GMC-Autopilot: Provably Safe Invariance of Hybrid Cyber-Navigation in Contested Topologies

Dr. Panda Garagara

Independent Researcher, Institute of Bamboo and Cyber-Aerodynamics

Secure Communications: panda_garagara@proton.me

Zenodo doi: https://doi.org/10.5281/zenodo.20710474 / https://doi.org/10.5281/zenodo.20772226 / https://doi.org/10.5281/zenodo.20805743 / https://doi.org/10.5281/zenodo.21195161

I. Control-Theoretic Paradigm vs. Heuristic Empiricism

Modern distributed edge network topologies rely heavily on high-capacity, non-convex empirical function approximators (e.g., neural network behavioral classifiers) to regulate stateful resource allocation and boundary access. When subjected to non-stationary parameters and adversarial step-shocks, these trial-and-error machine learning models suffer from inherent causal feedback latency, out-of-distribution hallucinations, and catastrophic boundary violation cascades.

We leave hyperparameter tuning, empirical backoff patching, and random-seed hunting to the heuristics community.

GMC-Autopilot establishes a mathematically rigorous alternative. We demonstrate that under closed-loop output feedback, any unmodeled, non-cooperative classification barrier can be projectively mapped into a set of trivial linear operators. Trajectory safety is formalized as a robust topological property of the system's state space—to be mathematically guaranteed via forward invariance, not a probabilistic gamble on gradient descent.

II. Dual-Loop Output-Feedback Architecture

Multi-node routing and transaction scheduling are formalized as an Asymmetric Temporally Decoupled Hybrid System. To resolve real-time safe control inputs while eliminating algebraic feedback loops within a single computational epoch, we deploy a hierarchical dual-loop control interface:

  ┌──────────────────────────────────────────────┐
  │      Outer Loop: Macro-Temporal Planner      │
  │  Resolves ATV-TSP via Monte Carlo Tree Search│
  └──────────────────────┬───────────────────────┘
                         │
                         │ Feedforward Command: u_nominal(k)
                         ▼
  ┌──────────────────────────────────────────────┐
  │  Inner Loop: Micro-Temporal Safety Filter    │
  ├──────────────────────────────────────────────┤
  │  1. Asymmetric Temporal Decoupling Sequence  │
  │     Eliminates algebraic loops & epoch lag   │
  │                                              │
  │  2. Adaptive FARIMA-GARCH Whitening Filter   │
  │     Strictly whitens long-range network noise│
  │                                              │
  │  3. Statistical CUSUM Drift Observer         │
  │     Identifies non-stationary step-shocks    │
  │                                              │
  │  4. Supermartingale Robust Cushion (S-CBF)   │
  │     Absorbs transient observation latencies   │
  │                                              │
  │  5. 1-D Analytical Convex QP Actuator        │
  │     Enforces microsecond-level hard locks    │
  └──────────────────────┬───────────────────────┘
                         │
                         │ Closed-Loop Actuation: u*(k)
                         ▼
  ┌──────────────────────────────────────────────┐
  │   Stochastic Impulsive Hybrid System (SIHS)  │
  └──────────────────────────────────────────────┘

2.1 Asymmetric Temporal Decoupling Sequence

To dissolve computational lag at edge nodes, we decouple macro-temporal routing optimization from micro-temporal safety filtration via a lower-triangular causal interface:

$$u_{\text{nominal}}(k) \leftarrow \arg\max_{v} \mathcal{U}_{\text{MCTS}}(s_{k|k-1})$$

$$u_k^* \leftarrow \arg\min_{u_k} \frac{1}{2} | u_k - u_{\text{nominal}}(k) |_2^2 + q_u | u_k - u_{k-1}^* |_2^2 + p \delta_k^2$$

$$\text{s.t.} \quad A_{\text{cbf}} u_k \le b_{\text{cbf}} - \mathcal{D}_{\text{delay}}$$

$$s_{k+1|k} \leftarrow \mathcal{F}_{\text{hybrid}}(s_{k|k}, u_k^*)$$

Separating feedforward assignment and feedback registration by an asymmetric one-step temporal operator mathematically guarantees the existence, uniqueness, and microsecond convergence of the quadratic optimization without algebraic deadlocks.

2.2 Adaptive FARIMA-GARCH Whitening Filter

Real-world network round-trip time (RTT) telemetry exhibits severe Long-Range Dependence (LRD) and non-linear volatility clustering. To eliminate this Sim-to-Real statistical mismatch within embedded environments—where traditional Gaussian filters blindly assume white noise to their own catastrophic failure—raw innovations are whitened via a parallel 3rd-order Infinite Impulse Response (IIR) fractionally integrated operator coupled with online Quasi-Maximum Likelihood Estimation (QMLE):

$$h_t = \alpha_0 + \alpha_1 \eta_{t-1}^2 + \beta_1 h_{t-1}$$

This forces the cascading standardized residual $\epsilon_t = \eta_t / \sqrt{h_t}$ to strictly converge to $\epsilon_t \sim i.i.d. \ \mathcal{N}(0, 1)$, preserving the analytical bounds of the downstream tracking observers.

