Conformal-net reconstruction of a VOA under generating and energy hypotheses

For a local algebra $M=\mathcal{A}(I)$, $a\in D(S)$ and $x^{\mathrm{\prime }}\in M^{\mathrm{\prime }}$, define $T_a^0{x}^{\mathrm{\prime }}\mathrm{\Omega }=x^{\mathrm{\prime }}a$. If $x_j\mathrm{\Omega }\to a$ and $x_j^{\ast }\mathrm{\Omega }\to Sa$, commutation with $x^{\mathrm{\prime }},y^{\mathrm{\prime }}\in M^{\mathrm{\prime }}$ and passage to these graph limits gives $⟨T_a^0{x}^{\mathrm{\prime }}\mathrm{\Omega },y^{\mathrm{\prime }}\mathrm{\Omega }⟩=⟨x^{\mathrm{\prime }}\mathrm{\Omega },T_{Sa}^0{y}^{\mathrm{\prime }}\mathrm{\Omega }⟩$. Thus $T_{Sa}^0\subset (T_a^0{)}^{\ast }$ on the dense commutant-vacuum domain, proving closability. A commutant unitary preserves that domain and commutes with $T_a^0$ there; closure and its inverse show affiliation of $\overline{T_a^0}$ with $M$. This proves the raw FJ operator assertion, with no unbounded commutator assumed from formal notation.

Fix $k\ge 0$, $v\in D(N^k)$, and an interval $J$ with $\bar{J}\subset I^{\mathrm{\prime }}$. By F1, $\mathcal{A}(J)\mathrm{\Omega }$ is dense in $H$. Choose a nonnegative $\chi \in C_c^{\mathrm{\infty }}(\mathbb{R})$ with $\int \chi (t)dt=1$, set ${\chi }_{ϵ}(t)={ϵ}^{-1}\chi (t\mathrm{/}ϵ)$, and define $K_{ϵ}\xi =\int {\chi }_{ϵ}(t)U(r_t)\xi dt$ as a Bochner integral in $H$ for each $\xi \in H$. Use the Fourier convention ${\widehat{\chi }}_{ϵ}(s)=\int {\chi }_{ϵ}(t)e^{-ist}dt$; the spectral theorem gives $K_{ϵ}={\widehat{\chi }}_{ϵ}(-L_0)$. Integration by parts gives ${\sup }_{\lambda \ge 0}(1+\lambda {)}^k\mathrm{\mid }{\widehat{\chi }}_{ϵ}(-\lambda )\mathrm{\mid }<\mathrm{\infty }$ for each fixed $ϵ$, so $K_{ϵ}$ maps $H$ boundedly to $D(N^k)$. Equip this domain with graph norm $\mathrm{\parallel }w{\mathrm{\parallel }}_{D(N^k)}=\mathrm{\parallel }w\mathrm{\parallel }+\mathrm{\parallel }{N}^{k}w\mathrm{\parallel }$. The multiplier commutes with $N$, and strong continuity of rotations gives $K_{ϵ}v\to v$ in this graph norm as $ϵ\downarrow 0$. Choose ${ϵ}_m\downarrow 0$ small enough that $r_{t}J\subset I^{\mathrm{\prime }}$ on $\mathrm{supp}{\chi }_{{ϵ}_m}$. For each $m$, density of $\mathcal{A}(J)\mathrm{\Omega }$ and boundedness of the fixed map $K_{{ϵ}_m}:H\to D(N^k)$ allow a choice $X_m\in \mathcal{A}(J)$ with $\mathrm{\parallel }{K}_{{ϵ}_m}(X_m\mathrm{\Omega }-v){\mathrm{\parallel }}_{D(N^k)}<1\mathrm{/}m$. Set $B_m$ equal to the weak-operator integral $\int {\chi }_{{ϵ}_m}(t)U(r_t)X_{m}U(r_t{)}^{\ast }dt$, meaning its matrix coefficients are the integrals of those of the integrand. Covariance and isotony put every integrand in $\mathcal{A}(I^{\mathrm{\prime }})$, so weak-operator closure gives $B_m\in \mathcal{A}(I^{\mathrm{\prime }})\subset M^{\mathrm{\prime }}$. Since rotations fix $\mathrm{\Omega }$, $B_m\mathrm{\Omega }=K_{{ϵ}_m}{X}_m\mathrm{\Omega }\in H^{\mathrm{\infty }}$. The two approximation bounds imply $B_m\mathrm{\Omega }\to v$ in graph norm. Thus $M^{\mathrm{\prime }}\mathrm{\Omega }\cap H^{\mathrm{\infty }}$ is dense in every $D(N^k)$. Apply the energy inequality to differences and use closedness of the FJ operator: its domain contains $D(N^k)$ and the same bound holds there.

