Supplementary Code: Identifiable Uniqueness of Impulsively Forced, Inert, and Stabilized Body Trajectories

Overview

This repository contains the code needed to reproduce the numerical illustration (Figures 1–2) and the finite-sample Monte Carlo tables in the paper. The paper asks when trajectory data can uniquely recover the dynamical class of a moving body among three reference regimes — impulsively forced motion (C), inert propagation (I), and active stabilization such as a controlled spacecraft (S) — and when the data instead leave a set of compatible classes. It formulates this as a classical identification problem: the regimes are told apart by the way their trajectories respond to disturbances, and identification succeeds only when the regimes leave distinct observable signatures and the data are precise enough to resolve the difference. The code follows the paper's diagnostic framework of four observables — O1 (forecast-residual regularity), O2 (disturbance attenuation), O3 (excess-kurtosis reduction), and O4 (impulse response) — referred to as O1–O4 throughout; the Mathematical Background section below gives each diagnostic's role in the two-stage identification.

All numerical experiments use synthetic data generated from fixed random seeds; no observational or experimental data are used or generated. The scripts here draw the two schematic/illustrative figures and run the Monte Carlo that documents the finite-sample behavior of the diagnostics.

Key Results

Repository Structure

Figures

fig_authority.pdf — Figure 1: actuator geometry

Three planar unidirectional jets spaced at 120° produce zero net baseline force when fired equally; differential modulation around that baseline supplies correction authority, whose symmetric headroom depends on both the lower and upper actuator bounds (and is not maximized by firing at the upper bound).

Every arrow is computed from the three jet directions and nonnegative commands; the residual scales are schematic. Geometry alone supplies neither a stochastic covariance law, a centered control policy, nor an O1 (forecast-residual regularity) or O4 (impulse-response) diagnostic result — Figure 1 motivates the actuation model but does not stand in for the statistical arguments.

fig_regimes.pdf — Figure 2: illustrative inert and stabilized reference models

All parameters are illustrative; no observed object is fitted.

Analysis Scripts

Requirements: Python 3.10 or later with NumPy and Matplotlib. The reference build used Python 3.10.6, NumPy 2.2.6 and Matplotlib 3.10.x, and both generated macro files matched the deposited values byte for byte. Fixed random seeds and suppressed volatile PDF metadata support repeatability in a fixed environment; identical output across untested library versions is not promised.

fig_authority.py — Figure 1

Draws the actuator geometry and computed correction arrows. Key functions: sym_set / inradius (symmetric-headroom geometry of the nonnegative command set), jets_lower (directional commands built by lowering jets), draw_jets / panel (rendering). A built-in guard asserts no label overlaps plotted data.

fig_regimes.py — Figure 2 (and macros_fig.tex)

Draws the inert/stabilized reference-model panels and emits the figure/caption macros. Key functions: pulse_draw (fixed-seed disturbance draws), ged_pdf / ged_kurt / ged_lam (generalized-error densities matched by kurtosis to the variance-share identity), mix_big (component scaling for unit total variance).

mc_diagnostics.py — Monte Carlo (and macros_mc.tex)

Simulates the three reference structures at arc lengths \(T = 60, 120, 250\) and writes one \newcommand per reported number. Key functions:

• vh_test — restricted residual-bootstrap test on the accumulated conditional-variance response, with the null \(v_h = 0\) imposed in the resampling (size under the inert limit I).
• cusum_t0 — CUSUM mean-shift detector that locates the disturbance onset.
• vh_detect_full — the full detect-then-test pipeline that re-runs the detector inside every resample, so the selection effect enters both the observed statistic and the null distribution.
• adf_t / adf_robust — Dickey-Fuller clean-trajectory screen and its outlier-robust variant (leakage under impulsive forcing C, the Franses–Haldrup spurious-rejection effect).
• levelshift_flag, xsec_flag — level-shift and cross-sectional detection flags.

make_all.py

Runs the three generators from the package root, then strictly verifies SHA256SUMS.

Mathematical Background

Reference classes

The three admitted regimes are impulsively forced motion (C), inert propagation without correction (I), and active corrective stabilization (S). Classification is relative to the specified reference models and the sampling cadence: the inert label is a cadence-relative operational class, since a sufficiently slow modulation is indistinguishable from inert propagation at a fixed sampling rate.

The two-stage identification and the four diagnostics

Stage 1 excludes impulsive forcing (C) via the O3 excess-kurtosis gap; Stage 2 then distinguishes active stabilization (S) from inert propagation (I).

Resolvability margin and identifiable uniqueness

Two conditions must both hold for a unique recovery: the classes must leave distinct observable signatures (separability), and the data must be precise enough to resolve that difference (a resolvability margin — the separation between the true signature and the nearest competing one must exceed the achievable estimation tolerance, and the detection margin must exceed the observation noise). When both hold, the nearest-signature classifier is consistent for the true class. When the margin fails, the result is set-identification: the estimate belongs to the inert–stabilized compatible pair, and the criterion flags the unresolved contrast for targeted follow-up rather than forcing a unique label.

Reproducibility

Run in an extracted copy, because generation overwrites the figures and macros:

1. Install Python 3.10+ with the required packages:

   pip install numpy matplotlib

2. Run the master script:

   python make_all.py

   The Monte Carlo can take a few minutes. Any missing or changed manifest entry fails the run.
3. To check integrity without regenerating:

   python -c "from pathlib import Path; from make_all import verify_manifest; verify_manifest(Path('.'), Path('SHA256SUMS'))"

The generated macro files (macros_fig.tex, macros_mc.tex) and figures reproduce byte-identically in a fixed environment.

Citation

The paper and this software archive are separate works; please cite both.

Paper:

Andrée, B. P. J. (2026). Identifiable Uniqueness of Impulsively Forced, Inert, and Stabilized Body Trajectories. Working paper.

Software (this package):

Andrée, B. P. J. (2026). Identifiable Uniqueness of Impulsively Forced, Inert, and Stabilized Body Trajectories — replication code. Zenodo. https://doi.org/10.5281/zenodo.22876267

License

This project is licensed under the Creative Commons Attribution 4.0 International License. See the LICENSE file for details.