Phase decomposition and Wightman reconstruction · Part 2

of. For the general filter pass to any finer ultrafilter. Each bounded scalar coordinate has a compact-disc limit, giving such a holomorphic. It agrees with on the accumulating set and hence on by the identity theorem. Every finer ultrafilter thus converges to), proving the original filter convergence. This applies to boundary heights and phase-character limits with only local tail bounds. The lift retains the slice, filter and local boundedness data. Recovery preserves the tested analytic functions and their accumulation-set values, transporting this uniqueness and convergence argument without imposing a single global tail bound.

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Every-character vacuum and slice-limit recovery.
Keep the same compact character space
and full-support probability in the source record. For each χ let Vχ be the section unitary and
xZ
χ = Vχxχ, ΩZ
χ = VχΩχ. For a fixed compact-word sum and s ≥1, the recovered scalar function is

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qZ
s (χ) = ⟨xZ
χ, e−sHZ
χ xZ
χ⟩−|⟨ΩZ
χ, xZ
χ⟩|2 = qs(χ).

Its continuity is the scalar heat/word continuity established above; it requires no topology on the family of phase Hilbert spaces. Consequently the source and physical closed sets are identical. For each fixed full support makes this set all characters. Fixing afterwards and taking the strong heat limit therefore recovers

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P Z
χ xZ
χ = ⟨ΩZ
χ, xZ
χ⟩ΩZ
χ,
P Z
χ = VχPχV −1
χ
= |ΩZ
χ⟩⟨ΩZ
χ|.

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The second equality extends from compact words through the same compact test cutoffs, Schwartz-
word continuity and cyclic density. Inverse unitary recovery gives exactly the source’s rank-one
zero-energy conclusion and its heat-limit input to weak irreducibility.
On a complex slice retain the original indexing filter F, connected domain U, local boundedness
balls and accumulation set. Scalar word recovery gives F Z
a (z) = Fa(z) and fZ(z) = f(z) at every
interior point. It therefore preserves the local tail sets and Cauchy bounds with exactly their stated
ball dependence. For every finer ultrafilter U the compact-disc coordinate reader obeys

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lim
a→U F Z
a (z) = lim
a→U Fa(z) = f(z),
z ∈U.

Local equicontinuity, holomorphy and accumulation-set uniqueness are thus recovered on that same slice; convergence for all finer ultrafilters recovers the original limit. No selection of a new height sequence or character subsequence enters this readout.

19.9 Bounded slab operators by powers

Let realize the algebraic future-word span in its reflection-positive completion. Suppose obeyand put. For 0 assume each has finite with (ST1)

The Gram adjoint relation makes and its powers symmetric. Set and finite = sup. Symmetry and Cauchy–Schwarz give

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d2
n = D−4n|⟨Xv, XT 2nv⟩| ≤∥Xv∥d2n ≤∥Xv∥s.

Taking suprema yields, hence. Consequently

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∥XAv∥2 = |⟨Xv, XTv⟩| ≤D2∥Xv∥2.
(ST2)

Thus descends through null vectors and extends boundedly with norm at most, removing the individual. Apply to slab insertion and reflected insertion; hierarchy translation invariance supplies their adjoint relation. If adjoins successive reflected/unreflected slab pairs, it suffices that Gram( (ST3)

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A scalar multiple has norm at most |a|√QvD2n; triangle inequality extends (ST1) to every finite
word combination. Hence a diagonal estimate on individual words really does produce a bounded
operator on their entire dense span.
This bounds the entire dense span without presupposing
boundedness, in every character. The lift retains the algebraic insertion maps and Gram adjoint
relation. Isometric recovery preserves powers and null spaces, transporting (ST1)–(ST3) and the
bounded extension with the same norm.

19.10 Exact slab norm transfer to every character

For unselected slab operator and ground-space word maps, concatenation and reflection give

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Xenter(w,v) = UXv,
Xvn = (U∗U)nXv,
∥Xvn∥≤∥Xv∥∥U∥2n.

The character Gram is the reading of the compressed two-sided word:

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| Gramχ(v, v)| = |χ(X∗
vXv)| ≤∥χ∥∥Xv∥2.

