# Constructing spatial energy from finite observations ## OA-MOD-SC-01. Conventions and the actual prerequisites Fix a concrete unital von Neumann algebra \(M\subseteq B(H)\), put \(N=M'\), and let \(\psi\) be a normal semifinite faithful weight on \(N\). Both \(H\) and the algebras are arbitrary. Inner products are linear in the first variable. Write \[ (H_\psi,\pi_\psi,\Lambda_\psi),\qquad \mathfrak n_\psi=\{x\in N:\psi(x^*x)<\infty\} \] for the semicyclic construction. We use the direct commutant convention: the denominator is a weight on \(M'\), not on an unnamed opposite algebra. For \(\xi\in H\), say \(\xi\in D_\psi\) when \[ \|x\xi\|\le C_\xi\|\Lambda_\psi(x)\| \qquad(x\in\mathfrak n_\psi) \tag{SC.1} \] for some finite constant. Then \(R(\xi):H_\psi\to H\) is the bounded extension of \(\Lambda_\psi(x)\mapsto x\xi\), and \[ \theta(\xi,\eta)=R(\xi)R(\eta)^*\in M. \tag{SC.2} \] The extension, the intertwining identity, and \(M\)-covariance are proved in [OA-MOD-SD-02](../../reader/orbit-proof-route/sddirect.html#OA-MOD-SD-02). In particular \(D_\psi\) is linear and invariant under \(M\), and \(R(a\xi)=aR(\xi)\). The coefficient ideal and its positive cone are proved in **SD-04**; the finite energy polarization is **SD-05**. Here are the further exact inputs. - [CP-06–08](../../../OA-MOD/concrete-preduals.html#OA-MOD-CP-06): the concrete Banach predual; its identification with ultraweakly continuous functionals; the positive norm formula; and domination of a positive normal functional by a square-summable vector sum. The exact positive vector-sum representation needed here is proved in SC-02 below. - [NW-11](../../reader/orbit-proof-route/nw.html#OA-MOD-NW-11): every normal weight \(\chi\) satisfies \(\chi(a)=\sup_{\omega\in N_*^+,\,\omega\le\chi}\omega(a)\) on its whole positive cone. Its proof uses the convex-analysis foundations stated in NW-01. - **WG-003–006 and WG-008:** finite-domain algebra, GNS construction, and finite positive contraction nets for semifinite weights. **WS-02–06** provide the finite-domain projection \(e_\varphi\), the largest null projection \(f_\varphi\le e_\varphi\), and their exact corner conventions for a normal numerator \(\varphi\). - **BK-02–06:** bicommutants, bounded monotone nets, bounded strong-to-ultraweak convergence, continuous functional calculus, inverse order and support cutoffs. The positive-operator square root, bounded-form representation, completion and orthogonal projection inputs are the course's stated Hilbert and bounded-calculus prerequisites. - [FC-02–05](../../reader/orbit-proof-route/target-fc.html#OA-MOD-FC-02): the completion test for closability, relative lower semicontinuity and exact form cores. [QF-03](../../../OA-MOD/closed-positive-forms.html#OA-MOD-QF-03) represents a closed positive form on the closure of its domain. The required arbitrary-dimensional spectral calculus is supplied by [SK-04–09](../../../OA-MOD/spectral-calculus-kernel.html#OA-MOD-SK-04), including exact integral domains, square roots and spectral transport. These proofs use their stated scalar, Hilbert-space and bounded-calculus prerequisites. No standard form, modular conjugation, relative Tomita map, normal-representation image theorem or weight on a GNS commutant is used in the proof. In particular, the bounded normal-map criterion is not substituted for NW's infinite-valued weight theorem. The arguments below use these specified mathematical prerequisites. ## OA-MOD-SC-02. Positive observations can be implemented by vector sums **Lemma.