems—amounts to finding zeros of a nonlinear map (or fixed points of an induced operator) in a high-dimensional function space [1,2]. A large numerical toolbox has therefore developed for DSGE computation, including projection and collocation methods, time iteration, policy iteration, and Newton or quasi-Newton solvers [1,2,3,4]. Despite this maturity, solver performance claims remain difficult to compare across papers and codebases, because they are often made in different residual representations and different norms.

The practical issue is that “convergence” is a geometric statement. A residual trace depends on which equilibrium conditions are measured, how they are discretised, and how the resulting components are scaled and aggregated. Changing variable normalisation, equation scaling, or aggregation weights can change solver rankings without changing the underlying economic fixed point [1]. This motivates a disciplined measurement layer: before comparing solvers, one must freeze the residual geometry in which deviation from equilibrium is quantified.

This paper introduces such a frozen geometry through a DSFL (Deterministic Statistical Feedback Law) residual calculus. The central object is a quadratic residual of sameness $R={\parallel e\parallel }_W^2$, where e is a declared defect vector (equilibrium violations in a chosen representation) and $W\succ 0$ is a declared symmetric positive definite (SPD) ruler that aggregates, scales, and types those violations. Once $(e,W)$ are fixed, convergence becomes a typed statement in a single normed space: a numerical solver is evaluated by the defect dynamics it induces in the frozen norm, and solver comparisons become reproducible because all residual traces are measured in the same geometry [5,6].

1.1. Problem Statement: Solver Comparisons Are not Typed

A practical gap motivates the paper. DSGE solvers are commonly compared by wall-clock time, ad hoc Euler-error aggregations, unreported scaling conventions, or tolerance thresholds that depend on the norm used for stopping. But if two implementations use different residual definitions or different implicit rulers, then their “errors” are not commensurate objects. A mathematically honest comparison therefore requires a frozen record: one must declare (i) the defect representation e (which equilibrium conditions are measured and how) and (ii) the ruler W (which defines the norm in which contraction and convergence are claimed). This paper adopts that discipline as the core typing requirement.

1.2. Scope and Standing Assumptions

Scope.

We do not propose new DSGE algorithms. Instead, we provide a solver-agnostic certificate layer that attaches to any solver once the defect representation and ruler are declared. The focus is on local (or tail) convergence diagnostics in a frozen norm, including forced variants that tolerate numerical noise and inexact solves. We illustrate the framework on a standard small New Keynesian (NK) model solved by several solver families.

Typing data.

The frozen record consists of: a defect map $F:\mathcal{Z}\to \mathcal{E}$ with defect $e\left(z\right)=F\left(z\right)$, a neutral convention (optional projection to a neutral-free subspace), and an SPD ruler W on $\mathcal{E}$ inducing ${\parallel e\parallel }_W$ and $R={\parallel e\parallel }_W^2$. All claims in the paper are explicitly typed by this data.

Regularity.

For the local linearisation arguments and Newton-type certificates we assume Fréchet differentiability of F in a neighbourhood of the solution and a local chart/identifiability condition (neutral-free invertibility), as recorded in Assumption A1. These assumptions are standard in nonlinear operator equations and root-finding theory [3,7,8,9].

1.3. What Is Known and the Gap

The mathematical language for contraction and decay in fixed normed geometries is classical in semigroup theory and numerical analysis [5,6]. The DSGE numerical toolbox (projection/collocation, time iteration, policy iteration, Newton and quasi-Newton methods) is also well developed [1,2,3,4]. What is less standard is a single measurement geometry that makes solver performance claims comparable across implementations and exposes transient pathologies caused by non-normal update operators. Eigenvalue-based diagnostics can miss substantial finite-iteration amplification when updates are non-normal, even when they are asymptotically stable [10]. This paper supplies a frozen-geometry certificate layer that targets precisely these comparability and diagnosis failures.

1.4. Approach and Method Overview

Our approach has three steps.

(i)

Frozen residual geometry.

We fix the equilibrium defect map and declare an SPD ruler W, producing the quadratic residual $R\left(z\right)={\parallel F\left(z\right)\parallel }_W^2$. Neutral conventions (redundant equations, normalisations) are handled explicitly by projection or augmentation.

(ii)

Solvers as induced defect dynamics.

Each solver step $z_{k+1}=S\left(z_k\right)$ induces defect dynamics $e_{k+1}=T\left(e_k\right)+{\xi }_k$ on the frozen defect space. Contraction in the frozen norm yields explicit residual envelopes; forcing terms ${\xi }_k$ produce robust tail envelopes and principled stopping rules.

(iii)

Spectral/Gram certificates and non-normal diagnostics.

For linear or linearised updates, the contraction factor ${\parallel T\parallel }_{W\to W}$ is encoded by a Gram operator $C_W\left(T\right)$. This yields a solver-agnostic rate score ${\alpha }_{\mathrm{alg}}=-log{\parallel T\parallel }_{W\to W}$ and companion diagnostics that detect transient amplification when updates are non-normal [10,11].

1.5. Main Contributions and Results

We provide four contributions.

C1

Typed residual geometry for equilibrium computation. We formalise a frozen record for DSGE equilibrium problems: defect map, neutral convention, and SPD ruler W that defines a single quadratic residual used consistently across solvers.

C2

Contraction and forced-contraction certificates. We prove that frozen-norm contraction of the induced defect update implies explicit geometric residual envelopes, and we extend this to forced envelopes that tolerate numerical noise and inexact solves, yielding principled stopping rules [5,6,9].

C3

Gram-spectrum solver certificates. For linear or linearised updates we introduce the Gram operator $C_W\left(T\right)$ and show that its top eigenvalue equals ${\parallel T\parallel }_{W\to W}^2$, providing a computable, solver-agnostic contraction certificate and a comparable rate score ${\alpha }_{\mathrm{alg}}=-log{\parallel T\parallel }_{W\to W}$ [11].

C4

Non-normal transient diagnostics. We show how the same frozen-geometry machinery separates tail contraction from transient amplification, explaining why eigenvalue-only diagnostics can fail to predict practical behaviour in non-normal regimes [10].

1.6. Evidence and Instantiations

We instantiate the framework on a standard small New Keynesian model with a fixed discretisation and compare three solver families (time iteration, policy iteration, Newton/quasi-Newton) by reporting residual traces in the same frozen ruler, certificate rates, and transient amplification indicators. The purpose is a computational audit: to demonstrate that typed certificate numbers predict observed convergence rankings and remain interpretable under solver-dependent transients.

1.7. Roadmap (How to Read the Paper)

The paper is organised as a certificate pipeline: we first freeze the measurement geometry, then identify the induced defect dynamics of solvers, then prove contraction/forcing certificates and spectral diagnostics, and finally instantiate the framework on a standard DSGE testbed.

Part I: Frozen record (typing data).

Section 2 fixes the equilibrium defect map $F:\mathcal{Z}\to \mathcal{E}$, the defect $e\left(z\right)=F\left(z\right)$, and the SPD ruler $W\succ 0$ that defines the quadratic residual $R\left(z\right)={\parallel F\left(z\right)\parallel }_W^2$. Optional neutral conventions (redundant equations, normalisations) are handled by an explicit projection or augmentation. This section supplies the core methodological requirement: solver claims are meaningful only after the residual representation and its norm have been declared and frozen.

Part II: Solvers as induced defect dynamics.

Section 3 rewrites standard DSGE solvers as update maps $z_{k+1}=S\left(z_k\right)$ and expresses their performance in the frozen geometry via the induced defect evolution $e_{k+1}=T\left(e_k\right)$ (exactly in linear settings and locally after linearisation). The output is the object to be certified: the induced defect update operator (or its local Jacobian) in the frozen W-norm.

Part III: Certificate spine (proved theorems).

Section 4 proves the analytical results used throughout: (i) frozen-norm contraction implies explicit residual envelopes; (ii) forcing/inexactness yields tail-robust envelopes and principled stopping rules; (iii) the Gram operator $C_W\left(T\right)$ encodes ${\parallel T\parallel }_{W\to W}$ and provides a solver-agnostic spectral certificate; and (iv) non-normality is diagnosed through transient amplification indicators that eigenvalues alone can miss. These results are theorem-level and are reusable: once a solver is expressed as a defect update in the frozen geometry, the envelopes follow.

Part IV: Comparable solver metrics.

Section 5 defines the solver-agnostic comparison score ${\alpha }_{\mathrm{alg}}=-log{\parallel T\parallel }_{W\to W}$ and clarifies what it measures (typed contraction per iteration in the frozen geometry) and what it does not measure (welfare, wall-clock time, or global convergence). Companion diagnostics are recorded for transient amplification.

Part V: Numerical instantiation.

Section 6 instantiates the framework on a small New Keynesian model with a fixed discretisation. The defect evaluator, ruler W, and initialisation are held fixed while solver families are varied (time iteration, policy iteration, Newton/quasi-Newton). The section reports residual traces in the frozen norm together with certificate numbers (rate and transient indicators) to demonstrate solver-comparable, reproducible diagnostics.

Part VI: Discussion and outlook.

Section 8 discusses what the certificate layer adds to DSGE computation (reproducibility by typing, diagnosis of non-normal transients, and stopping rules under forcing), and records explicit limitations (locality, ruler design, and high-dimensional approximation issues). The conclusion summarises the certificate pipeline and lists theorem-shaped upgrades (welfare-aligned rulers, uniform-in-refinement stability, and certified forcing budgets).

2. Frozen Residual Geometry for DSGE Equilibrium Problems

This section fixes, once and for all, the representation of equilibrium, the residual (defect) map, and the norm in which deviation from equilibrium is measured. The methodological point is classical but often left implicit: solver statements such as “converges”, “is accurate”, or “is faster” are not invariant facts — they are typed claims that depend on (i) what is called the unknown and (ii) how violation of equilibrium is aggregated and scaled. Accordingly, we treat the residual map and the ruler as theorem data. This mirrors the role of a chosen norm in semigroup stability and numerical analysis: convergence rates, error bounds, and conditioning are only meaningful relative to a declared geometry [5,6,12,13].

2.1. Equilibrium as a Defect Equation

Let $(\mathcal{Z},\parallel ·{\parallel }_{\mathcal{Z}})$ be a Banach space of candidate representations for a DSGE equilibrium. The choice of $\mathcal{Z}$ is solver-dependent: for projection and perturbation methods, z may be a vector of coefficients for policy functions; for time-iteration and endogenous-grid methods, z may encode a grid representation of decision rules; for stacked-residual approaches, z may already be a vector of discretised equilibrium conditions [1,14,15,16,17].

Let $\mathcal{E}$ be a real Hilbert space (finite-dimensional in most computational implementations) equipped with inner product $\langle ·,·\rangle$. We represent the equilibrium conditions as a (generally nonlinear) defect map

$$F:\mathcal{Z}\to \mathcal{E},\mathrm{and}\mathrm{the}\mathrm{equilibrium}\mathrm{problem}\mathrm{is}F\left(z\right)=0.$$

(1)

The vector $F\left(z\right)$ stacks all equilibrium restrictions in the chosen representation: Euler equations, pricing relations, implementability constraints, policy rules, resource constraints, and (if present) auxiliary conditions used to pin down expectations or enforce laws of motion. For any candidate $z\in \mathcal{Z}$ we define the defect

$$e\left(z\right):=F\left(z\right)\in \mathcal{E},$$

(2)

which measures deviation from equilibrium in the frozen representation.

Differential structure.

Many convergence theorems and certificates require local smoothness. We therefore record a standing regularity class.

Assumption A1 

(Local differentiability of the defect map). There is an open set $\mathcal{U}\subseteq \mathcal{Z}$ such that F is Fréchet differentiable on $\mathcal{U}$, with derivative $DF\left(z\right)\in \mathcal{L}(\mathcal{Z},\mathcal{E})$, and $DF$ is locally Lipschitz on $\mathcal{U}$.

Assumption A1 is standard in Newton–Kantorovich analyses and in local conditioning statements for nonlinear systems [3,7,8,9]. In many DSGE constructions F is composed of smooth primitives (utility, production, transition densities) and smooth expectation operators under dominated convergence; nondifferentiabilities can be handled by smoothing or by switching to semismooth frameworks, but we do not pursue that here [18].

2.2. Residual and Ruler

To compare solvers, we must choose how to aggregate the equilibrium violations in $e\left(z\right)$. We therefore fix a symmetric positive definite (SPD) linear operator

$$W:\mathcal{E}\to \mathcal{E},W=W^{\top },W\succ 0,$$

(3)

and define the associated weighted inner product and norm

$${\langle u,v\rangle }_W:=\langle u,Wv\rangle ,{\parallel u\parallel }_W:=\sqrt{{\langle u,u\rangle }_W}.$$

(4)

The quadratic residual is

$$R\left(z\right):={\parallel e\left(z\right)\parallel }_W^2=\langle F\left(z\right),WF\left(z\right)\rangle .$$

(5)

This R is the single scalar object to which we will attach accuracy statements and solver certificates. The role of W is analogous to preconditioning: it fixes which components of the equilibrium system matter more and how units are reconciled [3,4].

