Oracle integrity under predatory attack: a calibrated jump-diffusion model with public-data evidence and a positivity-preserving numerical scheme

1 Introduction

Decentralised finance has made oracle systems part of settlement infrastructure. A lending protocol does not observe collateral value directly, a perpetual exchange does not observe an asset’s spot price directly, and a liquidation engine does not observe a borrower’s solvency directly. Each contract reads a feed and then executes. When the feed is stale, manipulable, or too slow for prevailing market conditions, a formally correct contract can produce an economically wrong outcome. The Mango Markets incident provides a concrete example: the United States Commodity Futures Trading Commission charged a single agent with manipulating reference prices used by the protocol’s perpetual index, enabling withdrawals exceeding USD 110 million within a few minutes (Commodity Futures Trading Commission, 2023). The analysis targets this class of failure rather than a single incident.

The object of interest is the joint dynamics of feed integrity and attack pressure. Feed integrity is a renewable reliability stock depleted by profitable adversarial extraction. Attack pressure increases when integrity remains exploitable and decays when defensive responses, governance, or the absence of opportunity removes the incentive. The predator–prey terminology is a modelling analogy for these coupled growth, depletion, and recovery dynamics (Strogatz, 2018; Oksendal, 2003; Applebaum, 2009; Mao and Yuan, 2006).

The model couples adversarial stress to a documented incident corpus through a controlled positive jump-diffusion for oracle integrity and attack pressure. It yields two structural implications: multiplicative jumps with γ i > − 1 preserve the admissible integrity state, whereas unrestricted additive jumps do not; and the equilibrium depletion flow α P * D * is invariant to α. The latter implication is assessed within the stated calibration ranges by Latin-hypercube sensitivity analysis; it is not a general empirical claim about all oracle systems.

Three public-data inputs anchor the scenarios: 24 documented oracle-related DeFi incidents over 2020–2023 (cumulative loss $867.37 m), an oracle-consumer TVL panel of 2721 protocols (combined TVL $106.37 bn), and Chainlink metadata for 1,669 feeds across 29 networks. Cached public extracts are processed into the tables and figures reported below.

1.1 Worked real-world example: Mango Markets

The information flow can be read against the Mango Markets incident. The CFTC record and the project post-mortem describe a price-manipulation episode in which the reference price used by a perpetual index was moved, the protocol consumed that reported price in its account and collateral calculations, and the resulting balance-sheet state permitted withdrawals exceeding USD 110 million before the position was unwound (Commodity Futures Trading Commission, 2023; Distributed Networks Institute, 2022). In the notation used here, the sequence is as follows: (i) an exploitable feed condition lowers effective integrity D t; (ii) profitable manipulation raises attack pressure P t; (iii) the contract applies the reported value to collateral or settlement; and (iv) an intervention, market correction, or exhaustion of the opportunity changes the subsequent path. The sequence is illustrative; it does not replay the Mango feed.

2 Related literature

2.1 Oracle design

The oracle problem arises because smart contracts execute in a closed computational environment and require an external bridge to observe market prices, reserve balances, or other off-chain state. The original Ethereum white paper (Buterin, 2014) and later systematisations (Eskandari et al., 2021; Liu et al., 2021; Beniiche, 2020; Caldarelli, 2020) treat oracles as institutional interfaces with explicit trust assumptions. Recent design work covers push and pull oracle architectures (Mammadzada et al., 2020), optimistic oracles with dispute resolution windows (UMA Project, 2026), threshold-signature aggregation in Chainlink and similar networks (Breidenbach et al., 2021; Chainlink, 2026), low-latency pull-based feeds in Pyth (Pyth Network, 2023), and reserve-attestation oracles (Chainlink, 2025). Mechanism design for oracle markets is discussed by Adler et al. (2018) and Kamiya (2020). Chainlink (2024) described Smart Value Recapture (SVR), while Obadia et al. (2021) formalised cross-domain maximal extractable value. These sources motivate the integrity observables and the interpretation of protocol controls.

2.2 Oracle attacks and DeFi incidents

Flash-loan-enabled atomic transactions (Qin et al., 2021; Wang et al., 2021) make oracle manipulation economically plausible because an attacker can fund, execute, and unwind a position within one block. The bZx, Harvest Finance, Cream Finance, Inverse Finance, Mango Markets, Euler, and Beanstalk incidents in the coded corpus involve mechanisms ranging from spot-AMM manipulation to stale or pull-oracle timing and TWAP-window manipulation. The aggregate corpus is anchored to the systematic attack taxonomy and dated incident records (Zhou et al., 2023; Rekt News, 2026; Commodity Futures Trading Commission, 2023; Distributed Networks Institute, 2022; Beanstalk Farms, 2022). Related analyses include the general DeFi-crisis study of Gudgeon et al. and the scam-token study of Xia et al. (Werner et al., 2022; Gudgeon et al., 2020; Xia et al., 2021). Time-weighted average price (TWAP) mechanisms are analysed by Adams et al. (2021) and Angeris et al. (2020), while oracle-related MEV is analysed by Daian et al. (2020) and Qin et al. (2022). This literature informs the incident corpus and the multiplicative jump specification.

