{"id":"396edf98-e110-4fb0-9a8c-77d936c497bb","arxiv_id":"2504.21448","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper introduces a signed scaled graph, using the Hilbert transform to give nonlinear system phase a sign, so phase-lead and phase-lag systems can be distinguished in feedback stability analysis.","lead":"The paper defines a signed version of the scaled graph, a graphical tool for nonlinear control systems, by using the Hilbert transform to give phase a sign. This lets engineers tell phase-lead from phase-lag behavior, which the original scaled graph cannot do, and yields stability tests for feedback loops.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Appendix A proves a reciprocal separation condition not equivalent to (16); the central stability theorem is supported only for the variant stated as 'still holds,' so Theorem 2 is unproven as stated.","rationale":"I read Theorem 2 as the paper's central claim: a signed scaled-graph separation test that distinguishes lead from lag. The definitions, Lemma 1, and the motivating lead/lag examples are clear, and the idea of using the Hilbert transform to sign the phase is sensible. The proof, however, appears to analyze a different separation condition. In Appendix A, z1 is defined as an element of SSG(H1) and z2 as an element of SSG†(-τH2), with magnitudes ||y1||/||u1|| and ||-τy2||/||u2||. That is the reciprocal of the theorem's condition (16), which uses SSG†(H1) and SSG(-τH2). The case inequalities (21)-(23) only make sense under the reciprocal pairing: for instance, |z1| ≤ |z2|-ε yields the small-gain product bound only because z1 is the forward gain and z2 the inverse gain. The paper's parenthetical that Theorem 2 'still holds if condition (16) is replaced by' the reciprocal condition signals the mismatch, but no equivalence proof is supplied. Reciprocal-distance conditions are not equivalent in general: with a = 0.05 and b = 10, dist(1/a, τb) ≥ 10 for all τ ∈ (0,1], yet dist(a, 1/(τb)) = 0.05. Thus Theorem 2 is unsupported as stated. I also acknowledge the reader's uniform-epsilon concern: even in the corrected reciprocal variant, the step from a positive distance lower bound to a uniform δ in (25)-(31) requires bounds on the magnitude set or a more careful partition; finite-gain assumptions provide some bounds but the proof does not state or use them. These are repairable proof gaps rather than evidence that the result is false, so the appropriate verdict remains conditional pending revision.","tokens_in":13414,"tokens_out":18016,"duration_ms":180871,"concrete_test":"Re-derive Appendix A using the theorem's actual sets: take z1 = (||u1||/||y1||) e^{-jφ(u1,y1)} ∈ SSG†(H1) and z2 = (τ||y2||/||u2||) e^{jφ(u2,-τy2)} ∈ SSG(-τH2), and check whether equations (21)-(27) still follow; then test whether dist(SSG†(H1), SSG(-τH2)) ≥ r implies dist(SSG(H1), SSG†(-τH2)) ≥ r' for a uniform r' > 0. A scalar counterexample (a = 0.05, b = 10, τ ∈ (0,1]) shows the implication can fail, so unless additional structure is identified, Theorem 2 should be amended to the reciprocal separation condition or supplied with a new proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is Theorem 2, whose hypothesis (16) is dist(SSG†(H1), SSG(-τH2)) ≥ r. Appendix A, however, defines z1 ∈ SSG(H1) with magnitude ||y1||/||u1|| and z2 ∈ SSG†(-τH2) with magnitude ||-τy2||/||u2||, i.e. it analyzes dist(SSG(H1), SSG†(-τH2)) — exactly the condition the paper says the theorem 'still holds if' (16) is replaced by. The case inequalities (21)-(23) only cohere under this reciprocal pairing: for example, |z1| ≤ |z2|-ε is converted into a small-gain product bound only when z1 is the forward gain of H1 and z2 is the inverse gain of H2. No argument shows that (16) implies this reciprocal separation, and the two conditions are not equivalent for arbitrary sets: with a = 0.05 and b = 10, dist(1/a, τb) ≥ 10 for all τ ∈ (0,1], while dist(a, 1/(τb)) = 0.05. The proof never uses (16) with z1 ∈ SSG†(H1) and z2 ∈ SSG(-τH2), so the theorem as stated lacks support. Separately, even for the corrected variant, the passage from a positive distance lower bound to a uniform ε for magnitude/angle separation is not justified by compactness or explicit magnitude bounds, compounding the gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a 'signed scaled graph' (SSG) for nonlinear systems, replacing the unsigned singular angle of the scaled graph with a phase defined through the Hilbert transform of the input. The main stability claim, Theorem 2, states that if the SSG of the inverse of H1 and the SSG of -tau H2 are separated by a positive distance for all tau in (0,1], then the negative feedback interconnection is finite-gain stable for all inputs outside a set W. The paper also proposes SSG-based definitions of passive and SSG-negative-imaginary systems, and argues that the signed phase distinguishes phase-lead from phase-lag, recovering passivity and negative-imaginary interconnection results and reducing conservatism relative to the original scaled graph. The lead/lag example and Theorem 3 illustrate the intended advantage of the new construction.","tokens_in":13767,"tokens_out":11921,"duration_ms":115339,"significance":"If Theorem 2 were established, the SSG would be a valuable extension of scaled-graph theory: it would provide a graphical stability test that separates phase lead from phase lag, and would unify passivity and negative-imaginary results in a single framework. The paper is clearly written, the construction is self-contained, and the motivating examples are compelling. Lemma 1 and Theorem 3 are elementary and correct. However, the main stability theorem is not soundly proven as stated, and the proof contains a load-bearing mismatch between the hypothesis and the argument. The significance of the paper is therefore prospective rather than established in its current form.","major_comments":[{"comment":"The proof of Theorem 2 does not address the hypothesis (16). Condition (16) is dist(SSG†(H1), SSG(−τH2)) ≥ r, but the proof's opening paragraph takes z1 ∈ SSG(H1) and z2 ∈ SSG†(−τH2), and the inequalities (21) and (23) are built from the forward gain of H1 paired with the inverse gain of H2. This is the reciprocal separation condition dist(SSG(H1), SSG†(−τH2)) ≥ r, which the paper only notes as an alternative under which the theorem 'still holds.' No argument shows that (16) implies this reciprocal condition, and it is not equivalent for arbitrary sets: for a=0.05 and b=10, dist({1/a}, {τb : τ∈(0,1]}) ≥ 10 while dist({a}, {1/(τb) : τ∈(0,1]}) = 0.05. Thus Theorem 2 as stated lacks proof support.","section":"Theorem 2 and Appendix A"},{"comment":"The claim that |z1−z2| ≥ r is 'equivalent' to a uniform dichotomy—equal-angle points separated in magnitude by at least ε, or equal-magnitude points separated in angle by at least ε, for a single ε > 0—is not justified. The sets SSG(H1) and SSG(−τH2) are not shown to be compact, nor to be bounded away from zero and infinity, and trajectories may have arbitrarily large or small gains. A positive distance between two noncompact sets does not imply a uniform threshold ε (or later δ) independent of the trajectory magnitudes. Since the constants c1,...,c4 in (22), (24), (29), and (32) depend on this ε or δ, the homotopy argument requires an additional hypothesis (for example, boundedness of the SSG sets away from 0 and ∞) or a different proof technique.","section":"Appendix A, uniform-epsilon step"},{"comment":"The excluded set W is defined in (15) through conditions on the internal signals u1, u2, and y2, so the conclusion 'finite-gain stable for all inputs w ∈ L2 \\ W' is not an easily checkable input-space statement: membership of w in W depends on the solution of the feedback equations and on the resulting trajectories, not on w alone. The paper notes that W is empty in some classical cases and that W ≠ L2 in Example 2, but it does not quantify how large W can be. This weakens the practical force of the theorem; if W is intended as a technical exception, it should be characterized more directly or the theorem should be reformulated with an explicit condition on the data of the problem.","section":"Equation (15) and Theorem 2"}],"minor_comments":[{"comment":"In the proof of Theorem 2, the displayed expression for z2 ∈ SSG†(−τH2) uses the forward gain ||−τy2||/||u2||, which is the magnitude of a point in SSG(−τH2), not in its inverse. The notation should be aligned with the hypothesis of the theorem or explicitly corrected.","section":"Appendix A, notation"},{"comment":"The statement that SSG(H1) and −SSG†(τHlead) do not intersect is argued from the Nyquist diagram bounding SG(H1). Since the SSG is defined for all L2 inputs, the example should clarify why broadband nonlinear trajectories cannot leave the claimed half-plane; otherwise the graphical conclusion is only heuristic.","section":"Example 2"},{"comment":"In the proof of Theorem 4, the sentence asserting that the separation of SSG(H1) \\ X and SSG†(H2) \\ X 'also holds for all scalings τ∈(0,1]' is stated without argument. Because scaling by τ changes magnitudes, a short justification would improve