Phase decomposition and Wightman reconstruction · Part 6

full-measure sets just obtained. For each character in that intersection, the projections of all those word vectors lie in the vacuum line. Their projected span is dense in the heat-fixed space, because the ground projection is continuous and onto. The vacuum line is finite dimensional and closed, so the heat-fixed space equals it. This proves the almost-everywhere vacuum property and supplies an admissible selected character. The earlier full-support continuity argument upgrades the corresponding final reconstructed uniqueness statement to every character.

Spatial invariance of the heat ground space

Write = lim) strongly. The existence of this limit and its identification with the heat-fixed projection follow from the positive contraction spectral calculus: on the spectrum of the integer powers decrease to 1, and strong dominated convergence applies; general times follow from the semigroup law and contraction bounds. Its range is exactly the vectors fixed by all

For any two distinct Euclidean words, place their reflected and untranslated copies in the positive half-space. Their correlation with separation, for sufficiently large time component, has the form x (GI1)

for fixed future-word vectors and fixed offsets. This is an exact translation and reflection identity. Along any sequence with, adding a fixed purely temporal displacement changes (GI1) by a quantity tending to zero: both heat vectors converge to, and the intervening, possibly varying spatial unitaries preserve the norm of their difference.

The same limiting insensitivity holds when adding any fixed with 0, along the pure-time separation Choose a Euclidean orthogonal transformation sending to whose transformed has positive time component. Such a transformation is obtained by reflection in the bisecting hyperplane between and; preservation of their inner product gives the positive time component. Rotate both clusters and their displacements. The previous purely temporal comparison applies because the rotated large separation still has time tending to infinity. Euclidean covariance then gives (GI2)

Given a spatial displacement, apply (GI2) to, and subtract the purely temporal comparison for adding to the sequence. This proves that adding has no effect on the limiting correlation. Taking heat limits in (GI1) yields

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⟨v, U(a)P0w⟩= ⟨v, P0w⟩

for all future-word vectors. Density implies, so every ground vector is spatially invariant. This argument does not assume spatial invariance while comparing the varying large separations.

Long cluster products and commutativity

Let be distinct-point cluster words, and let their displacements) have pairwise ordered time separations tending to +. Then

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r
Y
!
S
k=1
τdk(N)wk
−→⟨Ω0, M(w1) · · · M(wr)Ω0⟩.
(GI3)

To prove this, first place each word in the positive half-space by a fixed shift, compensated in its displacement. Induct from the last cluster backwards. Between successive cluster insertions there is a heat duration tending to infinity and an arbitrary spatial translation. If vectors and durations, contractivity and the just-proved spatial invariance give

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U(aN)E(tN)vN −→P0v.

Indeed its difference from has norm at most. Apply the fixed bounded insertion operator and continue the induction. Vacuum pairing gives (GI3). The large time separations also ensure that the combined word has distinct points eventually. The four-cluster construction proves the needed exchange of two compressed factors. In the plane of Euclidean time and one spatial coordinate choose displacement slopes

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(0, 0),
(1, 2),
(2, 1),
(3, 3)

for clusters, respectively, and multiply all slopes by. Their time order is terchange those two Euclidean axes: the time order is now At finite, permutation symmetry and Euclidean covariance identify the two combined expectations. Taking (GI3) gives equality between the first compressed product and the rotated second product. Remove the simultaneous rotation using another application of (GI3), placing the four clusters along a direction with both axis components positive, such as; its time order remains the same before and after the interchange. Therefore

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⟨Ω0, M(a)M(b)M(c)M(d)Ω0⟩= ⟨Ω0, M(a)M(c)M(b)M(d)Ω0⟩.
(GI4)

Choose the exterior words in (GI4) as the reversed reflected word of a future probe and as another future probe. Equation (PH3) and the adjoint relation turn (GI4) into equality of matrix elements of) and) between projected future-word vectors. Those vectors span densely in, so the two bounded operators are equal. Compressed generators are closed under adjoints: reflection in the slab sends one generator to its adjoint. Thus all generators and their adjoints commute. Algebraic star polynomials in them commute, and norm continuity of multiplication extends this equality to their norm closure. This proves commutativity of the actual compression algebra used in (PH1).

