# Local Fourier and closed-form foundations for the state core

*Independent local reconstruction by GPT-6 Astra (OpenAI), Ultra, 2026-10-04. Any rights in the new exposition are dedicated under CC0-1.0. Self-checked by the writing AI.*

The earlier scalar measure inputs are proved in [SC-01–02](OA-FLOW-SC.md#sc-01), [SC-03–07](OA-FLOW-SC.md#sc-03) and [SC-08–09](OA-FLOW-SC.md#sc-08): outer measure and Lebesgue measure, simple approximation, MCT/DCT, Hölder/Minkowski, Lp completeness, and one-variable calculus. Their precise replacements for the historical bindings are recorded in the input ledger. The additional input is the complete local [scalar interchange proof](#scalar-interchange): finite and sigma-finite Borel measures on real Borel sets, proved by the rectangle pi-lambda argument and MCT. Its complex version is used only when the integral of the absolute value is finite. Repeated application gives the finite iterated integrals used below; all nonnegative functions are Borel, and absolute integrability is checked before complex interchanges.

The ordinary GNS representation used in FF-5 is proved in [GNS Theorem 5.1](OA-FLOW-GNS.md#gns-theorem-5-1); its normality for a normal positive functional is proved locally in FF-5. Hilbert projection and the adjoint construction are supplied by [SF-0 and SB-0 of the operator foundations](OA-FLOW-SF.md#oa-flow.shared-foundations.sf-0); SB-1–SB-6 there supply spectral calculus with full domains. No general Fourier, Bochner, Riesz–Markov or closed-form representation theorem is imported below.

For human context, [Jerison's freely accessible MIT Fourier handout](https://math.mit.edu/~jerison/103/handouts/fourierint1.13.pdf) states the Fourier conclusions, and [Hiai's author notes, Appendix A.2 and §8.1](https://arxiv.org/pdf/2004.02383v1) discuss forms and extended-positive elements. These references replace none of the following proofs.

<a id="scalar-interchange"></a>

<a id="oa-flow.ff.1"></a>

## Scalar interchange of nonnegative integrals

We record the precise scalar interchange needed below. Let $X,Y$ be Borel subsets of the real line, with finite Borel measures $m,n$. If $h:X\times Y\to[0,\infty]$ is Borel, then both section integrals are measurable and

$$
 \int_X\!\left(\int_Y h(x,y)\,dn(y)\right)dm(x)
 =\int_Y\!\left(\int_X h(x,y)\,dm(x)\right)dn(y).
 \tag{SC1}
$$

The same statement holds for sigma-finite measures. If either iterated integral of $|h|$ is finite, it also holds for complex $h$. We prove the statement from countable additivity, simple integration and monotone convergence; no product-measure existence theorem is needed.

First, a family $\mathcal D$ of subsets of a set $Z$ is called a lambda-system if it contains $Z$, is closed under complements in $Z$, and is closed under countable disjoint unions. Such a family is closed under $B\setminus A$ whenever $A\subset B$ and both belong to the family: use the complement of the disjoint union $A\cup(Z\setminus B)$.

Here is the elementary closure argument we will use. Suppose that a family $\mathcal P$ contains $Z$ and is closed under finite intersections, and let $\mathcal D$ be the smallest lambda-system containing $\mathcal P$, obtained by intersection of all such lambda-systems. For fixed $A\in\mathcal P$, the family

$$
 \{B\in\mathcal D:A\cap B\in\mathcal D\}
$$

is a lambda-system: the complement step uses $A\setminus(A\cap B)$ and the nested-difference property just proved. It contains $\mathcal P$, so it is all of $\mathcal D$. Now fix any $B\in\mathcal D$ and run the same argument on $\{A\in\mathcal D:A\cap B\in\mathcal D\}$; the first step shows it contains $\mathcal P$. Thus $\mathcal D$ is closed under intersections. Complements and intersections give finite unions and differences; disjointifying a countable union then proves closure under arbitrary countable unions. Hence $\mathcal D$ is a sigma-algebra, and equals the sigma-algebra generated by $\mathcal P$.