2.3 Stochastic ecology and predator–prey dynamics

Threshold persistence in classical predator–prey systems is studied by Lotka (1925); Volterra (1926); Strogatz (2018); and Murray (2002). Stochastic extensions with diffusion (Mao and Yuan, 2006; Du et al., 2016; Yang and Zhong, 2014) and Lévy jumps (Bao et al., 2011; Bao and Yuan, 2012; Zhu and Li, 2016) provide the existence, uniqueness, positivity, and stationary-distribution results used here. Jump-diffusion models are also standard in mathematical finance (Merton, 1976; Kou, 2002; Cont and Tankov, 2004; Applebaum, 2009). The stochastic-ecology literature supports the coupled-state analysis; the finance literature informs the jump intensities and amplitudes.

2.4 Positivity-preserving numerical schemes

Standard Euler–Maruyama discretisations of geometric SDEs can violate positivity because the diffusion increment is symmetric (Higham, 2001; Higham et al., 2002). Log-Euler and Milstein schemes are positivity-preserving for geometric Brownian motion (Lord et al., 2010; Higham and Kloeden, 2021), and corresponding schemes for jump-diffusions are analysed by Platen and Bruti-Liberati (2010) and Bruti-Liberati and Platen (2007). We use a log-Euler discretisation in which each Poisson arrival applies a deterministic multiplicative shock log ( 1 + γ i ) to the log-state, which is exact at each jump and preserves positivity by construction.

2.5 Positioning

The oracle literature identifies architectures, attack mechanisms, and protocol-level defences, while stochastic predator–prey and jump-diffusion studies provide tools for analysing positive interacting state variables under continuous and discontinuous shocks. The analysis combines these strands in an incident-anchored model of oracle integrity and attack pressure and interprets its stability, positivity, and simulation results as scenario diagnostics for a representative exposed protocol. It does not propose a new oracle architecture or estimate a causal effect of feed design on losses.

Table 1 makes the evidence-to-method links explicit: the oracle and incident sources motivate the observables and scenarios, while the stochastic and numerical sources support the model and discretisation.

TABLE 1. Literature map linking the main evidence strands to the present analysis.

3 State variables and observable construction

Let D t ∈ ( 0 , ∞ ) denote oracle integrity and P t ∈ ( 0 , ∞ ) denote attack pressure at time t, with time measured in months throughout. The integrity state is the latent stock of reliable, timely, manipulation-resistant data available to contracts. It is not a token price, an oracle-provider market capitalisation, or a protocol TVL: those quantities measure something else. A useful empirical construction iswhere C t is feed coverage (feeds per network or share of collateral assets with direct feeds), H t is update discipline (heartbeat intervals and deviation thresholds), R t is source redundancy (independent providers or venues per feed), S t is feed staleness or cross-source disagreement, and L t is reference-market liquidity. The pressure state iswhere V t is realised volatility, A t is arbitrage or attack profitability, B t is borrowable capital such as flash-loan capacity, M t is reference-market manipulability, and E t is exploit attention or incident clustering.

D t = g C t , H t , R t , S t , L t , (1)

P t = h V t , A t , B t , M t , E t , (2)

Definition 1 (Oracle integrity). Oracle integrity is the positive latent state D t summarising the capacity of an oracle system to deliver timely, accurate, manipulation-resistant data to on-chain contracts.

Definition 2 (Attack pressure). Attack pressure is the positive latent state P t summarising the intensity of profitable manipulation incentives, technical feasibility, and available capital for exploiting oracle-dependent contracts.

The observable design follows from these definitions. Confounding integrity with TVL would conflate capital exposure and data reliability; using incident counts alone for P t would conflate realised events with latent pressure. Section 4 reports observable proxies for each component.

The quantities in (Equations 1, 2) are calibration anchors rather than direct measurements of D t or P t. Feed metadata characterise observable design features, TVL characterises potential exposure, and documented incidents characterise realised stress events. The functions g and h are not estimated, and no proxy is treated as interchangeable with either latent state.

3.1 Observable dictionary

Table 2 consolidates the model quantities, their substantive interpretation, and the role assigned to each public-data input. It separates the latent states from their observable calibration anchors.

TABLE 2. : Observable dictionary for the model states, parameters, and public-data inputs.

4 Public-data layer

The data workflow is deterministic. It caches public extracts, records loose name matches excluded from the oracle panel, parses Chainlink feed metadata, constructs an oracle-consumer TVL panel, and writes the tables used in the analysis. The DefiLlama protocol endpoint and Chainlink feed page were retrieved on 2026-05-04 UTC; the cached extracts, cleaned tables, and generation script are included with the source files. The incident corpus is a separately coded, fixed 2020–2023 sample and is not contemporaneous with the 2026 metadata snapshot. The three inputs are described below according to their distinct roles.

4.1 Oracle-consumer exposure panel

Total Value Locked is not a meaningful proxy for oracle integrity. It is, however, a meaningful proxy for the exposure that depends on oracle integrity. Lending, CDP, derivatives, synthetics, liquid-staking, yield, and RWA-lending protocols all use price feeds for collateral valuation, liquidation triggers, or settlement. A representative oracle failure does not destroy integrity at the protocol issuing the feed; it transmits losses to the consumers that read the feed. We, therefore, retain the DefiLlama categories: lending, CDP, liquid staking, derivatives, synthetics, yield, yield aggregator, leveraged farming, options, options vault, insurance, algo-stables, and RWA lending (DefiLlama, 2026); protocols merely containing the string “oracle” outside the exact Oracle category are recorded and excluded. The resulting panel contains 2,721 protocols with a combined TVL of $106.37 billion.

