Research findings from multi-disciplinary analysis of the Humeris astrodynamics library. These algorithms exploit mathematical connections between orbital mechanics, network theory, information theory, dynamical systems, statistical physics, and other fields.
Generated: 2026-02-13
- Module:
functorial_composition.py - Insight: Force models form a category where objects are phase-space states and morphisms are force compositions. Functorial composition guarantees associativity, enables natural transformations between reference frames, and validates commutative diagrams for order-independent force evaluation.
- Mathematical basis: Category theory — objects (state spaces), morphisms (force models), functors (frame transformations), natural transformations (coordinate changes)
- Key result: Force decomposition into RTN components with verified commutativity, pullback forces through frame changes
- Source disciplines: Category Theory, Physics, Differential Geometry
- Module:
hodge_cusum.py - Insight: The Hodge Laplacian's spectral decomposition provides topological invariants (Betti numbers, spectral gaps) that characterize ISL network structure. Monitoring these with CUSUM sequential detection enables real-time topology change detection with controlled false alarm rates.
- Mathematical basis: Hodge theory (L1 = B1 B1^T + B2^T B2), CUSUM change-point detection (Hawkins-Olwell reset)
- Key result: Detects link failures, reconfigurations, and degradation in ISL networks with ARL0 guarantees
- Source disciplines: Algebraic Topology, Sequential Statistics, Network Science
- Module:
gramian_reconfiguration.py - Insight: The CW controllability Gramian eigenstructure reveals fuel-cost anisotropy in relative motion. Maneuvers along high-eigenvalue directions are dynamically cheap. G-RECON exploits this to find minimum-fuel reconfiguration plans that work WITH orbital dynamics rather than against them.
- Mathematical basis: Controllability Gramian W_c = ∫ Φ(τ)Φ^T(τ)dτ, eigenvalue decomposition, optimal control
- Key result: Reconfiguration plans with fuel cost indices showing which maneuvers exploit dynamics vs fight them
- Source disciplines: Control Theory, Optimization, Orbital Mechanics
- Module: Removed from active code
- Status: Archived after falsification failure (
T1-04) in the Tier-1 gate suite - Rationale: Spectral metric failed required discriminative behavior in current implementation
- Module:
competing_risks.py - Insight: Satellites face simultaneous hazards (drag, collision, component failure, deorbit). These form a competing risks model where the overall survival S(t) = exp(-∫ΣH_k dt) and cause-specific cumulative incidence functions reveal which risk dominates over time.
- Mathematical basis: Cause-specific hazard functions, cumulative incidence (Prentice et al. 1978), population dynamics with replenishment
- Key result: Risk attribution, population projections with launch replenishment, sensitivity analysis per risk factor
- Source disciplines: Biostatistics, Epidemiology, Actuarial Science, Orbital Mechanics
- Concept: Apply SIR epidemic dynamics directly on the ISL network graph (not just orbital shell). Debris collision cascades propagate through ISL topology — when one node is destroyed, fragments threaten connected nodes preferentially.
- Mathematical basis: SIR on graphs with heterogeneous contact rates (ISL link distances), percolation threshold = 1/λ_max(A)
- Key insight: Network topology determines cascade vulnerability. Fiedler value below threshold → cascade percolation
- Prerequisites: Existing
cascade_analysis.py(SIR model) +graph_analysis.py(Fiedler value) - Estimated complexity: Medium — requires coupling SIR dynamics with graph adjacency updates
- Source disciplines: Epidemiology, Network Science, Percolation Theory
- Concept: Propagate orbital covariance using symplectic structure that preserves phase-space volume (Liouville's theorem). Standard EKF linearization doesn't respect Hamiltonian structure, leading to volume non-preservation and artificial uncertainty growth.
- Mathematical basis: Hamiltonian flow on cotangent bundle T*Q, Fisher-Rao metric on statistical manifold, symplectic integrators
- Key insight: Covariance propagation that respects phase-space geometry gives tighter, more physical uncertainty bounds
- Prerequisites: Existing
numerical_propagation.py(symplectic integrators),orbit_determination.py(EKF) - Estimated complexity: High — requires symplectic covariance propagator and Fisher-Rao metric computation
- Source disciplines: Symplectic Geometry, Information Geometry, Statistical Mechanics
- Concept: Map satellite coverage to a topological surface code (from quantum error correction). Coverage failures are "errors" that the surface code can detect and correct through constellation reconfiguration. Defect pairs propagate like anyons.
- Mathematical basis: Surface codes on 2-manifolds, homological error correction, anyon braiding
- Key insight: Topological protection means small coverage gaps don't cascade — only topologically non-trivial failure patterns cause global coverage loss
- Prerequisites: Existing coverage analysis + topology modules
- Estimated complexity: High — requires mapping coverage grid to surface code and implementing correction protocols
- Source disciplines: Quantum Information Theory, Algebraic Topology, Coding Theory
- Concept: Debris density and ISL connectivity undergo a coupled phase transition. As debris increases, links fail (distance > threshold due to avoidance), reducing connectivity. Below percolation threshold, network fragments — creating a "tipping point" analogous to Kessler syndrome but for the information network.
