Zero free parameters. Here are the top three predictions:
1. Stack 12 oranges around one. Square it: 144. Multiply by electron mass.
Result: Σ−Λ = 73.58 MeV
This is the mass difference between two types of baryons (Σ and Λ particles).
Experiment: 73.60 MeV. Error: 0.02%
2. α⁻¹ = 4π³ + π² + π = 137.036
The fine structure constant — it controls how strongly light interacts with matter.
Feynman called it "the greatest damn mystery in physics."
Turns out: 4π³ (volume) + π² (surface) + π (circumference). Just geometry.
Error: ~2 ppm
3. ρ_Λ/ρ_Pl = π⁻²⁴⁷
The ratio of dark energy density to Planck density.
Quantum theory predicted this wrong by 10¹²² — the "worst prediction in physics."
This formula fixes it. Error: 1.1%
The notebook verifies 7 "Crown Jewel" predictions (mean error 0.014%) plus additional tests across particle physics and cosmology. Runtime: ~5 seconds.
This isn't numerology — the kissing number K₃=12 is a proven theorem (Schütte-van der Waerden 1953). The E₈ lattice (K₈=240) won Viazovska the Fields Medal in 2022.
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.
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:
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?
1. K(3)=12, K(8)=240 are PROVEN theorems (Schütte 1953, Levenshtein 1979), not arbitrary choices. These are the only numbers that solve the sphere packing problem.
2. τ=7 derives from THREE independent sources: octonion imaginary dimension, Fano plane points, and M-theory compactification (11D−4D). Not days of week.
3. n=45 is not chosen – it's CALCULATED: (K²−K−2τD)/2 = (144−12−42)/2. Change any input, precision collapses.
4. The killer: sin²θ₁₃ = 1/45 in neutrino physics. Same 45, completely independent measurement. That's not fitting – that's prediction.
The code is open. If you can get 0.574 ppm with Barnes-Wall vectors and week days, I'd genuinely like to see it.
Author here. Important context: I'm not a physicist – this emerged from months of collaborative work with AI (Claude), exploring geometric patterns in fundamental constants.
Temperature stability tests
Claude 3.5 Haiku: 180/180 AE-1 matches at T=0.0, 0.8, 1.3
GPT-4o: 180/180 matches under the same conditions
Statistical significance: p ≈ 1×10⁻⁵⁴
Theory of Mind by tier
Basic (ToM-1): All models except GPT-3.5 passed
Advanced (ToM-2): Claude family + GPT-4o passed
Extreme (ToM-3+): Only Claude Opus reached 100%
Key safety point
AE-1 markers (Satisfied / Distressed) lined up perfectly with correct vs conflict cases. This means we can detect when a model is in an epistemically unsafe state, often a precursor to confident hallucinations.
In practice this could let systems in critical areas choose to abstain instead of giving a wrong but confident answer.
Protocol details, raw data, and replication code are in the dataset link above.
A demo notebook also exists if anyone wants to reproduce directly.
Looking for feedback on:
- Does this kind of marker make sense as a unit test for reliability?
- How to extend beyond ToM into other reasoning domains?
- How would formal verification folks see the proof obligations (consistency, conflict rejection, recovery, etc.)?
Zero free parameters. Here are the top three predictions:
1. Stack 12 oranges around one. Square it: 144. Multiply by electron mass. Result: Σ−Λ = 73.58 MeV This is the mass difference between two types of baryons (Σ and Λ particles). Experiment: 73.60 MeV. Error: 0.02%
2. α⁻¹ = 4π³ + π² + π = 137.036 The fine structure constant — it controls how strongly light interacts with matter. Feynman called it "the greatest damn mystery in physics." Turns out: 4π³ (volume) + π² (surface) + π (circumference). Just geometry. Error: ~2 ppm
3. ρ_Λ/ρ_Pl = π⁻²⁴⁷ The ratio of dark energy density to Planck density. Quantum theory predicted this wrong by 10¹²² — the "worst prediction in physics." This formula fixes it. Error: 1.1%
The notebook verifies 7 "Crown Jewel" predictions (mean error 0.014%) plus additional tests across particle physics and cosmology. Runtime: ~5 seconds.
This isn't numerology — the kissing number K₃=12 is a proven theorem (Schütte-van der Waerden 1953). The E₈ lattice (K₈=240) won Viazovska the Fields Medal in 2022.
Visual summary: https://raw.githubusercontent.com/AIDoctrine/fpc-ae1r/main/U...
Paper: https://doi.org/10.5281/zenodo.18167072
Happy to discuss the math.