We argue that the transcendental constants π and e, and the universal probability distributions of mathematical physics — the Gaussian, the Poisson, the Boltzmann, and the Fourier transform — are not independently discovered mathematical tools that happen to be applicable to physics. They are structural fingerprints of a universe whose fundamental geometry is the seven-sphere S7 (the space of unit octonions) governed by a one-way valve on the Fano plane. Specifically: π is the geodesic half-length of S7 — the angular separation between the matter pole (+1) and the antimatter pole (−1) of the universe. It appears in every formula involving a traversal or measurement of this fundamental sphere. e is the exponential map of the G2 one-way valve — the unique accumulated effect of a continuous proportional, memoryless advance in the preferred direction on S7. It appears in every formula involving time-directed, proportional, or history-independent processes. The Gaussian distribution is the unique fixed point of the valve's convolution operation, which is the true content of the Central Limit Theorem. The Poisson distribution counts discrete valve crossings. The Fourier transform is G2 harmonic analysis on S7. All four distributions are therefore views of the same object: the Fano one-way valve operating on S7. Euler's identity eiπ + 1 = 0 is the master equation of this picture, stating that the valve (e) applied through an imaginary crossing (i) over the full geodesic angle (π) maps matter (+1) to the pre-nucleation state (0). The pattern of appearances of these constants and distributions in specific physical formulae constitutes a falsifiable fingerprint of the octonionic structure of spacetime, and provides a structural resolution of Wigner's puzzle about the unreasonable effectiveness of mathematics. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.
Bharathi Jagadeesan (Fri,) studied this question.