The 12 protocols in the exact oracle category report $0 of TVL in this snapshot. This is consistent with oracle networks being thin-balance-sheet infrastructure rather than custodial venues. The economic stake at risk from oracle failure should therefore be measured at the consumer side, as in Table 3, not at the issuer side.

TABLE 3. Oracle-consumer DeFi TVL by category. The lending, CDP, derivatives, and synthetics categories depend on price feeds for collateral valuation, liquidation, and settlement.

4.2 Chainlink feed metadata

Table 4 reports core feed-rule statistics. The median heartbeat is 24.00 h, indicating that update timing is governed primarily by deviation-based triggers rather than scheduled refresh, with the median deviation threshold of 0.50% defining the typical price-move trigger. The pipeline identifies 96 SVR-enabled feeds.

TABLE 4. Summary of public feed-rule metadata used in the empirical design.

The quantitative calibration uses Chainlink because the reproducible public pull supplies feed-level metadata in a common schema. Pyth and UMA are included as qualitative comparators; the three systems are not treated as interchangeable. A pooled multi-provider estimate would require matched feed histories, update timestamps, and incident labels that are not available in this snapshot.

Tables 5, 6 show that feed coverage is broad but uneven across networks and asset classes. This heterogeneity matters because the same model parameters may not apply across networks: the reliability capacity K and the depletion intensity α can differ between deep, well-covered chains and thin, peripheral ones. Section 9 treats this with a sensitivity analysis rather than asserting a single calibrated parameter set.

TABLE 5. Top Chainlink networks by visible feed count in the public-data pull.

TABLE 6. Visible Chainlink feeds by asset class.

Figure 1 visualises the network-level coverage summarised in Table 5.

Figure 2 provides the distributional view corresponding to the feed-rule statistics in Table 4.

4.3 Oracle-incident corpus

We assemble 24 oracle-related DeFi incidents covering February 2020 to September 2023, with a cumulative reported loss of $867.37 m. The corpus is compiled from the SoK by Zhou et al. (2023), the Rekt News database (Rekt News, 2026), the CFTC Mango action (Commodity Futures Trading Commission, 2023), and the project-specific post-mortems cited in Section 2. The implied descriptive incident rate is 6.55 per year across the universe, or 0.545 per month; this rate provides the jump-intensity anchor in Section 7.

Table 7 lists the coded events and loss anchors used in the rate calculation; Figure 3 shows the same corpus on a log-loss scale.

TABLE 7. Documented oracle-related DeFi incidents used to calibrate the jump intensity. Loss figures are in USD millions at the time of the event.

5 Deterministic predator–prey skeleton

The deterministic skeleton is with all parameters strictly positive: r is the integrity rebuild rate under calm conditions, K is the reliability capacity, α is the depletion intensity of pressure on integrity, β is the pressure response coefficient, and δ is the pressure decay rate.

D ̇ t = r D t 1 − D t K − α P t D t , (3)

P ̇ t = β P t D t − δ P t , (4)

5.1 Equilibrium structure

Proposition 1 (Equilibria). The system (Equations 3, 4) has exactly three equilibria:

Proof. At an equilibrium D ̇ = D [ r ( 1 − D / K ) − α P ] = 0 and P ̇ = P ( β D − δ ) = 0. The four sign cases on ( D , P ) provide exactly the listed solutions, and positivity of P * requires δ < β K.

Proposition 2 (Boundary instability). E 0 is a saddle with eigenvalues { r , − δ } and unstable for every parameter choice. E K is a saddle with eigenvalues { − r , β K − δ } whenever δ < β K (i.e., whenever E * exists), locally asymptotically stable when δ > β K , and non-hyperbolic when δ = β K.

Proof. Direct evaluation of the Jacobian (computed below in Theorem 1) at E 0 and E K.

Proposition 3 (Positive invariance). The open positive orthant { ( D , P ) : D > 0 , P > 0 } is forward invariant for (Equations 3, 4).

Proof. On { D = 0 }, the right-hand side of (Equation 3) vanishes, so D = 0 is invariant. Similarly, { P = 0 } is invariant. Local Lipschitz continuity of the vector field provides uniqueness; hence, trajectories starting in the open positive orthant cannot cross the boundary in finite time.

5.2 Local stability and oscillation discriminant

The deterministic result isolates the system’s baseline behaviour before diffusion and attack-window shocks are introduced. Local stability means that small departures from the interior integrity–pressure balance contract back toward that balance; the discriminant distinguishes a direct return from damped overshooting. This distinction matters operationally because an oscillatory return creates repeated periods of elevated pressure even when the deterministic equilibrium is stable.

Theorem 1 (Local stability of E * and oscillation discriminant). Assume δ < β K . The Jacobian of (Equations 3, 4) at E * is with tr ( J * ) = − r δ / ( β K ) < 0 and det ( J * ) = r δ 1 − δ / ( β K ) > 0. Hence, E * is locally asymptotically stable. The linearised eigenvalues are real (so the linear modes are non-oscillatory) if and only ifand form a complex–conjugate pair with damped oscillatory modes otherwise.

