{"id":"cf3586e2-3d03-4f2c-967b-b927b700f7a4","arxiv_id":"2412.19356","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The susceptibility of edges in a trained resistor network contains all physical information of the cost landscape and identifies the edges responsible for the learned function.","lead":"Trained electrical resistor networks carry a hidden fingerprint of the learned task: a susceptibility, computable from the network's own voltage responses, that reveals which edges matter most. The paper derives an exact decomposition of the learning cost landscape into task information and this physical susceptibility, then shows in simulations that the most susceptible edges are the functionally important ones.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exact Hessian factorization requires cr=0; finite-cost corrections are unquantified, yet the classification example runs at C≈0.05 where sqrt(C)~0.2 makes corrections potentially non-negligible.","rationale":"The reader's weakest-assumption analysis correctly identifies the zero-cost requirement as the most fragile point of the derivation. I could not find a more fundamental flaw: the matrix calculus leading to Eq. (16) is consistent, the bordered-Laplacian treatment is standard, and the susceptibility vectors are well-defined physical observables. The finite-cost correction is the single load-bearing concern because the paper's headline claim is an exact statement ('all the physical information ... captured by a susceptibility'), while all simulation evidence comes from networks with residual cost. For the low-cost regression and allostery examples the correction is plausibly negligible, but the classification example, which is prominently used to demonstrate applicability to 'arbitrarily complex tasks', sits at C≈0.05 where the correction is not guaranteed to be small. The paper provides only visual correlations and no quantitative test of the omitted term. This warrants the existing CONDITIONAL verdict: the theory is sound at C=0, but the burden is on the authors to show that the finite-C measurements used as evidence remain in the regime where the exact identity is a good approximation. A direct computation of the exact Hessian in the classification case would settle the point. My read therefore does not change the reader's verdict.","tokens_in":13669,"tokens_out":7589,"duration_ms":66233,"concrete_test":"For the classification network of Fig. 5e, compute the exact cost Hessian including the omitted second-derivative term Σ_r c_r^T ∂²c_r/∂k_i∂k_j (by finite differences of the constraint gradients or automatic differentiation) and diagonalize it. Compare the top three stiff eigenvectors with those of the factorized approximation H_approx = L....S from Eq. (12); if the squared overlap of the stiff eigenspaces is below 0.9, or if the correlation of ||s_i||² with the exact stiff-mode entries drops markedly from the value implied by Fig. 5h, the finite-cost correction materially changes the conclusion. As a complementary check, repeat for the regression tasks at several intermediate costs (e.g., stop training early to reach C = 10^-9, 10^-6, 10^-3, 10^-1) and plot the eigenvector overlap versus C to map where the zero-cost approximation breaks down.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central identity, Eq. (12), is derived only under Assumption I of Section III: the trained network satisfies every task constraint exactly (cr=0 for all r). In that case the second-derivative term in the cost Hessian vanishes and Eq. (7) reduces to the product of first derivatives. For any actual trained network with finite cost, the exact Hessian contains an additional term Σ_r c_r^T ∂²c_r/∂k_i∂k_j, which is not captured by L....S. The magnitude of this omitted term scales roughly as ||c_r||, i.e. as sqrt(C) for the quadratic cost, so for C≈10^-9 to 10^-5 it is likely small relative to the retained term. However, the classification example uses a cosine-similarity cost with C≈0.05, corresponding to ||c_r||~0.2 in the quadratic sense; at this level the omitted correction can be comparable to the leading term, yet the paper still reports a strong susceptibility–stiff-mode correlation in Fig. 5h without estimating the correction. No bound or empirical check is provided to show that the C=0 factorization remains a good approximation at the finite costs actually achieved. Since the abstract claims without qualification that 'all the physical information relevant to the trained input-output relation can be captured by a susceptibility', the unquantified finite-cost gap is a load-bearing weakness in the support for that claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies trained adaptive resistor