** If \(A\subseteq B(L)\) is a concrete unital von Neumann algebra and \(\omega\in A_*^+\), there are vectors \(v_j\in L\), indexed by positive integers, such that \[ \sum_j\|v_j\|^2=\omega(1),\qquad \omega(a)=\sum_j\langle av_j,v_j\rangle \quad(a\in A). \tag{SC.3} \] The series is absolutely convergent. Every component functional \(\omega_{v_j}\) is positive normal and satisfies \(\omega_{v_j}\le\omega\). No separability hypothesis on \(L\) is needed. **Proof.** CP-08 gives square-summable vectors \(\zeta=(\zeta_j)\in\ell^2(\mathbb N,L)\) such that \[ \omega(a)\le\vartheta(a):= \sum_j\langle a\zeta_j,\zeta_j\rangle \qquad(a\ge0). \] Let \(\rho(a)=\operatorname{diag}(a,a,\ldots)\), a bounded *-representation on \(\ell^2(\mathbb N,L)\), and put \(K=\overline{\rho(A)\zeta}\). The subspace is invariant under \(\rho(A)\) and its adjoints, so it reduces this representation. Also \(\zeta\in K\), because \(A\) is unital. On the dense subspace \(\rho(A)\zeta\subseteq K\), define \[ B(\rho(a)\zeta,\rho(b)\zeta)=\omega(b^*a). \] The inequality \(\omega(a^*a)\le\vartheta(a^*a)=\|\rho(a)\zeta\|^2\), together with the positive-functional Cauchy–Schwarz inequality, proves that this rule is well-defined and \[ |B(\rho(a)\zeta,\rho(b)\zeta)| \le\|\rho(a)\zeta\|\,\|\rho(b)\zeta\|. \] It extends to a positive bounded sesquilinear form on \(K\). The bounded-form theorem gives a unique positive contraction \(h\in B(K)\) with \(B(u,v)=\langle hu,v\rangle\). For \(c\in A\), the equality \[ B(\rho(c)\rho(a)\zeta,\rho(b)\zeta) =B(\rho(a)\zeta,\rho(c^*)\rho(b)\zeta) \] holds because both sides equal \(\omega(b^*ca)\). Extension from the dense cyclic subspace shows that \(h\) commutes with \(\rho(c)|_K\). Its square root commutes as well. Set \(v=h^{1/2}\zeta\in K\), and write its components as \(v_j\in L\). For every \(a\in A\), \[ \omega(a)=B(\rho(a)\zeta,\zeta) =\langle\rho(a)v,v\rangle =\sum_j\langle av_j,v_j\rangle. \] Evaluation at \(1\) gives the asserted sum of squared norms. Absolute convergence follows from \(|\langle av_j,v_j\rangle|\le\|a\|\|v_j\|^2\). For positive \(a\), each nonnegative summand is at most the sum, proving \(\omega_{v_j}\le\omega\). Single-vector functionals are ultraweakly continuous by CP. If \(\omega=0\), the construction may be replaced by the zero sequence. ∎ The use of a countable series concerns one predual functional. The family of all normal functionals can be arbitrarily large. We have not chosen a countable family which detects the entire algebra. ## OA-MOD-SC-03. The exact closure of bounded vectors For this result alone let \(\chi\) be any normal weight on a concrete unital von Neumann algebra \(A\subseteq B(L)\). Do not assume semifiniteness or faithfulness. Define \(D_\chi(L)\) by the estimate (SC.1), using \(\chi\), its finite left ideal, and the original action on \(L\). Let \(f_\chi\) be the largest \(\chi\)-null projection from WS-04. **Theorem.** \[ \overline{D_\chi(L)}^{\,\|\cdot\|} =(1-f_\chi)L. \tag{SC.4} \] In particular, a faithful normal weight has dense bounded vectors. For a normal semifinite weight, \(1-f_\chi\) is its support in the WS-06 convention. **Proof.