Manuscript IX-D@3 · IX-D-v3.tex · assumption · 3ad4586e

Assumption: Copied carrier, charge, and classical closure

Release conventions and inherited inputs › Geometry and actual functionals · lines 347–391

IX-D@3#ass:carrier, IX-D@3#eq:boundary

The carrier $P_{Σ}$ consists of actual finite polynomial functionals with compact distributional kernels satisfying

(D_{η} F)_{h}^{'} (ψ) \in S_{1} = M_{1} + Q_{Σ}

for every smooth configuration and finite smooth direction list. Here

q_{a} = L^{*} a = C^{*} (θ G_{A}^{0} a), Q_{Σ} = {q_{a} : a \in C_{c}^{\infty} (M_{+}; T^{*} M)_{C}},

and the intrinsic charge is

b (m + q_{a}) = (G_{A}^{0} a) ∣_{Σ} .

It is well defined on actual linear kernels and vanishes on $M_{1}$ .

The locally convex topology $τ_{Σ}$ records strong compact-distribution coefficients and all intrinsic trace coordinates. Classical multiplication, directional differentiation, and conjugation preserve the carrier. Classical multiplication is separately continuous.

The sharp adjoint retains the surface term

C^{*} (θ b) = θ C^{*} b + δ (t) B (b ∣_{Σ}) .

For nonzero real $f$ , the inherited sharp approximants satisfy

B_{f, ε} \neq ⟶ B_{f} in τ_{Σ} .

This nonconvergence is not used as a substitute for a charge witness.

Smooth Cauchy and Green-operator mapping properties and finite propagation are available for the fixed operators in (eq:L).

Displayed formulas in this passage

IX-D@3#eq:boundary · line 375
$C^{*} (θ b) = θ C^{*} b + δ (t) B (b ∣_{Σ}) .$
unlabeled · line 351
$(D_{η} F)_{h}^{'} (ψ) \in S_{1} = M_{1} + Q_{Σ}$
unlabeled · line 357
$q_{a} = L^{*} a = C^{*} (θ G_{A}^{0} a), Q_{Σ} = {q_{a} : a \in C_{c}^{\infty} (M_{+}; T^{*} M)_{C}},$
unlabeled · line 362
$b (m + q_{a}) = (G_{A}^{0} a) ∣_{Σ} .$
unlabeled · line 381
$B_{f, ε} \neq ⟶ B_{f} in τ_{Σ} .$

Assumption: Copied carrier, charge, and classical closure

Displayed formulas in this passage

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