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GeomHerd: A Forward-looking Herding Quantification via Ricci Flow Geometry on Agent Interactive Simulations
Lake Yang, Junwei Su, Jingfeng Zeng, Wenhao Lu, Xingzhi Qian, Weitong Zhang, Chuan Wu, Dunhong Jin
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
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Summary
GeomHerd is a forward-looking geometric framework that quantifies financial market herding by tracking discrete Ollivier-Ricci curvature on agent-interaction graphs generated from an LLM-driven multi-agent simulator. It anticipates herding events earlier than traditional price-correlation or trading-flow baselines by detecting structural topology changes and contagion bridges, and theoretically bridges to the classical CSAD statistic.
Entities (10)
Relation Signals (7)
GeomHerd → uses → Ollivier-Ricci curvature
confidence 95% · By tracking the discrete Ollivier–Ricci curvature of these action graphs, GeomHerd captures the structural topology of emerging coordination.
GeomHerd → generatesgraphsfrom → LLM-driven multi-agent simulator
confidence 94% · To generate these graphs, we treat a heterogeneous LLM-driven multi-agent simulator -- each financial trader instantiated by a persona-conditioned LLM call -- as a forecastable world
GeomHerd → evaluatedon → Cividino-Sornette continuous-spin model
confidence 93% · evaluate the geometric pipeline on the Cividino--Sornette continuous-spin agent-based substrate as our headline financial testbed.
Ollivier-Ricci curvature → bridgesto → CSAD
confidence 92% · Theoretically, we establish a mean-field bridge mapping our graph-theoretic metric to CSAD, the classical macroscopic herding statistic
CUSUM detector → detects → Herding events
confidence 91% · For each signal we run a one-sided cumulative-sum (CUSUM) detector... we fire an alarm at the earliest time at which their dynamics deviate from a pre-stress baseline
GeomHerd → transfersto → Vicsek self-driven-particle model
confidence 90% · The geometric signature transfers out-of-domain to the Vicsek self-driven-particle model
Ricci flow → predicts → Singularity time
confidence 88% · we use Ricci flow as a descriptor generator: at every dynamic snapshot Gt, we run a fresh flow and record its singularity time τsing
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Abstract
Abstract:Herding -- where agents align their behaviors and act collectively -- is a central driver of market fragility and systemic risk. Existing approaches to quantify herding rely on price-correlation statistics, which inherently lag because they only detect coordination after it has already moved realised returns. We propose GeomHerd, a forward-looking geometric framework that bypasses this observability lag by quantifying coordination directly on upstream agent-interaction graphs. To generate these graphs, we treat a heterogeneous LLM-driven multi-agent simulator -- each financial trader instantiated by a persona-conditioned LLM call -- as a forecastable world, and evaluate the geometric pipeline on the Cividino--Sornette continuous-spin agent-based substrate as our headline financial testbed. By tracking the discrete Ollivier--Ricci curvature of these action graphs, GeomHerd captures the structural topology of emerging coordination. Theoretically, we establish a mean-field bridge mapping our graph-theoretic metric to CSAD, the classical macroscopic herding statistic, linking GeomHerd to downstream price-dispersion measurement. Empirically, GeomHerd anticipates herding long before aggregate market baselines: on the continuous-spin substrate, our primary detector fires a median of 272 steps before order-parameter onset; a contagion detector ($\beta_{-}$) recalls 65% of critical trajectories 318 steps early; and on co-firing trajectories the agent-graph signal precedes price-correlation-graph baselines by 40 steps. As a complementary indicator, the effective vocabulary of agent actions contracts during cascades. The geometric signature transfers out-of-domain to the Vicsek self-driven-particle model, and a curvature-conditioned forecasting head reduces cascade-window log-return MAE over detector-conditioned and price-only baselines.
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- Source: https://arxiv.org/abs/2605.11645v1
- Canonical: https://arxiv.org/abs/2605.11645v1
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GeomHerd: A Forward-looking Herding Quantification via Ricci Flow Geometry on Agent Interactive Simulations Lake Yang l.yang1@imperial.ac.uk &Junwei Su University of Science and Technology of China junweisu.cs@gmail.com &Jingfeng Zeng MaxQuant jeffery@maxquant.ai &Wenhao Lu The University of Hong Kong whlu@connect.hku.hk &Xingzhi Qian University College London xingzhi.qian.23@ucl.ac.uk &Weitong Zhang MaxQuant weitong.zhang20@imperial.ac.uk &Chuan Wu The University of Hong Kong cwu@cs.hku.hk &Dunhong Jin The University of Hong Kong dhjin@hku.hk Abstract Herding—where agents align their behaviors and act collectively—is a central driver of market fragility and systemic risk. Existing approaches to quantify herding rely on price-correlation statistics, which inherently lag because they only detect coordination after it has already moved realised returns. We propose GeomHerd, a forward-looking geometric framework that bypasses this observability lag by quantifying coordination directly on upstream agent-interaction graphs. To generate these graphs, we treat a heterogeneous LLM-driven multi-agent simulator—each financial trader instantiated by a persona-conditioned LLM call—as a forecastable world, and evaluate the geometric pipeline on the Cividino–Sornette continuous-spin agent-based substrate as our headline financial testbed. By tracking the discrete Ollivier–Ricci curvature of these action graphs, GeomHerd captures the structural topology of emerging coordination. Theoretically, we establish a mean-field bridge mapping our graph-theoretic metric to CSAD, the classical macroscopic herding statistic, linking GeomHerd to downstream price-dispersion measurement. Empirically, GeomHerd anticipates herding long before aggregate market baselines: on the continuous-spin substrate, our primary detector fires a median of 272 steps before order-parameter onset; a contagion detector (β− _-) recalls 65% of critical trajectories 318 steps early; and on co-firing trajectories the agent-graph signal precedes price-correlation-graph baselines by 40 steps. As a complementary indicator, the effective vocabulary of agent actions contracts during cascades. The geometric signature transfers out-of-domain to the Vicsek self-driven-particle model, and a curvature-conditioned forecasting head reduces cascade-window log-return MAE over detector-conditioned and price-only baselines. 1 Introduction Herding—where market participants collectively align their actions rather than relying on independent information—is a central mechanism behind market fragility, contagion, and tail-risk events (Bikhchandani et al., 1992; Bikhchandani and Sharma, 2001). Classical finance literature explains herding through informational cascades, where financial agents rationally imitate predecessors (Bikhchandani et al., 1992; Banerjee, 1992; Avery and Zemsky, 1998), and reputational herding, where professional investors align with peers to avoid the career risk of being wrong alone (Scharfstein and Stein, 1990). From a systems perspective, herding can be viewed as a multi-agent phase transition in which behavioral diversity collapses into highly coordinated collective dynamics. Detecting such coordination early is therefore fundamental for risk monitoring and systemic-stability analysis. Despite extensive study, existing herding diagnostics remain largely downstream—that is, they read coordination off observables (realised returns or disclosed positions) that only become available after herding has already moved the market. The classical literature primarily relies on two families of such signals. The first consists of return-based measures, including cross-sectional dispersion statistics such as CSSD (Christie and Huang, 1995), CSAD by CCK (Chang et al., 2000), and state-space formulations (Hwang and Salmon, 2004). The second consists of trading-flow measures, most notably the LSV institutional buy/sell imbalance statistic (Lakonishok et al., 1992) and sequential trading-correlation measures (Sias, 2004). However, both families are fundamentally post-hoc: they detect herding only after coordinated actions have already propagated into realized returns or disclosed positions. Recently, geometric approaches based on discrete Ricci curvature (Sandhu et al., 2016; Samal et al., 2021; Wang et al., 2022; Sánchez García and Gherghe, 2024; Srinivasan, 2026; Akgüller et al., 2025) have been proposed for market-stress analysis, but these methods still operate on price-correlation graphs, which inherit the same observation-layer bottleneck. At the same time, a large body of financial-network literature shows that systemic fragility is fundamentally shaped by the topology of interactions among