** The bounded-vector space is linear. If \(b\in A'\) and \(\xi\) is bounded, then \[ \|x b\xi\|=\|b x\xi\| \le\|b\|C_\xi\chi(x^*x)^{1/2}, \] so \(b\xi\) is bounded. The closed subspace \(\overline{D_\chi(L)}\) therefore reduces \(A'\), and its projection \(P\) lies in \(A''=A\). Since \(\chi(f_\chi)=0\), the projection \(f_\chi\) lies in \(\mathfrak n_\chi\). Testing the bounded-vector estimate with \(x=f_\chi\) gives \(f_\chi\xi=0\). Consequently \(P\le1-f_\chi\). For the reverse inequality, let \(\omega\in A_*^+\) satisfy \(\omega\le\chi\), and implement it by the vector sum in SC-02. For every \(x\in\mathfrak n_\chi\), \[ \|xv_j\|^2=\omega_{v_j}(x^*x) \le\omega(x^*x)\le\chi(x^*x). \] Thus each \(v_j\in D_\chi(L)\), with constant one. Hence \[ \omega(1-P)=\sum_j\|(1-P)v_j\|^2=0. \] NW-11, applied at the positive element \(1-P\), gives \(\chi(1-P)=0\). Maximality of \(f_\chi\) implies \(1-P\le f_\chi\), or \(1-f_\chi\le P\). Combining the inequalities proves (SC.4). ∎ Faithfulness gives \(f_\chi=0\), so the denominator \(\psi\) of SC-01 has \(D_\psi\) dense in \(H\). Semifiniteness is still needed elsewhere: it makes the bounded-vector map \(\xi\mapsto R(\xi)\) injective by finite cutoffs. SC-03 alone does not assert that injectivity for an arbitrary normal weight. ## OA-MOD-SC-04. A coefficient ideal with an explicit approximate identity Return to the setting of SC-01, and set \[ \mathcal J=\operatorname{span}\{\theta(\xi,\eta):\xi,\eta\in D_\psi\}. \] By SD-04 this is an algebraic two-sided *-ideal of \(M\), and \[ \mathcal J_+=\mathcal J\cap M_+ =\left\{\sum_{j=1}^m\theta(\xi_j,\xi_j): \xi_j\in D_\psi,\ m<\infty\right\}. \tag{SC.5} \] **Proposition.** The ideal is ultraweakly dense. More precisely, direct \(\mathcal J_+\) by the usual positive order and put \[ u_a=a(1+a)^{-1}\qquad(a\in\mathcal J_+). \tag{SC.6} \] Then \(u_a\in\mathcal J_+\), \(0\le u_a\le1\), and \(u_a\uparrow1\) strongly. For any \(x\in M_+\), \[ x^{1/2}u_a x^{1/2}\in\mathcal J_+, \qquad x^{1/2}u_a x^{1/2}\uparrow x. \tag{SC.7} \] **Proof.** Addition makes the index set directed. Functional calculus and the ideal property put \(u_a\) in \(\mathcal J_+\). Inverse order gives \(a\le b\Rightarrow u_a\le u_b\), so there is a positive contraction \(u\in M\) which is their strong supremum. For every fixed \(a\in\mathcal J_+\), all \(ta\), \(t>0\), are indices, and the support-cutoff theorem gives \(u_{ta}\uparrow s(a)\). Thus \(u\ge s(a)\), and a positive contraction dominating a projection is the identity on its range. These support ranges span \(H\). Indeed, a vector \(v\) orthogonal to all of them is annihilated by every \(\theta(\xi,\xi)\). Therefore \(R(\xi)^*v=0\). Choose the finite positive contraction net \(e_i\uparrow1\) for \(\psi\). Each \(e_i\in\mathfrak n_\psi\), and \[ \langle e_i\xi,v\rangle =\langle R(\xi)\Lambda_\psi(e_i),v\rangle=0. \] Strong convergence gives \(v\perp\xi\) for every \(\xi\in D_\psi\); SC-03 gives \(v=0\). Thus \(u\) acts as the identity on a dense subspace, and \(u=1\). The ideal property gives \(xu_a\in\mathcal J\) for every \(x\in M\). This norm-bounded net converges strongly and ultraweakly to \(x\), proving density. Finally, compression of the increasing net by \(x^{1/2}\) preserves positivity and order, gives the strong limit in (SC.7), and stays in the ideal. ∎ This proof supplies actual increasing positive approximants. Ultraweak density of a linear space, by itself, would not supply (SC.7) for passing limits through an infinite-valued normal weight. ## OA-MOD-SC-05. Finite energy and its Hilbert-space closure Let \(\varphi\) be