arxiv: 2604.05567 · v1 · submitted 2026-04-07 · 🧮 math.OC · cs.SY· eess.SY

Recognition: 2 theorem links

· Lean Theorem

Scaled Graph Containment for Feedback Stability: Soft-Hard Equivalence and Conic Regions

Eder Baron-Prada , Julius P. J. Krebbekx , Adolfo Anta , Florian D\"orfler

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Pith reviewed 2026-05-10 19:15 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords scaled graphsfeedback stabilitymultiplier methodscircular regionsconic regionssoft-hard equivalencehyperbolic convexitystability certification

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The pith

Soft and hard scaled graph containment are equivalent for circular regions under positive-negative multipliers

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops containment conditions for scaled graphs inside multiplier-defined regions for feedback stability analysis. For circular regions it proves that soft and hard containment coincide exactly when the multiplier is positive-negative. This equivalence lets hard stability certificates be computed directly from soft conditions, removing the positive-semidefinite storage requirement and the homotopy step used in earlier methods. The work also identifies which conic regions are hyperbolically convex and shows they can give tighter bounds than circles when the operator graph is nonsymmetric.

Core claim

Scaled graphs provide a geometric framework for feedback stability. For circular regions the paper shows soft and hard SG containment are equivalent whenever the multiplier is positive-negative. This equivalence yields hard stability certification from soft computations alone, bypassing both the positive-semidefinite storage constraint and the homotopy condition of existing methods. The paper further characterizes hyperbolically convex conic regions and demonstrates that such regions supply tighter SG bounds than circles whenever the operator SG is nonsymmetric.

What carries the argument

Scaled-graph containment inside multiplier-defined circular and conic regions, with soft-hard equivalence when the multiplier is positive-negative

Load-bearing premise

The multiplier must be positive-negative for soft-hard equivalence to hold in circular regions.

What would settle it

A positive-negative multiplier together with a scaled graph that satisfies soft containment but violates hard containment, or a frequency-domain certificate that holds for a conic region that is not hyperbolically convex.

Figures

Figures reproduced from arXiv: 2604.05567 by Adolfo Anta, Eder Baron-Prada, Florian D\"orfler, Julius P. J. Krebbekx.

**Figure 1.** Figure 1: Negative feedback interconnection of H1 and H2. Theorem 4 (Soft SG separation [2], [4]). Let H1, H2 : L2 → L2 be causal, L2-stable systems, and assume that their feedback interconnection as in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: (a) For system H1, soft SG (dark gray) and hard SG (light gray), together with the multiplier region S(Π1) (blue) and its inverse S −1 (Π1) (yellow). (b) For system H2, soft SG (dark gray) and hard SG (light gray), along with the multiplier region S(Π2) (blue). (c) Corresponding sets used for stability analysis: −SG(H2) (dark gray), −SGe(H2) (light gray), −S(Π2) (blue), and S −1 (Π1) (yellow). (a) (b) [PI… view at source ↗

**Figure 4.** Figure 4: Scalability results for systems with state dimension [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 3.** Figure 3: (a) System H1: soft SG (gray), circular region S(Π1) (blue), and ellipsoidal region C(Θ1) (red). (b) System H2: soft SG (gray), circular region S(Π2) (blue), and ellipsoidal region C(Θ2) (red). C. Scalability Demonstration To quantify the computational advantages of Corollary 1, we compare the soft and hard LMI formulations across systems of increasing dimension. We consider block-diagonal MIMO systems co… view at source ↗

read the original abstract

Scaled graphs (SGs) offer a geometric framework for feedback stability analysis. This paper develops containment conditions for SGs within multiplier-defined regions, addressing both circular and conic geometries. For circular regions, we show that soft and hard SG containment are equivalent whenever the associated multiplier is positive-negative. This enables hard stability certification from soft computations alone, bypassing both the positive semidefinite storage constraint and the homotopy condition of existing methods. Numerical experiments on systems with up to 300 states demonstrate computational savings of 15-44 % for the circular containment framework. We further characterize which conic regions are hyperbolically convex, a condition our frequency-domain certificate requires, and demonstrate that such regions provide tighter SG bounds than circles whenever the operator SG is nonsymmetric.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The soft-hard equivalence for circular regions under positive-negative multipliers is the main practical advance, and the hyperbolic convexity characterization for conics looks like a clean geometric extension.