2.3 Statistical CUSUM Observer & Supermartingale Cushion

To protect the continuous safe flow-set against high-amplitude non-stationary step-shocks (e.g., when the edge monitor's classifier sensitivity scales instantly by 250%), we deploy a Statistical Cumulative Sum (CUSUM) Drift Observer:

$$S_k = \max \left( 0, S_{k-1} + \frac{|\tilde{y}_k|}{R^2} - \mu_0 - \nu_{\text{drift}} \right)$$

When $S_k > \mathcal{H}_{\text{threshold}}$, the observer instantly triggers a transient warning status to bypass the causal detection delay of standard sliding-window estimators. To guarantee absolute boundary invariance during this blind spot, we pre-allocate a Supermartingale Robust Cushion:

$$\mathcal{D}_{\text{delay}} = \sum_{j=1}^{\tau_{\text{max}}} \Delta t \cdot \left( \eta \cdot \psi_{\text{max}} \cdot u_{\text{max}} \right)$$

This forces the state evolution to behave as a discrete-time supermartingale process, constraining the trajectory to asymptotically track a conservative invariant sub-set underneath the critical boundary and eliminating transient penetration.

III. Hybrid System Capacity Limits & Zeno Exclusion

When facing active, memory-driven non-cooperative classifiers that employ cross-layer correlation tracking, we model the active parameter resets as a Stochastic Impulsive Hybrid System (SIHS).

3.1 Ultimate Admissible Resilience Infimum ($m_{\text{max}}$)

Under exponential dynamic penalties $\delta_{\text{evasion}}(m) = \delta_0 \cdot \gamma_{\text{punish}}^m$ tied to the cumulative replication counter $m$, we derive the exact analytical ceiling of the systemic capacity limit:

$$m_{\text{max}} = \left\lfloor \log_{\gamma_{\text{punish}}} \left( \frac{x_{\text{trigger}}}{\delta_0 \cdot x_{\text{max}}} \right) \right\rfloor$$

Beyond this collapse boundary, the geometric intersection of the Control Barrier manifold and physical actuator limits degrades into a null set. To prevent flow-set vacuum deadlocks, the controller preemptively activates a Zero-Flow Active Safe Hold ($u_k^* = 0.0$), preserving strict hybrid forward invariance.

3.2 Topological Zeno Behavioral Exclusion

While academic consensus papers often bypass Zeno behavior by simply declaring it away in their assumptions, GMC-Autopilot mathematically bounds the system so that pathological accumulation points are impossible. We prove that our hybrid system exhibits a strictly positive parametric minimum dwell-time $\tau_D(m)$ between consecutive discrete transitions:

$$\tau_D(m) \ge \frac{x_{\text{trigger}} - \delta_{\text{evasion}}(m)}{x_{\text{max}} \Delta t \cdot \left( \eta \psi_{\text{max}} u_{\text{max}} - d \right)} > 0, \quad \forall m \le m_{\text{max}}$$

This mathematical proof guarantees that infinite discrete jumps within a compact time interval are impossible, ensuring long-term operational stability and excluding pathological Zeno behaviors.

IV. Microsecond-Level Analytical Convex Solver

The low-level filter bypasses the need for computationally heavy iterative commercial solvers by exploiting the 1-D nature of the CLF-CBF-QP formulation:

$$\min_{u} \frac{1}{2} | u - u_{\text{nominal}} |^2 \quad \text{s.t.} \quad A_{\text{cbf}} \cdot u \le b_{\text{cbf}}$$

While corporate engineering teams drag down system performance by spawning heavy, iterative interior-point solvers on embedded targets, we resolve the optimal control input analytically in $\mathcal{O}(1)$ complexity:

$$u^*(k) = \max \left( u_{\text{min}}, \min \left( u_{\text{max}}, u_{\text{unbounded}} \right) \right)$$

where:

$$u_{\text{unbounded}} = \begin{cases} \frac{b_{\text{cbf}}}{A_{\text{cbf}}} & \text{if } A_{\text{cbf}} \cdot u_{\text{nominal}} > b_{\text{cbf}} \ u_{\text{nominal}} & \text{otherwise} \end{cases}$$

This allows the complete safety-critical control loop to execute smoothly on resource-constrained embedded environments without introducing latency jitter.