Proof. The Jacobian of (Equations 3, 4) isAt E * = ( D * , P * ) with D * = δ / β and P * = ( r / α ) ( 1 − δ / ( β K ) ),

J D , P = r − 2 r D / K − α P − α D β P β D − δ .

J 11 * = r − 2 r δ β K − α r α 1 − δ β K = − r δ β K , J 12 * = − α δ β , J 21 * = β P * = β r α 1 − δ β K , J 22 * = β D * − δ = 0 .

Therefore, tr ( J * ) = − r δ / ( β K ) < 0 and

det J * = 0 − − α δ β β r α 1 − δ β K = r δ 1 − δ β K > 0 .

The two-dimensional Routh–Hurwitz criterion yields local asymptotic stability. The discriminant of μ 2 − tr ( J * ) μ + det ( J * ) = 0 is tr ( J * ) 2 − 4 det ( J * ), which yields (Equation 6).

Thus, the deterministic model identifies a restoring mechanism only under δ < β K. The subsequent stochastic analysis does not turn this local result into a global safety guarantee; instead, it quantifies the neighbourhood around the equilibrium that remains plausible when noise and jumps persist.

Corollary 1 (Equilibrium depletion flow is α-invariant). At E *, the depletion flow is given by Equation 7:which depends on r , β , δ , K but not on α. A protocol intervention that lowers α raises P * by exactly the offsetting factor and leaves the equilibrium harmful flow unchanged. Welfare-relevant comparisons must therefore use first-passage times to the collapse threshold and tail-risk probabilities, not the equilibrium pressure coordinate alone.

Proof. Substitute D * = δ / β and P * = ( r / α ) ( 1 − δ / ( β K ) ) into α P * D * and observe that the α factors cancel.

Remark 1 (Empirical implication). Corollary 1 shows that controls that only attenuate α leave the equilibrium loss flow unchanged. It therefore motivates comparing them with controls that raise δ or reduce jump amplitudes using first-passage and tail-risk outcomes. The Latin-hypercube sensitivity in Section 9 examines this implication within the selected parameter ranges: δ has Spearman correlation −0.79 with collapse probability, while α has correlation near zero.

6 Controlled positive jump-diffusion model

The deterministic skeleton encodes thresholds but cannot represent the discrete arrival of attack windows. We embed it in the stochastic model. where the integrated form uses the log-state representation that makes positivity manifest. Equivalently, in the standard pre-jump notation,with X D = D, X P = P, μ D ( D , P ) = r ( 1 − D / K ) − α P + u t, and μ P ( D , P ) = β D − δ + v t. The − λ i γ i drift acts as the explicit jump compensator. The processes W D , W P are independent standard Brownian motions; N D , N P are independent Poisson counting processes with constant intensities λ D , λ P > 0 and constant jump amplitudes γ D , γ P satisfying γ i > − 1. The controls u t , v t are progressively measurable and uniformly bounded. Independence of N D and N P is the canonical specification; a coupled-shock variant is examined in Section 9 as a robustness check.

d ⁡ log D t = r 1 − D t − K − α P t − + u t − λ D γ D − 1 2 σ D 2 d t + σ D d W t D + log 1 + γ D d N t D , (8)

d ⁡ log P t = β D t − − δ + v t − λ P γ P − 1 2 σ P 2 d t + σ P d W t P + log 1 + γ P d N t P , (9)

d X t i = X t − i μ i D t − , P t − − λ i γ i d t + σ i X t − i d W t i + X t − i γ i d N t i , (10)

Assumption 1 (Admissibility). ( W D , W P ) and ( N D , N P ) are mutually independent and defined on a filtered probability space ( Ω , F , ( F t ) t ≥ 0 , P ) satisfying the usual conditions. The control processes u t , v t are F t -progressively measurable with sup t , ω ( | u t | + | v t | ) ≤ U < ∞ . The jump amplitudes satisfy γ i > − 1 for i ∈ { D , P }.

6.1 Existence, uniqueness, and positivity

For an integrity state, crossing zero is not a benign numerical or modelling event: it would make a reliability stock economically uninterpretable. The following theorem, therefore, establishes both that the model is well-posed and that the multiplicative shock specification preserves the state space without imposing an artificial boundary correction.

Theorem 2 (Global solution and positivity). Under Assumption 1 and for every initial condition ( D 0 , P 0 ) ∈ ( 0 , ∞ ) 2 , system (Equations 8, 9) admits a pathwise unique strong solution ( D t , P t ) defined for all t ≥ 0 , and P ( D t , P t ) ∈ ( 0 , ∞ ) 2 for all t ≥ 0 = 1.

Proof. Let X t = log D t and Y t = log P t. Applying Itô’s formula with jumps to log yields

d X t = r 1 − e X t / K − α e Y t + u t − λ D γ D − 1 2 σ D 2 d t + σ D d W t D + log 1 + γ D d N t D , d Y t = β e X t − δ + v t − λ P γ P − 1 2 σ P 2 d t + σ P d W t P + log 1 + γ P d N t P .

The drift coefficients are locally Lipschitz on every set { ( X , Y ) : max ( | X | , | Y | ) ≤ R }, and the diffusion coefficients are constant. By [4, Thm. 6.2.3], for every ( X 0 , Y 0 ) ∈ R 2, the system has a unique strong solution up to τ ∞ ≔ lim n τ n, where τ n ≔ inf { t : max ( | X t | , | Y t | ) > n }.

