Phase decomposition and Wightman reconstruction · Part 7

empty word, give one countable list before any character chamber is chosen. In every smooth symmetric reflection-positive hierarchy its planted-vector map on each parameter space is norm-continuous: the Gram pairing is the hierarchy on the reflected joined configurations, which remain distinct because the blocks lie on opposite sides of the plane. Fixed-basis expansion makes coefficient dependence a finite sum of continuous products and smooth fixed-label readings; adjoint coefficients are continuous too. Diagonal and mixed pairings are continuous, hence the squared norm difference is continuous. Every future word is approximated from its parameter subset; the full future-word span is dense. The same countable list therefore spans densely in every component chamber, justifying exactly the simultaneous countable intersection above.

At a distinct configuration the component moment is

Searchable transcription

Sχ,d(X; v) = χ(md(X; v)).
(CH1)

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Outside that set use the specified zero extension. Here md is the same whole-word compression
M(w) of PH2. Permutation, unit insertion, common translation, adjoint and quotient-label iden-
tities pass through χ; individual rotation and reflection identities pass once proved for these same
elements. For smoothness, extend the bounded character from the closed compression subspace to
the ambient bounded-operator space by the norm-preserving extension theorem. Composition with
the smooth ambient operator-valued moment is smooth and agrees with CH1 on the compression
subspace. No differentiability structure on the character space is imposed. Reflection positivity
follows by applying χ to one positive quadratic expression. For finitely many future words bi and
scalars ci, set F = P
i ciXbi. PH3 gives

Searchable transcription

a =
X
cicjM(b∗
i bj),
a|H0 = F ∗F.
i,j

This is positive in the ambient operator algebra. Spectral permanence of the closed unital- subalgebra makes its spectrum in nonnegative. Since) belongs to that spectrum, it is real and nonnegative. Linearity gives the component reflection-positive Gram sum at every character, without an exceptional set. In the source lift, is retained before any character operation. The bounded linear reader transports smoothness; the positive algebra element transports Gram positivity. All character readings together recover the compression element, while the retained common probes recover the completed chamber separately at each phase.

Common-probe and positive character recovery. Retain one countable dense parameter subset for each () before choosing a phase. Its realized probe words are lifted by the same word map in every chamber. For a parameter limit in a fixed parameter space and its phase section, the norm reader is

Searchable transcription

∥[bkr]Z
h(χ) −[b]Z
h(χ)∥= ∥[bkr]χ −[b]χ∥−→0.

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The limit follows from the retained diagonal and mixed smooth Gram readings. Thus the same
enumerated list recovers dense span in every phase; the parameter approximation may depend on
the word but the probe list does not depend on the phase.
For the compression star isomorphism α and character map h, set mZ
d (X; v) = α(md(X; v)).
The whole-word reader gives

Searchable transcription

h(χ)(mZ
d (X; v)) = χ(md(X; v)) = Sχ,d(X; v).

The zero extension and all finite-label identities are retained before this scalar reading. For the finite Gram sum on ground inputs, its source element satisfies

Searchable transcription

aZ = α(a) = (F Z)∗F Z,
X
cicjh(χ)(MZ(b∗
i bj)) = χ(a) ≥0.
i,j

The ambient positive operator, its spectrum in the closed compression algebra and its character value therefore recover together, at every character. Smoothness uses the same norm-preserving bounded-functional extension composed with the actual smooth ambient moment. The extension is used for that differentiation argument only; all moment readings remain on the retained compression subspace.

19.39 Compression-valued multilinearity, separability and common bounds

The derivative-label identity is retained in its prescribed form:

Searchable transcription

∂Xa
i DjSχ,v = DjSχ,v[i←∂avi].

It follows for iterated derivatives by differentiating the local identity at distinct configurations. If every fixed-label reading has bound, induction supplies such a bound for every fixed derivative order. Bound the norm of a linear map on joint position space by the sum of its values’ norms on coordinate unit vectors; at each step take the maximum of the finitely many replacementlabel exponents and the sum of their constants. For 01 that maximum bounds every term. Quotient compatibility transfers this to raw component labels. Constants depend on fixed arity, labels and order; no uniformity in those data is added. The following subsections 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.

