Response Renormalization for Critical Deep Equilibrium Models

# Response Renormalization for Critical Deep Equilibrium Models
Source: [https://arxiv.org/html/2608.23725](https://arxiv.org/html/2608.23725)
Jose Luis Lima de Jesus SilvaAffiliation:Federal University of Bahia, Department of GeophysicsAffiliation:Salvador, BA 40170\-115, BrazilAffiliation:Grupo de Estudos e Aplicação de Inteligência Artificial em Geofísica \(GAIA\)Affiliation:Federal University of Bahia, Salvador, BA 40170\-115, BrazilEmail:[jseluis\.silva@gmail\.com](mailto:)

###### Abstract

Deep Equilibrium Models \(DEQs\) compute predictions by finding a hidden representation that remains unchanged under the model’s update\. Training through this equilibrium uses implicit differentiation, which requires solving an adjoint system built from the residual Jacobian\. If this Jacobian is nearly singular along directions to which the loss is sensitive, small perturbations can be strongly amplified in the adjoint response, producing large and highly sensitive gradients that can make optimization unreliable\. We introduce Response Renormalization, a backward\-pass framework that lifts selected near\-pole denominators while leaving unlifted response channels unchanged\. Collective Mode Response Renormalization \(CMR\) applies this correction in a low\-dimensional critical subspace, while Phi\-adaptive CMR computes a bounded response mass from a prescribed positive susceptibility rule\. We derive dense and matrix\-free collective formulations, distinguish exact gradients of a modified frozen\-anchor residual from backward\-response surrogates, and extend the construction to Structured Implicit Layers and Vector Attractors \(SILVA\)\. Across 23 multiphysics families spanning partial differential equations, three\-dimensional fields, operator maps, complex geometries, and particle systems, models trained with CMR and Phi\-CMR have test errors no more than five percent higher than those trained with exact implicit differentiation in more than 98% of static and 95% of transient family–seed comparisons\. Solver\-index experiments additionally show finite responses converging toward the static adjoint, while physical\-time rollouts retain predictive fidelity under the evaluated conditions\. These results demonstrate that selective response renormalization can control near\-critical adjoint amplification without globally damping well\-conditioned sensitivity\. Therefore, the method can make parameter updates more reliable while preserving the useful gradient information needed for learning\.

*Keywords*deep equilibrium models⋅\\cdotimplicit differentiation⋅\\cdotresponse theory⋅\\cdotsingular values⋅\\cdotrenormalization

## 1Introduction

Modern implicit layers replace a finite computation graph by the solution of an equation, and Deep Equilibrium Models \(DEQs\) instantiate this idea through the hidden fixed pointz⋆=fθ​\(z⋆,x\)z^\{\\star\}=f\_\{\\theta\}\(z^\{\\star\},x\)\. Effective depth is therefore determined by a solver rather than by an explicitly stored stack of layers\([Bai et al\., 2019](https://arxiv.org/html/2608.23725#bib.bib1)\), placing DEQs within the broader implicit deep learning framework of[El Ghaoui et al\. \(2021\)](https://arxiv.org/html/2608.23725#bib.bib8)\. The same fixed\-point principle now supports multiscale vision models\([Bai et al\., 2020](https://arxiv.org/html/2608.23725#bib.bib2)\), steady\-state neural operators for partial differential equation \(PDE\) solution maps\([Marwah et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib10)\), and equilibrium algorithmic reasoners for iterative computation\([Georgiev et al\., 2024](https://arxiv.org/html/2608.23725#bib.bib11)\), making implicit layers attractive because useful features can emerge from many recurrent interactions without declaring a finite depth in advance\.

This modeling advantage creates a training bottleneck because fixed\-point learning still depends on a linearized sensitivity problem whose conditioning can dominate parameter updates\. Existing approaches use Jacobian penalties\([Bai et al\., 2021](https://arxiv.org/html/2608.23725#bib.bib3)\), monotone networks\([Winston and Kolter, 2020](https://arxiv.org/html/2608.23725#bib.bib9)\), initialization analysis\([Agarwala and Schoenholz, 2022](https://arxiv.org/html/2608.23725#bib.bib7)\), phantom gradients\([Geng et al\., 2021](https://arxiv.org/html/2608.23725#bib.bib4)\), standardized TorchDEQ solvers\([Geng and Kolter, 2023](https://arxiv.org/html/2608.23725#bib.bib5)\), or reversibility\([McCallum et al\., 2025](https://arxiv.org/html/2608.23725#bib.bib6)\), but do not directly control the inverse response while preserving informative equilibrium sensitivities\.

The unresolved issue in this setting is geometric because implicit differentiation avoids storing the forward solver trajectory by replacing unrolled backpropagation with the adjoint equation, whose conditioning depends on how the residual Jacobian stretches, compresses, and rotates directions in the hidden\-state space\. Letz⋆z^\{\\star\}denote the equilibrium hidden state for inputxx,fθf\_\{\\theta\}the model update with parametersθ\\theta, andℒ\\mathcal\{L\}the loss\. The residual JacobianKK, equilibrium\-state loss gradientgg, and adjointvvsatisfy

K⊤​v=g,K=I−∂zfθ​\(z⋆,x\),g=∇z⋆ℒ\.K^\{\\top\}v=g,\\qquad K=I\-\\partial\_\{z\}f\_\{\\theta\}\(z^\{\\star\},x\),\\qquad g=\\nabla\_\{z^\{\\star\}\}\\mathcal\{L\}\.\(1\)In the singular\-value decompositionK=U​Σ​V⊤K=U\\Sigma V^\{\\top\}, the columnsuiu\_\{i\}ofUUandviv\_\{i\}ofVVare the left and right singular vectors, and the diagonal entriesσi\\sigma\_\{i\}ofΣ\\Sigmaare the singular values\. The inverse response is

v=U​Σ−1​V⊤​g=∑ivi⊤​gσi​ui,v=U\\Sigma^\{\-1\}V^\{\\top\}g=\\sum\_\{i\}\\frac\{v\_\{i\}^\{\\top\}g\}\{\\sigma\_\{i\}\}u\_\{i\},\(2\)Equation \([2](https://arxiv.org/html/2608.23725#S1.E2)\) identifiesvi⊤​gv\_\{i\}^\{\\top\}gas the loss\-source projection onto modeiiand1/σi1/\\sigma\_\{i\}as its amplification factor\. For a non\-normal linearized operator withJ​J⊤≠J⊤​JJJ^\{\\top\}\\neq J^\{\\top\}J, eigenvalues do not capture this source\-pole geometry because distinct input and output directions can strongly amplify perturbations even when the eigenvalues appear well behaved\([Trefethen and Embree, 2005](https://arxiv.org/html/2608.23725#bib.bib19)\)\. The same geometry connects DEQ training to classical ill\-posed inverse problems and numerical linear algebra, since pseudoinverses make singular linear responses well defined\([Ben\-Israel and Greville, 2003](https://arxiv.org/html/2608.23725#bib.bib24)\), Tikhonov regularization gives a finite inverse by filtering every singular direction with a common regularization parameter\([Tikhonov, 1963](https://arxiv.org/html/2608.23725#bib.bib27)\), and Krylov methods such as the generalized minimal residual method \(GMRES\) make nonsymmetric adjoint solves practical without forming dense matrices\([Saad and Schultz, 1986](https://arxiv.org/html/2608.23725#bib.bib21)\)\. In this DEQ setting, the resulting finite adjoint response may also attenuate informative long\-range sensitivity together with the near\-singular component\.

We introduce Response Renormalization by identifyingG0​\(0\)=K−1G\_\{0\}\(0\)=K^\{\-1\}as the zero\-frequency response of the discrete fixed\-point trajectory andv=G0​\(0\)⊤​gv=G\_\{0\}\(0\)^\{\\top\}gas its adjoint action to control loss\-visible near\-singular amplification while preserving the remaining response directions\. Appendix[A\.1](https://arxiv.org/html/2608.23725#A1.SS1), Eqs\. \([24](https://arxiv.org/html/2608.23725#A1.E24)\)–\([35](https://arxiv.org/html/2608.23725#A1.E35)\), derives the finite\-horizon limit and separates solver index from physical time\. Response theory names the loss gradient the source, small singular denominators response poles, and a controlled denominator change a counterterm\([Martin et al\., 1973](https://arxiv.org/html/2608.23725#bib.bib12)\)\. The source\-response constructions of[Janssen \(1976\)](https://arxiv.org/html/2608.23725#bib.bib13)and[De Dominicis \(1976\)](https://arxiv.org/html/2608.23725#bib.bib14), together with their use in analyses of neural dynamics\([Schuecker et al\., 2016](https://arxiv.org/html/2608.23725#bib.bib15)\), motivate this terminology\. Applied to a DEQ, they provide a finite and explicit procedure that identifies loss\-aligned singular response directions and modifies only their unstable denominators, while retaining explicit forms inKK,K⊤​v=gK^\{\\top\}v=g, singular vectors, source projections, and modified denominators\.

Response Renormalization aims to preserve the original implicit response in stable directions while making only source\-visible critical response finite\. It implements this objective through Collective Mode Response Renormalization \(CMR\), a pole\-selective denominator lift, and through Phi\-adaptive CMR \(Phi\-CMR\) and Delta\-Phi \(Δ​Φ\\Delta\\Phi\) structured finite\-response and source\-conditioned gates, with dense and matrix\-free forms distinguishing exact from surrogate gradients\. We test selectivity through controlled mechanisms and source rotations, learned physical prediction through Darcy and native SILVA experiments, and transfer across equations and representations through 13 public datasets\.111The extension includes PDEArena\([Gupta and Brandstetter, 2023](https://arxiv.org/html/2608.23725#bib.bib37)\)\(Navier–Stokes 2D, Shallow Water 2D, Maxwell 3D, and Kuramoto–Sivashinsky 1D\), DynaBench\([Dulny et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib38)\)\(Advection, Burgers, Gas Dynamics, Kuramoto–Sivashinsky, Reaction–Diffusion, and Wave\), PDEGym\([Herde et al\., 2024](https://arxiv.org/html/2608.23725#bib.bib39)\)\(Poisson–Gauss\), CFDBench\([Luo et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib40)\)\(Cylinder Geometry\), and LagrangeBench\([Toshev et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib41)\)\(Taylor–Green Vortex 2D\)\.Consistent with this objective, the CMR and Phi\-CMR error ratios are at most 1\.05 in all 65 static comparisons and in 59 and 57 of 60 horizon\-seven comparisons, respectively\.

## 2Methods

Response Renormalization treats the DEQ backward pass as a controlled response in which, at equilibriumz⋆∈ℝdz^\{\\star\}\\in\\mathbb\{R\}^\{d\}, the adjoint propagates loss sourceg∈ℝdg\\in\\mathbb\{R\}^\{d\}through the transpose inverse of the residual Jacobian\. By leaving stable channels exact and lifting only resolved critical denominators, the construction recovers the original implicit adjoint when no critical response is present\. Appendix[B](https://arxiv.org/html/2608.23725#A2), Eqs\. \([36](https://arxiv.org/html/2608.23725#A2.E36)\)–\([39](https://arxiv.org/html/2608.23725#A2.E39)\), derives the adjoint and singular compatibility condition, and Appendix[C](https://arxiv.org/html/2608.23725#A3)derives the causal response construction\.

Vectors are columns,A⊤A^\{\\top\}denotes transpose,‖q‖2\\\|q\\\|\_\{2\}the Euclidean norm,diag⁡\(qi\)\\operatorname\{diag\}\(q\_\{i\}\)a diagonal matrix, and\[q\]\+=max⁡\{q,0\}\[q\]\_\{\+\}=\\max\\\{q,0\\\}\. All SVDs use the fixed state representation and train\-only normalization, while Appendix[D](https://arxiv.org/html/2608.23725#A4)gives the coordinate\-aware metric form\. Indicesnn,ii, andjjlabel solver steps, full singular modes, and computed low\-rank modes\. Euclidean interval projection is denoted by𝒫\[ℓ,u\]​\(q\)=arg⁡miny∈\[ℓ,u\]​12​\(y−q\)2\\mathcal\{P\}\_\{\[\\ell,u\]\}\(q\)=\\arg\\min\_\{y\\in\[\\ell,u\]\}\\tfrac\{1\}\{2\}\(y\-q\)^\{2\}, with its piecewise form and the full symbol and index registry in Appendix[E](https://arxiv.org/html/2608.23725#A5), Table[5](https://arxiv.org/html/2608.23725#A5.T5)\.

#### Response poles in DEQ adjoints\.

Letzn∈ℝdz\_\{n\}\\in\\mathbb\{R\}^\{d\}be the hidden state at solver indexnn, letxxbe the input, and letθ\\thetacontain the trainable parameters\. After adding a small sourcehn∈ℝdh\_\{n\}\\in\\mathbb\{R\}^\{d\}, writezn=z⋆\+ϕnz\_\{n\}=z^\{\\star\}\+\\phi\_\{n\}and defineJ=∂zfθ​\(z⋆,x\)∈ℝd×dJ=\\partial\_\{z\}f\_\{\\theta\}\(z^\{\\star\},x\)\\in\\mathbb\{R\}^\{d\\times d\}\. Linearization and a discrete Fourier transform give

zn\+1\\displaystyle z\_\{n\+1\}=fθ​\(zn,x\)\+hn,\\displaystyle=f\_\{\\theta\}\(z\_\{n\},x\)\+h\_\{n\},\(3\)ϕn\+1\\displaystyle\\phi\_\{n\+1\}=J​ϕn\+hn,\\displaystyle=J\\phi\_\{n\}\+h\_\{n\},\(4\)G0​\(ω\)\\displaystyle G\_\{0\}\(\\omega\)=\(ei​ω​I−J\)−1,G0​\(0\)=\(I−J\)−1=K−1\.\\displaystyle=\(e^\{\\mathrm\{i\}\\omega\}I\-J\)^\{\-1\},\\qquad G\_\{0\}\(0\)=\(I\-J\)^\{\-1\}=K^\{\-1\}\.\(5\)Hereϕn\\phi\_\{n\}is the equilibrium displacement,ω∈\[−π,π\]\\omega\\in\[\-\\pi,\\pi\]is frequency, and the bare propagatorG0​\(ω\)G\_\{0\}\(\\omega\)maps the transformed sourceh⁡\(ω\)h\(\\omega\)toϕ⁡\(ω\)\\phi\(\\omega\)\. Thus the residual JacobianK=I−JK=I\-Jis the inverse static response\. The loss sourceg=∇z⋆ℒ∈ℝdg=\\nabla\_\{z^\{\\star\}\}\\mathcal\{L\}\\in\\mathbb\{R\}^\{d\}entersK⊤​v=gK^\{\\top\}v=g, and the adjointv∈ℝdv\\in\\mathbb\{R\}^\{d\}converts state sensitivity into every parameter derivative\. In particular,

v=G0​\(0\)⊤​g=K−⁣⊤​g\.v=G\_\{0\}\(0\)^\{\\top\}g=K^\{\-\\top\}g\.\(6\)Appendix Eqs\. \([44](https://arxiv.org/html/2608.23725#A3.E44)\)–\([46](https://arxiv.org/html/2608.23725#A3.E46)\) and Eqs\. \([36](https://arxiv.org/html/2608.23725#A2.E36)\)–\([37](https://arxiv.org/html/2608.23725#A2.E37)\) give the Taylor, Fourier, and adjoint substitutions, including why one adjoint solve replaces one direct solve per parameter, while response\-field theory supplies this source\-response language\([Martin et al\., 1973](https://arxiv.org/html/2608.23725#bib.bib12);[Janssen, 1976](https://arxiv.org/html/2608.23725#bib.bib13);[De Dominicis, 1976](https://arxiv.org/html/2608.23725#bib.bib14)\)\. For a DEQ,ggis the source,K−1K^\{\-1\}is the response operator, and the singular values ofKKare its denominators\. Appendix[C](https://arxiv.org/html/2608.23725#A3)connects the same DEQ derivativesF\(p\)=∂zpfθ​\(z⋆,x\)F^\{\(p\)\}=\\partial\_\{z\}^\{p\}f\_\{\\theta\}\(z^\{\\star\},x\)to direct perturbations, the generating functional, Feynman expansion, Dyson dressing, and the finite counterterm\. Equations \([47](https://arxiv.org/html/2608.23725#A3.E47)\)–\([48](https://arxiv.org/html/2608.23725#A3.E48)\) then project these insertions onto the singular channels ofKK\. The method implements the resulting finite denominator through Eq\. \([9](https://arxiv.org/html/2608.23725#S2.E9)\), without numerically summing diagrams\.

For the singular\-value decompositionK=U​Σ​V⊤K=U\\Sigma V^\{\\top\}, columnsuiu\_\{i\}andviv\_\{i\}are output and source directions, whileΣ=diag⁡\(σ1,…,σd\)\\Sigma=\\operatorname\{diag\}\(\\sigma\_\{1\},\\ldots,\\sigma\_\{d\}\)has0≤σ1≤⋯≤σd0\\leq\\sigma\_\{1\}\\leq\\cdots\\leq\\sigma\_\{d\}\. Equation \([2](https://arxiv.org/html/2608.23725#S1.E2)\) therefore assigns channeliithe source amplitudevi⊤​gv\_\{i\}^\{\\top\}g, gain1/σi1/\\sigma\_\{i\}, and output directionuiu\_\{i\}, with its exact, regularized, and singular limits given by Appendix Eqs\. \([39](https://arxiv.org/html/2608.23725#A2.E39)\)–\([40](https://arxiv.org/html/2608.23725#A2.E40)\)\.

The following non\-normal example shows why eigenvalues alone do not contain this source\-to\-response geometry,

J=\(1−εgapM01−εgap\),K=I−J=\(εgap−M0εgap\),J=\\begin\{pmatrix\}1\-\\varepsilon\_\{\\rm gap\}&M\\\\ 0&1\-\\varepsilon\_\{\\rm gap\}\\end\{pmatrix\},\\qquad K=I\-J=\\begin\{pmatrix\}\\varepsilon\_\{\\rm gap\}&\-M\\\\ 0&\\varepsilon\_\{\\rm gap\}\\end\{pmatrix\},\(7\)the spectral gapεgap\>0\\varepsilon\_\{\\rm gap\}\>0fixes both eigenvalues at1−εgap1\-\\varepsilon\_\{\\rm gap\}, while the scalar non\-normal couplingMMcreates the response scaleM​εgap−2M\\varepsilon\_\{\\rm gap\}^\{\-2\}inK−1K^\{\-1\}, a susceptibility exposed by the singular vectors\([Trefethen and Embree, 2005](https://arxiv.org/html/2608.23725#bib.bib19)\)\.

#### Response\-renormalization construction\.

At thresholdκ\\kappa, the critical set𝒞=\{i∣σi<κ\}\\mathcal\{C\}=\\\{i\\mid\\sigma\_\{i\}<\\kappa\\\}contains modes with potentially large exact gains1/σi1/\\sigma\_\{i\}, whereas all remaining gains are bounded by1/κ1/\\kappa\. A critical mode is loss\-visible ifvi⊤​g≠0v\_\{i\}^\{\\top\}g\\neq 0, and affects parameteraaonly if additionallyui⊤​Ba≠0u\_\{i\}^\{\\top\}B\_\{a\}\\neq 0\. CMR obtains this partition from the singular\-value decomposition ofKK\([Golub and Kahan, 1965](https://arxiv.org/html/2608.23725#bib.bib20)\)and chooses positive effective denominatorsσieff\\sigma\_\{i\}^\{\\rm eff\}only fori∈𝒞i\\in\\mathcal\{C\}\. Appendix[B](https://arxiv.org/html/2608.23725#A2)derives all responses in common singular coordinates and separates the exact modified\-forward gradient from the backward surrogate\.