any normal weight on \(M\). Write \(e=e_\varphi\) and \(f=f_\varphi\) for its finite-domain and null projections, so \(f\le e\). Define \[ F_\varphi(\xi)=\varphi(\theta(\xi,\xi)) \quad(\xi\in D_\psi),\qquad E_\varphi=\{\xi\in D_\psi:F_\varphi(\xi)<\infty\}. \tag{SC.8} \] SD-05 proves that \(E_\varphi\) is linear and that \[ q_\varphi^0(\xi,\eta) =\widetilde\varphi(\theta(\xi,\eta)) \quad(\xi,\eta\in E_\varphi) \tag{SC.9} \] is a nonnegative sesquilinear form. Here \(\widetilde\varphi\) is the finite linear extension on \(\mathfrak m_\varphi\); its argument is in that domain by SD-05. The proof there needs no semifiniteness of the numerator, so it applies with the present hypotheses. **Proposition.** \[ \overline{E_\varphi}^{\,\|\cdot\|}=eH. \tag{SC.10} \] Consequently this form is densely defined on \(H\) exactly when \(\varphi\) is semifinite. **Proof.** If \(\xi\in E_\varphi\), the positive operator \(\theta(\xi,\xi)\) has finite weight. WS-02 therefore gives \(\theta(\xi,\xi)=e\theta(\xi,\xi)e\). For every \(v\in H\), \[ \|R(\xi)^*(1-e)v\|^2 =\langle\theta(\xi,\xi)(1-e)v,(1-e)v\rangle=0. \] Thus \((1-e)R(\xi)=0\). Testing at \(\Lambda_\psi(e_i)\), with the finite denominator cutoffs \(e_i\uparrow1\), gives \((1-e)e_i\xi=0\). Since \(e\in M\) commutes with \(e_i\in N\), passage to the strong limit gives \(\xi=e\xi\). Hence \(E_\varphi\subseteq eH\). Conversely, let \(a\in M_+\) have finite \(\varphi\)-weight, and let \(\xi\in D_\psi\). Covariance and the bound \(\theta(\xi,\xi)\le\|R(\xi)\|^2 1\) give \[ 0\le\theta(a^{1/2}\xi,a^{1/2}\xi) =a^{1/2}\theta(\xi,\xi)a^{1/2} \le\|R(\xi)\|^2a. \] Its weight is finite, so \(a^{1/2}\xi\in E_\varphi\). Density of \(D_\psi\) shows that the closure of \(E_\varphi\) contains \(a^{1/2}H\), and hence its closure \(s(a)H\). By WS-02, the ranges of all these finite positive elements span \(eH\). This proves the reverse inclusion and (SC.10). Finally \(e=1\) is exactly semifiniteness by WS-02. ∎ No invariance of \(E_\varphi\) under arbitrary elements of \(M\) has been asserted. The displayed estimate uses a finite positive element on the outside of a bounded coefficient; a general weight does not satisfy a tracial conjugation bound. ## OA-MOD-SC-06. Lower semicontinuity on bounded vectors **Theorem.** The function \(F_\varphi:D_\psi\to[0,\infty]\) is lower semicontinuous for the topology inherited from the Hilbert norm of \(H\). The form \(q_\varphi^0\) on \(E_\varphi\) is closable. **Proof.** For each \(\omega\in M_*^+\) with \(\omega\le\varphi\), choose a representation (SC.3). Since \(R(\xi)\) is bounded, \[ \omega(\theta(\xi,\xi)) =\sum_j\|R(\xi)^*v_j\|^2 \qquad(\xi\in D_\psi). \tag{SC.11} \] For a fixed vector \(v\in H\), the norm of \(R(\xi)^*v\) has the variational description \[ \|R(\xi)^*v\|^2 =\sup_{x\in\mathfrak n_\psi} \left(2\operatorname{Re}\langle v,x\xi\rangle -\|\Lambda_\psi(x)\|^2\right). \tag{SC.12} \] Indeed, the expression equals \(2\operatorname{Re}\langle R(\xi)^*v,\Lambda_\psi(x)\rangle-\|\Lambda_\psi(x)\|^2\), and \(\Lambda_\psi(\mathfrak n_\psi)\) is dense in \(H_\psi\). Completing the square gives the supremum of the squared norm. Each expression inside the supremum in (SC.12) is norm continuous in \(\xi\) on all of \(H\): \(x\) is one fixed bounded operator. Thus this squared norm is lower semicontinuous on \(D_\psi\). Finite sums of these nonnegative lower semicontinuous functions are lower