Apex AnalyticaAPEX ANALYTICA
// FRAMEWORKmanifold.omega_f

A canonical scoring framework for systemic fragility.

Ω-Fragility (ΩF) is a 0–10 score for any node in a critical system, built from five canonical pillars under configurable weights.

FORMULAΩF = wI·I + wR·R + wJ·J + wC·C + wT·Twhere Σw = 1.0
ΩF PROFILE · SAMPLE NODE
NODE_ID 0x7A3E
246810IIRREPLACEABILITY9.0RRESTORATION LATENCY8.0JJURISDICTIONAL HAZARD4.0CCASCADE LOAD7.0TTAIL DEPTH6.0ΩF7.42
// THE FIVE PILLARS
I · R · J · C · T
I
PILLAR · I
IRREPLACEABILITY

How impossible to substitute, technically or commercially.

Captures the true cost of substitution: technical compatibility, certification windows, contractual lock-in, and supplier concentration. A node with I=10 cannot be replaced on any meaningful timeline.

BACKED · PEARL
R
PILLAR · R
RESTORATION LATENCY

Time to restore equivalent capacity after catastrophic failure.

Models the recovery clock: lead times, capital requirements, regulatory approvals, and skill availability. A node with R=10 has effectively no near-term recovery path — failure is permanent on operational horizons.

BACKED · PEARL
J
PILLAR · J
JURISDICTIONAL HAZARD

Sanctions, conflict, export controls, and regulatory exposure.

Encodes geopolitical and regulatory exposure: sanctions risk, conflict zones, export controls, dual-use restrictions. A node with J=10 sits at the intersection of multiple hostile regimes or under a single decisive lever.

BACKED · TARSKI
C
PILLAR · C
CASCADE LOAD

Downstream impact depth: GDP, sectors, nonlinear propagation.

Quantifies the systemic blast radius: how many downstream nodes lose function, how deep the cascade runs, what nonlinear amplifiers (panic, hoarding, deleveraging) kick in. A node with C=10 is load-bearing for the system itself.

BACKED · SPIRTES + PARETO
T
PILLAR · T
TAIL DEPTH

Distributional depth beyond VaR — fat-tail severity.

Goes past Value-at-Risk to characterize the depth of the tail: how bad is the bad case, conditional on being bad. Heavy-tailed processes routinely produce realizations multiple sigmas beyond historical VaR.

BACKED · PARETO
// COMPOSING ΩF
WEIGHTS PER DOMAIN
PILLAR WEIGHTS · EXAMPLE
Σw = 1.00
I
R
J
C
T
I
w = 0.25
R
w = 0.20
J
w = 0.15
C
w = 0.25
T
w = 0.15

A reinsurer evaluating sovereign credit cares about J and T. A supply-chain operator cares about I and R. The same five pillars, reweighted to your risk model.

// SYSTEM-LEVEL METRICS
ΩSF · ΩSX · CONTAGION · BUFFER
ΩSF
SYSTEM FRAGILITY
Σᵢ αᵢ·ΩFᵢ

Throughput-weighted system fragility. Each node contributes proportional to its share of system flow.

ΩSX
SYSTEM EXPOSURE
Σᵢ eᵢ·ΩFᵢ

Exposure-weighted system fragility. Weighted by sanctions, geographic concentration, capital risk.

Ω-Contagion
CASCADE REACH
|{j : Lⱼ(failᵢ) > τ}|

Number of downstream nodes whose loss exceeds threshold τ when node i fails.

Ω-Buffer
TIME TO FAILURE
τ_buffer(i)

Time between a node's failure and the onset of systemic failure. The operational window for response.

// SEE IT IN PRACTICE

Watch ΩF scoring on a live graph.

Trial accounts include the full pillar breakdown, configurable weights, and system-level aggregation.