clarity.","section":"Theorem 4"}],"recommendation":"major_revision","confidential_remarks":"The paper proposes an interesting and potentially useful extension of scaled graphs, and the lead/lag example is convincing. However, the main theorem's proof is currently inconsistent with its statement: the proof establishes a reciprocal separation condition, not condition (16), and it relies on an unjustified uniform-epsilon step. These issues are fixable either by changing the theorem to the reciprocal condition with added boundedness assumptions or by supplying a proof of (16), so major revision is appropriate rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know about this paper: it defines the signed scaled graph (SSG), a variant of the scaled graph that uses Hilbert-transform-based phase to distinguish lead from lag. That is a real idea, and the paper does several good things with it. It shows SSG separates lead and lag filters that are identical under the ordinary scaled graph, recovers passivity results, and gives a scaled-graph route to negative imaginary interconnections. The motivating examples are clear and honestly worked.\n\nThe soft spot is Theorem 2, the main stability result. The stated hypothesis (16) is dist(SSG†(H1), SSG(-τH2)) ≥ r, with the dagger on H1. But Appendix A defines z1 in SSG(H1) and z2 in SSG†(-τH2) — that is, it proves separation for dist(SSG(H1), SSG†(-τH2)), the condition the paper notes only in the sentence \"Theorem 2 still holds if condition (16) is replaced by...\" That sentence is doing real work: the proof supports only the variant, not the advertised theorem. The two conditions are not equivalent; a simple scalar example shows the distance can be positive one way and zero the other. So Theorem 2 as stated lacks support.\n\nThere is also a secondary gap. Even for the corrected variant, the proof jumps from a fixed distance lower bound r to a uniform epsilon (and later delta) controlling magnitude or angle separation. That inference requires compactness or explicit magnitude bounds, and none are supplied. The SSG sets can be unbounded or have accumulation points at zero, so the uniform constant may not exist.\n\nThe definition of the exceptional set W is honest — the paper says it is a technical restriction. But W is effectively defined as the set where the proof's hardest case appears, which is a mild bootstrap. Not fatal by itself, but it compounds the main issue.\n\nOverall: the conceptual contribution is solid and worth engaging with. The proof needs a major repair. I would send it to peer review because the idea is novel and the examples are valuable, but I would expect the referee to demand a corrected theorem statement and a complete proof. If I were citing it, I would cite the SSG definition and the lead/lag separation example, not Theorem 2 as it stands.","headline":"The signed scaled graph is a genuinely useful new object, but the main stability theorem is not proven as stated because the proof analyzes a different separation condition than the one in the theorem.","tokens_in":14261,"tokens_out":1831,"would_cite":true,"duration_ms":18807,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C10","93D25","93C80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The signed scaled graph, using Hilbert-transform phase, separates phase lead from phase lag in nonlinear feedback systems and yields a stability test that is never more conservative than the original scaled-graph test.","keywords":["signed scaled graph","Hilbert transform","phase lead and lag","nonlinear feedback stability","scaled graph","passivity","negative imaginary systems","graphical stability test"],"falsifier":"Construct two finite-gain stable systems whose signed graphs are always separated by, say, distance 1 for every $\\tau \\in (0,1]$, but along a sequence of inputs the equal-magnitude points have phase differences that shrink to zero while norms grow, so no uniform $\\delta$ exists for the small-phase case. If such a pair were also a counterexample to finite-gain stability, Theorem 2's conclusion would fail; if stability still held, the proof's uniformity premise would still be refuted.","tokens_in":13210,"feed_emoji":"📈","tokens_out":4827,"duration_ms":48034,"temperature":0.7,"pith_summary":"The paper proposes changing the phase notion inside the scaled graph (SG), a graphical tool that plots gain and phase of nonlinear input-output trajectories like a Nyquist plot. The SG's phase is an unsigned singular angle, so a lead filter and a lag filter with the same gain profile look identical; the paper's signed scaled