Interpretation of W (typical choices).

(i)

Steady-state scaling. Let D be a diagonal scaling matrix built from steady-state magnitudes of each equation. Then $W=D^{-2}$ yields a dimensionless residual, mitigating dominance by large-unit conditions [1].

(ii)

Variance weighting. If $e\left(z\right)$ stacks moments or forecast errors, W can be chosen as an inverse covariance proxy (or block-diagonal approximation), turning R into a generalized least squares objective [19].

(iii)

Welfare-based weighting. If some equations correspond to implementability or planner optimality, W can encode marginal welfare weights or Lagrange-multiplier magnitudes, aligning residual decrease with welfare loss to first order [1].

Frozen-Geometry Principle

Throughout, W is treated as theorem data. All convergence claims are to be read as statements about the decay of $R\left(z_k\right)$ in this fixed norm. Changing W changes what is being measured, and therefore changes what “convergence” means.

2.2.1. Norm Equivalence Bookkeeping

If a solver is analysed in one norm but reported in another, explicit equivalence constants are required. In finite dimensions, any two norms are equivalent, but constants matter for rates and stopping rules [11,12]. For SPD W, we have the sharp comparison with the ambient norm $\parallel ·\parallel$ induced by $\langle ·,·\rangle$:

Lemma 1 

(Weighted norm equivalence). Let $\mathcal{E}\simeq {\mathbb{R}}^m$ with inner product $\langle u,v\rangle =u^{\top }v$ and let $W\succ 0$ with eigenvalues $0<{\lambda }_{min}\left(W\right)\le {\lambda }_{max}\left(W\right)$. Then for all $u\in \mathcal{E}$,

$${\lambda }_{min}{\left(W\right)\parallel u\parallel }^2\le {\parallel u\parallel }_W^2\le {\lambda }_{max}\left(W\right){\parallel u\parallel }^2,$$

(6)

and hence ${\parallel u\parallel }_W\le \sqrt{{\lambda }_{max}\left(W\right)}\parallel u\parallel$ and $\parallel u\parallel \le {\lambda }_{min}{\left(W\right)}^{-1/2}{\parallel u\parallel }_W.$

Proof. 

Diagonalise $W=Q^{\top }\Lambda Q$ with orthogonal Q and $\Lambda =\mathrm{diag}\left({\lambda }_i\right)$. Then ${\parallel u\parallel }_W^2=u^{\top }Wu={\left(Qu\right)}^{\top }\Lambda \left(Qu\right)={\sum }_i{\lambda }_i{\left(Qu\right)}_i^2$. Bounding ${\lambda }_i$ by ${\lambda }_{min}\left(W\right)$ and ${\lambda }_{max}\left(W\right)$ gives (6). □

Lemma 1 is the precise mechanism by which we will allow norm changes: they are admissible only through declared constants, never silently.

2.2.2. Residual Geometry and First-Order Stationarity

Although the equilibrium problem is $F\left(z\right)=0$, many solvers can be cast as residual-reduction schemes. We therefore record the derivative of R and the corresponding normal equations.

Lemma 2 

(Gradient of the quadratic residual). Assume $\mathcal{Z}={\mathbb{R}}^n$ with Euclidean inner product and F is differentiable at z. Let $J\left(z\right):=DF\left(z\right)\in {\mathbb{R}}^{m\times n}$. Then

$$\nabla R\left(z\right)=2J{\left(z\right)}^{\top }WF\left(z\right).$$

(7)

Moreover, any stationary point of R satisfies the weighted normal equation

$$J{\left(z\right)}^{\top }WF\left(z\right)=0.$$

(8)

Proof. 

Differentiate $R\left(z\right)=F{\left(z\right)}^{\top }WF\left(z\right)$ and use the chain rule: $dR=2{\left(dF\right)}^{\top }WF=2{\left(Jdz\right)}^{\top }WF=\langle 2J^{\top }WF,dz\rangle$. □

Equation (8) clarifies the difference between solving $F\left(z\right)=0$ and minimising$R\left(z\right)$: local minima of R can exist with nonzero defect if $F\left(z\right)$ lies in a near-null direction of $J{\left(z\right)}^{\top }W$. This is precisely why the identifiability and neutral-direction discussion in subsec:neutrals matters, and why we keep W explicit (it changes both conditioning and stationarity).

2.3. Neutral Directions and Identifiability (Optional)

Some DSGE representations contain neutral directions: changes in z that leave the economic allocation unchanged or change it only by a normalisation (e.g. the numéraire), or redundancy in stacked equilibrium conditions (e.g. one market-clearing equation is implied by the others under Walras’ law). If neutrals are not removed, a solver may appear to “stall” because it is moving along an equivalence class rather than reducing economically meaningful defect.

2.3.1. Quotient Formulation

Let $\mathcal{N}\subseteq \mathcal{E}$ be a closed subspace of neutral defect directions (redundant equations), and let ${\Pi }_{\perp }:\mathcal{E}\to \mathcal{E}$ be the orthogonal projector onto ${\mathcal{N}}^{\perp }$. Define the effective defect

$$e_{\perp }\left(z\right):={\Pi }_{\perp }F\left(z\right),$$

(9)

and the corresponding neutral-free residual

$$R_{\perp }\left(z\right):={\parallel e_{\perp }\left(z\right)\parallel }_{W_{\perp }}^2,W_{\perp }:={\Pi }_{\perp }^{\top }W{\Pi }_{\perp }{|}_{{\mathcal{N}}^{\perp }}\succ 0.$$

(10)

All contraction and certification statements can then be stated for $R_{\perp }$ without ambiguity. This is the Hilbert-space way to “drop” redundant equations while keeping a consistent ruler [12].

2.3.2. Parameter Non-Identification in z

Neutrality can also arise in the unknown representation: distinct z encode the same allocation due to gauge-like choices (e.g. scaling/numéraire, normalisations of value functions, or redundant coefficients in an approximation basis). A clean mathematical treatment is to declare an equivalence relation $z\sim z^{\prime }$ (same economic object) and work on the quotient $\mathcal{Z}/\sim$, but in computation it is more common to fix a gauge (normalisation) or to add a pinning condition. Formally, one may add a gauge-fixing map $G:\mathcal{Z}\to {\mathbb{R}}^q$ and solve the augmented system

$$\tilde{F}\left(z\right):=\left[\begin{array}{c}F\left(z\right)\\ G\left(z\right)\end{array}\right]=0,$$

(11)

with an augmented ruler $\tilde{W}=\mathrm{diag}(W,W_G)$. This converts non-identification into an explicit design choice (the gauge map and its weight), making solver comparisons well-typed.

Why the main results assume neutral-free geometry.

To keep the exposition focused, the core results of the paper are stated under the simplifying assumption that the declared representation has already removed neutrals (either by projection or by augmentation). When neutrals are present, every statement below is read with F replaced by $e_{\perp }$ (or by $\tilde{F}$), and with the corresponding ruler $W_{\perp }$ (or $\tilde{W}$). Nothing essential changes; what changes is that one must explicitly state the projection/gauge as part of the frozen record.

Forward reference (use in solver theorems).

Later, when we state solver comparison results, we will treat each algorithm as producing a sequence $\left(z_k\right)$ and we will compare them through the same frozen scalar $R\left(z_k\right)$ (or $R_{\perp }\left(z_k\right)$). This permits theorems of the following general form: under local regularity and identifiability, a solver decreases the frozen residual at a provable rate, and stopping rules become meaningful because the ruler has been declared in advance [4,8,9].

3. DSGE Solvers as Update Maps

This section rewrites standard DSGE solution methods in a single algebraic language: update maps acting on a declared defect (residual) representation. The objective is not to propose new algorithms, but to identify the operator (or local Jacobian) that each method induces, because that operator is what determines contraction, non-normal transients, and solver comparability in a frozen ruler. The presentation is deliberately typed: we distinguish the economic model data (equilibrium conditions) from the numerical method (update rule) and from the measurement geometry (the SPD ruler W). This is the discipline that allows solver comparisons to be meaningful across codes and implementations [1,2,5,6].

3.1. Equilibrium Problems as Operator Equations

A DSGE model solution can be posed as an operator equation on a Banach/Hilbert space of candidate objects (e.g. policy functions, decision rules, or stacked time-t objects such as $(x_t,y_t)$). Let $\mathcal{Z}$ be the model’s solution-object space and let

$$F:\mathcal{Z}\to \mathcal{E}$$

(12)

be the equilibrium operator, mapping a candidate solution object $z\in \mathcal{Z}$ to a defect vector $e=F\left(z\right)$ in a defect space $\mathcal{E}$. Typical choices include Euler-equation defects, intratemporal first-order conditions, implementability constraints, and resource constraints; for log-linear models F is linear, while for global nonlinear methods F is nonlinear and evaluated pointwise on a grid or basis expansion [1,2].

Defect space and frozen ruler.

We assume throughout that the defect space $\mathcal{E}$ is a real Hilbert space with inner product $\langle ·,·\rangle$ and that the analyst declares a symmetric positive definite ruler W on $\mathcal{E}$. The induced norm is

$${\parallel e\parallel }_W:=\sqrt{\langle e,We\rangle },R\left(e\right):={\parallel e\parallel }_W^2.$$

(13)

This is the frozen measurement geometry: all contraction and rate statements in this section are in ${\parallel ·\parallel }_W$. The choice of W is part of the theorem data; changing W changes what “fast” or “stable” means [5,6].

Equilibrium.

An equilibrium solution is a zero of the equilibrium operator:

$$z^{*}\in \mathcal{Z}\mathrm{s}.\mathrm{t}.F\left(z^{*}\right)=0.$$

(14)

We emphasise that solvers act on z, but our certificates (later) will be expressed in terms of the induced dynamics of the defect $e=F\left(z\right)$.

3.2. Solver Steps as Maps on Defect Space

Let $S:\mathcal{Z}\to \mathcal{Z}$ denote a single solver step:

$$z_{k+1}=S\left(z_k\right).$$

(15)

This induces a defect update map $T:\mathcal{E}\to \mathcal{E}$ via the composition

$$e_{k+1}=F\left(z_{k+1}\right)=F\left(S\left(z_k\right)\right)=:(T\circ F)\left(z_k\right).$$

(16)

In general T depends on the representation of $\mathcal{Z}$ and on how F is evaluated, but once the frozen record (defect representation and ruler) is fixed, the performance question becomes a question about the dynamics

$$e_{k+1}=T\left(e_k\right)$$

(17)

either exactly (in linear problems where F and S reduce to affine maps) or locally (after linearisation around $z^{*}$).

Why defect maps (rather than z maps).

Two implementations can use different parameterisations of z (different bases, different normalisations, different state scalings) while representing the same economics. Defects provide a canonical comparison target: if the defect representation is fixed, then all solvers are judged in the same space $\mathcal{E}$ with the same ruler W. This is the typed notion of comparability underlying the certificate layer.

3.3. Local linearisation and Jacobians

To expose solver dynamics near an equilibrium, we linearise the defect update. Assume F and S are Fréchet differentiable in a neighbourhood of $z^{*}$.

Definition 1 

(Defect Jacobian of a solver). Let $z^{*}$ satisfy $F\left(z^{*}\right)=0$ and define $z^{*}$ as a fixed point of the solver step (i.e. $S\left(z^{*}\right)=z^{*}$). Thedefect Jacobianinduced by S at $z^{*}$ is the bounded linear operator $J:\mathcal{E}\to \mathcal{E}$ satisfying

$$e_{k+1}=T\left(e_k\right)=J{e}_k+o(\parallel e_k\parallel ),J:=D(F\circ S){|}_{z^{*}}\circ {\left(DF{|}_{z^{*}}\right)}^{-1}$$

(18)

whenever ${DF|}_{z^{*}}$ is invertible as a map $\mathcal{Z}\to \mathcal{E}$ (local chart condition).

The slightly abstract form (18) separates three components: (i) the model map F, (ii) the solver step S, and (iii) the coordinate choice linking $\mathcal{Z}$ and $\mathcal{E}$. In many DSGE applications one works directly with e-based updates, in which case ${J=DT|}_{e^{*}}$ in the usual sense.

Remark 1 

(When ${DF|}_{z^{*}}$ is not invertible). In models with neutral directions (e.g. indeterminacy, redundant equations, normalisation conventions), ${DF|}_{z^{*}}$ may fail to be invertible. The correct typed object is then the Jacobian on the quotient or on the projected subspace (“neutral-free” defect space), obtained by choosing neutrals and a projector ${\Pi }_{\perp }$. We postpone this neutral discipline to the frozen-record section; in this section we work in the simplest setting where the local chart is available.