σieff=\{σi,i∉𝒞,max⁡\(σi,mi\),i∈𝒞,mi\>0\.\\sigma\_\{i\}^\{\\rm eff\}=\\begin\{cases\}\\sigma\_\{i\},&i\\notin\\mathcal\{C\},\\\\ \\max\(\\sigma\_\{i\},m\_\{i\}\),&i\\in\\mathcal\{C\},\\end\{cases\}\\qquad m\_\{i\}\>0\.\(8\)Hereκ\\kappais the critical resolution,mim\_\{i\}the response mass, andrκ=\|𝒞\|r\_\{\\kappa\}=\|\\mathcal\{C\}\|the active pole rank\. Dense evaluation finds𝒞\\mathcal\{C\}directly, while matrix\-free evaluation requestsr≥rκr\\geq r\_\{\\kappa\}smallest triplets\. WithU𝒞=\[ui\]i∈𝒞U\_\{\\mathcal\{C\}\}=\[u\_\{i\}\]\_\{i\\in\\mathcal\{C\}\}andV𝒞=\[vi\]i∈𝒞V\_\{\\mathcal\{C\}\}=\[v\_\{i\}\]\_\{i\\in\\mathcal\{C\}\}, the response lift is

δi\\displaystyle\\delta\_\{i\}=σieff−σi,Δ​K𝒞=U𝒞​diag⁡\(δi\)​V𝒞⊤,\\displaystyle=\\sigma\_\{i\}^\{\\rm eff\}\-\\sigma\_\{i\},\\qquad\\Delta K\_\{\\mathcal\{C\}\}=U\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\(\\delta\_\{i\}\)V\_\{\\mathcal\{C\}\}^\{\\top\},\(9\)RθR​\(z,x∣𝒜\)\\displaystyle R^\{R\}\_\{\\theta\}\(z,x\\mid\\mathcal\{A\}\)=z−fθ​\(z,x\)\+Δ​K𝒞​\(z−zref\),KR=∂zRθR=K\+Δ​K𝒞,\\displaystyle=z\-f\_\{\\theta\}\(z,x\)\+\\Delta K\_\{\\mathcal\{C\}\}\(z\-z\_\{\\rm ref\}\),\\qquad K^\{R\}=\\partial\_\{z\}R^\{R\}\_\{\\theta\}=K\+\\Delta K\_\{\\mathcal\{C\}\},\(10\)\(KR\)⊤​vR\\displaystyle\(K^\{R\}\)^\{\\top\}v^\{R\}=g,∇θRℒ=∂θℒ\+\(∂θfθ\)⊤​vR\.\\displaystyle=g,\\qquad\\nabla\_\{\\theta\}^\{R\}\\mathcal\{L\}=\\partial\_\{\\theta\}\\mathcal\{L\}\+\(\\partial\_\{\\theta\}f\_\{\\theta\}\)^\{\\top\}v^\{R\}\.\(11\)Here𝒜=\(zref,U𝒞,V𝒞,δ\)\\mathcal\{A\}=\(z\_\{\\rm ref\},U\_\{\\mathcal\{C\}\},V\_\{\\mathcal\{C\}\},\\delta\)is held fixed during one update\. Atzrefz\_\{\\rm ref\}, substitution of Eq\. \([9](https://arxiv.org/html/2608.23725#S2.E9)\) into Eq\. \([10](https://arxiv.org/html/2608.23725#S2.E10)\) givesKR=U​Σeff​V⊤K^\{R\}=U\\Sigma\_\{\\rm eff\}V^\{\\top\}, and hence

vCMR=U​diag⁡\(\(σieff\)−1\)​V⊤​g\.v\_\{\\rm CMR\}=U\\operatorname\{diag\}\(\(\\sigma\_\{i\}^\{\\rm eff\}\)^\{\-1\}\)V^\{\\top\}g\.\(12\)Equations \([9](https://arxiv.org/html/2608.23725#S2.E9)\)–\([11](https://arxiv.org/html/2608.23725#S2.E11)\) define the modified DEQ and its exact frozen\-anchor gradient, whereas applying only Eq\. \([12](https://arxiv.org/html/2608.23725#S2.E12)\) at the original equilibrium gives the separately evaluated surrogate\. Appendix Eqs\. \([49](https://arxiv.org/html/2608.23725#A3.E49)\)–\([50](https://arxiv.org/html/2608.23725#A3.E50)\) give the finite operator matching, while Eqs\. \([41](https://arxiv.org/html/2608.23725#A2.E41)\)–\([42](https://arxiv.org/html/2608.23725#A2.E42)\) give the exact and surrogate boundaries, and Algorithm[2](https://arxiv.org/html/2608.23725#algorithm2)implements the complete construction\. For comparison, generalized inverses define the singular reference\([Ben\-Israel and Greville, 2003](https://arxiv.org/html/2608.23725#bib.bib24)\), while Tikhonov regularization\([Tikhonov, 1963](https://arxiv.org/html/2608.23725#bib.bib27)\)uses

Fμ​\(σ\)=σσ2\+μ2,F\_\{\\mu\}\(\\sigma\)=\\frac\{\\sigma\}\{\\sigma^\{2\}\+\\mu^\{2\}\},\(13\)Stable\-critical filtering leaves the exact gain on stable modes and applies ridge only within𝒞\\mathcal\{C\}, providing a critical\-only ridge comparator\. CMR instead lifts each critical denominator and leaves stable response exact\.

Ifσieff≥mmin\>0\\sigma\_\{i\}^\{\\rm eff\}\\geq m\_\{\\min\}\>0on𝒞\\mathcal\{C\}andσi≥κ\\sigma\_\{i\}\\geq\\kappaoff𝒞\\mathcal\{C\}, then

‖vCMR‖2≤\(∑i∉𝒞\|vi⊤​g\|2σi2\+∑i∈𝒞\|vi⊤​g\|2mmin2\)1/2\.\\\|v\_\{\\rm CMR\}\\\|\_\{2\}\\leq\\left\(\\sum\_\{i\\notin\\mathcal\{C\}\}\\frac\{\|v\_\{i\}^\{\\top\}g\|^\{2\}\}\{\\sigma\_\{i\}^\{2\}\}\+\\sum\_\{i\\in\\mathcal\{C\}\}\\frac\{\|v\_\{i\}^\{\\top\}g\|^\{2\}\}\{m\_\{\\min\}^\{2\}\}\\right\)^\{1/2\}\.\(14\)The stable\-sector response is unchanged, while the induced bias remains confined to the critical sector,

vCMR−v=∑i∈𝒞\(vi⊤​g\)​\(\(σieff\)−1−σi−1\)​ui\.v\_\{\\rm CMR\}\-v=\\sum\_\{i\\in\\mathcal\{C\}\}\(v\_\{i\}^\{\\top\}g\)\\left\(\(\\sigma\_\{i\}^\{\\rm eff\}\)^\{\-1\}\-\\sigma\_\{i\}^\{\-1\}\\right\)u\_\{i\}\.\(15\)

#### Collective and matrix\-free algorithm\.

Dense singular\-value decompositions \(SVDs\)\([Golub and Kahan, 1965](https://arxiv.org/html/2608.23725#bib.bib20)\)expose the full\-state mechanism, whereas a structured specialization acts on a small collective denominator when a local–global factorization is available\. For

K=KL−A​B⊤,K=K\_\{L\}\-AB^\{\\top\},\(16\)with inexpensiveKL−1K\_\{L\}^\{\-1\}products and low\-rankA,BA,B, the Woodbury identity gives\([Woodbury, 1950](https://arxiv.org/html/2608.23725#bib.bib28)\)

K−1=KL−1\+KL−1​A​\(I−B⊤​KL−1​A\)−1​B⊤​KL−1\.K^\{\-1\}=K\_\{L\}^\{\-1\}\+K\_\{L\}^\{\-1\}A\(I\-B^\{\\top\}K\_\{L\}^\{\-1\}A\)^\{\-1\}B^\{\\top\}K\_\{L\}^\{\-1\}\.\(17\)The small matrix

Γ=I−B⊤​KL−1​A\\Gamma=I\-B^\{\\top\}K\_\{L\}^\{\-1\}A\(18\)is the collective denominator whose critical singular values CMR lifts\. Its singular values need not equal those of the fullKKbecause Woodbury only localizes the low\-rank correction\. The rank\-one identity of[Sherman and Morrison \(1950\)](https://arxiv.org/html/2608.23725#bib.bib29)is the one\-mode case\. Jacobian\-vector products \(JVPs\), vector\-Jacobian products \(VJPs\), and the generalized minimal residual method \(GMRES\)\([Saad and Schultz, 1986](https://arxiv.org/html/2608.23725#bib.bib21)\)then require onlyz⋆z^\{\\star\}, products withKKandK⊤K^\{\\top\}, and a low\-rank critical subspace\. Appendix[C](https://arxiv.org/html/2608.23725#A3)derives the matrix\-free products and collective adjoint in Eq\. \([43](https://arxiv.org/html/2608.23725#A3.E43)\)\. Appendix[F](https://arxiv.org/html/2608.23725#A6)gives Algorithms[1](https://arxiv.org/html/2608.23725#algorithm1)–[3](https://arxiv.org/html/2608.23725#algorithm3), covering the conceptual pass, full DEQ procedure, and Structured Implicit Layers and Vector Attractors \(SILVA\) local–global specialization\([Silva, 2026a](https://arxiv.org/html/2608.23725#bib.bib32)\)\. Dense or matrix\-free low\-rank estimates\([Golub and Kahan, 1965](https://arxiv.org/html/2608.23725#bib.bib20);[Lehoucq et al\., 1998](https://arxiv.org/html/2608.23725#bib.bib22)\)and bounded parametersm0,κ,r,αmax,λ,cmaxm\_\{0\},\\kappa,r,\\alpha\_\{\\max\},\\lambda,c\_\{\\max\}complete the numerical specification\.

#### Phi\-CMR and Delta\-Phi\.

CMR still needs a finite response scale, which fixed CMR declares directly and Phi\-CMR parameterizes through a finite\-response target,

Gi​\(0\)=χR,miΦ=1χR,δ​miΦ=\[miΦ−σi\]\+\.G\_\{i\}\(0\)=\\chi\_\{R\},\\qquad m\_\{i\}^\{\\Phi\}=\\frac\{1\}\{\\chi\_\{R\}\},\\qquad\\delta m\_\{i\}^\{\\Phi\}=\[m\_\{i\}^\{\\Phi\}\-\\sigma\_\{i\}\]\_\{\+\}\.\(19\)HereGi​\(0\)G\_\{i\}\(0\),χR\>0\\chi\_\{R\}\>0,miΦ=χR−1m\_\{i\}^\{\\Phi\}=\\chi\_\{R\}^\{\-1\}, andδ​miΦ\\delta m\_\{i\}^\{\\Phi\}are the channel response, target susceptibility, inverse response mass, and nonnegative lift aboveσi\\sigma\_\{i\}, respectively\. The positive monotone per\-mode and collective parameterizations used in the experiments are

miΦ\\displaystyle m\_\{i\}^\{\\Phi\}=m0​\(1\+m0σi\+m0\),m0≤miΦ≤2​m0,\\displaystyle=m\_\{0\}\\left\(1\+\\frac\{m\_\{0\}\}\{\\sigma\_\{i\}\+m\_\{0\}\}\\right\),\\qquad m\_\{0\}\\leq m\_\{i\}^\{\\Phi\}\\leq 2m\_\{0\},\(20\)m𝒞Φ\\displaystyle m\_\{\\mathcal\{C\}\}^\{\\Phi\}=m0​\[1\+\(αmax−1\)​p𝒞\],p𝒞=𝒫\[0,1\]​\(κ−σ𝒞κ\),\\displaystyle=m\_\{0\}\\left\[1\+\(\\alpha\_\{\\max\}\-1\)p\_\{\\mathcal\{C\}\}\\right\],\\qquad p\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[0,1\]\}\\\!\\left\(\\frac\{\\kappa\-\\sigma\_\{\\mathcal\{C\}\}\}\{\\kappa\}\\right\),\(21\)whereσ𝒞=mini∈𝒞⁡σi\\sigma\_\{\\mathcal\{C\}\}=\\min\_\{i\\in\\mathcal\{C\}\}\\sigma\_\{i\},p𝒞∈\[0,1\]p\_\{\\mathcal\{C\}\}\\in\[0,1\],m0\>0m\_\{0\}\>0, andαmax≥1\\alpha\_\{\\max\}\\geq 1are the smallest selected denominator, pole pressure, base mass, and maximum multiplier\. For a structured operator,σ𝒞\\sigma\_\{\\mathcal\{C\}\}is the smallest selected singular value ofΓ\\Gamma\. These are prescribed positive DEQ parameterizations, not unique derivations or automatically learned corrections\. Appendix[G](https://arxiv.org/html/2608.23725#A7)gives their Hartree\-style motivation\([Baym, 1962](https://arxiv.org/html/2608.23725#bib.bib30);[Cornwall et al\., 1974](https://arxiv.org/html/2608.23725#bib.bib31)\)and proves positivity, boundedness, and monotonicity\.

Delta\-Phi adds the source geometry that CMR uses diagnostically through a gate that measures the critical source fractiona𝒞a\_\{\\mathcal\{C\}\}, pole pressurep𝒞p\_\{\\mathcal\{C\}\}, and signed source gates𝒞s\_\{\\mathcal\{C\}\}before projecting the response mass onto a positive admissible interval\.

a𝒞\\displaystyle a\_\{\\mathcal\{C\}\}=∑i∈𝒞\(vi⊤​g\)2∑i=1d\(vi⊤​g\)2\+εden,\\displaystyle=\\frac\{\\sum\_\{i\\in\\mathcal\{C\}\}\(v\_\{i\}^\{\\top\}g\)^\{2\}\}\{\\sum\_\{i=1\}^\{d\}\(v\_\{i\}^\{\\top\}g\)^\{2\}\+\\varepsilon\_\{\\rm den\}\},\(22\)p𝒞\\displaystyle p\_\{\\mathcal\{C\}\}=𝒫\[0,1\]​\(κ−σ𝒞κ\),s𝒞=𝒫\[−1,1\]​\(2​a𝒞−1\),\\displaystyle=\\mathcal\{P\}\_\{\[0,1\]\}\\\!\\left\(\\frac\{\\kappa\-\\sigma\_\{\\mathcal\{C\}\}\}\{\\kappa\}\\right\),\\qquad s\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[\-1,1\]\}\(2a\_\{\\mathcal\{C\}\}\-1\),mΔ​Φ\\displaystyle m^\{\\Delta\\Phi\}=𝒫\[m0,cmax​m0\]​\(mΦ\+m0​λ​p𝒞​s𝒞\)\.\\displaystyle=\\mathcal\{P\}\_\{\[m\_\{0\},c\_\{\\max\}m\_\{0\}\]\}\\\!\\left\(m^\{\\Phi\}\+m\_\{0\}\\lambda p\_\{\\mathcal\{C\}\}s\_\{\\mathcal\{C\}\}\\right\)\.\(23\)
Herea𝒞∈\[0,1\]a\_\{\\mathcal\{C\}\}\\in\[0,1\],s𝒞∈\[−1,1\]s\_\{\\mathcal\{C\}\}\\in\[\-1,1\],λ≥0\\lambda\\geq 0, andcmax≥1c\_\{\\max\}\\geq 1are critical source energy, signed activation, correction strength, and upper\-mass multiplier, whileεden\>0\\varepsilon\_\{\\rm den\}\>0prevents division by zero\. This bounded source\-conditioned rule replaces a free multilayer perceptron \(MLP\), and Appendix[G](https://arxiv.org/html/2608.23725#A7)derives its source\-conditioned potential, bounds, and activation limits\. The per\-mode formulation usesmi=max⁡\(miΦ,mΔ​Φ\)m\_\{i\}=\\max\(m\_\{i\}^\{\\Phi\},m^\{\\Delta\\Phi\}\), while the collective formulation uses the scalarmΔ​Φm^\{\\Delta\\Phi\}directly\. The per\-mode field formulation is therefore lift\-only relative to Phi\-CMR, whereas the collective formulation can reduce the Phi mass toward the positive floor when the source avoids the pole\. In either case Eq\. \([8](https://arxiv.org/html/2608.23725#S2.E8)\) turns the selected mass into the effective denominator\. Appendix[H](https://arxiv.org/html/2608.23725#A8)defines the metrics and parameters, and Algorithms[2](https://arxiv.org/html/2608.23725#algorithm2)and[3](https://arxiv.org/html/2608.23725#algorithm3)give the per\-mode DEQ and scalar collective SILVA procedures\.

![Refer to caption](https://arxiv.org/html/2608.23725v1/phase49_pole_response_panel.png)Figure 1:Pole\-response maps with \(a\) dense pole strength−log10⁡σmin\-\\log\_\{10\}\\sigma\_\{\\min\}, \(b\) the Phi\-CMR mass multiplier, \(c\)RΦ−CMR−RTikR\_\{\\Phi\{\\rm\-CMR\}\}\-R\_\{\\rm Tik\}at critical rank one, and \(d\) the same difference at critical rank two over target gap and state dimension\.

## 3Results and Discussion

![Refer to caption](https://arxiv.org/html/2608.23725v1/phase49_mechanism_panel.png)Figure 2:Selective renormalization of the near\-pole response with \(a\) exact and filtered response amplification versus singular value, \(b\) original and renormalized denominators for stable and critical modes, \(c\) Phi\-CMR mass versus pole pressure by state dimension and critical rank, and \(d\) retained response versus residual against the original adjoint equation\.The experiments test whether pole strength predicts the adjoint while a selective lift preserves resolved stable response, whether source rotation activates the effect throughvi⊤​gv\_\{i\}^\{\\top\}g, and whether transfer retains fidelity through physical\-time rollouts and new representations\. Table[1](https://arxiv.org/html/2608.23725#S3.T1)quantifies the first two mechanisms, with Phi\-CMR improving amplitude and direction over ridge on the collective grid and CMR improving norm ratio and relative error under source alignment\. Appendix[H](https://arxiv.org/html/2608.23725#A8)and Table[6](https://arxiv.org/html/2608.23725#A8.T6)give the metrics, protocols, extended results, and validity conditions, while native SILVA links the controlled tests to physical data\. Together they evaluate directional selectivity and predictive fidelity for deliberately biased backward interventions\.

Table 1:Directional dense\-adjoint evidence\. Entries are mean±\\pmstandard deviation over configurations\.RmR\_\{m\}is the norm ratio,CmC\_\{m\}is directional cosine,PmP\_\{m\}is signed projection onto the exact adjoint, andEmE\_\{m\}is relative adjoint error\.The first sweep tests Eq\. \([2](https://arxiv.org/html/2608.23725#S1.E2)\) against the exact inverse, ridge\([Tikhonov, 1963](https://arxiv.org/html/2608.23725#bib.bib27)\), stable\-critical filtering, CMR, and Phi\-CMR\. In Fig\.[1](https://arxiv.org/html/2608.23725#S2.F1), yellow at the smallest gaps in panel \(a\) marks the1/σi1/\\sigma\_\{i\}pole, while the matching yellow in panel \(b\) is the bounded rise ofm𝒞Φm\_\{\\mathcal\{C\}\}^\{\\Phi\}prescribed by Eq\. \([21](https://arxiv.org/html/2608.23725#S2.E21)\)\. The positive dark\-to\-yellow fields in panels \(c,d\) show greater response retention than Tikhonov at critical ranks one and two, and Table[1](https://arxiv.org/html/2608.23725#S3.T1)confirms that this gain maintains higher mean directional alignment with the exact adjoint than Tikhonov\. Appendix[I](https://arxiv.org/html/2608.23725#A9), Fig\.[6](https://arxiv.org/html/2608.23725#A9.F6), provides the rank\-two pole and mass maps\.

Figure[2](https://arxiv.org/html/2608.23725#S3.F2)summarizes a controlled singular\-spectrum experiment at target gaps0\.020\.02,0\.010\.01, and0\.0050\.005, state dimensions 12, 16, and 24, and critical ranks one and two, comparing how each backward method treats stable and near\-singular modes\. In panel \(a\), the exact inverse diverges as the singular value approaches zero, Tikhonov suppresses that direction toward zero, and CMR and Phi\-CMR instead cap it at a finite nonzero value before recovering the exact response outside the critical region\. Panel \(b\) makes this selectivity explicit because the critical denominator increases from 0\.005 to 0\.030 under CMR and 0\.0581 under Phi\-CMR, while the stable denominator remains fixed at 0\.125\. Panel \(c\) visualizes the prescribed dependence of Phi\-CMR mass on pole pressure, and its consistency across dimensions and critical ranks verifies that the prescribed response law does not explicitly depend on model size\. Panel \(d\) then shows that CMR and Phi\-CMR retain more of the exact response than stable\-critical filtering and Tikhonov as a function of residual against the original equation after the deliberate lift, while the renormalized\-equation residual stays at solver tolerance, confirming that the displacement records the intended modification rather than solver failure\. Together, panels \(a–d\) show that Response Renormalization removes singular amplification without globally damping the backward signal, consistent with Eq\. \([14](https://arxiv.org/html/2608.23725#S2.E14)\) and Table[1](https://arxiv.org/html/2608.23725#S3.T1)\.

![Refer to caption](https://arxiv.org/html/2608.23725v1/phase64_darcy_bridge_main_panel.png)Figure 3:Darcy field bridge with \(a\) target field, \(b\) Phi\-CMR absolute error, \(c\) Tikhonov\-minus\-Phi\-CMR absolute\-error difference, \(d\) source\-weighted pole mode, \(e\) coefficient field with solution contours, \(f\) Delta\-Phi absolute error with pole contours\.At fixed operator, the gain over Tikhonov increases withvi⊤​gv\_\{i\}^\{\\top\}g, and the controlled source intervention directly tests this loss visibility \(Appendix Fig\.[7](https://arxiv.org/html/2608.23725#A9.F7)\)\. Structured holdouts further identify pole geometry for CMR, monotone mass for Phi\-CMR, and bounded source activation for Delta\-Phi as the necessary components, whereas free multilayer perceptrons drift off\-axis \(Appendix Figs\.[8](https://arxiv.org/html/2608.23725#A9.F8)and[9](https://arxiv.org/html/2608.23725#A9.F9)\)\.

Figure[3](https://arxiv.org/html/2608.23725#S3.F3)connects the modal law to a physical field by aligning the solution and coefficient in panels \(a,e\), showing that Phi\-CMR error is spatially localized, and panel \(c\) uses red for locations where Tikhonov error is larger and blue where Phi\-CMR error is larger\. Panel \(d\) visualizes the source visibility\|vi⊤​g\|\|v\_\{i\}^\{\\top\}g\|applied to the near\-pole output structure in Eq\. \([2](https://arxiv.org/html/2608.23725#S1.E2)\), while the contours in panel \(f\) relate the bounded correction in Eq\. \([23](https://arxiv.org/html/2608.23725#S2.E23)\) to that geometry\. The corresponding relative errors are 1\.21 for Phi\-CMR and Delta\-Phi, 1\.31 for CMR, 1\.56 for Tikhonov, and 1\.58 for stable\-critical filtering\. The response–suppression analysis places CMR and Phi\-CMR between exact response and stronger spectral damping, confirming selective rather than global attenuation \(Table[7](https://arxiv.org/html/2608.23725#A8.T7), Figs\.[7](https://arxiv.org/html/2608.23725#A9.F7)–[8](https://arxiv.org/html/2608.23725#A9.F8)\)\. Errors above the unit zero\-prediction reference indicate a shared forward approximation limit, whereas spatial differences identify the source\-aligned backward effect tested in Appendix[J\.1](https://arxiv.org/html/2608.23725#A10.SS1)\.