semicontinuous. Their increasing supremum over finite initial segments is (SC.11), so that function is also lower semicontinuous. Finally NW-11 gives, including infinite values, \[ F_\varphi(\xi) =\sup_{\omega\in M_*^+,\,\omega\le\varphi} \omega(\theta(\xi,\xi)). \tag{SC.13} \] A supremum of lower semicontinuous functions is lower semicontinuous. Restricting to \(E_\varphi\) proves relative lower semicontinuity of the diagonal of \(q_\varphi^0\); FC-04 then proves closability. ∎ There is no assumption that \(\|R(\xi_n)\|\) stays bounded when \(\xi_n\to\xi\). Formula (SC.12) was used precisely to make every test continuous without such a bound. Also, the normal-functionals supremum need not be a directed or countable supremum. ## OA-MOD-SC-07. Completion gives the exact core and energy formula Let \(q_\varphi\) be the canonical closure of \(q_\varphi^0\), constructed by completing its form norm as in FC-02. Write \(V_\varphi=D(q_\varphi)\subseteq eH\). **Theorem.** The form \(q_\varphi\) is closed and densely defined on \(eH\). Its original domain is exactly \[ E_\varphi=D_\psi\cap V_\varphi. \tag{SC.14} \] It is a form core: every \(\xi\in V_\varphi\) is the limit of a sequence \(\xi_n\in E_\varphi\) in the norm \((\|\xi\|^2+q_\varphi[\xi])^{1/2}\). On all bounded vectors the exact extended energy identity is \[ \varphi(\theta(\xi,\xi))= \begin{cases} q_\varphi[\xi],&\xi\in V_\varphi,\\ +\infty,&\xi\notin V_\varphi, \end{cases} \qquad\xi\in D_\psi. \tag{SC.15} \] These properties determine the closed form uniquely. **Proof.** Closability is SC-06. FC-02 supplies the closed extension and makes \(E_\varphi\) a form core by construction. Hilbert limits of its form-Cauchy sequences stay in \(eH\) by SC-05, and its domain contains \(E_\varphi\), which is dense there. Thus it is densely defined on \(eH\). The inclusion \(E_\varphi\subseteq D_\psi\cap V_\varphi\) and equality of its old and new energies follow from extension. Conversely, let \(\xi\in D_\psi\cap V_\varphi\), and choose the core sequence \(\xi_n\in E_\varphi\). Its energies converge to \(q_\varphi[\xi]\). One way to see this is to apply Cauchy–Schwarz for the form to the difference: \(|q_\varphi[\xi_n]^{1/2}-q_\varphi[\xi]^{1/2}|\le q_\varphi[\xi_n-\xi]^{1/2}\). Relative lower semicontinuity from SC-06 gives \[ \varphi(\theta(\xi,\xi)) \le\liminf_n\varphi(\theta(\xi_n,\xi_n)) =q_\varphi[\xi]<\infty. \] Therefore \(\xi\in E_\varphi\). This proves (SC.14); agreement of the extension then gives equality, rather than merely the preceding inequality. The remaining bounded vectors have infinite original energy by the definition of \(E_\varphi\), proving (SC.15). If another closed form has the same diagonal on \(E_\varphi\) and has \(E_\varphi\) as a form core, polarization gives the same form there. Both forms are the completion of that same form norm with the same inclusion into \(H\), so FC-02 identifies them. ∎ The sequence in this statement approximates one vector in a Hilbert norm. It does not impose a countable exhaustion of \(M\) or a separable Hilbert space. The core condition is necessary for the uniqueness statement; agreement on a merely Hilbert-dense subspace is insufficient. ## OA-MOD-SC-08. All and only the null directions survive at zero energy **Theorem.** \[ \{\xi\in V_\varphi:q_\varphi[\xi]=0\}=fH. \tag{SC.16} \] **Proof of the inclusion from the null projection.