Raw FJ operators are covariant under the given Möbius representation: conjugation transports $M^{\mathrm{\prime }}\mathrm{\Omega }$, the Tomita graph and the vector $a(f)$, and hence their closures. For rotations this follows directly termwise from the vacuum series, and for the three Möbius infinitesimals it follows from the lowest-weight ladder relations and the change of circle test density; differentiating these series gives $[L_j,Y(a,z)]=z^{j+1}{\mathrm{\partial }}_{z}Y(a,z)+(j+1)d{z}^{j}Y(a,z)$ for $j=-1,0,1$ and quasi-primary $a\in H_d$. The integrated identities follow first on the convergent lowest-weight vectors and then on the commutant-vacuum graphs by the same conjugation. In particular, differentiating rotation covariance gives $[L_0,Y(a,f)]=iY(a,f^{\mathrm{\prime }})$. Repeated commutation with $N$ and the preceding extended energy bound gives every Sobolev output bound, so these smearings act continuously $H^{\mathrm{\infty }}\to H^{\mathrm{\infty }}$. Finite partitions and a joint refinement now glue them globally by F2.

Global Fourier tests give energy modes $a_n$ with $a_n{H}_k\subset H_{k-n}$ and polynomial estimates $\mathrm{\parallel }{a}_{n}v\mathrm{\parallel }\le C(1+\mathrm{\mid }n\mathrm{\mid }{)}^s\mathrm{\parallel }{N}^{r}v\mathrm{\parallel }$ on $D$. The state-field modes are $a_{(m)}=a_{m-d+1}$. The vacuum series is exactly $Y(a,z)\mathrm{\Omega }=e^{z{L}_{-1}}a$, and the preceding Möbius relation gives $[L_{-1},Y(a,z)]={\mathrm{\partial }}_{z}Y(a,z)$. Applying the graph identity in the first step to the corrected Tomita relation, then the smooth commutant graph density, gives $a_n^{†}=(-1{)}^d(\theta a{)}_{-n}$ on the finite graded blocks. If $\theta a$ has initially unspecified quasi-primary weight, comparison of this adjoint identity with the $L_1,L_{-1}$ covariance of its two nonzero fields forces that weight to equal $d$: the coefficients in $[L_{-1},(\theta a{)}_{-n}]$ differ by precisely the two weights. Zero states cause no restriction. Thus Hermitian quasi-primary combinations $a+(-1{)}^d\theta a$ and $i(a-(-1{)}^d\theta a)$ have Hermitian energy modes and span the same generating fields.

For two generating fields, their polynomial mode energy estimates bound every commutator coefficient on $H^{\mathrm{\infty }}$ by a fixed polynomial in the two mode numbers times one Sobolev norm of the input. Consequently their operator-valued distribution has finite order bounded independently of the input vector. Affiliation and net locality make it vanish for disjoint test supports, so its support is the diagonal. In a coordinate normal to that diagonal, an order-$q$ distribution supported there is killed by every smooth function vanishing to order $q+1$: multiply a test function by a cutoff of width $ϵ$ in the normal coordinate, subtract its normal Taylor polynomial through degree $q$, and bound every derivative through order $q$ by $O(ϵ)$; support permits this cutoff and the limit is zero. Since $z-w$ is a nonvanishing unit times that normal coordinate, $(z-w{)}^{q+1}$ kills the commutator. This proves formal locality with one finite exponent, not an input-dependent formal assertion.

