Fundamental Constants μ=1836 and α⁻¹=137
Definition
ODTOE derives the proton-to-electron mass ratio μ = m_p/m_e ≈ 1836.15 and the inverse fine-structure constant α⁻¹ ≈ 137.036 from first principles using π, φ and integers with zero free parameters. Both formulas reflect the strange-loop fixed point Ψ* = Φ(Ψ*) and reproduce CODATA values to high precision (μ to 0.002%).
Formula
Related Terms
Self-Observation Fixed Point
The self-observation fixed point Ψ* is the unique stable state satisfying Ψ* = Φ(Ψ*), where Φ is the self-observation map. Schauder's theorem guarantees its existence and Banach's contraction theorem its uniqueness, closing the bootstrap problem of how an observer can constitute itself without pre-existing.
φ-Resonance (Golden Ratio)
φ-resonance is the role of the golden ratio φ = 1.618… as the unique stable resonance frequency in ODTOE, selected from the potentiality field by the KAM theorem because φ is the «most irrational» number. φ is the fixed point of the self-referential map f(x) = 1 + 1/x and appears in fundamental constants, nested φ-tori and recursive structure of reality.
π as Structural Invariant
In ODTOE π is the continuous-phase invariant of self-consistent observation — the form of the observation cycle. Its transcendence (Lindemann 1882) guarantees that the φ-torus trajectory never closes, which produces the arrow of time and the eternal expansion of the universe.
Planck Quantum as Observation
In ODTOE the Planck quantum h is one full revolution of the strange loop Φ — the minimum action portion required for a single act of observation. h is derived in closed form h(d,S) = 2π(π−3)²φ^(d+1)·Σ(d)·(1−S)^(−1/2)·A₀ from π, φ, observer dimensionality d and coherence S, with zero free parameters; the numerical value matches CODATA to six significant digits.
Source Articles
Two Fundamental Constants from First Principles: μ=1836 and α⁻¹=137 in ODTOE
Self-referential formulae for μ=mp/me and α⁻¹ derived from π, φ, integers with zero free parameters. Formula for μ contains four layers. Result: μ=1836.15267. Formula for α⁻¹ contains three layers. Result: α⁻¹=137.035999. Both formulae reflect the strange loop fixed point Ψ*=Φ(Ψ*).
Architecture of the Quantum: π, φ and the Spiral Gap as the Foundation of Reality
A unified architecture of the quantum linking π (observation cycle form), φ (discrete step between cycles) and spiral gap (π−3)² (incomplete loop closure energy). The quantum = one full revolution of strange loop Φ (circumference 2π). Planck's constant h interpreted as minimum action portion. α⁻¹=137.036 derived from first principles with nine significant digits. Ternary architecture π: 1×π=3 (act), 2×π=6 (cycle), 3×π=9 (self-observation).