Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Derivative and sign checks

Transposition and execution audit › Derivative and sign checks · lines 3236–3298

The line-only derivative in $K_{6}$ ,

\partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z),

is not

\partial_{a^{α}} (R (b - a) δ_{D}^{(4)} (a - z)) .

The response rows, by contrast, contain derivatives of the full incidence seed as proved in (eq:response).

The chart derivatives are

(c, s, t) = (b, b - a, b - z) : (c, s, r) = (a, b - a, z - a) : \partial_{b} = \partial_{c} + \partial_{s} + \partial_{t}, \partial_{a} = - \partial_{s}, \partial_{z} = - \partial_{t}, \partial_{b} = \partial_{s}, \partial_{z} = \partial_{r}, \partial_{a} = \partial_{c} - \partial_{s} - \partial_{r} .

Their coefficients are constant and the charts have unit absolute determinant.

Row	Sign mechanism checked in the seed proof
$47$	Outer minus derivative gives a plus test derivative. Three- and two-derivative seed terms then produce their distinct signs in (eq:P47).
$5 +$ , $5 -$	Two line signs cancel. The outer plus derivative and the three seed derivatives have cancelling transposition signs.
$6$	The line-only $a$ -derivative has a minus sign; the resulting two transfers give the minus sign in (eq:P6).
$8$	The response-incidence sign cancels on transfer; the outer minus derivative gives a plus test derivative.
$9 d$	The same incidence cancellation and outer minus-derivative transfer give (eq:P9d).
$9 m$	There is no outer derivative. The original mass minus sign remains.
Localized vertex	The selected-family $i$ times the local crossed-line $i$ gives $- 1$ ; output transposition gives (eq:Ucombined).

Displayed formulas in this passage

unlabeled · line 3239
$\partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z),$
unlabeled · line 3243
$\partial_{a^{α}} (R (b - a) δ_{D}^{(4)} (a - z)) .$
unlabeled · line 3251
$(c, s, t) = (b, b - a, b - z) : (c, s, r) = (a, b - a, z - a) : \partial_{b} = \partial_{c} + \partial_{s} + \partial_{t}, \partial_{a} = - \partial_{s}, \partial_{z} = - \partial_{t}, \partial_{b} = \partial_{s}, \partial_{z} = \partial_{r}, \partial_{a} = \partial_{c} - \partial_{s} - \partial_{r} .$