Apply this with $Z=X\times Y$ and the rectangles $A\times B$, where $A,B$ are Borel. These generate the Borel sigma-algebra of $X\times Y$: open sets are countable unions of intersections with open rectangles having rational endpoints. Every Borel set $E$ has Borel sections $E_x$ and $E^y$, since the sets with that property form a sigma-algebra containing the rectangles.

Let $\mathcal C$ be the Borel sets $E$ for which $x\mapsto n(E_x)$ and $y\mapsto m(E^y)$ are measurable and their integrals agree. A rectangle belongs to $\mathcal C$, with common integral $m(A)n(B)$. The whole product also does. Complements preserve $\mathcal C$, because the section measures are subtracted from the finite constants $n(Y)$ and $m(X)$, and both iterated integrals are subtracted from $m(X)n(Y)$. Countable disjoint unions preserve $\mathcal C$, by countable additivity on each section and monotone convergence for the partial sums. Therefore $\mathcal C$ is a lambda-system, and the preceding closure argument proves that it contains all Borel sets.

This proves (SC1) for indicators, and finite linear combinations prove it for nonnegative simple functions. Choose nonnegative simple $h_k\uparrow h$. Monotone convergence in the inner and outer integrals proves their measurability and (SC1) for $h$.

For sigma-finite $m,n$, choose disjoint Borel partitions $X=\bigsqcup_i X_i$, $Y=\bigsqcup_jY_j$ into finite-measure pieces. Apply the finite result on every $X_i\times Y_j$ and sum. Monotone convergence permits the sums through the integrals. The order of two nonnegative countable sums is immaterial, since both equal the supremum of the sums over finite subsets of pairs. This proves the sigma-finite version. Finally apply it to $|h|$, then to the positive and negative parts of the real and imaginary parts of an absolutely integrable complex $h$, to prove the last assertion. This also justifies the triangle inequality under all iterated scalar integrals below.

The earlier scalar foundations are [CF Section 1](OA-FLOW-CF.md#OA-FLOW.CF.1) and [SC-01–05](OA-FLOW-SC.md#sc-01): construction and normalization of Lebesgue measure, continuity of measures, simple integration and monotone/dominated convergence. These actual earlier proofs supply every scalar input of the additional interchange argument proved above.

<a id="oa-flow.ff.2"></a>

## FF-1. Scalar density, translations and Gaussian approximate identities

Continuous compactly supported functions are dense in $L^p(\mathbb R)$ for $1\leq p<\infty$. Here are the measure details. First use value and support truncation and simple approximation from the scalar input, reducing to an indicator of a bounded Borel set. The interval-cover construction of Lebesgue outer measure supplies open outer approximations with arbitrarily small measure excess. On a fixed compact interval, complementing an outer approximation gives a compact inner approximation. Thus choose compact $F\subseteq E\subseteq O$, with $O$ bounded and open and $|O\setminus F|<\varepsilon$. A continuous function equal to one on $F$, zero off $O$, and between zero and one is given by the distance quotient
$$
\frac{d(x,\mathbb R\setminus O)}
 {d(x,F)+d(x,\mathbb R\setminus O)}.
$$
For empty $F$ use zero. Its $L^p$ error from $1_E$ is at most $\varepsilon^{1/p}$. Linear combinations finish the density proof. The same approximants and truncations give simultaneous $L^1$ and $L^2$ approximation for a member of their intersection.

Translations $T_tf(s)=f(s-t)$ are isometries on each of these spaces, by scalar change of variables. For a compactly supported continuous function, uniform continuity and a common bounded support for small translations give $\|T_tf-f\|_p\to0$. Approximation and the isometry extend this to every $f\in L^p$.