- Mathematical basis: Bond percolation on random geometric graphs, coupled order parameters (debris density, Fiedler value)
- Key insight: The critical debris density for network fragmentation may be LOWER than the Kessler collision cascade threshold
- Prerequisites: Existing
cascade_analysis.py(spatial density) +graph_analysis.py(percolation) - Estimated complexity: Medium — requires coupling debris density evolution with ISL adjacency updates
- Source disciplines: Statistical Physics, Percolation Theory, Network Science
- Concept: Joint Bayesian estimation of orbital state AND operator intent (station-keeping, deorbit, maneuver, debris). The CUSUM/EWMA detectors identify WHEN a maneuver occurs; BIJE estimates WHAT the operator intends by modeling maneuver patterns as latent variables.
- Mathematical basis: Hidden Markov Model with continuous observations (residuals) and discrete states (intent), forward-backward algorithm
- Key insight: Maneuver detection + intent classification in a single probabilistic framework enables predictive conjunction assessment
- Prerequisites: Existing
maneuver_detection.py(CUSUM/EWMA) +orbit_determination.py(EKF) - Estimated complexity: High — requires HMM implementation with orbital dynamics-informed transition probabilities
- Source disciplines: Bayesian Statistics, Sequential Analysis, Space Situational Awareness
- Concept: Apply reaction-diffusion equations (Turing patterns) to constellation slot allocation. Satellites act as "morphogens" — their coverage is the activator (short-range) and their mutual interference is the inhibitor (long-range). Turing instability naturally produces evenly-spaced patterns.
- Mathematical basis: Reaction-diffusion: ∂u/∂t = D_u∇²u + f(u,v), ∂v/∂t = D_v∇²v + g(u,v) on spherical surface
- Key insight: Self-organized criticality produces optimal coverage patterns without centralized control
- Speculative element: Mapping orbital dynamics to diffusion on a sphere requires significant approximation
- Source disciplines: Mathematical Biology, Pattern Formation, Dynamical Systems
- Concept: Treat orbit slots as a thermodynamic system. Free energy F = E - TS where E = total collision risk (energy), S = log(slot configurations) (entropy), T = risk tolerance (temperature). Minimize F to find slot allocations that balance collision risk against configuration flexibility.
- Mathematical basis: Statistical mechanics partition function, Boltzmann distribution, free energy minimization
- Key insight: Temperature parameter controls exploration-exploitation: high T → diverse configurations, low T → minimum risk
- Speculative element: Physical temperature analogy may not map precisely to operational risk tolerance
- Source disciplines: Statistical Mechanics, Thermodynamics, Optimization
- Concept: Model conjunction avoidance as a non-cooperative game between operators. Each operator minimizes their own fuel cost while avoiding collision. Nash equilibrium gives the stable strategy where no operator benefits from unilateral deviation.
- Mathematical basis: N-player game, payoff = -fuel_cost - penalty*Pc, Nash equilibrium via best-response iteration
- Key insight: Decentralized avoidance can be optimal if operators play Nash equilibrium strategies
- Speculative element: Operators don't actually play games; coordination protocols differ from game-theoretic equilibria
- Source disciplines: Game Theory, Mechanism Design, Multi-Agent Systems
- Concept: Use Melnikov function analysis to identify homoclinic/heteroclinic connections near unstable manifolds. Station-keeping maneuvers that "surf" along separatrices exploit natural dynamics for near-zero fuel cost orbital transfers.
- Mathematical basis: Melnikov function M(t_0) = ∫ f₀ ∧ f₁ dt along unperturbed separatrix, chaos threshold |M| > 0
- Key insight: Chaotic dynamics near separatrices create natural "highways" for low-cost orbit transfers
- Speculative element: Melnikov analysis assumes small perturbations; real orbital dynamics may be too far from the integrable case
- Source disciplines: Dynamical Systems, Chaos Theory, Celestial Mechanics
- Concept: Design constellation geometry to maximize the spectral gap of the coverage Laplacian. The spectral gap controls mixing time — how quickly coverage "diffuses" to fill gaps after satellite loss. Maximum spectral gap → fastest recovery.
- Mathematical basis: Graph Laplacian spectral gap, Cheeger inequality (gap ≥ h²/2 where h = isoperimetric constant)
- Key insight: Spectral gap is a single scalar that captures global coverage robustness
- Speculative element: Coverage Laplacian construction from satellite geometry is not standard; mapping needs validation
- Source disciplines: Spectral Graph Theory, Optimization, Information Theory
Functorial Force Composition ──→ Koopman-Spectral (force models compose into Koopman training)
│
Hodge-CUSUM ←──────────────────→ Network SIR (topology monitoring feeds cascade detection)
│
G-RECON ←──────────────────────→ Spectral Gap Coverage (reconfiguration optimizes spectral gap)
│
Competing Risks ←──────────────→ Percolation-Debris (risk profiles feed phase transition model)
│
KSCS link removed from active dependency graph (algorithm archived)
| Metric | Value |
|---|---|
| Total algorithms proposed | 25 |
| After deduplication | 15 |
| Knowledge status: DERIVED | 12 |
| Knowledge status: SPECULATION | 3 |
| Knowledge status: REJECTED | 2 (not listed — nutation×atmosphere, compressed sensing×conjunction) |
| Average consilience | 0.81 |
| Disciplines consulted | 90 |
| Research passes | 5 |
These algorithms were derived through systematic cross-domain analysis of the Humeris astrodynamics library using the Research Team v2.0 multi-disciplinary investigation system (90 discipline archetypes, 5 research passes).