Let $W$ be the algebraic span of finite generating-mode words on $\mathrm{\Omega }$. It is graded and stable under every generating mode and its graded adjoint. The orthogonal projection $P$ onto its Hilbert closure preserves each finite-dimensional $H_k$, and $P{H}_k=W\cap H_k$, since that finite-dimensional subspace is closed. Hence $PD=W$. Adjoint invariance makes $P$ commute with all generating modes on $D$. As $P$ commutes with $L_0$, it preserves $H^{\mathrm{\infty }}$; energy bounds and Fourier convergence show commutation with every generating smearing on that core and hence with its closure and adjoint. By the net's F-generating hypothesis $P$ lies in the global net commutant. Irreducibility and $P\mathrm{\Omega }=\mathrm{\Omega }$ give $P=1$. Therefore $W=D$, establishing the algebraic spanning hypothesis that does not follow merely by writing the net generating condition.

The creative translation-covariant local generating fields satisfy every hypothesis of F4, so they reconstruct the unique graded vertex algebra on $D$ and its full Jacobi identity. For the energy bound fix homogeneous states $A,B$ of weights $d_A,d_B$, an integer $n$, and $C=A_{(n)}B$. If $C\mathrm{\ne }0$, its weight is $d_C=d_A+d_B-n-1$. Keep the convention from 4.1: $A_{(r)}=A_{r-d_A+1}$ and $B_{(r)}=B_{r-d_B+1}$, where parenthesized indices are vertex modes and unparenthesized indices are energy modes. The Jacobi coefficient formula in F4, with its first index zero, is the vertex-mode iterate identity. To compute the energy mode $C_k$, its vertex-mode index is $k+d_C-1$. Substitution in each of the four mode indices gives $(A_{(n)}B{)}_k={\sum }_{j\ge 0}(-1{)}^j\left(\genfrac{}{}{0px}{}{n}{j}\right)(A_{n-j-d_A+1}{B}_{k+j+d_A-n-1}-(-1{)}^n{B}_{k-j+d_A-1}{A}_{j-d_A+1})\mathrm{.}$ The notation on the left means the energy mode of the state $A_{(n)}B$. Both products on the right have total energy index $k$, as they must. For $A=B=\omega$, $d_A=d_B=2$ and $n=0$, this is $[L_{-1},L_{k+1}]=-(k+2)L_k$, the energy mode of $L_{-1}\omega$; it is not $[L_0,L_k]$. The zero-state case has zero field and needs no estimate.

On an input $v\in H_h$, the first product vanishes when its rightmost mode index $k+j+d_A-n-1$ exceeds $h$, and the second vanishes when its rightmost index $j-d_A+1$ exceeds $h$, because there are no negative-energy output spaces. Hence every nonzero term has $0\le j\le \max \{0,h-k-d_A+n+1,h+d_A-1\}\le h+\mathrm{\mid }k\mathrm{\mid }+C_{A,n}\mathrm{.}$ This proves vectorwise truncation of the correctly reindexed formula, including negative $n$. For fixed integer $n$, $\mathrm{\mid }\left(\genfrac{}{}{0px}{}{n}{j}\right)\mathrm{\mid }$ grows at most polynomially in $j$: for $n\ge 0$ it is zero for $j>n$, and for $n<0$ its absolute value is $\left(\genfrac{}{}{0px}{}{j-n-1}{-n-1}\right)$. Suppose the fields of $A$ and $B$ have polynomial mode bounds. In each surviving product, all mode indices and the intermediate nonnegative energy are at most a constant times $1+h+\mathrm{\mid }k\mathrm{\mid }$ in absolute value. Apply the two mode bounds successively, then sum the polynomially many terms with their polynomial binomial coefficients. This gives $\mathrm{\parallel }{C}_{k}v\mathrm{\parallel }\le C(1+h+\mathrm{\mid }k\mathrm{\mid }{)}^M\mathrm{\parallel }v\mathrm{\parallel }\le C^{\mathrm{\prime }}(1+\mathrm{\mid }k\mathrm{\mid }{)}^M(1+h{)}^M\mathrm{\parallel }v\mathrm{\parallel }$, with constants depending on $A,B,n$ but not $h,k,v$.