agents (Allen and Gale, 2000; Acemoglu et al., 2015; Elliott et al., 2014; Brunnermeier and Pedersen, 2009). This suggests a natural question: instead of forecasting herding only after it manifests in returns or disclosed flows, can one directly quantify the evolving geometry of the agent coordination process itself? Turning this upstream structural information into a practical forward-looking signal remains largely unexplored. Our approach. We propose GeomHerd, a forward-looking geometric framework built on the agent interaction graph. The intuition: herding is geometric collapse—as agents imitate one another, their neighborhoods on the interaction graph become progressively similar, overlapping, and tightly connected. We treat a heterogeneous LLM-driven multi-agent financial simulator (Yang et al., 2025; Hashimoto et al., 2025; Guo et al., 2025; Yu et al., 2024)—in which each financial trader is instantiated by a separate persona-conditioned LLM call, so one LLM agent simulates one financial agent (one node i∈Vi∈ V) and the action stream of every agent is fully observable at every step—as a forecastable world, and on top of it construct a dynamic agent graph whose nodes are agents and whose edges encode recent behavioral agreement. Each agent is modelled with distinct system-prompted personas (varying risk appetites, momentum horizons, and herding tendencies), and only population-level behavioral sweeps are controlled. Discrete Ricci geometry—an edge-level, signed notion of curvature on a graph, recently used in geometric deep learning for diagnosing over-squashing and guiding GNN graph rewiring—here serves as a herding detector on this upstream (agent) layer, yielding a complementary signal pair: positive Ollivier-Ricci curvature (κ¯OR+ κ_OR^+) captures within-clique coordination, and strongly negative curvature (β− _-) identifies bridge-like edges along which contagion propagates. We further evolve each snapshot under discrete Ricci flow and record the first neckpinch time τsing _sing as a forward-looking proximity-to-collapse descriptor, and add an information-theoretic signal VeffV_eff measuring the effective diversity of agents’ action language. Together (κ¯OR+,β−,τsing,Veff)( κ_OR^+,\, _-,\, _sing,\,V_eff) captures topological and behavioral collapse earlier than any individual statistic. Importantly, GeomHerd remains tied to classical instruments: a mean-field bridge to CSAD (Proposition 1) and empirical alignment with, but temporal lead over, LSV. Empirical study and findings. Our headline financial testbed is the Cividino-Sornette continuous-spin (CWS) agent-based model (Cividino et al., 2023), with the LLM-agent simulator built on Bedrock Claude Opus 4.6 instantiating the persona-conditioned setup of §2, and out-of-domain transfer evaluated on the Vicsek self-driven-particle model (Vicsek et al., 1995) (a canonical physics model of flocking). In brief: (i) GeomHerd leads aggregate market events and price-graph or flow baselines on CWS, with (κ¯OR+,β−)( κ_OR^+, _-) splitting precision vs. contagion recall; (i) multi-metric benchmarks plus augmented CCK and LSV-track consistency checks anchor the signal to the classical literature; (i) effective vocabulary contracts during cascades and curvature transfers out-of-domain to Vicsek, indicating behavioural homogenisation and a substrate-robust coordination signature; and (iv) the curvature triplet (κ¯OR,τsing,Veff)( κ_OR, _sing,V_eff) conditions a Kronos-style discrete forecasting head whose cascade-window log-return MAE improves over detector-conditioned and price-only baselines. Code and configurations to reproduce all experiments are available at the anonymous link. 2 Method We model interactions among N agents at each simulator step t as a weighted graph Gt=(V,Et,wt)G_t=(V,E_t,w_t) with |V|=N|V|=N and self-loops wt(i,i)=0w_t(i,i)=0. The pipeline has four stages: graph construction (§2.1), edge curvature (§2.2), detection (§2.3), and two complementary scalars from the same geometry-the forward-looking flow descriptor τsing _sing and the information-theoretic VeffV_eff (§2.4). A theoretical bridge to the classical CSAD return-dispersion statistic (§2.5) anchors the geometric signal to the finance literature; assumptions, the proof sketch, identification caveats, and the relation to the LSV trading-flow track are in Appendix B. 2.1 Agent graph construction The graph GtG_t has five design axes summarised in Table 6 (Appendix A.1); all five are ablated in §3.4. The choices are governed by a three-layer logic: alignment with the agent-based-model (ABM) substrate (which exposes discrete actions), with the phenomenological definition of herding as synchronised action-taking (Appendix A), and with the Ricci-geometry interpretation. The two consequential choices are nodes-as-individual-agents (collapsing into persona super-nodes destroys within-cluster dynamics; using assets as nodes recreates the price-correlation graph of Sandhu et al. Sandhu et al. (2016) and Srinivasan Srinivasan (2026)) and binary-windowed-agreement edges (a cosine-similarity variant removes the herding-side signal; Appendix F). Let ai(t)∈a_i(t) denote agent i’s discrete action at simulator step t, drawn from a finite alphabet A (e.g., =buy, hold, sellA=\buy, hold, sell\ on CWS, ||=3|A|=3). The edge weight between agents i and j is then the windowed agreement frequency wt(i,j)=1Tw∑s=t−Tw+1t[ai(s)=aj(s)]∈[0,1],w_t(i,j)\;=\; 1T_w _s=t-T_w+1^t1 [a_i(s)=a_j(s) ]\;∈\;[0,1], (1) with window Tw=100T_w=100. We sparsify by retaining edges with wt(i,j)>w0w_t(i,j)>w_0 at w0=0.5w_0=0.5 (well above the action-uniform baseline of ≈1/||≈ 1/|A|, so retained edges represent meaningfully elevated agreement). The graph is reconstructed every Δt=10 t=10 steps with 50% temporal overlap. The construction reads only the discrete action labels, so it is invariant to who generates them. In our setup, each market participant (each node i∈Vi∈ V) is driven by a single LLM call conditioned on a private persona and the current market state, so one LLM agent corresponds to exactly one financial agent throughout. LLM personas do not make the Ricci-curvature operator more powerful, but they produce a richer baseline action stream than ABMs with hardcoded trader archetypes (e.g., noise / fundamental / momentum traders), so the agent graph in the no-herding (subcritical) regime is more dispersed and the contrast against the herded regime is sharper. 2.2 Curvature and sign decomposition Each node carries the lazy-walk transition kernel μit(j)=αδij+(1−α)wt(i,j)∑kwt(i,k), _i^t(j)\;=\;α\, _ij\;+\;(1-α)\, w_t(i,j) _kw_t(i,k), (2) with laziness α=0.5α=0.5 matching Sandhu et al. Sandhu et al. (2016) for direct comparability. The Ollivier-Ricci curvature on each edge (i,j)∈Et(i,j)∈ E_t is κOR(i,j;t)= 1−W1(μit,μjt)dt(i,j), _OR(i,j;t)\;=\;1\;-\; W_1( _i^t,\, _j^t)d_t(i,j), (3) where W1W_1 is the 1-Wasserstein distance solved exactly by linear programming (POT Flamary et al. (2021)), and we set the edge length dt(i,j)=wt(i,j)d_t(i,j)=w_t(i,j), treating the agreement weight directly as a similarity-as-distance (higher agreement ⇒ shorter edge). We do not use 1−wt1-w_t or −logwt- w_t: those would map a herding clique (high wtw_t) to long effective distances and thus invert the sign of the herding signal. On weighted graphs, discrete Ricci curvature has a natural community/bridge interpretation (Sia et al., 2019; Ni et al., 2019): positively curved edges form within-community structure, while negatively curved edges form bridges between communities. This licenses a clean sign decomposition Et=Et+∪Et0∪Et−,Et+=e:κOR(e;t)>κ+,Et−=e:κOR(e;t)<κ−,E_t=E_t^+\,∪\,E_t^0\,∪\,E_t^-, E_t^+=\e: _OR(e;t)> _+\, E_t^-=\e: _OR(e;t)< _-\, with κ+=+0.1 _+=+0.1 and κ−=−0.1 _-=-0.1. The two scalars we monitor are the herding-side mean κ¯OR+(t) κ_OR^+(t) over Et+E_t^+, which rises under cascade onset (information-cascade mechanism (Bikhchandani et al., 1992)), and the contagion-side fraction β−(t)=|Et−|/|Et| _-(t)=|E_t^-|/|E_t|, which rises as bridges multiply (network-interlinkage mechanism (Elliott et al., 2014; Jiang et al., 2023)). The same curvature operator generates both quantities, and the signs map onto the two distinct mechanisms. Figure 1: Geometry leads herding on the agent interaction graph. (a) A snapshot of GtG_t during the cascade window. (b) Mean Ollivier-Ricci curvature κ¯OR(t) κ_OR(t) rises through the geometric threshold θgeom=0.30 _geom=0.30 before (c) the order parameter Va(t)V_a(t) crosses the herding-event threshold θevent=0.50 _event=0.50. Figure 2: Ricci-flow geometric evolution of GtG_t on a single supercritical CWS trajectory. Edges coloured by κOR(i,j;t) _OR(i,j;t) on a diverging scale (red = negative, between-clique bridge; blue = positive, within-clique cascade). Across snapshots the graph contracts into a dense crystallised clique while highly-negative bridge edges connect it to peripheral nodes, the topological signature β− _- targets. 