For anythis gives (ST3) with. The power proof removes this prefactor and constructs the phase slab with norm at most. Operators for different agree on the dense insertion-prescribed word span and hence everywhere. Takinggives

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∥Uχ∥≤∥U∥.
(ST4)

This includes zero norm and transfers all unselected arbitrary-copy constants uniformly to every character. Construct each character chamber, slabs and inherited derivative calculus; these bounds supply contour and strict-wave slab laws for field reconstruction with common labels. Apply the field reconstruction to those data to obtain its Hilbert space and fields, with the common conjugation and rotation labels. This is legitimate because they satisfy the same requirements; consequently no comparison between independently chosen copies is needed there. At the previously selected character retain its already constructed objects, which satisfy those requirements. This keeps the actual chosen realization and supplies phasewise polynomial bounds for tested boundaries. Contour and field construction remain below; vacuum uniqueness is separate. Their proof does not require discarding exceptional characters. The operator estimates needed in reconstruction hold at every character of the compression algebra. The lift retains ground-space word maps and whole-word compression before character reading. Recovery preserves concatenation, Gram evaluation and the dense-span extension, transporting (ST4) with its exact constant.

Algebraic powers, null descent and slab recovery. Let be the word lift, and let be its word readout with. Define the source Gram form and insertion maps by

For every ; the zeroth power is the identity, with the same recovered Gram norm. The Gram adjoint relation, all and their finite supremum are consequently identical to the physical ones for. The power estimate recovers

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∥AZζ∥gZ ≤D∥ζ∥gZ,
ker gZ = R−1(ker X).

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It therefore defines a bounded map on the source null quotient before completion. For v = P
i aivi,
the explicit finite-combination constant is Cv = P
i |ai|
pQvi, so the descent covers the whole
algebraic word span. The induced completion unitary VH gives AZ = VHAV −1
H
and preserves its
norm.
For whole-word compression let V0 identify the unselected ground spaces, and put CZ = V0CV −1
0
on the closed compression algebra. Recover a character by χZ(CZ) = χ(C). These are isometric
algebra identifications and preserve the character norm. Hence, for the lifted ground-space word
map, the retained positive Gram reader is exactly

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χZ((XZ
v )∗XZ
v ) = χ(X∗
vXv),
QZ
v = ∥χZ∥∥XZ
v ∥2 = ∥χ∥∥Xv∥2.

The preceding null-descent construction removes this prefactor on each character chamber. Its bounded extension is independent of the chosenbecause its action agrees on the same dense word span. Letting decrease tothus recovers (ST4), including= 0, with the same constant in every character. The already selected chamber and its slab maps remain the prescribed members of this family.

19.11 Centered insertions and buffered strict-time words

Let) be chamber heat contraction,) its holomorphic contraction semigroup on and) commuting spatial translations fixing Ω. Define

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Bp(v) = place(2p, [(0, p; v)]),
p > 0.
(SW1)

Its future-word matrix elements are hierarchy values with insertion between reflected left spectators and right spectators shifted by 2. Distinct positions and hierarchy multilinearity make linear. Finite-level representatives of the same union label have equal matrix elements by derivative-plan inclusion, hence equal bounded operators on the dense word span. Thus this is a map on the common field union. Arbitrary-copy slab bounds give, for each label,

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∥Bp(v)∥≤Qvp−dv,
0 < p < 1.
(SW2)

Both buffers have width; unused width zero may be assigned zero. Set ) for0. Its norm is at most one. For strict imaginary heights choose

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0 < p < 1,
4p < Izj −Izi
(i < j).
(SW3)

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The ordered buffered matrix element is

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Mp(z, x; v) = ⟨Ω,Bp(v1)R(z2 −z1, 2p, x2 −x1)Bp(v2) · · ·
R(zn −zn−1, 2p, xn −xn−1)Bp(vn)Ω⟩.
(SW4)

Empty words giveΩ Ω; all factors are defined under (SW3) and multilinear in union labels. The lift retains boundary buffers, representatives and the exact ordered semigroup factors. Unitary recovery preserves the dense matrix-element identification and (SW1)–(SW4), including the same polynomial norm bounds and strict-height domain.

19.12 Width independence and strict-wave bounds

Slab stretching and concatenation give

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Bp+u(v) = E(u)Bp(v)E(u).
(SW5)

For admissible larger width, semigroup composition and spatial commutation also give

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R(w, 2p, x) = E(u)R(w, 2(p + u), x)E(u).
(SW6)

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Insert (SW5) into the larger-width product.
Interior heat factors combine by (SW6), exterior
ones fix the vacuum, proving Mp+u = Mp.
Comparison of any two admissible widths proves
independence. For zi = iyi, recursively build the ket backwards: centered insertion plants at time
p, and heat gap yi+1 −yi −2p with spatial translation places the remaining word. Vacuum pairing
gives the complete Euclidean word up to a common translation, removed by invariance. Quotient-
label identity identifies its union hierarchy value. Define Fn = Mp at strict times. An admissible
p exists because there are finitely many positive imaginary gaps; the choice is irrelevant by width
independence. Bounded composition makes Fn holomorphic in complex times and continuous in
spatial positions. No regularity of a global width choice is required. Contractions and Cauchy–
Schwarz give |Fn| ≤∥Ω∥2 Q
i ∥Bp(vi)∥. For gaps at least 0 < k ≤1, choose p = k/8 to obtain