F3 supplies $L_n$ and $\omega =L_{-2}\mathrm{\Omega }$ with creative field $\mathcal{T}(z)=\sum L_n{z}^{-n-2}$. Before duality, real supported stress smearings are affiliated only with $\mathcal{A}(I^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$: their local diffeomorphism flows fix $\mathcal{A}(I^{\mathrm{\prime }})$ pointwise. This suffices for commutation with opposite supported FJ fields. The same finite-order diagonal argument therefore makes $\mathcal{T}$ formally local with every generating field. It is translation covariant and creative with state $\omega$, so the reconstruction uniqueness argument identifies it with $Y(\omega ,z)$. The Virasoro identities and grading in F3 make this the required conformal vector; finite-dimensionality and nonnegative grading are inherited from $H$.

Each term of $C_k$ sends $H_h$ to $H_{h-k}$. For a finite sum $v={\sum }_h{v}_h$, these outputs are orthogonal; squaring the last estimate and summing therefore gives the full energy inequality $\mathrm{\parallel }{C}_{k}v\mathrm{\parallel }\le C^{\mathrm{\prime }}(1+\mathrm{\mid }k\mathrm{\mid }{)}^M\mathrm{\parallel }(1+L_0{)}^{M}v\mathrm{\parallel }$. Starting from the generating fields and the identity field, induction on the finite bracketed field-product expressions in F4 proves polynomial energy bounds for every reconstructed state, rather than inserting unshifted modes into an energy estimate. The same estimates with additional powers of $1+L_0$ give continuity on $H^{\mathrm{\infty }}$; closability follows after the adjoint identity below.

Apply the Hermitian-generators lemma F4 to the real combinations above and the given positive Möbius form. It constructs an intrinsic antiunitary involution $\mathrm{\Theta }$, all descendant adjoints and the invariant bilinear form, fixes $\omega$ and gives $\mathrm{\Theta }a=\theta a$ on the original family $F$. Only this equality on $F$ is used; global mode intertwining by the originally supplied $\theta$ is not presumed. For $v\in V_0$, positivity gives $\mathrm{\parallel }{L}_{-1}v{\mathrm{\parallel }}^2=⟨v,L_1{L}_{-1}v⟩=0$. Thus $Y(v,z)$ is a constant field commuting with every generating field. The product $v\ast w=v_{(-1)}w$ makes $V_0$ a finite-dimensional commutative associative positive star algebra, faithfully represented on itself, with $v\mapsto \mathrm{\Theta }v$ as star. Simultaneous diagonalization gives self-adjoint primitive idempotents. Their constant fields are self-adjoint idempotents on $D$, hence bounded orthogonal projections, since $\mathrm{\parallel }Eu{\mathrm{\parallel }}^2=⟨Eu,u⟩\le \mathrm{\parallel }Eu\mathrm{\parallel }\mathrm{\parallel }u\mathrm{\parallel }$. They commute with $L_0$ and all closed generating fields, hence with the irreducible net. Only one primitive idempotent can remain; consequently $V_0=\mathbb{C}\mathrm{\Omega }$.

The resulting VOA is simple. An ideal is graded by its $L_0$ stability and finite spectral interpolation on each vector. Its graded orthogonal complement is stable under every mode because the full invariant adjoint is a finite sum of state modes preserving the ideal. A proper ideal has zero grade-zero space, so its orthogonal complement contains $\mathrm{\Omega }$ and hence every generated state. It is therefore all of $V$, and the ideal is zero. This proves simplicity using positivity, rather than assuming it in reconstruction.