graph (SSG) replaces that angle with a signed phase computed from the Hilbert transform of the input, which carries the sign of the imaginary part of the frequency response in the linear case. The main result (Theorem 2) says negative feedback is finite-gain stable whenever the SSG of the forward system and the inverse SSG of the scaled negative feedback system are separated by a positive distance, for all gain scalings, except for a technical set of inputs. This recovers passivity and negative-imaginary interconnection results as special cases and, when applied to a lead-versus-lag example, reproduces the Nyquist stability boundary without conservatism.","feed_headline":"Signed scaled graphs separate lead from lag","feed_subtitle":"A Hilbert-transform phase sharpens nonlinear feedback stability tests and recovers passivity and negative-imaginary results.","key_machinery":"The signed scaled graph (SSG): for $u,y$, gain $\\rho = \\|y\\|/\\|u\\|$, unsigned singular angle $\\theta = \\arccos(\\langle u,y\\rangle/(\\|u\\|\\|y\\|))$, and signed phase $\\phi = \\operatorname{sgn}(\\langle\\!\\langle\\!\\langle \\hat{u}, E(y)\\rangle\\!\\rangle\\!\\rangle)\\,\\theta$, with a two-valued convention when the Hilbert inner product vanishes. The Hilbert-inner-product sign replaces the unsigned singular angle, so $\\operatorname{SSG}(H) \\subset \\operatorname{SG}(H)$ and $\\operatorname{SSG}(H) \\cup \\operatorname{SSG}^*(H) = \\operatorname{SG}(H)$. The proof of Theorem 2 partitions trajectory pairs into small-gain, large-gain, and small-phase cases, using the graph-separation distance to obtain uniform $\\epsilon$ or $\\delta$ constants in each case, then invokes a homotopy and well-posedness argument to convert the uniform bounds into a finite-gain estimate.","core_discovery":"The paper's central claim is that phase information in a nonlinear operator can be made signed by defining phase from the Hilbert transform: for input $u$ and output $y$, the signed phase is the unsigned singular angle multiplied by the sign of the cross-correlation between the Hilbert transform of $u$ and the output. The resulting SSG is no longer symmetric about the real axis, and a positive-distance separation condition between $\\operatorname{SSG}(H_1)$ and $\\operatorname{SSG}^\\dagger(-\\tau H_2)$ for all $\\tau \\in (0,1]$ implies finite-gain stability of the negative feedback loop for all $L_2$ inputs outside the exceptional set $W$. Because the signed graph distinguishes phase lead from phase lag, the test is never more conservative than Theorem 1's unsigned separation test, and it recovers the classical passivity theorem and a negative-imaginary interconnection result as special cases. The authors define SSG-negative imaginary systems through a Hilbert-transform inequality and state a corresponding positive-feedback stability theorem.","pith_inferences":["If the uniform $\\epsilon/\\delta$ gap can be shown to follow from a fixed set distance under mild compactness assumptions, Theorem 2 would hold for all $L_2$ inputs, removing the exceptional set $W$; the paper itself suggests $W$ is technical.","Because the Hilbert transform is an $L_2$ isometry up to sign, the SSG inequality defining SSG-negative imaginary systems may connect to integral quadratic constraint analyses for nonlinear systems, giving a frequency-domain interpretation that derivative-based negative-imaginary definitions lack.","For multi-input/multi-output operators the Hilbert inner product is still well-defined componentwise, so the construction appears to extend beyond scalar or single-channel settings, though only scalar signals are treated here.","The signed-phase construction could be used for synthesis, e.g., shaping a nonlinear controller's SSG to lie in the right half plane, which the introduction flags as a motivation for distinguishing lead from lag."],"forward_implications":["For LTI systems, the SSG separates the positive-frequency and negative-frequency halves of the Nyquist diagram, so a lead filter and a lag filter with the same gain curve no longer produce the same graphical object.","Any feedback pair that passes the original scaled-graph separation test also passes the SSG test, and some pairs that fail the original test pass the SSG test; stability certificates are therefore at least as sharp.","The strict-passivity-plus-passivity theorem follows from the SSG separation condition with $W$ empty, giving finite-gain stability for all square-integrable inputs.","SSG-negative imaginary systems, defined via a Hilbert-transform inequality rather than time derivatives, admit