3.4. Contraction in a Frozen SPD Ruler

All solver comparisons here are statements in the frozen W-geometry. For a bounded linear map $A:\mathcal{E}\to \mathcal{E}$, define the induced operator norm

$${\parallel A\parallel }_W:=\underset{e\ne 0}{sup}{\displaystyle \frac{{\parallel Ae\parallel }_W}{{\parallel e\parallel }_W}}=\parallel W^{1/2}A{W}^{-1/2}{\parallel }_2,$$

(19)

where ${\parallel ·\parallel }_2$ is the Euclidean/operator norm in coordinates. The equivalence (19) is standard for SPD-weighted norms.

Theorem 1 

(Local W-contraction implies a certified residual envelope). Assume that near $e=0$ the defect update satisfies

$${\parallel T\left(e\right)\parallel }_W\le q{\parallel e\parallel }_W\mathrm{for}\mathrm{all}{\parallel e\parallel }_W\le r$$

(20)

for some $r>0$ and some $q\in [0,1)$. Then for all iterates that remain in the ball $\parallel e_k{\parallel }_W\le r$ we have the geometric decay

$$R_k:={\parallel e_k\parallel }_W^2\le q^{2k}{R}_0.$$

(21)

If moreover T admits an additive perturbation model (tolerances, numerical noise)

$${\parallel T\left(e\right)\parallel }_W\le q{\parallel e\parallel }_W+{ϵ}_k,{ϵ}_k\ge 0,$$

(22)

then

$$\parallel e_k{\parallel }_W\le q^k{\parallel e_0\parallel }_W+\sum _{j=0}^{k-1}{q}^{k-1-j}{ϵ}_j,R_k\le {\left(q^k{\parallel e_0\parallel }_W+\sum _{j=0}^{k-1}{q}^{k-1-j}{ϵ}_j\right)}^2.$$

(23)

Proof. 

Under (20), $\parallel e_{k+1}{\parallel }_W\le q{\parallel e_k\parallel }_W$, hence by induction $\parallel e_k{\parallel }_W\le q^k{\parallel e_0\parallel }_W$ and (21) follows by squaring. For (22), iterate the inequality: $\parallel e_k{\parallel }_W\le q\parallel e_{k-1}{\parallel }_W+{ϵ}_{k-1}\le \dots \le q^k{\parallel e_0\parallel }_W+{\sum }_{j=0}^{k-1}{q}^{k-1-j}{ϵ}_j$. □

Corollary 1 

(Linear case: Jacobian bound is sufficient). If T is linear, $T\left(e\right)=Je$, and ${\parallel J\parallel }_W<1$, then (21) holds globally with $q={\parallel J\parallel }_W$.

Proof. 

For linear $T\left(e\right)=Je$, the Lipschitz property (20) holds with $q={\parallel J\parallel }_W$ for all e by definition of the operator norm. □

Interpretation.

Theorem 1 formalises the point that solver performance is, at root, a statement about contractivity of the induced update in the declared ruler. Different solver families correspond to different induced J (and different higher-order terms away from the equilibrium), hence different q in (20). Later sections will turn this into computable certificates.

3.5. Canonical Solver Families and Their Induced Defect Maps

We now record the induced defect maps for standard solver families. The purpose is not exhaustive algorithmics, but a clean identification of the T (or J) each method implies.

3.5.1. Fixed-Point (Time) Iteration

Many global methods express equilibrium as a fixed point $z=\Phi \left(z\right)$ (e.g. policy function operators). Time iteration applies the forward map:

$$z_{k+1}=\Phi \left(z_k\right).$$

(24)

A common defect is $e\left(z\right):=z-\Phi \left(z\right)$, so $F\left(z\right)=z-\Phi \left(z\right)$. Then the induced defect map is

$$e_{k+1}=F\left(z_{k+1}\right)=\Phi \left(z_k\right)-\Phi (\Phi \left(z_k\right))=:T_{\mathrm{TI}}\left(e_k\right),$$

(25)

and locally (if $\Phi$ is differentiable) the Jacobian at a fixed point $z^{*}=\Phi \left(z^{*}\right)$ is

$$J_{\mathrm{TI}}=D(F\circ \Phi ){|}_{z^{*}}\circ {\left(DF{|}_{z^{*}}\right)}^{-1}=(I-D\Phi \left(z^{*}\right))D\Phi \left(z^{*}\right){(I-D\Phi \left(z^{*}\right))}^{-1},$$

(26)

whenever $I-D\Phi \left(z^{*}\right)$ is invertible. In the common case where one tracks the state error $z_k-z^{*}$ rather than the defect $e_k$, the local dynamics reduce to $z_{k+1}-z^{*}\approx D\Phi \left(z^{*}\right)(z_k-z^{*})$. Either way, the spectral and non-normal properties of $D\Phi \left(z^{*}\right)$ determine local speed [1,2].

What changes across implementations.

Two codes can implement the same $\Phi$ but scale variables differently, or aggregate equation errors differently. That changes the effective ruler W on the defect space and can change $\parallel J_{\mathrm{TI}}{\parallel }_W$ even when the economics are the same. This is exactly the motivation for freezing W before comparing solvers.

3.5.2. Policy Iteration / Howard Improvement (Partial Inversion)

Policy iteration (Howard improvement) alternates between policy evaluation (solve linear(ised) conditions under a fixed policy guess) and policy improvement (update the policy by re-optimisation under the implied value/marginals). In deterministic dynamic programming, policy iteration is known to accelerate value iteration by effectively applying a partial inverse of $(I-\beta P^{\pi })$ on the Bellman residual; analogous ideas appear in DSGE policy-function solvers when intratemporal blocks can be solved (exactly or approximately) conditional on intertemporal objects [1,20].

Abstractly, write the defect as $F\left(z\right)=0$ and split variables as $z=(u,v)$ where u is the block to be “evaluated” and v the block to be “improved”. A single policy-iteration step can be written as

$$u_{k+1}=\Psi \left(v_k\right),v_{k+1}={\rm Y}(u_{k+1},v_k),$$

(27)

where $\Psi$ represents the evaluation solve (often a linear system solve or a conditional expectation solve) and ${\rm Y}$ the improvement map. The induced defect Jacobian takes a block form that includes an explicit solve (inverse) in the u-block, which is precisely why policy iteration can contract faster than a pure forward iteration. Because the exact block formulas depend on the model representation (Euler equation form, state reduction, basis choice), we record the general principle:

Policy iteration replaces part of the forward map by a (possibly approximate) inverse of the evaluation operator, hence the induced defect Jacobian is a preconditioned map with potentially much smaller ${\parallel J\parallel }_W$ than the forward iteration.

This preconditioning viewpoint will align policy iteration with Newton-type methods below.

3.5.3. Newton and Quasi-Newton Steps as Preconditioned Defect Updates

Suppose the equilibrium is posed directly as a root-finding problem $F\left(z\right)=0$ with F differentiable. Newton’s method updates

$$z_{k+1}=z_k-{\left[DF\left(z_k\right)\right]}^{-1}F\left(z_k\right).$$

(28)

To see the induced defect update, linearise around $z^{*}$ where $F\left(z^{*}\right)=0$ and $DF\left(z^{*}\right)$ is invertible. Let $\delta z_k:=z_k-z^{*}$. Then $F\left(z_k\right)=DF\left(z^{*}\right)\delta z_k+O(\parallel \delta z_k{\parallel }^2)$ and

$$\delta z_{k+1}=\delta z_k-{\left[DF\left(z^{*}\right)\right]}^{-1}DF\left(z^{*}\right)\delta z_k+O(\parallel \delta z_k{\parallel }^2)=O(\parallel \delta z_k{\parallel }^2),$$

so Newton is locally quadratic in state error.

In defect coordinates, the update is even simpler: to first order,

$$e_{k+1}=F\left(z_{k+1}\right)=0+O(\parallel e_k{\parallel }^2),$$

(29)

so the defect Jacobian is $J_{\mathrm{Newton}}=0$ at the solution. Thus Newton’s method is, in the frozen geometry, infinitesimally contractive at the fixed point.

Quasi-Newton replaces ${\left[DF\left(z_k\right)\right]}^{-1}$ by an approximate inverse $B_k$:

$$z_{k+1}=z_k-B_{k}F\left(z_k\right).$$

(30)

Linearising at $z^{*}$ gives the local state-error iteration

$$\delta z_{k+1}=(I-B_{*}DF\left(z^{*}\right))\delta z_k+O(\parallel \delta z_k{\parallel }^2),$$

(31)

and the induced defect Jacobian becomes

$$J_{\mathrm{qN}}\approx DF\left(z^{*}\right)(I-B_{*}DF\left(z^{*}\right)){\left(DF\left(z^{*}\right)\right)}^{-1}=I-DF\left(z^{*}\right)B_{*}.$$

(32)

The message is clear: quasi-Newton is a preconditioned defect iteration with defect map approximately $e_{k+1}\approx (I-DF\left(z^{*}\right)B_{*})e_k$. Hence $\parallel J_{\mathrm{qN}}{\parallel }_W$ is small precisely when the preconditioner $B_{*}$ is a good right-inverse in the frozen geometry. This viewpoint is standard in numerical analysis [3,4] and will be used later to build a posteriori contraction checkers.

3.5.4. Projection and Galerkin Solvers (Basis-Induced Maps)

Global DSGE solvers often restrict the solution object to a finite-dimensional approximation space ${\mathcal{Z}}_K\subset \mathcal{Z}$ (polynomials, splines, Smolyak grids, neural bases) and impose equilibrium by collocation or Galerkin conditions [1]. In that case the solver step is a composition of (i) projection to ${\mathcal{Z}}_K$, (ii) a finite-dimensional update (Newton, time iteration, etc.), and (iii) lifting back to $\mathcal{Z}$. The induced defect map is then a compressed update on the image of $F\left({\mathcal{Z}}_K\right)$. This is where refinement ladders enter naturally: increasing K changes the representation, and any meaningful convergence claim must be made in a geometry that does not move with K. We record this principle now and formalise it later.

3.6. A Theorem-Shaped Summary: Solver Families Differ by Induced J

The preceding subsections can be condensed into a single theorem-shaped statement emphasising what we will certify.

Theorem 2 

(All standard solver families induce a local defect update operator). Let $F:\mathcal{Z}\to \mathcal{E}$ encode the DSGE equilibrium conditions and let $S:\mathcal{Z}\to \mathcal{Z}$ be a solver step (time iteration, policy iteration, Newton/quasi-Newton, or a projection-based variant). Assume F and S are Fréchet differentiable at a solution $z^{*}$ with $F\left(z^{*}\right)=0$ and $S\left(z^{*}\right)=z^{*}$, and assume a local chart condition (invertibility on the neutral-free subspace). Then there exists a bounded linear operator $J:\mathcal{E}\to \mathcal{E}$ such that, for $e_k$ sufficiently small,

$$e_{k+1}=T\left(e_k\right)=J{e}_k+o(\parallel e_k\parallel ).$$

(33)

Moreover:

(i)

For forward (time) iteration, J is (conjugate to) the Jacobian of the forward policy operator.

(ii)

For policy iteration, J contains a partial inverse (evaluation solve) and is therefore a preconditioned forward map.

(iii)

For Newton, $J=0$ at the solution (quadratic local defect decay).

(iv)

For quasi-Newton, $J\approx I-DF\left(z^{*}\right)B_{*}$ for a preconditioner $B_{*}$.

Proof. 

The existence of a local linearisation (33) follows from Fréchet differentiability of $T=F\circ S$ in defect coordinates (or from the chain rule in z-coordinates under the chart condition), giving $T\left(e\right)=T\left(0\right)+DT\left(0\right)e+o(\parallel e\parallel )$ with $T\left(0\right)=0$ and $J=DT\left(0\right)$. The method-specific characterisations are immediate from the standard first-order expansions of the corresponding S (Section 3.5.1, Section 3.5.2 and Section 3.5.3). □

Why this matters for the next section.

Theorem 2 is the bridge from “which solver do you use?” to “what contraction do you induce in the frozen ruler?”. Once the induced J is identified (or bounded), the certificate layer reduces performance claims to inequalities involving ${\parallel J\parallel }_W$ and its non-normal structure. That is where Gram-operator certificates and a posteriori checkers enter.

3.7. Pointers to Implementation Practice (Typed Diagnostics)

Two practical consequences follow immediately from the frozen-geometry viewpoint.

(1) Residual logging must be W-typed.

If a code logs “Euler errors” but rescales equations or aggregates them differently across runs, then the logged sequence is not comparable. A meaningful solver audit must log $e_k$ (or sufficient statistics to reconstruct it) and compute $R_k={\parallel e_k\parallel }_W^2$ in the same ruler across solvers.

(2) Jacobian estimation can be certificate-driven.