networks and derives an exact factorization of the cost Hessian at the learned solution. In Section III, under the assumption that all task constraints are satisfied exactly (C=0), the cost Hessian is written as a contraction of a task-dependent training tensor L and a physical susceptibility tensor S. Section IV simplifies the susceptibility tensor to products of per-edge susceptibility vectors si = Δ^T_i H^{-1}, where H is the physical (bordered Laplacian) Hessian, and shows that susceptibility norms are dominated by soft physical modes. The paper then presents numerical correlations between susceptibility and stiff cost modes for linear regression, classification, and allosteric tasks, and interprets highly susceptible edges as current-blocking 'walls' or 'corridors' that implement the learned function. The authors argue that susceptibility provides a task-agnostic, experimentally measurable probe of learned functionality.","tokens_in":13910,"tokens_out":5651,"duration_ms":51836,"significance":"The algebraic core of the paper is elegant and, as far as I can verify, correct: differentiating H^{-1} gives -2 si⊗si, and the contraction in Eq. (12) follows from the chain rule. If the finite-cost and correlation issues are resolved, this factorization usefully unifies earlier observations about low-dimensional physical responses, stiff-mode correspondence, and topological sectors under a single susceptibility-based framework. The proposed quantity is experimentally accessible and independent of the training protocol, which is a genuine strength. However, the demonstrated support for the central 'susceptibility captures functional importance' claim is currently qualitative, and the exactness of the factorization is tied to a zero-cost limit that the simulations only approximately reach. The paper therefore has solid potential but needs strengthening of the demonstrative layer before its claims are fully supported.","major_comments":[{"comment":"The exact factorization in Eq. (12) is derived under Assumption I (cr=0 for every task), which eliminates the second-derivative term from the cost Hessian. The simulations in Appendix D reach finite costs: C≈10^-5, 10^-8, and 1×10^-5 for the regression and allosteric tasks, but C≈0.05 for the cosine-similarity classification cost of Eq. (23). For the classification example, ||cr|| ~ sqrt(C) ~ 0.2, so the omitted term Σ_r c_r^T ∂²c_r/∂k_i∂k_j can be comparable to the retained first-derivative term. The paper does not bound this correction or show empirically that the C=0 factorization remains accurate at C=0.05. Please quantify the correction (for instance, by comparing the exact cost Hessian with the factorization on a trained network) or explicitly qualify the central claim as holding only in the C→0 limit.","section":"III (Eq. (7), Assumption I) and Appendix D"},{"comment":"The claim that edge susceptibility 'correlates positively' with stiff-mode entries is supported only by scatter plots. No correlation coefficient, confidence interval, or significance test is reported, and the ensemble of 50 initial conditions used in Fig. 7 is not used to assess the variability of the claimed correlations across tasks. Please report quantitative correlation measures (for example, Spearman rank correlation with bootstrap uncertainties) for each task, together with the number of edges used in each comparison. This is necessary to judge the strength of the demonstrated correspondence and to support the abstract's unqualified statement that susceptibility 'captures' the relevant physical information.","section":"V, Figs. 4d, 5d, 5h, 6c"},{"comment":"The abstract and Discussion state that 'all the physical information relevant to the trained input-output relation can be captured by a susceptibility.' This statement is exact only at C=0 and for responses that are linear perturbations around a local minimum (conditions II and III in Section IV). For nonlinear networks, the factorization holds only to quadratic order in the physical landscape, as acknowledged in Eq. (21) and the surrounding text. Please qualify the scope of the claim in the abstract and Discussion, particularly because one of the demonstrated examples (classification) operates at a finite cost where the exact identity does not hold.","section":"VIII and Abstract"}],"minor_comments":[{"comment":"The physical Hessian is denoted H in Eq. (2) and the cost Hessian is also denoted H in Eq. (7); please use distinct symbols (e.g., H^phys and H^cost) or consistent boldface to avoid confusion.","section":"II and III (Eqs. (2) and (7))"},{"comment":"The sentence 'we obtain Sij∼H^{-4}, and ultimately H∼H^{-4}' uses the same symbol H for both the susceptibility tensor and the cost Hessian; please