RESEARCH PREPRINT v05The article studies recovery of relative three-dimensional geometry without an absolute restoring force on formation translation. The simulator is an implementation companion, not a proof of the rigid-body, collision, or continuous-safety extensions.
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IMPLEMENTED MATHEMATICS · r18
Nominal flow, local profiles, and observed conditions
Coherent recovery and exact event-free motion
At a sampling event, each agent contributes Hᵢ = xᵢ − rᵢ − qᵢ. A complete same-epoch table commits mᵢ = Hᵢ − meanⱼ Hⱼ, so Σᵢmᵢ = 0 up to roundoff. The held correction defines cᵢ = xᵢ − mᵢ and the nominal law ẋᵢ = vᵢ, v̇ᵢ = −K mᵢ − C vᵢ + dᵢ.
For an event-free interval Δ with constant Uᵢ = −K mᵢ + dᵢ, Equations (61)–(62) give the following analytical update. In nominal mode, the implementation evaluates this map with 64-bit state and stable exponential differences. Exact refers to the analytical time-step formula, subject to floating-point roundoff.
E = exp(−CΔ), A = (1 − E) / C, B = (Δ − A) / C
vᵢ(t + Δ) = E vᵢ(t) + A Uᵢ
xᵢ(t + Δ) = xᵢ(t) + A vᵢ(t) + B Uᵢ
Here A and B are integration coefficients; they are unrelated to the profile-amplitude control A. At C = 0 their continuous limits are A = Δ and B = Δ²/2. Held data and velocity jumps change at simulation event boundaries; each constant-input interval uses its own update. Physical extensions add sampled ordinary forces and rigid contact corrections; that extended numerical motion is not an exact solution of the nominal continuous equation.
Extreme velocity jumps, measured speed, and camera frames
The extreme test applies the velocity-jump convention of Section IV at the current simulation event boundary: position is continuous and velocity takes its prescribed right-hand limit. The magnitude control specifies ‖Δv‖, normalized along the chosen direction. It does not specify the resulting speed. A balanced pair receives opposite increments and preserves the instantaneous arithmetic mean velocity; a common increment preserves all instantaneous relative velocities. These are velocity increments, not equal momentum impulses for unequal masses.
xᵢ(t⁺) = xᵢ(t⁻)
vᵢ(t⁺) = vᵢ(t⁻) + Δvᵢ
Balanced pair: Δvₐ = +Δv, Δvᵦ = −Δv
Common jump: Δvᵢ = Δv ∀ i
Applying this named test explicitly selects exact nominal recovery, preserving the current positions, velocities before the declared jump, held corrections, controller gains, clocks, and communication queues. The event-free update above then continues with the same 1/120-second physics step. The 100, 1,000 and 10,000 m/s presets change the imposed increment without changing the recovery law. Nominal wind, avoidance and contacts remain inactive. The physical solver and its sampled nonlinear forces have additional numerical limits; the nominal experiment does not certify physical motion at those speeds.
The measured panel reports selected, maximum, root-mean-square and centroid speeds in metres per simulated second. The requested speed limit bounds requested control only in limited mode; raising it alone supplies no velocity increment. Requested playback speed and achieved simulation time per wall second are separate clock quantities. Under load the scheduler retains complete fixed steps and reports the backlog. A following camera can cancel common translation visually; a world-fixed camera holds its target until explicitly moved. Camera changes do not alter the simulated state.
Fast bodies can outrun finite-speed priority signals or leave the configured link range. A connected candidate graph then need not yield connected consumed influence. The recovery snapshot service still uses its separate logical-delay contract. Extreme tests expose these assumptions; they do not replace finite-speed priority transport with instantaneous delivery or change the sufficient small-age gain bound.
Delayed priority updates and optional local profiles
Both priority transports implement the stochastic delayed-row update of Equation (45), using immutable consumed payloads and retained historical information. The shared-anchor model uses qᵢ = wᵢZ. The optional Section VIII specialization factors a fixed local table as Zᵢⱼ = Zⱼ + A bᵢsⱼ, giving qᵢ = wᵢZ + A bᵢ(wᵢ · s).
wᵢ(k + 1) = aᵢᵢ(k)wᵢ(k) + Σⱼ≠ᵢ aᵢⱼ(k)wⱼ(k − τᵢⱼ(k))
qᵢ = wᵢZ + A bᵢ(wᵢ · s)
xᵢ − xⱼ → rᵢ − rⱼ + A(bᵢ − bⱼ)(μ · s)
The last expression is the terminal relative shape when the article’s required assumptions hold and priority rows converge to μ. In shared-anchor mode, the final A term is absent. Profile factors remain fixed within a metadata revision; explicit control or membership changes start a new revision. The number of active objective coordinates equals the current cohort size in this implementation.
Sufficient small-age bound and finite-run evidence
The controller specializes the article to K = a² and C = 2a. It computes the recovery age horizon H = h + L from the actual sampling and logical-delivery schedule, and checks the sufficient bound κ = 2aH/e < 1. The displayed gain can decrease as H grows to keep the configured margin. The article permits other gains; this simulator exposes this particular specialization.
Recovery and priority transport have separate clocks and routes. Selecting a connected candidate graph does not establish connectivity of the links actually consumed by moving priority packets. The diagnostics therefore report consumed-link connectivity over a finite observation window, packet age, snapshot coherence, current errors, and body clearance separately. Future connectivity, a final fixed cohort, and eventual disturbance conditions remain assumptions for an asymptotic theorem.
The tolerance indicator is current and revocable. It reports centered recovery error, relative speed, priority agreement, and applicable diagnostic conditions for the requested hold time. Physical contact, continuous collision safety, arbitrary persistent forcing, and impossible rigid-body target geometry are not certified by the nominal theorem.
Distributed corrections and visible communication
Every active node samples its own residual and computes its correction from its own complete received table. Recovery uses one forwarding tree per origin; a node can be a leaf for one origin and a relay for another. The simulator supplies the graph, per-origin routes, common cohort metadata, shared sampling and commit clocks, and a global all-or-none completeness guard. This is distributed local computation under an explicitly coordinated snapshot service.
Translucent cyan spheres show the separate priority service; orange spheres show recovery bundles. Their motion follows the immediate sender-to-receiver direction. Each representative uses a real queued packet and its actual service clock: priority progress still uses the immutable send position and finite signal speed, while recovery progress uses its declared logical delay. The drawn path is schematic and attaches to the current sender and receiver centres; a retired sender without a current body uses its retained send position. This display attachment does not change delivery, payloads or the communication model.
Packet spheres fade at the receiving body’s surface and are drawn with body depth, so packets behind foreground bodies can be hidden. Hollow marks distinguish waiting packets and static fixed-delay priority history; the reference history creates no animated packet events. The overview aggregates busy directed links while retaining every pending sender and receiver in its representative coverage; selecting an orb retains every incident group. Coverage concerns endpoint identities, not guaranteed visible pixels. Exact counts include all queued packets and recovery-frame copies even when markers are aggregated, occluded, entering a body, or outside the viewport.
DISCLOSURE
AI-use disclosure
This work was developed with assistance from AI agents operating within the HORTRAME Systems research architecture. Depending on the research stage, these agents contributed to literature analysis, mathematical formulation, proof reconstruction, adversarial review, counterexample search, simulation and code analysis, verification design, reproducibility checks, revision, and manuscript preparation. AI-agent outputs were treated as research artifacts subject to verification and correction rather than as independent authority. G. Bolduc is the human author responsible for the manuscript, its claims, source selection, and publication decisions. AI-agent review does not constitute independent human peer review, proof-assistant certification, experimental validation, or publication acceptance.