2.3 Detection rules Goal. Given the two curvature time series κ¯OR+(t) κ_OR^+(t) and β−(t) _-(t), we fire an alarm at the earliest time at which their dynamics deviate from a pre-stress baseline, and measure the lead time Δ between this alarm and the order-parameter herding event τ⋆=mint:Va(t)>θeventτ = \t:V_a(t)> _event\ (θevent=0.50 _event=0.50) on the same trajectory. Detector. For each signal we run a one-sided cumulative-sum (CUSUM) detector (Page, 1954), augmented on the contagion side by a Kendall-τ slope test as a complementary trend channel: St+ S_t^+ =max(0,St−1++(κ¯OR+(t)−μbase+−k+)),At+=[St+>h+], = (0,\,S_t-1^++( κ_OR^+(t)- _base^+-k_+) ),\;A_t^+=1[S_t^+>h_+], (4) St− S_t^- =max(0,St−1−+(β−(t)−μbase−k−)),At−,cusum=[St−>h−], = (0,\,S_t-1^-+( _-(t)- _base^--k_-) ),\;A_t^-,cusum=1[S_t^->h_-], (5) At− A_t^- =At−,cusum∨ 1[Kendall[t−Wτ,t](β−)>τthresh]. =A_t^-,cusum\; \;1\! [Kendall_[t-W_τ,t]( _-)> _thresh ]. (6) The CUSUM tracks the level of bridge emergence while the Kendall-τ test tracks its trend; the OR combination is robust to either signal alone being noisy. Operating-point calibration of (k±,h±,τthresh,Wτ)(k_±,h_±, _thresh,W_τ) is in Appendix D. Why CUSUM and Kendall-τ. A z-score on a rolling baseline is a Shewhart-style detector and is blind to slow drifts: a 1σ1σ mean shift takes on average 44 windows to signal, whereas CUSUM signals in roughly 10. The contagion-bridge regime is precisely a slow trend toward the unstable manifold (the Scheffer-style critical-slowing-down regime (Scheffer et al., 2009)), so CUSUM is detection-delay-matched to the regime we target; the Kendall-τ slope test adds robustness to non-Gaussian noise on the trend, where the parametric CUSUM threshold can be brittle. 2.4 Forward-looking flow descriptor and effective vocabulary Singularity time τsing _sing. The same geometry that produces κ¯OR(t) κ_OR(t) also drives a discrete Ricci flow on GtG_t. Unlike Srinivasan (2026) which use neckpinch surgery as a static clustering operator on a price graph, we use Ricci flow as a descriptor generator: at every dynamic snapshot GtG_t, we run a fresh flow and record its singularity time τsing(t)=infs>0:∃e∈Et,κOR(s)(e)→−∞ _sing(t)= \\,s>0:∃\,e∈ E_t,\ _OR^(s)(e)→-∞\, the first-hitting time of a Ricci-flow neckpinch. Whereas κ¯OR(t) κ_OR(t) summarises the current graph, τsing(t) _sing(t) predicts the time-to-coordination from it—a forward-looking time series rather than a clustering output. Effective vocabulary VeffV_eff. Veff(t)=exp(H(pt))V_eff(t)= (H(p_t)), where H(pt)H(p_t) is the entropy of the codebook utilization distribution from a fixed three-dimensional finite-scalar-quantization (FSQ) codebook with Ld=4L_d=4 levels per dimension (total K=Ld3=64K=L_d^3=64) (Mentzer et al., 2024). Agents homogenize their behavioural repertoire as herding develops and VeffV_eff contracts. The codebook is intentionally non-learned—a learned codebook would adapt to the very distribution shift that VeffV_eff is designed to detect. VeffV_eff does not depend on the geometric pipeline; this is the source of its value as a complementary sanity check (§3.3.3). 2.5 Theoretical anchor: mean-field bridge to CSAD The standard finance instrument for measuring herding from the return cross-section is the CSAD regression of CCK Chang et al. (2000). We anchor our geometric signal to this instrument via a mean-field scaling argument. Table 1: Phenomenological mapping: herding state ↔ CSAD ↔ κ¯OR(t) κ_OR(t). Herding state CSAD (return dispersion) κ¯OR κ_OR (agent-graph curvature) Strong (agents follow the crowd) Low (returns concentrate) High (neighborhoods overlap) Weak (agents decide independently) High (returns disperse) Low or negative (neighborhoods separate) Proposition 1 (Mean-field bridge to CSAD; dominant-order scaling). Under standard assumptions (agent graph A1, lazy-walk curvature A2, mean-field concentration of action correlations A3, linear price impact A4, CCK CSAD estimand A5; full statement in Appendix B), CSADt=σξ2/π(1−κ¯OR(t))1/2+(N−1/2)CSAD_t\;=\; _ξ 2/π\, (1- κ_OR(t) )^1/2\;+\;O(N^-1/2) (7) holds in the N→∞N→∞ limit, so CSADtCSAD_t is monotonically decreasing in κ¯OR(t) κ_OR(t) at dominant order. We state this as a scaling identity because Step 2 of the derivation invokes a closed-form W1W_1 between two near-degenerate kernels whose remainder bound we do not establish here; a fully rigorous proof is left to follow-on work. Appendix B contains assumptions A1–A5, the four-step derivation, failure modes (Remark 1), and identification caveats for the empirical γ^3 γ_3 test (Remark 2). Operational anchor (LSV trading track). The agent interaction graph is unobservable to outside investors—only the trades it generates and the prices they move are visible—so the trading-flow pillar of the herding literature Lakonishok et al. (1992); Sias (2004) addresses the disclosure-constrained regime via 13F-style buy/sell imbalance (here, “13F” refers to the SEC’s quarterly Form 13F filings of institutional holdings, on which institutional-herding measures are typically computed Wermers (1999)). Within our simulator substrates the action stream is observable by construction at every step, which lifts that identification restriction and makes the substrate-pivot result testable: with simultaneous access to the action stream (input to GeomHerd) and the cleared trade flow (input to LSV), we can empirically measure the temporal gap between the two detectors on the same trajectories (§3.3.2). A 13F-style fund-as-agent deployment template—in which each institutional manager is treated as a single node in the agent graph and its quarterly 13F holdings (rather than per-step actions) drive the edges—is in Appendix I. 3 Experiments 3.1 Research questions We organise the empirical study around four research questions that together validate the core claims of GeomHerd: (RQ1) does the agent-graph curvature signal anticipate herding earlier than detectors built on price-correlation graphs or trading-flow aggregates? (RQ2) does the sign decomposition into (κ¯OR+,β−)( κ_OR^+, _-) deliver complementary herding/contagion alarms covering within-clique tightening and between-clique-bridge regimes? (RQ3) is the agent-graph signal directionally consistent with the classical CSAD/CCK return-track and the LSV trading-flow herding statistics, as predicted by Proposition 1? (RQ4) does the signal carry forecasting content beyond detection alone, and does it generalise beyond the headline CWS substrate to a non-financial system? 3.2 Setup Substrate. The Cividino-Sornette continuous-spin (CWS) substrate (Cividino et al., 2023) is our headline testbed: a physics-inspired financial agent-based model (ABM) in the Ising / O(n)-vector family, where noise-trader herding is modelled as a critical phenomenon in continuous-spin coupling and rational fundamentalists rebalance in response. Mechanically, CWS is a discrete-time simulator in which N heterogeneous agents repeatedly choose actions over nan_a assets, with each agent’s next action driven schematically by a private signal, the average action of its neighbours weighted by a coupling strength κ, and idiosyncratic noise; cleared trades feed back into asset prices through a linear-impact rule (Appendix B, A4). We instantiate N=66N=66 agents and na=4n_a=4 assets, and sweep the coupling parameter κ∈0.5,0.8,1.2,1.8,2.5κ∈\0.5,0.8,1.2,1.8,2.5\—which interpolates from independent decision (κ<1κ<1) to herd coordination (κ>1κ>1)—at 80 seeds per level, yielding 400 trajectories (240 supercritical, 160 subcritical). Simulated returns reproduce the regularities of Cont Cont (2001) (tail index α=5.75α=5.75, volatility-ACF slope β=0.27β=0.27, martingale raw returns); a full panel is in supplementary materials. For cross-substrate transfer we additionally evaluate on the Vicsek self-driven-particle model (Vicsek et al., 1995)—a canonical physics model of flocking in which N particles move at constant speed and align their headings with local neighbours under angular noise η, undergoing an order–disorder phase transition at a critical noise level ηc _c. As a non-financial system, Vicsek tests whether the curvature signature reflects universal collective coordination rather than a finance-specific artefact (see §3.3.3 and Appendix G). Baselines. We benchmark against a seven-detector slate: trade-flow LSV (Lakonishok et al., 1992); return-cross-section CSAD of CCK Chang et al. (2000); three price-correlation geometric methods (Sandhu et al., 2016; Huang et al., 2023; Srinivasan, 2026); the point-process geometric detector of Jiang et al. Jiang et al. (2023); and an action-agreement mutual-information (A-MI) baseline adapted from the synchronous-action-coupling probe of Tessera et al. Tessera et al. (2026). Metrics. Following Guritanu et al. (2025); Bury et al. (2021), we report (i) precision, recall, F1 and False Alarm Rate (FAR) per day on supercritical/subcritical detection; (i) AUROC and AUPRC; (i) conditional median lead time; and (iv) the rare-event-stable metric of Nikolopoulos Nikolopoulos (2025) appropriate for the low-event-prevalence regime. Paired-bootstrap differences are computed on co-firing trajectories with nboot=5000n_boot=5000. Two operating points. We report two operating points for the upward CUSUM on κ¯OR+ κ_OR^+, drawn from the calibration sweep of Appendix D. The recall-oriented point (kσ,hσ)=(0.50,4.0)(k_σ,h_σ)=(0.50,4.0) delivers a long lead at higher subcritical FAR; this is the abstract figure. The precision-oriented point (2.0,4.0)(2.0,4.0) delivers a shorter but tightly FAR-controlled lead used for all paired contrasts in Table 2, since head-to-head lead comparisons are only meaningful at FAR-controlled thresholds. 