Define η = α / β and V ( D , P ) = D + η P + log ( 1 / D ) + η ⁡ log ( 1 / P ). Then, V ≥ 1 + η on ( 0 , ∞ ) 2 and diverges as D + P + | log ⁡ D | + | log ⁡ P | → ∞. Along the original dynamics,where μ D = r ( 1 − D / K ) − α P + u and μ P = β D − δ + v. The choice η = α / β cancels the D P cross term, the compensators cancel the linear jump expectations, and − r D 2 / K is bounded above. Thus, there are constants c 0 , c 1, depending only on the parameters and U, such that L V ≤ c 0 + c 1 V outside a compact set.

L V D , P = μ D D − μ D + η μ P P − μ P + 1 2 σ D 2 + 1 2 η σ P 2 + λ D γ D − log 1 + γ D + η λ P γ P − log 1 + γ P ,

Dynkin’s formula at t ∧ τ n yields

E V D t ∧ τ n , P t ∧ τ n ≤ V D 0 , P 0 + c 0 t + c 1 ∫ 0 t E V D s ∧ τ n , P s ∧ τ n d s .

Gronwall’s inequality yields the uniform bound ( V ( D 0 , P 0 ) + c 0 t ) e c 1 t. Since V diverges on the boundary of every compact set, P ( τ n ≤ t ) → 0, and hence τ ∞ = ∞ almost surely. Finally, X t , Y t ∈ R for every finite t almost surely, while each jump is the finite shift log ( 1 + γ i ). Thus D t = e X t > 0 and P t = e Y t > 0 for all t ≥ 0.

The result concerns the internal consistency of the state model. It does not imply that a real protocol cannot suffer economic failure; it ensures that such deterioration is represented by low positive integrity, rather than by an invalid negative state created by the model or discretisation.

Lemma 1 (Necessity of multiplicative jumps). Suppose (Equation 8) were modified to use additive jumps X t D = X t − D + η D , with η D ∈ R unrestricted. Then, for any ε > 0 , there exists a probability space and an initial condition with P ( D t < 0 for some t ∈ [ 0 , ε ] ) > 0.

Proof. Fix D 0 = 1, and let the jump distribution put positive mass on { η D < − 2 }. Then, with positive probability, the first arrival of N D occurs in [ 0 , ε ] and produces D t = 1 + η D < − 1 < 0. Multiplicative jumps with γ D > − 1 avoid this failure because D t = D t − ( 1 + γ D ) > 0.

6.2 Local practical mean-square stability

Set ξ t = ( D t − D * , P t − P * ), and let J * be as in (Equation 5). Because Brownian and jump shocks remain active at the deterministic equilibrium, the linearised stochastic system cannot converge in mean square to ξ = 0 unless all noise intensities are zero. The relevant local statement is, therefore, a practical mean-square bound around the deterministic equilibrium. The exact first-order stochastic expansion contains both additive terms evaluated at E * and multiplicative terms proportional to ξ t. For the additive-noise diagnostic used below, we freeze the stochastic coefficients at E * and write, with N ̃ t i = N t i − λ i t,where Σ = diag ( σ D D * , σ P P * ) and Γ = diag ( γ D D * , γ P P * ). For the symmetric positive-definite Lyapunov matrix M solving the continuous-time Lyapunov equation J * ⊤ M + M J * = − I 2, define the constants in Equation 12:

d ξ t = J * ξ t d t + Σ d W t + Γ d N ̃ t , (11)

C ∗ = ‖ M ‖ 2 max D * 2 , P * 2 σ D 2 + σ P 2 + λ D γ D 2 + λ P γ P 2 , a ∗ = λ max M − 1 . (12)

By Theorem 1, J * is Hurwitz whenever δ < β K, so M exists.

The next result makes the operational content of practical stability explicit: it bounds expected squared deviation from the deterministic balance by a contraction term plus a residual noise radius. It, therefore, distinguishes a design that reduces persistent attack uncertainty from one that only changes a transient trajectory.

Theorem 3 (Quantitative practical mean-square bound). Under Assumption 1 and for u t ≡ v t ≡ 0 , the frozen-coefficient additive-noise system (Equation 11) satisfies and therefore In particular, the stochastic linearisation remains in a mean-square neighbourhood of E * whose radius is proportional to the aggregate diffusion and jump variance.

Proof. Freeze the stochastic coefficients of (Equation 10) at E * with controls held at zero and write ξ = ( D − D * , P − P * ). The additive diffusion matrix is Σ = diag ( σ D D * , σ P P * ) and the additive compensated jump matrix is Γ = diag ( γ D D * , γ P P * ). The multiplicative fluctuation terms omitted by this diagnostic are outside the scope of the bound proved here.

By Theorem 1, J * is Hurwitz, so for Q = I 2, the Lyapunov equation J * ⊤ M + M J * = − I 2 has a unique symmetric positive-definite solution M. Applying Itô’s formula with jumps to V ( ξ ) = ξ ⊤ M ξ along (Equation 11) yieldsSince ‖ ξ ‖ 2 ≥ V ( ξ ) / λ max ( M ), L V ( ξ ) ≤ − a ∗ V ( ξ ) + C ∗. Dynkin’s formula and Gronwall’s inequality yield (Equation 13). Finally, λ min ( M ) ‖ ξ t ‖ 2 ≤ V ( ξ t ) yields (Equation 14).