We calculate the Gaussian normalization without a planar change-of-variables theorem. Put $I=\int_{\mathbb R}e^{-x^2}\,dx$, finite by comparison outside a compact interval with $e^{-|x|}$. The scalar interchange theorem gives
$$
I^2=\int_{\mathbb R^2}e^{-(x^2+y^2)}\,dx\,dy
=\int_0^1 |\{(x,y):x^2+y^2<-\log t\}|\,dt.
$$
The second equality applies interchange twice to the nonnegative indicator $1_{\{0<t<e^{-(x^2+y^2)}\}}$. Disk sections give area
$$
\int_{-R}^R2\sqrt{R^2-x^2}\,dx=\pi R^2:
$$
substitute $x=R\sin u$ and integrate $2R^2\cos^2u$ from $-\pi/2$ to $\pi/2$, using the already proved one-variable substitution. The case $R=0$ is immediate. Hence $I^2=\pi\int_0^1-\log t\,dt=\pi$, so $I=\sqrt\pi$.

For real $r$, differentiate $\int e^{-x^2}e^{irx}\,dx$ under the integral; $|x|e^{-x^2}$ is an integrable bound on the difference quotients. Integration by parts on bounded intervals, then the Gaussian tail bound, gives $G'(r)=-(r/2)G(r)$. Since $G(0)=\sqrt\pi$, differentiation of $e^{r^2/4}G(r)$ shows it is constant. Thus
$$
\int_{\mathbb R}e^{-x^2}e^{irx}\,dx=\sqrt\pi e^{-r^2/4}.
\tag{FF1}
$$

For $\varepsilon>0$, put
$$
\gamma_\varepsilon(t)=(4\pi\varepsilon)^{-1/2}e^{-t^2/(4\varepsilon)}.
$$
It has integral one and its mass outside any fixed neighborhood of zero tends to zero as $\varepsilon\downarrow0$, by rescaling and the integrable Gaussian tails. Convolution by this probability density is contractive on $L^1$, by the triangle inequality and scalar interchange, and on $L^2$, by
$$
|(\gamma_\varepsilon*f)(s)|^2
\leq\int\gamma_\varepsilon(t)|f(s-t)|^2\,dt
$$
and interchange. The same estimates for $\gamma_\varepsilon*f-f$, using the translation continuity just proved, yield
$$
\|\gamma_\varepsilon*f-f\|_p\longrightarrow0\quad(p=1,2).
\tag{FF2}
$$
Indeed split the probability integral into $|t|<\delta$, where the translation norm is small, and its complement, bounded by $2\|f\|_p$; use the squared version of the bound for $p=2$. These arguments define the convolutions almost everywhere and justify every integral involved.

<a id="oa-flow.ff.3"></a>

## FF-2. Fourier uniqueness and the full unitary transform

For $f\in L^1(\mathbb R)$, write
$$
\mathcal F_+f(r)=(2\pi)^{-1/2}\int e^{irs}f(s)\,ds,
\qquad \mathcal F_-f(r)=(2\pi)^{-1/2}\int e^{-irs}f(s)\,ds.
$$
They are bounded continuous functions by DCT. Formula (FF1) implies the explicit identity
$$
\gamma_\varepsilon(t)=\frac1{2\pi}\int e^{-\varepsilon r^2}e^{irt}\,dr.
\tag{FF3}
$$
If either sign of the Fourier integral of $f\in L^1$ is identically zero, substitute (FF3) in $f*\gamma_\varepsilon$. The double absolute integral is bounded by $\|f\|_1(2\pi)^{-1}\int e^{-\varepsilon r^2}\,dr<\infty$. Scalar interchange shows $f*\gamma_\varepsilon=0$ everywhere; (FF2) then gives $f=0$ in $L^1$. This proves Fourier uniqueness, not only its continuous-function special case.