Fix a distinct configuration, a translation and a strict containing slab. The same choices work on a neighborhood, and for every coefficient vector of each fixed label degree. A slab matrix element between future spectators is the hierarchy of the reflected left word, the inserted slab word and translated right word; all positions are distinct. Move one insertion to the front by permutation. Its reading is an exterior-field functional with these actual spectators; the all-spectator null quotient makes it linear in the projected label and independent of its representative. Dense future-word spans on both sides then identify the bounded slab operators themselves. Bounded linear compression preserves this identity. Repeat at every slot to prove multilinearity of and replace labels one at a time to prove equality for equal projected tuples. Zero extension offdistinct configurations preserves these identities. The same matrix-element argument proves permutation and unit deletion before compression. The empty compressed word is the identity because its heat evolution fixes ground inputs. Slab reflection gives the adjoint, and placement independence gives common translation. These structural identities precede character evaluation and do not assume commutativity. Locally the moment is exactly

Searchable transcription

X 7−→P0 place(d, τsw(X))|H0.

Slab regularity at fixed, smooth position translation and bounded linear compression give operator-norm smoothness. Move any chosen slot to the front to obtain this result at every slot. Placement independence identifies these local functions with the canonical moment everywhere on the distinct set.

For each arity and degree tuple fix basis labels in each slot. The distinct configuration space is separable, and its norm-continuous moment image is separable in operator norm. The countable union over basis tuples, degrees and arities is separable; multilinearity puts all compressed words in its complex span. Continuity of multiplication makes finite products separable, with countable union over lengths. Adjoints, finite complex linear combinations and norm closure preserve separability and generate precisely. This proves the countable norm density used in character-space metrizability. Characters obey. Translate any word of separation 1] into a slab padded by one unit on both sides, preserving. Compression cannot increase norm. The unselected arbitrary-copy slab estimate therefore gives the same copy constants uniformly over characters and receiving hierarchies with the specified word values. For fixed arity, bounded degrees and coefficient norms it reads

Searchable transcription

|Sχ,d(X; v)| ≤KnnBnr−Bn,

with the source’s empty-word convention. This is the scalar bound required by the derivative induction. It comes from the whole-word operator norm; character evaluation introduces no further constants. The lifted moment source retains the local slab, both actual spectator slots and quotient representatives. Equality of their dense matrix elements recovers the bounded operator, and its norm-continuous compression recovers. Thus multilinearity, null-label deletion, smoothness and phase-uniform bounds are operations of the same source object rather than independently assigned scalar properties.

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Local moment, separability and bound recovery.
For a fixed local containing slab write
Ad(X; v) = place(d, τsw(X; v)). With chamber and ground sections VR, V0, its lifted operator is
AZ
d = VRAdV −1
R . The compressed moment is therefore

Searchable transcription

mZ
d (X; v) = P Z
0 AZ
d (X; v)|HZ
0 = V0md(X; v)V −1
0
= α(md(X; v)).

The two actual spectator slots recover every matrix element before compression. Hence their equality preserves the projected-label multilinearity and all structural word identities in the stated order. Since is fixed bounded linear and isometric, local norm smoothness recovers for every positional derivative:

Searchable transcription

Dj
XmZ
d = α(Dj
Xmd),
∥mZ
d (X; v) −mZ
d (Y ; v)∥= ∥md(X; v) −md(Y ; v)∥.

The source moment consequently has the same separable norm images. Choose countable dense subsets of those images and form finite star words and finite linear combinations with rational real and imaginary coefficients. Their countable collection is norm dense in the generated algebra, and its image under is dense in. This explicitly recovers the countable norm-dense sequence from the same compressed generators.

Finally the unchanged padded slab and isometric compression reader give

Searchable transcription

|h(χ)(mZ
d (X; v))| ≤∥mZ
d (X; v)∥= ∥md(X; v)∥≤∥Ad(X; v)∥.

The source arbitrary-copy bill thus supplies exactly the stated character bound with its fixed-data and empty-word conventions. There is no extra character-dependent constant in this recovery or in the subsequent fixed-order derivative induction.

19.40 Reflection squares and individual Euclidean invariance

For a generic grid word with real coefficient labels, write for its geometrically time-reflected word with reflected labels, and put

Searchable transcription

D = M(w) −M(wθ),
a = Re⟨Ω0, D2Ω0⟩.

