Hello Wolfram Community,

The concept of branchial space—introduced in the Wolfram Physics Project—offers a transformative way to model multi-way relationships between system states. While profound in theoretical physics, I believe it also holds immense potential for applied engineering.

In this post, I’ll demonstrate a simple proof-of-concept using Wolfram Language: visualizing satellite close approaches ("conjunctions") through the lens of branchial space. This approach could redefine how we handle space traffic management.

The Challenge: Uncertainty in Orbital Dynamics

A satellite’s orbit isn’t deterministic; it’s a cloud of possibilities due to measurement errors, solar radiation pressure, and atmospheric drag. Traditional Monte Carlo simulations model many paths, but branchial space allows us to explore the relationships between all possible future states.

A Simple Branchial Model in Wolfram Language

Let’s model two satellites on near-collision courses. We’ll represent each satellite’s state by its orbital elements and simulate their evolution under uncertainty.

Step 1: Define orbital propagation with uncertainty:

(* Propagate an orbit: add uncertainty to mean anomaly *)
propagateOrbit[state_, dt_, uncertainty_] := Module[{a, e, v, newV},
  {a, e, v} = state;
  newV = v + dt/1000 + RandomVariate[NormalDistribution[0, uncertainty]];
  {a, e, newV}
];

(* Initial states for two satellites *)
sat1Initial = {7000, 0.01, 0};       (* {semi-major axis (km), eccentricity, mean anomaly} *)
sat2Initial = {7000, 0.01, Pi + 0.001}; (* Second satellite starts near the first *)

(* Generate possible states over 10 timesteps *)
numSteps = 10;
uncertainty = 0.0005;
sat1States = NestList[propagateOrbit[#, 1, uncertainty] &, sat1Initial, numSteps];
sat2States = NestList[propagateOrbit[#, 1, uncertainty] &, sat2Initial, numSteps];

Step 2: Construct a branchial graph of conjunctions: We define a node for each combination of satellite states at a given time.An edge connects two nodes if the satellites are dangerously close (a conjunction) at that point, linking possible futures.

(* Calculate distance in parameter space *)
stateDistance[state1_, state2_] := Abs[state1[[3]] - state2[[3]]];

(* Build graph of conjunctions: nodes are (time, branch1, branch2) *)
conjunctionThreshold = 0.002;
branchialGraph = {};
Do[
  dist = stateDistance[sat1States[[t, b1]], sat2States[[t, b2]]];
  If[dist < conjunctionThreshold,
   AppendTo[branchialGraph, 
    Labeled[ {t, b1, b2} -> {t+1, b1, b2}, Round[dist, 0.0001]] ]
  ],
  {t, 1, numSteps-1}, {b1, Length[sat1States]}, {b2, Length[sat2States]}
];

(* Visualize the graph *)
Graph[branchialGraph,
  GraphLayout -> "SpringElectricalEmbedding",
  VertexLabels -> Placed["Index", Center],
  EdgeLabels -> "EdgeTag",
  PlotLabel -> "Branchial Graph of Satellite Conjunction Scenarios",
  ImageSize -> 800
]

Step 3: Interpretation:

· Each node represents a possible system state at time t. · Edges indicate potential conjunctions—paths the system could traverse through branchial space. · Clusters show high-risk scenarios where many branches lead to close encounters.

Why This Matters

This model moves beyond probability estimates to reveal the connectivity of risk:

· Identify critical decision points to avoid entire branches of collisions. · Plan maneuvers that minimize risk across multiple possible futures. · Extend this to cybersecurity, finance, or logistics—any domain with branching futures.

Conclusion: Toward Applied Rulial Thinking

This is a basic illustration, but it highlights a powerful application of computational irreducibility and branchial space. At Rulial Space Agency, we’re working to operationalize these concepts from the Wolfram Physics Project into scalable solutions for space, defense, and finance.

We’re building the tools to navigate the ruliad—and we welcome discussions, collaborations, or questions from the community.

To learn more about our work, visit us at Rulial Space Agency.