Manuscript X-D@3 · X-D-v3.tex · theorem · b7774529

Theorem: Sharp action-level identification

Identification of the sharp prescription › Transmission equation · lines 1021–1058 · proof included or linked

X-D@3#thm:identification, X-D@3#eq:transmission

The limiting coordinates are the unique formal piecewise smooth continuous coordinates equal to $x$ for $t < 0$ and satisfying

\partial_{μ} (A_{sh}^{μν} \partial_{ν} X_{sh}^{ρ}) = 0 (mod κ^{3})

distributionally. Their normal flux is continuous across $Σ$ .

Proof. The flux is a locally integrable piecewise smooth function. Its distributional divergence has a surface delta with coefficient equal to the normal-flux jump.

The negative-side flux is $δ_{0}^{ρ}$ . The positive-side initial flux is

δ_{0}^{ρ} + κ (v_{1}^{ρ} + a_{Σ}^{0 ρ}) + κ^{2} (v_{2}^{ρ} + A_{Σ}^{0 ρ} + a_{Σ}^{00} v_{1}^{ρ}) .

Tangential positive-order coordinate derivatives vanish initially because the initial positions vanish. The jump is zero exactly for (eq:velocities). The positive-time bulk equations are (eq:limit-cauchy), so (eq:transmission) holds.

Conversely, continuity and normal-flux matching force the same initial data. Coefficientwise Cauchy uniqueness gives uniqueness.

Displayed formulas in this passage

X-D@3#eq:transmission · line 1025
$\partial_{μ} (A_{sh}^{μν} \partial_{ν} X_{sh}^{ρ}) = 0 (mod κ^{3})$
unlabeled · line 1043
$δ_{0}^{ρ} + κ (v_{1}^{ρ} + a_{Σ}^{0 ρ}) + κ^{2} (v_{2}^{ρ} + A_{Σ}^{0 ρ} + a_{Σ}^{00} v_{1}^{ρ}) .$

Theorem: Sharp action-level identification

Displayed formulas in this passage

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