Good catches!

1. You're right I should clarify: the formula uses Σ⁺ (1189.37 MeV), not Σ⁰ (1192.642 MeV). Different isospin states.

   Σ⁺ - Λ = 73.69 MeV
   UCT: 12² × mₑ = 73.58 MeV  
   Error: 0.14%

2. Two parts:

   a) "Any number expressible in base π" true, but the claim isn't 
      that 137 ≈ something×π. It's that α⁻¹ = 4π³ + π² + π with 
      INTEGER coefficients {4, 1, 1} arising from kissing number geometry.
      
   b) "Units issue" these are dimensionless coefficients, not literal 
      m³ + m² + m. The "volume/surface/circumference" is structural 
      (π³, π², π¹ powers), not dimensional.

Fair questions thanks for the rigor check!

What if I like Euler's number better than pi?

    e^4 + 5*e^2 + 16*e + 2*e^0 = 137.03594

That's a lot closer to 137.035999177 than the pi approximation (137.03630)

EDIT

better yet:

            3*e^4 -  5*e^3 + 20*e^2 - 28*e +  2*e^0 = 137.035996
            5*e^4 -  9*e^3 + 11*e^2 -  9*e - 12*e^0 = 137.035998
            2*e^4 + 22*e^3 - 61*e^2 -  6*e + 53*e^0 = 137.03599937
    2*e^5 + 6*e^4          - 53*e^2 - 47*e + 32*e^0 = 137.03599922

Excellent numerology! But here's the key question: can your e-polynomial derive OTHER constants? UCT's π-formula α⁻¹ = 4π³ + π² + π isn't chosen because it's "close" it's chosen because the SAME geometric framework derives:

Proton mass: m_p/m_e = 6π⁵ (0.0017% error) Muon mass: m_μ/m_e = 2π⁴ + 12 (0.024% error) Tau mass: m_τ/m_e = (π⁷·ln10)/2 (0.0003% error) Weinberg angle: sin²θ_W = φ/7 (0.027% error) Cosmological constant: ρ_Λ/ρ_P = π^(-247) (1.1% error)

The difference between numerology and physics:

Numerology: Find ONE formula that fits ONE number Physics: Find ONE framework that predicts MANY numbers

Your e⁴ + 5e² + 16e + 2 = 137.036 is impressive! Now use those same coefficients (1, 5, 16, 2) to predict the proton mass. If you can't, it's a coincidence. If you can, publish immediately. UCT coefficients (4, 1, 1) come from π-exponents in the Duality Theorem connecting α to E₈ geometry. They're not fitted they're derived.

This is not my area. I've never seen powers of pi used in geometry or anywhere else for that matter. Where is a good introductory resource for geometry that uses powers of pi? Why does the tau mass need the natural log of 10?