The same response law transfers to SILVA\([Silva, 2026a](https://arxiv.org/html/2608.23725#bib.bib32)\)on PDEBench and The Well\([Takamoto et al\., 2022](https://arxiv.org/html/2608.23725#bib.bib25);[Ohana et al\., 2024](https://arxiv.org/html/2608.23725#bib.bib34)\)\. Appendix[A](https://arxiv.org/html/2608.23725#A1)gives the matched design, baseline roles, and shared forward protocol, with CMR and Phi\-CMR alone modifying the small collective denominator while Delta\-Phi remains the source\-conditioned ablation\. Figure[5](https://arxiv.org/html/2608.23725#A1.F5)and Table[2](https://arxiv.org/html/2608.23725#S3.T2)quantify fidelity, activation, convergence, and cost: CMR has the lowest one\-step error, JFB the lowest cost, and Phi\-CMR lower observed error than Tikhonov in 27 of 50 paired comparisons\.

Table 2:All native SILVA baselines over ten one\-step families and nine dynamic families\. Errors and time are normalized by exact implicit differentiation\.Appendix[A\.1](https://arxiv.org/html/2608.23725#A1.SS1)distinguishes and derives the solver\-index and physical\-time limits, while the Phi\-CMR error ratio is at most 1\.05 in 43 of 45 horizon\-eight comparisons and its finite causal\-response error decreases from0\.4110\.411at one solver step to2\.50×10−42\.50\\times 10^\{\-4\}at 64 steps without assigning the modified backward endpoint a new PDE time scale\.

Across 13 further datasets, the Phi\-CMR error ratio is at most 1\.05 in all 65 static and 57 of 60 horizon\-seven comparisons, with sparse activation in every dataset \(Appendix[J](https://arxiv.org/html/2608.23725#A10)\)\. It also lowers error in all 30 high\-error capacity cases \(Appendix[J\.1](https://arxiv.org/html/2608.23725#A10.SS1)\), and Appendices[K](https://arxiv.org/html/2608.23725#A11)and[L](https://arxiv.org/html/2608.23725#A12)give the validity conditions and field–pole realizations\.

## 4Conclusion

Response Renormalization formulates near\-critical DEQ/SILVA sensitivity as a source\-visible pole problem by decomposing the adjoint into local and low\-dimensional collective terms\. CMR lifts only critical denominators, Phi\-CMR selects their positive masses, and Delta\-Phi bounds source conditioning\. The implicit response is recovered without a source\-visible critical mode, while dense CMR preserves resolved stable singular channels ofKKand SILVA preserves unliftedΓ\\Gammamodes\. Controlled source rotations verify the predictedvi⊤​gv\_\{i\}^\{\\top\}gdependence, which the Darcy field bridge localizes against the shared forward error\.

Pole sweeps, source rotations, and eight\-method comparisons establish directional fidelity, predictive preservation, cost, and solver\-index convergence across equations and representations\. Capacity scaling associates part of predictive error with forward capacity, while joint adjoint–rollout analyses distinguish solver from PDE time\. Source\-visible activation and\(σ𝒞,a𝒞,ρR\)\(\\sigma\_\{\\mathcal\{C\}\},a\_\{\\mathcal\{C\}\},\\rho\_\{R\}\)respectively diagnose backward criticality and its pole pressure, loss coupling, and response consistency\.

Together, these results define an operating regime where modifying only loss\-visible poles addresses near\-criticality without damping the full adjoint response or disturbing resolved modes\. Across 23 families of PDE grids, three\-dimensional fields, operator maps, irregular geometries, and particles, capacity extensions reduce field error while solver\-index and physical\-time tests separate backward response from forward dynamics, thereby making singular inversion a finite response problem shared by dense DEQs and SILVA\.

## Data and Software Availability

The physical benchmark data used in this study are publicly available\. Native experiments use PDEBench and The Well\([Takamoto et al\., 2022](https://arxiv.org/html/2608.23725#bib.bib25);[Ohana et al\., 2024](https://arxiv.org/html/2608.23725#bib.bib34)\), whereas cross\-benchmark experiments use PDEArena, DynaBench, PDEGym Poisson–Gauss, CFDBench, and LagrangeBench\([Gupta and Brandstetter, 2023](https://arxiv.org/html/2608.23725#bib.bib37);[Dulny et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib38);[Herde et al\., 2024](https://arxiv.org/html/2608.23725#bib.bib39);[Luo et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib40);[Toshev et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib41)\)\. Controlled mechanism studies use procedurally generated synthetic systems as specified in the experimental descriptions\. Dataset versions, data partitions, temporal offsets, train\-only normalization, random seeds, model settings, and solver tolerances are specified in Appendix[A\.2](https://arxiv.org/html/2608.23725#A1.SS2)\. SILVA Networks is archived on Zenodo\([Silva, 2026b](https://arxiv.org/html/2608.23725#bib.bib33)\)\.

## Acknowledgments

The broader research program from which this study emerged was initiated within the Swedish National Infrastructure for Computing \(SNIC\) Small Compute project*Artificial Intelligence for Physics and Engineering, Modeling and Simulation*, Project No\. SNIC 2022/22\-843, conducted at Linköping University under the author’s principal investigatorship\. This work was supported by the Brazilian National Council for Scientific and Technological Development \(CNPq\) under grant No\. 445344/2024\-5\. The author also acknowledges financial support provided through the Program Talentos Brasil, as Project Investigator\.

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## Appendix ASILVA Experimental Design

The SILVA study compares backward response geometry within a shared forward architecture and numerical protocol\. CMR and Phi\-CMR enter only as automatic\-differentiation backward operators, with matrix\-free VJP/GMRES products for local adjoints and a modification confined to the small collective denominator\. A no\-lift CMR case therefore recovers the original implicit gradient, whereas a lifted case is explicitly labeled as a backward\-response surrogate\. The eight matched modes are exact implicit, Phantom, Neumann, JFB, SHINE, full\-operator Tikhonov, CMR, and Phi\-CMR, allowing predictive differences to be interpreted as consequences of the backward rule under this controlled design\.

The evaluation contains five Darcy regimes and nine additional families, namely Advection, Burgers, Reaction\-Diffusion 1D, Diffusion\-Sorption 1D, Diffusion\-Reaction 2D, Shallow Water 2D, compressible CFD 1D, incompressible Navier–Stokes 2D, and The Well turbulent radiative layer 2D\. Five Darcy regimes count as one family, giving ten physical families, five seeds, and 560 method evaluations\. The selected data subsets occupy 120\.685 GB in total, with the individual storage footprints reported in Table[3](https://arxiv.org/html/2608.23725#A1.T3)\.

Table 3:Storage footprints of the selected data subsets\. Values are decimal gigabytes \(1​GB=1091~\\mathrm\{GB\}=10^\{9\}bytes\) and exclude unused portions of the complete source collections\.Native datasetGBCross\-benchmark datasetGBPDEBench Darcy 2D \(five regimes\)6\.554PDEArena Navier–Stokes 2D0\.826PDEBench Advection 1D8\.233PDEArena Shallow Water 2D0\.089PDEBench Burgers 1D8\.233PDEArena Maxwell 3D1\.887PDEBench Reaction–Diffusion 1D4\.137PDEArena Kuramoto–Sivashinsky 1D0\.931PDEBench Diffusion–Sorption 1D4\.217DynaBench Advection2\.558PDEBench Diffusion–Reaction 2D13\.244DynaBench Burgers5\.090PDEBench Shallow Water 2D6\.626DynaBench Gas Dynamics10\.156PDEBench compressible CFD 1D12\.411DynaBench Kuramoto–Sivashinsky2\.558PDEBench incompressible Navier–Stokes 2D9\.906DynaBench Reaction–Diffusion5\.090The Well turbulent radiative layer 2D7\.149DynaBench Wave5\.090PDEGym Poisson–Gauss2\.624CFDBench Cylinder Geometry2\.621LagrangeBench Taylor–Green Vortex 2D0\.454Native subtotal80\.710Cross\-benchmark subtotal39\.975Combined total120\.685Every method receives the same initialization, ordered 16 minibatches, a fixed 64\-update direct or 128\-update residual budget, and identical optimizer, forward\-solver, and tolerance settings\. Because the backward rule changes parameter updates, trained parameters and equilibria may differ across methods\. Training and test ranges are disjoint, and The Well uses its dataset\-provided training and test partitions, and normalization is fitted only on training data\. The error metric inverse\-standardizes the prediction and target, flattens all stored output entries per sample, and computes‖y^i−yi‖2/max⁡\(‖yi‖2,10−12\)\\\|\{\\widehat\{y\}\}\_\{i\}\-y\_\{i\}\\\|\_\{2\}/\\max\(\\\|y\_\{i\}\\\|\_\{2\},10^\{\-12\}\), then averages samples\. For mixed\-unit channels this is a descriptive numerical score that can be dominated by larger\-scale variables rather than a dimensionally homogeneous physical norm\.

### A\.1All\-method baselines and transient derivations

Tikhonov is the primary*mechanistic*reference because it regularizes the same inverse response, exact implicit differentiation supplies the fidelity reference, and Phantom and Neumann\[[Geng et al\., 2021](https://arxiv.org/html/2608.23725#bib.bib4)\], JFB\[[Fung et al\., 2022](https://arxiv.org/html/2608.23725#bib.bib35)\], and SHINE\[[Ramzi et al\., 2022](https://arxiv.org/html/2608.23725#bib.bib36)\]provide published inexact\-backward or Jacobian\-free training references for comparison with the CMR and Phi\-CMR structured inverses\. Table[2](https://arxiv.org/html/2608.23725#S3.T2)separates the resulting accuracy–cost profiles, with CMR giving the lowest mean one\-step error ratio \(0\.986\) and JFB remaining substantially faster than the structured methods\. This distinction matters because agreement with Tikhonov tests the regularization mechanism, whereas agreement with the broader baseline set tests whether that mechanism remains competitive as a training rule\.

Two transient scales leave the static SILVA backward operator unchanged, with the first describing convergence along the*solver index*\. Linearizing the original fixed\-point map at a converged state gives

δ​zn\+1=J​δ​zn,J=∂zfθ​\(z⋆,x\),K=I−J\.\\delta z\_\{n\+1\}=J\\,\\delta z\_\{n\},\\qquad J=\\partial\_\{z\}f\_\{\\theta\}\(z^\{\\star\},x\),\\qquad K=I\-J\.\(24\)For a state\-space loss sourcegg, define the causal finite\-horizon adjoint by the zero\-initialized recurrence

v0=0,vn\+1=g\+J⊤​vn\.v\_\{0\}=0,\\qquad v\_\{n\+1\}=g\+J^\{\\top\}v\_\{n\}\.\(25\)Substitution givesv1=gv\_\{1\}=gandv2=g\+J⊤​gv\_\{2\}=g\+J^\{\\top\}g\. Induction therefore yields

vN=∑n=0N−1\(J⊤\)n​g,vN→N→∞\(I−J⊤\)−1​g=K−⁣⊤​g=:v∞,v\_\{N\}=\\sum\_\{n=0\}^\{N\-1\}\(J^\{\\top\}\)^\{n\}g,\\qquad v\_\{N\}\\xrightarrow\[N\\to\\infty\]\{\}\(I\-J^\{\\top\}\)^\{\-1\}g=K^\{\-\\top\}g=:v\_\{\\infty\},\(26\)when the iteration is stable\. The convergence statement follows directly from the finite geometric\-series identity

K⊤​vN\\displaystyle K^\{\\top\}v\_\{N\}=\(I−J⊤\)​∑n=0N−1\(J⊤\)n​g=\[I−\(J⊤\)N\]​g,\\displaystyle=\(I\-J^\{\\top\}\)\\sum\_\{n=0\}^\{N\-1\}\(J^\{\\top\}\)^\{n\}g=\\bigl\[I\-\(J^\{\\top\}\)^\{N\}\\bigr\]g,v∞−vN\\displaystyle v\_\{\\infty\}\-v\_\{N\}=\(J⊤\)N​v∞,\\displaystyle=\(J^\{\\top\}\)^\{N\}v\_\{\\infty\},\(27\)K⊤​vN−g\\displaystyle K^\{\\top\}v\_\{N\}\-g=−\(J⊤\)N​g\.\\displaystyle=\-\(J^\{\\top\}\)^\{N\}g\.\(28\)Consequently,

‖vN−v∞‖2‖v∞‖2≤‖\(J⊤\)N‖2,‖K⊤​vN−g‖2‖g‖2≤‖\(J⊤\)N‖2\.\\frac\{\\\|v\_\{N\}\-v\_\{\\infty\}\\\|\_\{2\}\}\{\\\|v\_\{\\infty\}\\\|\_\{2\}\}\\leq\\\|\(J^\{\\top\}\)^\{N\}\\\|\_\{2\},\\qquad\\frac\{\\\|K^\{\\top\}v\_\{N\}\-g\\\|\_\{2\}\}\{\\\|g\\\|\_\{2\}\}\\leq\\\|\(J^\{\\top\}\)^\{N\}\\\|\_\{2\}\.\(29\)These bounds also explain why a non\-normalJJcan show finite\-NNtransient amplification even when its spectral radius is below one\. Equations \([25](https://arxiv.org/html/2608.23725#A1.E25)\)–\([29](https://arxiv.org/html/2608.23725#A1.E29)\) are the quantities evaluated in Fig\.[4](https://arxiv.org/html/2608.23725#A1.F4)\(b,c\)\. The independent causal\-kernel derivation appears in Eqs\. \([45](https://arxiv.org/html/2608.23725#A3.E45)\)–\([46](https://arxiv.org/html/2608.23725#A3.E46)\)\.

SILVA enters only at the static backward endpoint, whereb=KL−⁣⊤​gb=K\_\{L\}^\{\-\\top\}g,Y=KL−⁣⊤​BY=K\_\{L\}^\{\-\\top\}B,h=A⊤​bh=A^\{\\top\}b, and collective liftΓeff\\Gamma\_\{\\rm eff\}give the CMR/Phi\-CMR response

vR=b\+Y​Γeff−⁣⊤​h,Γ=I−B⊤​KL−1​A\.v^\{R\}=b\+Y\\Gamma\_\{\\rm eff\}^\{\-\\top\}h,\\qquad\\Gamma=I\-B^\{\\top\}K\_\{L\}^\{\-1\}A\.\(30\)Because this expression is a collective backward surrogate unless no lift occurs, it does not define a full\-stateU​Σeff​V⊤U\\Sigma\_\{\\rm eff\}V^\{\\top\}, while the transient evaluation instead constructsvNv\_\{N\}andv∞v\_\{\\infty\}from the*original*JJfor each trained model\. By neither iterating a renormalizedJR=I−KRJ^\{R\}=I\-K^\{R\}nor interpreting an effective denominator as a physical time constant, the evaluation tests the causal approach to the converged static adjoint rather than a new finite\-frequency SILVA law\.

The second scale concerns*physical PDE time*\. Let𝒫\\mathcal\{P\}be the reference one\-step evolution and let𝒫θ\\mathcal\{P\}\_\{\\theta\}be the learned one\-step map, evaluated with the inner SILVA equilibrium solved to the stated tolerance\. A free rollout is

yt\+1=𝒫⁡\(yt\),y^t\+1=𝒫θ​\(y^t\),et=y^t−yt\.y\_\{t\+1\}=\\mathcal\{P\}\(y\_\{t\}\),\\qquad\\widehat\{y\}\_\{t\+1\}=\\mathcal\{P\}\_\{\\theta\}\(\\widehat\{y\}\_\{t\}\),\\qquad e\_\{t\}=\\widehat\{y\}\_\{t\}\-y\_\{t\}\.\(31\)Define the local one\-step modeling defectdt=𝒫θ​\(yt\)−𝒫⁡\(yt\)d\_\{t\}=\\mathcal\{P\}\_\{\\theta\}\(y\_\{t\}\)\-\\mathcal\{P\}\(y\_\{t\}\)and tangentAt=D​𝒫θ​\(yt\)A\_\{t\}=D\\mathcal\{P\}\_\{\\theta\}\(y\_\{t\}\)\. Taylor expansion around the reference state gives the first\-order error recurrence

et\+1=At​et\+dt\+rt,‖rt‖2=𝒪⁡\(‖et‖22\)\.e\_\{t\+1\}=A\_\{t\}e\_\{t\}\+d\_\{t\}\+r\_\{t\},\\qquad\\\|r\_\{t\}\\\|\_\{2\}=\\mathcal\{O\}\(\\\|e\_\{t\}\\\|\_\{2\}^\{2\}\)\.\(32\)For𝒯b:a:=Ab−1⋯Aa\\mathcal\{T\}\_\{b:a\}:=A\_\{b\-1\}\\cdots A\_\{a\}and𝒯a:a:=I\\mathcal\{T\}\_\{a:a\}:=I, repeated substitution produces

eH\\displaystyle e\_\{H\}=𝒯H:0e0\+∑j=0H−1𝒯H:j\+1\(dj\+rj\),\\displaystyle=\\mathcal\{T\}\_\{H:0\}e\_\{0\}\+\\sum\_\{j=0\}^\{H\-1\}\\mathcal\{T\}\_\{H:j\+1\}\(d\_\{j\}\+r\_\{j\}\),\(33\)‖eH‖2\\displaystyle\\\|e\_\{H\}\\\|\_\{2\}≤∥𝒯H:0∥2∥e0∥2\+∑j=0H−1∥𝒯H:j\+1∥2\(∥dj∥2\+𝒪\(∥ej∥22\)\)\.\\displaystyle\\leq\\\|\\mathcal\{T\}\_\{H:0\}\\\|\_\{2\}\\\|e\_\{0\}\\\|\_\{2\}\+\\sum\_\{j=0\}^\{H\-1\}\\\|\\mathcal\{T\}\_\{H:j\+1\}\\\|\_\{2\}\\left\(\\\|d\_\{j\}\\\|\_\{2\}\+\\mathcal\{O\}\(\\\|e\_\{j\}\\\|\_\{2\}^\{2\}\)\\right\)\.\(34\)This separates local one\-step error from its amplification by the learned physical\-time tangent products\. With a shared reference initial condition,e0=0e\_\{0\}=0, only the accumulated defect terms remain\. The horizon\-hhmetric is

ϵh\(m\)=‖y^t\+h\(m\)−yt\+h‖2‖yt\+h‖2\+ε,Rh\(m\)=1\|𝒟\|​∑\(f,s\)∈𝒟ϵh,f,s\(m\)ϵh,f,s\(implicit\),\\epsilon\_\{h\}^\{\(m\)\}=\\frac\{\\\|\\widehat\{y\}\_\{t\+h\}^\{\(m\)\}\-y\_\{t\+h\}\\\|\_\{2\}\}\{\\\|y\_\{t\+h\}\\\|\_\{2\}\+\\varepsilon\},\\qquad R\_\{h\}^\{\(m\)\}=\\frac\{1\}\{\|\\mathcal\{D\}\|\}\\sum\_\{\(f,s\)\\in\\mathcal\{D\}\}\\frac\{\\epsilon\_\{h,f,s\}^\{\(m\)\}\}\{\\epsilon\_\{h,f,s\}^\{\(\\mathrm\{implicit\}\)\}\},\(35\)where𝒟\\mathcal\{D\}contains the equally weighted family–seed pairs\. This is the quantity plotted in Fig\.[4](https://arxiv.org/html/2608.23725#A1.F4)\(a\), and the backward method changes the learned model, while every rollout evaluates its inner SILVA equilibrium with the same solver and stated tolerance\.

Figure 4:Native SILVA transient analysis with \(a\) family\-weighted free physical\-time rollout error relative to models trained with exact implicit differentiation on nine dynamic datasets and five seeds, \(b\) finite causal\-response error relative to the converged static adjoint, and \(c\) fixed\-point state error relative to equilibrium, where panels \(b,c\) contain 45 dataset–seed models per method\. Shading in panel \(a\) denotes error ratios at most 1\.05\.The physical\-time study contains9×5×8×8=28809\\times 5\\times 8\\times 8=2880repeated evaluations from 45 independently seeded family–model units and shows no consistent long\-horizon penalty from backward response control under the evaluated conditions\. At horizon eight, Phi\-CMR has mean error ratio 1\.025, with ratios at most 1\.05 in 43 of 45 family–seed pairs, compared with all 45 for CMR, 43 of 45 for Phantom and Tikhonov, 42 of 45 for Neumann and JFB, and 31 of 45 for SHINE\. Because Reaction–Diffusion 1D is the hardest long\-horizon case, both mean and median are shown\. Independently, 945 solver\-horizon evaluations test the causal response identity, with the Phi\-CMR mean finite\-response error falling from 0\.411 atN=1N=1to2\.50×10−42\.50\\times 10^\{\-4\}atN=64N=64as state error falls from 0\.387 to1\.24×10−41\.24\\times 10^\{\-4\}\. The sampled final\-state sources do not activate a lift, although training traces activate in every physical family, which supports the precise interpretation that CMR/Phi\-CMR changes the static backward endpoint when a pole is active, while physical evolution and the causal approach to that endpoint remain properties of the learned SILVA map\.

Table 4:SILVA convergence, fidelity, and activation diagnostics\. Statistics use physical\-family weighting, with the five Darcy regimes contributing one family\-level unit\.The residuals lie below the stated numerical tolerances, while the Phi\-CMR test\-error ratio to the exact\-implicit training baseline is at most 1\.05 in 48 of 50 family–seed pairs and the structured mechanism activates in every family\. Its error is lower than Tikhonov in 27 of 50 paired comparisons, all dataset\-level paired 95% intervals against implicit include zero, and 12 of 14 intervals against Tikhonov include zero, with Reaction–Diffusion 1D and Shallow Water 2D resolving in opposite directions\. Numerical convergence, mechanism activation, and predictive fidelity therefore support distinct parts of the argument rather than serving as interchangeable success measures\.