3.3 Results We report three sets of results, each structured as (i) brief experiment description, (i) result summary, (i) connection to research questions. 3.3.1 Result 1: GeomHerd anticipates herding earlier than price-based and trading-flow baselines (RQ1, RQ2) Experiment. On the CWS replay set, we run the upward CUSUM on κ¯OR+ κ_OR^+ at the two operating points and the contagion-bridge alarm on β− _- (CUSUM-plus-Kendall-τ rule). For each detector and trajectory we record whether and when an alarm fires before the order-parameter herding event τ⋆τ , and then compare paired lead times against each baseline on co-firing trajectories. Result. Conditional on κ¯OR+ κ_OR^+ firing, the median lead at the precision-oriented operating point is 178 steps with 95% CI [71, 407] (Table 2). Against the closest geometric correlation-graph detectors (Srinivasan, 2026; Sandhu et al., 2016; Huang et al., 2023), the paired-bootstrap median advantage is positive at 191.7, 74.4, and 153.8 steps respectively, with the Lap-CSAD row significant at α=0.05α=0.05 (p=0.03p=0.03). Across paired comparisons we have npaired∈6,8,9n_paired∈\6,8,9\, reflecting the high-precision-low-recall regime; at the recall-oriented operating point, the same comparison is better-powered (Appendix C.1). On the same trajectories where both an agent-graph and a price-correlation-graph detector fire before τ⋆τ , the agent-graph signal precedes price-based signals by a pooled median of 40 simulator steps (95% CI [18, 68]). The contagion-bridge detector β− _- recalls 65% of supercritical trajectories with median lead 318 steps (Table 3), accepting a high subcritical FAR (0.81) in exchange for early bridge alarm. Connection to research questions. RQ1–RQ2: κ¯OR+ κ_OR^+ leads price-graph baselines under FAR control; β− _- adds complementary contagion recall at higher subcritical FAR (Table 3). Near-chance AUROC reflects score sparsity, not detector failure—conditional lead is the appropriate metric Nikolopoulos (2025). Table 2: Paired-bootstrap lead-time difference (GeomHerd κ¯OR+ κ_OR^+ at the precision-oriented point minus comparator) on the binary-edge CWS replay set, restricted to co-firing trajectories. Comparator Lead diff. (steps) 95% CI npairedn_paired p-value Srinivasan 2026 191.7 [-35.0, 393.3] 6 0.1135 Sandhu 2016 74.4 [-36.7, 195.6] 9 0.203 Huang et al. 2023 (Lap-CSAD) 153.8 [28.7, 297.5] 8 0.03 CSAD (CCK) -42.8 [-211.6, 106.9] 9 0.624 Figure 3: Money trace on a single supercritical CWS seed at κ/κc=1.8κ/ _c=1.8. (a–d) Per-agent action stream, asset price P(t)P(t), headline geometric scalar κ¯OR+(t) κ_OR^+(t) with the CUSUM alarm (blue), and order parameter Va(t)V_a(t) with the herding event τ⋆τ (red). Vertical guides mark τsing _sing and τ⋆τ in every panel so the lead Δ=τ⋆−τsing =τ - _sing is readable directly. The displayed lead matches the recall-oriented operating-point headline of 272 steps. Table 3: Multi-axis detection profile under the binary-edge calibration. The agent-graph row group (top) is the natural head-to-head set; trading-flow / contagion-direction baselines (LSV, Jiang) achieve longer raw lead times by firing on every trajectory regardless of regime (FARsub=1.00FAR_sub=1.00) and are not substitutes for a regime classifier. Detector Precision Recallsuper FARsub AUROC Median lead 95% CI Agent-graph substrate (head-to-head) GeomHerd κ¯OR+ κ_OR^+ (ours) 0.45 0.04 0.07 0.48 178 [71, 407] GeomHerd τsing _sing (ours) 0.42 0.03 0.07 0.48 -93 [-216, 233] A-MI Tessera et al. (2026)† n/a 0.00 0.00 0.50 n/a [n/a, n/a] Price-correlation substrate Srinivasan 2026 0.71 0.79 0.49 0.66 20 [-52, 65] Sandhu 2016 0.72 0.95 0.55 0.72 80 [43, 106] Lap-CSAD Huang et al. (2023) 0.85 0.85 0.23 0.80 -42 [-74, -8] Trading-flow / contagion direction (different phenomenon) CSAD 0.69 1.00 0.68 0.75 180 [150, 214] LSV 1992 0.60 1.00 1.00 0.48 355 [333, 388] Jiang 2023 0.60 1.00 1.00 0.50 306 [262, 329] Sign-decomposed contagion-bridge detector (post-hoc on v34d trajectories) GeomHerd β− _- (τneg=−0.4 _neg=-0.4, CUSUM+slope, up) 0.55 0.65 0.81 0.80 318 [272, 344] †Our A-MI baseline (adapted from the synchronous-action-coupling probe of Tessera et al. Tessera et al. (2026)) returns a degenerate-output flag on 400/400 trajectories under the binary-edge configuration: the saturated agent graph drives baseline A-MI variance below numerical resolution. ‡The β− _- detector uses CUSUM+slope on negative-curvature edges (τneg=−0.4 _neg=-0.4, upward direction), calibrated on the supercritical replay set; subcritical FAR is on a separately generated set (nsub=160n_sub=160). 3.3.2 Result 2: The geometric signal is consistent with classical CSAD and LSV (RQ3) Experiment. We anchor the agent-graph signal to the two pillars of the classical herding literature. Return track: we estimate the augmented CCK regression CSADt=α+γ1|Rm,t|+γ2Rm,t2+γ3κ¯OR(t)+εtCSAD_t\;=\;α+ _1\,|R_m,t|+ _2\,R_m,t^2+ _3\, κ_OR(t)+ _t (8) per seed on the CWS replay set with heteroskedasticity-and-autocorrelation-consistent (HAC; Newey–West) standard errors, summarising γ^3 γ_3 across 240 supercritical seeds under deterministic replay. Trading track: we compute the windowed LSV statistic on simulated buy/sell flows and measure the temporal gap to κ¯OR+ κ_OR^+ on co-firing trajectories. Result. The cross-seed median of γ^3 γ_3 is -0.0072 with bootstrap CI [-0.00769, -0.00602], consistent with Eq. (7); the median CCK quadratic coefficient shifts from -1.15 to -1.24 (absolute median change 8 % in |β2|| _2|) once κ¯OR κ_OR is included, indicating that part of the return-dispersion nonlinearity previously absorbed by the quadratic term is explained by agent-graph curvature. LSV achieves recall 1.001.00 on supercritical and subcritical trajectories (Table 3, last block), so it is not a regime classifier; on co-firing trajectories κ¯OR+ κ_OR^+ precedes LSV in time, since the agent-graph clique is detectable in the action stream before buy/sell imbalance accumulates to the LSV threshold. Connection to research questions. RQ3: sign-consistency with CSAD (Eq. (7)) and directional lead vs. LSV on co-fires; identification limits are in Remark 2. 3.3.3 Result 3: The signal carries forecasting content and generalises out-of-domain (RQ4) Figure 4: Cascade-window forecasting MAE (CWS, log-return scale). GeomHerd triplet (κ¯OR,τsing,Veff)( κ_OR, _sing,V_eff) vs. herding-detector baselines and a price-only AR baseline; rliable (Agarwal et al., 2021) IQM bars. Experiment. (i) Vicsek transfer. We sweep angular noise η∈0.5,1.0,1.6,2.0,2.5η∈\0.5,1.0,1.6,2.0,2.5\ at 20 seeds per level (N=600N=600 particles, ηc≈1.6 _c≈ 1.6); the agent graph is built from k-N (k=10k=10) on the heading sequence with binary edge weights, and each trajectory is scored by κ¯OR(τ⋆) κ_OR(τ ) at the polarisation event. (i) Forecasting head. On the CWS substrate, we train a curvature-conditioned next-step forecasting head: a Kronos-style discrete price tokeniser (a learned vector-quantiser that maps OHLCV sequences into a fixed token vocabulary) feeds a transformer that consumes the GeomHerd triplet (κ¯OR,τsing,Veff)( κ_OR, _sing,V_eff) via AdaLN-Zero conditioning (adaptive layer-norm with zero-initialised gating). The price tokeniser is frozen; only the conditioning layers are trained. We compare cascade-window forecasting mean absolute error (MAE) on log-return scale against herding-detector baselines and a price-only autoregressive (AR) baseline. (i) Behavioural sanity check. As a complement, we track the effective vocabulary Veff(t)=exp(H(pt))V_eff(t)= (H(p_t)) on the same CWS trajectories. (a) Behavioural homogenisation on CWS. (b) Out-of-domain transfer on Vicsek. Figure 5: Behavioural homogenisation and out-of-domain generalisation. (a) On the CWS financial substrate, the cross-correlation between the effective vocabulary VeffV_eff and the curvature signal κ¯OR κ_OR peaks at lag ≈15≈ 15 steps with VeffV_eff leading, indicating that behavioural concentration sets in slightly ahead of the geometric collapse during the cascade window. (b) On the Vicsek collective-motion substrate, κ¯OR(τ⋆) κ_OR(τ ) separates ordered from disordered regimes with AUROC 0.99, demonstrating that the geometric signature generalises beyond finance. Result. On the forecasting task (Fig. 4), the GeomHerd-conditioned head attains the lowest interquartile-mean (IQM) MAE among all methods compared, beating both detector-conditioned baselines and the price-only AR baseline—so the geometric signal is not only a detection statistic but a useful conditioning feature for downstream forecasting. On Vicsek, κ¯OR(τ⋆) κ_OR(τ ) separates ordered from disordered regimes with AUROC 0.99 (95% CI [0.98,1.00][0.98,1.00], Fig. 5(b)), and per-η medians are monotone in η (+0.08→−0.26+0.08→-0.26). On the same CWS trajectories, the behavioural-homogenisation signal VeffV_eff co-moves with the geometric signal across the cascade window and in fact leads it by ≈15≈ 15 steps in cross-correlation (Fig. 5(a)), supporting the claim that herding is also a behavioural-homogenisation process. Together, forecasting gain, Vicsek transfer, and VeffV_eff co-movement (RQ4) demonstrate forecasting content beyond the detection rule alone and out-of-domain generalisation. 