It remains to prove equality of local algebras for all reconstructed fields. Fix the upper semicircle $I_0$ and its standard data $(M,\mathrm{\Omega },S,J,\mathrm{\Delta })$ from F5. In $x=\tan (\vartheta \mathrm{/}2)$ coordinates the positive parabolic groups have velocities $1+\cos \vartheta$ and $1-\cos \vartheta$. Their generators on $H^{\mathrm{\infty }}$ are $P_{\pm }=L_0\pm B$, where $B=(L_1+L_{-1})\mathrm{/}2$. Put $D_0=(L_1-L_{-1})\mathrm{/}(2i)$ and $V(t)=e^{it\overline{D_0}}$, the supplied Möbius dilation group. The brackets give $i[D_0,L_0]=B$ and $i[D_0,B]=L_0$; equivalently, the adjoint action of this genuine group on its Lie algebra gives $V(t)L_{0}V(t{)}^{\ast }=\cosh (t)L_0+\sinh (t)B$ on its common invariant smooth core. Thus, for every $\xi \in H^{\mathrm{\infty }}$, $2e^{-t}V(\pm t)L_{0}V(\pm t{)}^{\ast }\xi =P_{\pm }\xi +e^{-2t}{P}_{\mp }\xi \to P_{\pm }\xi$ in norm as $t\to +\mathrm{\infty }$. Each left-hand quadratic form is nonnegative, since $L_0\ge 0$ and $V(\pm t{)}^{\ast }\xi \in D(L_0)$. Hence $⟨P_{\pm }\xi ,\xi ⟩\ge 0$. F3 makes these real stress smearings essentially self-adjoint on $H^{\mathrm{\infty }}$; approximating any vector in a closure domain in graph norm preserves this inequality. Their self-adjoint generators are therefore nonnegative. Positive translations preserve $I_0$; negative special-conformal parameters preserve it. Apply the half-sided lemma to $M$ for translations, and to $M^{\mathrm{\prime }}$ for the positive special-conformal group (the reversed inclusion follows by taking commutants). The modular operator for $M^{\mathrm{\prime }}$ is ${\mathrm{\Delta }}^{-1}$. It follows that ${\mathrm{\Delta }}^{it}$ has the same conjugation on both parabolic groups as $e^{itK}$, where $K=i\pi (L_1-L_{-1})$ dilates $x$ by $e^{-2\pi t}$; $J$ implements their parameter reflection. The two groups generate the Möbius group, so these assertions hold for that group, including $J{L}_{n}J=L_n$ for $n=-1,0,1$ on the invariant core.

Set $Z(t)={\mathrm{\Delta }}^{it}{e}^{-itK}$. The preceding conjugation relations imply that $Z(t)$ commutes with the Möbius representation, forms a strongly continuous unitary group and fixes $\mathrm{\Omega }$. Both factors normalize $M$ by F5 and Möbius covariance; since every local interval is a Möbius image of $I_0$, $Z(t)$ normalizes every local algebra. It is therefore a global net gauge, not yet assumed trivial. It commutes with $L_0$ and preserves the finite-dimensional grades. The conjugated generating fields $Z(t)Y(a,z)Z(t{)}^{\ast }$ retain their energy bounds, creativity, translation covariance and net locality, and have state $Z(t)a$. Formal uniqueness with the original generating fields identifies them with $Y(Z(t)a,z)$. Mode-word spanning extends this to all states; hence $Z(t)$ is a unitary vertex automorphism commuting with the Möbius modes. It fixes the conformal vector by the same positive weight-two uniqueness argument in the PCT lemma, so it is a VOA automorphism.

Write $Z(t)=e^{itQ}$. Every unitary VOA automorphism commutes with the intrinsic $\mathrm{\Theta }$: conjugate its opposite-adjoint field construction by that unitary and take the vacuum state. Thus $\mathrm{\Theta }Q\mathrm{\Theta }=-Q$. Also $JZ(t)J=Z(t)$, by the polar identity $J{\mathrm{\Delta }}^{it}J={\mathrm{\Delta }}^{it}$ and the reflected Möbius action on $K$, giving $JQJ=-Q$. Consequently $W=J\mathrm{\Theta }$ is a unitary commuting with $Q$. All these statements restrict to ordinary finite matrices on each $H_d$, so no unbounded exponential is moved through an unrelated domain.

