λEnter Calculus CompanionRegistered manuscripts, kept distinct

Manuscript V@1 · V.tex · figure · e7ddf37d

Compact representatives on the TT plane

Restricted compact representatives on a two-dimensional TT plane › The full finite-plane annihilator · lines 1882–1908

V@1#fig:curvature-dual-plane

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metricscalarrepresented quantumdefect / unresolvedsupport / scope
v1=e, v2=rv_1=e,\ v_2=rVfin=Span⁡R{e,r}\mathcal V_{\mathrm{fin}}=\operatorname{Span}_{\mathbb R}\{e,r\}
→
Eel(va)=−12Δva\mathcal E_{\mathrm{el}}(v_a)=-\tfrac12\Delta v_a
→
compact ρ≥0\rho\ge 0Gab=∫ρ Eel(va)ijEel(vb)ijG_{ab}=\int\rho\,\mathcal E_{\mathrm{el}}(v_a)_{ij}\mathcal E_{\mathrm{el}}(v_b)_{ij}
Ca=ρ∑b(G−1)abEel(vb)⟶Ja,FJa(vc)=δacC_a=\rho\sum_b (G^{-1})_{ab}\mathcal E_{\mathrm{el}}(v_b)\longrightarrow J_a,\quad F_{J_a}(v_c)=\delta_{ac}
ba=Bf(va,ϕ),Jc=b1J1+b2J2b_a=B_f(v_a,\phi),\quad J_c=b_1 J_1+b_2 J_2
E′=Ann⁡(Vfin)⊕Span⁡{J1,J2}\mathcal E'=\operatorname{Ann}(\mathcal V_{\mathrm{fin}})\oplus\operatorname{Span}\{J_1,J_2\}
Two interpolation conditions. Not a positive-current theorem. Not a dense-TT theorem.

Compact representatives on the fixed TT plane. Curvature separation yields a positive-definite finite Gram matrix and compact conserved currents dual to the two basis waves. Their response-weighted combination represents the actual restricted functional. The construction imposes two interpolation conditions and leaves a codimension-two annihilator; it does not establish dense-TT representability.

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