The grid construction supplies real measurable approximants to widely translated copies of this difference, with and pair errors at most0. Fix, a finite + 1 copies and their displacement parameter before the ordered mesh/volume limit. Each difference-product’s four terms converge to the continuum two-cluster readings. Then send displacement to infinity; GI3 makes every off-diagonal comparison tend to. The ordinary real square at finite lattice parameters gives

Searchable transcription

q
X
!2
,
0 ≤(q + 1)
R2
k + q(a + εk)

.
(RS1)
i=0
f(k)
i
0 ≤E

The second inequality follows after bounding diagonals and paired off-diagonals and taking limits in that order. At each finitely many eventual pair tails can be intersected. If 0, large contradicts RS1; hence0. Real-label adjoint and permutation identities give

Searchable transcription

∥DΩ0∥2 = −Re⟨Ω0, D2Ω0⟩= −a.

Nonnegativity forcesΩ = 0, and the separating vacuum gives = 0. Approximate a fixed generic configuration by canonical addresses in increasingly fine grids from the common funded sequence. Norm continuity gives equality there. Generic configurations have full measure in the distinct region; continuity and full support of Lebesgue measure extend equality throughout it. Expand multilinearly in real coordinate basis labels fixed by coefficient conjugation to cover arbitrary complex labels. Thus individual time reflection fixes each compressed observable.

GI3 gives simultaneous rotation invariance of pair stagesΩ. If an orthogonal sends to positive time component, escape along; if that component is zero, escape along , positive in time before and after rotation. If negative, first compose with time reflection and remove it afterwards by the individual invariance just proved. These three cases prove simultaneous pair invariance for every Euclidean orthogonal map. Now rotate a pair by, reflect only its second factor, and rotate back. Its second factor changes by while all pair readings stay equal. Test with the adjoint words of that factor and its transformed copy: the Gram identity gives zero norm for their vacuum-vector difference, and separation gives equality of the elements. Every hyperplane reflection is such a conjugate (zero normal is the trivial case), and orthogonal maps are finite products of hyperplane reflections. The label representation law therefore gives

Searchable transcription

M(r · w) = M(w)
(r ∈O(4)).

The lifted source retains the bounded Wilson approximants, copy counts, pair tolerances and ordered limits. Its square reader RS1 recovers individual reflection before the pair-stage rotation reader recovers individual orthogonal invariance. Character evaluation thus uses the same whole-word operators throughout.

Bounded square and individual invariance recovery. At each finite lattice cutofflift every real approximant by the actual configuration map: . Normalized Wilson pushforward and surjectivity give exactly

Searchable transcription

EZ
 q
X
i=0
fZ,(k)
i
!2
= E
q
X
!2
≥0,
∥fZ,(k)
i
∥∞≤Rk.
i=0
f(k)
i

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All pair-error bills and finite intersections of eventual tails are therefore preserved. Keep k, q and
displacement fixed through the ordered inner mesh and outer volume limits, then take the large
separation limit. The source square reader recovers exactly (RS1). For each k, if a + εk < 0, any
integer q > R2
k/[−(a + εk)] contradicts that inequality. Only afterwards is k →∞taken to recover
a ≥0.
For DZ = α(D) the compressed algebra reader satisfies

Searchable transcription

(DZ)∗= −DZ,
∥DZΩZ
0 ∥2 = −R⟨ΩZ
0 , (DZ)2ΩZ
0 ⟩= −a.

The separating vacuum and the isomorphism recover = 0 and = 0. Grid-address approximation, norm continuity and multilinear real-basis expansion are unchanged under this isometric map. Likewise each pair-stage rotation and conjugated reflection reads the same matrix elements, so the zero vacuum-vector difference recovers the same compressed element. Finite reflection generation then gives

This establishes individual Euclidean invariance under the same reflection-square and whole-word pair-stage operations used by the component character readers.

19.41 Derivative and stress-window inheritance at every character

Searchable transcription

Fix T = Sχ. Extend χ only as a bounded complex-linear functional to all ground-space operators
and compose with compression.
This yields a single bounded linear functional φ on chamber
operators with
S useAt T(X; v) = φ
place(D, τsw(X; v))

.
(HW1)

No positivity or multiplicativity is required of this extension outside. One common contains a finite configuration and any compact insertion set with strict time margins, persisting on a neighborhood. Pairing with arbitrary actual future spectators proves the all-spectator inclusion identity for the slab operator; applying gives component inclusion compatibility. Norm differentiation at fixed width and shift gives