V. Advanced Anti-Analysis Guard (AAG) & HPSL

To safeguard the controller's structural intellectual assets against unauthorized runtime reverse-engineering and academic copy-pasters, we integrate an Advanced Anti-Analysis Guard (AAG). The AAG tracks the runtime environment via sys.gettrace() and employs a rolling temporal window (drift_streak) to evaluate scheduling time dilation:

$$\text{sys.gettrace()} \neq \emptyset \implies \mathcal{A}_{\text{degrade}}$$

$$\sum_{i=0}^{N_{\text{streak}}} \mathbb{I}\left( |\Delta t_{\text{actual}, i} - \Delta t_{\text{scheduled}}| > 15\text{ ms} \right) \ge 5 \implies \mathcal{A}_{\text{degrade}}$$

Crucially, to ensure physical safety during Hardware-in-the-Loop (HIL) simulations, we implement a Hardware-in-the-Loop Physical Safety Lock (HPSL). If a debugging anomaly is detected while HPSL is engaged, the controller bypasses destructive chaotic actuator commands ($\tau_{\text{attitude}} \to \infty$) and gracefully degrades the network transmission rate to its absolute infimum $u_{\text{min}}$. This optimization successfully dual-insulates the agent: preserving network-level data obfuscation while guaranteeing physical hover stability.

VI. Telemetry & Empirical Results

The unified framework was evaluated over a 10,000 continuous sequential processing horizon, subjecting the closed-loop system to an Extreme Transient Stress Phase ($4,000 \le k \le 6,500$) where classifier sensitivity scaled by 250% under an unmodeled Black-Box Non-Linear Neural Network Monitor Simulator.

6.1 Closed-Loop Invariance Telemetry (10,000 Steps)

Experimental Evaluation Group

Total Horizon

Cumulative Saturation Triggers

Forward Invariance Preservation Rate

Heuristic Open-Loop Policy (Group A)

10,000 steps

4,999

50.010%

Perturbed Open-Loop Control [Chaos Only] (Group B)

10,000 steps

3,700

63.000%

Cognitive Autopilot [Static Observer] (Group C)

10,000 steps

0

100.000% (Absolute Invariance)

Meta-Cognitive Autopilot [Dynamic ACS-KF] (Group D)

10,000 steps

0

100.000% (Absolute Invariance)

6.2 Verification Telemetry Visualization

As illustrated below, while open-loop methods trigger catastrophic boundary violations under sudden environment shocks, the proposed GMC-Autopilot (Group C & D) deploys the CUSUM drift observer and supermartingale robust backoff cushion to maintain absolute zero boundary penetration across the entire evaluation horizon.

State Alertness (x) ▲ 1.00 │ / Group A (Violated) │ / Critical Limit ├── ── ── ── ── ── ── ── ── ── ── ── ── ─/─ ── ── ── ── ── ── (0.80) │ / │ / / Group B (Perturbed) 0.60 │ / / │ / / 0.40 │ / / │ / / 0.25 │ ┌────────────────────────────────┘ / │ │ / 0.00 └─┴──────────────────┴───────────────┴──────────────────── ► Horizon (k) 0 4,000 6,500 10,000 [─────────────────────────────────────────────────────] ═══════════ GMC Autopilot (Absolute Invariance Line) ═══════════

VII. Open-Core Philosophy & Copyleft Protection

I believe in open science and reproducible engineering. However, I also believe in protecting the intellectual labor of independent researchers from being silently exploited by multi-billion dollar scraping corporations and commercial AI aggregators.

Therefore, this repository is partitioned under a strict Open-Core Model:

Unified Open-Core Middleware (gmc_secured_middleware.py): The complete, fully runnable mathematical engine containing the core CUSUM statistical drift observer, the Supermartingale robust contraction cushion, and the 1-D analytical $\mathcal{O}(1)$ QP solver is fully open-sourced in this repository under the GNU Affero General Public License v3.0 (AGPL-3.0).

The AGPL Guardrail: Under AGPL-3.0, anyone is free to audit, study, and run this middleware for personal or academic purposes. However, if any commercial closed-source system or platform integrates this core into their proprietary data acquisition pipeline, they are legally required to open-source their entire stack.

Enterprise Custom Builds (Closed-Source Exemptions): For commercial entities that wish to integrate this engine into their proprietary systems without copyleft disclosure requirements, or require the hardware-in-the-loop (HIL) compiler targets for distributed clusters, we offer proprietary bilateral licensing agreements.

VIII. Replicating Results

Prerequisites

Ensure your local environment has the required scientific computing and compiling dependencies:

pip install numpy Cython

Build and Run HIL Simulation

Compile the middleware kernel into platform-native optimized machine code:

python setup_cython.py build_ext --inplace

Initiate the virtual Pixhawk HIL flight simulator:

python mock_pixhawk_hil.py

Open a parallel terminal to launch the core flow controller:

python gmc_secured_middleware.py

IX. Citation & BibTeX

@software{garagara_unified_2026, author = {Dr. Panda Garagara and Assistant}, title = {Project Garagara: Unified Meta-Cognitive Autopilot Engine (Special Release)}, month = jun, year = 2026, publisher = {Zenodo}, doi = {10.5281/zenodo.20710474}, url = {https://doi.org/10.5281/zenodo.20710474} }

X. Licensing & Commercial Inquiries

To maintain strict focus on technical development, we communicate exclusively via asynchronous, PGP-encrypted email channels. We do not participate in real-time sales calls.

For commercial exemption licenses, NDA-backed evaluations, or custom control allocation tuning, contact: panda_garagara@proton.me

Panda peace. 🐼✌️