Bohr radius a_0

The Bohr radius a_0 is, in the τ-framework, a Layer-4 verification readout of the cascade — derived as a_0 = ℏ/(α m_e c) from the τ-predicted α and m_e. It inherits the 0.025 ppm precision of the mass-ratio chain (Tier A).

Categorical invariant. a_0 = ℏ / (α m_e c) — the natural atomic length scale, a length-sector readout of α and m_e.

Primary registry anchor: IV.T238

Supporting items: IV.P126, IV.D407, IV.D255, V.T157

1. I.K0 — Universe Postulate
2. IV.D255 — Master constant ι_τ
3. IV.D407 — α = (11/15)² ι_τ⁴
4. IV.P126 — Bohr radius from ι_τ — proposition
5. IV.T238 — Bohr Radius — a_0 = ℏ / (α m_e c)
6. V.T157 — Calibration Sufficiency — a_0 is a Layer-4 verification readout

Lean modules referenced: TauLib.BookIV.Calibration.SIReference, TauLib.BookV.Coda.ConstantsLedger

Numerical value: 5.29177210903e-11 ± 8e-21 m

Calibration anchor: PG-P01-neutron

Calibration chain:

1. Layer 0: ι_τ = 2/(π + e_math)
2. Layer 1: α = (11/15)² ι_τ⁴
3. Layer 2: m_e = m_n / R
4. Layer 4: a_0 = ℏ / (α m_e c) = 5.291 772 109 × 10⁻¹¹ m
5. τ-predicted vs CODATA: 0.025 ppm (Tier A)

Manuscript reference: manuscript-sources/book-04/part06/ch45-hydrogen-atom-bohr-rydberg.tex

Status: Formalized

Module: TauLib.BookIV.Calibration.SIReference

Lean kind: def

Related glossary entries

• PG-C02-iota-tau Master constant ι_τ
• PG-C05-fine-structure-alpha Fine-structure constant α
• PG-C09-electron-mass Electron mass m_e
• PG-C11-rydberg-constant Rydberg constant R_∞
• PG-P01-neutron τ-Neutron