Searchable transcription

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

The common translation has identity position derivative. Passing this through bounded proves the derivative identity used above. Let be a compact smooth window constant near the distinguished insertion and zero near every spectator. Contain its entire closed support in that same slab. For continuous scalar put

Searchable transcription

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

This is a norm-integrable operator integral. The scalar factor has compact support disjoint from all collision points: its derivative vanishes near the distinguished point and the window vanishes near spectators. The insertion is norm-smooth there. Offthat support the complete integrand is locally zero, regardless of an arbitrary bare collision convention; hence the full integrand is continuous and compactly supported. Pair with two arbitrary future-word vectors. Their additional exterior spectators are strictly outside the slab and entire window. Apply the original stress-window identity with these actual spectators, pass the bounded matrix-element functional through the integral and

use density to get the operator identity. Passing through the integral and using HW1 gives the numerical identity in. In particular,

Searchable transcription

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

For a skew Euclidean generator, take and sum to obtain

Searchable transcription

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

The degree-four stress label, generator labels and geometric window are unchanged. The lifted source retains the actual operator integral and exterior spectators; its bounded linear character extension commutes with derivatives and this norm integral. It therefore recovers both stress laws at every character with the original window and label data.

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One bounded functional for derivative and window recovery.
Let eχ be the source’s chosen
bounded linear extension on ground-space operators. Define

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φ(A) = eχ(P0A|H0),
φZ(AZ) = φ(V −1
R AZVR).

This retains one extension for the fixed character and gives. On an actual compressed moment it recovers the character reading; outside the compression algebra only bounded complex linearity is used. For the common locally fixed slab and shift,

Searchable transcription

φZ(∂Xa
i AZ(X)) = ∂Xa
i φ(A(X)),
AZ(X) = VRA(X)V −1
R .

Thus inclusion and derivative-label recovery use the same actual spectator-tested slab identity before scalar evaluation. Keep the compact window, continuous weight and its entire support in that same slab. For its full norm-integrable operator integrand), unitary conjugation preserves both the collision-free support and the local zero extension. Hence

Searchable transcription

IZ
a,u =
Z
VRJa,u(y)V −1
R
dy = VRIa,uV −1
R ,
φZ(IZ
a,u) =
Z
φ(Ja,u(y)) dy.

Bounded matrix-element readers commute with this same integral and retain the actual exterior spectators outside the slab. Their dense pairings therefore recover the operator stress identity first. The bounded scalar reader then recovers (HW3)–(HW4), including their minus signs, the weight and the unchanged degree-four stress and generator labels. No multiplicative or positive extension premise is added to this derivative-and-integral passage.

19.42 Smeared positivity and bounded reflection witnesses

A signed coordinate permutation taking a chosen axis to time intertwines its word reflection with the temporal reflection, preserves the positive side and preserves readings by real-spin covariance. It transports the complete finite point Gram sum to every coordinate plane. For joint tests fix such an axis and its positive future-word Gram Hilbert space. For compact supported on positive mutually distinct configurations set)] there and zero elsewhere. Smooth diagonal and mixed smeared Gram kernels are continuous, also across the cutoffsupport boundary. Expanding

the squared norm difference proves strong continuity of; compact support gives Hilbert-valued integrability. Fubini gives

Searchable transcription

ZZ
Ksmear(f, g; X, Y ) dX dY =
Z
ζf(X)dX,
Z
ζg(Y )dY

.

Every finite smeared positivity sum is thus a squared norm. This constructs joint-test positivity directly from the point kernel, without assuming a full hierarchy in order to prove one of its clauses. For a real observable with bounded local complex approximant of rooted discrepancy at most and, use Re. Indeed

Searchable transcription

Re O′ −O = 1
2(O′ −O) + 1
2(O′ −O).

Subadditivity, homogeneity and conjugation invariance of the rooted seminorm preserve the error bound. Measurability and locality persist, and Re. The actual cutoffobservable need not itself be uniformly bounded. For a fixed finite degree-bounded label list, form products of these real local approximants at the translated lattice sites. The product comparison applies to the concatenated two-cluster list. For distinct integer displacement slopes, the relevant coordinate difference has a fixed part plus); clusters are eventually disjoint. Wrapping remains injective on the period tail. The same extraction order therefore controls all four ordinary/reflected cluster combinations. If their product approximants are , use the real witness , bounded by the sum of the two product bounds. Expanding the pair product gives total comparison error

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εk = ck(A00 + A01 + A10 + A11) −→0.