### A\.2Datasets and Reproducibility

The learning tasks consume stored benchmark fields rather than regenerating the PDE solutions, with governing equations, domains, boundary conditions, numerical discretizations, and generation parameters following the cited dataset releases\[[Takamoto et al\., 2022](https://arxiv.org/html/2608.23725#bib.bib25),[Ohana et al\., 2024](https://arxiv.org/html/2608.23725#bib.bib34),[Gupta and Brandstetter, 2023](https://arxiv.org/html/2608.23725#bib.bib37),[Dulny et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib38),[Herde et al\., 2024](https://arxiv.org/html/2608.23725#bib.bib39),[Luo et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib40),[Toshev et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib41)\]\. PDEBench supplied five Darcy coefficient\-to\-solution regimes and temporal state\-to\-next\-state tasks for Advection, Burgers, Reaction–Diffusion, Diffusion–Sorption, Diffusion–Reaction, Shallow Water, compressible CFD, and incompressible Navier–Stokes, while The Well supplied the turbulent radiative\-layer multichannel transition\. The cross\-benchmark evaluation used PDEArena fields, DynaBench trajectories, PDEGym Poisson–Gauss source\-to\-solution pairs, CFDBench geometry\-conditioned velocity fields, and LagrangeBench particle trajectories\. For temporal data, “next” denotes the subsequent frame in the released trajectory at the selected offset rather than a newly computed PDE solution\.

Dataset\-provided training, validation, and test partitions define the splits when available, with PDEBench and PDEGym using deterministic disjoint sample or time partitions and CFDBench using disjoint case\-level 80/10/10 partitions, while normalization is fitted only on training data\. Each experiment specifies the dataset version, selected variables, temporal offsets, method, seed, response parameters, architecture, optimization budget, and forward/backward tolerances\. Hardware configurations and library versions are unavailable for a subset of the native and cross\-benchmark experiments\. Accordingly, the reproducibility claim concerns the numerical conclusions under the stated design rather than bitwise cross\-platform identity\. The five independent seeds are 123, 456, 789, 1011, and 2027, and none of the SILVA experiments uses stochastic data augmentation, gradient clipping, a learning\-rate schedule, or early stopping\.

The matched native and cross\-benchmark studies usem0=0\.08m\_\{0\}=0\.08,κ=0\.12\\kappa=0\.12, collective rank one, local and global initialization scales 0\.65 and 0\.34, andμ=0\.05\\mu=0\.05for Tikhonov\. Phi\-CMR applies the per\-mode rule in Eq\. \([20](https://arxiv.org/html/2608.23725#S2.E20)\) to the singular values ofΓ\\Gamma\(γ=Φmode\\gamma=\\Phi\_\{\\rm mode\}in Algorithm[3](https://arxiv.org/html/2608.23725#algorithm3)\)\. Adam uses learning rate10−310^\{\-3\}, and the native and cross\-benchmark hidden widths are 48 and 32\. The native study allows 200 forward and 200 backward iterations, while the cross\-benchmark study allows 120 of each, and both use forward and backward tolerances10−710^\{\-7\}and10−810^\{\-8\}\. Minibatches contain four or eight samples and the held\-out evaluations contain 16 or 32 samples, according to the field representation\.

Figure 5:Native SILVA multiphysics evaluation with \(a\) final inverse\-standardized sample\-relative Euclidean error normalized by exact implicit differentiation and paired\-seed 95% intervals, \(b\) paired Phi\-CMR\-minus\-Tikhonov differences, \(c\) the fraction of training updates with a lifted collective response, and \(d\) measured forward\-solve plus backward wall time relative to implicit differentiation, with the Darcy regimes aggregated as one physical family\. Shading in panel \(a\) denotes error ratios at most 1\.05\. The Darcy confidence interval in \(a\) extends beyond the displayed range\.

## Appendix BImplicit Gradients, Poles, and Baseline Filters

Applying the implicit function theorem toRθ​\(z⋆,x\)=0R\_\{\\theta\}\(z^\{\\star\},x\)=0\[[Krantz and Parks, 2002](https://arxiv.org/html/2608.23725#bib.bib26),[Bai et al\., 2019](https://arxiv.org/html/2608.23725#bib.bib1)\]gives the sensitivity relation directly\. Holdingxxfixed and varying one scalar parameterθa\\theta\_\{a\}, the total differential of the equilibrium condition is

0=d​Rθ=∂zRθ​d​z⋆\+∂θaRθ​d​θa=K​d​z⋆−Ba​d​θa,0=dR\_\{\\theta\}=\\partial\_\{z\}R\_\{\\theta\}\\,dz^\{\\star\}\+\\partial\_\{\\theta\_\{a\}\}R\_\{\\theta\}\\,d\\theta\_\{a\}=K\\,dz^\{\\star\}\-B\_\{a\}\\,d\\theta\_\{a\},becauseBa=∂θafθB\_\{a\}=\\partial\_\{\\theta\_\{a\}\}f\_\{\\theta\}and therefore∂θaRθ=−Ba\\partial\_\{\\theta\_\{a\}\}R\_\{\\theta\}=\-B\_\{a\}\. Dividing byd​θad\\theta\_\{a\}gives

K​∂z⋆∂θa=Ba,∂z⋆∂θa=K−1​Ba\.K\\frac\{\\partial z^\{\\star\}\}\{\\partial\\theta\_\{a\}\}=B\_\{a\},\\qquad\\frac\{\\partial z^\{\\star\}\}\{\\partial\\theta\_\{a\}\}=K^\{\-1\}B\_\{a\}\.\(36\)The chain rule then yieldsd​ℒ/d​θa=∂θaℒ\+g⊤​K−1​Bad\\mathcal\{L\}/d\\theta\_\{a\}=\\partial\_\{\\theta\_\{a\}\}\\mathcal\{L\}\+g^\{\\top\}K^\{\-1\}B\_\{a\}\. Defining the adjoint byK⊤​v=gK^\{\\top\}v=gmoves the inverse outside the parameter dimension and gives

d​ℒd​θa=∂θaℒ\+v⊤​Ba\.\\frac\{d\\mathcal\{L\}\}\{d\\theta\_\{a\}\}=\\partial\_\{\\theta\_\{a\}\}\\mathcal\{L\}\+v^\{\\top\}B\_\{a\}\.\(37\)Here∂θaℒ\\partial\_\{\\theta\_\{a\}\}\\mathcal\{L\}is the direct derivative at fixed equilibrium state, whereasd​ℒ/d​θad\\mathcal\{L\}/d\\theta\_\{a\}also contains the change ofz⋆z^\{\\star\}induced byθa\\theta\_\{a\}\. The adjoint solve is shared by all parameters, and the remaining parameter dependence enters through the contractionsv⊤​Bav^\{\\top\}B\_\{a\}\. In singular coordinates,K⊤​v=gK^\{\\top\}v=gbecomesΣ​U⊤​v=V⊤​g\\Sigma U^\{\\top\}v=V^\{\\top\}g, which gives each nonsingular coefficient asui⊤​v=\(vi⊤​g\)/σiu\_\{i\}^\{\\top\}v=\(v\_\{i\}^\{\\top\}g\)/\\sigma\_\{i\}, and hence

d​ℒd​θa=∂θaℒ\+∑i\(vi⊤​g\)​\(ui⊤​Ba\)σi\.\\frac\{d\\mathcal\{L\}\}\{d\\theta\_\{a\}\}=\\partial\_\{\\theta\_\{a\}\}\\mathcal\{L\}\+\\sum\_\{i\}\\frac\{\(v\_\{i\}^\{\\top\}g\)\(u\_\{i\}^\{\\top\}B\_\{a\}\)\}\{\\sigma\_\{i\}\}\.\(38\)At an exact pole, solvability of the adjoint requires

V0⊤​g=0,V\_\{0\}^\{\\top\}g=0,\(39\)whereV0V\_\{0\}spansNull⁡\(K\)\\operatorname\{Null\}\(K\)\. When this condition holds, the generalized inverse\[[Ben\-Israel and Greville, 2003](https://arxiv.org/html/2608.23725#bib.bib24)\]gives the minimum\-norm response\(K⊤\)\+​g=U​Σ\+​V⊤​g\(K^\{\\top\}\)^\{\+\}g=U\\Sigma^\{\+\}V^\{\\top\}g\. When it fails, the unchanged deterministic derivative is not finite, which separates a source\-visible pole from a removable null direction\.

The exact, Tikhonov, and denominator\-lifted modal responses are

vexact=U​diag⁡\(σi−1\)​V⊤​g,v\_\{\\rm exact\}=U\\operatorname\{diag\}\(\\sigma\_\{i\}^\{\-1\}\)V^\{\\top\}g,\(40\)vμ=U​diag⁡\(σiσi2\+μ2\)​V⊤​g,vTSVD=U​diag⁡\(𝟏σi≥κσi\)​V⊤​g,vR=U​diag⁡\(1σ^i\)​V⊤​g\.v\_\{\\mu\}=U\\operatorname\{diag\}\\\!\\left\(\\frac\{\\sigma\_\{i\}\}\{\\sigma\_\{i\}^\{2\}\+\\mu^\{2\}\}\\right\)V^\{\\top\}g,\\quad v\_\{\\rm TSVD\}=U\\operatorname\{diag\}\\\!\\left\(\\frac\{\\mathbf\{1\}\_\{\\sigma\_\{i\}\\geq\\kappa\}\}\{\\sigma\_\{i\}\}\\right\)V^\{\\top\}g,\\qquad v\_\{R\}=U\\operatorname\{diag\}\\\!\\left\(\\frac\{1\}\{\\widehat\{\\sigma\}\_\{i\}\}\\right\)V^\{\\top\}g\.The Tikhonov expression\[[Tikhonov, 1963](https://arxiv.org/html/2608.23725#bib.bib27)\]follows by differentiating‖K⊤​v−g‖22\+μ2​‖v‖22\\\|K^\{\\top\}v\-g\\\|\_\{2\}^\{2\}\+\\mu^\{2\}\\\|v\\\|\_\{2\}^\{2\}, obtaining\(K​K⊤\+μ2​I\)​v=K​g\(KK^\{\\top\}\+\\mu^\{2\}I\)v=Kg, and substituting the SVD, whereas CMR setsσ^i=σi\\widehat\{\\sigma\}\_\{i\}=\\sigma\_\{i\}on𝒮\\mathcal\{S\}andσ^i=max⁡\(σi,mi\)\\widehat\{\\sigma\}\_\{i\}=\\max\(\\sigma\_\{i\},m\_\{i\}\)on𝒞\\mathcal\{C\}, thereby preserving every resolved stable channel exactly\.

#### Worked two\-mode check\.

LetK=diag⁡\(1,10−4\)K=\\operatorname\{diag\}\(1,10^\{\-4\}\),g=\(1,1\)⊤g=\(1,1\)^\{\\top\}, and parameter sourcesB1=\(1,0\)⊤B\_\{1\}=\(1,0\)^\{\\top\},B2=\(0,1\)⊤B\_\{2\}=\(0,1\)^\{\\top\}, for which the exact adjoint is\(1,104\)⊤\(1,10^\{4\}\)^\{\\top\}and Eq\. \([38](https://arxiv.org/html/2608.23725#A2.E38)\) gives parameter contributions\(1,104\)\(1,10^\{4\}\)\. A CMR massm=0\.05m=0\.05changes these to\(1,20\)\(1,20\), whereas TSVD withκ=10−3\\kappa=10^\{\-3\}gives\(1,0\)\(1,0\), with both biased rules retaining the first resolved mode\. If insteadB2=\(1,0\)⊤B\_\{2\}=\(1,0\)^\{\\top\}, the critical loss projection remains nonzero but its contribution to parameter 2 vanishes, illustrating whyvi⊤​gv\_\{i\}^\{\\top\}galone is insufficient\. For the stated CMR mass, the counterterm isΔ​K=diag⁡\(0,0\.05−10−4\)=diag⁡\(0,0\.0499\)\\Delta K=\\operatorname\{diag\}\(0,0\.05\-10^\{\-4\}\)=\\operatorname\{diag\}\(0,0\.0499\)\. Consequently,KR=K\+Δ​K=diag⁡\(1,0\.05\)K^\{R\}=K\+\\Delta K=\\operatorname\{diag\}\(1,0\.05\), and direct substitution into\(KR\)⊤​vR=g\(K^\{R\}\)^\{\\top\}v^\{R\}=greturnsvR=\(1,20\)⊤v^\{R\}=\(1,20\)^\{\\top\}\. This verifies in one calculation the denominator lift, the counterterm, the lifted adjoint, and its parameter\-source contributions\. If the second denominator lies outside the critical set or already exceeds its target mass, thenΔ​K=0\\Delta K=0and the same calculation recovers the exact implicit adjoint\.

#### Exact modified model and backward\-only rule\.

Let

Rθ,κR​\(z,x\)=Rθ​\(z,x\)\+Δ​K𝒜​\(z−zref,𝒜\),KR=K\+Δ​K𝒜,R^\{R\}\_\{\\theta,\\kappa\}\(z,x\)=R\_\{\\theta\}\(z,x\)\+\\Delta K\_\{\\mathcal\{A\}\}\(z\-z\_\{\{\\rm ref\},\\mathcal\{A\}\}\),\\qquad K^\{R\}=K\+\\Delta K\_\{\\mathcal\{A\}\},with the anchor𝒜\\mathcal\{A\}fixed during one derivative\. The lifted equation

\(KR\)⊤​vR=gR\(K^\{R\}\)^\{\\top\}v^\{R\}=g^\{R\}\(41\)is the exact adjoint of the modified equilibrium, and because∂θaRR=−Ba\\partial\_\{\\theta\_\{a\}\}R^\{R\}=\-B\_\{a\}at a frozen anchor, implicit differentiation repeats the preceding steps and gives

d​ℒRd​θa=∂θaℒR\+\(vR\)⊤​Ba\.\\frac\{d\\mathcal\{L\}^\{R\}\}\{d\\theta\_\{a\}\}=\\partial\_\{\\theta\_\{a\}\}\\mathcal\{L\}^\{R\}\+\(v^\{R\}\)^\{\\top\}B\_\{a\}\.\(42\)Keeping the original forward equilibrium while solving the lifted adjoint produces a backward surrogate, with the original and modified residuals distinguishing these two uses\.

## Appendix CMatrix\-Free, Causal, and Renormalized Response

#### Dense and matrix\-free realizations\.

The matrix\-free formulation changes how the required linear\-algebra objects are obtained but leaves the response operator unchanged\. When the state dimension permits, one may formJJandK=I−JK=I\-Jexplicitly, compute a dense SVD, and compare the modal reconstruction with a direct solve of the lifted adjoint equation\. In the matrix\-free formulation, automatic differentiation supplies only the actionsw↦K​ww\\mapsto Kwandw↦K⊤​ww\\mapsto K^\{\\top\}w\. An iterative singular solver then uses both actions to recover the smallest triplets, which must satisfy the two triplet\-residual conditions in Algorithm[1](https://arxiv.org/html/2608.23725#algorithm1)\. The requested rank is sufficient when the largest returned small singular value is at or above the cutoff, with the rank increased until this condition is met\. This ensures that every resolved denominator belowκ\\kappahas been included and avoids replacing the required singular geometry by eigenvalue information fromJJ\.

Automatic differentiation\[[Baydin et al\., 2018](https://arxiv.org/html/2608.23725#bib.bib23)\]supplies the actions

K​w=w−J​w,K⊤​w=w−J⊤​w,Kw=w\-Jw,\\qquad K^\{\\top\}w=w\-J^\{\\top\}w,These identities allow partial singular solvers and GMRES\[[Golub and Kahan, 1965](https://arxiv.org/html/2608.23725#bib.bib20),[Lehoucq et al\., 1998](https://arxiv.org/html/2608.23725#bib.bib22),[Saad and Schultz, 1986](https://arxiv.org/html/2608.23725#bib.bib21)\]to operate without materializingJJ\. For a SILVA operator\[[Silva, 2026a](https://arxiv.org/html/2608.23725#bib.bib32)\]K=KL−A​B⊤K=K\_\{L\}\-AB^\{\\top\}, the Woodbury identity\[[Woodbury, 1950](https://arxiv.org/html/2608.23725#bib.bib28),[Sherman and Morrison, 1950](https://arxiv.org/html/2608.23725#bib.bib29)\]gives

K−1=KL−1\+KL−1​A​\(I−B⊤​KL−1​A\)−1​B⊤​KL−1\.K^\{\-1\}=K\_\{L\}^\{\-1\}\+K\_\{L\}^\{\-1\}A\(I\-B^\{\\top\}K\_\{L\}^\{\-1\}A\)^\{\-1\}B^\{\\top\}K\_\{L\}^\{\-1\}\.WritingΓ=I−B⊤​KL−1​A\\Gamma=I\-B^\{\\top\}K\_\{L\}^\{\-1\}Aand transposing the identity yields

v=KL−⁣⊤​g\+KL−⁣⊤​B​Γ−⁣⊤​A⊤​KL−⁣⊤​g\.v=K\_\{L\}^\{\-\\top\}g\+K\_\{L\}^\{\-\\top\}B\\,\\Gamma^\{\-\\top\}A^\{\\top\}K\_\{L\}^\{\-\\top\}g\.\(43\)Only the small collective denominatorΓ\\Gammais decomposed and lifted\. If the state dimension isddand the structured correction has rankrr, then

KL∈ℝd×d,A,B,X,Y∈ℝd×r,Γ∈ℝr×r,b,vR∈ℝd,h,xR∈ℝr,K\_\{L\}\\in\\mathbb\{R\}^\{d\\times d\},\\quad A,B,X,Y\\in\\mathbb\{R\}^\{d\\times r\},\\quad\\Gamma\\in\\mathbb\{R\}^\{r\\times r\},\\quad b,v\_\{R\}\\in\\mathbb\{R\}^\{d\},\\quad h,x\_\{R\}\\in\\mathbb\{R\}^\{r\},whereX=KL−1​AX=K\_\{L\}^\{\-1\}A,Y=KL−⁣⊤​BY=K\_\{L\}^\{\-\\top\}B,b=KL−⁣⊤​gb=K\_\{L\}^\{\-\\top\}g,h=A⊤​bh=A^\{\\top\}b, andxR=Γeff−⁣⊤​hx\_\{R\}=\\Gamma\_\{\\rm eff\}^\{\-\\top\}h\. These dimensions givevR=b\+Y​xRv\_\{R\}=b\+Yx\_\{R\}and expose the reduction from add\-dimensional adjoint to anrr\-dimensional collective solve\. The factorization can be checked without forming the full operator by verifyingK​w=KL​w−A⁡\(B⊤​w\)Kw=K\_\{L\}w\-A\(B^\{\\top\}w\)on independent probe vectors, together with the local and collective residuals\. The determinant lemma gives

detK=detKL​detΓ,\\det K=\\det K\_\{L\}\\,\\det\\Gamma,but it does not equate the singular spectra\. A useful conditioning bound is

‖K−1‖2≤‖KL−1‖2​\(1\+‖A‖2​‖Γ−1‖2​‖B‖2​‖KL−1‖2\)\.\\\|K^\{\-1\}\\\|\_\{2\}\\leq\\\|K\_\{L\}^\{\-1\}\\\|\_\{2\}\\left\(1\+\\\|A\\\|\_\{2\}\\\|\\Gamma^\{\-1\}\\\|\_\{2\}\\\|B\\\|\_\{2\}\\\|K\_\{L\}^\{\-1\}\\\|\_\{2\}\\right\)\.Accordingly, collective, local, and full\-KKdiagnostics are evaluated separately\. For factor\-gauge stability, computeA​B⊤=P​Σ​Q⊤AB^\{\\top\}=P\\Sigma Q^\{\\top\}and use balanced factorsA¯=P​Σ1/2\\bar\{A\}=P\\Sigma^\{1/2\},B¯=Q​Σ1/2\\bar\{B\}=Q\\Sigma^\{1/2\}\. This leavesKKunchanged and prevents arbitrary rescaling of raw factor norms\.

To derive the response interpretation, perturb the fixed\-point iteration by a sourcesns\_\{n\},

zn\+1=fθ​\(zn,x\)\+ϵ​sn\.z\_\{n\+1\}=f\_\{\\theta\}\(z\_\{n\},x\)\+\\epsilon s\_\{n\}\.\(44\)Setzn=z⋆\+ϵ​ϕn\+O⁡\(ϵ2\)z\_\{n\}=z^\{\\star\}\+\\epsilon\\phi\_\{n\}\+O\(\\epsilon^\{2\}\)\. Taylor expansion at the equilibrium gives the causal recurrence

ϕn\+1=J​ϕn\+sn,\\phi\_\{n\+1\}=J\\phi\_\{n\}\+s\_\{n\},\(45\)hence

ϕn=∑m<nJn−1−m​sm,χn,m=∂zn∂sm=Jn−1−m​𝟏n\>m\.\\phi\_\{n\}=\\sum\_\{m<n\}J^\{n\-1\-m\}s\_\{m\},\\qquad\\chi\_\{n,m\}=\\frac\{\\partial z\_\{n\}\}\{\\partial s\_\{m\}\}=J^\{n\-1\-m\}\\mathbf\{1\}\_\{n\>m\}\.For a constant source andρ⁡\(J\)<1\\rho\(J\)<1, summing the geometric series produces

G0​\(0\)=∑ℓ=0∞Jℓ=\(I−J\)−1=K−1\.G\_\{0\}\(0\)=\\sum\_\{\\ell=0\}^\{\\infty\}J^\{\\ell\}=\(I\-J\)^\{\-1\}=K^\{\-1\}\.\(46\)The discrete Fourier transform gives the same limit throughG0​\(ω\)=\(ei​ω​I−J\)−1G\_\{0\}\(\\omega\)=\(e^\{\\mathrm\{i\}\\omega\}I\-J\)^\{\-1\}\.

