Mercury orbital geometry, Earth's gravity, CMB register boundary, and the 21 cm hydrogen line — one {3,5,π} arithmetic
21cm Identity
2⁴×3⁸×π/5⁶ cm
Mercury orbit / 100
·
Surface Gravity
25π/8
from Balmer pipeline
·
Propositions
8
P-MHG-1 to P-MHG-8
The Chain
Four steps: Lyman α to 21 cm hydrogen line
The algebraic chain from the hydrogen Lyman wavelength through Mercury's spin and orbital periods to the 21 cm hydrogen line is exact in four arithmetic steps, no free parameters.
P-MHG-1
Lyman α × 4/(2π) → Mercury spin → ×3/2 → Mercury orbital → ÷100 → H 21 cm
H 21 cm FOT = 2⁴×3⁸×π/5⁶ = 21.10667715 cm | NIST: 21.10611405 cm (26.7 ppm)
Registered Propositions
Key Results
P-MHG-1
Lyman α (3⁵/2 nm) → ×4/(2π) → Mercury spin → ×3/2 → Mercury orbital hours → ÷100 = H 21 cm wavelength in cm = 2⁴×3⁸×π/5⁶ cm.
P-MHG-5
R_CMB = 3⁷×5/π km = 10,935/π ≈ 3,480.72 km. The atomic ceiling 3⁷=2187 defines the CMB register boundary.
P-MHG-7
The {3,5,π} sector of the Tau-lattice generates Mercury periods, Earth gravity, CMB boundary, and the 21 cm line simultaneously. Four domains, one lattice node.
A note on “constants.” Within the Universal Force of Time there are no universal constants. A quantity like the Rydberg is not one fixed number but a small family of register faces — each an exact {2, 3, 5, π} value, each reproducing the spectrum on its own scale of Τ. The Rydberg alone carries at least three: 10,966,227.11 m⁻¹ (= 10⁷π²/9), 10,967,215.73, and 10,973,936.9 m⁻¹. What conventional physics records as the constant — the CODATA 10,973,731.568157 m⁻¹ — is not a fourth fundamental number; it is a single measurement sitting between those faces, in the band they define, read from the one register our instruments occupy: the Earth-surface node, g₁. Every wavelength, and the speed of light, Planck’s value, and the fine-structure ratio with it, behaves the same way — each shifts from g₀ to g₁ to g₂ to g₃ by the lattice step δG, not by error. These are not constants; they are the values Τ wears at the register where we stand.