L V ξ = ξ ⊤ J * ⊤ M + M J * ξ + tr Σ ⊤ M Σ + ∑ i ∈ D , P λ i γ i 2 X i , * 2 M i i ≤ − ‖ ξ ‖ 2 + ‖ M ‖ 2 σ D 2 D * 2 + σ P 2 P * 2 + λ D γ D 2 D * 2 + λ P γ P 2 P * 2 ≤ − ‖ ξ ‖ 2 + C ∗ .

The radius in (Equation 14) is a local model diagnostic, not an estimated confidence region for a particular oracle network. Its value is useful because it shows separately how diffusion intensity, jump frequency, and jump size enter the persistent deviation from the deterministic benchmark.

Corollary 2 (Comparative statics of the practical radius). Each of the following protocol-side moves reduces the noise-radius term C ∗ in (Equation 14): (i) lowering either diffusion intensity σ D or σ P ; (ii) lowering either jump intensity λ D or λ P ; and (iii) shrinking the jump amplitudes | γ D | or | γ P | . Changes in δ affect the contraction matrix J * and the Lyapunov matrix M ; in the calibrated numerical ranges of Section 9 , larger δ strengthens mean reversion and lowers simulated collapse risk.

Proof. (i)–(iii) reduce C ∗ directly. The statement about δ is not asserted as a global monotonic theorem because J * depends on δ nonlinearly; it is evaluated numerically in Section 9.

7 Calibration to the oracle-incident corpus

The 24 incidents over 44 months imply a corpus-wide descriptive rate of 0.545 per month or 6.55 per year. To construct a representative high-exposure protocol scenario, we allocate roughly thirty percent of corpus-wide events to one protocol. This allocation is a modelling assumption, not an estimate for a named protocol. It yields the Baseline value λ D = λ P = 0.18 per month; the Stress and Controlled values are 0.40 and 0.10, respectively. The sensitivity analysis varies both jump intensities over a wider range.

The corpus records loss mechanisms and amounts, not direct integrity loss. We, therefore, use γ D = − 0.20 as a moderate integrity-erosion scenario and γ P = + 0.55 as an elevated post-incident pressure scenario, consistent with reported copy-cat activity within 30 days (Zhou et al., 2023). The diffusion intensities σ D , σ P reflect variation in the median-heartbeat range. These are reproducible stress-test inputs rather than fitted structural parameters.

7.1 Bayesian calibration roadmap

The current public extracts do not contain a continuous, provider-labelled panel of feed deviations, staleness, cross-source disagreement, and attack outcomes. The scenario values are, therefore, not presented as Bayesian estimates. With a longitudinal panel, provider- and asset-class-specific observations Y j , t could be used to estimate θ j = ( α j , β j , δ j , σ D , j , σ P , j , λ D , j , λ P , j , γ D , j , γ P , j ) from the posterior in Equation 15:where ϕ contains hierarchical provider and asset-class effects. A Poisson likelihood for event arrivals, a positive-support likelihood for jump magnitudes, and observation models for deviation and staleness would allow informative priors from the incident corpus while preserving uncertainty in the latent states. Posterior predictive checks could compare collapse-probability coverage, first-passage distributions, and feed-level stale-update rates across providers. This analysis requires a longitudinal multi-provider panel and remains outside the scope of the present snapshot.

p θ , ϕ ∣ Y ∝ p Y ∣ θ p θ ∣ ϕ p ϕ , (15)

8 Numerical methods

A standard Euler–Maruyama discretisation of (Equation 10) is not positivity-preserving: the symmetric Brownian increment can drive the integrity state below zero in a single step, requiring an artificial reflection that is structurally incorrect. We use the log-Euler updates in Equations 16, 17: where Z n D , Z n P are independent standard normal draws, Δ N n i ∼ Poisson ( λ i Δ t ), and the per-step drifts, given in Equations 18, 19, include the explicit compensator and Itô correction: The scheme is exact for the diffusion of geometric Brownian motion and applies each Poisson arrival as an exact deterministic multiplicative shock, so the discrete trajectory is positivity-preserving by construction without any clipping. Strong convergence is of order 1 / 2 jointly in the diffusion and jump components (Platen and Bruti-Liberati, 2010; Bruti-Liberati and Platen, 2007).

log D t n + 1 = log D t n + μ n D Δ t + σ D Δ t Z n D + log 1 + γ D Δ N n D , (16)

log P t n + 1 = log P t n + μ n P Δ t + σ P Δ t Z n P + log 1 + γ P Δ N n P , (17)

μ n D = r 1 − e log D t n / K − α e log P t n + u t n − λ D γ D − 1 2 σ D 2 , (18)

μ n P = β e log D t n − δ + v t n − λ P γ P − 1 2 σ P 2 . (19)

We use a step size Δ t = 0.1 months (≈ 3 days), a horizon of 120 months (1,200 steps), and 4,000 sample paths per scenario. Initial conditions are D 0 = 0.72 and P 0 = 0.28 with capacity K = 1. Collapse is defined as the event { D t < 0.25 for some t ≤ T }. The seed is fixed at 20260501 for reproducibility. Collapse-probability point estimates are reported with two-sided 95% Clopper–Pearson exact binomial confidence intervals.