The following construction supplies a response\-theoretic interpretation of the same operator identities\. Its generating functional and diagrammatic expansion are not required to evaluate the dense or matrix\-free denominator lift\. A response\-field representation\[[Martin et al\., 1973](https://arxiv.org/html/2608.23725#bib.bib12),[Janssen, 1976](https://arxiv.org/html/2608.23725#bib.bib13),[De Dominicis, 1976](https://arxiv.org/html/2608.23725#bib.bib14)\]inserts the trajectory constraint with a Fourier multiplierz~n\\widetilde\{z\}\_\{n\},

1=∫Dz~exp\[−i∑nz~n⊤\(zn\+1−fθ\(zn,x\)−sn\)\]\.1=\\int D\\widetilde\{z\}\\,\\exp\\\!\\left\[\-\\mathrm\{i\}\\sum\_\{n\}\\widetilde\{z\}\_\{n\}^\{\\top\}\(z\_\{n\+1\}\-f\_\{\\theta\}\(z\_\{n\},x\)\-s\_\{n\}\)\\right\]\.Differentiating the resulting generating functional once with respect to a state probe and once with respect to the source returns the mixed correlator⟨zn​z~m⊤⟩=χn,m\\langle z\_\{n\}\\widetilde\{z\}\_\{m\}^\{\\top\}\\rangle=\\chi\_\{n,m\}\. Expanding the static equationK​ϕ=ϵ​s\+12​F\(2\)​\[ϕ,ϕ\]\+⋯K\\phi=\\epsilon s\+\\frac\{1\}\{2\}F^\{\(2\)\}\[\\phi,\\phi\]\+\\cdotsgives

ϕ=ϵ​G0​s\+ϵ22​G0​F\(2\)​\[G0​s,G0​s\]\+O⁡\(ϵ3\),G0=K−1\.\\phi=\\epsilon G\_\{0\}s\+\\frac\{\\epsilon^\{2\}\}\{2\}G\_\{0\}F^\{\(2\)\}\[G\_\{0\}s,G\_\{0\}s\]\+O\(\\epsilon^\{3\}\),\\qquad G\_\{0\}=K^\{\-1\}\.\(47\)Thus every perturbative line carries a factor ofG0G\_\{0\}, matching the propagator structure of path\-integral and diagrammatic perturbation theory\[[Feynman, 1948](https://arxiv.org/html/2608.23725#bib.bib16),[Feynman, 1949](https://arxiv.org/html/2608.23725#bib.bib17)\], and nonlinear insertions dress the inverse through the Dyson identity\[[Dyson, 1949](https://arxiv.org/html/2608.23725#bib.bib18)\]

G=G0\+G0​Π​G,G−1=G0−1−Π\.G=G\_\{0\}\+G\_\{0\}\\Pi G,\\qquad G^\{\-1\}=G\_\{0\}^\{\-1\}\-\\Pi\.Projecting onto a selected singular channel replaces the bare denominator byσi−πi​\(0\)\\sigma\_\{i\}\-\\pi\_\{i\}\(0\)\. Response renormalization imposes the finite matching condition

vi⊤​\(K\+Δ​K\)⊤​ui=σi\+δi=σ^i\>0,v\_\{i\}^\{\\top\}\(K\+\\Delta K\)^\{\\top\}u\_\{i\}=\\sigma\_\{i\}\+\\delta\_\{i\}=\\widehat\{\\sigma\}\_\{i\}\>0,\(48\)which is realized by the minimum\-rank counterterm

Δ​K=U𝒞​diag⁡\(\[σ^i−σi\]\+\)​V𝒞⊤\.\\Delta K=U\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\\\!\\left\(\[\\widehat\{\\sigma\}\_\{i\}\-\\sigma\_\{i\}\]\_\{\+\}\\right\)V\_\{\\mathcal\{C\}\}^\{\\top\}\.\(49\)Substitution into\(K\+Δ​K\)⊤​vR=g\(K\+\\Delta K\)^\{\\top\}v\_\{R\}=ggives

vR=U𝒞​diag⁡\(σ^i−1\)​V𝒞⊤​g\+U𝒮​diag⁡\(σi−1\)​V𝒮⊤​g\.v\_\{R\}=U\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\(\\widehat\{\\sigma\}\_\{i\}^\{\-1\}\)V\_\{\\mathcal\{C\}\}^\{\\top\}g\+U\_\{\\mathcal\{S\}\}\\operatorname\{diag\}\(\\sigma\_\{i\}^\{\-1\}\)V\_\{\\mathcal\{S\}\}^\{\\top\}g\.\(50\)This chain connects the causal DEQ response, the zero\-frequency pole, the Dyson denominator, and the low\-rank lift used here\.

## Appendix DCoordinate Metric and Threshold Stability

Under a nonsingular state changey=S​zy=Sz, use

Ry=S​R,Ky=S​K​S−1,gy=S−⁣⊤​g,By,a=S​Ba\.R\_\{y\}=SR,\\quad K\_\{y\}=SKS^\{\-1\},\\quad g\_\{y\}=S^\{\-\\top\}g,\\quad B\_\{y,a\}=SB\_\{a\}\.Althoughg⊤​K−1​Ba=gy⊤​Ky−1​By,ag^\{\\top\}K^\{\-1\}B\_\{a\}=g\_\{y\}^\{\\top\}K\_\{y\}^\{\-1\}B\_\{y,a\}, Euclidean singular values ofKKare not similarity invariants\. To declare the state geometry, letH≻0H\\succ 0be a metric,H=L⊤​LH=L^\{\\top\}L, and apply CMR toK~=L​K​L−1\\widetilde\{K\}=LKL^\{\-1\}andg~=L−⁣⊤​g\\widetilde\{g\}=L^\{\-\\top\}g\. Under the coordinate change,Hy=S−⁣⊤​H​S−1H\_\{y\}=S^\{\-\\top\}HS^\{\-1\}, and mapping the adjoint back preservesv⊤​Bav^\{\\top\}B\_\{a\}, which makes the construction coordinate\-consistent under the declared metric\.

For threshold stability, select a subspace rather than an arbitrary basis in a near\-degenerate cluster\. Modes belowκ−η\\kappa\-\\etaare critical, those aboveκ\+η\\kappa\+\\etastable, and the intervening band is classified as unresolved\. Perturbation checks varyκ\\kappa,η\\eta, the requested rank, solver tolerance, and floating\-point precision, while discontinuous decisions inside the unresolved interval are not interpreted as physical transitions\.

## Appendix ENotation and Response Identities

For an equilibriumz⋆=fθ​\(z⋆,x\)z^\{\\star\}=f\_\{\\theta\}\(z^\{\\star\},x\), define the residual, Jacobian denominator, loss source, and parameter source by

Rθ=z−fθ​\(z,x\),K=∂zRθ=I−J,g=∇z⋆ℒ,Ba=∂θafθ​\(z⋆,x\)\.R\_\{\\theta\}=z\-f\_\{\\theta\}\(z,x\),\\qquad K=\\partial\_\{z\}R\_\{\\theta\}=I\-J,\\qquad g=\\nabla\_\{z^\{\\star\}\}\\mathcal\{L\},\\qquad B\_\{a\}=\\partial\_\{\\theta\_\{a\}\}f\_\{\\theta\}\(z^\{\\star\},x\)\.The singular\-value decomposition isK=U​Σ​V⊤K=U\\Sigma V^\{\\top\}, with0≤σ1≤⋯≤σd0\\leq\\sigma\_\{1\}\\leq\\cdots\\leq\\sigma\_\{d\}\. A critical set𝒞=\{i:σi<κ\}\\mathcal\{C\}=\\\{i:\\sigma\_\{i\}<\\kappa\\\}has basesU𝒞U\_\{\\mathcal\{C\}\}andV𝒞V\_\{\\mathcal\{C\}\}, while𝒮\\mathcal\{S\}denotes its stable complement\. A critical mode is loss\-visible whenvi⊤​g≠0v\_\{i\}^\{\\top\}g\\neq 0and parameter\-visible foraawhenui⊤​Ba≠0u\_\{i\}^\{\\top\}B\_\{a\}\\neq 0\. The remaining notation is\[a\]\+=max⁡\(a,0\)\[a\]\_\{\+\}=\\max\(a,0\),𝒫\[ℓ,u\]​\(a\)=min⁡\{max⁡\(a,ℓ\),u\}\\mathcal\{P\}\_\{\[\\ell,u\]\}\(a\)=\\min\\\{\\max\(a,\\ell\),u\\\}, anddiag⁡\(ai\)\\operatorname\{diag\}\(a\_\{i\}\)for the diagonal matrix with entriesaia\_\{i\}\. The unscriptedvvdenotes the adjoint, whileviv\_\{i\}denotes theii\-th right singular vector\. For a vector\-valued parameterθ\\theta, the scalar sourcesBaB\_\{a\}define the components of the parameter derivative but need not be assembled as a matrix, because their contraction is the single reverse\-mode product\(∂θfθ\)⊤​v\(\\partial\_\{\\theta\}f\_\{\\theta\}\)^\{\\top\}v\.

Table 5:Response\-renormalization notation\.
## Appendix FResponse\-Renormalization Algorithms

The procedures useγ\\gammaas the gate selector,rras the requested number of singular triplets,κ\>0\\kappa\>0as the critical cutoff, andτsvd,τK\>0\\tau\_\{\\rm svd\},\\tau\_\{K\}\>0as singular\-triplet and linear\-solve tolerances\.

#### Reference dense construction\.

The complete dense calculation can be summarized by the operator sequence

z⋆⟶\(J,K,g\)⟶\(U,Σ,V\)⟶\(𝒞,mi,σ^i\)⟶Δ​K⟶vR⟶∇θRℒ\.z^\{\\star\}\\longrightarrow\(J,K,g\)\\longrightarrow\(U,\\Sigma,V\)\\longrightarrow\(\\mathcal\{C\},m\_\{i\},\\widehat\{\\sigma\}\_\{i\}\)\\longrightarrow\\Delta K\\longrightarrow v\_\{R\}\\longrightarrow\\nabla\_\{\\theta\}^\{R\}\\mathcal\{L\}\.After solving the forward equilibrium, one evaluatesJ=∂zfθ​\(z⋆,x\)J=\\partial\_\{z\}f\_\{\\theta\}\(z^\{\\star\},x\),K=I−JK=I\-J, andg=∇z⋆ℒg=\\nabla\_\{z^\{\\star\}\}\\mathcal\{L\}\. An SVD ofKK, with its singular values arranged in ascending order, identifies𝒞=\{i:σi<κ\}\\mathcal\{C\}=\\\{i:\\sigma\_\{i\}<\\kappa\\\}\. The chosen gate determines the target masses, andσ^i=max⁡\(σi,mi\)\\widehat\{\\sigma\}\_\{i\}=\\max\(\\sigma\_\{i\},m\_\{i\}\)is applied only to selected modes\. The counterterm then follows fromΔ​K=U𝒞​diag⁡\(σ^i−σi\)​V𝒞⊤\\Delta K=U\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\(\\widehat\{\\sigma\}\_\{i\}\-\\sigma\_\{i\}\)V\_\{\\mathcal\{C\}\}^\{\\top\}, after which the corresponding adjoint equation is solved and the parameter gradient is obtained by one reverse\-mode product\. The dense calculation is internally verified by the forward residual, the two singular\-triplet residuals, the lifted\-adjoint residual, and exact recovery when the active set is empty\. The two\-mode calculation in Appendix[B](https://arxiv.org/html/2608.23725#A2)supplies a closed\-form reference case\.

The forward and backward choices determine which objective is differentiated:

z⋆​from​Rθ=0,K⊤​v=g,exact gradient of the original DEQ,z⋆​from​Rθ=0,\(K\+Δ​K\)⊤​vR=g,backward\-response surrogate,zR​from​Rθ,κR=0,\(KR\)⊤​vR=gR,exact gradient of the frozen\-anchor modified DEQ\.\\begin\{array\}\[\]\{ccl\}z^\{\\star\}\\text\{ from \}R\_\{\\theta\}=0,&K^\{\\top\}v=g,&\\text\{exact gradient of the original DEQ\},\\\\ z^\{\\star\}\\text\{ from \}R\_\{\\theta\}=0,&\(K\+\\Delta K\)^\{\\top\}v\_\{R\}=g,&\\text\{backward\-response surrogate\},\\\\ z^\{R\}\\text\{ from \}R^\{R\}\_\{\\theta,\\kappa\}=0,&\(K^\{R\}\)^\{\\top\}v^\{R\}=g^\{R\},&\\text\{exact gradient of the frozen\-anchor modified DEQ\}\.\\end\{array\}WhenΔ​K=0\\Delta K=0, these alternatives coincide with the original implicit adjoint\. When the lift is active, retaining both the original\-equation and lifted\-equation residuals distinguishes the intended response change from linear\-solve error\.

Algorithm 1Critical\-subspace implicit backward pass for CMR, Phi\-CMR, and Delta\-PhiInput:Equilibrium

z⋆z^\{\\star\}, actions

w↦K​ww\\mapsto Kwand

w↦K⊤​ww\\mapsto K^\{\\top\}w, source

gg, requested rank

rr, cutoff

κ\\kappa, base mass

m0m\_\{0\}, gate

γ∈\{CMR,Φ,Δ​Φ\}\\gamma\\in\\\{\\mathrm\{CMR\},\\Phi,\\Delta\\Phi\\\}, parameters

\(αmax,λ,cmax,εden\)\(\\alpha\_\{\\max\},\\lambda,c\_\{\\max\},\\varepsilon\_\{\\rm den\}\), tolerances

\(τsvd,τK\)\(\\tau\_\{\\rm svd\},\\tau\_\{K\}\), and optional exact adjoint

vexactv\_\{\\rm exact\}
Output:Gated adjoint

vgatev\_\{\\rm gate\}, selected triplets, effective denominators, residuals, and response diagnostics

1Set

d=dim\(g\)=dim\(z⋆\)d=\\dim\(g\)=\\dim\(z^\{\\star\}\)
2Estimate the

rrsmallest triplets

\(ui,σi,vi\)\(u\_\{i\},\\sigma\_\{i\},v\_\{i\}\)satisfying

K​vi=σi​uiKv\_\{i\}=\\sigma\_\{i\}u\_\{i\}and

K⊤​ui=σi​viK^\{\\top\}u\_\{i\}=\\sigma\_\{i\}v\_\{i\}by dense or iterative partial SVD

3Compute

ηsvd=maxi⁡\{‖K​vi−σi​ui‖2,‖K⊤​ui−σi​vi‖2\}\\eta\_\{\\rm svd\}=\\max\_\{i\}\\\{\\\|Kv\_\{i\}\-\\sigma\_\{i\}u\_\{i\}\\\|\_\{2\},\\\|K^\{\\top\}u\_\{i\}\-\\sigma\_\{i\}v\_\{i\}\\\|\_\{2\}\\\}
4if*ηsvd\>τsvd\\eta\_\{\\rm svd\}\>\\tau\_\{\\rm svd\}*then

5Return an unresolved\-triplet diagnostic and tighten the singular solver

6end if

7Sort

0≤σ1≤⋯≤σr0\\leq\\sigma\_\{1\}\\leq\\cdots\\leq\\sigma\_\{r\}and set

𝒞=\{i≤r:σi<κ\}\\mathcal\{C\}=\\\{i\\leq r:\\sigma\_\{i\}<\\kappa\\\}
8if*r<dr<dandσr<κ\\sigma\_\{r\}<\\kappa*then

9Return an incomplete\-rank diagnostic and repeat with larger

rr
10end if

11if*𝒞=∅\\mathcal\{C\}=\\varnothing*then

12Solve

K⊤​vgate=gK^\{\\top\}v\_\{\\rm gate\}=gto tolerance

τK\\tau\_\{K\}
13Set

σ^i=σi\\widehat\{\\sigma\}\_\{i\}=\\sigma\_\{i\}and

ρ0=ρR=‖K⊤​vgate−g‖2\\rho\_\{0\}=\\rho\_\{R\}=\\\|K^\{\\top\}v\_\{\\rm gate\}\-g\\\|\_\{2\}, then return

14else

15Form

U𝒞=\[ui\]i∈𝒞U\_\{\\mathcal\{C\}\}=\[u\_\{i\}\]\_\{i\\in\\mathcal\{C\}\},

V𝒞=\[vi\]i∈𝒞V\_\{\\mathcal\{C\}\}=\[v\_\{i\}\]\_\{i\\in\\mathcal\{C\}\}, and

c𝒞=V𝒞⊤​gc\_\{\\mathcal\{C\}\}=V\_\{\\mathcal\{C\}\}^\{\\top\}g
16Set

PU=I−U𝒞​U𝒞⊤P\_\{U\}=I\-U\_\{\\mathcal\{C\}\}U\_\{\\mathcal\{C\}\}^\{\\top\},

PV=I−V𝒞​V𝒞⊤P\_\{V\}=I\-V\_\{\\mathcal\{C\}\}V\_\{\\mathcal\{C\}\}^\{\\top\},

g𝒞=V𝒞​c𝒞g\_\{\\mathcal\{C\}\}=V\_\{\\mathcal\{C\}\}c\_\{\\mathcal\{C\}\}, and

g𝒮=PV​gg\_\{\\mathcal\{S\}\}=P\_\{V\}g
17Compute

a𝒞=‖c𝒞‖22/\(‖g‖22\+εden\)a\_\{\\mathcal\{C\}\}=\\\|c\_\{\\mathcal\{C\}\}\\\|\_\{2\}^\{2\}/\(\\\|g\\\|\_\{2\}^\{2\}\+\\varepsilon\_\{\\rm den\}\),

σ𝒞=mini∈𝒞⁡σi\\sigma\_\{\\mathcal\{C\}\}=\\min\_\{i\\in\\mathcal\{C\}\}\\sigma\_\{i\}, and

p𝒞=𝒫\[0,1\]​\(\(κ−σ𝒞\)/κ\)p\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[0,1\]\}\(\(\\kappa\-\\sigma\_\{\\mathcal\{C\}\}\)/\\kappa\)
18if*γ=CMR\\gamma=\\mathrm\{CMR\}*then

19Set

mi=m0m\_\{i\}=m\_\{0\}for every

i∈𝒞i\\in\\mathcal\{C\}
20else

21Set

miΦ=m0​\(1\+m0/\(σi\+m0\)\)m\_\{i\}^\{\\Phi\}=m\_\{0\}\(1\+m\_\{0\}/\(\\sigma\_\{i\}\+m\_\{0\}\)\)for every

i∈𝒞i\\in\\mathcal\{C\}
22if*γ=Φ\\gamma=\\Phi*then

23Set

mi=miΦm\_\{i\}=m\_\{i\}^\{\\Phi\}
24else

25Set

s𝒞=𝒫\[−1,1\]​\(2​a𝒞−1\)s\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[\-1,1\]\}\(2a\_\{\\mathcal\{C\}\}\-1\)and

m𝒞Φ=m0​\[1\+\(αmax−1\)​p𝒞\]m\_\{\\mathcal\{C\}\}^\{\\Phi\}=m\_\{0\}\[1\+\(\\alpha\_\{\\max\}\-1\)p\_\{\\mathcal\{C\}\}\]
26Set

mΔ​Φ=𝒫\[m0,cmax​m0\]​\(m𝒞Φ\+m0​λ​p𝒞​s𝒞\)m^\{\\Delta\\Phi\}=\\mathcal\{P\}\_\{\[m\_\{0\},c\_\{\\max\}m\_\{0\}\]\}\(m\_\{\\mathcal\{C\}\}^\{\\Phi\}\+m\_\{0\}\\lambda p\_\{\\mathcal\{C\}\}s\_\{\\mathcal\{C\}\}\)
27Set

mi=max⁡\(miΦ,mΔ​Φ\)m\_\{i\}=\\max\(m\_\{i\}^\{\\Phi\},m^\{\\Delta\\Phi\}\)for every

i∈𝒞i\\in\\mathcal\{C\}
28end if

29end if

30Set

σ^i=max⁡\(σi,mi\)\\widehat\{\\sigma\}\_\{i\}=\\max\(\\sigma\_\{i\},m\_\{i\}\),

δi=σ^i−σi\\delta\_\{i\}=\\widehat\{\\sigma\}\_\{i\}\-\\sigma\_\{i\}, and

Δ​K=U𝒞​diag⁡\(δi\)​V𝒞⊤\\Delta K=U\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\(\\delta\_\{i\}\)V\_\{\\mathcal\{C\}\}^\{\\top\}
31Solve

PV​K⊤​PU​v𝒮=g𝒮P\_\{V\}K^\{\\top\}P\_\{U\}v\_\{\\mathcal\{S\}\}=g\_\{\\mathcal\{S\}\}, constrained by

v𝒮=PU​v𝒮v\_\{\\mathcal\{S\}\}=P\_\{U\}v\_\{\\mathcal\{S\}\}, to tolerance

τK\\tau\_\{K\}
32Reconstruct

v𝒞=U𝒞​diag⁡\(σ^i−1\)​c𝒞v\_\{\\mathcal\{C\}\}=U\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\(\\widehat\{\\sigma\}\_\{i\}^\{\-1\}\)c\_\{\\mathcal\{C\}\}and set

vgate=v𝒮\+v𝒞v\_\{\\rm gate\}=v\_\{\\mathcal\{S\}\}\+v\_\{\\mathcal\{C\}\}
33Compute

ρ0=‖K⊤​vgate−g‖2\\rho\_\{0\}=\\\|K^\{\\top\}v\_\{\\rm gate\}\-g\\\|\_\{2\}and

ρR=‖\(K\+Δ​K\)⊤​vgate−g‖2\\rho\_\{R\}=\\\|\(K\+\\Delta K\)^\{\\top\}v\_\{\\rm gate\}\-g\\\|\_\{2\}
34Retain

\(σi,σ^i,δi\)\(\\sigma\_\{i\},\\widehat\{\\sigma\}\_\{i\},\\delta\_\{i\}\),

ηsvd\\eta\_\{\\rm svd\},

a𝒞a\_\{\\mathcal\{C\}\}, lifted rank, Krylov iterations,

ρ0\\rho\_\{0\}, and

ρR\\rho\_\{R\}
35if*vexactv\_\{\\rm exact\}is available*then

36Compute the norm ratio, cosine, relative error, signed projection, and critical magnitude suppression defined in Appendix[H](https://arxiv.org/html/2608.23725#A8)