All four integrals exist by local approximant estimates and finite lattice moment bounds. This is precisely the common witness family used in the finite-square argument. Fix the approximation index and number of copies, retain the prescribed ordered mesh/period limits and then the large-displacement comparison as in RS1. The approximants are obtained by fixing reporting scale and finite atlas, choosing a sufficiently late auxiliary scale, and transferring the local estimates back to the original mesh and period tails on the same chamber. The following gain-chart estimates supply this choice quantitatively.

Joint smeared positivity and real witness recovery. For each coordinate-reflection Gram space, use its section unitary and keep the same positive-side joint test. Then

Searchable transcription

ζZ
f (X) = Vaζf(X),
Z
ζZ
f (X) dX = Va
Z
ζf(X) dX.

The scalar Gram kernel, compact support and strong continuity are unchanged. Consequently the complete finite smeared reader is

Searchable transcription

ZZ
Ksmear(fi, fj; X, Y ) dX dY =
Z
ζZ
fi(X) dX
2
≥0.
X
X
cicj
i
ci
i,j

The signed coordinate permutation recovers this same construction at each axis, preserving both the reflection and positive-side domain. For the bounded local witnesses, configuration pullback satisfies

Searchable transcription

(RO′) ◦π = R(O′ ◦π),
(RO′ −O) ◦π = 1
2((O′ −O) ◦π) + 1
2(O′ −O) ◦π.

Thus the retained rooted seminorm laws give the same error bound after real-part extraction. Measurability, local physical read sets and the uniform approximant bound are retained. Product and subtraction readers then recover every real reflection witness and its four-term pair expansion before integration. Normalized measure pushforward gives identical four integrals, with exactly the summed bill). The finite-copy tails, eventual unwrapped separation and period wrapping injectivity are the original physical conditions read through the same local configuration maps. Hence the source witnesses recover the family used by (RS1), with the reporting scale and atlas fixed before the late auxiliary scale and with the stated subsequent cutoffand separation order.

19.43 Coarse lenses, coefficient inversion and rooted approximation

At coarse scale the local lens is the explicit function

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

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Here ak is field scaling, L1 the constant coefficient, F the stencil footprint and Ck,L the scaled
nonconstant candidate polynomial. All other regular activity terms of the primary lens are zero in
this specialization. On the healthy compact chart the candidate is bounded by Ak,L uniformly in
finer terminal cutoffand period; outside it the product is zero. The lens bound is |akL1|+Ak,L. Its
read set lies in the coarse window of stencil radius plus four, since every inspected edge is within
four sites of a stencil point. For recovered coefficient columns Jbi assume |ei −Jbi|1 ≤1/4 for every
coordinate basis vector, and set Av = P
i viJbi. Expansion and the triangle inequality give

Searchable transcription

|v −Av|1 ≤1
4|v|1,
|v|1 ≤2|Av|1.
(AP2)

Thus is injective and, in equal finite dimensions, onto. Solve for a terminal label and use injectivity of to recover

Searchable transcription

L =
X
i
vi terminal(bi),
|v|1 ≤2ecolumnLogs(k)|L|1 = stockk(L).
(AP3)

The last bound is the backward coefficient budget. Each basis lens back-lifted to the fine lattice approximates its recovered terminal observable with common rooted error stakes. The AP3 linear combination therefore obeys

Searchable transcription

∥B∥∞≤stockk(L)Ak,
∥B −OL∥root ≤stockk(L)stakesJ,m
w
.
(AP4)

Back transport preserves the sup bound and the atlas room condition puts its read set in the common fine chamber. The exact terminal label decomposition, subadditivity and homogeneity give the rooted bound for this same. For the basis bound insert separated atlas copies of the difference, with a compulsory mark on each, and extract the coefficient containing every copy once. Exact RG transport preserves its Wilson integral. For atlas cells and discrepancy moment insertion estimates give

Searchable transcription

|M|wq ≤eqc,
c = (aJ/am)4treasurym + 5.

The scale power comes from the exact cell census; the five units are the insertion and vault costs. Theth root gives = stakes. The coefficient identity cancels the same positive Wilson partition function, preserving normalization. The admissible mark price satisfies

Searchable transcription

w|am|4(1 + columnsk)
gainChart(k, m) ≤1
2ζd,m.
(AP5)