Planck mass m_P
The Planck mass m_P is, in the τ-framework, the τ-effective Layer-2 readout m_P = m_n / ι_τ — the gravity-sector partner of the neutron-mass anchor. It exhibits the gauge hierarchy as a single power of ι_τ⁻¹ and is one of the three Tier-B (∼3 ppm) entries of the precision ladder.
τ-Definition
The Planck mass m_P is, in the τ-framework, the τ-effective Layer-2 readout m_P = m_n / ι_τ — the gravity-sector partner of the neutron-mass anchor. It exhibits the gauge hierarchy as a single power of ι_τ⁻¹ and is one of the three Tier-B (∼3 ppm) entries of the precision ladder.
Categorical invariant. m_P = m_n / ι_τ — the Planck character of the gravity sector, anchored to m_n.
Primary registry anchor:
V.R103
τ-Derivation Chain
-
I.K0— Universe Postulate -
IV.D255— Master constant ι_τ -
V.R103— Planck mass as a derived quantity — m_P = m_n / ι_τ -
V.D45— Newton's gravitational constant G -
V.T154— G–α Bridge entails m_P/m_n ≈ α⁻⁹ · 3^{1/4} · (1 − (3/π)α)^{−1/2}
Lean modules referenced:
TauLib.BookV.GravityField.CalibrationTriangle,
TauLib.BookV.Coda.ConstantsLedger
SI Translation
Numerical value: 2.176434e-8 ± 2.4e-13 kg
Calibration anchor: PG-P01-neutron
Calibration chain:
- Layer 0: ι_τ = 2/(π + e_math) ≈ 0.341
- Layer 2: m_P = m_n / ι_τ = 1.674 927 × 10⁻²⁷ / 0.341 = 4.91 × 10⁻²⁷ kg (Route 3 — Planck character)
- Equivalently m_P = √(ℏ c / G) (standard SI); the cascade enforces G = ℏ c ι_τ² / m_n²
- τ-predicted vs CODATA: ∼3 ppm (Tier B — closing-identity chain)
Manuscript reference: manuscript-sources/book-05/part07-closure/ch-closure-constants.tex