read the letter

The paper's central result is that soft and hard scaled-graph containment are equivalent for circular regions exactly when the multiplier is positive-negative. This removes the need for PSD storage and homotopy steps in the hard case, which is a direct win for computation. The authors also give a characterization of hyperbolically convex conic regions so the frequency-domain certificate can be applied, and they show these regions can give tighter bounds than circles when the operator graph is nonsymmetric. Numerical runs on systems up to 300 states report 15-44% savings, which is concrete evidence that the equivalence translates to practice. The derivations appear to come straight from the geometry of the scaled graph and the sign pattern of the multiplier, without obvious reductions to earlier fitted quantities. The positive-negative condition and the hyperbolic-convexity restriction are stated up front, so the scope is clear. The only real limitation is that the equivalence and the conic test both depend on those specific multiplier and region properties; outside them the old bottlenecks remain. This work is aimed at researchers who already use scaled-graph or multiplier methods for feedback stability, especially on large or nonsymmetric systems. It is a focused, incremental improvement rather than a wholesale new framework, but the claims are specific enough and the savings are measured, so it deserves a serious referee. I would send it out for review.

Referee Report

0 major / 2 minor

Summary. The paper claims to establish an equivalence between soft and hard scaled graph (SG) containment for circular regions in the context of feedback stability analysis when the multiplier is positive-negative. This equivalence allows for hard stability certification using only soft computations, avoiding positive semidefinite storage constraints and homotopy conditions. Additionally, it characterizes hyperbolically convex conic regions for which frequency-domain certificates are valid and demonstrates that these regions provide tighter bounds than circular ones for nonsymmetric operator SGs. Numerical experiments show computational savings of 15-44% for systems with up to 300 states.

Significance. The results, if correct, represent a meaningful advance in the scaled graph framework for stability analysis by providing a direct bridge between soft and hard containment methods for circular regions and extending the approach to conic regions with improved tightness. The explicit if-and-only-if condition based on multiplier sign pattern and the hyperbolic convexity characterization are valuable theoretical contributions. The reported numerical savings indicate potential for practical use in large-scale systems, strengthening the applicability of geometric methods in control theory.

minor comments (2)

The numerical experiments would benefit from additional details on the specific system dimensions, types of systems tested, and the baseline methods used to compute the reported 15-44% savings for better reproducibility and assessment of the claims.
Consider adding a figure or table comparing the tightness of conic regions versus circular regions for nonsymmetric SGs to visually support the theoretical claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The referee summary accurately reflects the paper's contributions on soft-hard equivalence for circular scaled graph containment and the characterization of hyperbolically convex conic regions. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The central results—the soft-hard SG containment equivalence for circular regions under positive-negative multipliers, and the characterization of hyperbolically convex conic regions—are derived directly from the geometry of scaled graphs and multiplier sign patterns as stated in the abstract and skeptic analysis. No load-bearing self-citations, fitted inputs renamed as predictions, or self-definitional reductions appear; the equivalence is presented as an if-and-only-if geometric statement, and the frequency-domain certificate is explicitly conditioned on hyperbolic convexity with an independent characterization supplied. Numerical savings are reported as consequences of the framework rather than inputs to the proof. The work extends prior scaled-graph literature but the new claims remain independent of any self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the prior scaled graph framework for feedback stability and introduces new containment conditions and convexity properties without introducing new free parameters or invented entities in the abstract.

axioms (2)

domain assumption Scaled graphs offer a geometric framework for feedback stability analysis
Invoked in the opening sentence as the foundation for the containment conditions.
domain assumption The frequency-domain certificate requires hyperbolic convexity of the region
Stated as a necessary condition for the conic analysis to apply.

pith-pipeline@v0.9.0 · 5448 in / 1267 out tokens · 36555 ms · 2026-05-10T19:15:21.828567+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Every positive-negative multiplier admits a J-spectral factorization Π = Ψᵀ J Ψ ... Lemma 1 (Soft–hard IQC equivalence)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 from linking) echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

C(Θ) is h-convex ⇔ Θ₁₁ ≥ Θ₂₂ ... Beltrami–Klein coordinate representation

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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