Since $Q$ commutes with the Möbius representation, ${\mathrm{\Delta }}^{1\mathrm{/}2}=e^{K\mathrm{/}2}{e}^{Q\mathrm{/}2}$ on its joint spectral domain. For a generating quasi-primary $a\in H_d$, $e^{Q\mathrm{/}2}a$ is defined in the finite-dimensional $H_d$, and the upper-semicircle identity F5 applies to its entire lowest-weight descendants. The corrected Tomita identity for arbitrary complex upper-supported $f$ therefore gives $\theta a=J{e}^{Q\mathrm{/}2}a$: explicitly $e^{K\mathrm{/}2}b(f)=(-1{)}^{d}b(f\circ j)$ and $Jb(f\circ j)=(Jb)(\bar{f})$, with $j(z)=z^{-1}$. Equality of these smeared vacuum vectors determines the underlying state by its Fourier coefficients. As $\mathrm{\Theta }a=\theta a$ on $F$, this reads $Wa=e^{Q\mathrm{/}2}a$. Decompose the finite-dimensional $H_d$ into $Q$ eigenspaces; $W$ is unitary and commutes with $Q$, so each nonzero component $a_q$ satisfies $\mathrm{\parallel }{a}_q\mathrm{\parallel }=e^{q\mathrm{/}2}\mathrm{\parallel }{a}_q\mathrm{\parallel }$. Hence $Qa=0$ for every $a\in F$. The VOA automorphism $Z(t)$ fixes all generators and the vacuum, thus every mode word. Density gives $Z(t)=1$, ${\mathrm{\Delta }}^{it}=e^{itK}$, and $Ja=\mathrm{\Theta }a$ on $F$. This proves the geometric modular action instead of adding it as an axiom.

Möbius covariance and $JMJ=M^{\mathrm{\prime }}$ give $J\mathcal{A}(I)J=\mathcal{A}((jI{)}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$ for every interval $I$, without Haag duality. For a generating field form its reflected conjugate by $J$: on energy modes these are $J{a}_{n}J$, and its smearing with $f$ is $JY(a,\bar{f}\circ j)J$, affiliated with $\mathcal{A}(I^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$ for support in $I$. It therefore commutes with the original FJ fields in $I^{\mathrm{\prime }}$. It has the same energy bounds, translation covariance and creative state $Ja=\mathrm{\Theta }a$. Formal uniqueness gives $J{a}_{n}J=(\mathrm{\Theta }a{)}_n$ for every generating mode. Applying this to all mode words yields $J=\mathrm{\Theta }$ on $D$ and hence on $H$. On the other hand the intrinsic PCT maps the F-generated algebra $\mathcal{A}(I)$ onto $\mathcal{A}(jI)$. Combining these two conjugation formulas gives Haag duality $\mathcal{A}(I^{\mathrm{\prime }})=\mathcal{A}(I{)}^{\mathrm{\prime }}$ for all $I$.

The net is inner additive directly from its F-generating definition: each compactly supported test in $I$ is a finite sum of tests supported in intervals compactly contained in $I$, so $\mathcal{A}(I)={\bigvee }_{K⋐I}\mathcal{A}(K)$. Taking commutants and using Haag duality gives outer regularity $\mathcal{A}(I)={\bigcap }_{J\supset \bar{I}}\mathcal{A}(J)$. Here $J$ ranges over slightly larger proper intervals; the complementary smaller intervals exhaust $I^{\mathrm{\prime }}$. These are algebra equalities, not assumptions about strong additivity at a removed interior point.

For a nonempty interval $I$ let ${\mathcal{P}}_F(I)\mathrm{\Omega }$ be the linear span of finite products of smooth F-smearings compactly supported there, applied to $\mathrm{\Omega }$. This space is Hilbert dense. Indeed, if $v$ is orthogonal to it, pair $v$ with $e^{i{t}_1{L}_0}{A}_1(f_1)e^{i{t}_2{L}_0}\cdots A_n(f_n)\mathrm{\Omega }$. Each relative time variable extends holomorphically into its upper half-plane and continuously to the boundary: positive energy gives contractions on every Sobolev space, and the field-product energy bounds lose only finitely many Sobolev orders starting from $\mathrm{\Omega }\in H^{\mathrm{\infty }}$. For all sufficiently small real times the transported supports remain in $I$, so the boundary function is zero. In one variable, reflection across this zero boundary interval (or Morera on the upper function glued to zero) and the identity theorem make it vanish throughout the half-plane and its real boundary. Apply this successively to each relative variable, fixing the others first in their small real intervals; it then vanishes for all real relative times. Independent rotations and a finite partition of arbitrary tests into short supports show orthogonality to every global F-mode word. The already proved word spanning implies $v=0$.