The resulting spatial ground invariance, cluster limit and commutative compression algebra will now supply one compact phase space and one countable family of future probes for all its characters.

One countable family of future probes for every phase

The countable family used in heat-vacuum selection can be chosen before any character chamber is given. For each arity and tuple of finite label degrees), use the parameter space consisting of distinct positions in the positive half-space and raw coefficient vectors in the finite-dimensional spaces of those degrees. This is a separable metrizable space; choose a countable dense subset. There are only countably many. The union of the realized future words from these subsets, together with the empty word, is therefore a single countable list.

For any smooth symmetric hierarchy and any of its reflection-positive realizations, the plantedvector map on each such parameter space is norm-continuous. Its Gram pairing at two parameters is the hierarchy evaluated on their reflected and joined configurations. These configurations remain distinct because the two blocks are on opposite sides of the reflection plane. To see joint continuity in the coefficients, expand in fixed bases of the finite label spaces. The reading is a finite sum of products of continuous coordinate coefficients and smooth fixed-label hierarchy readings. The

adjoint coefficients in the reflected block are continuous as well. Thus both the diagonal Gram value and the mixed pairing with a fixed planted vector are continuous; expanding the squared norm difference proves norm continuity.

Every future word has some arity, degree tuple and parameter in this list of parameter spaces. Its vector is therefore a norm limit of vectors from the corresponding countable dense subset. Since all future-word vectors have dense linear span, the chosen common countable list has dense linear span in every such chamber. This proves exactly the simultaneous density needed when intersecting the countably many full-measure conditions in phase selection.

Character readings, regularity, and positive quadratic forms

The source constructs the component moment at a distinct configuration by first forming the compression-algebra-valued multilinear moment), and then applying the character:

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Sχd(X; v) = χ(md(X; v)).
(CH1)

Outside the distinct-point set it uses the specified zero extension. The algebra-valued moment is the same whole-word compression) used in (PH2), not a product of separate point insertions. Its permutation, unit insertion, common translation, adjoint and quotient-label identities therefore transfer through the character. Once individual Euclidean rotation and reflection invariance of these compression elements is established, those identities transfer too.

Smoothness in is also a consequence of the actual algebra-valued moment. A continuous character on the closed compression subspace extends to a bounded linear functional on the ambient bounded-operator space by the norm-preserving extension theorem. Compose that extension with the smooth ambient operator-valued moment. On the compression subspace it agrees with the original character, proving smoothness of (CH1) on distinct configurations. This argument avoids imposing an extra differentiability structure on the character algebra.

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Reflection positivity follows by applying χ to one positive quadratic expression. For finitely
many future words bi and coefficients ci, set F = P
i ciXbi. Equation (PH3) gives the algebra
element
X

cicjM(b∗
H0 = F ∗F.
a =
i bj),
a

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i,j

It is positive in the ambient bounded-operator algebra. Spectral permanence for a closed unital- subalgebra makes its spectrum in the compression algebra nonnegative as well. Since) belongs to that spectrum, it is real and nonnegative. Expanding) by linearity gives the component reflection-positive Gram sum. This proves positivity at each character, with no exceptional set.

For derivative estimates, suppose positional differentiation is represented by replacing a label by its prescribed derivative label At distinct configurations it then holds for iterated derivatives as well, by differentiating the local identity:

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∂Xa
i DjSχ,v = DjSχ,v[i←∂avi].

If each fixed-label scalar reading has a separation bound, induction gives such a bound for every fixed derivative order. Specifically the operator norm of a linear map on joint position space is at most the sum of its norms on the coordinate unit vectors. At the induction step take the maximum of the finitely many exponents supplied by the replacement labels and the sum of their constants. Since 01, all terms are bounded by that common exponent. Quotient compatibility transfers the result to the raw component labels. All constants remain dependent on the fixed arity, labels and derivative order; no unproved uniformity in those data is introduced.

The next sections construct the compression-valued moments, prove norm separability and individual Euclidean invariance, and transfer the derivative and shell identities. These supply the premises used in the character formulas above.

Constructing the compression-valued multilinear moments

Fix a distinct configuration and choose one translation and one strict slab containing it. The same translation and slab contain all sufficiently nearby configurations. When varying a label at a fixed position, they also work for every coefficient vector of that label degree, since the slab condition depends on positions alone.

