feat(coq): Wave 8.3 e^2 origin investigation in m_H formula - #45
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…nature-check]
Investigate the e^2 factor in Trinity formula m_H = 4*phi^3*e^2.
FINDING: e = exp(1) (Napier's constant), not elementary charge, not
an H4 Coxeter exponent. Three paths examined:
PATH A (e^2 as H4 invariant): RULED OUT
- phi^4 = 6.854, off by 7.24% from exp(1)^2 = 7.389
- Lucas L(4) = phi^4 + phi^{-4} = 7.000, off 5.56%
- 3*pi^2/4 = 7.402, off 0.18% but not H4-geometric
- No H4 integer combination reproduces 7.389
PATH B (e^2 = 4*pi*alpha_QED): RULED OUT
- 4*pi/127.94 = 0.0982, factor 75x too small
- Required alpha ~ 0.588 has no physical interpretation
PATH C (empirical fit): ACCEPTED
- exp(1)^2 matches required factor to within 0.001
- No first-principles derivation from H4 / spectral triple
- Declared in theorem e2_empirical_fit (Qed)
- [phenomenological_fit] header tag satisfies anti-numerology gate
New file: proofs/trinity/HiggsE2Origin.v
Updated: docs/status/LAGRANGIAN_HONEST_STATUS.md (Wave 8.3 section)
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Wave 8.3: e^2 Origin Investigation
Summary
Investigates the origin of the
e^2factor in the Trinity Higgs mass formula:Finding
The
ein this formula isexp(1)— Napier's mathematical constant (Euler's number, 2.71828...), as confirmed by inspection ofHiggsPrediction.v:Definition H01_theoretical : R := 4 * phi^3 * (exp 1)^2.Three paths investigated
PATH A — e^2 as H4 geometric invariant: RULED OUT
phi^4 = 6.854, off by 7.24% fromexp(1)^2 = 7.389phi^4 + phi^{-4} = L(4) = 7, off by 5.56%3*pi^2/4 = 7.402, closest at 0.18% but not H4-geometricPATH B — e^2 = 4pialpha_QED at some scale: RULED OUT
exp(1), not the elementary electric charge4*pi*alpha_em(M_Z) = 0.0982— factor 75x too smallalpha ~ 0.588(i.e.,1/alpha ~ 1.70) has no physical interpretationPATH C — empirical fit: ACCEPTED
exp(1)^2 = 7.38906, required factor= 125.20 / (4*phi^3) = 7.38893exp(1)e2_empirical_fit(Qed)Files changed
proofs/trinity/HiggsE2Origin.v— new file, 372 linespath_A_no_H4_derivation,path_B_alpha_ruled_out,e2_empirical_fitdocs/status/LAGRANGIAN_HONEST_STATUS.md— Wave 8.3 section appendedStatus
The numerical formula accuracy (0.02 sigma vs PDG 2024) is unaffected. The e^2 factor is now formally classified as empirical, joining the project's honesty-pass tradition.