Manuscript X-B-ST@1 · X-B-ST.tex · section · d03c150c

The retained edge and switch mechanisms

First-order background identity and retained defects › The retained edge and switch mechanisms · lines 1256–1297

X-B-ST@1#eq:defectclass, X-B-ST@1#eq:edgegap, X-B-ST@1#eq:edgewitness

Passing (eq:variation) to the free algebra gives

δ_{ξ} A_{1} = \int f (ξ - w [K ξ])^{μ} \partial_{μ} ϕ d^{4} x - ϕ (Σ_{g, ξ} E_{A} f),

where the integral denotes its compact test action.

For an initial-trivial compact parameter,

ξ - w [K ξ] = E_{0, R} ((1 - σ) □ ξ) + E_{0, R} (σ^{'} a_{1}^{0 \cdot} [K ξ]) .

Indeed,

□ w^{μ} [K ξ] = σ □ ξ^{μ} - σ^{'} a_{1}^{0 μ} [K ξ], ξ = E_{0, R} □ ξ .

The inherited edge witness is

ξ = α (t) β (x) \partial_{1}, 0 \neq = α \geq 0, supp α \subset (0, ε), \int β d^{3} x = 1,

for which

\int_{R^{3}} (w^{1} [K ξ] - ξ^{1}) (t, x) d^{3} x = - \int_{0}^{ε} σ^{'} (s) α (s) d s < 0, t \geq ε .

It is outside $g_{p}$ .

Displayed formulas in this passage

X-B-ST@1#eq:defectclass · line 1259
$δ_{ξ} A_{1} = \int f (ξ - w [K ξ])^{μ} \partial_{μ} ϕ d^{4} x - ϕ (Σ_{g, ξ} E_{A} f),$
X-B-ST@1#eq:edgegap · line 1269
$ξ - w [K ξ] = E_{0, R} ((1 - σ) □ ξ) + E_{0, R} (σ^{'} a_{1}^{0 \cdot} [K ξ]) .$
X-B-ST@1#eq:edgewitness · line 1290
$\int_{R^{3}} (w^{1} [K ξ] - ξ^{1}) (t, x) d^{3} x = - \int_{0}^{ε} σ^{'} (s) α (s) d s < 0, t \geq ε .$
unlabeled · line 1276
$□ w^{μ} [K ξ] = σ □ ξ^{μ} - σ^{'} a_{1}^{0 μ} [K ξ], ξ = E_{0, R} □ ξ .$
unlabeled · line 1283
$ξ = α (t) β (x) \partial_{1}, 0 \neq = α \geq 0, supp α \subset (0, ε), \int β d^{3} x = 1,$