For future spectator words, the matrix element of this slab operator is the hierarchy reading of the reflected left word, the inserted slab word, and the translated right word. All these positions are distinct. Move a distinguished insertion to the front by permutation symmetry. Its reading is then an exterior-field functional, with those actual two spectator words as exterior data. By definition of the all-spectator null quotient it depends only on the projected field label and is linear in that label.

Equality of every such matrix element implies equality of bounded slab operators, because the future-word vectors have dense span on both sides. Thus the slab operator itself depends linearly on the projected distinguished label. Compression is a bounded linear operation, so the same is true for its element of. Move any chosen slot to the front to obtain this result at every slot. It proves the multilinearity of). Replacing labels one at a time proves that equal projected tuples give equal compression elements. Offthe distinct-point set the zero convention preserves these algebraic identities.

The identical matrix-element argument proves permutation symmetry and deletion of a unit label before compression. The empty compressed word is the identity because its slab operator is heat evolution and acts identically on ground inputs. Reflection of the slab gives the adjoint of its compressed word. Common-translation invariance was proved by placement independence. These establish the structural identities used in (CH1) without assuming commutativity or applying a character.

Smoothness in position uses the locally fixed translation and slab. On that neighborhood the compression-valued moment is exactly

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X 7−→P0 place
d, τsw(X)


H0.

The slab regularity estimate gives smoothness in operator norm at fixed; the position translation is smooth and the compression map is bounded linear. The resulting smooth function agrees locally with the canonical compression moment by placement independence, proving its smoothness on the entire distinct-point set.

Norm separability and common phase bounds

For each arity and degree tuple, fix basis labels in every slot. The image of the distinct configuration space under its compression moment is separable in operator norm, since that space is separable and the moment map is norm-continuous. The union of these images over all basis tuples, degrees and arities is still separable. Multilinear expansion puts every compressed word in its complex linear span. Finite products of a separable set form a separable set, by continuity of multiplication and countable union over the product length. Include adjoints and finite complex linear combinations, and take the norm closure. These operations preserve separability and generate exactly. This supplies the countable norm-dense sequence used for metrizability of its character space.

Characters have the uniform bound. A word with separation1, can be translated into a slab with one unit of padding on both sides; its mutual separation remains Compression has norm at most that of the slab operator. Applying the unselected arbitrary-copy slab estimate therefore bounds the entire character reading with precisely the same copy constants, uniformly over characters and all receiving hierarchies with the same specified word values. For fixed arity and bounded degrees and coefficient norms, the source’s radius estimate is of the form

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|Sχd(X; v)| ≤KnnBnr−Bn,

with its empty-word convention understood. This gives the fixed-label scalar separation bound required for the derivative induction above. Thus the component bounds come from whole-word operator norms, not from separate uncontrolled character norms.

The reflection square from bounded lattice approximants

For a fixed generic grid word with real coefficient labels, let be its geometrically time-reflected word with the prescribed reflected labels, and set

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D = M(w) −M(wθ),
a = R⟨Ω0, D2Ω0⟩.

The grid construction supplies real measurable approximants to widely translated copies of this reflection difference, with , and pair correlation errors at most0. Fix the approximation index, a finite number + 1 of copies, and their displacement parameter before taking the ordered mesh/volume limit. The four terms of each difference-product converge to the corresponding continuum two-cluster readings. Next send the displacement parameter to infinity. Equation (GI3) makes each off-diagonal comparison tend to

At finite lattice parameters positivity of an ordinary real square gives

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q
X
!2
.
i=0
f(k)
i
0 ≤E

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Bound the diagonal terms by R2
k, and the two symmetric off-diagonal terms by the prescribed pair
comparisons. Passing through the limits in the stated order yields

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0 ≤(q + 1)
R2
k + q(a + εk)

.
(RS1)

Only finitely many pair conditions are needed at each, so their eventual tails can be intersected before the limit. If 0, choosing large contradicts (RS1). Thus0 for every, and

For real labels, the adjoint and permutation identities give follows that Ω ΩΩ

Both sides’ nonnegativity and0 forceΩ The separating vacuum property of the commutative compression algebra then gives

To extend the identity beyond the generic grids, choose increasingly fine grids from the common funded approximation sequence and approximate a fixed generic configuration by their canonical grid addresses. The grid points are eventually generic and converge to it. Norm continuity of both compression moments gives equality there. Generic configurations have full measure in the distinct-point region; continuity and full support of Lebesgue measure extend equality throughout that open region. Finally expand the multilinear compression moment in real coordinate basis labels, each fixed by coefficient conjugation. Equality on those basis tuples proves equality for arbitrary complex labels. Thus individual geometric time reflection fixes each compressed observable.

