mobius-units

Fundamental constants and unit conversions from the eigenvalue tower — one measurement derives them all.

from mobius_units import alpha_inv, m_e, G, l_P

alpha_inv   # 137.03599917656376 (tower-derived, not CODATA lookup)
m_e         # 9.109383714e-31 kg (from tower alpha + R_inf)
G           # 6.67436e-11 m^3 kg^-1 s^-2 (tower-derived, 0.37 sigma)
l_P         # 1.61626e-35 m (tower-derived)

The Problem

Existing unit libraries (pint, astropy.units, scipy.constants) carry 20+ empirical conversion factors from CODATA lookup tables. When CODATA updates, the tables must be refreshed. No algebraic relationship connects the constants — they are independent numbers from independent measurements.

The Fix

Three algebraic equations produce three dimensionless quantities with zero measured inputs:

Equation	Produces	Agreement with experiment
a^-1 + S*a = 4*pi^3 + pi^2 + pi	alpha (fine structure constant)	0.3 sigma
6*pi^5 * (1 + a^2/(2*sqrt(2)) - (22/27)*a^4)	mu (proton-electron mass ratio)	0.03 sigma
ln(mP/mp) = 14*pi + pi^2*sqrt(a)/28	eta (Planck hierarchy)	4.2 ppm

Combined with five SI-exact constants (c, h, e, k_B, N_A) and one measurement (the Rydberg constant R_inf), these produce the entire CODATA table.

Install

pip install mobius-units

Requires Python >= 3.9. One dependency: mobius-constant.

Constants

Every alpha-dependent constant uses the tower value, not a CODATA lookup.

Dimensionless (tower-derived)

Name	Symbol	Value
alpha_inv	1/alpha	137.035999177
alpha	alpha	7.2973526e-3
mu	mp/me	1836.15267343
eta	ln(mP/mp)	44.0124

Atomic scale (tower alpha + R_inf)

Name	Symbol	Value	alpha-dep
m_e	electron mass	9.109e-31 kg	alpha^-2
m_p	proton mass	1.673e-27 kg	alpha^-2
a_0	Bohr radius	5.292e-11 m	alpha^1
r_e	classical e- radius	2.818e-15 m	alpha^3
lambda_C	Compton wavelength	3.862e-13 m	alpha^2
mu_B	Bohr magneton	9.274e-24 J/T	alpha^2
E_h	Hartree energy	4.360e-18 J	alpha^0 (exact)
sigma_T	Thomson cross-section	6.652e-29 m^2	alpha^6

Planck scale (tower hierarchy)

Name	Symbol	Value	Agreement
G	gravitational constant	6.674e-11	0.37 sigma
m_P	Planck mass	2.176e-8 kg	0.36 sigma
l_P	Planck length	1.616e-35 m	0.38 sigma
t_P	Planck time	5.391e-44 s	0.37 sigma
T_P	Planck temperature	1.417e32 K	0.36 sigma

Unit Systems

Three natural unit systems, all algebraically connected:

from mobius_units import AtomicUnits, PlanckUnits, ParticleUnits

# Atomic -> SI
AtomicUnits.to_si(1.0, 'energy')   # 1 Hartree in joules
AtomicUnits.to_si(1.0, 'length')   # 1 Bohr in metres
AtomicUnits.to_si(1.0, 'time')     # 1 atomic time in seconds

# SI -> Atomic
AtomicUnits.from_si(9.11e-31, 'mass')  # electron mass in atomic units (= 1)

# Planck -> SI
PlanckUnits.to_si(1.0, 'length')   # 1 Planck length in metres
PlanckUnits.to_si(1.0, 'temperature')  # 1 Planck temperature in kelvin

# Particle physics (SI-exact, no tower needed)
ParticleUnits.to_si(125.0, 'energy')  # 125 GeV in joules

# Cross-system
from mobius_units import atomic_to_planck
atomic_to_planck(1.0, 'energy')  # 1 Hartree in Planck energy units

The Alpha Source

All alpha-dependent constants trace to mobius-constant:

from mobius_constant import Alpha_inv
float(Alpha_inv)  # 137.03599917656376

This is the tower-derived value, not the CODATA adjusted average. It sits 0.3 sigma from the best measurement in physics (Fan et al., 2023). The ~4.5 sigma offset from CODATA 2018 adjusted constants traces entirely to the documented Cs-133 recoil anomaly.

See Also

• mobius-constant — Exact irrational constants (sqrt(2)**2 == 2, by construction)
• mobius-number — Complementary residue arithmetic (0.1 + 0.2 = 0.3, exactly)
• mobius-integer — Dual-strand integer: machine i64 + exact BigInt (Rust)
• One Measurement Derives Them All — companion paper (Carpenter, 2026)

Same pattern. Same anatomy. Same fix. Different domain.

License

MIT — Jay Carpenter, 2026