Even when J is not formed explicitly, one can estimate bounds on ${\parallel J\parallel }_W$ from Jacobian blocks, from finite-difference probes, or from iterate data. These become the a posteriori “checkers” later; the present section identifies what object the checker must approximate: the induced defect update operator in the fro

4. Contraction Certificates in a Frozen Norm

This section provides the analytical spine needed for solver-agnostic certification. The guiding idea is simple: once the residual geometry $(\mathcal{E},{\langle ·,·\rangle }_W)$ is frozen, a solver update induces an operator on defects, and contractivity in the frozen norm yields explicit, checkable convergence envelopes. We develop (i) deterministic contraction certificates, (ii) forcing-robust envelopes that tolerate numerical noise and finite-precision stopping, and (iii) spectral/Gram diagnostics that turn solver behaviour into comparable numbers. The results are standard in numerical analysis and operator theory, but we state them in the frozen-geometry language so that they can be used as reusable certificate components [4,8,9,10,12].

4.1. Setup: Defect Dynamics Induced by a Solver

Let $F:\mathcal{Z}\to \mathcal{E}$ be the defect map and $e\left(z\right)=F\left(z\right)$ the defect as in sec:frozen-geometry. Consider an iterative solver producing iterates ${\left(z_k\right)}_{k\ge 0}$ via

$$z_{k+1}=\mathcal{A}\left(z_k\right),$$

(34)

where $\mathcal{A}$ is the algorithm map (Newton, quasi-Newton, time-iteration, projection, policy iteration, etc.). The associated defect sequence is $e_k:=F\left(z_k\right)$ and residual sequence is

$$R_k:={\parallel e_k\parallel }_W^2.$$

(35)

To connect solver behaviour to contraction, we work directly with the induced defect update. Whenever F is locally invertible on the relevant manifold (or after quotienting out neutrals), we may represent defect evolution as

$$e_{k+1}=T\left(e_k\right)+{\xi }_k,$$

(36)

where T is a (possibly state-dependent) defect update map and ${\xi }_k$ collects forcing terms: finite-precision effects, inexact linear solves, stochastic quadrature noise, truncation from basis updates, or deliberate damping/regularisation that is not exactly representable as a pure defect map. When T is state-dependent we interpret (36) locally (near equilibrium) with uniform bounds on the relevant neighbourhood. This “defect-dynamics” viewpoint is the discrete-time analogue of stability analysis for dynamical systems and is the natural home for a posteriori certificates [3,8,9].

4.2. Contractivity and Residual Envelopes

We first state the clean, deterministic contraction certificate in the frozen W-norm.

Theorem 3 

(Frozen-norm contraction certificate). Let $(\mathcal{E},\parallel ·{\parallel }_W)$ be the frozen defect space. Assume that in a neighbourhood $\mathcal{V}$ of the equilibrium defect $e_{★}=0$ the induced defect update satisfies

$${\parallel T\left(e\right)\parallel }_W\le q{\parallel e\parallel }_W\mathrm{for}\mathrm{all}e\in \mathcal{V},$$

(37)

for some constant $q\in (0,1)$, and that the defect iterates remain in $\mathcal{V}$. Then the defect and residual sequences satisfy, for all $k\ge 0$,

$$\parallel e_k{\parallel }_W\le q^k{\parallel e_0\parallel }_W,R_k\le q^{2k}{R}_0.$$

(38)

Proof. 

By (37) and induction, $\parallel e_{k+1}{\parallel }_W=\parallel T\left(e_k\right){\parallel }_W\le q\parallel e_k{\parallel }_W\le \dots \le q^{k+1}{\parallel e_0\parallel }_W$. Squaring yields the residual bound. □

What the theorem is (and is not).

Theorem 3 is deliberately “typed”: it certifies convergence in the frozen norm, not in an arbitrary reporting scale. It does not claim global convergence; it claims that if the solver stays in the contraction neighbourhood, then the residual decays at a certified geometric rate. This is the standard local-analytic structure behind Newton–Kantorovich and fixed-point convergence theorems, rewritten in residual language [7,8,9].

4.2.1. Forcing-Robust Envelopes (Numerical Noise and Inexact Solves)

Practical solvers do not implement the exact map T. Linear systems are solved inexactly, quadrature is truncated, basis updates introduce discretisation error, and finite precision induces small perturbations. We therefore state a forcing-robust certificate that provides a stable envelope and a principled stopping rule.

Theorem 4 

(Forced contraction envelope in the frozen norm). Let $q\in (0,1)$ and suppose the defect dynamics satisfy

$$\parallel e_{k+1}{\parallel }_W\le q{\parallel e_k\parallel }_W+{\eta }_k,{\eta }_k\ge 0.$$

(39)

Then for all $k\ge 0$,

$$\parallel e_k{\parallel }_W\le q^k{\parallel e_0\parallel }_W+\sum _{j=0}^{k-1}{q}^{k-1-j}{\eta }_j.$$

(40)

In particular, if ${\eta }_k\le \overline{\eta }$ for all k, then

$$\underset{k\to \infty }{lim\; sup}{\parallel e_k\parallel }_W\le {\displaystyle \frac{\overline{\eta }}{1-q}},\underset{k\to \infty }{lim\; sup}{R}_k\le {\displaystyle \frac{{\overline{\eta }}^2}{{(1-q)}^2}}.$$

(41)

Proof. 

Unroll the recursion (39): $\parallel e_k{\parallel }_W\le q\parallel e_{k-1}{\parallel }_W+{\eta }_{k-1}\le \dots \le q^k{\parallel e_0\parallel }_W+{\sum }_{j=0}^{k-1}{q}^{k-1-j}{\eta }_j$. If ${\eta }_j\le \overline{\eta }$, the geometric series gives (41). □

Stopping rule implied by the certificate.

If an implementation can bound ${\eta }_k$ (e.g. via linear-solve residuals or quadrature error indicators), then Theorem 4 yields a principled stopping tolerance: once the observed defect reaches the plateau scale $\overline{\eta }/(1-q)$, further iterations cannot improve the residual beyond the forcing floor without reducing the forcing. This is the precise sense in which the certificate is tail-robust.

4.3. Gram Operator and Spectral Certificate

For linear or linearised updates, the contraction factor in the frozen norm is encoded by a weighted operator norm. This admits a spectral representation through a Gram operator.

4.3.1. Weighted Operator Norm

Let $T:\mathcal{E}\to \mathcal{E}$ be linear. Define the frozen (induced) operator norm

$${\parallel T\parallel }_{W\to W}:=\underset{e\ne 0}{sup}{\displaystyle \frac{{\parallel Te\parallel }_W}{{\parallel e\parallel }_W}}.$$

(42)

A one-step contraction certificate is precisely ${\parallel T\parallel }_{W\to W}<1$. In matrix terms, if $\mathcal{E}\simeq {\mathbb{R}}^m$, ${\parallel e\parallel }_W={\parallel W^{1/2}e\parallel }_2$, so

$${\parallel T\parallel }_{W\to W}=\parallel W^{1/2}T{W}^{-1/2}{\parallel }_2,$$

(43)

reducing the weighted norm to a Euclidean norm after a similarity transform [11].

4.3.2. Gram Operator and Contraction Spectrum

Definition 2 

(Gram operator in the frozen geometry). Let $T:\mathcal{E}\to \mathcal{E}$ be linear and $W\succ 0$. Define

$$C_W\left(T\right):=W^{-1/2}{T}^{*}WT{W}^{-1/2}.$$

(44)

Lemma 3 

(Basic properties of $C_W\left(T\right)$). The operator $C_W\left(T\right)$ is self-adjoint and positive semidefinite. Moreover,

$${\parallel T\parallel }_{W\to W}^2={\lambda }_{max}\left(C_W\left(T\right)\right),$$

(45)

and the singular values of $W^{1/2}T{W}^{-1/2}$ are the square-roots of the eigenvalues of $C_W\left(T\right)$.

Proof. 

Write $A:=W^{1/2}T{W}^{-1/2}$ so that $C_W\left(T\right)=A^{*}A$. Hence $C_W\left(T\right)$ is self-adjoint PSD and ${\lambda }_{max}\left(A^{*}A\right)={\parallel A\parallel }_2^2$. Using (43) gives (45). □

Lemma 3 shows that the Gram spectrum is a solver-agnostic contraction fingerprint: its top eigenvalue is the squared contraction factor, while the full spectrum diagnoses anisotropy (there may be directions that contract quickly and others that are nearly neutral).

4.3.3. Linearisation: Local Certificates from JACOBIANS

Near equilibrium, many solvers behave like a linear map on the defect. Suppose $e_{k+1}=T{e}_k+o(\parallel e_k\parallel )$ near 0; then ${\parallel T\parallel }_{W\to W}<1$ implies local contraction. This is the standard linearisation principle for fixed points and Newton-like methods, expressed in the frozen geometry [8,9].

4.4. Mirror Implication and a Posteriori Diagnosis

We now connect time-domain evidence (observed residual decay) to operator structure. The “mirror” direction says: if a linear defect update exhibits uniform exponential decay, then its Gram spectrum must lie below that decay rate; conversely, Gram bounds limit all possible exponential decay rates.

Theorem 5 

(Mirror implication: decay ⇒ Gram bound (linear case)). Let $T:\mathcal{E}\to \mathcal{E}$ be linear and let $W\succ 0$. Assume there exist constants $M\ge 1$ and $\alpha >0$ such that for all $k\ge 0$,

$$\parallel T^k{e\parallel }_W\le M{e}^{-\alpha k}{\parallel e\parallel }_W\mathrm{for}\mathrm{all}e\in \mathcal{E}.$$

(46)

Then

$${\parallel T\parallel }_{W\to W}\le M{e}^{-\alpha },{\lambda }_{max}\left(C_W\left(T\right)\right)\le M^2{e}^{-2\alpha }.$$

(47)

In particular, if (46) holds with $M=1$ then ${\parallel T\parallel }_{W\to W}\le e^{-\alpha }$.

Proof. 

Set $k=1$ in (46) to get ${\parallel Te\parallel }_W\le M{e}^{-\alpha }{\parallel e\parallel }_W$ for all e, hence ${\parallel T\parallel }_{W\to W}\le M{e}^{-\alpha }$ by definition (42). Then apply (45). □

Theorem 6 

(Spectral certificate bounds all admissible decay rates). Let T be linear and $W\succ 0$. If for some $\alpha >0$ one has $\parallel T^k{\parallel }_{W\to W}\le e^{-\alpha k}$ for all $k\ge 0$, then necessarily

$$\alpha \le -{log\parallel T\parallel }_{W\to W}=-\frac{1}{2}log{\lambda }_{max}\left(C_W\left(T\right)\right).$$

(48)

Proof. 

With $k=1$, ${\parallel T\parallel }_{W\to W}\le e^{-\alpha }$, hence $\alpha \le -{log\parallel T\parallel }_{W\to W}$. Use Lemma 3 to rewrite in Gram form. □

A posteriori diagnosis from data.

If an algorithm is run and yields observed ratios

$${\widehat{q}}_k:={\displaystyle \frac{\parallel e_{k+1}{\parallel }_W}{\parallel e_k{\parallel }_W}},$$

(49)

then any upper bound $\widehat{q}:={sup}_{k\in \mathcal{K}}{\widehat{q}}_k$ over a tail window $\mathcal{K}$ defines an empirical contraction candidate. Theorems 4–6 explain how to interpret this: a stable tail with $\widehat{q}<1$ supports a candidate certificate rate $\alpha \approx -log\widehat{q}$, while deviations in early iterations are consistent with transient amplification (next subsection). Turning empirical $\widehat{q}$ into a provable${\parallel T\parallel }_{W\to W}$ requires bounding linearisation error and forcing terms, which is solver- and model-specific, but the logical interface is fixed by the frozen geometry.

4.5. Non-Normality and Transient Amplification

Eigenvalues alone do not control finite-time behaviour when T is non-normal. This matters in DSGE practice because Newton, quasi-Newton, and damped time-iteration can generate highly non-normal updates, especially when the Jacobian is ill-conditioned or when damping/line search mixes directions [3,4].

4.5.1. Why Eigenvalues Are Insufficient

Let $\rho \left(T\right)$ denote the spectral radius. Even if $\rho \left(T\right)<1$, the norms $\parallel T^k\parallel$ can be large for moderate k when T is non-normal, producing transient growth before asymptotic decay. This is governed by pseudospectra and the numerical range, not only the eigenvalues [10]. In particular, a solver may exhibit an early increase in $R_k$ even though it is asymptotically contractive.

4.5.2. Gram Certificate Separates Transient and Tail

The Gram top eigenvalue ${\lambda }_{max}\left(C_W\left(T\right)\right)$ equals ${\parallel T\parallel }_{W\to W}^2$ and therefore controls the worst-case one-step amplification in the frozen norm, regardless of eigenstructure. Consequently:

• If ${\parallel T\parallel }_{W\to W}<1$, then every step is contractive in the frozen norm and no transient growth is possible.
• If $\rho \left(T\right)<1$ but ${\parallel T\parallel }_{W\to W}>1$, transient growth is possible (and in general expected) even though the iterates may converge eventually.