For $f\in C_c(\mathbb R)$, direct expansion and (FF3) give
$$
\int e^{-\varepsilon r^2}|\mathcal F_+f(r)|^2\,dr
=\int f(s)\overline{(\gamma_\varepsilon*f)(s)}\,ds.
\tag{FF4}
$$
The triple absolute integral before the $r$-integration is bounded by
$(2\pi)^{-1}\|f\|_1^2\int e^{-\varepsilon r^2}\,dr$; hence all interchanges are justified. The right side tends to $\|f\|_2^2$ by (FF2) and Cauchy–Schwarz. On the left, MCT as $\varepsilon\downarrow0$ gives
$$
\|\mathcal F_+f\|_2=\|f\|_2.
\tag{FF5}
$$
Completeness and the proved $C_c$ density extend $\mathcal F_+$ to an isometry on $L^2$; the same argument gives $\mathcal F_-$. On $L^1\cap L^2$, simultaneous approximation gives uniform convergence of the integral transforms and $L^2$ convergence of the extended transforms. A subsequence of an $L^2$-convergent sequence converges almost everywhere: choose a subsequence with summable $L^2$ increments, apply Minkowski and MCT to the sum of their absolute values, and use its finite value almost everywhere. Consequently the integral and $L^2$ definitions agree almost everywhere.

We prove surjectivity. For $f\in C_c$, $\gamma_\varepsilon*f\in L^1\cap L^2$, and absolutely convergent scalar interchange proves the convolution identity
$$
\mathcal F_+(\gamma_\varepsilon*f)(r)
=e^{-\varepsilon r^2}\mathcal F_+f(r).
\tag{FF6}
$$
Suppose $u\in L^2$ is orthogonal to the closed range of $\mathcal F_+$. Its inner product with (FF6) is zero. The function $u(r)e^{-\varepsilon r^2}$ is in $L^1\cap L^2$. Expanding the inner product against $\mathcal F_+f$ and interchanging, with absolute bound $(2\pi)^{-1/2}\|f\|_1\|u e^{-\varepsilon r^2}\|_1$, shows that the continuous function
$$
G_\varepsilon(s)=(2\pi)^{-1/2}
 \int u(r)e^{-\varepsilon r^2}e^{-irs}\,dr
$$
has integral against every $\overline f$, $f\in C_c$, equal to zero. A continuous function with that property vanishes: if its value at a point is nonzero, multiply it by a phase so that its real part is positive near that point, and test a nonnegative nonzero continuous bump there. Fourier uniqueness therefore gives $u e^{-\varepsilon r^2}=0$ almost everywhere, so $u=0$. The isometric range is all of $L^2$. Reflection gives the same result for the other sign.

Finally for $f,g\in C_c$, the absolutely convergent double integral proves
$$
\langle\mathcal F_+ f,g\rangle
=\langle f,\mathcal F_-g\rangle.
$$
Density extends this adjoint identity to $L^2$. Surjective isometries have adjoint equal to inverse, so $\mathcal F_-=\mathcal F_+^{-1}$. This fixes both sign and normalization. Translation and modulation formulas follow by a scalar substitution on $L^1\cap L^2$, then by density on $L^2$.

If $k\in L^1$, convolution by $k$ is bounded on $L^2$ with norm at most $\|k\|_1$. For a nonnegative normalized $|k|$, apply the $L^2$ estimate in FF-1 with its phase included; for general mass rescale. Approximation from $L^1\cap L^2$ and scalar interchange give
$$
\mathcal F_+(k*f)=\sqrt{2\pi}\,(\mathcal F_+k)(\mathcal F_+f)
\quad(f\in L^2).
\tag{FF7}
$$
No general Fourier inversion theorem was assumed to obtain these identities.

<a id="oa-flow.ff.4"></a>

## FF-3. Vector and predual integrals used in the core

For a separable Hilbert space $H$ with orthonormal basis $(v_n)$, measurable $H$-valued functions have measurable coordinates; conversely measurable coordinates with square-summable values define a measurable vector function by their finite coordinate sums. Their $L^2$ norm is the sum of their coordinate $L^2$ norms, by MCT. The complete scalar $L^2$ spaces and the complete Hilbert direct sum therefore prove vector $L^2$ completeness. Applying FF-2 coordinatewise gives a unitary Fourier transform on $L^2(\mathbb R,H)$. Countable nonnegative interchange shows that its norm is the scalar sum of the coordinate norms. For an integrable square-integrable vector function the transform agrees with its vector integral: approximate by finite coordinates and then simple integrable functions, using the integral norm bound below. Thus the vector Plancherel identity used later has been proved from scalar results.

