Applied Physics & Mathematical Frameworks

Mathematical Physics & Catastrophe Actuarial Engineering.

Nine mathematical models mapping physical control loops, thermodynamic phase transitions, epidemiological vulnerability spread, and heavy-tailed catastrophe risk.

Explore 9 Physics FrameworksActuarial Engine Specification
Actuarial Risk Framework

Actuarial Cyber Catastrophe Engine

Portfolio risk modeling built on 4-module catastrophe frameworks, Exceedance Probability (EP) loss curves, Rotated 90° Clayton Copulas, and Lloyd's Market Bulletin Y5381 war exclusion filters.

9 Applied Physics Models

9 Core Physics & Mathematical Frameworks

Comprehensive catalogue of open-access physics models, equations, and control loop equations spanning cyber-physical systems.

Actuarial Modules
4-Stage Quantitative Pipeline
Physics Suite
9 Mathematical Frameworks
Underwriting Standard
Lloyd's Bulletin Y5381 Aligned
Open scientific models & reproducible LaTeX derivations
Actuarial Catastrophe Modeling

Catastrophe-Grade Cyber-Physical Actuarial Engine

Portfolio risk modeling built on 4-module catastrophe frameworks, Exceedance Probability (EP) loss curves, Rotated 90° Clayton Copulas, and Lloyd's Market Bulletin Y5381 war exclusion filters.

Actuarial Copula Engine Specification

Clayton Copula Tail Dependence & ALE Algorithm

Cθ(u,v)=(u−θ+v−θ−1)−1/θ,λL=2−1/θC_\theta(u, v) = \left(u^{-\theta} + v^{-\theta} - 1\right)^{-1/\theta}, \quad \lambda_L = 2^{-1/\theta}

Conventional cyber insurance relies on independent Bernoulli trials. In physical infrastructure, operational breaches exhibit lower tail dependence. When one turbine or SCADA controller trips under stress, adjacent control loops experience immediate cascading failure.

Clayton Copula Tail Index
λL=2−1/θ>0\lambda_L = 2^{-1/\theta} > 0
War Exclusion Standard
Lloyd's Bulletin Y5381 Enforced
ATTACHMENT POINTLOSS AMOUNTP(EXCEEDANCE)
ALE · exceedance curve
EP loss vs. attachment

4-Module Catastrophe Architecture

  • 1. Hazard Module

    Simulates adversary perturbation parameters and physical threat vectors.

  • 2. Vulnerability Module

    Maps telemetry against EPSS vulnerability scoring and SCADA topology.

  • 3. Financial Exposure Module

    Calculates asset replacement values and business interruption outage costs.

  • 4. Insurance Loss Engine

    Outputs Exceedance Probability (EP) curves, attachment points, and policy limits.

Open Scientific Research Catalogue

Applied Physics & Mathematical Frameworks

9 non-linear physical risk models, algebraic topology frameworks, and statistical thermodynamics algorithms implemented across the Eigenia digital twin engine.

MODEL 01

GGNN Directed Graph Topology

Graph Neural Networks & Physical Topology
GRU UPDATE
GGNN-01 · directed graph
7-layer propagation
hi(t)=GRU(hi(t−1),∑j∈N(i)Weij)h_i^{(t)} = \text{GRU}\left(h_i^{(t-1)}, \sum_{j \in \mathcal{N}(i)} W e_{ij}\right)

7-layer directed graph neural network mapping physical SCADA/PLC control loops, lateral movement trajectories, and real-time vulnerability propagation across OT topologies.

Deliverables & Proofs:
•7-Layer Gated Graph Neural Network Topology
•Dynamic Lateral Movement Trajectory Sampling
•Real-Time SCADA/PLC Vulnerability Mapping
Read Mathematical Specification
MODEL 02

L0/L1 KL Divergence Asset Drift

Information Divergence & Asset Drift
SENSOR VALUEDENSITYDRIFT GAP
L0/L1-02 · KL divergence
design vs. live telemetry
DKL(PL1∥PL0)=∑xPL1(x)log⁡PL1(x)PL0(x)D_{\text{KL}}(P_{\text{L1}} \parallel P_{\text{L0}}) = \sum_{x} P_{\text{L1}}(x) \log \frac{P_{\text{L1}}(x)}{P_{\text{L0}}(x)}

Kullback-Leibler divergence metric measuring calibration drift between Platonic design datasheets (L0) and live physical sensor telemetry (L1).

Deliverables & Proofs:
•DEXPI 2.0 Platonic Datasheet Schema Validation
•Sensors & Actuators Calibration Drift Detection
•Minimum Operational Requirements (MOR) Index
Read Mathematical Specification
MODEL 03

McKenney-Lacan Psychometric Tensor

Lacanian Four Discourses & RSI Triad
REALSYMBOLICIMAGINARYADVERSARY TARGET PROFILE
LACAN-03 · RSI triad
adversary targeting
Pi=[DISC]⊗[OCEAN]=[DISC]⊗[OCEAN]P_i = [\text{DISC}] \otimes [\text{OCEAN}] = \begin{bmatrix} D & I \\ S & C \end{bmatrix} \otimes \begin{bmatrix} O & C & E & A & N \end{bmatrix}

The McKenney-Lacan Calculus forms the mathematical core of L4, mapping topological cognitive dissonance across Real, Symbolic, and Imaginary registers via a 20-dimensional Kronecker psychometric tensor.

