Manuscript XII-F@1 · XII-F.tex · section · 54cb3169

Conclusion

F6: decision, exclusions, and successor package › Conclusion · lines 1383–1408

Δ_{ξ} Φ^{(2)}_{D}_{C_{sol}} = - A_{1} (K ρ_{ξ}; ϕ) - 2 \int (D L_{ξ} β_{f, g} [ϕ]) \cdot b, (CP) ⟹ Δ_{ξ} Φ^{(2)}_{D}_{C_{sol}} = - 2 \int (D L_{ξ} β_{f, g} [ϕ]) \cdot b, D^{*} b = 0 near S_{ξ} ⟹ Δ_{ξ} Φ^{(2)}_{D} = 0, explicit free-shell witness under (CP): Δ_{ξ} Φ^{(2)}_{D} = - 8 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x < 0.

The local vanishing conditions are sufficient, not asserted to be necessary for every individual zero value. What is excluded is invariance without an additional configuration condition or a separately constructed cancelling completion. A classical admitted class has not been renamed as a retarded product.

Displayed formulas in this passage

unlabeled · line 1385
$Δ_{ξ} Φ^{(2)}_{D}_{C_{sol}} = - A_{1} (K ρ_{ξ}; ϕ) - 2 \int (D L_{ξ} β_{f, g} [ϕ]) \cdot b, (CP) ⟹ Δ_{ξ} Φ^{(2)}_{D}_{C_{sol}} = - 2 \int (D L_{ξ} β_{f, g} [ϕ]) \cdot b, D^{*} b = 0 near S_{ξ} ⟹ Δ_{ξ} Φ^{(2)}_{D} = 0, explicit free-shell witness under (CP): Δ_{ξ} Φ^{(2)}_{D} = - 8 \int f (t, x) \int_{0}^{t} (t - s)^{2} a (s) d s d^{4} x < 0.$