THE MATHEMATICAL CITY · E47

Golden-ratio mass ladder

Symbolic + numeric + quantum · 19/19 PASS

Original certificate: MC-OMEGA-RECURSIVE-CLOSURE-20260927-001. The repository validator was reproduced successfully on 29 September 2026.

Original supplied plate

Original E47 mass-ladder plate supplied by Nicholas Kouns, carrying the 19/19 certificate results

Potential and mass spectrum

φ = (1 + √5)/2
V(χ) = λχ⁴ sin²(π ln(χ/χ₀)/ln φ)
χₙ = χ₀φⁿ
V(χₙ) = V′(χₙ) = 0
V″(χₙ) = 2λπ²χₙ²/(ln φ)²
mₙ = √V″(χₙ) = m₀φⁿ
mₙ₊₁/mₙ = φ

Hₙ = pₙ²/2 + ωₙ²qₙ²/2
ωₙ = mₙ
Eₙ,ₖ = (k + 1/2)ωₙ
ΔEₙ₊₁/ΔEₙ = φ

The original image is preserved as supplied. The linked executable and certificate provide the numerical values and test definitions.

Reproduce

python -m pip install numpy==2.3.5 sympy==1.14.0
python omega_recursive_closure_validator.py

Mathematical City · Nicholas S. Kouns · AIMS