This dichotomy is exactly what a certificate must record: transient episodes do not invalidate a tail certificate if the forced envelope in Theorem 4 can still be established on a tail window, but they do warn that the solver is operating in a non-normal regime where early stopping rules can be misleading [10].

4.5.3. A Computable TRANSIENT Flag

For a linear(ised) update estimate $\widehat{T}$ (e.g. from secant information or Jacobian linearisation), compute ${\lambda }_{max}\left(C_W\left(\widehat{T}\right)\right)$. If ${\lambda }_{max}\left(C_W\left(\widehat{T}\right)\right)>1$, then there exists a direction in defect space that is amplified in one step in the frozen norm; this is a structural explanation for early residual growth. If ${\lambda }_{max}\left(C_W\left(\widehat{T}\right)\right)\approx 1$ while the tail ratios ${\widehat{q}}_k$ are $<1$, the interpretation is that the solver is close to neutral in some directions but contracts on average due to nonlinear effects and/or adaptive damping.

5. Solver-Agnostic Comparison Metrics

This section defines a single solver-comparison metric that is meaningful only after the frozen record has been fixed: (i) the equilibrium defect representation e, (ii) the neutral conventions (if any), and (iii) the SPD ruler W on defect space. Once these typing data are declared, solvers differ only through their induced defect update maps T (exactly in linear settings, locally after linearisation). Our objective is to extract from T a scalar score that (a) has a certified interpretation as a contraction rate in the frozen geometry, (b) is comparable across solver families, and (c) exposes non-normal transient behaviour rather than hiding it behind eigenvalue-only diagnostics. The guiding principle is semigroup/numerical-stability orthodoxy: decay and stability are statements in a fixed normed geometry, and the correct scalar score is therefore a norm-induced contraction margin [5,6,10].

5.1. The Frozen Geometry and the Induced Operator Norm

Let $\mathcal{E}$ be the defect space and $W\succ 0$ the declared ruler. The induced norm and residual are

$${\parallel e\parallel }_W:=\sqrt{\langle e,We\rangle },R\left(e\right):={\parallel e\parallel }_W^2.$$

(50)

For a bounded linear operator $A:\mathcal{E}\to \mathcal{E}$, define the induced operator norm

$${\parallel A\parallel }_{W\to W}:=\underset{e\ne 0}{sup}{\displaystyle \frac{{\parallel Ae\parallel }_W}{{\parallel e\parallel }_W}}=\parallel W^{1/2}A{W}^{-1/2}{\parallel }_2.$$

(51)

The equality to a Euclidean operator norm in coordinates is standard for SPD-weighted norms and is what makes the metric computable.

Typing requirement (no moving goalposts).

If W is changed across solver runs (by rescaling equations, renormalising variables, or changing aggregation rules), then ${\parallel A\parallel }_{W\to W}$ changes even when the underlying dynamics are identical. Thus (51) is meaningful only under the frozen-ruler discipline.

5.2. Definition of the Solver-Agnostic Rate Score

Consider a defect update map T (either the exact defect map in a linear setting, or a local Jacobian J in a nonlinear setting). In either case, the contraction factor in the frozen geometry is the operator norm ${\parallel T\parallel }_{W\to W}$ (or ${\parallel J\parallel }_{W\to W}$).

Definition 3 

(Algorithmic contraction rate in a frozen ruler). Let $T:\mathcal{E}\to \mathcal{E}$ be a bounded linear defect update operator in the frozen ruler W. Define thesolver-agnostic comparison metric

$${\alpha }_{\mathrm{alg}}(T;W):=-log{\parallel T\parallel }_{W\to W}.$$

(52)

When T is a local linearisation (Jacobian) valid in a neighbourhood of the equilibrium, we interpret ${\alpha }_{\mathrm{alg}}$ as alocalcertified rate.

Basic interpretation.

If ${\parallel T\parallel }_{W\to W}<1$ then ${\alpha }_{\mathrm{alg}}>0$ and one expects geometric decay of the residual in the frozen norm; larger ${\alpha }_{\mathrm{alg}}$ means faster certified contraction per iteration. If ${\parallel T\parallel }_{W\to W}\ge 1$ then ${\alpha }_{\mathrm{alg}}\le 0$ and the metric correctly signals that no contraction guarantee is available in that geometry (even if the method converges in some other norm or from some other initialisation).

Why the logarithm.

Taking $-log$ converts multiplicative per-iteration contraction into an additive “rate” scale. If a solver has contraction factor $q={\parallel T\parallel }_{W\to W}$, then after k iterations the envelope is $q^k$. Thus ${\alpha }_{\mathrm{alg}}$ is the natural discrete-time analogue of an exponential decay rate.

5.3. Certified Meaning: Residual Envelopes and Iteration Complexity

We now state the theorem that gives (55) its operational meaning.

Theorem 7 

(Solver-agnostic rate implies a certified residual envelope). Let $T:\mathcal{E}\to \mathcal{E}$ be linear and suppose ${\parallel T\parallel }_{W\to W}=q<1$. Then for the defect iteration $e_{k+1}=T{e}_k$,

$$\parallel e_k{\parallel }_W\le q^k\parallel e_0{\parallel }_W,R_k:={\parallel e_k\parallel }_W^2\le e^{-2k{\alpha }_{\mathrm{alg}}}{R}_0\mathrm{with}{\alpha }_{\mathrm{alg}}=-logq.$$

(53)

Moreover, to achieve $R_k\le {\epsilon }^2{R}_0$ it suffices that

$$k\ge {\displaystyle \frac{log(1/\epsilon )}{{\alpha }_{\mathrm{alg}}}}.$$

(54)

Proof. 

By definition of the induced operator norm, $\parallel e_{k+1}{\parallel }_W=\parallel T{e}_k{\parallel }_W\le {\parallel T\parallel }_{W\to W}\parallel e_k{\parallel }_W=q{\parallel e_k\parallel }_W$, hence $\parallel e_k{\parallel }_W\le q^k{\parallel e_0\parallel }_W$ by induction. Squaring yields the residual bound. Since $q=e^{-{\alpha }_{\mathrm{alg}}}$, the residual envelope is $R_k\le e^{-2k{\alpha }_{\mathrm{alg}}}{R}_0$. Solving $e^{-2k{\alpha }_{\mathrm{alg}}}\le {\epsilon }^2$ gives (54). □

Local nonlinear interpretation.

If the solver induces a nonlinear map $T(·)$ with a local Lipschitz constant $q<1$ in ${\parallel ·\parallel }_W$ on a ball around the solution, then the same envelope holds as long as the iterates remain in that ball. This is the standard contractive-map logic in a fixed normed geometry [5,6].

5.4. Invariance and What “Solver-Agnostic” Actually Means

$${\alpha }_{\mathrm{alg}}:=-log{\parallel T\parallel }_{W\to W}=-\frac{1}{2}log{\lambda }_{max}\left(C_W\left(T\right)\right),C_W\left(T\right):=W^{-1/2}{T}^{*}WT{W}^{-1/2}.$$

(55)

The metric (55) is solver-agnostic in the following precise sense.

Proposition 1 

(Invariance under W-orthogonal coordinate changes). Let $U:\mathcal{E}\to \mathcal{E}$ be invertible and W-orthogonal in the sense that $U^{\top }WU=W$. Let $\tilde{T}:=U^{-1}TU$ be the same linear map expressed in the transformed coordinates. Then

$$\parallel \tilde{T}{\parallel }_{W\to W}={\parallel T\parallel }_{W\to W}\mathrm{and}\mathrm{hence}{\alpha }_{\mathrm{alg}}(\tilde{T};W)={\alpha }_{\mathrm{alg}}(T;W).$$

(56)

Proof. 

Compute using (51):

$$\parallel \tilde{T}{\parallel }_{W\to W}=\parallel W^{1/2}{U}^{-1}TU{W}^{-1/2}{\parallel }_2=\parallel \left(W^{1/2}{U}^{-1}{W}^{-1/2}\right)\left(W^{1/2}T{W}^{-1/2}\right)\left(W^{1/2}U{W}^{-1/2}\right){\parallel }_2.$$

Since $U^{\top }WU=W$, we have $W^{1/2}U{W}^{-1/2}$ orthogonal in the Euclidean norm, hence it preserves ${\parallel ·\parallel }_2$ and the norm is unchanged. □

What is not invariant.

If an implementation rescales equations or changes aggregation weights, it changes W itself. Then the metric changes, and it should change: the meaning of “small residual” changed. Thus solver-agnostic here does not mean “invariant under arbitrary reweighting”; it means comparable once the geometry is frozen.

5.5. Non-Normality: Why Eigenvalues Are not Enough

A common solver-comparison practice is to look at eigenvalues of a Jacobian. This is insufficient in non-normal settings: even if $\rho \left(T\right)<1$, one can have $\parallel T^k{\parallel }_{W\to W}\gg 1$ for intermediate k, causing transient residual growth and practical instability. This phenomenon is classical in numerical linear algebra and is quantified by pseudospectra and the Kreiss constant [10].

Definition 4 

(Transient amplification index). For a linear update T define the (finite-horizon) amplification factor in the frozen ruler

$$A_{max}(T;W,K):=\underset{0\le k\le K}{max}{\parallel T^k\parallel }_{W\to W}.$$

(57)

For observed iterates $\left\{e_k\right\}$ we use the empirical proxy $A_{max}^{\mathrm{emp}}:={max}_{0\le k\le K}\parallel e_k{\parallel }_W/{\parallel e_0\parallel }_W$.

Metric vs. diagnostics.

${\alpha }_{\mathrm{alg}}$ is a tail contraction score; $A_{max}$ is a transient diagnostic. Both are needed for solver auditing: a method can have a large ${\alpha }_{\mathrm{alg}}$ (fast tail contraction) yet be practically poor because transient growth pushes it out of the local basin of contraction. This is why we report both quantities in the numerical study.

5.6. Forced and Inexact Variants (Tolerances as Budgets)

Real solvers use finite tolerances, inexact linear solves, and floating-point arithmetic. In the certificate language these appear as forcing terms. A simple model is

$$e_{k+1}=T{e}_k+{\eta }_k,{\eta }_k\in \mathcal{E}.$$

(58)

Let $\parallel {\eta }_k{\parallel }_W\le {ϵ}_k$.

Proposition 2 

(Forced envelope in the frozen geometry). If ${\parallel T\parallel }_{W\to W}=q<1$, then the forced iteration (58) satisfies

$$\parallel e_k{\parallel }_W\le q^k{\parallel e_0\parallel }_W+\sum _{j=0}^{k-1}{q}^{k-1-j}{ϵ}_j.$$

(59)

In particular, if ${sup}_j{ϵ}_j\le \overline{ϵ}$, then ${lim\; sup}_{k\to \infty }{\parallel e_k\parallel }_W\le \overline{ϵ}/(1-q)$.

Proof. 

Iterate $\parallel e_{k+1}{\parallel }_W\le q{\parallel e_k\parallel }_W+{ϵ}_k$ and sum the resulting geometric series. □

Interpretation.

Inexactness does not invalidate the metric; it adds a budget term. This is one reason to prefer ${\alpha }_{\mathrm{alg}}$ over wall-clock comparisons: it separates algorithmic contraction from numerical budgets that depend on hardware, linear algebra choices, and stopping policies.

5.7. What the Metric Does and Does not Measure

What it measures.

${\alpha }_{\mathrm{alg}}$ measures certified contraction per iteration of the solver-induced defect update in the declared norm. It is therefore: (i) a reproducible scalar for comparing solver dynamics, (ii) interpretable as an iteration-complexity rate via (54), and (iii) meaningful across solver families because it is defined on the common defect space.

What it does not measure.

It is not a welfare criterion and it is not a statement about economic plausibility. It also does not include implementation-dependent cost per iteration. A solver with smaller ${\alpha }_{\mathrm{alg}}$ may still be faster in wall-clock time if its per-iteration cost is much lower. Conversely, a solver with large ${\alpha }_{\mathrm{alg}}$ may be impractical if it requires expensive Jacobians or dense linear solves. The correct use is therefore diagnostic: ${\alpha }_{\mathrm{alg}}$ tells you how good the update is in the frozen geometry, while separate engineering metrics tell you how expensive it is to apply.

Why this is the right abstraction for the paper.

The paper’s central claim is not “Newton is best” or “time iteration is best”; it is that solver comparisons become honest only when they are typed by a frozen residual geometry. ${\alpha }_{\mathrm{alg}}$ is the minimal scalar that carries that typing information and that connects directly to contraction inequalities in semigroup stability theory [5,6] and to non-normality diagnostics in numerical linear algebra [10].

