Choptyuk Formula

The unified Choptyuk formula combining b-C and a-C corrections.

b-C Correction (Berry Phase)

The b-C correction is the first-order spinor correction, interpreted as a Berry phase:

\[ \Delta_{bC} = \lambda_1(D^2_{\sigma_0}) + \frac{\delta_C^2}{2} = 3.338 + 0.100710 = 3.438710 \]

a-C Braking (2nd Order)

The a-C braking is the second-order correction:

\[ \delta_{\mathrm{eff}} = \frac{\delta_C^5}{22} \approx \frac{1}{1200} = 0.000828 \]

This "braking" term reduces the b-C correction. The denominator 22 is not arbitrary — it equals \(b_2(K3)/\hat{A}(K3)\) where \(b_2 = 22\) is the second Betti number of the K3 surface and \(\hat{A}(K3) = 2\) is the Dirac index.

Unified Formula (Base)

\[ \Delta_{\mathrm{Ch}}^{\mathrm{base}} = \lambda_1(D^2_{\sigma_0}) + \frac{\delta_C^2}{2} - \frac{\delta_C^5}{22} = 3.437883 \]

With Higher-Order Corrections

Including the \(\delta_C^4\) and \(\delta_C^6\) terms:

\[ \Delta_{\mathrm{Ch}} = \Delta_{\mathrm{Ch}}^{\mathrm{base}} + \frac{\delta_C^4}{8} + \frac{\delta_C^6}{2} = 3.447040 \]

Choptyuk Constant

\[ b_{\mathrm{Ch}} = 1 - \cos\left(\frac{2\pi}{7}\right) = 2\sin^2\left(\frac{\pi}{7}\right) \approx 0.376510 \]

Verification

Quantity	Computed	Observed	Deviation
\(\Delta_{bC}\)	3.438710	3.443	0.125%
\(\Delta_{\mathrm{Ch}}^{\mathrm{base}}\)	3.437883	3.443	0.149%
\(\Delta_{\mathrm{Ch}}\)	3.447040	3.443	0.117%
\(b_{\mathrm{Ch}}\)	0.376510	0.377	0.130%

Computational Access

from src.core.choptyuk_formula import ChoptyukFormula

formula = ChoptyukFormula()
print(f"Δ_bC = {formula.delta_bC:.6f}")          # 3.438710
print(f"δ_eff = {formula.delta_eff:.6f}")        # 0.000828
print(f"Δ_Ch (base) = {formula.delta_Ch_base:.6f}")  # 3.437883
print(f"Δ_Ch (full) = {formula.delta_Ch:.6f}")       # 3.447040
print(f"b_Ch = {formula.b_Ch:.6f}")              # 0.376510