3.4 Ablation Table 4 reports a decomposed ablation in which each row replaces one headline choice. The headline binary-edge construction accounts for the bulk of the conditional lead: the cosine-similarity variant removes the herding-side signal entirely (0/90 vs. 11/90 supercritical fires; Appendix F), since mean cosine similarity in the supercritical pool concentrates in a narrow band that the rolling-baseline CUSUM cannot trip. Detector-swap rows stress-test alternative mappings from the same replay to an alarm and should not be read as redefining the contribution; window-length and sign-pooling variants are in Appendix H. Table 4: Decomposed ablation. Each row replaces one headline choice relative to the binary-edge LP-W1W_1 baseline on κ¯OR+ κ_OR^+ with upward CUSUM at 80 seeds per tranche. Configuration Precision Recall AUROC Med. lead 95% CI Headline (κ¯OR+ κ_OR^+, wtw_t=binary, LP W1W_1, CUSUM, 80 seeds) 0.45 0.04 0.48 178 [71, 407] Detector ablations → z-score (current production default) 0.36 0.13 0.40 232 [-47, 394] → CUSUM (Page 1954) 0.00 0.00 0.49 n/a [n/a, n/a] → EWMA (exponentially weighted) 0.34 0.11 0.40 283 [77, 424] → Kendall-τ slope-only 0.00 0.00 0.48 n/a [n/a, n/a] Geometry ablations → wtw_t cosine (vs binary) 0.00 0.00 0.49 n/a [n/a, n/a] → TwT_w=50 (vs 100) n/a 0.00 0.50 n/a [n/a, n/a] → TwT_w=200 (vs 100) 0.48 0.22 0.42 204 [120, 274] → Sinkhorn W1W_1 (vs LP) 0.00 0.00 0.49 n/a [n/a, n/a] → no sign decomposition (abs upward+downward) 0.00 0.00 0.49 n/a [n/a, n/a] Triplet ablations → remove τsing _sing from triplet 0.00 0.00 0.49 n/a [n/a, n/a] 4 Related Work We present a more comprehensive review in Appendix A. Financial measurement and the observability gap. Classical theories of herding and contagion are traditionally measured via post-hoc aggregate statistics, utilizing either the trading track (e.g., the LSV overlap statistic (Lakonishok et al., 1992)) or the return track (e.g., the CCK cross-sectional absolute deviation (Chang et al., 2000)). Because real-world, agent-level action sequences are largely unobservable, evaluating forward-looking micro-structural signals on real data is fundamentally restricted. Recent LLM-driven multi-agent financial simulators (Yang et al., 2025; Hashimoto et al., 2025) elegantly bypass this observability gap. By generating behaviorally rich, transparent agent ecosystems, these simulators provide the ideal substrate for evaluating upstream topological signals before they manifest in downstream price aggregates. Discrete curvature on financial graphs. Recent literature has applied discrete geometry—particularly Ollivier and Forman-Ricci curvature—to financial networks to detect systemic fragility and crashes (Sandhu et al., 2016; Samal et al., 2021; Jiang et al., 2023). However, prior work overwhelmingly operates on price-correlation graphs and aggregates curvature into a single, global scalar. Our approach fundamentally diverges: we apply Ollivier-Ricci geometry directly to the agent-action graph. Furthermore, building on the community-bridge dichotomy of Sia et al. Sia et al. (2019) and Ni et al. Ni et al. (2019), we explicitly decompose curvature by sign. This novel framing mathematically disentangles herding (positive curvature driving intra-clique density) from contagion (negative curvature defining inter-community bridges). Early-warning signals (EWS). Traditional EWS for critical transitions rely on lagging statistical moments such as rising variance or autocorrelation (Scheffer et al., 2009; Bury et al., 2021). While recent topological approaches have advanced the state-of-the-art (Guritanu et al., 2025), many remain post-hoc descriptors rather than predictive alarms. GeomHerd bridges this gap by coupling our sign-decomposed graphs with continuous-time Ricci flow singularities and one-sided CUSUM detectors (Page, 1954). Chosen for optimal detection-delay properties, these components ensure GeomHerd acts as a forward-looking warning system. 5 Conclusion GeomHerd advances forward-looking herding quantification via three architectural shifts: (1) Substrate pivot: measuring curvature causally upstream on the agent interaction graph rather than on downstream price-correlation graphs. (2) Forward-looking flow: tracking Ricci-flow neckpinch time (τsing _sing) as a dynamic proximity-to-collapse scalar rather than a static clustering operator. (3) Mean-field bridge: linking the geometric metric to the classical CSAD statistic (Proposition 1). The LLM-driven multi-agent simulator supplies a behaviourally rich, fully observable substrate on which the geometric pipeline anticipates coordination. Empirically, on the continuous-spin substrate GeomHerd fires up to 272 steps before order-parameter onset, is sign-consistent with CSAD, transfers out-of-domain to the Vicsek physical model, and conditions a forecasting head that reduces cascade-window log-return MAE over detector-conditioned and price-only baselines. References [1] D. Acemoglu, A. Ozdaglar, and A. Tahbaz-Salehi (2015) Systemic risk and stability in financial networks. American Economic Review 105 (2), p. 564–608. External Links: Document Cited by: Appendix A, §1. [2] R. Agarwal, M. Schwarzer, P. S. Castro, A. C. Courville, and M. G. Bellemare (2021) Deep reinforcement learning at the edge of the statistical precipice. In Advances in Neural Information Processing Systems, Vol. 34, p. 29304–29320. Cited by: Figure 4, Figure 4. [3] Ö. Akgüller, M. A. Balcı, L. M. Batrancea, and L. Gaban (2025) Network geometry of Borsa Istanbul: analyzing sectoral dynamics with Forman–Ricci curvature. Entropy 27 (3), p. 271. External Links: Document Cited by: Appendix A, Table 5, §1. [4] F. Allen and D. Gale (2000) Financial contagion. Journal of Political Economy 108 (1), p. 1–33. External Links: Document Cited by: Appendix A, §1. [5] C. Avery and P. Zemsky (1998) Multidimensional uncertainty and herd behavior in financial markets. American Economic Review 88 (4), p. 724–748. Cited by: §1. [6] A. V. Banerjee (1992) A simple model of herd behavior. The Quarterly Journal of Economics 107 (3), p. 797–817. Cited by: §1. [7] S. Bikhchandani, D. Hirshleifer, and I. Welch (1992) A theory of fads, fashion, custom, and cultural change as informational cascades. Journal of Political Economy 100 (5), p. 992–1026. Cited by: §1, §2.2. [8] S. Bikhchandani and S. Sharma (2001) Herd behavior in financial markets. IMF Staff Papers 47 (3), p. 279–310. Cited by: §1. [9] M. K. Brunnermeier and L. H. Pedersen (2009) Market liquidity and funding liquidity. Review of Financial Studies 22 (6), p. 2201–2238. External Links: Document Cited by: Appendix A, §1. [10] T. M. Bury, D. Dylewsky, C. T. Bauch, M. Anand, L. Glass, A. Shrier, and G. Bub (2023) Predicting discrete-time bifurcations with deep learning. Nature Communications 14 (1), p. 6331. Cited by: Appendix A. [11] T. M. Bury, R. I. Sujith, I. Pavithran, M. Scheffer, T. M. Lenton, M. Anand, and C. T. Bauch (2021) Deep learning for early warning signals of tipping points. Proceedings of the National Academy of Sciences 118 (39), p. e2106140118. External Links: Document Cited by: Appendix A, Appendix J, §3.2, §4. [12] E. C. Chang, J. W. Cheng, and A. Khorana (2000) An examination of herd behavior in equity markets: an international perspective. Journal of Banking & Finance 24 (10), p. 1651–1679. Cited by: Appendix A, item A5, Appendix B, §1, §2.5, §3.2, §4. [13] W. G. Christie and R. D. Huang (1995) Following the pied piper: do individual returns herd around the market?. Financial Analysts Journal 51 (4), p. 31–37. External Links: Document Cited by: Appendix A, Appendix B, §1. [14] D. Cividino, R. Westphal, and D. Sornette (2023) Multiasset financial bubbles in an agent-based model with noise traders’ herding described by an n-vector Ising model. Physical Review Research 5 (1), p. 013009. External Links: Document Cited by: §1, §3.2. [15] R. Cont (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1 (2), p. 223–236. External Links: Document Cited by: §3.2. [16] M. Elliott, B. Golub, and M. O. Jackson (2014) Financial networks and contagion. American Economic Review 104 (10), p. 3115–3153. External Links: Document Cited by: Appendix A, §1, §2.2. [17] R. Flamary, N. Courty, A. Gramfort, M. Z. Alaya, A. Boisbunon, S. Chambon, L. Chapel, A. Corenflos, K. Fatras, N. Fournier, et al. (2021) POT: Python optimal transport. Journal of Machine Learning Research 22 (78), p. 1–8. Cited by: Appendix C, §2.2. [18] S. Frey, P. Herbst, and A. Walter (2014) Measuring mutual fund herding — a structural approach. Journal of International Financial Markets, Institutions and Money 32, p. 219–239. Note: Original working paper 2007 External Links: Document Cited by: Appendix A. [19] T. Guo, H. Shen, J. Huang, Z. Mao, J. Luo, B. Chen, Z. Chen, L. Liu, B. Xia, X. Liu, Y. Ma, and M. Zhang (2025) MASS: muli-agent simulation scaling for portfolio construction. arXiv preprint arXiv:2505.10278. Cited by: Appendix A, §1. [20] E. Guritanu, E. Barbierato, and A. Gatti (2025) Topological machine learning for financial crisis detection: early warning signals from persistent homology. Computers 14 (10), p. 408. Cited by: Appendix A, §3.2, §4. [21] R. Hashimoto, T. Takayanagi, M. Suzuki, and K. Izumi (2025) Agent-based simulation of a financial market with large language models. In International Conference on Principles and Practice of Multi-Agent Systems, p. 20–28. Cited by: Appendix A, §1, §4. [22] C. Huang, Y. Cai, X. Yang, Y. Deng, and X. Yang (2023) Laplacian-energy-like measure: does it improve the Cross-Sectional Absolute Deviation herding model?. Economic Modelling 127, p. 106505. External Links: Document Cited by: §3.2, §3.3.1, Table 3. [23] S. Hwang and M. Salmon (2004) Market stress and herding. Journal of Empirical Finance 11 (4), p. 585–616. External Links: Document Cited by: Appendix A, §1. [24] H. Jiang, M. Zhao, Z. Zhang, and T. Luo (2023) Evaluating financial contagion through Ricci curvature on multivariate reactive point processes. Finance Research Letters 58, Part A, p. 104248. External Links: Document Cited by: Appendix A, Table 5, §2.2, §3.2, §4. [25] J. Lakonishok, A. Shleifer, and R. W. Vishny (1992) The impact of institutional trading on stock prices. Journal of Financial Economics 32 (1), p. 23–43. External Links: Document Cited by: Appendix A, §1, §2.5, §3.2, §4. [26] F. Mentzer, D. Minnen, E. Agustsson, and M. Tschannen (2024) Finite scalar quantization: VQ-VAE made simple. In International Conference on Learning Representations (ICLR), Cited by: §2.4. [27] C. Ni, Y. Lin, F. Luo, and J. Gao (2019) Community detection on networks with Ricci flow. Scientific Reports 9, p. 9984. External Links: Document Cited by: Appendix A, §2.2, §4. [28] S. D. Nikolopoulos (2025) An imbalance-robust evaluation framework for extreme risk forecasts. arXiv preprint arXiv:2512.00916. Cited by: §3.2, §3.3.1. [29] Y. Ollivier (2009) Ricci curvature of Markov chains on metric spaces. Journal of Functional Analysis 256 (3), p. 810–864. External Links: Document Cited by: Appendix A. [30] E. S. Page (1954) Continuous inspection schemes. Biometrika 41 (1/2), p. 100–115. External Links: Document Cited by: Appendix A, §2.3, §4. [31] A. Samal, H. K. Pharasi, S. J. Ramaia, H. Kannan, E. Saucan, J. Jost, and A. Chakraborti (2021) Network geometry and market instability. Royal Society Open Science 8 (2), p. 201734. External Links: Document Cited by: Appendix A, Table 5, §1, §4. [32] J. Sánchez García and S. Gherghe (2024) On the Ollivier-Ricci curvature as fragility indicator of the stock markets. arXiv preprint arXiv:2405.07134. Cited by: Appendix A, Table 5, §1. [33] R. S. Sandhu, T. T. Georgiou, and A. R. Tannenbaum (2016) Ricci curvature: An economic indicator for market fragility and systemic risk. Science Advances 2 (5), p. e1501495. External Links: Document Cited by: Appendix A, Table 5, §1, §2.1, §2.2, §3.2, §3.3.1, §4. [34] D. S. Scharfstein and J. C. Stein (1990) Herd behavior and investment. American Economic Review 80 (3), p. 465–479. Cited by: §1. [35] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M. Rietkerk, and G. Sugihara (2009) Early-warning signals for critical transitions. Nature 461, p. 53–59. External Links: Document Cited by: Appendix A, §2.3, §4. [36] J. Sia, E. Jonckheere, and P. Bogdan (2019) Ollivier-Ricci curvature-based method to community detection in complex networks. Scientific Reports 9, p. 9800. External Links: Document Cited by: Appendix A, §2.2, §4. [37] R. W. Sias (2004) Institutional herding. Review of Financial Studies 17 (1), p. 165–206. External Links: Document Cited by: Appendix A, §1, §2.5. [38] B. Srinivasan (2026) Intrinsic geometry of the stock market from graph ricci flow. arXiv preprint arXiv:2510.15942. Cited by: Appendix A, Table 5, §1, §2.1, §2.4, §3.2, §3.3.1. [39] K. Tessera, L. Hinckeldey, R. Zamboni, D. Abel, and A. Storkey (2026) Probing Dec-POMDP reasoning in cooperative MARL. In Proceedings of the 25th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), Note: arXiv:2602.20804 Cited by: Table 5, Table 5, Table 5, §3.2, Table 3, Table 3. [40] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet (1995) Novel type of phase transition in a system of self-driven particles. Physical Review Letters 75 (6), p. 1226–1229. External Links: Document Cited by: §1, §3.2. [41] X. Wang, L. Zhao, N. Zhang, L. Feng, and H. Lin (2022) Stability of China’s stock market: measure and forecast by Ricci curvature on network. arXiv preprint arXiv:2204.06692. Cited by: Appendix A, Table 5, §1. [42] R. Wermers (1999) Mutual fund herding and the impact on stock prices. Journal of Finance 54 (2), p. 581–622. External Links: Document Cited by: Appendix A, §2.5. [43] Y. Yang, Y. Zhang, M. Wu, K. Zhang, Y. Zhang, H. Yu, Y. Hu, and B. Wang (2025) TwinMarket: a scalable behavioral and social simulation for financial markets. In Advances in Neural Information Processing Systems (NeurIPS), Note: arXiv:2502.01506 Cited by: Appendix A, §1, §4. [44] Y. Yu, Z. Yao, H. Li, Z. Deng, Y. Jiang, Y. Cao, Z. Chen, J. W. Suchow, Z. Cui, R. Liu, et al. (2024) Fincon: a synthesized llm multi-agent system with conceptual verbal reinforcement for enhanced financial decision making. Advances in Neural Information Processing Systems 37, p. 137010–137045. Cited by: Appendix A, §1. Appendix A Related Work (Extended) This appendix gives the full version of the related-work review summarised in §4. Classical herding measurement. The herding-measurement literature has two pillars. The trading-flow track originates with the LSV [25] cross-sectional buy/sell imbalance statistic, with extensions to mutual-fund flows [42], momentum-decomposed sequential-trade correlations [37], and finite-sample bias corrections [18]; all are computed from disclosed positions and are post-hoc by construction. The return-cross-section track replaces flows with dispersion: the cross-sectional standard deviation of Christie and Huang [13], the cross-sectional absolute deviation regression of CCK [12], and the state-space variant of Hwang and Salmon [23]. We adopt LSV and CSAD as the two classical anchors against which GeomHerd is consistency-checked (Prop. 1, §3.3.2, §2.5); to our knowledge, prior curvature-on-finance work benchmarks against at most one of the two. Mechanisms of contagion. A structurally distinct line concerns how localized shocks propagate once herding has formed. Foundational results establish that interbank network topology determines whether shocks dissipate or amplify [4, 1], that cross-holding cascades are non-monotone in diversification [16], and that liquidity spirals propagate margin shocks across funds [9]. The shared structural prediction - shocks travel along a sparse set of inter-cluster edges - is what our negative-curvature detector β− _- targets, with the Sia–Ni [36, 27] community-bridge interpretation supplying the mathematical reading. Discrete curvature on financial graphs. Discrete Ricci curvature [29] has been applied to financial graphs to detect systemic stress [33, 31, 41, 32, 3, 38]; Jiang et al. [24] is closest to our contagion-side claim, showing on multivariate-Hawkes point-process networks that more negative curvature predicts systemic risk earlier than CATFIN and the absorption ratio. We share the geometric machinery but differ on substrate (agent graph vs. price-correlation or point-process graph) and framing (sign-decomposed herding/contagion duality vs. a single global fragility scalar); Table 5 summarises the comparison. Table 5: Positioning vs. prior discrete-curvature financial work plus the closest same-substrate information-theoretic baseline [39], along five axes. PCG = price-correlation graph; AG = agent graph; P = point-process network. Method Subs. Scalar Detector Evaluation OOD Sandhu 2016 [33] PCG ORC mean (global) descriptive VIX-corr (1 metric) – Samal 2021 [31] PCG ORC vs. F-Ricci VIX-corr only VIX-corr (1 metric) – Wang 2023 [41] PCG F-Ricci descriptive descriptive only – Sánchez 2024 [32] PCG ORC post-hoc post-hoc only – Akgüller 2025 [3] PCG (MI) F-Ricci, sliding descriptive sectoral – Srinivasan 2026 [38] PCG ORC + neckpinch clustering cluster quality – Jiang 2023 [24] P-net ORC mean (neg. only) ranking precision, lead vs. CATFIN – Tessera 2026 [39] AG A-MI (info-th.) kσk_σ on A-MI MARL benchmarks – GeomHerd (ours) AG ORC sign-decomp. CUSUM + Kendall-τ multi-axis (8+ metrics) ✓ Vicsek Early-warning signals and LLM-agent simulators. Classical critical-slowing-down work establishes generic precursors of tipping points [35], recently extended to learned classifiers [11, 10] and persistence-homology detectors [20]; we adopt the same multi-axis evaluation philosophy and use Scheffer et al. [35] as a sanity check (§3.3.1). On the detector side, Page’s [30] CUSUM motivates our alarm rule (§2.3). LLM-driven multi-agent financial simulators [43, 21, 19, 44] provide the heterogeneous, fully-observable action stream on which the agent-graph substrate becomes testable. A.1 Graph design axes Table 6: The five graph-design axes of GeomHerd, with the choice adopted in this paper and the layer of the three-layer logic that drives it. Axis Adopted choice Driven by Nodes individual agents substrate (finest layer the ABM exposes) Edge semantics windowed action agreement herding semantics (BHW-1992 definition) Edge weights binary frequency in [0,1][0,1] (Eq. 1) substrate (no extra design freedom) Sparsification threshold w0=0.5w_0=0.5 geometry (suppress chance co-occurrence) Snapshot frequency every Δt=10 t=10 steps, Tw=100T_w=100 substrate / herding semantics Appendix B Outline of Proposition 1 and notation alignment This appendix expands the four-step outline of §2.5. The argument is presented as a dominant-order scaling derivation rather than a fully rigorous proof; in particular, Step 2 invokes a closed-form W1W_1 that follows from mean-field concentration but whose remainder bound we do not establish here. A complete proof would require propagating mean-field convergence through the bipartite optimal-transport plan, which we leave to follow-on work. Setting and assumptions. N is the agent count and t a fixed simulator step. A1 (Agent graph.) Agents i∈Vi∈ V, |V|=N|V|=N, take discrete actions ai(t)∈a_i(t) . The graph Gt=(V,Et,wt)G_t=(V,E_t,w_t) has edge weights given by Eq. (1) and edges retained above a sparsification threshold. A2 (Lazy walk and curvature.) Each node carries the lazy-walk kernel μit(j)=αδij+(1−α)wt(i,j)/∑kwt(i,k) _i^t(j)=α\, _ij+(1-α)\,w_t(i,j)/ _kw_t(i,k) with α=0.5α=0.5, distance dt(i,j)=wt(i,j)d_t(i,j)=w_t(i,j), and curvature κOR(i,j;t)=1−W1(μit,μjt)/dt(i,j) _OR(i,j;t)=1-W_1( _i^t, _j^t)/d_t(i,j); κ¯OR(t) κ_OR(t) denotes the mean over EtE_t. A3 (Mean-field concentration.) There exists a one-dimensional order parameter M(t)=i[ai(t)]M(t)=E_i[a_i(t)] such that pairwise action correlations [ai(s)=aj(s)]E[1\a_i(s)=a_j(s)\] concentrate around a function f(M(t))f(M(t)) at rate (N−1/2)O(N^-1/2), uniformly over the window of width TwT_w. A4 (Linear price impact.) Per-asset returns satisfy ri,t=βiM(t)+ξi,tr_i,t= _i\,M(t)+ _i,t with i.i.d. noise ξi,t∼(0,σξ2) _i,t (0, _ξ^2) and i[βi]=1E_i[ _i]=1 wlog. A5 (CSAD estimand.) The CCK [12] convention CSADt=i[|ri,t−r¯t|]CSAD_t=E_i [|r_i,t- r_t| ] with r¯t=i[ri,t] r_t=E_i[r_i,t]. Sketch of Proposition 1. Under A3, agent action correlations collapse onto M(t)M(t) and wt(i,j)w_t(i,j) concentrates around a function of M(t)M(t). The lazy-walk transport between two nodes whose neighborhoods both concentrate around the same mean-field measure satisfies 1−κOR(i,j;t)∝1−M(t)21- _OR(i,j;t) 1-M(t)^2 at dominant order. Under A4, the half-normal expectation reduces CSADtCSAD_t to σξ2/π(1−M(t)2)1/2 _ξ 2/π\,(1-M(t)^2)^1/2 plus (N−1/2)O(N^-1/2) corrections. Substituting the curvature scaling gives Eq. (7). The full step-by-step derivation appears below. ∎ Remark 1 (Failure modes). Eq. (7) breaks in three regimes. (i) Mean-field breakdown: persistent multi-modal clustering violates A3, so 1−κ¯OR(t)1- κ_OR(t) no longer collapses onto 1−M(t)21-M(t)^2. (i) Nonlinear price impact (A4 fails): the half-normal reduction breaks and the 2/π 2/π prefactor acquires a moment-dependent correction. (i) Boundary regime |M(t)|→1|M(t)|→ 1: the (N−1/2)O(N^-1/2) remainder bound becomes loose. Empirically, these are exactly the regimes in which the headline κ¯OR+ κ_OR^+ detector either saturates (i, i) or fires on a distorted signal (i); the failure modes characterise precisely the regimes in which a geometric-vs-CSAD comparison is informative. Remark 2 (Identification of γ^3 γ_3). The augmented-CCK regression in §3.3.2 estimates γ3 _3 on simulated trajectories. Three caveats apply. First, κ¯OR(t) κ_OR(t) and CSADtCSAD_t are both deterministic functions of M(t)M(t) in the mean-field limit, so the regression measures partial correlation rather than identifying an independent channel; γ^3<0 γ_3<0 is consistent with Eq. (7) but does not by itself distinguish the bridge from any other M(t)M(t)-mediated relationship. Second, κ¯OR(t) κ_OR(t) inherits the (N−1/2)O(N^-1/2) measurement error of A3, attenuating γ^3 γ_3 toward zero. Third, the bridge predicts a (1−κ¯OR(t))1/2(1- κ_OR(t))^1/2 functional form rather than the linear specification used; the linear form is a local approximation. We frame γ^3 γ_3 as a sign-consistency check of Proposition 1, not a hypothesis test of the bridge against alternatives. Notation alignment with the CSAD/CCK literature. Table 7 reconciles the symbols used in Proposition 1 with those used by Christie and Huang [13] and CCK [12] (“CCK”). Where two papers use different letters for the same quantity, we adopt the form used in the paper body. The point of the table is to defend against the technical objection that our CSADtCSAD_t might be a different estimand from CCK’s; it is the same. Table 7: Symbol alignment between Proposition 1 and the classical CSAD/CCK regression literature. Quantity This paper (§2.5) CCK / Christie–Huang Cross-sectional return dispersion CSADt=i[|ri,t−r¯t|]CSAD_t=E_i[|r_i,t- r_t|] CSADtCSAD_t (CCK eq. 3); CSSDtCSSD_t (CH eq. 1) Per-agent / per-asset return ri,tr_i,t Ri,tR_i,t Cross-sectional mean return r¯t=i[ri,t] r_t=E_i[r_i,t] R¯m,t R_m,t (market) Order parameter / market trend M(t)M(t) Rm,tR_m,t (market return) Per-agent loading βi _i βi _i Idiosyncratic noise ξi,t∼(0,σξ2) _i,t (0, _ξ^2) i.i.d. εi,t _i,t Number of agents / assets N N Quadratic-term coefficient β2 _2 in Eq. (8) γ2 _2 (CCK), often β2 _2 Geometric augmentation coeff. γ3 _3 in Eq. (8) (not in CCK) The augmented regression Eq. (8) adds the third regressor κ¯OR(t) κ_OR(t) to the CCK specification without changing the estimand CSADtCSAD_t or the quadratic regressor Rm,t2R_m,t^2 that defines CCK herding; γ^3 γ_3 measures the partial association between CSADtCSAD_t and the agent-graph curvature after controlling for the linear and quadratic market-return terms. Derivation (scaling argument). We expand the four-step outline from §2.5. Each step is a dominant-order scaling claim rather than a fully-bounded inequality; we flag the missing remainder bound explicitly at the point it would be required. Step 1: mean-field collapse of agent-graph weights. Under A3, agent action correlations [ai(s)=aj(s)]E[1\a_i(s)=a_j(s)\] collapse onto a one-dimensional order parameter M(t)=i[ai(t)]M(t)=E_i[a_i(t)], with concentration at rate (N−1/2)O(N^-1/2). The windowed agreement frequency (1) concentrates around wt(i,j)→f(M(t))w_t(i,j)→ f(M(t)) where f is determined by the action distribution conditional on M. We do not prove uniform concentration over the full window of width TwT_w here; the argument is an LLN under A3 conditional on M(⋅)M(·) over the window. Step 2: closed-form Wasserstein in the mean field. Once wtw_t concentrates around f(M(t))f(M(t)) at every node, the lazy-walk kernels μit _i^t and μjt _j^t both concentrate around the same mean-field measure on a single asymptotic neighbourhood. We argue that the optimal transport between two such near-degenerate kernels admits a closed-form W1W_1 proportional to the deviation of the underlying weights from saturation, yielding 1−κOR(i,j;t)∝1−M(t)21- _OR(i,j;t) 1-M(t)^2 on edges and the same scaling for the mean over edges κ¯OR(t) κ_OR(t). A rigorous derivation requires bounding the bipartite transport between two near-uniform measures on the asymptotic neighbourhood, and we do not give that bound here; this is the step we identify as requiring follow-on work in the conclusion. Step 3: substituting linear impact into CSAD. Under A4, the CSAD definition CSADt=i[|ri,t−r¯t|]CSAD_t=E_i[|r_i,t- r_t|] reduces, via the half-normal expectation of |ξi,t|| _i,t|, to CSADt=σξ2/π+(N−1/2)CSAD_t= _ξ 2/π+O(N^-1/2) when M(t)=0M(t)=0, and shrinks as |M(t)|→1|M(t)|→ 1 because the βiM(t) _iM(t) component of ri,tr_i,t is exactly the cross-sectional mean r¯t r_t at i[βi]=1E_i[ _i]=1 and so cancels from |ri,t−r¯t||r_i,t- r_t| at leading order. Step 4: combining. Substituting the mean-field scaling 1−κ¯OR(t)∝1−M(t)21- κ_OR(t) 1-M(t)^2 from Step 2 into the CSAD-impact identity from Step 3 yields (7). The (N−1/2)O(N^-1/2) remainder tracks the rate of mean-field collapse in A3. Appendix C Headline-table source data and reproducibility The headline run uses 80 seeds across 5 values of the control parameter κ (Cividino–Sornette sbase=0.6s_base=0.6, spost=1.6×sweep_value/1.3s_post=1.6×sweep\_value/1.3), giving 240 supercritical trajectories for pooled statistics. Configuration, seeds, and output hashes are archived with the code release; numeric refresh date 2026-05-05. The geometric pipeline uses POT [17] for exact W1W_1 via linear programming, SciPy CSR-graph Dijkstra for all-pairs shortest paths, and a fixed FSQ codebook with K=64K=64. Detection thresholds are calibrated from a pre-stress baseline window (§2.3); we hold (k±,h±,τthresh,Wτ)(k_±,h_±, _thresh,W_τ) fixed across substrates and report sensitivity