disambiguate the notation.","section":"IV"},{"comment":"The caption refers to 'full susceptibility' without defining it; please define it as s_i^{(N+1)} and state that n is the number of modes retained in Eq. (24).","section":"Fig. 7 caption"},{"comment":"Figure references are inconsistent in capitalization ('fig.', 'Fig.', 'Figure'); please standardize to a single style.","section":"Throughout"},{"comment":"The text says that susceptibility is 'even better at capturing the important edges for weak training signals,' but Fig. 12 appears only to show overlap with the single stiff mode; please clarify what 'better' is compared with (e.g., persistent homology sectors) and define the metric used for the comparison.","section":"Appendix B"},{"comment":"Reference [15] is cited as an arXiv preprint; please update it if a journal version has appeared, and check that all references contain complete publication data.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper builds directly on the authors' prior work [15,17], and the tensor-factorization result is a meaningful extension of that framework. The main gap between the abstract's sweeping claim and the evidence is the finite-cost issue combined with the purely qualitative correlation demonstrations. Adding a quantitative bound or empirical check for the C>0 corrections, and reporting correlation statistics, would make the paper's central claim much more robust. I would also encourage the authors to include a data/code availability statement, as the simulations are otherwise hard to reproduce."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this paper has a clean, correct mathematical core and a somewhat under-supported empirical layer. The core is the decomposition of the cost Hessian for a trained linear resistor network into a training tensor L and a susceptibility tensor S, with S built from per-edge susceptibility vectors s_i = Δ_i^T H^{-1}. The derivation is elementary matrix calculus—differentiating H^{-1} gives -2 s_i ⊗ s_i—but it's nice to see it written out, and it genuinely unifies earlier results on mode correspondence, low-dimensional response, and topological sectors.\n\nThe paper does well at showing that high-susceptibility edges match the stiff modes of the cost Hessian in several tasks, and that these edges act as current blockers. The Fig. 8/9 physical interpretation (walls and corridors) is suggestive and a useful guide for experiments.\n\nWhere I'd push back: the factorization (Eq. 12) is derived assuming C=0 exactly. In the simulations, costs are finite; the classification example sits at C≈0.05, where sqrt(C)≈0.2 and the omitted second-derivative term is not obviously negligible. The paper reports a strong correlation in Fig. 5h but gives no estimate of the correction or a bound. That's a load-bearing gap for the unqualified abstract claim. Also, the correlations are shown as scatter plots without correlation coefficients or error bars, and no code or data are provided to reproduce the figures. These are exactly the kinds of things a referee should ask for.\n\nThe math itself holds up. The self-citation to [15] is fine—the Hessian formula at C=0 is re-derived here, and the susceptibility construction is checked against independently computed eigenvectors in simulation.\n\nWho's it for: anyone working on physical learning, especially experiment groups wanting a cheap probe of which edges matter. The susceptibility is measurable in principle, so the paper has real practical value.\n\nMy verdict: worth a serious referee. I'd send it out with a request for finite-cost analysis or at least an empirical check that the classification result survives a bound on the omitted term, plus correlation statistics and code/data.\n\nCandidly, if the finite-cost correction turns out to matter at C≈0.05, the classification panel weakens but the regression/allostery panels (C≈10^-5 or lower) likely survive. The core result is still worth publishing, possibly with a more careful claim.\n\nBring it to reading group.