For completeness, a strongly measurable vector function $F$ with $\int\|F\|<\infty$ is integrated as follows. It has simple approximants converging pointwise with norm error tending to zero and dominated by an integrable function, after first truncating its support and values. This follows from a countable dense subset of the separable range and measurable nearest-point choices. Their vector integrals are Cauchy because the norm of a simple integral is at most the integral of its norm; DCT passes to the limit. The result is independent of the approximants, has the same norm bound, and commutes with bounded linear maps. For continuous functions on compact intervals it agrees with Riemann integration, because both use uniform step approximation. For a continuous function with integrable norm it agrees with the limit of its compact integrals.

Vector $L^2$ convergence also has an almost-everywhere convergent subsequence: choose summable $L^2$ norm increments; the scalar sum of the pointwise vector norms belongs to scalar $L^2$ by Minkowski and MCT. Its finiteness almost everywhere makes the vector series Cauchy there; Hilbert completeness gives the limit and the $L^2$ tail estimate. This is the subsequence assertion needed for closed integral forms.

When $E$ is a Banach space and $F:\mathbb R\to E$ is norm continuous with integrable norm, the same compact Riemann and tail construction defines its integral without a global separability assumption. A compact parameter image is separable by uniform continuity and rational meshes, and the real line is a countable union of compact intervals. In particular this applies to norm-continuous paths in a von Neumann predual. Their integrals remain in that Banach predual and therefore give normal functionals. Arbitrary measurable predual paths needed below are limits in integral norm of finite sums of such paths, by approximating the Hilbert vectors by simple functions; the displayed Cauchy–Schwarz estimates in the main proof justify that limit.

<a id="oa-flow.ff.5"></a>

## FF-4. Closed nonnegative forms, including a nondense finite domain

A nonnegative extended form $q$ on a Hilbert space is a function into $[0,\infty]$ whose finite domain $D_q$ is linear, and whose restriction there is a nonnegative quadratic form with the usual complex polarization. It is closed when $D_q$, equipped with
$$
\langle v,w\rangle_q=\langle v,w\rangle+q(v,w),
$$
is complete. We now prove its spectral representation, allowing $D_q$ not to be dense.

Put $H_0=\overline{D_q}$, let $V=D_q$ with this graph inner product, and let $j:V\to H_0$ be inclusion. It is bounded, injective, contractive, and has dense range. Thus $B=jj^*$ is a positive contraction on $H_0$ with zero kernel. Its square root is provided by the proved spectral calculus. On $\operatorname{Ran}j^*$ the rule
$$
j^*u\longmapsto B^{1/2}u
$$
is an isometry, since both squared norms equal $\langle Bu,u\rangle$. The first range is dense in $V$, because $(\operatorname{Ran}j^*)^\perp=\ker j=0$; the second is dense in $H_0$, because $B^{1/2}$ has zero kernel. The isometry extends to a unitary $W:V\to H_0$. Taking adjoints of $Wj^*=B^{1/2}$ gives
$$
j=B^{1/2}W.
\tag{FF8}
$$
This is a proof of the particular polar factorization needed here, not a polar-decomposition import. It yields
$$
D_q=\operatorname{Ran}B^{1/2},\qquad
\|B^{-1/2}v\|^2=\|v\|^2+q(v).
$$
The exact inverse-range and spectral-domain calculus therefore gives a positive self-adjoint operator $A=B^{-1}-I$ on $H_0$, with
$$
D(A^{1/2})=D_q,\qquad q(v)=\|A^{1/2}v\|^2.
\tag{FF9}
$$
Declare $q(v)=\infty$ off this domain, including whenever $v$ has a nonzero component in $H_0^\perp$. The pair $(A,H_0^\perp)$ is uniquely determined: the graph Hilbert space and its inclusion determine $B$, and then spectral calculus determines $A$.