Individual Euclidean invariance of compressed words

Long two-cluster limits first give simultaneous rotation invariance of pair stages

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⟨Ω0, M(p)M(q)Ω0⟩.

If a Euclidean orthogonal map sends to a vector with positive time component, use clusters escaping along in (GI3). If that component is zero, use instead the direction, which has positive time component both before and after rotation. If it is negative, first compose with time reflection; individual time-reflection invariance removes the added reflection from each compressed factor afterwards. These three cases prove simultaneous pair invariance for every Euclidean orthogonal map.

Conjugate the individual time reflection by an arbitrary orthogonal map. Rotate a pair stage by, reflect its second compressed factor only, and rotate both factors back. The first factor returns to itself and the second has been transformed by. Hence its pair readings with every first word agree with their originals. Test with the adjoint word of the second factor and of its transformed copy. The Gram identity then makes the norm of their vacuum-vector difference zero. The separating vacuum makes the compressed elements equal.

Every hyperplane reflection is a conjugate of, except the identity case of a zero normal vector. Orthogonal transformations in finite dimension are finite products of hyperplane reflections. The individual invariance just proved is closed under multiplication by the label representation law. Therefore

This completes the Euclidean rotation and reflection identities used for every component hierarchy in (CH1). The reflection-square argument uses its bounded Wilson approximants before taking limits, and the rotation argument uses the same compressed whole words throughout.

Inheritance of derivatives and stress-window identities

Fix a character component. Extend, as a bounded complex-linear functional only, from the closed compression algebra to all operators on the ground space, then compose with compression. This gives a single bounded linear functional on chamber operators such that

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S useAt T(X; v) = φ(place(D, τsw(X; v)))
(HW1)

whenever the translated distinct configuration lies strictly in the slab (0 No positivity or multiplicativity is used for outside the original algebra.

One common can contain any given finite distinct configuration together with any specified compact set of potential insertion points: take lower and upper bounds for their time coordinates and add a strict margin. This placement persists on a neighborhood of the finite configuration. Inclusion identities for finite field labels hold for the slab operator itself, by pairing with arbitrary future words on both sides and using the original all-spectator inclusion identity. Applying proves inclusion compatibility in

Likewise norm differentiability of the fixed-width slab insertion gives

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∂Xa
i place(D, τsw(X; v)) = place(D, τsw(X; v[i ←∂avi])).

The common translation has derivative the identity on position space. Differentiate (HW1) through the bounded linear functional, keeping that width and translation fixed throughout its neighborhood. This proves the component derivative identity used in the preceding separation estimates.

For stress identities, let be a compact smooth window equal to a constant near the distinguished insertion and zero near every spectator insertion. Pack its entire closed support in the same slab. For a continuous scalar weight, form the operator integral

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Ia,u =
Z
(∂aχ0)(y) u(y) place(D, τsw(y, X)) dy.
(HW2)

This is a norm-integrable operator-valued function. Its support is compact. On the support of its scalar factor, cannot collide with the distinguished point, because the window is locally constant there; it cannot collide with any spectator, because the window vanishes near each of them. The insertion is therefore norm-smooth locally on that support. Offit the complete integrand is locally zero, even if the bare insertion at a collision was defined arbitrarily. These observations prove continuity and compact support of the full integrand.

