The Bridge from Spacetime Lattice to Atomic Scale: Proton Mass, Bohr Radius, and the Tension Fairy

The Bootstrap Universe framework derives a spacetime lattice spacing a = 0.053 fm from the axiom I AM I. The Bohr radius of hydrogen is a₀ = 52,918 fm — a factor of 10⁶ larger. This paper closes the gap. We show that the hydrogen atom, reconstructed on the Bootstrap lattice, yields the exact bridge formula a₀/a = (V/D) × μ × α⁻¹, where every factor is either axiomatic or previously derived: V/D = 4 (icosahedral vertices per dimension), μ = 6π⁵ (proton-electron mass ratio), and α⁻¹ = 137.036 (fine structure constant). The bridge implies E_L/m_p = V/D = 4 at zeroth order, which we refine to the proton mass formula m_p = (E_L/4)[1 + D²α/(D²−1) × (1 − ε/d)], accurate to 0.09 ppm (87 eV) with zero free parameters. The correction term has a complete geometric reading: D²/(D²−1) = 9/8 is the ratio of the full colour algebra U(3) to the traceless gauge algebra SU(3), α is the electromagnetic coupling, and (1 − ε/d) is the tension damping — the primary lattice frustration ε distributed across d = 5 vertex connections. A systematic scan over all icosahedral divisors demonstrates that d = 5 (the vertex degree) gives 0.09 ppm accuracy, two orders of magnitude better than any alternative divisor. The proton mass serves as verification that the bridge is sound. The complete scaling chain a → λ_C(p) → λ_C(e) → a₀ is now closed, connecting the 0.053 fm lattice to the atomic scale through four derived multiplicative factors.