** For \(\xi\in D_\psi\), covariance gives \[ \theta(f\xi,f\xi)=f\theta(\xi,\xi)f. \] This positive element has \(\varphi\)-weight zero by WS-04. Thus \(fD_\psi\subseteq E_\varphi\) and its energy vanishes. Since \(D_\psi\) is dense, \(fD_\psi\) is dense in \(fH\). For any \(\eta\in fH\), a Hilbert-norm approximating sequence from this space is also form-Cauchy, because all its differences have zero energy. The closure construction puts \(\eta\) in \(V_\varphi\) with zero energy. **Proof that there are no further null directions.** Choose one vector-sum implementation \((v_{\omega,j})_j\) for every \(\omega\in M_*^+\) dominated by \(\varphi\), and let \[ L=\overline{\operatorname{span}} \{y v_{\omega,j}:y\in N,\ j\ge1,\ \omega\le\varphi\}. \] This space reduces \(N\), so its projection \(g\) lies in \(M\). For every such \(\omega\), \[ 0\le\omega(f)\le\varphi(f)=0, \qquad \omega(f)=\sum_j\|fv_{\omega,j}\|^2. \] Thus every implementing vector is in \((1-f)H\), and this space is \(N\)-invariant because \(f\in M\). Hence \(g\le1-f\). On the other hand, all implementing vectors lie in \(gH\), so \(\omega(1-g)=0\) for every dominated \(\omega\). NW-11 gives \(\varphi(1-g)=0\), hence \(1-g\le f\). Therefore \[ L=(1-f)H. \tag{SC.17} \] Now let \(\xi\in V_\varphi\) have zero energy. Choose \(\xi_n\in E_\varphi\) converging to it in form norm. Then \(q_\varphi^0[\xi_n]\to0\). For fixed \(\omega,j\) and \(x\in\mathfrak n_\psi\), (SC.11) gives \[ |\langle x\xi_n,v_{\omega,j}\rangle| \le\|\Lambda_\psi(x)\|\, \|R(\xi_n)^*v_{\omega,j}\| \le\|\Lambda_\psi(x)\|\,q_\varphi^0[\xi_n]^{1/2} \longrightarrow0. \] The fixed bounded operator \(x\) preserves Hilbert-norm convergence, so \(\langle x\xi,v_{\omega,j}\rangle=0\). For any \(y\in N\), the denominator's finite cutoffs satisfy \(ye_i\in\mathfrak n_\psi\) and \(ye_i\xi\to y\xi\). It follows that \(\langle y\xi,v_{\omega,j}\rangle=0\) for all \(y\in N\). Replacing \(y\) by its adjoint shows that \(\xi\) is orthogonal to every generator of \(L\). Equation (SC.17) gives \(\xi\in fH\), as required. ∎ The argument uses \(1-f\) as a null-carrier projection during testing. The effective support of the closed form on its actual Hilbert space \(eH\) is \(e-f\). Equating these two projections before imposing semifiniteness would lose the infinite-energy directions. ## OA-MOD-SC-09. The representing operator and its domains Apply QF-03 to \(q_\varphi\) on \(K=eH\). There is a unique nonnegative self-adjoint operator \(A_\varphi\) on \(K\) such that \[ D(A_\varphi^{1/2})=V_\varphi, \qquad q_\varphi(\xi,\eta) =\langle A_\varphi^{1/2}\xi,A_\varphi^{1/2}\eta\rangle. \tag{SC.18} \] Its operator domain is exactly \[ \begin{split} D(A_\varphi)=\{\xi\in V_\varphi:\ &\text{there is }z\in K\text{ with}\\ &q_\varphi(\xi,\eta)=\langle z,\eta\rangle \text{ for every }\eta\in V_\varphi\}, \end{split} \qquad A_\varphi\xi=z. \tag{SC.19} \] Uniqueness of \(z\) follows from density in \(K\). The initial finite-energy domain \(E_\varphi\) is an operator core for \(A_\varphi^{1/2}\), because its graph norm is exactly the form norm. Thus the restriction of \(A_\varphi^{1/2}\) to \(E_\varphi\) is essentially self-adjoint on \(K\). The kernel of \(A_\varphi\) is \(fH\), and its support, viewed as a projection on \(H\), is \(e-f\). To verify the kernel without confusing the two domains, if \(q_\varphi[\xi]=0\), form Cauchy–Schwarz gives \(q_\varphi(\xi,\eta)=0\) for every \(\eta\); (SC.19) then gives \(\xi\in