\n\nBest.","headline":"Clean math, under-supported empirics, worth peer review with a request for finite-cost bounds.","tokens_in":14456,"tokens_out":2612,"would_cite":true,"duration_ms":22227,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A trained resistor network leaves its whole learned task imprinted in one experimentally measurable susceptibility tensor.","keywords":["physical learning","resistor networks","cost Hessian","susceptibility tensor","coupled learning","allosteric networks","persistent homology","low-dimensional response"],"falsifier":"Simulate training on one fixed regression task but stop at a ladder of final costs from $C\\approx10^{-9}$ to $C\\approx1$; compute each network's susceptibility norms and the stiff modes of the true cost Hessian, and plot their correlation versus $C$. If the correlation collapses well before the costs reached in realistic training, then susceptibility carries the relevant physical information only at the idealized zero-cost point, not for trained networks as realized.","tokens_in":13438,"feed_emoji":"⚡","tokens_out":6798,"duration_ms":59902,"temperature":0.7,"pith_summary":"This paper establishes that, in a trained adaptive resistor network, every task-relevant physical imprint of learning is contained in a single quantity: the edge susceptibility, defined from the network's physical Hessian alone. The authors prove that the cost Hessian, the matrix governing how the learned cost responds to small conductance changes, factorizes exactly into a task-dependent training tensor and a susceptibility tensor built from per-edge susceptibility vectors. Because the susceptibility is measurable from voltage responses to unit currents, it can reveal which edges actually perform the trained function without knowing the inputs, outputs, or task. Simulations across linear regression, classification, and allosteric voltage-drop tasks show that high-susceptibility edges coincide with the stiff modes of the cost Hessian, cluster in a few soft physical modes, and act as current-blocking walls and corridors that implement the response.","feed_headline":"One tensor stores a resistor network's learned physics","feed_subtitle":"Edges with high susceptibility reveal the functional walls and corridors, no need to know the task.","key_machinery":"The load-bearing object is the per-edge susceptibility vector $s_i = \\bar{\\Delta}_i^T H^{-1}$, where $\\bar{\\Delta}_i$ is the extended incidence row of edge $i$ and $H$ is the extended physical Hessian (bordered Laplacian) of the grounded resistor network. Its entries are voltage drops across edge $i$ induced by unit currents at each node, and its squared norm is a scalar susceptibility per edge. The identity $S_{ij} = 4\\,s_i \\otimes s_i \\otimes s_j \\otimes s_j$ reduces the entire susceptibility tensor to these vectors, after the derivative $\\partial H^{-1}/\\partial k_i = -2\\,s_i \\otimes s_i$ is computed from the matrix-inverse rule. Because $\\|s_i\\|^2 = \\sum_\\alpha \\nu_\\alpha^{-2}(\\bar{\\Delta}_i^T W_\\alpha)^2$, the susceptibility is dominated by soft modes, which is what ties it to low-dimensional responses and to the stiff modes of the cost Hessian.","core_discovery":"The paper's central claim is an exact factorization of the cost Hessian for a successfully trained physical network: $\\mathcal{H}_{ij} = \\mathcal{L}\\,\\cdots\\,S_{ij}$ (full contraction), where $\\mathcal{L}$ is a fourth-rank training tensor built only from the input currents and output projectors, and the susceptibility tensor is $S_{ij} = 4\\,s_i \\otimes s_i \\otimes s_j \\otimes s_j$, with $s_i = \\bar{\\Delta}_i^T H^{-1}$ the susceptibility vector of edge $i$. Each component of $s_i$ is the voltage drop across edge $i$ when a unit current is injected at one node, so the whole tensor is a linear-response observable. The paper shows numerically that the norm $\\|s_i\\|^2$ is dominated by the softest modes of the physical Hessian and that, across linear regression (two and five outputs), three-class classification, and allosteric edge tasks, highly susceptible edges coincide with the entries of the stiff eigenmodes of the cost Hessian. It further shows that these edges have low relative conductance and act as current-blocking walls or corridors, and it argues that the same derivation applies to any physical network whose response minimizes a Lyapunov function, with all physical information captured up to quadratic order in the physical landscape.","pith_inferences":["A natural test the paper leaves open is the finite-cost regime: measuring how the susceptibility-to-stiff-mode correlation degrades as the final cost rises from $10^{-9}$ to $O(1)$ would map the practical validity of the exact factorization.","The wall-and-corridor picture suggests a design heuristic not stated in the paper: placing low-conductance barriers along high-susceptibility edges before training may bias learning toward solutions that are easier to interpret and possibly more robust.","By analogy to the allosteric discussion, a susceptibility-like response matrix could serve as an experimental probe of global epistasis in proteins, connecting slow physical modes to non-additive mutation effects without needing the task to be a single-mode response.","Because the susceptibility is a property of the physical Hessian alone, one could compute it in untrained networks as well; if high-susceptibility edges are also enriched