This local polynomial space is also a core for every $N^k$ after an arbitrarily small enlargement of its interval. Choose $I_1⋐I$ and average common small rotations of ${\mathcal{P}}_F(I_1)\mathrm{\Omega }$ with a smooth time kernel. For fixed kernel, the resulting Fourier multiplier of $L_0$ is bounded $H\to D(N^k)$. Hilbert density just proved, followed by an approximate identity in time, gives graph density. Each averaged product can itself be approximated in every Sobolev norm by finite sums of products with tests supported in $I$: write its common-rotation integral as a compact smooth test function in all field variables, approximate that function in every smooth seminorm by finite sums of separable tests (local Fourier series times cutoffs suffice), and use continuity of the energy-bounded product on $H^{\mathrm{\infty }}$. Thus ${\mathcal{P}}_F(I)\mathrm{\Omega }$ has the claimed graph density. Since every state field has an energy bound, it is a graph core for its closure as well.

Fix a quasi-primary state $a\in V_d$ and a test $f$ compactly supported in $I$. By the upper-semicircle identity, $J=\mathrm{\Theta }$ and Möbius covariance, $a(f)\in D(S_I)$ and $S_{I}a(f)=Y(a,f{)}^{†}\mathrm{\Omega }=(-1{)}^d(\mathrm{\Theta }a)(\bar{f})$. This is a proved relation for every reconstructed quasi-primary, not an extension of the original FJ hypothesis by notation. Fix $J_1\supset \bar{I}$ and a bounded $A\in \mathcal{A}(J_1^{\mathrm{\prime }})$. Smooth its conjugation orbit under sufficiently small rotations to obtain $A_s\in \mathcal{A}(I^{\mathrm{\prime }})$, converging strongly with adjoints to $A$ and mapping $H^{\mathrm{\infty }}$ to itself. Derivatives of that orbit are bounded after smoothing; commuting powers of $L_0$ then gives every asserted Sobolev-domain preservation.

Put $M=\mathcal{A}(I)$, and let $X_1,X_2\in {\mathcal{P}}_F(I^{\mathrm{\prime }})$. On $H^{\mathrm{\infty }}$ set $\eta =X_1^{†}{A}_s{X}_2\mathrm{\Omega }$ and ${\eta }^{\mathrm{\prime }}=X_2^{†}{A}_s^{†}{X}_1\mathrm{\Omega }$. The closed polynomial operators are affiliated with $M^{\mathrm{\prime }}$ by locality and duality. For $B\in M$, bounded $B$ and $B^{\ast }$ preserve their domains and commute with them; all displayed vectors are in the smooth domain. Moving these operators in the Hilbert pairing gives $⟨S_{I}B\mathrm{\Omega },\eta ⟩=⟨B^{\ast }\mathrm{\Omega },X_1^{†}{A}_s{X}_2\mathrm{\Omega }⟩=⟨X_2^{†}{A}_s^{†}{X}_1\mathrm{\Omega },B\mathrm{\Omega }⟩$. The defining antilinear adjoint relation and the Tomita graph core therefore prove $\eta \in D(S_I^{\ast })$ and $S_I^{\ast }\eta ={\eta }^{\mathrm{\prime }}$. This is the exact domain assertion used next.