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|Fn| ≤Cvk−mv,
mv =
X
i
dvi,
Cv = ∥Ω∥2 Y
i
(Qvi8dvi).
(SW7)

This is uniform in real times and positions. Empty gap conditions for arities zero and one still permit that choice; constants may depend on arity and labels. In particular this argument requires no continuity of the arbitrarily chosen width as a function of. At purely imaginary times the same product is exactly the original Euclidean hierarchy at the points (). The lift retains buffered words and the exact heat composition. Unitary recovery preserves width independence, Euclidean identification and the same local analytic construction and gap bound.

Buffered product recovery on the strict-time domain. Fix the section unitary for the chamber. Its bounded-operator source maps are

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BZ
p (v) = VHBp(v)V −1
H ,
F Z(ζ) = VHF(ζ)V −1
H ,
UZ(x) = VHU(x)V −1
H .

Set) and Ω Ω. The field-union reader uses the same quotient labels, so linearity and equality of finite-level representatives pass through these maps. Norm recovery gives the exact bound, with both boundary buffers equal to. The propagated factor and the whole buffered word recover as

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RZ(w, s, x) = VHR(w, s, x)V −1
H ,
M Z
p (z, x; v) = Mp(z, x; v).

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These equalities use the ordered product in (SW4), cancelling only adjacent V −1
H VH factors. The
admissibility condition remains Iw > s and (SW3); the empty-word reader is the same unit norm.
The two exact source composition identities are

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BZ
p+u(v) = EZ(u)BZ
p (v)EZ(u),
RZ(w, 2p, x) = EZ(u)RZ(w, 2(p + u), x)EZ(u).

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They recover (SW5)–(SW6) before taking the vacuum matrix element. Thus width independence,
the backwards Euclidean plant recursion and its common translation are all read from the same
word. In particular F Z
n = MZ
p = Fn for every admissible width.

Conjugation is a bounded linear map on the operator spaces and preserves holomorphic dependence on every local strict-time neighborhood with a fixed admissible. Hence the source wave has exactly the same local holomorphy and spatial continuity. Choosing8 recovers (SW7) with the identical; its uniformity remains over real positions and times for the fixed word. Arity zero and one retain the empty gap condition and the same allowed width choice.

19.13 Tested Cauchy–Riemann identities

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For real spacetime X define
fv,y(X) = Fn((ti(X) + iyi)i, (xi(X))i; v),
Ψv(y; φ) =
Z
fv,y(X)φ(X) dX.

Strict heights have a neighborhood with one positive gap lower bound; (SW7) bounds all nearby integrands uniformly in noncompact. This follows by decreasing the finitely many strictly positive gaps slightly; equivalently subtract a sufficiently small multiple of the increasing height vector. Schwartz decay gives absolute integrability and dominated height continuity. For direction let have time coordinates and zero spatial coordinates. The slice

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hX,u(ζ) = fv,u+(Iζ)a(X + (Rζ)ea)

is holomorphic on the strict-height domain. One disc of radius 0 works for all and nearby with common bound. Hence (CR1)

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Slice holomorphy gives
d
dsfv,y+sa = i∂eafv,y+sa. The uniform derivative bound permits differentia-
tion under the test integral; real-direction integration by parts yields

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d
dsΨv(y + sa; φ) = −iΨv(y + sa; ∂eaφ).
(CR2)

Bounded original and derivative functions with Schwartz decay justify this on the whole real space. It holds for joint tests and every height direction remaining strict, supplying the earlier Taylor boundary proof. A single disc is admissible for every and all in a sufficiently small height neighborhood. The lift retains joint tests, real/time direction correspondence and the uniform slice radius. Recovery preserves the holomorphic slice and its integrable derivative bound, so the sign and test action in (CR1)–(CR2) are recovered before taking boundary limits.