Deliverables & Proofs:
•Lacanian Four Discourses Threat Typology Profiling
•Real/Symbolic/Imaginary Structural Target Prediction
•Psychometric Tensor Alignment Matrix
Read Mathematical Specification
MODEL 04

Ising Phase Transition Security Model

Statistical Thermodynamics & Crisis Culture
DISORDEREDORDERED PHASE
ISING-04 · phase transition
culture order parameter
Hinteraction=−∑⟨i,j⟩Jijσiσj−h∑iσi\mathcal{H}_{\text{interaction}} = -\sum_{\langle i,j \rangle} J_{ij} \sigma_i \sigma_j - h \sum_i \sigma_i

Statistical thermodynamics Ising spin model measuring correlated team dynamics, security culture phase transitions, and Granovetter cascading threshold failure.

Deliverables & Proofs:
•Ising Phase Transition Security Culture Model
•Emergency Response Team Consonance Metrics
•Granovetter Cascading Critical Cut Identification
Read Mathematical Specification
MODEL 05

Kramers Potential Barrier Escape Model

Topological Risk & Time-to-Compromise
ΔEREACTION COORDINATEPOTENTIAL ENERGY
KRAMERS-05 · ΔE barrier
MTTC = 1/k · escape rate
k=Aexp⁡(−ΔEkBT),MTTC=1kk = A \exp\left(-\frac{\Delta E}{k_B T}\right), \quad \text{MTTC} = \frac{1}{k}

Transition state theory modeling stochastic attacker escape across topological energy barriers (ΔE) under threat temperature (k_B T), yielding Mean Time to Compromise (MTTC).

Deliverables & Proofs:
•Mean-Time-to-Compromise (MTTC) Epoch Calculation
•Topological Defensive Energy Barrier ΔE Hardening
•Stochastic Monte Carlo Walk Weighting
Read Mathematical Specification
MODEL 06

SIR Compartmental Vulnerability Spreading

Epidemic Kinetics & Vulnerability Spreading
SSUSCEPTIBLEIINFECTEDRRECOVEREDβγ
SIR-06 · contagion flow
R₀ = βS₀/γ
dSdt=−βSI,dIdt=βSI−γI,R0=βS0γ\frac{dS}{dt} = -\beta S I, \quad \frac{dI}{dt} = \beta S I - \gamma I, \quad R_0 = \frac{\beta S_0}{\gamma}

Epidemiological kinetic model quantifying rapid vulnerability propagation across interconnected SCADA networks, where R_0 > 15 triggers tipping state alerts.

Deliverables & Proofs:
•Network Contagion Threshold R_0 Rate Calculation
•Interconnected Asset Infection Path Mapping
•Tipping Point Early Warning Trigger
Read Mathematical Specification
MODEL 07

Clayton Copula Tail Dependence & ALE

Heavy-Tailed Reinsurance & Catastrophe Underwriting
0uvJOINT TAIL RISK
CLAYTON-07 · tail dependence
λL = 2⁻¹ᐟᶿ
Cθ(u,v)=(u−θ+v−θ−1)−1/θ,λL=2−1/θ>0C_\theta(u, v) = (u^{-\theta} + v^{-\theta} - 1)^{-1/\theta}, \quad \lambda_L = 2^{-1/\theta} > 0

Bivariate lower-tail copula capturing non-linear joint risk dependence and Aggregate Loss Exceedance (ALE) under Lloyd's Y5381 physical war exclusions.

Deliverables & Proofs:
•Clayton Copula Systemic Dependency Loss Matrix
•Aggregate Loss Exceedance (ALE) Catastrophe Curve
•Lloyd's Y5381 Physical State Exclusion Validation
Read Mathematical Specification
MODEL 08

Hawkes Self-Exciting Cascade Process

Correlated Incident Cascades & Aftershocks
t₀t₁TIMEINTENSITY λ(t)
HAWKES-08 · self-excitation
λ(t) triggered cascade
λ(t)=μ(t)+∑ti<tαe−β(t−ti)\lambda(t) = \mu(t) + \sum_{t_i < t} \alpha e^{-\beta(t - t_i)}

Self-exciting point process modeling temporal clustering of physical security breaches, where one incident elevates the probability of subsequent cascade events.

Deliverables & Proofs:
•Temporal Incident Cluster Intensity λ(t)
•Secondary Aftershock Impact Probability
•Cascading Control Loop Failure Forecasting
Read Mathematical Specification
MODEL 09

Pareto Fat-Tailed Peak-Over-Threshold (POT)

Extreme Value Theory & Black Swan Risk
THRESHOLDBLACK SWANLOSS MAGNITUDEPROBABILITY
PARETO-09 · fat tail
EVT vs. Gaussian
TVaRq(X)=αα−1xm(1−q)1/α,X=xmin⁡(1−U)−1/α\text{TVaR}_q(X) = \frac{\alpha}{\alpha - 1} \frac{x_m}{(1-q)^{1/\alpha}}, \quad X = x_{\min} (1 - U)^{-1/\alpha}

Asymptotic tail loss quantification overriding Gaussian assumptions with fat-tailed Pareto distributions where 80% of damage stems from 1% of Black Swan events.

Deliverables & Proofs:
•Tail Value-at-Risk (TVaR) Asymptotic Exceedance
•Heavy-Tailed Pareto Loss Index α Computation
•Extreme Value Theory (EVT) Black Swan Simulation
Read Mathematical Specification
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