An interactive diagnostic for complex systems.
Three inputs — error rate, process depth, oversight allocation — produce a failure mode classification, numerical intervention paths, and a phase diagram showing where your system sits relative to a mathematically derived safety threshold.
Every complex system splits its resources between doing work and checking work. This tool computes the optimal split, compares it to your system's actual allocation, and classifies the failure mode — the specific way the system is likely to break.
The diagnostic separates three structurally different situations:
- Under-maintained — the system spends less on oversight than its complexity requires. Expect sudden, correlated failures.
- Over-maintained — the system spends more on coordination than necessary. Overhead consumes productive capacity.
- Structurally stressed — the system is too complex to be made safe by any overhead allocation. The only fix is to simplify.
The tool computes exact numerical targets: how much to increase oversight, or how much to reduce error rates or process depth to cross below the safety threshold.
Most risk frameworks — FMEA, safety culture assessments, risk matrices — implicitly assume that with enough oversight, any system can be made safe. This tool identifies the threshold where that assumption breaks. Above it, adding more checking stops helping and simplifying the system becomes the only option.
Eight real-world systems are included as calibration points, with confidence tiers:
| System | λ | Classification | Confidence |
|---|---|---|---|
| Soviet Nuclear Program (1986) | 0.900 | Critical | Low |
| Boeing 737 MAX (pre-grounding) | 0.660 | Critical | Moderate |
| Ottoman Timar System (late) | 0.500 | Fragile | Low |
| German Hospitals (CIRS) | 0.360 | Under-maintained | Moderate |
| TCP/IP Protocol Stack | 0.060 | Optimal | High |
| Big Tech (mature) | 0.105 | Optimal | Moderate |
| Queuing System (ρ=0.7) | 0.125 | Optimal | High |
| Commercial Aviation (nominal) | 0.160 | Optimal | High |
Parameters are analytical estimates from published data, not direct measurements. The diagnostic is calibrated against observed outcomes.
The model optimizes effective throughput:
T_eff(κ) = (1 − κ) · exp(−λ/κ)
where λ = ε₀ · d is failure pressure (error rate × depth) and κ is maintenance overhead.
The optimal overhead η*(λ) = (−λ + √(λ² + 4λ)) / 2 and the phase transition at λ = 1/e ≈ 0.368 emerge from this equation. The exponential reliability form is the unique function satisfying four natural axioms (no free lunch, perfect verification, scale invariance, constant log-efficiency).
Validated across 75 systems. Full derivation, proofs, and empirical validation in the Semantic Tax research program (Brandes, 2025 — SSRN).
git clone https://github.com/AMBRA7592/semantic-stress-diagnostic.git
cd semantic-stress-diagnostic
npm install
npm run devConnected to Vercel for automatic deployment from main.
MIT
Amadeus Brandes — Independent analyst. Systems theory and complexity science applied to organizational diagnostics and infrastructure dependencies.