Manuscript X-C-ST@1 · X-C-ST.tex · section · b679954e

Factorial, signs, and diagonal nonadditivity

Direct construction of the two new maps › Factorial, signs, and diagonal nonadditivity · lines 822–863

X-C-ST@1#eq:scalar-map, X-C-ST@1#eq:cross-terms

For arbitrary smooth $u$ , define

r_{q} u = u - κ E_{R} L_{1, g} [h] u + κ^{2} (E_{R} L_{1, g} [h] E_{R} L_{1, g} [h] u - E_{R} L_{2, g} [h, k] u) .

The transpose of this scalar response is precisely (eq:moller). The second vertex therefore enters with the same negative response sign as the first vertex, while two first responses have positive sign.

For the first vertex alone,

\frac{d ^{2}}{d λ ^{2}} (1 + λ L_{1, g} [h] E_{A})^{- 1} c_{λ = 0} = 2 L_{1, g} [h] E_{A} L_{1, g} [h] E_{A} c .

This explains the factor $1/2$ in (eq:iteration). No factorial is assigned after the gauge calculation.

The diagonal coefficient is quadratic, not additive, in its Euler operator. For two specified admissible operators $L, M$ , the direct identity is

(L + M) E_{A} (L + M) E_{A} c - L E_{A} L E_{A} c - M E_{A} M E_{A} c = L E_{A} M E_{A} c + M E_{A} L E_{A} c .

These mixed terms may not be discarded under the name of polarization or diagonal additivity. Equation (eq:cross-terms) is ordinary operator bookkeeping; it does not install a general mixed retarded-product family.

Displayed formulas in this passage

X-C-ST@1#eq:scalar-map · line 825
$r_{q} u = u - κ E_{R} L_{1, g} [h] u + κ^{2} (E_{R} L_{1, g} [h] E_{R} L_{1, g} [h] u - E_{R} L_{2, g} [h, k] u) .$
X-C-ST@1#eq:cross-terms · line 852
$(L + M) E_{A} (L + M) E_{A} c - L E_{A} L E_{A} c - M E_{A} M E_{A} c = L E_{A} M E_{A} c + M E_{A} L E_{A} c .$
unlabeled · line 839
$\frac{d ^{2}}{d λ ^{2}} (1 + λ L_{1, g} [h] E_{A})^{- 1} c_{λ = 0} = 2 L_{1, g} [h] E_{A} L_{1, g} [h] E_{A} c .$