Here, “positivity-preserving” refers to preservation of the model’s analytically admissible state space: the simulated integrity and pressure states remain strictly positive without clipping or reflection. It does not mean that raw feed data are error-free or that the numerical output preserves every data feature. This distinction matters for interpretation because a negative state produced by an unconstrained discretisation would be a numerical artefact, whereas a low positive state represents severe but economically interpretable integrity deterioration.

9 Numerical experiments

The simulations assess whether the combined Stress configuration raises collapse incidence and shortens first-passage time, whether the Controlled configuration reduces collapse incidence, and whether pressure decay δ has a stronger rank association with collapse probability than depletion intensity α. They are scenario diagnostics rather than causal estimates from protocol-level data.

9.1 Three-scenario comparison

Table 8 reports the three scenarios. The Baseline produces a 10-year collapse probability of 10.1% (95% CI [9.2, 11.1]%); the Stress regime, which doubles the jump intensity, raises depletion sensitivity, and lowers pressure decay, raises the collapse probability to 93.0% (CI [92.2, 93.8]%); the Controlled regime, which halves the jump intensity, raises pressure decay to δ = 0.82, and adds a small positive integrity-side control drift, produces no realised collapses among 4,000 paths. Because zero observed events do not imply zero risk, the one-sided 95% upper bound for the Controlled regime is reported as 0.09%.

TABLE 8. Monte Carlo jump-diffusion experiment with the structure-preserving log-Euler scheme. Collapse is defined as D t < 0.25 at least once over the simulated horizon. CIs are 95% Clopper–Pearson exact intervals.

The first-passage moments in Table 8 provide the scenarios on an operational scale. Conditional on collapse, the median first-passage time falls from 62 months in the Baseline to 30 months in the combined Stress configuration. These numbers are not loss forecasts or service-level commitments for a named protocol. They indicate that, under the stated inputs, a monitoring and response process designed for a multi-year deterioration can be inadequate when attack opportunities cluster and pressure decays more slowly.

Figures 4, 5 are the principal output views: they show path distributions, the collapse threshold, and uncertainty intervals rather than only algorithmic specifications. The workflow diagram in Figure 6 connects these outputs to the public data and simulation stages.

9.2 Latin-hypercube sensitivity analysis

We sample 256 designs over the eight-dimensional box α ∈ [ 0.20 , 0.70 ], β ∈ [ 0.60 , 1.20 ], δ ∈ [ 0.40 , 1.00 ], r ∈ [ 0.40 , 1.00 ], σ D ∈ [ 0.05 , 0.20 ], σ P ∈ [ 0.08 , 0.25 ], λ D ∈ [ 0.05 , 0.40 ], and λ P ∈ [ 0.05 , 0.40 ], holding controls at zero and using the same log-Euler scheme on a reduced inner sample (600 paths, 600 steps) to keep total compute tractable. We then compute Spearman rank correlations between each parameter and (i) the collapse probability and (ii) the mean minimum integrity over the horizon.

Table 9 reports the parameter–outcome rank correlations that are plotted in Figure 7.

TABLE 9. Latin-hypercube sensitivity. Spearman rank correlations between each parameter and (i) collapse probability and (ii) mean minimum integrity. 256 designs.

The pressure-decay rate δ has a Spearman correlation of −0.79 with collapse probability and + 0.74 with mean minimum integrity; within this design, it is the largest-ranked driver. The pressure-response coefficient β ranks second. The integrity rebuild rate r and the diffusion intensities σ D , σ P contribute moderately. The depletion intensity α has a Spearman correlation near zero with both outcomes, while the prey-side jump intensity λ D is weakly associated with outcomes when λ P is also varied. This supports P3 within the specified parameter box and equilibrium-centred simulation design. It should not be read as evidence that α is irrelevant outside that setting.

9.3 Robustness

The robustness checks leave the scenario ordering unchanged. Replacing independent jumps with a coupled shock, in which one Poisson process drives both D and P, raises the Baseline collapse probability by roughly 1.5 percentage points and the Stress probability by roughly 0.5 percentage points; both differences are within twice the Clopper–Pearson half-width. Halving the diffusion intensities while doubling the jump intensities, thereby approximately preserving the variance contribution, also preserves the ordering but narrows the gap between Baseline and Controlled.

Table 10 reports the month-level resampling intervals. We bootstrap the 44 monthly incident counts with 100,000 fixed-seed month-level resamples. The observed corpus-wide rate is 0.545 incidents per month, with a 95% bootstrap interval of [0.318, 0.841]; applying the stated 30% representative-protocol allocation yields [0.095, 0.252] per month, which contains the Baseline value λ D = λ P = 0.18. The interval quantifies uncertainty in the calibration anchor, not predictive validity. An out-of-sample analysis would fit an earlier window, hold out later months, and score posterior-predictive event probabilities and first-passage outcomes; the current corpus lacks the feed-level outcome labels needed for that exercise.

TABLE 10. Month-level bootstrap robustness for the incident-rate calibration. The 44 monthly counts are resampled with replacement for 100,000 fixed-seed replicates; the 30% allocation is the scenario assumption used for the representative protocol.

