Manuscript IX-Free@1 · IX-Free.tex · assumption · 33b20e96

Assumption: Carrier, trace, and inherited closure

Contract, inherited inputs, and domains › Prerequisite ledger · lines 311–353

IX-Free@1#ass:carrier, IX-Free@1#eq:admission, IX-Free@1#eq:boundary

The carrier consists of actual finite polynomial functionals with compact symmetric distributional kernels satisfying

P_{Σ} = {F \in P_{dist} : (D_{η} F)_{h}^{'} (ψ) \in S_{1} for every smooth ψ and finite η} .

Here

S_{1} = M_{1} + Q_{Σ}, q_{a} = L^{*} a = C^{*} (θ G_{A}^{0} a), Q_{Σ} = {q_{a} : a \in C_{c}^{\infty} (M_{+}; T^{*} M)} .

The space $M_{1}$ contains compact smooth linear metric kernels and has no nonzero nonspacelike wavefront covectors.

The topology $τ_{Σ}$ records strong compact-distribution coefficients and every intrinsic trace coordinate

b (m + q_{a}) = (G_{A}^{0} a) ∣_{Σ} .

Classical multiplication, configuration-direction differentiation, and conjugation preserve $P_{Σ}$ . Classical multiplication is separately continuous. No quantum multiplication is included.

The generating test maps are injective. Their fixed-support test-family continuity and the fixed-support first-stage continuity reported in IX-C are inherited. For nonzero real $f$ ,

B_{f, ε} \neq ⟶ B_{f} in τ_{Σ} .

The sharp adjoint identity is

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

Displayed formulas in this passage

IX-Free@1#eq:admission · line 315
$P_{Σ} = {F \in P_{dist} : (D_{η} F)_{h}^{'} (ψ) \in S_{1} for every smooth ψ and finite η} .$
IX-Free@1#eq:boundary · line 348
$C^{*} (θ b) = θ C^{*} b + δ (t) B (b ∣_{Σ}) .$
unlabeled · line 324
$S_{1} = M_{1} + Q_{Σ}, q_{a} = L^{*} a = C^{*} (θ G_{A}^{0} a), Q_{Σ} = {q_{a} : a \in C_{c}^{\infty} (M_{+}; T^{*} M)} .$
unlabeled · line 334
$b (m + q_{a}) = (G_{A}^{0} a) ∣_{Σ} .$
unlabeled · line 344
$B_{f, ε} \neq ⟶ B_{f} in τ_{Σ} .$

Assumption: Carrier, trace, and inherited closure

Displayed formulas in this passage

Nearby in this section