a positive-feedback stability theorem under a DC-gain product condition.","The graphical separation condition can be checked by plane geometry, so the theorem functions as a nonlinear analogue of the Nyquist test."],"supporting_citations":[{"why":"Introduces scaled relative graphs, the precursor object whose unsigned gain-phase pairs the SSG modifies.","marker":"[3]"},{"why":"Defines the scaled graph for nonlinear systems and its graphical stability framework, which the SSG extends.","marker":"[6]"},{"why":"Supplies the unsigned separation theorem (Theorem 1) that the SSG result is designed to sharpen.","marker":"[7]"},{"why":"Gives the Hilbert transform background and Plancherel identities underlying the signed phase definition.","marker":"[16]"},{"why":"Introduces phase of nonlinear systems via the Hilbert transform, the sign mechanism used in the SSG.","marker":"[17]"},{"why":"Provides the small-gain, large-gain, and small-phase partition of trajectory pairs used in the proof of Theorem 2.","marker":"[12]"},{"why":"Supplies the homotopy and well-posedness result that turns uniform bounds into finite-gain stability.","marker":"[27]"}],"fun_headline_variants":["Hilbert-transform phase sharpens nonlinear stability","Signed phase via Hilbert transform tightens feedback bounds","Lead-lag separation in scaled graphs through signed phase","Signed scaled graph: nonlinear Nyquist with phase direction","Hilbert phase separates lead from lag in scaled graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's case distinction relies on the assumption that a fixed positive distance between the two signed graph sets forces a uniform gap in magnitude for equal-angle pairs or a uniform gap in angle for equal-magnitude pairs, with the same gap size for every trajectory magnitude; the paper does not show the graph sets are compact or bounded away from zero and infinity, so this uniformity is not automatic.","fun_headline_variants_meta":{"raw":{"variants":["Hilbert-transform phase sharpens nonlinear stability","Signed phase via Hilbert transform tightens feedback bounds","Lead-lag separation in scaled graphs through signed phase","Signed scaled graph: nonlinear Nyquist with phase direction","Hilbert phase separates lead from lag in scaled graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000393,"raw_usage":{"total_tokens":2031,"prompt_tokens":879,"completion_tokens":1152,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":1077}},"tokens_in":495,"tokens_out":1152,"duration_ms":10601,"temperature":1.0,"reasoning_tokens":1077,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:03:56.254832+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct two finite-gain stable systems whose signed graphs are always separated by, say, distance 1 for every $\\tau \\in (0,1]$, but along a sequence of inputs the equal-magnitude points have phase differences that shrink to zero while norms grow, so no uniform $\\delta$ exists for the small-phase case. If such a pair were also a counterexample to finite-gain stability, Theorem 2's conclusion would fail; if stability still held, the proof's uniformity premise would still be refuted.","supporting_citations":[{"cited_title":"Scaled relative graphs: Non- expansive operators via 2D Euclidean geometry,","cited_arxiv_id":null,"evidence_quote":"Introduces scaled relative graphs, the precursor object whose unsigned gain-phase pairs the SSG modifies."},{"cited_title":"Graphical Nonli near System Analysis,","cited_arxiv_id":null,"evidence_quote":"Defines the scaled graph for nonlinear systems and its graphical stability framework, which the SSG extends."},{"cited_title":"Scaled graphs for reset control system analysis,","cited_arxiv_id":null,"evidence_quote":"Supplies the unsigned separation theorem (Theorem 1) that the SSG result is designed to sharpen."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Hilbert transform background and Plancherel identities underlying the signed phase definition."},{"cited_title":"Soft and hard scal ed relative graphs for nonlinear feedback stability","cited_arxiv_id":null,"evidence_quote":"Provides the small-gain, large-gain, and small-phase partition of trajectory pairs used in the proof of Theorem 2."},{"cited_title":"Freeman, ‘On the Role of Well-Posedness in Homotop y Methods for the Stability Analysis of non-linear Feedback Systems’ , In Lecture Notes in Control and Information Sciences, pp","cited_arxiv_id":null,"evidence_quote":"Supplies the homotopy and well-posedness result that turns uniform bounds into finite-gain stability."}],"review_version":1}