Manuscript XI-E@1 · XI-E.tex · section · 46e55514

The affine witness retains a layer term

Route 2: the complete word and the affine recomputation › The affine witness retains a layer term · lines 1057–1101

XI-E@1#eq:Seps, XI-E@1#eq:bareaffine, XI-E@1#eq:completeaffine

Let $ϑ$ be an odd Grassmann generator and take

c = 2 ϑt \partial_{t}, ϕ = ϕ_{m} (t) = {sin (m t) / m, t, m > 0, m = 0.

Then

(R c)_{00} = 2 ϑ, a_{1}^{00} [R c] = - ϑ,

and

(L_{ε} R c)^{0} = ϑ S_{ε} (t), S_{ε} (t) = \int_{0}^{t} σ_{ε} (s) d s .

Ramp symmetry $χ (1 - s) = 1 - χ (s)$ gives

S_{ε} (t) = t - ε /2, t \geq ε .

Therefore

s_{0} B_{f} s_{0} B_{f} + d_{1} F_{f} = - ϑ \int f S_{ε} (t) cos (m t) d^{4} x, = ϑ \frac{ε}{2} \int f cos (m t) d^{4} x, supp f \subset {t \geq ε} .

A nonnegative test supported where $cos (m t)$ has a fixed nonzero sign detects both formulas.

The complete-word coefficient does not vanish on this witness. It has exactly the stated width-dependent factor, while the bare insertion retains $S_{ε} (t)$ . There is no limit statement. The witness is not in the compact above-ramp ghost class of Theorem thm:plateau.

Displayed formulas in this passage

XI-E@1#eq:Seps · line 1077
$S_{ε} (t) = t - ε /2, t \geq ε .$
XI-E@1#eq:bareaffine · line 1082
$s_{0} B_{f} s_{0} B_{f} + d_{1} F_{f} = - ϑ \int f S_{ε} (t) cos (m t) d^{4} x, = ϑ \frac{ε}{2} \int f cos (m t) d^{4} x, supp f \subset {t \geq ε} .$
XI-E@1#eq:completeaffine · line 1082
$s_{0} B_{f} s_{0} B_{f} + d_{1} F_{f} = - ϑ \int f S_{ε} (t) cos (m t) d^{4} x, = ϑ \frac{ε}{2} \int f cos (m t) d^{4} x, supp f \subset {t \geq ε} .$
unlabeled · line 1060
$c = 2 ϑt \partial_{t}, ϕ = ϕ_{m} (t) = {sin (m t) / m, t, m > 0, m = 0.$
unlabeled · line 1066
$(R c)_{00} = 2 ϑ, a_{1}^{00} [R c] = - ϑ,$
unlabeled · line 1071
$(L_{ε} R c)^{0} = ϑ S_{ε} (t), S_{ε} (t) = \int_{0}^{t} σ_{ε} (s) d s .$