To prove the required identity for (HW2) as an operator, pair it with two arbitrary future-word vectors. The added exterior spectators lie strictly below or above the whole chosen slab, and hence outside the entire window support. The original stress-window identity therefore applies with these genuine additional spectators. Passing the bounded matrix-element functional through the integrable operator expression proves the operator identity by density. Pass through the same integral and use (HW1); this gives exactly the numerical identity in

In particular, if denotes the prescribed degree-four stress label and the translation derivative of a distinguished label, the identity is

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Z
(∂aχ0)(y) T(Tab(y), B(X0), v(Xsp)) dy = T((YbB)(X0), v(Xsp)).
(HW3)
−
X
a

For a skew Euclidean generator, use in (HW2) and sum over. The corresponding identity is

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Z
(∂aχ0)(y)[g(y −X0)]b T(Tab(y), B(X0), v(Xsp)) dy = T((ZgB)(X0), v(Xsp)).
(HW4)
−
X
a,b

The geometric window and the same stress and generator labels are retained. Thus both translation and rotation stress laws transfer at every character.

Smeared reflection positivity of the components

The temporal point Gram positivity already proved transfers to any coordinate reflection plane by a signed coordinate permutation taking that axis to time. This permutation intertwines the two word reflections, preserves the positive side, and preserves the component readings by their real-spin identity. It therefore transports the complete finite Gram sum.

Point positivity also implies the required positivity for joint smearing tests. Fix an axis and construct the Gram Hilbert space from its positive future-word kernel. For a compact joint test ) supported on positive, mutually distinct configurations, let

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ζf(X) = f(X)[w(X)]

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there, and define it as zero elsewhere. Its diagonal and mixed Gram pairings are the corresponding
smooth smeared hierarchy kernels. They are continuous, including across the cutoffsupport bound-
ary. The squared-norm criterion proves strong continuity of ζf; its compact support then gives an
integrable Hilbert-valued function. Fubini and the Gram formula yield
Z Z
Ksmear(f, g; X, Y ) dX dY =
Z
ζf(X) dX,
Z
ζg(Y ) dY

.

Every finite smeared reflection-positive sum is consequently the squared norm of a finite linear combination of these integrated vectors. This proves nonnegativity for joint tests directly from point positivity, without assuming a full hierarchy in order to prove one of its clauses.

Assembling the bounded reflection witnesses

The real bounded witnesses in (RS1) are obtained from the same local observable approximants used in the continuum comparison. For a real label observable, suppose a local complex approximant has and rooted discrepancy at most. Replace it by. The observable is real for the prescribed real label, and

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RO′ −O = 1
2(O′ −O) + 1
2(O′ −O).

The rooted seminorm is subadditive, homogeneous and invariant under conjugation. Its discrepancy is therefore still at most. Local dependence and measurability are preserved, and This supplies genuinely real approximants, without requiring the cutoffobservables themselves to be uniformly bounded.

For a fixed finite list of degree-bounded labels, take products of their real local approximants at the prescribed translated lattice sites. The product comparison applies to the concatenated list of two clusters. Distinct clusters with integer displacement slopes are eventually disjoint: any coordinate difference has a fixed finite contribution plus), which tends to infinity. At the finite period, injectivity after wrapping also holds on the period tail. Thus the same product estimate bounds each of the four combinations of an ordinary cluster and its reflected cluster, on the same extraction order. Let and be the resulting real product approximants for these two clusters. Set the reflection witness to . Its uniform bound is the sum of their two product bounds. Expand the pair product into its four terms. If their individual comparison costs are the triangle inequality gives total error

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εk = ck(A00 + A01 + A10 + A11) −→0.

All four integrals exist by the local approximant estimates and finite lattice moment bounds. This is precisely the common witness family used in the finite-square argument. Fixed approximation index, fixed finite number of copies, large displacement, and the prescribed ordered mesh/period limits retain the ordering stated there.

The source obtains the individual local approximants by first fixing a reporting scale and an admissible finite atlas, then choosing a sufficiently late auxiliary scale. The gain-chart denominator makes the rooted approximation error small: its decaying scale factor dominates the subexponential column and stock factors. It then takes the real part as above and transfers the resulting estimates to the original mesh and period tails, using locality on the common chamber. The following coarselens and rooted-approximation argument supplies the gain-chart estimate, with the same local chamber and transported coefficients.

Coarse lenses, coefficient recovery, and rooted approximation

At a fixed coarse scale, the lens of a label is an explicit bounded local function:

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Lk,L(U) = akL1 + 1Healthy(θk,F) Ck,L(U).
(AP1)

Here is the prescribed field scaling factor, its constant coefficient, the local stencil footprint, and the scaled candidate polynomial of its nonconstant coefficients. The other regular activity