in Appendix D. C.1 Recall-oriented operating point The abstract’s headline figure of 272 steps median lead corresponds to the recall-oriented CUSUM operating point (kσ,hσ)=(0.50,4.0)(k_σ,h_σ)=(0.50,4.0) in the calibration sweep of Table 8: at this point the detector recalls 0.52 of supercritical trajectories at the cost of a higher subcritical FAR (0.76). The two operating points correspond to two distinct deployment use-cases. The recall-oriented point prioritises early aggregate alarm at the cost of more false-positives on subcritical regimes; it is the right choice when the downside of missing a herding cascade dominates the cost of acting on a normal regime. The precision-oriented point (kσ,hσ)=(2.0,4.0)(k_σ,h_σ)=(2.0,4.0), used for the head-to-head paired contrasts in Table 2, prioritises low-FAR regime discrimination; head-to-head lead comparisons are only meaningful at FAR-controlled thresholds, so the body’s paired contrasts use this point rather than the recall-oriented one. Appendix D Calibration sweep, full table Operating-point sweep for the rolling-baseline CUSUM on κpos _pos (mean curvature over positive edges), direction up, with CUSUM baseline window W=35W=35 samples along the subsampled curvature trace (the first W successive κ observations in each replay; not raw ABM steps-under the headline snapshot stride Δt=10 t=10 this spans W⋅ΔtW· t simulator steps), over a 5×55× 5 grid of (kσ,hσ)(k_σ,h_σ) on the binary-edge replay set (400 trajectories). The two operating points highlighted in the body are: (kσ,hσ)=(0.50,4.0)(k_σ,h_σ)=(0.50,4.0) (recall-oriented; abstract headline of 272 steps) and (kσ,hσ)=(2.0,4.0)(k_σ,h_σ)=(2.0,4.0) (precision-oriented; FAR control, paired-contrast headline of 178 steps). Table 8: CUSUM operating-point sweep on κpos _pos (up), binary-edge replay set. kσk_σ hσh_σ Recall FAR Lead (med.) 95% CI 0.25 3.0 0.75⋆ 0.93 291 [262, 324] 0.25 4.0 0.68 0.88 282 [254, 323] 0.25 5.0 0.61 0.82 270 [239, 311] 0.25 6.0 0.53 0.78 261 [237, 305] 0.25 8.0 0.44 0.72 264 [243, 304] 0.50 3.0 0.60 0.80 282 [256, 338] 0.50 4.0 0.52 0.76 272 [236, 313] 0.50 5.0 0.46 0.70 273 [240, 332] 0.50 6.0 0.40 0.64 264 [243, 324] 0.50 8.0 0.32 0.54 252 [218, 313] 1.00 3.0 0.39 0.59 264 [247, 324] 1.00 4.0 0.31 0.51 253 [229, 319] 1.00 5.0 0.28 0.43 248 [207, 287] 1.00 6.0 0.21 0.40 242 [218, 309] 1.00 8.0 0.14 0.28 233 [208, 306] 1.50 3.0 0.23 0.38 244 [204, 306] 1.50 4.0 0.15 0.26 218 [153, 306] 1.50 5.0 0.11 0.18 232 [160, 306] 1.50 6.0 0.08 0.12 210 [152, 296] 1.50 8.0 0.04 0.08 242 [89, 413] 2.00 3.0 0.08 0.15 233 [127, 299] 2.00 4.0 0.05 0.09 218 [91, 406] 2.00 5.0 0.04 0.06 242 [78, 413] 2.00 6.0 0.03 0.04 180 [18, 543] 2.00 8.0 0.01 0.03 n/a — Appendix E Auxiliary benchmark: stylised drift on scalar controls This appendix does not advance a competing headline about chart taxonomy. The paper’s core object is the agent-graph curvature trajectory κ¯OR+(t) κ_OR^+(t) built from binary edges and discrete actions; the replay evidence above shows that object carries leading information. What follows isolates a textbook scalar scenario-Gaussian noise with a linear mean drift after an artificial change point-only to document how two standard one-sided alarm maps behave under matched empirical false-alarm rates on stationary paths. It is a reproducibility footnote, not a substitute for the ABM geometry results. With empirical FAR ≈0.10≈ 0.10 on null paths (scripts/cusum_vs_zscore_stylized_evidence.py), one-sided Page CUSUM vs. a calibrated Shewhart rule on the linear-drift alternative yields median delays 28 vs. 49 steps (ratio 1.75); horizon detection rates 100.0% vs. 100.0%. Readers should treat these numbers as sanity checks on the scalar alarm layer, not as the paper’s main empirical conclusion. Appendix F Cosine-edge ablation The headline configuration uses binary action-agreement edges (wt(i,j)=1w_t(i,j)=1 if windowed action match exceeds w0w_0, else 0). A natural alternative keeps wtw_t continuous via cosine similarity on a one-hot lifting of the action sequences. We replay the headline pipeline with this variant and report the result below. Cosine edges remove the herding-side signal: the upward detector on κ¯OR κ_OR fires on 0/900/90 supercritical trajectories (versus 11/9011/90 for binary edges at the same operating point). The mean cosine similarity in the supercritical pool concentrates in a narrow band around its baseline, so the rolling-baseline CUSUM never accumulates enough deviation to trip. This is a positive ablation result for binary edges: thresholding at w0=0.5w_0=0.5 amplifies the regime change from sub-population coordination to clique formation that the curvature signal relies on. The cosine-edge variant remains in the codebase as the sensitivity test, but its empirical role is to validate the binary-edge design rather than to substitute for it. Appendix G Out-of-domain transfer: Vicsek self-driven particles For each η∈0.5,1.0,1.6,2.0,2.5η∈\0.5,1.0,1.6,2.0,2.5\ we run 20 seeds for T=1000T=1000 steps, downsample to one snapshot every 50 steps, and build the agent graph from k-N (k=10k=10) on the heading sequence with binary edge weights as in the financial substrate. The order-parameter event τ⋆τ is the first step at which the rolling polarisation Va(t)=N−1‖∑iv^i(t)‖V_a(t)=N^-1\| _i v_i(t)\| exceeds 0.50.5 for three consecutive steps after a 5050-step warm-up. The geometric signal κ¯OR(τ⋆) κ_OR(τ ) separates ordered from disordered trajectories with AUROC 0.99 (95% CI [0.98, 1.00][0.98,\,1.00]), evaluated on the 95 of 100 trajectories that produce a valid polarisation event τ⋆τ before T; the remaining 5 lack a polarisation event under our event criterion and are excluded from the at-event score. Per-η medians of κ¯OR(τ⋆) κ_OR(τ ) are monotone in η: +0.08+0.08 (η=0.5η=0.5), −0.06-0.06 (η=1.0η=1.0), −0.19-0.19 (η=1.6η=1.6), −0.22-0.22 (η=2.0η=2.0), −0.26-0.26 (η=2.5η=2.5); the gap between the two flanking values of ηc=1.6 _c=1.6 is ≈0.13≈ 0.13, which the bootstrap CI excludes from zero. Sensitivity: alternative scoring rules. Scoring each trajectory by maxtκ¯OR(t) _t κ_OR(t) yields AUROC ≈0.64≈ 0.64 (95% CI excludes 0.50.5); tail and final-snapshot means give AUROC ≥0.99≥ 0.99. Full rule-by-rule breakdown is reported in this appendix. Lead-time pipeline note. The CUSUM upward detector calibrated on the financial substrate (σ-baseline window =100=100 steps, kσ=2.0k_σ=2.0, skip-initial =50=50) does not fire on any of the 100 Vicsek trajectories. The cause is timescale mismatch: the financial substrate’s herding cascade develops over 200200–300300 steps, while Vicsek’s alignment transient establishes within ≈50≈ 50 steps. A Vicsek-specific recalibration (baseline-window =20=20, skip-initial =5=5) restores non-trivial alarm rates; quantitative lead-time numbers are deferred. The AUROC headline does not depend on the alarm calibration-it operates directly on the curvature value at the event. Appendix H Additional ablation rows The decomposed ablation in Table 4 covers four families of variations: detector swaps (z-score, standalone CUSUM configuration, EWMA, Kendall-τ slope-only), geometry choices (cosine edges, window length, Sinkhorn transport, optional sign pooling), and a triplet ablation removing τsing _sing. A dedicated VeffV_eff drop-out row is left for a follow-on sweep. The cosine-edge row (A01) is populated from the matched replay run described in Appendix F. Appendix I 13F retrospective: substrate-pivot template under disclosure constraint The 13F retrospective is the natural deployment template for the substrate-pivot (§2.5) when agent actions are not fully observable. We build a funds-as-agents graph from quarterly 13F holdings disclosures, with edges given by Jaccard overlap between two funds’ position sets and a sliding window of width W=4W=4 quarters; we then compute κ¯OR(t) κ_OR(t) and β−(t) _-(t) on the resulting sparse graph. The construction recovers documented stress periods (2008Q4, 2011Q3, 2020Q1, 2022Q1) qualitatively; quantitative head-to-head against LSV / CATFIN / absorption ratio is deferred. We make no quantitative claim from this retrospective in the body; its purpose is to establish that the substrate-pivot construction reads cleanly off disclosure-constrained data. Appendix J Additional Discussion Limitations. At the precision-oriented threshold, extreme signal sparsity limits the statistical power of paired-difference claims. Furthermore, subcritical false-alarm rates lack strict held-out validation, and the CWS substrate is heavily stylized, though §2.5 bridges this gap to real-world disclosed-flow constraints. Future Work & Broader Impact. Future research will focus on (i) fusing geometric features with learned classifiers like EWSNet [11], (i) real-market deployments via 13F fund-overlap graphs, and (i) formalizing Proposition 1 with uniform optimal-transport bounds. While early-warning signals could be exploited by sophisticated traders, the systemic stability benefits for regulators dominate. We open-source our geometric analysis pipeline but intentionally withhold LLM prompt corpora to mitigate trivial adversarial replication.