10 Protocol implications

Within the model, the α-invariant equilibrium flow and the sensitivity ranking direct attention to post-event pressure decay and jump size alongside local attenuation of integrity loss. Faster oracle-aware governance, liquidity backstops, and public-information defences can affect pressure decay; source independence, dispute windows, and circuit breakers can reduce integrity-shock size. These mechanisms involve trade-offs: source independence can increase per-update gas, dispute windows can delay valid updates, and circuit breakers can freeze settlement during genuine market moves. Protocol-specific evaluation is therefore required before implementation.

Governance is also asset-class specific. Stablecoin systems place greater weight on peg, reserve, and stale-price checks, whereas derivative systems add index construction, mark-price, funding, and liquidation-settlement dependencies. The present model treats these channels through common latent states, but the Bayesian extension should estimate class-specific parameters and controls rather than assume that a stablecoin and a derivative venue share the same attack-pressure decay or jump-amplitude distribution.

11 Threats to validity

11.1 Construct validity

The integrity and pressure states are latent. The public inputs are calibration anchors with different meanings: feed metadata describe design features, TVL describes potential exposure, and the incident corpus describes realised events. They need not move together and do not identify the functions g and h in (Equations 1, 2). Alternative coverage measures could weight collateral-asset value rather than feed count, and direct measures of staleness, cross-source disagreement, or borrowable manipulation capital could alter the mapping.

11.2 Internal validity

The practical mean-square theorem is local to the linearised model and gives a stochastic neighbourhood around the deterministic equilibrium rather than convergence to the deterministic equilibrium itself. Global stochastic stability would require a stronger Lyapunov function, possibly of the form V ( D , P ) = D + P + a ⁡ log ( 1 / D ) + b ⁡ log ( 1 / P ) with parameter-dependent ( a , b ). The numerical experiments confirm but do not prove global stability outside neighbourhoods of E *.

11.3 External validity

The incident corpus is restricted to 2020–2023 documented losses, almost entirely on EVM-compatible chains, whereas the feed metadata and TVL panels are 2026 snapshots. Calibration to the post-2024 oracle environment, in which pull-based oracles such as Pyth (Pyth Network, 2023) and risk-managed feeds such as Redstone gain market share, would alter the scenario inputs. Asset-class heterogeneity (Table 6) suggests that crypto-feed and RWA-feed regimes deserve separate calibration. The comparison in Table 11 limits transferability claims: Chainlink estimates are not assumed to apply mechanically to Pyth or UMA.

TABLE 11. Cross-provider oracle architecture comparison used to assess external validity.

11.4 Statistical validity

The Controlled-scenario point estimate of 0% collapse probability over 4,000 paths provides an upper Clopper–Pearson bound of 0.09%. We report this bound explicitly. The Latin-hypercube sensitivity uses 256 designs; doubling the design count would tighten Spearman estimates by approximately a factor of 2 but would not establish causal parameter importance. The reported ranking is conditional on the selected parameter ranges, simulation seed, and specification. In particular, the allocation of roughly thirty percent of corpus-wide events to the representative protocol is a scenario assumption, assessed through variation in λ D and λ P, not a statistically estimated share.

11.5 Numerical validity

The log-Euler scheme is positivity-preserving by construction and converges with strong order 1 / 2 in the joint diffusion-jump norm. We have checked that halving the time step changes mean-path summaries by less than one percent of the corresponding standard error.

11.6 Predictive validation

The bootstrap interval quantifies uncertainty in the descriptive incident-rate anchor, but it does not establish predictive validity. The proposed Bayesian and hold-out workflow is the appropriate next step once longitudinal multi-provider feed observations and labelled attack outcomes are available.

12 Conclusion

Oracle integrity is a security constraint because DeFi contracts execute against reported data. The model couples integrity and attack pressure in a positive jump-diffusion, with scenario inputs anchored to documented incidents and public oracle data. Within the sampled equilibrium-centred design, pressure decay δ has the largest rank association with simulated collapse probability, while the equilibrium harmful flow is invariant to α (Corollary 1). These results guide scenario analysis; they do not rank real-world controls causally.

The framework also separates two practical design margins. Measures that shorten the period in which manipulation remains profitable act on the decay of attack pressure, whereas source diversification, dispute mechanisms, and circuit breakers can reduce the size of an integrity shock. A protocol should assess these measures jointly, since faster intervention, higher redundancy, and more restrictive circuit breakers each impose operational costs. First-passage time and tail-risk measures provide a more informative basis for that assessment than the equilibrium pressure coordinate alone.

The next empirical step is to add per-feed deviation and staleness series, then estimate parameters separately by provider and asset class. This would allow the scenario inputs used here to be replaced progressively with feed-level measurements and would clarify whether the model’s comparative-static patterns persist across oracle designs and market regimes. The Mango sequence supplies an interpretable event trace, while provider-specific Bayesian estimation, month-level bootstrap calibration, and held-out posterior-predictive checks provide a route for testing whether the simulated analytics extend beyond the Chainlink-anchored scenarios.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fbloc.2026.1942124/full#supplementary-material

SUPPLEMENTRAY DATASHEET 1 Supplementary material at the source

Supplementary material Supplementary material at the source

Footnotes

Edited by: Xiaoqi Li, Hainan University, China

Reviewed by: Daitri Tiwary, Birla Institute of Technology and Science, India

R. Durga, Vels Institute of Science Technology & Advanced Studies, India