Three-Domain Numerical Simulation
Semantic · Byzantine · IC Coupling

Key Simulation Numbers

Synthetic scenario: 4-replica LLM cluster, N=5 network vantages, 300-tick joint timeline. All values reproducible via scripts/simulate_three_domains.py.

Three Domains, One Algebra

Part A — Semantic Coherence: C₄ Detects What D² Cannot

The defining result of Domain 2: during the COHERENT_BUT_FALSE (CBF) window (ticks 120–180, 60 minutes), the classic Mahalanobis distance \(D^2\) remains below the WATCH threshold (7.81), because \(C_1 \approx C_2^{W_2} \approx C_3^{CKA} \approx 1\) — all replicas agree. They agree on the wrong answer. Only \(C_4\) (falsifiability coherence) falls to 0.359, triggering the lateral CBF label.

Figure A. Semantic coherence simulation. N=4 LLM replicas, T=360 min. Red shading: CBF window (ticks 120–180). Orange: ECMP cache miss (ticks 200–240). Top panel: all four axes. Middle-left: D² — note it stays below WATCH during CBF. Middle-right: C₄ close-up — minimum 0.359. Bottom: phase labels; CBF (pink lateral band) is entirely invisible to the main Φ_K classifier.

Part B — Byzantine Robustness: Geometric Median vs Arithmetic Mean

When vantage V₅ turns Byzantine at \(t=60\) (injecting a concentrated distribution over AS64500, unseen in honest traffic), the arithmetic-mean estimator produces \(D^2 = 327\) — a massive false alarm. The geometric-median estimator \(C_2^{gm}\) stays near BAU because its breakdown point is \(\lfloor N/2 \rfloor / N = 0.50\) vs \(1/N = 0.20\) for the mean. The minimax detector then attributes the alarm to V₅ with ratio \(\Delta_{byz}/D^2 = 0.995 > \theta_{byz} = 0.60\).

Figure B. Byzantine robustness simulation. N=5 vantages, f=1 Byzantine (V₅ from t=60). Left: C₂ standard (red) collapses to 0.498 while C₂^gm (green) stays robust at 0.320. Middle: D² standard fires a CRITICAL alarm; D²^mm(1) stays near zero after removing V₅. Right: Δ_byz/D² ratio crosses θ=0.60 at t=60, triggering SUSPECTED_BYZANTINE. Cascade time τ_C(0.5) ≈ 27.5 s (BGP MRAI SIR model).

Part C — Infrastructure-Cognitive Coupling: The Poincaré Three-Body Problem

Poincaré showed that three gravitationally coupled bodies cannot be solved in closed form — the system exhibits sensitive dependence on initial conditions. The Infrastructure-Cognitive (IC) coupling has the same character: network and AI coherence vectors interact through \(R_{cross} \neq 0\), and Phase 3 (COUPLED) events arise that are invisible to standalone monitors watching only x_net or only x_AI. The joint monitor watching \(z(t) \in [0,1]^6\) detects them.

Figure C. IC Coupling simulation. T=300 ticks. Top-left: x_net(t) — network axes. Top-right: R_cross heatmap (‖R_cross‖_F = 0.193 > 0, rejecting independence). Middle-left: x_AI(t) — note C₂^W2 lags C₃^net by ≈10 ticks (drift transfer function). Middle-right: Drift transfer function — ΔC₂^W2 ≈ −σ²_drift·‖ΔQ‖₁·L̄_s/W₂_max. Bottom: D²_net (cyan), D²_AI (pink), D²_joint (gold) — Phase 3 events visible only in D²_joint.

IC Phase Diagram

BE-MVPS — The Bandwidth-Efficient Variant: Hard Numbers, Honest Trade-off (formerly labelled "FMVPS / Fast")

The natural follow-up question: could MVPS be made absurdly fast? Below are real wall-clock benchmarks of three architectures on N=1000 vantages across 6 scenarios. All values measured with time.perf_counter(), code in scripts/benchmark_fmvps_vs_ml.py.

Three Architectures Compared

Per-Scenario Results

Measured wall-clock latency, throughput, and detection lag. Detection lag = ticks between scenario onset (t=80) and first alarm fired. 1 tick = 60 s in operational time.

Latency Benchmark

Figure E. Left: mean wall-clock latency per tick (log scale) across 6 scenarios. Right: throughput in ticks/second. FMVPS sits between ML-classic (slowest) and MVPS-classic (fastest pure-compute), while MVPS-classic does not support edge gating or sparse bandwidth.

Detection Lead Time Matrix

Figure F. Detection lag in seconds per scenario per architecture. Greener = faster detection. ML-classic misses the events that matter most (BAU stability, network anomaly, Phase 3 COUPLED). MVPS-classic and FMVPS catch every detectable event. The Byzantine row remains MISSED for both MVPS variants in this scenario because f=1 vantage out of 1000 (0.1%) is below the minimax cell-removal threshold — a known limitation requiring adversary fraction f ≥ 1/cells ≈ 10% for the geometric-median estimator to engage (this matches the breakdown-point theorem).

Scaling Behaviour

Figure G. Wall-clock latency per tick vs vantage count N (BAU state, log-log). ML-classic scales O(N·W) — at N=10 000 it costs 7.2 ms per tick. MVPS-classic scales O(N·d²) — 448 μs at N=10 000. FMVPS scales O(N) amortized — 914 μs at N=10 000. Crucially, FMVPS bandwidth grows linearly only in the changed vantage subset, not in N itself.

Where BE-MVPS Actually Wins (Honest Summary)

Interpretation — without numbers, "fast MVPS" is speculation. With numbers, the picture is precise: FMVPS pays a small CPU overhead in exchange for 25× bandwidth reduction, deployability at the edge, and the same detection capabilities. The right architecture is not "FMVPS replaces MVPS-classic" — it is "FMVPS at the edge, MVPS-classic at the broker": edge agents gate and aggregate locally; broker runs the dense joint algebra only on the aggregated cell sketches.

How Much Earlier Does MVPS + LM Detect?

The fundamental difference: an LM alone observes what the system produces. MVPS observes the geometry of the space where the system operates. These are orthogonal observation planes — which is why the combination detects what neither sees alone.

All values from scripts/simulate_three_domains.py, seed fixed (42/7/99), deterministic. CBF and Phase 3 rows represent qualitative new detection capability, not just speed.

Research Venue Map

The three domains naturally map to three IETF/academic venues. Each builds on the previous without algebraic changes — only the type of vantage changes (Poincaré's "same name").

The mysterious here is the Phase 3 event: a failure that no single instrument can see, detectable only when the full \([0,1]^6\) joint space is monitored. This is not a paradox — it is what happens when two complex systems couple. The mathematics gives it a name. The simulation makes it visible.

Reproducibility

# Clone and run locally (Python 3.10+, numpy, matplotlib, scipy):
git clone https://github.com/melegassi/catellix  # or download scripts/
python scripts/simulate_three_domains.py

# Output:
#   docs/figures/sim_partA_semantic.png
#   docs/figures/sim_partB_byzantine.png
#   docs/figures/sim_partC_coupling.png
#   docs/figures/sim_joint_3d.png
#   docs/SIM_NUMERICAL_RESULTS.txt

# All random seeds are fixed (RNG seed=42,7,99).
# Results are deterministic across platforms (IEEE 754).

Simulation date: 2026-05-21 · Framework: draft-melegassi-mvps-ai-coherence-00 §5–§10 · All values synthetic; no real network or AI infrastructure used.