Bootstrap Foundations: The Geometric Origin of Dimensionless Constants

We derive the geometric foundations of physics from a single axiom: "I AM I" — self-reference that closes. This paper establishes the derivation chain from Grandi's series through spatial dimensions to icosahedral geometry, culminating in the dimensionless constants α (fine structure) and γ (Euler-Mascheroni).

Everything emerges from one fundamental quantity: e (Euler's number). Spatial dimension D=3 is derived via the unique arithmetic identity D(D+1) = 2×D!, rigorously justified by the Borwein theorem which proves exactly 6 = D! terms fit before unity breaks. The icosahedron (V=12, E=30, F=20) encodes D=3 geometry, with twin primes (5,7), (11,13), (17,19) at the boundary 19 = E−V+1.

The fine structure constant α⁻¹ = e⁵ − 6√3 − 1 + 1/66 + harmonics achieves 0.0σ match with theoretical uncertainty below experimental precision. The Euler-Mascheroni constant γ = (e² + 5)/(9e − 3) + corrections achieves 66-digit precision from the same icosahedral scaffold. Both formulas share 66 = D! × (V−1) as a key structural number.

The constants ℏ, c, G are not arbitrary: ℏ is the minimum resonance length (without which no stable structure could form), c is the wave propagation velocity (the rate of self-reference verification), and G is the mass-geometry conversion factor (derived from the hierarchy formula). In natural units all three equal 1; SI values reflect historical metrological conventions established in 18th-century France.

v3.4 updates: γ formula now displayed with convergence table, expanded explanation of ℏ/c/G physical meaning, formal SI terminology, mass derivations moved to companion paper [Particle Masses].