19.14 Absolute-frequency identity

Use the construction’s Schwartz transform H , including its fixed coordinate rescaling, so

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−i∂d(H g) = −H (⟨d, ·⟩g),
d = ey.
(FT1)

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For compact smooth g, put gs(q) = es⟨d,q⟩g(q). The same compact support and smooth exponential
divided differences show d
dsgs = ⟨d, ·⟩gs in Schwartz topology: all weighted derivatives converge uni-
formly on that support and vanish outside. This proves the asserted Schwartz derivative, including
at an arbitrary parameter value. For strict y, A(s) = Ψv(sy; H gs) has derivative zero by (CR2)
and (FT1), whose height and test terms cancel. Interior differentiation uses the local majorant.
Near the boundary, the tested Taylor estimate gives one continuous Schwartz seminorm p with
|Ψv(u; φ)| ≤p(φ); moving-test errors therefore vanish uniformly in strict u. Constancy and s ↓0
yield
c
Wv(g) := Wv(H g) = Ψv(y; H (e⟨ey,q⟩g(q))).
(FT2)

This is in absolute frequency coordinates. On the interior, the ordinary locally dominated derivative argument applies. The lift retains the Fourier rescaling, joint compact test and height direction. Recovery commutes with Schwartz differentiation and the uniform boundary estimate, preserving the cancellation and absolute-coordinate identity before the cone argument.

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Joint-test differential and Fourier recovery.
The strict scalar wave reader already gives
fZ
v,y(X) = fv,y(X) for every real configuration and strict height. Keep the joint Schwartz space
and its spacetime direction ea unchanged. The source slice hZ
X,u is therefore the same holomorphic
scalar slice with the same radius r, independently of X. Its tested reader obeys

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ΨZ
v (y; φ) = Ψv(y; φ),
∂eafZ
v,u(X)φ(X)
 ≤(C/r)|φ(X)|.

This integrable majorant preserves differentiation under the integral. The whole-space integrationby-parts reader is explicitly

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d
dsΨZ
v (y + sa; φ) = i
Z
∂eafv,y+sa(X)φ(X) dX = −iΨZ
v (y + sa; ∂eaφ).

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Thus the minus sign, joint-test domain and arbitrary strict-cone height direction in (CR2) are
recovered before any boundary approach.
For the frequency step retain the source’s scaled Schwartz transform H and absolute momentum
coordinates q. Put gs = es⟨ey,q⟩g on the same compact support. Recovery commutes with this fixed
test operation and its Schwartz derivative. Writing d = ey, the two recovered product-rule terms
cancel exactly:
d
dsΨZ
v (sy; H gs) = −Ψv(sy; H (⟨d, q⟩gs))

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+ Ψv(sy; H (⟨d, q⟩gs)) = 0.

The retained common boundary seminorm also yields

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|ΨZ
v (sy; H (gs −g))| ≤p(H (gs −g)) −→0
(s ↓0).

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Consequently the source boundary distribution recovers c
W Z
v (g) = Wv(H g) and the full identity
(FT2) in absolute coordinates. The readout keeps the Fourier rescaling and its single derivative
sign throughout this moving-test limit.

19.15 The ordered time-frequency cone

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For weakly increasing y and compact smooth g supported where ⟨ey, q⟩< −1, fix strict unit-gap r
and use (FT2) at Y (t) = ty + r, t ≥0. The compactly supported tests gt = e⟨eY (t),q⟩g tend to zero
in Schwartz topology: each order-j weighted derivative is bounded by a finite sum
 
j
l
!
e−t(t∥ey∥)lpi,j−l,
0 ≤l≤j,

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with fixed constants including the r factor. Every term vanishes. Unit-gap (SW7) bounds (FT2)’s
right side by a fixed continuous seminorm of H gt, so its constant left side is zero. If ⟨ey, q⟩< 0 at
a point, rescale y to make it −2; a neighborhood has pairing below −1 and all its compact tests
vanish. Therefore
supp c
Wv ⊂
\
y1≤···≤yn
{q : ⟨ey, q⟩≥0}.
(FT3)

Constant height vectors of both signs enforce total time-frequency conservation; weak ordering also supplies partial-energy inequalities. Relativistic spectrum additionally uses the rotation/covariance arguments below. The lift retains absolute frequencies, weak-height cone and compact support tests. Recovery preserves exponential test decay and distributional support, transporting (FT3) with both conservation and ordering conditions.

19.16 Removing the outer angular flux and obtaining all Euclidean rotations

The lattice symmetries give only signed coordinate permutations. Continuous rotations of the physical quotient require the conserved stress and its shell maps constructed below. Write for the translation shell map and) for the angular shell map, where is a real skew 4-by-4 matrix. Before their normalization, the angular Ward expression is

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AK(v; x) =
X
l
{S(v1, . . . , Y (K)vl, . . . ; x) + (Kxl)αS(v1, . . . , pαvl, . . . ; x)} .
(RT1)

Its vanishing cannot be inferred by simply enlarging a spherical cutoff: the correlations have subpower infrared bounds, rather than the decay that such an argument would require. The source removes the outer flux through tested spectral estimates. We give the details of that passage here. The tests in this argument have the following precise meaning. Fix a nonnegative even smooth compactly supported function, equal to one on [ 1], and let

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cw(p) = w−1c(p/w),
w > 0.