Conversely every such pair gives a closed form, because the graph norm of the closed operator $A^{1/2}$ is complete. It is lower semicontinuous for the Hilbert norm: its form is the supremum of the continuous quadratic functions of the bounded spectral truncations
$$
h_n=\min(A,n)\ \text{ on }H_0,\qquad h_n=nI\ \text{ on }H_0^\perp.
\tag{FF10}
$$
If a form is initially known to be lower semicontinuous, its graph norm is complete. A graph-Cauchy sequence $v_n$ converges in Hilbert norm to $v$; lower semicontinuity applied to $v_n-v_m$, with $m\to\infty$, gives $q(v_n-v)$ bounded by the graph-Cauchy tail. It follows that $v\in D_q$ and $v_n\to v$ in graph norm. Thus either characterization of closedness is available.

For the integral form $q(v)=\int\|T_t v\|^2\,dt$, where each $T_t$ is bounded and the vector functions are measurable, graph completeness follows directly. A graph-Cauchy sequence gives a Hilbert limit $v$ and a vector $L^2$ limit $F$. The a.e. subsequence result of FF-3 identifies $F(t)=T_tv$, since $T_t v_n\to T_tv$ for each fixed $t$. This proves completeness and retains the possible infinite part.

<a id="oa-flow.ff.6"></a>

## FF-5. The concrete extended-positive cone and its covariance

For a concrete von Neumann algebra $N$, define its extended-positive spectral elements to be the pairs of FF-4 whose finite-part spectral projections and infinite-part projection belong to $N$. Equivalently their closed forms are invariant under every unitary in $N'$: one direction follows by spectral cutoffs, and the other follows because invariance gives a unitary on the graph Hilbert space, intertwines its inclusion, and therefore commutes with $jj^*$ and all its spectral projections. The unitary-span and bicommutant argument is proved in SB-6.

Here are proofs that this cone has the required order, sum, sandwich and supremum operations and that faithful normal isomorphisms preserve them. They remove the unproved intrinsic-cone premise from the selected lesson. Regard
$$
R_q=(I+A)^{-1}\ \text{ on }H_0,\qquad R_q=0\ \text{ on }H_0^\perp
$$
as a bounded positive contraction in $N$. It determines $q$ completely. Its two variational identities are
$$
\begin{aligned}
\|v\|^2+q(v)
 &=\sup_w\bigl(2\operatorname{Re}\langle v,w\rangle
                       -\langle R_qw,w\rangle\bigr),\\
\langle R_qw,w\rangle
 &=\sup_{v\in D_q}\bigl(2\operatorname{Re}\langle w,v\rangle
                       -\|v\|^2-q(v)\bigr).
\end{aligned}
\tag{FF11}
$$
To verify the first, write $R=R_q$. If the supremum is finite, scaling $w$ by real and complex scalars shows that $w\mapsto\langle v,w\rangle$ vanishes on $\ker R$ and is bounded for the norm $\|R^{1/2}w\|$. Hilbert representation on the closure of that range gives a vector $z\perp\ker R$ with $v=R^{1/2}z$. Conversely such a vector makes the supremum $\|z\|^2$, by completing the square and using density of $\operatorname{Ran}R^{1/2}$. This equals the left side by FF-4. For the second write $v=R^{1/2}z$ with $z\perp\ker R$, and complete the square in $z$; the maximum is attained at $z=R^{1/2}w$, equivalently at $v=Rw$.

It follows that
$$
q\leq r\ \text{ pointwise as forms}\quad\Longleftrightarrow\quad R_q\geq R_r.
\tag{FF12}
$$
The right-to-left implication uses the first variational identity and the left-to-right implication the second. In particular form order is encoded by an ordinary bounded-operator inequality.