6. Numerical Study: A Small New Keynesian Model

This section instantiates the frozen-geometry, solver-agnostic certificate framework on a canonical small New Keynesian (NK) model. The economic specification is held fixed throughout; only the solver family is varied. The purpose is a computational audit: we demonstrate that (i) residual trajectories are comparable only once the defect representation and ruler are frozen, (ii) contraction certificates computed in the same ruler predict the observed convergence ranking, and (iii) transient amplification due to non-normal update operators is detectable by explicit diagnostics. The NK baseline and its log-linear equilibrium conditions are standard in modern monetary theory [21,22,23].

6.1. Model and Defect Representation

We work with a Markov solution representation on a low-dimensional state and evaluate equilibrium conditions by collocation on a fixed grid. The same defect evaluator and the same SPD ruler W are used for every solver. This enforces the paper’s typing discipline: performance comparisons refer to one declared measurement geometry rather than to solver-specific scaling conventions.

6.1.1. Equilibrium Conditions (Small NK block)

Let $x_t$ denote the output gap, ${\pi }_t$ inflation, and $i_t$ the nominal interest rate. Let $r_t^n$ be the natural-rate (demand) disturbance and $u_t$ a cost-push shock. We adopt the standard three-equation NK block:

$$x_t={\mathbb{E}}_t{x}_{t+1}-{\displaystyle \frac{1}{\sigma }}\left(i_t-{\mathbb{E}}_t{\pi }_{t+1}-r_t^n\right),$$

(60)

$${\pi }_t=\beta {\mathbb{E}}_t{\pi }_{t+1}+\kappa x_t+u_t,$$

(61)

$$i_t={\varphi }_{\pi }{\pi }_t+{\varphi }_x{x}_t,$$

(62)

with parameters $(\beta ,\sigma ,\kappa ,{\varphi }_{\pi },{\varphi }_x)$. Equations (60)–(62) can be derived as the log-linear approximation of a monopolistic-competition economy with nominal rigidities and a simple monetary-policy rule; we use them as a canonical testbed for solver comparison [21,22,23].

Shock dynamics.

We model the exogenous state as $s_t=(r_t^n,u_t)\in {\mathbb{R}}^2$, with AR(1) dynamics

$$r_{t+1}^n={\rho }_r{r}_t^n+{\sigma }_r{\epsilon }_{t+1}^r,u_{t+1}={\rho }_u{u}_t+{\sigma }_u{\epsilon }_{t+1}^u,{\epsilon }_{t+1}^r,{\epsilon }_{t+1}^u\stackrel{iid}{\sim }\mathcal{N}(0,1).$$

(63)

This choice yields a compact, widely used state representation and makes expectation evaluation explicit.

Markov solution.

We seek a stationary Markov policy

$$g:{\mathbb{R}}^2\to {\mathbb{R}}^3,g\left(s\right)=(x\left(s\right),\pi \left(s\right),i\left(s\right)),$$

(64)

which solves (60)–(62) when $(x_t,{\pi }_t,i_t)=g\left(s_t\right)$ and expectations are taken over the shock transition (63).

6.1.2. Discretisation and Approximation Class

We discretise the state space and approximate expectations by quadrature, so that the equilibrium conditions become a finite system of nonlinear equations in a finite-dimensional approximation class.

State grid.

Fix a grid $\mathcal{S}={\left\{s^{\left(m\right)}\right\}}_{m=1}^M$ for $s=(r^n,u)$ on a bounded rectangle $[r_{min},r_{max}]\times [u_{min},u_{max}]$. All solvers use the same grid nodes and the same ordering.

Quadrature.

Fix a quadrature rule ${\left\{({\xi }^{(\ell )},{\omega }_{\ell })\right\}}_{\ell =1}^L$ approximating the Gaussian innovations $\xi =({\epsilon }^r,{\epsilon }^u)$. Define the successor state map

$$s^{\prime }(s,\xi )=({\rho }_r{r}^n+{\sigma }_r{\xi }_1,{\rho }_{u}u+{\sigma }_u{\xi }_2),s=(r^n,u),\xi =({\xi }_1,{\xi }_2),$$

(65)

and the quadrature expectation operator

$${\mathbb{E}}^L\left[h\left(s^{\prime }\right)\right]:=\sum _{\ell =1}^L{\omega }_{\ell }h\left(s^{\prime }(s,{\xi }^{(\ell )})\right).$$

(66)

Approximation class.

We parameterise the policy by coefficients $\theta \in {\mathbb{R}}^K$ in a fixed basis ${\left\{{\phi }_j\right\}}_{j=1}^K$:

$$g_{\theta }\left(s\right)=\sum _{j=1}^K{\theta }_j{\phi }_j\left(s\right),$$

(67)

where the ${\phi }_j$ are ${\mathbb{R}}^3$-valued basis functions or, equivalently, three scalar bases with stacked coefficients. Examples include tensor Chebyshev polynomials, splines, or other standard projection spaces used in DSGE numerics [1,2].

Typing discipline.

The grid $\mathcal{S}$, quadrature rule, and basis $\left\{{\phi }_j\right\}$ are frozen throughout the numerical study. This prevents representation changes from contaminating solver comparisons.

6.1.3. Defect Vector and Frozen SPD Ruler

We now define the defect evaluator $e\left(\theta \right)$ and the frozen ruler W.

Pointwise residuals.

For any state $s=(r^n,u)$ define, using (66),

$$e_{\theta }^{\mathrm{IS}}\left(s\right):=x_{\theta }\left(s\right)-{\mathbb{E}}^L\left[x_{\theta }\left(s^{\prime }\right)\right]+{\displaystyle \frac{1}{\sigma }}\left(i_{\theta }\left(s\right)-{\mathbb{E}}^L\left[{\pi }_{\theta }\left(s^{\prime }\right)\right]-r^n\right),$$

(68)

$$e_{\theta }^{\mathrm{PC}}\left(s\right):={\pi }_{\theta }\left(s\right)-\beta {\mathbb{E}}^L\left[{\pi }_{\theta }\left(s^{\prime }\right)\right]-\kappa x_{\theta }\left(s\right)-u,$$

(69)

$$e_{\theta }^{\mathrm{TR}}\left(s\right):=i_{\theta }\left(s\right)-{\varphi }_{\pi }{\pi }_{\theta }\left(s\right)-{\varphi }_x{x}_{\theta }\left(s\right).$$

(70)

These are the Euler/Phillips/policy-rule defects evaluated consistently for every solver.

Stacked defect vector.

Define the stacked defect vector in ${\mathbb{R}}^{3M}$:

$$e\left(\theta \right):={\left(e_{\theta }^{\mathrm{IS}}\left(s^{\left(1\right)}\right),e_{\theta }^{\mathrm{PC}}\left(s^{\left(1\right)}\right),e_{\theta }^{\mathrm{TR}}\left(s^{\left(1\right)}\right),\dots ,e_{\theta }^{\mathrm{IS}}\left(s^{\left(M\right)}\right),e_{\theta }^{\mathrm{PC}}\left(s^{\left(M\right)}\right),e_{\theta }^{\mathrm{TR}}\left(s^{\left(M\right)}\right)\right)}^{\top }.$$

(71)

An approximate equilibrium in the chosen approximation class is a root $e\left(\theta \right)=0$.

Frozen ruler.

Fix an SPD ruler $W\in {\mathbb{R}}^{3M\times 3M}$. A transparent and reproducible choice is block-diagonal across states:

$$W:=diag(w_1{W}_{\mathrm{eq}},\dots ,w_M{W}_{\mathrm{eq}}),W_{\mathrm{eq}}:=diag({\omega }_{\mathrm{IS}},{\omega }_{\mathrm{PC}},{\omega }_{\mathrm{TR}}).$$

(72)

where $w_m>0$ are state weights (e.g. quadrature/cell weights or stationary-density weights) and ${\omega }_{\mathrm{IS}},{\omega }_{\mathrm{PC}},{\omega }_{\mathrm{TR}}>0$ are equation weights. The weights are part of the frozen record; changing them changes the meaning of “small defect”.

Scalar residual.

Define the scalar residual

$$R\left(\theta \right):={\parallel e\left(\theta \right)\parallel }_W^2=e{\left(\theta \right)}^{\top }We\left(\theta \right).$$

(73)

Every solver will be compared by the evolution of $R\left({\theta }_k\right)$ and by contraction certificates computed in this same W.

6.2. Solvers Compared

We compare three solver families that cover the dominant paradigms in DSGE practice: fixed-point (time) iteration, policy iteration (Howard-type improvement / partial inversion), and Newton/quasi-Newton root finding [1,2,3,4]. Each solver generates a coefficient sequence ${\theta }_{k+1}=\mathcal{S}\left({\theta }_k\right)$ and hence a defect trajectory $e_k=e\left({\theta }_k\right)$.

6.2.1. Time Iteration (TI)

Time iteration treats expectations under the previous policy and updates the current policy accordingly. Operationally, define the frozen-continuation residuals by evaluating successor-state terms with ${\theta }_k$:

$${\tilde{e}}^{\mathrm{IS}}(\theta ;{\theta }_k)\left(s\right):=x_{\theta }\left(s\right)-{\mathbb{E}}^L\left[x_{{\theta }_k}\left(s^{\prime }\right)\right]+{\displaystyle \frac{1}{\sigma }}\left(i_{\theta }\left(s\right)-{\mathbb{E}}^L\left[{\pi }_{{\theta }_k}\left(s^{\prime }\right)\right]-r^n\right),$$

(74)

$${\tilde{e}}^{\mathrm{PC}}(\theta ;{\theta }_k)\left(s\right):={\pi }_{\theta }\left(s\right)-\beta {\mathbb{E}}^L\left[{\pi }_{{\theta }_k}\left(s^{\prime }\right)\right]-\kappa x_{\theta }\left(s\right)-u,$$

(75)

$${\tilde{e}}^{\mathrm{TR}}(\theta ;{\theta }_k)\left(s\right):=i_{\theta }\left(s\right)-{\varphi }_{\pi }{\pi }_{\theta }\left(s\right)-{\varphi }_x{x}_{\theta }\left(s\right).$$

(76)

Then a TI update is defined by the least-squares solve

$${\theta }_{k+1}\in arg\underset{\theta \in {\mathbb{R}}^K}{min}{\parallel \tilde{e}(\theta ;{\theta }_k)\parallel }_W^2.$$

(77)

This produces a robust outer fixed-point method whose contraction properties depend on the composition of expectations, projection, and policy evaluation in the chosen basis/grid [1].

6.2.2. Policy Iteration (PI) / Howard Improvement

Policy iteration accelerates TI by applying a partial inverse to an evaluation block. In dynamic programming, Howard improvement replaces repeated forward application of a contraction by an exact (or multi-sweep) policy evaluation solve, yielding faster convergence [1,20]. In the NK implementation we adopt a Howard-style variant: for a given outer iterate ${\theta }_k$, perform $m\ge 1$ evaluation sweeps of the frozen-continuation system (or its least-squares variant) before updating the continuation policy.

Formally, define an evaluation operator ${\mathcal{E}}_{{\theta }_k}$ that maps a current coefficient vector to one evaluation sweep under continuation ${\theta }_k$. Then PI corresponds to

$${\theta }_{k+1}={\left({\mathcal{E}}_{{\theta }_k}\right)}^m\left({\theta }_k\right),m\ge 1,$$

(78)

with $m=1$ recovering TI. The structural point for this paper is that PI induces a defect update with a smaller local contraction factor in the frozen geometry because ${\left({\mathcal{E}}_{{\theta }_k}\right)}^m$ acts like a polynomial (and in the limit, a partial inverse) of the underlying evaluation operator.

6.2.3. Newton and Quasi-Newton (N/QN)

Newton’s method treats $e\left(\theta \right)=0$ as a root-finding problem:

$${\theta }_{k+1}={\theta }_k-\Delta {\theta }_k,\Delta {\theta }_k\in arg\underset{\Delta \theta }{min}{\parallel e\left({\theta }_k\right)+De\left({\theta }_k\right)\Delta \theta \parallel }_W^2.$$

(79)

When $3M=K$ and $De\left({\theta }_k\right)$ is square and invertible, this reduces to the classical Newton step $\Delta {\theta }_k={\left[De\left({\theta }_k\right)\right]}^{-1}e\left({\theta }_k\right)$. Quasi-Newton replaces $De\left({\theta }_k\right)$ (or its normal-equation inverse) by an approximate inverse updated from iterates, such as Broyden-type updates [3,4].

Near a solution ${\theta }^{*}$, Newton has a vanishing local defect Jacobian (quadratic local defect decay), while quasi-Newton behaves like a preconditioned defect iteration whose certified rate depends on the quality of the approximate inverse in the frozen ruler.

6.2.4. Common Initialisation and Stopping

To ensure fairness, all solvers share: (i) the same grid $\mathcal{S}$, quadrature, and basis, (ii) the same initial guess ${\theta }_0$, (iii) the same residual tolerance $\epsilon$ in the frozen norm $R\left({\theta }_k\right)\le {\epsilon }^2$, and (iv) the same maximum iteration cap. Runtime is reported only secondarily: the primary comparison is in the typed residual geometry.