37end if

38Return

vgatev\_\{\\rm gate\}and the diagnostics

39end if

Algorithm 2Dense or matrix\-free DEQ response\-renormalization procedureInput:Map

fθf\_\{\\theta\}, input

xx, target

yy, requested rank

rr, cutoff

κ\\kappa, base mass

m0m\_\{0\}, gate

γ∈\{CMR,Φ,Δ​Φ\}\\gamma\\in\\\{\\mathrm\{CMR\},\\Phi,\\Delta\\Phi\\\}, gate parameters

\(αmax,λ,cmax,εden\)\(\\alpha\_\{\\max\},\\lambda,c\_\{\\max\},\\varepsilon\_\{\\rm den\}\), and forward, singular\-triplet, and Krylov tolerances

Output:Equilibrium

zRz^\{R\}, gated adjoint

vgatev\_\{\\rm gate\}, parameter gradient, and numerical diagnostics

1Solve

zref=fθ​\(zref,x\)z\_\{\\rm ref\}=f\_\{\\theta\}\(z\_\{\\rm ref\},x\)and verify the forward residual

2Define

Kref​w=w−\[∂zfθ​\(zref,x\)\]​wK\_\{\\rm ref\}w=w\-\[\\partial\_\{z\}f\_\{\\theta\}\(z\_\{\\rm ref\},x\)\]wby a JVP

3Define

Kref⊤​w=w−\[∂zfθ​\(zref,x\)\]⊤​wK\_\{\\rm ref\}^\{\\top\}w=w\-\[\\partial\_\{z\}f\_\{\\theta\}\(z\_\{\\rm ref\},x\)\]^\{\\top\}wby a VJP

4Estimate the

rrsmallest triplets of

Kref=U​Σ​V⊤K\_\{\\rm ref\}=U\\Sigma V^\{\\top\}with dense or iterative partial SVD

5Sort the selected denominators and collect

Ur=\[u1,…,ur\]U\_\{r\}=\[u\_\{1\},\\ldots,u\_\{r\}\],

Vr=\[v1,…,vr\]V\_\{r\}=\[v\_\{1\},\\ldots,v\_\{r\}\]
6Apply the triplet\-residual and rank\-coverage conditions from Algorithm[1](https://arxiv.org/html/2608.23725#algorithm1), stopping if either condition is unresolved

7if*γ=CMR\\gamma=\\mathrm\{CMR\}*then

8Set

mi=m0m\_\{i\}=m\_\{0\}for

i=1,…,ri=1,\\ldots,r
9else

10Set

miΦ=m0​\(1\+m0/\(σi\+m0\)\)m\_\{i\}^\{\\Phi\}=m\_\{0\}\(1\+m\_\{0\}/\(\\sigma\_\{i\}\+m\_\{0\}\)\)
11if*γ=Φ\\gamma=\\Phi*then

12Set

mi=miΦm\_\{i\}=m\_\{i\}^\{\\Phi\}
13else

14Compute

gref=∇zℒ​\(zref,y\)g\_\{\\rm ref\}=\\nabla\_\{z\}\\mathcal\{L\}\(z\_\{\\rm ref\},y\),

𝒞r=\{j≤r:σj<κ\}\\mathcal\{C\}\_\{r\}=\\\{j\\leq r:\\sigma\_\{j\}<\\kappa\\\}, and

a^𝒞=∑j∈𝒞r\(vj⊤​gref\)2/\(‖gref‖22\+εden\)\\widehat\{a\}\_\{\\mathcal\{C\}\}=\\sum\_\{j\\in\\mathcal\{C\}\_\{r\}\}\(v\_\{j\}^\{\\top\}g\_\{\\rm ref\}\)^\{2\}/\(\\\|g\_\{\\rm ref\}\\\|\_\{2\}^\{2\}\+\\varepsilon\_\{\\rm den\}\)
15Set

p𝒞=𝒫\[0,1\]​\(\(κ−minj∈𝒞r⁡σj\)/κ\)p\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[0,1\]\}\(\(\\kappa\-\\min\_\{j\\in\\mathcal\{C\}\_\{r\}\}\\sigma\_\{j\}\)/\\kappa\)when

𝒞r≠∅\\mathcal\{C\}\_\{r\}\\neq\\varnothing, otherwise set

p𝒞=a^𝒞=0p\_\{\\mathcal\{C\}\}=\\widehat\{a\}\_\{\\mathcal\{C\}\}=0
16Set

s𝒞=𝒫\[−1,1\]​\(2​a^𝒞−1\)s\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[\-1,1\]\}\(2\\widehat\{a\}\_\{\\mathcal\{C\}\}\-1\)
17Set

mΔ​Φ=𝒫\[m0,cmax​m0\]​\(m0​\[1\+\(αmax−1\)​p𝒞\]\+m0​λ​p𝒞​s𝒞\)m^\{\\Delta\\Phi\}=\\mathcal\{P\}\_\{\[m\_\{0\},c\_\{\\max\}m\_\{0\}\]\}\(m\_\{0\}\[1\+\(\\alpha\_\{\\max\}\-1\)p\_\{\\mathcal\{C\}\}\]\+m\_\{0\}\\lambda p\_\{\\mathcal\{C\}\}s\_\{\\mathcal\{C\}\}\)
18Set

mi=max⁡\(miΦ,mΔ​Φ\)m\_\{i\}=\\max\(m\_\{i\}^\{\\Phi\},m^\{\\Delta\\Phi\}\)for the field gate

19end if

20end if

21Set

σ^i=mi\\widehat\{\\sigma\}\_\{i\}=m\_\{i\}when

σi<κ\\sigma\_\{i\}<\\kappaand

σi<mi\\sigma\_\{i\}<m\_\{i\}, otherwise set

σ^i=σi\\widehat\{\\sigma\}\_\{i\}=\\sigma\_\{i\}
22Set

δi=σ^i−σi≥0\\delta\_\{i\}=\\widehat\{\\sigma\}\_\{i\}\-\\sigma\_\{i\}\\geq 0and

Δ​K=Ur​diag⁡\(δi\)​Vr⊤\\Delta K=U\_\{r\}\\operatorname\{diag\}\(\\delta\_\{i\}\)V\_\{r\}^\{\\top\}
23if*the modified\-forward route is active*then

24Solve

zR=fθ​\(zR,x\)−Δ​K​\(zR−zref\)z^\{R\}=f\_\{\\theta\}\(z^\{R\},x\)\-\\Delta K\(z^\{R\}\-z\_\{\\rm ref\}\)
25else

26Set

zR=zrefz^\{R\}=z\_\{\\rm ref\}and label the backward response as a surrogate

27end if

28Compute

g=∇zℒ​\(zR,y\)g=\\nabla\_\{z\}\\mathcal\{L\}\(z^\{R\},y\), define

K​w=w−\[∂zfθ​\(zR,x\)\]​wKw=w\-\[\\partial\_\{z\}f\_\{\\theta\}\(z^\{R\},x\)\]w, and set

KR=K\+Δ​KK^\{R\}=K\+\\Delta K
29Solve

\(KR\)⊤​vgate=g\(K^\{R\}\)^\{\\top\}v\_\{\\rm gate\}=gusing dense algebra or GMRES with

\(KR\)⊤​w=K⊤​w\+Vr​diag⁡\(δi\)​Ur⊤​w\(K^\{R\}\)^\{\\top\}w=K^\{\\top\}w\+V\_\{r\}\\operatorname\{diag\}\(\\delta\_\{i\}\)U\_\{r\}^\{\\top\}w
30Return

∇θRℒ=∂θℒ\+\(∂θfθ​\(zR,x\)\)⊤​vgate\\nabla\_\{\\theta\}^\{R\}\\mathcal\{L\}=\\partial\_\{\\theta\}\\mathcal\{L\}\+\(\\partial\_\{\\theta\}f\_\{\\theta\}\(z^\{R\},x\)\)^\{\\top\}v\_\{\\rm gate\}
31Compute the original and modified residuals, denominators, shifts, solver counts, and, when available, exact\-adjoint response diagnostics

32Refresh the frozen anchor

\(zref,Ur,Vr,δ\)\(z\_\{\\rm ref\},U\_\{r\},V\_\{r\},\\delta\)only after the optimizer update

Algorithm 3SILVA local–global CMR, Phi\-CMR, or Delta\-Phi adjointInput:Local matrix

KLK\_\{L\}, factors

A,BA,B, source

gg, cutoff

κ\\kappa, base mass

m0m\_\{0\}, gate

γ∈\{CMR,Φmode,Φcoll,Δ​Φ\}\\gamma\\in\\\{\\mathrm\{CMR\},\\Phi\_\{\\rm mode\},\\Phi\_\{\\rm coll\},\\Delta\\Phi\\\}, parameters

\(αmax,λ,cmax,εden\)\(\\alpha\_\{\\max\},\\lambda,c\_\{\\max\},\\varepsilon\_\{\\rm den\}\), and local\-solve tolerances

Output:SILVA\-shaped gated response

vgatev\_\{\\rm gate\}, collective spectrum, effective denominators, residuals, and solver diagnostics

1for*j=1,…,rj=1,\\ldots,r*do

2Solve

KLX:j=A:jK\_\{L\}X\_\{:j\}=A\_\{:j\}by local GMRES and record its residual

3end for

4Form the collective denominator

Γ=Ir−B⊤​X\\Gamma=I\_\{r\}\-B^\{\\top\}X
5Compute the dense SVD

Γ=P​diag⁡\(si\)​Q⊤\\Gamma=P\\operatorname\{diag\}\(s\_\{i\}\)Q^\{\\top\}and verify the triplet residuals

6Solve

KL⊤​b=gK\_\{L\}^\{\\top\}b=gby local GMRES and set

h=A⊤​bh=A^\{\\top\}b
7Set

𝒞=\{i:si<κ\}\\mathcal\{C\}=\\\{i:s\_\{i\}<\\kappa\\\}and

p𝒞=𝒫\[0,1\]​\(\(κ−mini∈𝒞⁡si\)/κ\)p\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[0,1\]\}\(\(\\kappa\-\\min\_\{i\\in\\mathcal\{C\}\}s\_\{i\}\)/\\kappa\)when

𝒞≠∅\\mathcal\{C\}\\neq\\varnothing, otherwise set

p𝒞=0p\_\{\\mathcal\{C\}\}=0
8Compute

a𝒞=∑i∈𝒞\(qi⊤​h\)2/\(‖h‖22\+εden\)a\_\{\\mathcal\{C\}\}=\\sum\_\{i\\in\\mathcal\{C\}\}\(q\_\{i\}^\{\\top\}h\)^\{2\}/\(\\\|h\\\|\_\{2\}^\{2\}\+\\varepsilon\_\{\\rm den\}\), where

qi=Q:iq\_\{i\}=Q\_\{:i\}
9for*j=1,…,rj=1,\\ldots,r*do

10Solve

KL⊤Y:j=B:jK\_\{L\}^\{\\top\}Y\_\{:j\}=B\_\{:j\}by local GMRES and record its residual

11end for

12if*γ=CMR\\gamma=\\mathrm\{CMR\}*then

13Set

mi=m0m\_\{i\}=m\_\{0\}for

i=1,…,ri=1,\\ldots,r
14else if*γ=Φmode\\gamma=\\Phi\_\{\\rm mode\}*then

15Set

mi=m0​\(1\+m0/\(si\+m0\)\)m\_\{i\}=m\_\{0\}\(1\+m\_\{0\}/\(s\_\{i\}\+m\_\{0\}\)\)for

i=1,…,ri=1,\\ldots,r
16else

17Set

mΦ=m0​\[1\+\(αmax−1\)​p𝒞\]m^\{\\Phi\}=m\_\{0\}\[1\+\(\\alpha\_\{\\max\}\-1\)p\_\{\\mathcal\{C\}\}\]
18if*γ=Φcoll\\gamma=\\Phi\_\{\\rm coll\}*then

19Set

mi=mΦm\_\{i\}=m^\{\\Phi\}for

i=1,…,ri=1,\\ldots,r
20else

21Set

s𝒞=𝒫\[−1,1\]​\(2​a𝒞−1\)s\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[\-1,1\]\}\(2a\_\{\\mathcal\{C\}\}\-1\)
22Set

mΔ​Φ=𝒫\[m0,cmax​m0\]​\(mΦ\+m0​λ​p𝒞​s𝒞\)m^\{\\Delta\\Phi\}=\\mathcal\{P\}\_\{\[m\_\{0\},c\_\{\\max\}m\_\{0\}\]\}\(m^\{\\Phi\}\+m\_\{0\}\\lambda p\_\{\\mathcal\{C\}\}s\_\{\\mathcal\{C\}\}\)
23Set

mi=mΔ​Φm\_\{i\}=m^\{\\Delta\\Phi\}for

i=1,…,ri=1,\\ldots,r
24end if

25end if

26Set

s^i=mi\\widehat\{s\}\_\{i\}=m\_\{i\}only when

si<κs\_\{i\}<\\kappaand

si<mis\_\{i\}<m\_\{i\}, otherwise set

s^i=si\\widehat\{s\}\_\{i\}=s\_\{i\}
27Form

Γeff=P​diag⁡\(s^i\)​Q⊤\\Gamma\_\{\\rm eff\}=P\\operatorname\{diag\}\(\\widehat\{s\}\_\{i\}\)Q^\{\\top\}and

xR=Γeff−⁣⊤​hx\_\{R\}=\\Gamma\_\{\\rm eff\}^\{\-\\top\}h
28Set

vgate=b\+Y​xRv\_\{\\rm gate\}=b\+Yx\_\{R\}
29Compute the original residual

ρ0=‖\(KL−A​B⊤\)⊤​vgate−g‖2\\rho\_\{0\}=\\\|\(K\_\{L\}\-AB^\{\\top\}\)^\{\\top\}v\_\{\\rm gate\}\-g\\\|\_\{2\}
30Compute the lifted collective residual

ρR=‖Γeff⊤​xR−h‖2\\rho\_\{R\}=\\\|\\Gamma\_\{\\rm eff\}^\{\\top\}x\_\{R\}\-h\\\|\_\{2\}and verify

ρR\\rho\_\{R\}and every local\-solve residual against their tolerances

31Retain

sis\_\{i\},

s^i\\widehat\{s\}\_\{i\}, lifted rank, susceptibility

1/mini⁡s^i1/\\min\_\{i\}\\widehat\{s\}\_\{i\},

p𝒞p\_\{\\mathcal\{C\}\},

a𝒞a\_\{\\mathcal\{C\}\},

mim\_\{i\}, local residuals, Krylov iterations, solve count, and matrix\-\-vector products

32if*no collective denominator is lifted*then

33Verify that

vgatev\_\{\\rm gate\}matches the unmodified SILVA adjoint

34end if

35Backpropagate

vgatev\_\{\\rm gate\}through the common SILVA equilibrium map and return the response and diagnostics

## Appendix GPhi and Source\-Conditioned Finite Response

#### Finite\-response condition\.

The renormalization condition fixes a target susceptibilityG⁡\(0\)=χR\>0G\(0\)=\\chi\_\{R\}\>0\. Since a scalar static response isG⁡\(0\)=mR−1G\(0\)=m\_\{R\}^\{\-1\}, the response mass ismR=χR−1m\_\{R\}=\\chi\_\{R\}^\{\-1\}\. To make the lift\-only consequence explicit, consider one singular response channel with denominatorσ≥0\\sigma\\geq 0\. We restrict the correction to a nonnegative lift, with\[x\]\+=max⁡\(x,0\)\[x\]\_\{\+\}=\\max\(x,0\), which gives

mR=χR−1,δ​m=\[mR−σ\]\+,σeff=max⁡\(σ,mR\)\.m\_\{R\}=\\chi\_\{R\}^\{\-1\},\\qquad\\delta m=\[m\_\{R\}\-\\sigma\]\_\{\+\},\\qquad\\sigma^\{\\rm eff\}=\\max\(\\sigma,m\_\{R\}\)\.The resulting response is

GR​\(0\)=1σeff=min⁡\(1σ,χR\)\.G\_\{R\}\(0\)=\\frac\{1\}\{\\sigma^\{\\rm eff\}\}=\\min\\\!\\left\(\\frac\{1\}\{\\sigma\},\\chi\_\{R\}\\right\)\.The target therefore acts as an upper bound on susceptibility\. A channel whose original response already lies below this bound is unchanged, whereas a channel whose response exceeds it is reduced to the finite target\. This distinction is why the construction controls selected amplification without imposing damping on every response direction\.

#### Bounded Phi\-CMR parameterization\.

Rather than choosing a separate susceptibility for every channel, Phi\-CMR uses the bounded monotone target that we prescribe as

miΦ=m0​\(1\+m0σi\+m0\),m0≤miΦ≤2​m0,d​miΦd​σi<0\.m\_\{i\}^\{\\Phi\}=m\_\{0\}\\left\(1\+\\frac\{m\_\{0\}\}\{\\sigma\_\{i\}\+m\_\{0\}\}\\right\),\\qquad m\_\{0\}\\leq m\_\{i\}^\{\\Phi\}\\leq 2m\_\{0\},\\qquad\\frac\{dm\_\{i\}^\{\\Phi\}\}\{d\\sigma\_\{i\}\}<0\.Herem0\>0m\_\{0\}\>0sets the base denominator scale and has the same units asσi\\sigma\_\{i\}\. The dimensionless factorm0/\(σi\+m0\)m\_\{0\}/\(\\sigma\_\{i\}\+m\_\{0\}\)is largest near the pole and decreases smoothly as the channel becomes better conditioned\. In particular,

miΦ​\(0\)=2​m0,miΦ​\(m0\)=32​m0,limσi→∞miΦ=m0\.m\_\{i\}^\{\\Phi\}\(0\)=2m\_\{0\},\\qquad m\_\{i\}^\{\\Phi\}\(m\_\{0\}\)=\\frac\{3\}\{2\}m\_\{0\},\\qquad\\lim\_\{\\sigma\_\{i\}\\to\\infty\}m\_\{i\}^\{\\Phi\}=m\_\{0\}\.Forσi≥0\\sigma\_\{i\}\\geq 0, the bounds and monotonicity follow directly from

0<m0σi\+m0≤1,∂miΦ∂σi=−m02\(σi\+m0\)2<0\.0<\\frac\{m\_\{0\}\}\{\\sigma\_\{i\}\+m\_\{0\}\}\\leq 1,\\qquad\\frac\{\\partial m\_\{i\}^\{\\Phi\}\}\{\\partial\\sigma\_\{i\}\}=\-\\frac\{m\_\{0\}^\{2\}\}\{\(\\sigma\_\{i\}\+m\_\{0\}\)^\{2\}\}<0\.It therefore applies the strongest finite lift to the smallest selected denominator and approaches the base mass continuously away from the pole\. This rational form is a prescribed smooth interpolation satisfying positivity, boundedness, and stronger action near smaller denominators\. It is not a unique choice and its parameters are not automatically learned\. Moreover, a target mass does not by itself imply a modification\. A channel is lifted only whenσi<κ\\sigma\_\{i\}<\\kappaandmiΦ\>σim\_\{i\}^\{\\Phi\}\>\\sigma\_\{i\}\. The thresholdκ\\kappaidentifies the candidate critical set while the positive\-part rule determines whether a nonzero correction is actually applied\.

#### Collective pole pressure\.

For a collective pole,

p𝒞=𝒫\[0,1\]​\(κ−σ𝒞κ\),m𝒞Φ=m0​\[1\+\(αmax−1\)​p𝒞\]\.p\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[0,1\]\}\\left\(\\frac\{\\kappa\-\\sigma\_\{\\mathcal\{C\}\}\}\{\\kappa\}\\right\),\\qquad m\_\{\\mathcal\{C\}\}^\{\\Phi\}=m\_\{0\}\[1\+\(\\alpha\_\{\\max\}\-1\)p\_\{\\mathcal\{C\}\}\]\.Here𝒫\[0,1\]\\mathcal\{P\}\_\{\[0,1\]\}denotes projection onto\[0,1\]\[0,1\], andp𝒞p\_\{\\mathcal\{C\}\}is a prescribed dimensionless measure of proximity to the critical threshold that is zero whenσ𝒞≥κ\\sigma\_\{\\mathcal\{C\}\}\\geq\\kappa, increases asσ𝒞\\sigma\_\{\\mathcal\{C\}\}decreases within the critical region, and approaches one asσ𝒞\\sigma\_\{\\mathcal\{C\}\}approaches zero\. The collective response mass consequently interpolates fromm0m\_\{0\}at the critical threshold toαmax​m0\\alpha\_\{\\max\}m\_\{0\}as the collective denominator approaches zero\. Normalizing byκ\\kappaexpresses the depth of the pole relative to the same resolution used to define the critical set\. The parameterαmax≥1\\alpha\_\{\\max\}\\geq 1bounds the adaptive range, andαmax=1\\alpha\_\{\\max\}=1recovers the fixed collective massm0m\_\{0\}\. When no collective singular value lies belowκ\\kappa, the conventionp𝒞=0p\_\{\\mathcal\{C\}\}=0is used and no collective channel is lifted\.