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The source uses this central frequency profile and real Schwartz profiles with the same normalization.
Its j-th moment is
R pjf(p) dp; substitution gives the moment of fw as wj times that of f. In the
Fourier convention used here,

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∂j
x bf(0) = (−i)j
Z
pjf(p) dp.

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Differentiation under the integral is justified by the integrable Schwartz majorant |p|j|f(p)|. If
moments below m vanish, repeated integration of the derivative along a segment bounds | bf(x)| ≤
C|x|m on a fixed ball. This is the origin-jet cancellation used in the compensated profiles.
For four one-dimensional tests fa, the tested stress pair is the product-test extension of

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Rab,cd(f) =
Z
⟨Jab(x)Jcd(0)⟩
Y
i
fi(xi) d4x.
(RT0a)

When a factor vanishes sufficiently far in Taylor order at zero this is the ordinary integral. For arbitrary Schwartz factors, subtract the finite product Taylor polynomial near zero; the polynomial short-distance bound makes the remainder integrable. This defines the extension without presuming rotation invariance. Pair exchange, tensor symmetry and signed covariance follow by the corresponding changes of variables in the remainder integral.

Its tested conservation defect is

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∆b,cd(f) =
X
i
Rib,cd(f0, . . . , f′
i, . . . , f3).

Let!, and let have factor at and elsewhere. Set

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wab,cd = ∆b,cd(f(a)),
R◦
ab,cd(f) = Rab,cd(f) −wab,cd
Y
i
fi(0).
(RT0b)

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The subtracted term is supported at the single contact x = 0. Its differentiated value on f(a) is
exactly wab,cd, because g(l)
j (0) = 1{j=l}. The resulting defect therefore vanishes on all first-order
product jets. It vanishes on the order-zero and order-two jets by total inversion: the three remaining
tensor indices give an odd sign whereas an even total test order gives an even sign. These finite
contact subtractions preserve the pair and signed symmetries; reflection in a single axis kills a
coefficient unless its four tensor indices occur in pairs. Thus the compensated conservation defect
has no jets of total degree below three. This precise contact remainder, rather than an assumed
exact conservation law at coincident points, enters the spectral band estimates.
For a fixed exterior word on coordinate line l, put

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Fdb(y) = ⟨Jdb(y)
Y
l
Bl(xl)⟩.

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Realize that exterior word as a finite word vector and pair it with the one-stress vector.
The
hierarchy’s permutation and conjugation laws identify this pairing with Fdb whenever the points
are distinct. This is why the spectral bounds apply to an arbitrary exterior word, not only to the
stress two-point function. The actual outer term is
Z
(∂dφab,l
λ )(y)Fdb(y) d4y,
d ̸= a,
(RT0c)

for the compensated angular primitive . Its frequency widths are in axes on the remaining line axis, and otherwise; it carries the primitive amplitude. The band subtractions described next give the actual estimate for (RT0c), including its integrability. The finite excluded insertion set has Lebesgue measure zero and does not change this integral. Compact shell conservation identifies the difference between the summed local charges and this outer integral. Its vanishing therefore proves (RT1).

For each coordinate, reflection positivity supplies its own positive joint spectral measures , supported on0, for finite linear combinations of reflected words. They include atoms at = 0. Mixed readings are obtained by polarization and obey the weighted Cauchy–Schwarz inequality

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t
Z
X dνB
i + t−1
Z
Z dνA
i

if |f|2 ≤XZ,
t > 0.
(RT2)
|⟨B, f(P)A⟩| ≤1
2

This follows by applying 2 to the two spectral vectors; approximation by simple spectral functions proves it for the displayed measurable weights. No identification of spectral measures for different axes is needed.

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All transverse moments needed below are finite after positive time damping. To prove this,
take the characteristic function of the finite damped measure and apply k pairs of forward and
backward differences in a transverse direction j. Its integral multiplier, up to the cancelling sign of
the difference pair, is

2k
eihpj −1
.

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h

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The fundamental theorem of calculus bounds the difference expression by the corresponding 2k-th
derivative of the smooth separated correlation. Fatou’s lemma as h →0 gives the moment of |pj|2k.
The same argument for moved word clusters uses their actual positive time gaps. For a cluster
collapsed in the i direction this yields, for some finite D, C depending on the word and k,
Z
e−tE|pj|2k dνA
i ≤Ct−D,
0 < t ≤1.
(RT3)