D(A_\varphi)\) and \(A_\varphi\xi=0\). Conversely, an operator-null vector has zero form by (SC.18), or by pairing (SC.19) with itself. Now apply SC-08. For a self-adjoint operator on \(K\), the closure of its range is the orthogonal complement in \(K\) of its kernel, which gives \((e-f)H\). When \(\varphi\) is semifinite, \(e=1\). We denote this operator on all of \(H\) by \[ A_\varphi=\frac{d\varphi}{d\psi}. \tag{SC.20} \] Equations (SC.14)–(SC.19) give its full finite-coefficient construction and exact square-root core. Its support is \(1-f=s(\varphi)\), and if \(\varphi\) is also faithful, it has zero kernel. These conclusions do not assert that \(A_\varphi\) is affiliated with \(M\); a spatial derivative generally is not. When \(e\ne1\), retain the pair \((eH,A_\varphi)\), or equivalently the closed form extended by infinity outside \(V_\varphi\). Adding the zero operator on \((1-e)H\) would give finite zero energy there and violate (SC.15). We make no such extension. This construction for arbitrary normal numerators is stronger in domain generality than the ordinary semifinite-numerator operator statement, while its relative modular identification remains a separate obligation. ## OA-MOD-SC-10. Weight order is exactly closed-form order For any normal numerator, extend the diagonal of \(q_\varphi\) by \(+\infty\) off \(V_\varphi\). For two such forms, write \(q_\varphi\le q_\rho\) when \[ V_\rho\subseteq V_\varphi, \qquad q_\varphi[\xi]\le q_\rho[\xi] \quad(\xi\in V_\rho). \tag{SC.21} \] This is the order of these extended diagonals on \(H\). **Theorem.** For arbitrary normal weights \(\varphi,\rho\) on \(M\), with the same denominator \(\psi\), \[ \varphi\le\rho\quad\Longleftrightarrow\quad q_\varphi\le q_\rho. \tag{SC.22} \] For semifinite numerators this is the usual form order of their spatial derivatives. **Proof of the forward implication.** Pointwise weight order gives \(E_\rho\subseteq E_\varphi\) and \(q_\varphi^0[\xi]\le q_\rho^0[\xi]\) on \(E_\rho\). Let \(\xi\in V_\rho\), and approximate it by \(\xi_n\in E_\rho\) in the \(q_\rho\)-norm. The inequality applied to differences shows that this sequence is Cauchy in the closed form norm of \(q_\varphi\). Completeness gives a limit in \(V_\varphi\), whose Hilbert-space image must be \(\xi\). Passing to the limits of the two energies proves \(q_\varphi[\xi]\le q_\rho[\xi]\), and also proves the needed domain inclusion. **Proof of the reverse implication.** Let \(\xi\in D_\psi\). If \(\rho(\theta(\xi,\xi))\) is finite, (SC.14) puts \(\xi\in V_\rho\). The form-order assumption and (SC.15) give \[ \varphi(\theta(\xi,\xi))\le\rho(\theta(\xi,\xi)). \] If the right side is infinite, this inequality holds automatically. Thus it holds for every bounded vector. The positive-cone identity (SC.5) and finite additivity of weights extend it to every element of \(\mathcal J_+\), including infinite values. For \(a\in M_+\), use the increasing approximants (SC.7) and normality: \[ \begin{aligned} \varphi(a) &=\sup_b\varphi(a^{1/2}u_ba^{1/2})\\ &\le\sup_b\rho(a^{1/2}u_ba^{1/2}) =\rho(a). \end{aligned} \] This proves the weight inequality on the whole positive cone. ∎ The construction is therefore injective on normal weights: equal closed spatial forms imply equal weights. An energy comparison on a smaller test set would need an additional core or positive approximation argument; that requirement is met here by SC-04 and SC-07.