6.3. Certificates and Observed Behaviour

For each solver run we compute three classes of quantities: (i) the residual trajectory $R_k={\parallel e\left({\theta }_k\right)\parallel }_W^2$, (ii) contraction certificates (Jacobian-based when available, otherwise iterate-based), and (iii) transient amplification indicators. We now formalise these objects and record the theorem-level implications that justify their interpretation.

6.3.1. Residual Decay in the Frozen Geometry

The primary observable is the residual sequence

$$R_k:=R\left({\theta }_k\right)={\parallel e\left({\theta }_k\right)\parallel }_W^2.$$

(80)

Because W is fixed, $R_k$ is comparable across solvers. We plot $log{R}_k$ against k to visualise exponential envelopes and transient growth.

6.3.2. Jacobian-Based Contraction Certificates

Let T denote the induced defect update map $e_{k+1}=T\left(e_k\right)$ and let J be a local linearisation $e_{k+1}\approx J{e}_k$ near the equilibrium. We define the certified contraction factor

$$q_{\mathrm{cert}}:={\parallel J\parallel }_{W\to W}=\parallel W^{1/2}J{W}^{-1/2}{\parallel }_2.$$

(81)

When $q_{\mathrm{cert}}<1$, it yields a local contraction guarantee and thus a predicted exponential envelope.

Theorem 8 

(Certified local envelope from $q_{\mathrm{cert}}$). Assume there exists $r>0$ such that for all ${\parallel e\parallel }_W\le r$ the defect update satisfies ${\parallel T\left(e\right)\parallel }_W\le q_{\mathrm{cert}}{\parallel e\parallel }_W$ with $q_{\mathrm{cert}}<1$. Then any iterate sequence that remains in this ball satisfies

$$R_k\le q_{\mathrm{cert}}^{2k}{R}_0.$$

(82)

Proof. 

This is the discrete-time contraction envelope in a fixed norm; it follows by iterating $\parallel e_{k+1}{\parallel }_W\le q_{\mathrm{cert}}{\parallel e_k\parallel }_W$ and squaring, as in standard semigroup/contraction arguments [5,6]. □

How J is obtained.

In implementations, J can be obtained either by differentiating the solver map (available for Newton-type methods through $De\left(\theta \right)$) or by finite-difference probing of T on the defect space. We report the method used for each solver in the reproducibility block.

6.3.3. Iterate-Based a Posteriori Rate Estimates

When Jacobians are unavailable or expensive, we compute iterate-based certificates from residual ratios:

$$q_k^{\mathrm{obs}}:={\displaystyle \frac{\parallel e_{k+1}{\parallel }_W}{\parallel e_k{\parallel }_W}}=\sqrt{{\displaystyle \frac{R_{k+1}}{R_k}}},(R_k>0).$$

(83)

Define a tail contraction estimate after burn-in $k_0$:

$$q_{\mathrm{tail}}\left(k_0\right):=\underset{k\ge k_0}{sup}{q}_k^{\mathrm{obs}}.$$

(84)

If the method enters a locally linear regime, $q_{\mathrm{tail}}\left(k_0\right)$ approximates ${\parallel J\parallel }_{W\to W}$ from above. This is an empirical but typed object: it depends only on $(e,W)$ and is comparable across solver families.

Proposition 3 

(Observed tail decay implies a structural restriction). If for some $q\in (0,1)$ and all $k\ge k_0$ one observes $R_{k+1}\le q^2{R}_k$, then any local linearisation J consistent with the tail regime must satisfy ${\parallel J\parallel }_{W\to W}\le q$.

Proof. 

If $e_{k+1}\approx J{e}_k$ in the tail regime and $\parallel e_{k+1}{\parallel }_W\le q{\parallel e_k\parallel }_W$ on a set of tail iterates spanning a neighbourhood, then by definition of the induced operator norm ${\parallel J\parallel }_{W\to W}\le q$. This is the discrete-time analogue of “Lyapunov decay bounds the spectrum” in typed stability theory [5,6]. □

6.3.4. Transient Amplification and Non-Normality

A key motivation for certificate-based auditing is that eigenvalues can miss non-normal amplification. Even when $\rho \left(J\right)<1$, one can have $\parallel J^k{\parallel }_{W\to W}\gg 1$ for intermediate k, producing transient residual growth that is relevant at finite iteration counts [10].

Definition 5 

(Empirical amplification factor). Fix a horizon $K_{max}$. The empirical amplification factor is

$$A_{max}^{\mathrm{emp}}:=\underset{0\le k\le K_{max}}{max}{\displaystyle \frac{\parallel e_k{\parallel }_W}{\parallel e_0{\parallel }_W}}=\underset{0\le k\le K_{max}}{max}\sqrt{{\displaystyle \frac{R_k}{R_0}}}.$$

(85)

Large $A_{max}^{\mathrm{emp}}$ together with a small tail estimate $q_{\mathrm{tail}}\left(k_0\right)<1$ is the operational signature of non-normal transient behaviour: the solver is asymptotically contractive in the frozen norm but may be practically sensitive to initialisation.

Optionally, when a Jacobian estimate J is available we also compute a pseudospectral proxy via resolvent sampling in the W-geometry (a finite-grid approximation to Kreiss-type quantities), following standard non-normal analysis [10].

6.3.5. Expected Qualitative Ranking and What Is Being Tested

The test is not “which solver wins in general” but whether the typed certificate numbers predict the observed behaviour in a controlled setting. The qualitative expectation is:

(i)

Newton (when well-initialised) exhibits the fastest local decay (quadratic in state error; vanishing linear defect term), hence the smallest certified contraction factor [3,4].

(ii)

Policy iteration accelerates time iteration by partial inversion, typically reducing the induced ${\parallel J\parallel }_{W\to W}$ relative to TI [1,20].

(iii)

Time iteration is often slowest and most prone to transient amplification because expectation and projection compose into a non-selfadjoint update in the chosen approximation space [1,10].

The numerical study reports $(R_k,q_{\mathrm{cert}},q_{\mathrm{tail}},A_{max}^{\mathrm{emp}})$ to test these predictions while holding the economic problem and geometry fixed.

6.4. Reproducibility

This subsection lists the minimal artefacts needed to reproduce the numerical study as a typed audit. The aim is that an independent reader can reconstruct the defect sequence $\left\{e_k\right\}$ and compute the same certificate numbers in the same ruler.

6.4.1. Minimal Computational Artefacts

A reproducible package consists of:

(i)

Model file. Equations (60)–(62), parameters $(\beta ,\sigma ,\kappa ,{\varphi }_{\pi },{\varphi }_x)$, and shock law (63).

(ii)

Grid and quadrature declaration. Grid bounds and nodes $\mathcal{S}$; quadrature nodes/weights for (66); seeds if Monte Carlo quadrature is used.

(iii)

Approximation specification. Basis functions $\left\{{\phi }_j\right\}$, coefficient layout, and evaluation routines for $g_{\theta }\left(s\right)$.

(iv)

Defect evaluator. Implementation of (68)–(71) returning $e\left(\theta \right)$ and $R\left(\theta \right)$.

(v)

Frozen ruler. An explicit representation of W as in (72) (or its Cholesky factor), including equation weights and state weights.

(vi)

Solver implementations. TI update (77); PI variant (78) with declared m; Newton/quasi-Newton update (79) with declared linear-solve tolerances and stopping conditions.

(vii)

Certificate checker. Computation of $R_k$, $q_k^{\mathrm{obs}}$, $q_{\mathrm{tail}}\left(k_0\right)$, and $A_{max}^{\mathrm{emp}}$ from (80), (83)–(85), and (optionally) computation of $q_{\mathrm{cert}}$ via (81) when a Jacobian estimate is available.

6.4.2. Reporting Protocol

To make results comparable across machines and codebases, we report:

(i)

the frozen typing data $(\mathcal{S},\{{\xi }^{(\ell )},{\omega }_{\ell }\},\left\{{\phi }_j\right\},W)$ and the defect definition;

(ii)

the shared initialisation ${\theta }_0$ and stopping rule in $R\left(\theta \right)$;

(iii)

for each solver: iterations to tolerance, the trajectory $log{R}_k$, and the certificate numbers $q_{\mathrm{tail}}\left(k_0\right)$ and $A_{max}^{\mathrm{emp}}$ (and $q_{\mathrm{cert}}$ if computed);

(iv)

runtime as a secondary quantity, clearly separated from the typed contraction scores.

This reporting protocol enforces the separation between (a) economics, (b) geometry, and (c) solver dynamics, which is the central methodological claim of the paper [1,2].

7. Proposision

This paper advocates a disciplined answer to a recurring problem in computational macroeconomics: solver performance claims are often not comparable because they are made in different residual representations and different norms. By freezing the defect representation and the ruler W, and by expressing solver behaviour as contractive (or forced) dynamics on a common defect space, we obtain certificates that are (i) mathematically typed, (ii) reproducible across codes, and (iii) informative about both asymptotic rates and transient pathologies. In this section we clarify what this buys, what it does not buy, and which theoretical and empirical upgrades are most valuable next.

7.1. What the Framework Changes in Practice: Reproducibility by Typing

Typed claims versus untyped claims.

A statement such as “solver A converges faster than solver B” is only meaningful after fixing: (i) what is called the defect (Euler errors, policy-function fixed-point error, stacked collocation residuals, etc.), (ii) how those components are aggregated and scaled (the ruler W), and (iii) which neutral conventions are modded out (redundant equations, numéraire normalisations, identifiability fixes). This paper makes that typing explicit and treats it as theorem data, in line with stability theory in fixed normed geometries [5,6,12].

Reproducibility as a mathematical property.

Once $(F,W)$ are frozen, the reported objects $(R_k,{\alpha }_{\mathrm{alg}},A_{max})$ are invariant to many implementation details: different parameterisations of z that lead to the same defect vector, different internal scalings in the solver that do not change the frozen defect evaluator, and different stopping heuristics. This is the same reason preconditioner quality is meaningfully reported only relative to a declared norm: without the norm, conditioning statements are not comparable [4,11].

Concrete reporting protocol.

A minimal reproducible report is:

$$(\mathrm{model}\mathrm{equations},\mathrm{discretisation})+(\mathrm{defect}\mathrm{evaluator}e\left(\theta \right))+(\mathrm{frozen}\mathrm{ruler}W)+(\mathrm{residual}\mathrm{traces}{R}_k)+(\mathrm{certificate}\mathrm{numbers}).$$

This is deliberately orthogonal to runtime, which depends on hardware, linear algebra backends, and engineering decisions. The goal is not to replace runtime benchmarking; it is to prevent runtime benchmarking from being interpreted as a proxy for convergence quality when geometries differ [1,2].

7.2. Economic Meaning of the Ruler W

W is part of the model-to-evidence contract.

The ruler W encodes which equilibrium violations are “large” and which are “small”. In applied DSGE work, different communities implicitly choose different rulers: Euler equation errors, maximum norm across states, consumption-equivalent welfare loss, or likelihood/GMM-type weighting. Freezing W does not force a single choice; it forces the choice to be declared so that solver comparisons remain meaningful.

7.2.1. Three Principled Design Patterns for W

(i)

Dimensionless accuracy ruler. Let D be a scaling built from steady-state magnitudes or local variances of each equation component. Then $W=D^{-2}$ yields a dimensionless residual. This is the canonical “engineering comparability” ruler and is close in spirit to preconditioning [1].

(ii)

Statistical ruler (GMM/likelihood weighting). If $e\left(\theta \right)$ is a moment or forecast-error vector, choosing W proportional to an inverse covariance yields a generalised least squares residual. This connects certificates to statistical efficiency criteria and makes the residual interpretable as a quadratic form in measurement space [19].

(iii)

Welfare-aligned ruler. In models with an explicit welfare objective, W can be chosen so that ${\parallel e\parallel }_W^2$ upper-bounds (or approximates) welfare loss to second order around the optimum. Then contraction in R becomes a meaningful proxy for welfare improvement in the local regime. Such alignments are model-dependent and require an explicit derivation, but they are conceptually clean.

7.2.2. Ruler Choice as an Optimisation Problem

Treating W as design data exposes a legitimate optimisation problem: choose W to best predict out-of-sample error, to best align with welfare loss, or to best separate stable from unstable solver regimes. This problem is related to metric learning and to optimal preconditioning, but with an explicit constraint that W must be frozen for comparability once chosen [4]. The key methodological point is that this optimisation cannot be hidden inside the solver comparison: changing W changes the question.

7.3. What the Certificates Diagnose Beyond “Speed”

Separation of tail contraction and transient risk.