in networks that train faster, the quantity would double as a trainability predictor."],"forward_implications":["Key functional edges can be identified from physics alone: measuring voltage responses to unit currents gives $s_i$, and high-susceptibility edges are the ones whose perturbation hurts the trained task most.","The same susceptibility measurement works without knowing the task specification, so trained networks become interpretable in an experimentally non-invasive way.","Because the factorization is independent of the training protocol, it applies to any route to a zero-cost solution, including local learning rules, global optimization, and evolutionary or biological processes.","Previously observed phenomena: low-dimensional physical response, stiff-mode/soft-mode correspondence, and topological sector structure in allosteric networks, follow as corollaries of the susceptibility tensor.","For nonlinear physical networks, all physical information up to quadratic order in the physical landscape is still contained in the susceptibility tensor."],"supporting_citations":[{"why":"supplies the bordered physical Hessian and the earlier observation that stiff cost modes correspond to soft physical modes, which this paper formalizes into the factorization.","marker":"[15]"},{"why":"introduces the Coupled Learning rule and the adaptable resistor-network setup used for all simulations here.","marker":"[11]"},{"why":"documents the low-dimensional physical response of trained networks that the susceptibility norm reproduces through soft-mode domination.","marker":"[17]"},{"why":"defines the allosteric voltage-drop task and the jammed-packing network ensemble used for those simulations.","marker":"[23]"},{"why":"provides the persistent-homology algorithm whose topological sectors the high-susceptibility edges are found to delimit.","marker":"[25]"},{"why":"demonstrates nonlinear analog learning and classification in physical networks, motivating the harder classification test studied here.","marker":"[13]"}],"fun_headline_variants":["Resistor net's learned physics fits in one tensor","Susceptibility walls mark task-relevant edges in trained nets","Adaptive resistor network: soft modes encode learned task","One susceptibility tensor captures all learned resistor physics","Trained resistor net: high-susceptibility edges act as walls"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire derivation assumes the trained network has cost exactly zero, meaning every task constraint is satisfied perfectly; the simulations only reach small nonzero costs ($C\\approx10^{-5}$ to $10^{-9}$, and $C\\approx0.05$ for classification), and no bound is given for how the susceptibility-importance correlation behaves as the cost grows.","fun_headline_variants_meta":{"raw":{"variants":["Resistor net's learned physics fits in one tensor","Susceptibility walls mark task-relevant edges in trained nets","Adaptive resistor network: soft modes encode learned task","One susceptibility tensor captures all learned resistor physics","Trained resistor net: high-susceptibility edges act as walls"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000166,"raw_usage":{"total_tokens":1279,"prompt_tokens":994,"completion_tokens":285,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":205}},"tokens_in":610,"tokens_out":285,"duration_ms":3335,"temperature":1.0,"reasoning_tokens":205,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T00:40:25.961100+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate training on one fixed regression task but stop at a ladder of final costs from $C\\approx10^{-9}$ to $C\\approx1$; compute each network's susceptibility norms and the stiff modes of the true cost Hessian, and plot their correlation versus $C$. If the correlation collapses well before the costs reached in realistic training, then susceptibility carries the relevant physical information only at the idealized zero-cost point, not for trained networks as realized.","supporting_citations":[{"cited_title":"Stern, D","cited_arxiv_id":null,"evidence_quote":"introduces the Coupled Learning rule and the adaptable resistor-network setup used for all simulations here."},{"cited_title":"Stern, A","cited_arxiv_id":null,"evidence_quote":"documents the low-dimensional physical response of trained networks that the susceptibility norm reproduces through soft-mode domination."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the allosteric voltage-drop task and the jammed-packing network ensemble used for those simulations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the persistent-homology algorithm whose topological sectors the high-susceptibility edges are found to delimit."}],"review_version":1}