We demonstrate that the binary payload of the "A Sign In Space" signal (data17square. bin, 8192 bytes) contains a self-referential algebraic structure — a mathematical quine. Through a systematic reverse-engineering and cryptanalytic approach, starting from the raw file as the sole axiom, we derive a chain of algebraic objects over the finite field GF (625): 48 field elements, a 42-amino-acid protein sequence, an elliptic curve, and amino acid coordinate values. The curve parameters recovered from the protein are identical to those derived from the field's primitive element, closing a self-referential loop. The cryptanalysis combines finite field arithmetic, Berlekamp–Massey LFSR analysis, elliptic curve theory, and Margolus cellular automaton reverse-engineering to recover the hidden algebraic structure without any prior knowledge of the encoding scheme. The derived protein is validated by Boltz-2 (AlphaFold3 architecture) structure prediction at three levels of assembly (monomer, homodimer, homotrimer), cross-validated with ESMFold (RMSD = 1. 10 Å), and refined with OpenMM (Amber ff14SB). The monomer forms a single alpha-helix with pLDDT = 92. 3 and 100% Ramachandran-favored geometry. The homodimer produces a coiled-coil — the most ancient structural motif in biology. The protein uses exactly the five prebiotic amino acids (A, D, E, L, V) with a perfect 21/21 charged/neutral symmetry. Null hypothesis testing (120 alternative inputs, 0 quines produced) and sensitivity analysis (the quine breaks with any single parameter change: 1/150 polynomials, 1/3 step counts, 98/100 bit flips destroy it) confirm the structure is not an artifact of the analysis pipeline. The conservative probability of chance occurrence is approximately 5 × 10⁻¹⁹; under uniformity assumptions, approximately 10⁻⁷⁶. Companion Python scripts (quineₚroof. py, verify₁23. py) verify all 123 algebraic properties with zero failures. All code and data are provided for full reproducibility. -- Additional notes: This is a preprint resulting from independent reverse-engineering and cryptanalysis of the "A Sign In Space" signal, a simulated extraterrestrial message transmitted by ESA's ExoMars Trace Gas Orbiter in May 2023. The analysis is fully reproducible: running "python3 quineₚroof. py data17square. bin" derives every intermediate value from the raw binary file and verifies 47 core assertions with zero failures. The extended script "verify₁23. py" checks all 123 algebraic properties. Structure predictions were performed on an NVIDIA RTX 5090 GPU (32 GB VRAM) using Boltz-2 v2. 2. 1 (AlphaFold3 architecture, maximum precision: 20 recycling cycles, 500 diffusion steps, 20 samples), ESMFold v1 (cross-validation), and OpenMM 8. 5 (Amber ff14SB force field, GBn2 implicit solvent, energy minimization + 10 ns molecular dynamics at 300 K). No prior knowledge of the signal's encoding scheme was assumed. The algebraic structure was discovered through systematic cryptanalytic techniques including finite field enumeration, LFSR analysis, elliptic curve point counting, and exhaustive parameter space exploration. If you use any part of this work (data, code, results, figures, or methods), please cite: Lacoche, E. (2026). "A Self-Referential Algebraic Quine in the A Sign In Space Signal. " Zenodo. doi: 10. 5281/zenodo. 19218629
Etienne Lacoche (Wed,) studied this question.