The sum $q+r$, and the sandwich $v\mapsto q(xv)$ for bounded $x$, are lower semicontinuous nonnegative quadratic forms with linear finite domains. Their lower semicontinuity follows respectively from addition of nonnegative lower semicontinuous functions and from continuity of $v\mapsto xv$; FF-4 therefore represents them. Their commutant invariance puts them in the same extended cone when $q,r$ belong to it and $x\in N$. An increasing directed supremum of such forms is again a lower semicontinuous quadratic form on its finite domain: the parallelogram identities and homogeneity pass through the directed supremum, using a common index for the finitely many terms involved. It too belongs to the cone.

We need the exact convergence of their bounded resolvents. If $q_\alpha\uparrow q$, (FF12) gives $R_\alpha\downarrow R$ strongly for a positive contraction $R\in N$. To justify the strong limit, each scalar quadratic value has a decreasing limit; polarization and Hilbert representation give $R$, and
$\|(R_\alpha-R)v\|^2\leq\langle(R_\alpha-R)v,v\rangle$ because $0\leq R_\alpha-R\leq I$. Let $v_\alpha=R_\alpha w\to Rw=v$. By the second formula of (FF11), $v_\alpha$ minimizes
$$
J_\alpha(u)=\|u\|^2+q_\alpha(u)-2\operatorname{Re}\langle w,u\rangle
$$
and its minimum is $-\langle R_\alpha w,w\rangle$. For any fixed $\beta$, lower semicontinuity of $q_\beta$ and $\alpha\geq\beta$ give $J(v)\leq\liminf_\alpha J_\alpha(v_\alpha)$ after taking the supremum over $\beta$. Comparison with any fixed test $u$, followed by its monotone limit, gives the reverse minimum bound. Thus $v$ minimizes $J$, and (FF11) identifies $v=R_qw$. Consequently
$$
R_{q_\alpha}\downarrow R_q\quad\text{strongly}.
\tag{FF13}
$$
The argument also covers nets and nondense domains. The minimum is finite, since $u=0$ is an admissible test.

Let $\rho:N\to N_1$ be a normal unital \*-homomorphism. Transport the spectral resolution of a pair by $\rho$, including the infinite-part projection; bounded spectral transport was proved in SB-6. Its resolvent is $\rho(R_q)$. For a sum use the increasing bounded forms $h_n+k_n$, with truncations from (FF10); their resolvents are $(I+h_n+k_n)^{-1}$. Normality preserves their decreasing strong limit by the same positive-operator estimate, and a unital homomorphism preserves their bounded inverses. Formula (FF13) therefore proves transport of sums. The bounded forms $x^*h_nx$ prove transport of sandwiches in the same way. Formula (FF12) proves order preservation; if $\rho$ is faithful, its inverse on the image reflects the bounded inequalities, hence also form order. Formula (FF13) proves preservation of increasing suprema. These establish the asserted intrinsic operations under a faithful normal change of representation.

Finally, for a normal positive functional $\omega$, its spectral value is
$\widehat\omega(q)=\sup_n\omega(h_n)$. This is the vector form in its normal GNS representation. The needed GNS normality can be verified directly: for $a_\alpha\uparrow a$, the values $\omega(x^*(a-a_\alpha)x)$ decrease to zero on the dense GNS vectors; the positive inequality $D_\alpha^2\leq C D_\alpha$ and a common operator bound give strong convergence there and then everywhere. Thus the GNS representation is order normal. Its spectral projections transport by the preceding argument, even when the representation has a kernel. In that representation $\omega(h_n)=\langle\pi_\omega(h_n)\Omega,\Omega\rangle$, and MCT gives the vector-form value, including the infinite part. The proved transport of sums, sandwiches and suprema now proves the usual evaluation rules. In particular if $\omega$ is represented by a vector in the given representation, its extended evaluation is exactly that vector's closed-form value.

These are the precise cone properties used by the construction of the operator-valued average. They follow from explicit closed forms and bounded resolvents, and do not depend on a sketch of a general cone representation theorem.