#### Hartree motivation and its scope\.

The finite\-mass interpretation follows from a Hartree stationary response\[[Baym, 1962](https://arxiv.org/html/2608.23725#bib.bib30),[Cornwall et al\., 1974](https://arxiv.org/html/2608.23725#bib.bib31)\]\. In a scalar quartic closure, letmmdenote the bare denominator,g4≥0g\_\{4\}\\geq 0the quartic coupling,D≥0D\\geq 0the fluctuation scale, andϕ\\phithe scalar fluctuation\. The self\-consistent relations aremR=m\+3​g4​⟨ϕ2⟩m\_\{R\}=m\+3g\_\{4\}\\langle\\phi^\{2\}\\rangleand⟨ϕ2⟩=D/mR\\langle\\phi^\{2\}\\rangle=D/m\_\{R\}\. Substitution of the second relation into the first and multiplication bymRm\_\{R\}give

mR2−m​mR−3​g4​D=0,mR=m\+m2\+12​g4​D2\>0\.m\_\{R\}^\{2\}\-mm\_\{R\}\-3g\_\{4\}D=0,\\qquad m\_\{R\}=\\frac\{m\+\\sqrt\{m^\{2\}\+12g\_\{4\}D\}\}\{2\}\>0\.The second quadratic root is nonpositive and is excluded by the requirement of a positive susceptibility\. Atm=0m=0, the admissible root remains finite atmR=3​g4​Dm\_\{R\}=\\sqrt\{3g\_\{4\}D\}, illustrating how self\-consistency can replace a vanishing bare denominator by a positive effective one\. This calculation supplies the structural motivation for a positive response mass\. It does not identifyg4g\_\{4\}orDDfor a DEQ, nor does it uniquely determine the rational Phi\-CMR law,m0m\_\{0\},κ\\kappa, orαmax\\alpha\_\{\\max\}\.

#### Source\-conditioned extension\.

Source conditioning follows from the critical source energy\. Phi\-CMR depends on the denominator geometry but not on whether the loss excites the corresponding directions, and Delta\-Phi adds this information through the critical source fraction

a𝒞=∑i∈𝒞\(vi⊤​g\)2∑i\(vi⊤​g\)2\+εden,s𝒞=𝒫\[−1,1\]​\(2​a𝒞−1\)\.a\_\{\\mathcal\{C\}\}=\\frac\{\\sum\_\{i\\in\\mathcal\{C\}\}\(v\_\{i\}^\{\\top\}g\)^\{2\}\}\{\\sum\_\{i\}\(v\_\{i\}^\{\\top\}g\)^\{2\}\+\\varepsilon\_\{\\rm den\}\},\\qquad s\_\{\\mathcal\{C\}\}=\\mathcal\{P\}\_\{\[\-1,1\]\}\(2a\_\{\\mathcal\{C\}\}\-1\)\.Because the right singular vectors are orthonormal, the numerator is the squared norm of the loss source projected onto the critical right\-singular subspace\. The denominator is the corresponding total modal source energy, withεden\>0\\varepsilon\_\{\\rm den\}\>0preventing division by zero\. Hencea𝒞a\_\{\\mathcal\{C\}\}is close to zero when the critical subspace is nearly invisible to the loss and close to one when most source energy lies in that subspace\. The centered quantitys𝒞s\_\{\\mathcal\{C\}\}is negative below one\-half, zero at one\-half, and positive above one\-half, with projection protecting the admissible interval against numerical roundoff\.

Letλ≥0\\lambda\\geq 0control the source\-dependent adjustment and letcmax≥1c\_\{\\max\}\\geq 1cap the admissible mass relative tom0m\_\{0\}\. Define the constrained potential

Δ​Φ𝒞​\(m\)=12​\[m−mΦ−m0​λ​p𝒞​s𝒞\]2,m∈\[m0,cmax​m0\]\.\\Delta\\Phi\_\{\\mathcal\{C\}\}\(m\)=\\frac\{1\}\{2\}\[m\-m^\{\\Phi\}\-m\_\{0\}\\lambda p\_\{\\mathcal\{C\}\}s\_\{\\mathcal\{C\}\}\]^\{2\},\\qquad m\\in\[m\_\{0\},c\_\{\\max\}m\_\{0\}\]\.Its unconstrained stationary point ismunc=mΦ\+m0​λ​p𝒞​s𝒞m\_\{\\rm unc\}=m^\{\\Phi\}\+m\_\{0\}\\lambda p\_\{\\mathcal\{C\}\}s\_\{\\mathcal\{C\}\}, and projection onto the admissible interval yields

mΔ​Φ=𝒫\[m0,cmax​m0\]​\(munc\)\.m^\{\\Delta\\Phi\}=\\mathcal\{P\}\_\{\[m\_\{0\},c\_\{\\max\}m\_\{0\}\]\}\(m\_\{\\rm unc\}\)\.Before projection, increasing source visibility raises the target mass whenp𝒞\>0p\_\{\\mathcal\{C\}\}\>0, whereas low visibility lowers it relative to the source\-independent Phi\-CMR target\. The adjustment vanishes when the pole pressure is zero or whena𝒞=1/2a\_\{\\mathcal\{C\}\}=1/2\. Projection prevents either case from crossing the positive floor or the prescribed upper cap\. This quadratic potential is a transparent parameterization of the gate, not an independently derived physical effective action\. The correction vanishes when the pole pressure is zero, changes monotonically with source visibility before clipping, and cannot remove the positive response floor\.

#### Lift and no\-lift cases\.

The same shift has the equivalent closed form

δi=𝟏\{σi<κ\}\[mi−σi\]\+,KR=K\+U𝒞diag\(δi\)V𝒞⊤\.\\delta\_\{i\}=\\mathbf\{1\}\_\{\\\{\\sigma\_\{i\}<\\kappa\\\}\}\[m\_\{i\}\-\\sigma\_\{i\}\]\_\{\+\},\\qquad K^\{R\}=K\+U\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\(\\delta\_\{i\}\)V\_\{\\mathcal\{C\}\}^\{\\top\}\.Henceδi=0\\delta\_\{i\}=0leaves the corresponding singular channel unchanged, and an empty active set recoversKR=KK^\{R\}=Kand the exact implicit adjoint\.

The Delta\-Phi mass is the solution of a one\-dimensional constrained minimization\. Withmunc=mΦ\+m0​λ​p𝒞​s𝒞m\_\{\\rm unc\}=m^\{\\Phi\}\+m\_\{0\}\\lambda p\_\{\\mathcal\{C\}\}s\_\{\\mathcal\{C\}\}, the Lagrangian forminm⁡12​\(m−munc\)2\\min\_\{m\}\\frac\{1\}\{2\}\(m\-m\_\{\\rm unc\}\)^\{2\}on\[m0,cmax​m0\]\[m\_\{0\},c\_\{\\max\}m\_\{0\}\]is

𝒥⁡\(m,η−,η\+\)=12​\(m−munc\)2\+η−​\(m0−m\)\+η\+​\(m−cmax​m0\)\.\\mathcal\{J\}\(m,\\eta\_\{\-\},\\eta\_\{\+\}\)=\\frac\{1\}\{2\}\(m\-m\_\{\\rm unc\}\)^\{2\}\+\\eta\_\{\-\}\(m\_\{0\}\-m\)\+\\eta\_\{\+\}\(m\-c\_\{\\max\}m\_\{0\}\)\.Stationarity, complementary slackness, and nonnegative multipliers require

m−munc−η−\+η\+=0,η−\(m0−m\)=0,η\+\(m−cmaxm0\)=0,η−,η\+≥0\.m\-m\_\{\\rm unc\}\-\\eta\_\{\-\}\+\\eta\_\{\+\}=0,\\quad\\eta\_\{\-\}\(m\_\{0\}\-m\)=0,\\quad\\eta\_\{\+\}\(m\-c\_\{\\max\}m\_\{0\}\)=0,\\quad\\eta\_\{\-\},\\eta\_\{\+\}\\geq 0\.The Karush–Kuhn–Tucker conditions for the constrained minimization give

mΔ​Φ=\{m0,munc<m0,munc,m0≤munc≤cmax​m0,cmax​m0,munc\>cmax​m0\.m^\{\\Delta\\Phi\}=\\begin\{cases\}m\_\{0\},&m\_\{\\rm unc\}<m\_\{0\},\\\\ m\_\{\\rm unc\},&m\_\{0\}\\leq m\_\{\\rm unc\}\\leq c\_\{\\max\}m\_\{0\},\\\\ c\_\{\\max\}m\_\{0\},&m\_\{\\rm unc\}\>c\_\{\\max\}m\_\{0\}\.\\end\{cases\}The interior case retains the unconstrained target, while the other two cases select the nearest admissible endpoint\. This establishes that the clipping rule is the exact minimizer of the stated one\-dimensional problem\.

Because every active effective denominator is at leastm0m\_\{0\}, the critical response obeys the finite bound

‖v𝒞R‖22=∑i∈𝒞\|vi⊤​g\|2σ^i2≤‖V𝒞⊤​g‖22m02,\\\|v\_\{\\mathcal\{C\}\}^\{R\}\\\|\_\{2\}^\{2\}=\\sum\_\{i\\in\\mathcal\{C\}\}\\frac\{\|v\_\{i\}^\{\\top\}g\|^\{2\}\}\{\\widehat\{\\sigma\}\_\{i\}^\{2\}\}\\leq\\frac\{\\\|V\_\{\\mathcal\{C\}\}^\{\\top\}g\\\|\_\{2\}^\{2\}\}\{m\_\{0\}^\{2\}\},while the stable contribution remainsU𝒮​diag⁡\(σi−1\)​V𝒮⊤​gU\_\{\\mathcal\{S\}\}\\operatorname\{diag\}\(\\sigma\_\{i\}^\{\-1\}\)V\_\{\\mathcal\{S\}\}^\{\\top\}g\. The bound separates the two ingredients needed for problematic amplification\. Large pole pressure indicates a small denominator, while largea𝒞a\_\{\\mathcal\{C\}\}indicates that the loss excites its singular subspace\. A near\-pole direction with negligible source projection contributes little to the adjoint, and a strongly aligned but well\-conditioned direction does not require a pole correction\. The strongest intervention is reserved for cases in which both quantities are large\. Increasingm0m\_\{0\}lowers the maximum critical gain but can introduce more critical\-sector bias, while increasingκ\\kappaadmits more candidate directions and increasingαmax\\alpha\_\{\\max\},λ\\lambda, orcmaxc\_\{\\max\}enlarges the available adaptive range\.

The corresponding operator dimensions are

z⋆,g,v∈ℝd,U𝒞,V𝒞∈ℝd×r𝒞,Δ​K∈ℝd×d\.z^\{\\star\},g,v\\in\\mathbb\{R\}^\{d\},\\qquad U\_\{\\mathcal\{C\}\},V\_\{\\mathcal\{C\}\}\\in\\mathbb\{R\}^\{d\\times r\_\{\\mathcal\{C\}\}\},\\qquad\\Delta K\\in\\mathbb\{R\}^\{d\\times d\}\.The matrix\-free representation stores only the two bases and ther𝒞r\_\{\\mathcal\{C\}\}shifts\. A residual JVP returnsK​wKw, a residual VJP returnsK⊤​wK^\{\\top\}w, and the counterterm actions areΔ​K​w=U𝒞​diag⁡\(δi\)​V𝒞⊤​w\\Delta Kw=U\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\(\\delta\_\{i\}\)V\_\{\\mathcal\{C\}\}^\{\\top\}wandΔ​K⊤​w=V𝒞​diag⁡\(δi\)​U𝒞⊤​w\\Delta K^\{\\top\}w=V\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\(\\delta\_\{i\}\)U\_\{\\mathcal\{C\}\}^\{\\top\}w\. Thus the dense and matrix\-free procedures evaluate the same lifted operator when the requested triplets and Krylov solves meet their stated tolerances, although this storage identity is not an end\-to\-end complexity or runtime\-scaling result\.

The projected stable solve in Algorithm[1](https://arxiv.org/html/2608.23725#algorithm1)follows directly from the singular decomposition\. WithPU=I−U𝒞​U𝒞⊤P\_\{U\}=I\-U\_\{\\mathcal\{C\}\}U\_\{\\mathcal\{C\}\}^\{\\top\}andPV=I−V𝒞​V𝒞⊤P\_\{V\}=I\-V\_\{\\mathcal\{C\}\}V\_\{\\mathcal\{C\}\}^\{\\top\},

PV​K⊤​PU=V𝒮​diag⁡\(σi\)​U𝒮⊤\.P\_\{V\}K^\{\\top\}P\_\{U\}=V\_\{\\mathcal\{S\}\}\\operatorname\{diag\}\(\\sigma\_\{i\}\)U\_\{\\mathcal\{S\}\}^\{\\top\}\.Consequently,

v𝒮=U𝒮​diag⁡\(σi−1\)​V𝒮⊤​g,v𝒞=U𝒞​diag⁡\(σ^i−1\)​V𝒞⊤​g\.v\_\{\\mathcal\{S\}\}=U\_\{\\mathcal\{S\}\}\\operatorname\{diag\}\(\\sigma\_\{i\}^\{\-1\}\)V\_\{\\mathcal\{S\}\}^\{\\top\}g,\\qquad v\_\{\\mathcal\{C\}\}=U\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\(\\widehat\{\\sigma\}\_\{i\}^\{\-1\}\)V\_\{\\mathcal\{C\}\}^\{\\top\}g\.Substitution ofvR=v𝒮\+v𝒞v\_\{R\}=v\_\{\\mathcal\{S\}\}\+v\_\{\\mathcal\{C\}\}into the lifted adjoint equation gives

\(K\+Δ​K\)⊤​vR=V𝒮​V𝒮⊤​g\+V𝒞​V𝒞⊤​g=g\.\(K\+\\Delta K\)^\{\\top\}v\_\{R\}=V\_\{\\mathcal\{S\}\}V\_\{\\mathcal\{S\}\}^\{\\top\}g\+V\_\{\\mathcal\{C\}\}V\_\{\\mathcal\{C\}\}^\{\\top\}g=g\.The lifted residual therefore vanishes to linear\-solve tolerance\. The residual against the original equation is

K⊤​vR−g=−V𝒞​diag⁡\(δiσ^i\)​V𝒞⊤​g,K^\{\\top\}v\_\{R\}\-g=\-V\_\{\\mathcal\{C\}\}\\operatorname\{diag\}\\\!\\left\(\\frac\{\\delta\_\{i\}\}\{\\widehat\{\\sigma\}\_\{i\}\}\\right\)V\_\{\\mathcal\{C\}\}^\{\\top\}g,which measures the intended change in the selected response rather than a failed solve\. Consequently, the lifted\-equation residual measures numerical solution accuracy, whereas the residual against the original equation measures the strength of the deliberate response modification\. In the result panels, stable retention near one verifies selectivity, reduced critical amplitude records the intended control of the pole, and a nonzero original\-equation residual should not be interpreted as failure of the renormalized solve\.

#### Connection to the reported parameter scales\.

The controlled sweeps use target gaps0\.0050\.005,0\.010\.01, and0\.020\.02, a representative stable denominator0\.1250\.125,m0=0\.03m\_\{0\}=0\.03, andκ=0\.08\\kappa=0\.08\. The ordering

0\.005≤σ𝒞≤0\.02<m0<κ<0\.1250\.005\\leq\\sigma\_\{\\mathcal\{C\}\}\\leq 0\.02<m\_\{0\}<\\kappa<0\.125places every constructed critical denominator inside the candidate set and keeps the stable denominator outside it\. Atσ𝒞=0\.005\\sigma\_\{\\mathcal\{C\}\}=0\.005andαmax=2\\alpha\_\{\\max\}=2,

p𝒞=0\.08−0\.0050\.08=0\.9375,m𝒞Φ=0\.03​\(1\+0\.9375\)=0\.058125,p\_\{\\mathcal\{C\}\}=\\frac\{0\.08\-0\.005\}\{0\.08\}=0\.9375,\\qquad m\_\{\\mathcal\{C\}\}^\{\\Phi\}=0\.03\(1\+0\.9375\)=0\.058125,which is the0\.05810\.0581effective denominator reported in the mechanism experiment\. The native and cross\-benchmark SILVA studies instead holdm0=0\.08m\_\{0\}=0\.08andκ=0\.12\\kappa=0\.12fixed across datasets and seeds\. Their per\-mode Phi\-CMR targets lie between0\.080\.08and0\.160\.16, but a measured collective mode is modified only when it lies below0\.120\.12and its target mass exceeds its original denominator\. Sparse activation therefore means that Phi\-CMR often recovers the implicit adjoint exactly under the observed conditioning, rather than indicating that the response calculation failed\. These numerical scales specify the evaluated operating regimes and are not claimed to be universal or uniquely optimal\.

#### Numerical parameter interpretation\.

For a new operator, the response parameters are determined relative to a resolved small\-singular\-value spectrum rather than transferred as universal constants\. The requested rankrris increased until the largest returned small singular value lies at or above the proposed cutoff\. The cutoffκ\\kappais then placed above the resolved near\-critical cluster, with the unresolved interval of Appendix[D](https://arxiv.org/html/2608.23725#A4)used when classifications change under tolerance or precision perturbations\. Because every active effective denominator is at leastm0m\_\{0\}, the base mass imposes a maximum selected gain1/m01/m\_\{0\}\. It therefore expresses the largest critical response retained by the calculation\. The parametersαmax,λ,cmax\\alpha\_\{\\max\},\\lambda,c\_\{\\max\}control the bounded adaptive range and may be evaluated on calibration cases through stable\-sector retention, critical response, prediction error, and activation frequency\. A valid numerical realization requires the forward, singular\-triplet, local\-solve, and lifted\-equation residuals to satisfy their stated tolerances and must recover the exact implicit adjoint whenever no denominator is lifted\. These conditions separate a deliberate response modification from unresolved spectral or linear\-solve error\.

## Appendix HMetrics, Protocol, and Quantitative Evidence

For candidate adjointvmv\_\{m\}and exact adjointv⋆v\_\{\\star\}, define the amplitude, direction, projection, error, and residual by

Rm=‖vm‖‖v⋆‖\+ϵ,Cm=vm⊤​v⋆‖vm‖​‖v⋆‖\+ϵ,Pm=vm⊤​v⋆‖v⋆‖2\+ϵ,R\_\{m\}=\\frac\{\\\|v\_\{m\}\\\|\}\{\\\|v\_\{\\star\}\\\|\+\\epsilon\},\\quad C\_\{m\}=\\frac\{v\_\{m\}^\{\\top\}v\_\{\\star\}\}\{\\\|v\_\{m\}\\\|\\\|v\_\{\\star\}\\\|\+\\epsilon\},\\quad P\_\{m\}=\\frac\{v\_\{m\}^\{\\top\}v\_\{\\star\}\}\{\\\|v\_\{\\star\}\\\|^\{2\}\+\\epsilon\},Em=‖vm−v⋆‖‖v⋆‖\+ϵ,ρm=‖K⊤​vm−g‖\.E\_\{m\}=\\frac\{\\\|v\_\{m\}\-v\_\{\\star\}\\\|\}\{\\\|v\_\{\\star\}\\\|\+\\epsilon\},\\qquad\\rho\_\{m\}=\\\|K^\{\\top\}v\_\{m\}\-g\\\|\.Stable and critical amplitudes are computed after projection ontoU𝒮U\_\{\\mathcal\{S\}\}andU𝒞U\_\{\\mathcal\{C\}\}\. After inverting train\-only standardization, the sample score is

ei=‖y^i−yi‖2max⁡\(‖yi‖2,10−12\),e\_\{i\}=\\frac\{\\\|\\widehat\{y\}\_\{i\}\-y\_\{i\}\\\|\_\{2\}\}\{\\max\(\\\|y\_\{i\}\\\|\_\{2\},10^\{\-12\}\)\},with all stored channels and spatial entries flattened before tables average samples and equally weight family–seed units\. Field panels use pointwise absolute error, while mixed\-unit channels makeeie\_\{i\}a descriptive Euclidean score, not a physical invariant\. The descriptive upper\-threshold count records whether the ratio to the paired implicit score is at most 1\.05, including cases where the candidate error is lower\. It is not a prespecified noninferiority or equivalence test\. Paired intervals use the five declared seeds, never training updates, horizons, channels, or examples as independent replicates\.

For the SILVA response, letxR=Γeff−⁣⊤​hx\_\{R\}=\\Gamma\_\{\\rm eff\}^\{\-\\top\}handvR=b\+Y​xRv\_\{R\}=b\+Yx\_\{R\}\. Its residual against the original full operator isρ0=‖\(KL−A​B⊤\)⊤​vR−g‖2\\rho\_\{0\}=\\\|\(K\_\{L\}\-AB^\{\\top\}\)^\{\\top\}v\_\{R\}\-g\\\|\_\{2\}, whileρR=‖Γeff⊤​xR−h‖2\\rho\_\{R\}=\\\|\\Gamma\_\{\\rm eff\}^\{\\top\}x\_\{R\}\-h\\\|\_\{2\}verifies the lifted collective solve\. The local solves forXX,YY, andbbare checked separately, preventing these two quantities from concealing an inaccurate local inverse\.

The controlled response sweeps usem0=0\.03m\_\{0\}=0\.03andκ=0\.08\\kappa=0\.08, with the collective and source\-rotation sweeps evaluated at seeds 123 and 456\. The collective sweep setsαmax=2\\alpha\_\{\\max\}=2in Eq\. \([21](https://arxiv.org/html/2608.23725#S2.E21)\) \(γ=Φcoll\\gamma=\\Phi\_\{\\rm coll\}in Algorithm[3](https://arxiv.org/html/2608.23725#algorithm3)\)\. The source\-rotation sweep setsαmax=2\.5\\alpha\_\{\\max\}=2\.5and uses critical source fractions0,0\.05,0\.10,0\.25,0\.50,0\.75,0\.90,10,\\allowbreak 0\.05,\\allowbreak 0\.10,\\allowbreak 0\.25,\\allowbreak 0\.50,\\allowbreak 0\.75,\\allowbreak 0\.90,\\allowbreak 1\. The Delta\-Phi selection uses0,0\.10,0\.50,10,\\allowbreak 0\.10,\\allowbreak 0\.50,\\allowbreak 1, withλ∈\{0,0\.25,0\.50,0\.75,1\}\\lambda\\in\\\{0,\\allowbreak 0\.25,\\allowbreak 0\.50,\\allowbreak 0\.75,\\allowbreak 1\\\},αmax=2\.5\\alpha\_\{\\max\}=2\.5,cmax=3c\_\{\\max\}=3, andεden=10−14\\varepsilon\_\{\\rm den\}=10^\{\-14\}\. It selectsλ\\lambdausing seeds 123 and 456 at gaps 0\.005 and 0\.01, and every configuration with seed 789 or gap 0\.02 forms the 20\-unit held\-out set\. The field ablation usesm0=0\.08m\_\{0\}=0\.08,κ∈\{0\.08,0\.12\}\\kappa\\in\\\{0\.08,0\.12\\\},λ∈\{0,1,4\}\\lambda\\in\\\{0,1,4\\\}, andcmax∈\{3,5\}c\_\{\\max\}\\in\\\{3,5\\\}, with the per\-mode Phi mass combined with the scalar Delta\-Phi mass as stated after Eq\. \([23](https://arxiv.org/html/2608.23725#S2.E23)\)\.

