Fundamental Constants and Statistical Distributions from Fano Geometry: π, e, kB, the Gaussian, Poisson, and Exponential Distributions, the Random Walk in H3, and the Bose-Einstein Condensate as λ = 4 Mode Condensation

The fundamental constants and statistical distributions of physics are not free parameters or independent axioms they are geometric consequences of the Fano brane-bulk framework. We derive six interrelated results. (1) π from hyperbolic Fano geometry: π appears in the framework as the half-circumference of the unit sphere in rapidity space (H3), encoded in ve independent places: the Klein Quar- tic area (8π from Gauss-Bonnet), the (2, 3, 7) angular decit (π/42), the H3 phase space (2π) in the asymptotic freedom formula, the maximum tunneling geodesic on S3 (τtunnel ≤πℓP /c), and the CMB rst peak (ℓ1 = π/θs). (2) e from the Fano Poisson process: Euler's number e = limn→∞(1 + 1/n)n emerges as the base of the natural logarithm that governs the Fano delocalization ξλ(E) = (ℓP /λ) ln(mPl/E), the exponential decay from independent Bernoulli trials at each Planck timestep, and the QCD scale ΛQCD = mPl exp(−2π/7αG2). (3) kB from minimum Fano entropy: The Boltzmann constant is the conversion factor between the discrete Fano entropy unit kB ln 1 = 0 (Paper LXVIII) and the thermodynamic energy scale. At the Planck scale, kBTPl = ℏc/ℓP = mPlc2, conrming the identication. (4) The Gaussian dis- tribution from the Central Limit Theorem on Fano node uctuations: The quantum wavefunction |ψ(x)|2 is Gaussian because it is the xed point of the CLT applied to N →∞independent Planck-timestep Fano node orientation uctuations (Paper LXXIII). The coherent state of quantum mechanics IS the CLT distribution of the Fano node random walk. (5) The Poisson and exponential distributions from discrete Fano decay trials: Radioactive decay is a Poisson process with rate λdecay = Γ/ℏ, arising from independent Bernoulli trials at each Planck timestep (Pa- per LXXIII). The exponential distribution of decay times P(t) = Γe−Γt/ℏ/ℏis the contin- uous limit of the geometric distribution of Planck-step trials. (6) The Random Walk in H3 as the Feynman Path Integral: The sub-Planck Fano node trajectory is a ran- dom walk in hyperbolic 3-space H3 (the space of Lorentzian rapidity, Paper LXV). The Feynman path integral is the sum over all such H3 walks weighted by ei×H3 arc length/ℓP . The Gaussian wavefunction is the diusion kernel of this H3 random walk. (7) Bose- Einstein condensation as λ = 4 Fano mode condensation: The EW transition is a Bose-Einstein condensate of λ = 4 Fano phonons in the pre-stressed brane lattice. The condensate occupation number ⟨n4⟩= v2/(2m2 H) ≈1932, the critical... 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.