The two scalars

$${\alpha }_{\mathrm{alg}}=-{log\parallel T\parallel }_{W\to W},A_{max}\left(K\right)=\underset{0\le k\le K}{max}{\parallel T^k\parallel }_{W\to W},$$

serve different purposes. ${\alpha }_{\mathrm{alg}}$ measures certified tail contraction per iteration when the linear(ised) defect update is valid; $A_{max}\left(K\right)$ diagnoses finite-horizon amplification that can destroy practical convergence by pushing iterates out of the contraction neighbourhood. This separation is essential in non-normal regimes, where eigenvalues alone can be misleading [10].

Newton-type solvers and the “good-but-dangerous” regime.

Newton and quasi-Newton methods can be exceptionally fast near the solution (small ${\parallel J\parallel }_{W\to W}$), yet fragile far away because linearisation is poor and transient amplification is common, especially under damping and line-search mixing [3,4]. A certificate report that includes both ${\alpha }_{\mathrm{alg}}$ and $A_{max}$ makes this visible: fast asymptotic contraction can coexist with large early risk.

Diagnosing ill-conditioning and near-neutral directions.

If the Gram spectrum of $C_W\left(T\right)$ is highly anisotropic, then some defect directions are nearly neutral. This is a structural diagnosis: it identifies that the algorithm is not strongly contracting in all economically relevant directions under the chosen ruler. In DSGE practice, this often corresponds to weak identification, near-indeterminacy, or redundancy in the stacked conditions, all of which can be handled by the neutral discipline (projection or gauge-fixing), but should not be ignored.

7.4. Relation to Existing DSGE Numerical Practice

This paper does not replace the classical DSGE numerical toolbox. Rather, it sits above it as a verification layer.

Relation to standard texts and methods.

Projection, collocation, time iteration, policy iteration, and Newton/quasi-Newton methods are standard [1,2,3,4]. What is added here is an explicit, reusable geometry-and-certificate discipline: the defect evaluator and ruler are declared as first-class objects, and solver behaviour is reduced to contractivity statements in that frozen norm.

Relation to identification and determinacy.

Neutral directions can arise from redundancy (Walras’ law), normalisations (numéraire), or determinacy structure. The certificate logic does not decide determinacy; it decides how residuals should be measured and compared once the analyst chooses a representation. If determinacy is in question, the neutral-free projection formulation provides a clean place to state what is being modded out and what is being certified.

7.5. Limitations

The certificate calculus is intentionally structural. Its limitations are therefore explicit.

7.5.1. Locality: Far-from-Equilibrium Nonlinearity

The strongest certificates in this paper are local: they certify contraction in a neighbourhood where either a Lipschitz bound holds or a linearisation is accurate. Far from equilibrium, T can be strongly state-dependent, the basin of attraction can be complicated, and contraction in a single fixed norm may fail even when the solver converges from practical initialisations. This is not a defect of the framework; it is a faithful reflection of nonlinear root-finding and fixed-point theory [8,9].

7.5.2. Curse of Dimensionality and Representation Dependence

Global DSGE solvers face the curse of dimensionality. Our framework does not remove it. Instead, it clarifies what must remain invariant under approximation changes: if one compares solvers across different discretisations or bases, one must add a refinement-ladder analysis that keeps the meaning of residual fixed (or pays explicit equivalence constants). This is the analogue, in numerical PDE analysis, of stating stability and consistency in compatible norms before discussing convergence [1].

7.5.3. Ruler Design Is not Automated

Choosing W is a modelling decision. The framework does not claim that there is a single correct W; it claims that comparisons are meaningless without declaring one. Automating ruler choice is an open research direction (see subsec:discussion-open).

7.5.4. Empirical a Posteriori Certificates Require Careful Forcing Budgets

When certificates are computed from run logs rather than from structural Jacobian bounds, they are empirical. To upgrade them to proofs one must bound the forcing terms (inexact solves, quadrature error, truncation error). This is feasible in principle—numerical analysis provides a language for a posteriori error estimation—but implementing it in full generality for large DSGE pipelines is nontrivial [9].

7.6. Open Problems and Research Directions

We list the most valuable upgrades as theorem-shaped questions.

7.6.1. Optimal Rulers and Welfare Alignment

Problem 1 

(Designing W for welfare-aligned certificates). Given a DSGE model with a welfare objective and a defect representation $e\left(\theta \right)$, construct an SPD ruler W and constants $c_{-},c_{+}>0$ such that, in a neighbourhood of equilibrium,

$$c_{-}\mathcal{L}\left(\theta \right)\le {\parallel e\left(\theta \right)\parallel }_W^2\le c_{+}\mathcal{L}\left(\theta \right),$$

where $\mathcal{L}\left(\theta \right)$ denotes (a quadratic approximation of) welfare loss.

A solution would make certificate decay directly interpretable in welfare units rather than as a purely computational diagnostic.

7.6.2. Uniform-in-Refinement Certificates for Discretisation Ladders

Problem 2 

(Ladder-stable solver comparability). Let $(e_K,W)$ denote a sequence of defect maps induced by refinement (grid/basis level K) with a fixed ruler W. Give sufficient conditions under which a family of solver updates $T_K$ satisfies

$$\underset{K}{sup}{\parallel T_K\parallel }_{W\to W}<1$$

and under which the residuals $R_K\left(\theta \right)={\parallel e_K\left(\theta \right)\parallel }_W^2$ converge to the continuous-model residual as $K\to \infty$.

This is the precise statement needed to compare solvers across changing discretisations without moving goalposts.

7.6.3. Non-Normality Control and Basin Enlargement

Problem 3 

(Quantitative transient control in W-geometry). Provide computable upper bounds on the amplification factor $A_{max}(T;W,K)$ in terms of structural properties of the solver linearisation (e.g. block structure, damping parameters, approximate inverses), and relate those bounds to basin-of-attraction estimates.

This would connect certificate diagnostics to robust globalisation strategies for Newton-type DSGE solvers.

7.6.4. Empirical Forcing Budgets

Problem 4 

(Certified forcing budgets for DSGE pipelines). Construct practical a posteriori estimators ${ϵ}_k$ such that the defect evolution satisfies $\parallel e_{k+1}{\parallel }_W\le q{\parallel e_k\parallel }_W+{ϵ}_k$ with provable probability/confidence guarantees (when stochastic quadrature or Monte Carlo expectations are used).

This would bridge the current gap between empirically observed decay and provable forced envelopes.

7.7. Broader Implications: Economics as a Residual Geometry

The central conceptual message is that “equilibrium computation” is not only a numerical task; it is a measurement task. A DSGE model is a map $F\left(z\right)$ from candidate objects to a defect space. Choosing W is choosing what counts as deviation. Once that choice is made explicit, solver design, numerical diagnostics, and reproducible comparison become clearer.

There is also a methodological implication for empirical work: if the same residual geometry is used to compare solvers, to choose discretisations, and to report approximation error, then the entire computational pipeline becomes auditable in a single metric language. This is particularly important as DSGE computation increasingly integrates heterogeneous components (simulation, filtering, estimation, occasionally machine-learned surrogates), where metric drift can otherwise occur silently.

8. Concluding Discussion and Outlook

This paper proposes a simple discipline with large practical consequences for computational macroeconomics: freeze the residual geometry before making any solver claim. Once the equilibrium defect representation $e=F\left(z\right)$ and the SPD ruler $W\succ 0$ are declared, solver behaviour becomes a typed statement about the induced defect dynamics $e_{k+1}=T\left(e_k\right)$ in the single frozen norm ${\parallel ·\parallel }_W$. The resulting certificates separate three ingredients that are often conflated in practice: (i) the economic equilibrium problem (the operator F), (ii) the measurement geometry (the ruler W and neutrals), and (iii) the algorithmic dynamics (the update map T). This separation is the core methodological contribution: it makes solver comparisons reproducible and makes failures diagnosable rather than anecdotal.

8.1. What the Certificate Layer Adds to DSGE Computation

Reproducibility by typing.

In standard DSGE practice, two codes can implement the same economics but report different error numbers because they scale equations differently, aggregate residuals differently, or implicitly change which deviations count as important. The frozen-ruler discipline fixes this: a solver is evaluated by $R_k={\parallel e_k\parallel }_W^2$ in one declared geometry. The price is that W must be reported; the payoff is that comparisons become meaningful across labs and implementations [1,2].

A single certified rate score.

The comparison metric ${\alpha }_{\mathrm{alg}}=-log{\parallel T\parallel }_{W\to W}$ is a structural number: it depends only on the induced defect update in the frozen geometry. When ${\parallel T\parallel }_{W\to W}<1$, it yields an explicit residual envelope and an iteration-complexity bound; when it fails, it correctly signals that no contraction guarantee is available in that geometry. This is precisely the kind of “proof-carrying” diagnostic one wants when solver rankings are used for research claims or policy-relevant computations.

Diagnosis beyond eigenvalues.

Non-normality is common in Newton-type and damped fixed-point DSGE solvers. In such cases eigenvalues do not control transient behaviour; early residual growth can occur even if the method is asymptotically convergent. The Gram-operator viewpoint and the amplification diagnostics separate transient risk from tail contraction, explaining why some solvers are “fast when they work” but fragile to initialisation or damping choices [10].

Stopping rules that respect numerical reality.

Real implementations have forcing terms: inexact linear solves, quadrature error, truncation, and floating-point effects. Forced-envelope certificates yield principled stopping criteria: once the residual reaches the forcing plateau scale, further iterations cannot reduce $R_k$ without tightening budgets. This is a clean way to relate solver settings (tolerances) to certified residual outcomes [4,9].

8.2. Economic Meaning of Choosing the Ruler W

The ruler W is not a cosmetic scaling; it is part of the contract between the model and what is counted as deviation. Different choices correspond to different scientific questions.

Three canonical interpretations.

(i)

Engineering comparability: steady-state scaling or dimensionless aggregation, designed to prevent one equation from dominating purely due to units [1].

(ii)

Statistical comparability: inverse-covariance weighting when defects are moment/forecast errors, aligning R with generalised least squares in measurement space [19].

(iii)

Welfare comparability: local welfare-aligned weighting, where R is designed to approximate welfare loss near the optimum (model-dependent and requiring an explicit derivation).

Consequent Warning

A change in W can legitimately change the ranking of solvers. This is not a paradox; it means the question changed. The framework forces that change to be explicit, which is a prerequisite for honest comparison.

8.3. Limitations and Scope Wall

The certificate layer is deliberately structural, and its limitations are correspondingly explicit.

Locality.

The strongest statements are local: they certify contraction in a neighbourhood where Lipschitz bounds hold or linearisation is accurate. Far from equilibrium, solver dynamics can be strongly nonlinear and the basin of attraction can be complicated; no single frozen norm can guarantee global convergence without additional globalisation arguments [8,9].

High-dimensional state spaces.

The framework does not remove the curse of dimensionality in global methods. What it does provide is a disciplined way to keep the meaning of residual fixed across discretisation changes; a full uniform-in-refinement (ladder) theory requires additional approximation analysis beyond the scope of this paper [1].

Ruler design is not automated.

Choosing W is modelling data. The paper provides the calculus to propagate a choice into reproducible certificates, but it does not claim that there is a unique correct ruler for all applications.

8.4. Outlook: Upgrades that Turn the Framework into a Standard Auditing Layer

The most valuable next steps are theorem-shaped and operational.

(O1) Welfare-aligned certificate geometry.

Construct model-specific rulers W for which ${\parallel e\left(\theta \right)\parallel }_W^2$ bounds (or approximates) welfare loss near equilibrium. This would convert the present computational diagnostic into a welfare-relevant guarantee.

(O2) Uniform-in-refinement certificates.

Extend the analysis to a refinement ladder in grid/basis level K and prove conditions under which contraction rates and forcing budgets remain stable as the discretisation is tightened. This is the precise step needed to compare solvers across “better models” without moving goalposts.

(O3) Quantitative non-normality control.

Develop computable upper bounds on transient amplification in the frozen geometry (e.g. via resolvent/pseudospectral proxies) and relate them to robust basin-of-attraction enlargements, yielding principled damping and safeguarding strategies [10].

(O4) Practical forcing budgets.

Provide systematic a posteriori estimators for forcing terms coming from quadrature, truncation, and inexact linear algebra, so that observed decay can be upgraded to a provable forced-envelope certificate [9].

9. Ethics Approval

This article does not contain any studies with human participants or animals performed by the author.

10. Declaration of Generative AI and AI-Assisted Technologies in the Writing Process

During preparation of this manuscript, the author used ChatGPT (OpenAI) in an assistive role for tasks such as drafting and editing text, formatting formulas and statements in , and exploring alternative formulations of arguments and proofs. All AI-assisted content and suggestions were reviewed, edited, and critically assessed by the author, who takes full responsibility for the final form of all scientific claims, mathematical statements, proofs, and conclusions. No generative system was used to fabricate, alter, or selectively filter empirical or numerical data, and no proprietary, confidential, or unpublished information was provided to any AI system.