Table 6:Evaluation blocks and statistical units\.Table 7:Extended quantitative evidence\. Negative field\-error differences favor the first method\.#### Validity boundary\.

The method requires a converged equilibrium, accurate local products, resolved critical subspaces, the relevant loss and parameter projections, and a positive bounded effective denominator, with no\-lift cases recovering the exact implicit adjoint\. Lifted backward\-only cases are identified as surrogates, frozen\-anchor modified\-forward cases are exact gradients of the stated modified residual, and physical rollout time remains distinct from DEQ solver time\.

## Appendix ISupplementary Pole and Source Diagnostics

![Refer to caption](https://arxiv.org/html/2608.23725v1/phase49_pole_response_rank2_supplement.png)Figure 6:Rank\-two response with \(a\) pole strength and \(b\) the rank\-two minus rank\-one Phi\-CMR advantage over Tikhonov\.Figure 7:Source\-alignment experiment with \(a\) retention gain over Tikhonov and \(b\) response norm versus critical\-mode suppression\.Figure 8:Delta\-Phi sweep with \(a\) bounded mass, \(b\) strength utility, and \(c\) response norm versus critical\-mode suppression\.Figure 9:Delta\-Phi ablation with \(a\) gate activation, \(b\) mass correction, and \(c\) suppression gain\.
## Appendix JCross\-Benchmark Evaluation of SILVA Response Renormalization

The cross\-benchmark analysis tests whether response renormalization depends on a particular PDE family or representation by repeating the matched eight\-method SILVA protocol on 13 additional public benchmark datasets comprising four PDEArena systems\[[Gupta and Brandstetter, 2023](https://arxiv.org/html/2608.23725#bib.bib37)\], six DynaBench systems\[[Dulny et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib38)\], PDEGym Poisson–Gauss\[[Herde et al\., 2024](https://arxiv.org/html/2608.23725#bib.bib39)\], CFDBench cylinder geometry\[[Luo et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib40)\], and LagrangeBench Taylor–Green vortex particles\[[Toshev et al\., 2023](https://arxiv.org/html/2608.23725#bib.bib41)\]\. This extension changes the governing equations, spatial dimension, operator type, geometry, and discretization while retaining five seeds, 16 training minibatches, the same 64\-update direct or 128\-update residual budget, eight backward methods, train\-only normalization, disjoint held\-out splits, and the same SILVA forward solver and numerical tolerances at evaluation\.

Table 8:Cross\-benchmark evaluation\. Dynamic entries use one\-step training and seven\-step free physical\-time rollouts, while Poisson–Gauss is evaluated as a static source\-to\-solution operator\.Table[9](https://arxiv.org/html/2608.23725#A10.T9)gives the family\-weighted static and horizon\-seven comparisons for all eight backward methods, comprising 520 static method evaluations and 3,360 repeated physical\-time evaluations\. Phi\-CMR has mean error ratios 0\.9999 statically and 1\.0016 at horizon seven, with ratios at most 1\.05 in all 65 static and 57 of 60 horizon\-seven paired evaluations, and lower static error than Tikhonov in 31 of 65 paired comparisons, while CMR gives ratios 0\.9989 and 0\.9978 under the same evaluation\. Near\-identity error ratios across grids, a vector\-field volume, an operator map, geometry\-conditioned flow, and moving particles show that the response rule transfers without requiring a representation\-specific notion of the critical denominator\.

Table 9:All\-method cross\-benchmark results\. Static values use 65 family–seed pairs, while horizon seven uses 60 pairs from the 12 dynamic datasets\. Error and measured solve\-plus\-backward time are normalized by exact implicit differentiation\.Figure[10](https://arxiv.org/html/2608.23725#A10.F10)disaggregates these averages by dataset to test whether aggregate fidelity conceals representation\-specific failures\. Panel \(a\) shows task error relative to implicit, panel \(b\) gives the paired Phi\-CMR–Tikhonov differences and seed intervals, panel \(c\) shows the fraction of training measurements on which the collective denominator was lifted, and panel \(d\) shows measured cost\. The lift occurs at least once in each dataset while remaining sparse over updates, matching the selective counterterm in Eq\. \([12](https://arxiv.org/html/2608.23725#S2.E12)\) and showing that near\-identity error ratios do not arise from an always\-active global correction\.

Figure 10:Cross\-benchmark evaluation of SILVA response renormalization with \(a\) final inverse\-standardized sample\-relative Euclidean error normalized by exact implicit differentiation, \(b\) paired Phi\-CMR\-minus\-Tikhonov differences across seeds, \(c\) the fraction of training measurements with an active collective lift, and \(d\) measured equilibrium\-solve plus backward time relative to implicit differentiation\. Shading in panel \(a\) denotes error ratios at most 1\.05\.The two time variables derived in Appendix[A\.1](https://arxiv.org/html/2608.23725#A1.SS1)are tested separately in Fig\.[11](https://arxiv.org/html/2608.23725#A10.F11), where panel \(a\) propagates prediction errors through seven physical PDE steps for all eight methods and panels \(b,c\) vary the DEQ solver horizonN∈\{1,2,4,8,16,32,64\}N\\in\\\{1,2,4,8,16,32,64\\\}, reaching atN=64N=64a Phi\-CMR causal\-response error of5\.27×10−45\.27\\times 10^\{\-4\}relative to the static adjoint and a state error of1\.49×10−41\.49\\times 10^\{\-4\}relative to equilibrium over 60 trained models\.

Figure 11:Cross\-benchmark transient evidence with \(a\) seven\-step free physical\-time rollouts for all eight backward methods on 12 dynamic datasets, \(b\) finite causal\-response error relative to the static implicit adjoint, and \(c\) original DEQ state error relative to equilibrium, where panels \(b,c\) use implicit, CMR, and Phi\-CMR models at solver horizonsN∈\{1,2,4,8,16,32,64\}N\\in\\\{1,2,4,8,16,32,64\\\}\. Shading in panel \(a\) denotes error ratios at most 1\.05\.### J\.1Forward\-capacity extension for the highest\-error families

The matched compact predictor controls the forward design but leaves visible approximation error in PDEArena Navier–Stokes, Shallow Water, and Maxwell, PDEBench Diffusion–Reaction 2D and Darcy, and LagrangeBench Taylor–Green vortex, motivating a forward\-capacity extension that tests whether this error belongs to the shared representation rather than to response renormalization\. Because held\-out trajectories, offsets, equilibrium solver and tolerances, backward methods, response parameters, and five seeds remain fixed while every method receives the same larger family\-specific forward model, the extension does not alter CMR or Phi\-CMR\. Darcy and Navier–Stokes use 128\-state two\-layer encoders and readouts with three\-layer width\-32 2D adapters, Shallow Water uses 96 states with a three\-layer width\-24 dilated adapter, and Maxwell reconstructs two three\-component vector fields at16316^\{3\}with a three\-layer width\-16 Conv3D adapter rather than folding depth and vector components into a 2D channel axis\. LagrangeBench uses a 128\-state two\-layer particle readout, while the fixedt0→t1t\_\{0\}\\\!\\to t\_\{1\}Diffusion–Reaction stress test uses 96 states with a two\-layer width\-16 adapter\. All profiles selected on seed 123 remain fixed throughout the subsequent five\-seed, eight\-method evaluation, and because the same seed participates in both model selection and evaluation, this extension is exploratory and its paired intervals are descriptive rather than confirmatory\.

Table 10:Phi\-CMR inverse\-standardized sample\-relative Euclidean error in the forward\-capacity extension, with capacity\-to\-baseline ratios and paired 95% intervals for the capacity\-minus\-baseline difference\.Phi\-CMR improves in all 30 family–seed comparisons in Table[10](https://arxiv.org/html/2608.23725#A10.T10), with the corresponding familywise prediction geometry shown in Figs\.[13](https://arxiv.org/html/2608.23725#A11.F13)–[18](https://arxiv.org/html/2608.23725#A11.F18)\. The largest reductions occur for Darcy and LagrangeBench, while the more modest Diffusion–Reaction reduction indicates a limitation of the selected physical\-time mapping rather than of the backward response law\. These results attribute a substantial part of the largest absolute errors to forward representation capacity, while the response\-renormalization comparison remains the matched compact benchmark defined by Tables[2](https://arxiv.org/html/2608.23725#S3.T2)and[9](https://arxiv.org/html/2608.23725#A10.T9)\.

Prediction geometry in Fig\.[12](https://arxiv.org/html/2608.23725#A11.F12)distinguishes shape recovery from amplitude calibration, which a single relative\-error number cannot separate\. On the same first held\-out example across five seeds, the capacity model changes Navier–Stokes correlation from0\.7040\.704to0\.7860\.786and amplitude ratio from0\.6460\.646to0\.8300\.830, while the corrected Maxwell Conv3D changes them from0\.2480\.248to0\.6660\.666and from0\.5660\.566to0\.9060\.906and LagrangeBench reaches correlation0\.9960\.996\. Because Darcy improves pattern correlation to0\.9800\.980but overpredicts the amplitude of that first example, the atlas uses the held\-out example nearest the median Phi\-CMR error rather than the first or best example\. The remaining smoothing of sharp Navier–Stokes vortical features and the Darcy amplitude mismatch identify forward\-resolution and calibration limits that are not explained by the backward response law\.

The unusual Diffusion–Reaction first step remains a stress test because, fromt0t\_\{0\}tot1t\_\{1\}, its two fields have persistence errors of8\.318\.31and20\.6620\.66and input–target correlations of only0\.230\.23and0\.100\.10, whereas the separately evaluatedt1→t2t\_\{1\}\\\!\\to t\_\{2\}transient window has correlations near0\.940\.94\. A64264^\{2\}, per\-channel normalized, four\-layer width\-32 residual adapter reaches mean Phi\-CMR error0\.26330\.2633, representative\-field correlation0\.9640\.964, and amplitude ratio0\.9720\.972on the latter window, which is excluded from the paired capacity ratio in Table[10](https://arxiv.org/html/2608.23725#A10.T10)because it represents a different physical\-time task\.

Figures[13](https://arxiv.org/html/2608.23725#A11.F13)–[18](https://arxiv.org/html/2608.23725#A11.F18)show held\-out fields or particles for higher\-capacity models using the same five\-seed all\-method aggregate and denominator/pole diagnostics as the compact atlases, with panels \(a–e\) selecting under a fixed evaluation seed the example whose Phi\-CMR error is nearest the median of 16 held\-out examples\.

Figures[19](https://arxiv.org/html/2608.23725#A11.F19)–[31](https://arxiv.org/html/2608.23725#A11.F31)use a fixed evaluation seed and the first held\-out example, except for DynaBench Kuramoto–Sivashinsky, which uses the deterministic median\-error example\. They retain panels \(a–h\) from Appendix[L](https://arxiv.org/html/2608.23725#A12), with Maxwell shown as a central slice and LagrangeBench in native particle coordinates\.

## Appendix KScope and Validity Conditions

The exact mathematical statement is narrower than a generic training\-stability claim because, forK=I−J=U​Σ​V⊤K=I\-J=U\\Sigma V^\{\\top\}, parameteraareceives modal contribution\(vi⊤​g\)​\(ui⊤​Ba\)/σi\(v\_\{i\}^\{\\top\}g\)\(u\_\{i\}^\{\\top\}B\_\{a\}\)/\\sigma\_\{i\}, requiring both the loss source and parameter source\. CMR changes a selected denominator and is therefore exact only for the explicitly modified, frozen\-anchor residual, whereas its application at the original equilibrium is a biased backward surrogate\. The experiments evaluate finite response, selectivity, and predictive fidelity under conditions in which exact implicit differentiation remains numerically available as a reference, allowing each response modification to be measured against a resolved adjoint\.

The dense rule requires a resolved critical singular subspace because Euclidean singular modes depend on state coordinates, and near a repeated singular cluster only the subspace, not individual vectors, is identifiable\. We therefore fix representation and train\-only normalization in every comparison, interpret threshold crossings within numerical tolerance as ambiguous, and recommend an unresolved interval\[κ−η,κ\+η\]\[\\kappa\-\\eta,\\kappa\+\\eta\]with principal\-angle tracking for repeated clusters, as formalized by the coordinate\-consistent metric in Appendix[D](https://arxiv.org/html/2608.23725#A4)\.

The SILVA specialization instead modifies singular values of the collective matrixΓ=I−B⊤​KL−1​A\\Gamma=I\-B^\{\\top\}K\_\{L\}^\{\-1\}A, which need not equal those of the fullKK, and therefore assumes accurate local solves and an informative low\-rank factorization\. Raw norms ofA,BA,Bare gauge dependent underA↦A​QA\\mapsto AQ,B↦B​Q−⁣⊤B\\mapsto BQ^\{\-\\top\}, which motivates an SVD\-balanced factorization for diagnostics, while neither the dense nor collective rule repairs poor forward convergence, architecture mismatch, unresolved subspaces, or rapidly changing modes\.

Finally, the error\-ratio threshold of 1\.05 is a descriptive upper threshold rather than a noninferiority or equivalence margin, and the five seeds are independent model repetitions, whereas updates, horizons, channels, and examples are repeated measurements\. Field errors are computed as inverse\-standardized sample\-relative Euclidean errors rather than conservation errors or PDE residuals\. Accordingly, the quantitative conclusions concern response fidelity, selective activation, convergence, and cost at the evaluated sizes under matched compact architectures\. Governing\-equation residuals, invariant preservation, asymptotic scaling, and architecture\-matched comparisons with monotone, spectral, or reversible DEQ designs require dedicated evaluations with the corresponding forward models\.

Forward parameterization uses training data only, with families whose training\-split persistence error is at most0\.650\.65predictingu^t\+1=ut\+rθ​\(ut\)\\widehat\{u\}\_\{t\+1\}=u\_\{t\}\+r\_\{\\theta\}\(u\_\{t\}\)and the others using a direct readout\. Shared normalization makesrθ=0r\_\{\\theta\}=0recover persistence at initialization, while projection after each update enforces∥W∥2≤0\.995\\lVert W\\rVert\_\{2\}\\leq 0\.995\. Consequently,J=diag⁡\(sech2⁡\(⋅\)\)​WJ=\\operatorname\{diag\}\(\\operatorname\{sech\}^\{2\}\(\\cdot\)\)Wsatisfies∥J∥2≤0\.995\\lVert J\\rVert\_\{2\}\\leq 0\.995andσmin​\(I−J\)≥0\.005\\sigma\_\{\\min\}\(I\-J\)\\geq 0\.005\. The forward map is therefore contractive while inducing near\-critical response, and static operator maps use direct parameterization\. All methods share architecture, initialization, data order, equilibrium solver, and numerical tolerances, which allows the atlas to compare predictions within a matched forward design even though the backward rule may produce distinct trained parameters and equilibria\.

Figure 12:Held\-out prediction geometry over five seeds with \(a\) representative\-field relative error, \(b\) centered spatial correlation, and \(c\) prediction\-to\-target amplitude ratio, where points are individual seeds, short horizontal marks are means, and Diffusion–Reaction uses the separatet1→t2t\_\{1\}\\\!\\to t\_\{2\}transient task\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_capacity_atlas_darcy_2d.png)Figure 13:Higher\-capacity Darcy 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Panels \(a–e\) use the median\-errorβ=1\\beta=1example, while panel \(f\) aggregates all five Darcy regimes\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_capacity_atlas_diffusion_reaction_2d.png)Figure 14:Transient\-aligned Diffusion–Reaction 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first physical channel for the separately evaluatedt1→t2t\_\{1\}\\\!\\to t\_\{2\}task\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_capacity_atlas_pdearena_ns2d.png)Figure 15:Higher\-capacity PDEArena Navier–Stokes 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first vorticity/state channel\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_capacity_atlas_pdearena_sw2d.png)Figure 16:Higher\-capacity PDEArena Shallow Water 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first pressure/height channel\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_capacity_atlas_pdearena_maxwell3d.png)Figure 17:Higher\-capacity PDEArena Maxwell 3D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels show the central slice of the first electric\-displacement component after vector\-aware Conv3D reconstruction\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_capacity_atlas_lagrangebench_tgv.png)Figure 18:Higher\-capacity LagrangeBench Taylor–Green vortex 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Panels \(a–e\) retain the native particle coordinates\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_pdearena_ns2d.png)Figure 19:PDEArena Navier–Stokes 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first vorticity/state channel\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_pdearena_sw2d.png)Figure 20:PDEArena Shallow Water 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first pressure/height channel\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_pdearena_maxwell3d.png)Figure 21:PDEArena Maxwell 3D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels show the central slice of the first electric\-displacement component\.Figure 22:PDEArena Kuramoto–Sivashinsky 1D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the scalar state\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_dynabench_advection.png)Figure 23:DynaBench Advection with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the advected scalar\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_dynabench_burgers.png)Figure 24:DynaBench Burgers with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the conserved scalar\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_dynabench_gas.png)Figure 25:DynaBench Gas Dynamics with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display density, while panel \(f\) aggregates all predicted variables\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_dynabench_ks.png)Figure 26:DynaBench Kuramoto–Sivashinsky with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the scalar state for the deterministic example nearest the Phi\-CMR median error \(offset 3 of 16\)\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_dynabench_rd.png)Figure 27:DynaBench Reaction–Diffusion with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first reaction–diffusion channel\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_dynabench_wave.png)Figure 28:DynaBench Wave with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first wave\-state channel\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_pdegym_poisson.png)Figure 29:PDEGym Poisson–Gauss with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Panels \(a–e\) show the source\-to\-solution operator example\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_cfdbench_cylinder.png)Figure 30:CFDBench cylinder geometry with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display velocity for a held\-out geometry case\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_extension_atlas_lagrangebench_tgv.png)Figure 31:LagrangeBench Taylor–Green vortex 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Panels \(a–e\) retain the native particle coordinates, and panel \(d\) is position error\.
## Appendix LNative SILVA Per\-Family Field and Pole Atlas

Figures[32](https://arxiv.org/html/2608.23725#A12.F32)–[41](https://arxiv.org/html/2608.23725#A12.F41)connect aggregate error ratios to recognizable physical predictions and the proposed response mechanism, with each visualization using a fixed seed and an a priori held\-out example\. Panels \(a–e\) show the first physical channel as input or initial condition, target, Phi\-CMR prediction, absolute error, and pointwise Tikhonov\-minus\-Phi\-CMR error, with positive values in panel \(e\) favoring Phi\-CMR, while panel \(f\) displays the all\-channel, five\-seed aggregate for all eight methods normalized by exact implicit differentiation and, for Darcy, also averages five conditioning regimes while the displayed field usesβ=1\\beta=1\.

Panel \(g\) traces the measured and effective Phi\-CMR collective denominators, the cutoffκ\\kappa, and active\-lift updates, while panel \(h\) evaluates the normalized modal contribution\|vi⊤​g\|/σi\|v\_\{i\}^\{\\top\}g\|/\\sigma\_\{i\}of the final 48\-dimensional hidden residualK=I−JK=I\-Jfor the same held\-out source\. The associated adjoint amplification is determined by the loss\-source projection onto each mode, while training traces activate the collective lift in every family even when a displayed final source does not\. Panel \(g\) therefore resolves the SILVA mechanism and panel \(h\) resolves full\-state source visibility\.

Table 11:Physical systems and displayed quantities in the native SILVA per\-family analysis\. Darcy is evaluated as an elliptic coefficient\-to\-solution map, while the remaining entries are one\-step dynamical predictions\.Figure 32:Advection 1D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the scalar state\.Figure 33:Burgers 1D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the scalar state\.Figure 34:Reaction–Diffusion 1D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first state channel\.Figure 35:Diffusion–Sorption 1D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display concentration\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_family_atlas_diffusion_reaction_2d.png)Figure 36:Diffusion–Reaction 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first physical channel\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_family_atlas_shallow_water_2d.png)Figure 37:Shallow Water 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display fluid height\.Figure 38:Compressible CFD 1D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display density, while panel \(f\) aggregates density, pressure, and velocity\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_family_atlas_incompressible_ns_2d.png)Figure 39:Incompressible Navier–Stokes 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display horizontal velocity\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_family_atlas_the_well_radiative_2d.png)Figure 40:The Well turbulent radiative layer 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Field panels display the first physical channel from the held\-out split\.![Refer to caption](https://arxiv.org/html/2608.23725v1/silva_family_atlas_darcy_2d.png)Figure 41:Darcy 2D with \(a\) input, \(b\) target, \(c\) Phi\-CMR prediction, \(d\) absolute error, \(e\) Tikhonov\-minus\-Phi\-CMR error, \(f\) five\-seed all\-method relative\-L2L\_\{2\}, \(g\) original and effective collective denominators with active lifts, and \(h\) source\-weighted singular response\. Panels \(a–e\) use a fixedβ=1\\beta=1coefficient\-to\-solution example, while panel \(f\) aggregates five Darcy regimes\.