Use the preceding identity, the just-proved $S_{I}a(f)=Y(a,f{)}^{†}\mathrm{\Omega }$, and formal commutation of the reconstructed fields with $X_i$ on $H^{\mathrm{\infty }}$. The pairings are $⟨Y(a,f)A_s{X}_2\mathrm{\Omega },X_1\mathrm{\Omega }⟩=⟨\eta ,S_{I}a(f)⟩=⟨a(f),{\eta }^{\mathrm{\prime }}⟩=⟨A_{s}Y(a,f)X_2\mathrm{\Omega },X_1\mathrm{\Omega }⟩$. Hilbert density of the polynomial test vectors gives $Y(a,f)A_{s}u=A_{s}Y(a,f)u$ on the local polynomial core $u\in {\mathcal{P}}_F(I^{\mathrm{\prime }})\mathrm{\Omega }$. Let $s\to 0$: strong convergence gives $A_{s}u\to Au$ and $A_{s}Y(a,f)u\to AY(a,f)u$, so closedness gives $Au\in D(Y(a,f))$ and $Y(a,f)Au=AY(a,f)u$. The polynomial graph-core result extends this to the whole closed field domain. Applying it to every commutant unitary $A\in \mathcal{A}(J_1^{\mathrm{\prime }})=\mathcal{A}(J_1{)}^{\mathrm{\prime }}$ proves affiliation with $\mathcal{A}(J_1)$. Outer regularity then gives affiliation with $\mathcal{A}(I)$.

Every state is a finite sum of $L_{-1}$ descendants of quasi-primaries. On each finite grade use the orthogonal decomposition $V_d=\ker L_1\oplus L_{-1}{V}_{d-1}$, since $L_1=L_{-1}^{†}$, and recurse to lower grades; at weight zero translation is zero. Translation of fields transfers $L_{-1}$ to a test derivative, with the corresponding finite circle-coordinate density factors. These operations preserve support and smoothness. The same bounded-commutant/core argument therefore gives affiliation with $\mathcal{A}(I)$ for every reconstructed smooth state field supported in $I$, not only for the original F-generators.

Fix an interval $I$ and a reconstructed closed field $T=Y(a,f)$ with $f$ supported in $I$. By Step 22.1 it is affiliated with $M=\mathcal{A}(I)$. Thus for every unitary $u\in M^{\mathrm{\prime }}$, $uT{u}^{\ast }=T$ as closed operators; taking adjoints gives $u{T}^{\ast }{u}^{\ast }=T^{\ast }$. Consequently $u{T}^{\ast }T{u}^{\ast }=T^{\ast }T$, and uniqueness of the positive square root gives $u\mathrm{\mid }T\mathrm{\mid }{u}^{\ast }=\mathrm{\mid }T\mathrm{\mid }$. Uniqueness of the spectral resolution then gives $u{E}_{\mathrm{\mid }T\mathrm{\mid }}(B)u^{\ast }=E_{\mathrm{\mid }T\mathrm{\mid }}(B)$ for every Borel $B\subseteq [0,\mathrm{\infty })$, and uniqueness of polar decomposition gives $u{V}_T{u}^{\ast }=V_T$. Each bounded generator used to define ${\mathcal{A}}_V(I)$ therefore commutes with every unitary in $M^{\mathrm{\prime }}$. Every element of a unital von Neumann algebra is a linear combination of unitaries: a self-adjoint contraction $h$ is the real part of the unitary $h+i(1-h^2{)}^{1\mathrm{/}2}$, and arbitrary elements split into real and imaginary self-adjoint parts and are rescaled. Hence these generators commute with all of $M^{\mathrm{\prime }}$, so lie in $M^{\mathrm{\prime }\mathrm{\prime }}=M$ by the concrete double-commutant theorem in F1. Taking the generated weak-operator closed algebra gives ${\mathcal{A}}_V(I)\subseteq \mathcal{A}(I)$. Conversely the original FJ fields equal the reconstructed F-fields on their common smooth core and hence as closed operators; the F-generating hypothesis gives $\mathcal{A}(I)\subseteq {\mathcal{A}}_V(I)$. Thus equality holds for every interval. Original net locality gives strong locality of the reconstructed VOA. It is simple, unitary and of CFT type by the earlier steps, and its associated net is the original one, proving the full unchanged Statement.