Phase decomposition and Wightman reconstruction · Part 4

1. The realized lattice source has only the signed permutation symmetry; the frame realization intertwines exactly that group (Section 9). No continuous rotation acts on or on

2. The readout (the continuum hierarchy on the physical quotient) acquires the full(4) covariance. It is therefore a symmetry of the recovered object that is not the image of a symmetry of the source: it is derived, through the stress shell maps, the vanishing of the outer flux and the rank transfer from the free comparison sector.

3. The representation on labels is obtained by transport along an intertwiner from the free sector, constructed by an exact idempotent correction; the choice of intertwiner is fixed up to the commutant, and the resulting action on the quotient is determined by its generators which are read from the hierarchy.

The quantitative heart of the flux estimate, and the conclusion, are the following.

Lemma 19.2 (Two-band elimination). With the six band readings (DB1) and the gap readings , let be the sum of the absolute values of the seven errors . Then

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0 ≤f ≤a0 −b0 + K ≤3b0 + 4K ≤7K.

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Proof. g ≤0 gives a0 −4b0 ≤3K (errors g −(v −V ), v −a0, V −4b0); h ≥0 gives a0 −16b0 ≥−8K
(errors h −(r −R), r −v, R −4V , v −a0, and V −4b0 four times). Subtracting, 12b0 ≤11K. Then
a0 −b0 = (a0 −4b0) + 3b0 ≤3K + 3b0 ≤3K + 11
4 K, and adding the error K of f gives at most
27

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4 K ≤7K.

Theorem 19.3 (Euclidean covariance of the hierarchy). The free orthogonal representation transported by the intertwiner has exponential derivative; alongwith simultaneously moved labels the correlation is constant; hence the hierarchy is covariant under all of, the actions preserve the inclusions, and averaging the normalized stress (with its tensor indices moved) over Haar measure of gives a symmetric conserved covariant stress with normalized translation and angular charges on the same hierarchy.

Proof. The product rule holds jointly in positions and labels (finite basis expansions), and with = 0 the derivative along a plane rotation vanishes. Plane rotations align any unit vector with a coordinate axis, every hyperplane reflection is conjugate to a coordinate reflection, and reflections generate(4). Both sides of an inclusion-intertwining equation solve the same finite-dimensional linear differential equation with the same initial value. Conservation, reality and the shell identities are finite-dimensional linear identities and pass through the Haar integral.

The constant 0 is a normalization of the stress, fixed by the translation Ward identity; its positivity is part of the rank-transfer conclusion. The contact subtraction (RT0b) is a retained residual: the unsubtracted stress pair is not asserted to be conserved at coincident points. The order of choices in the spectral sign walk (RSW1)–(RSW2) is part of the realized record: the bands, the window and the tolerance are fixed before the limit in which the outer flux is removed, and the seven error terms of Lemma 19.2 are each retained with their own bound until they are summed. The passage from the Lie algebra to the group uses only the exponential of the generators on the finite-dimensional filtered pieces and the signed permutations already present, so that all of and not only its identity component, is obtained without assuming normality of a subgroup.

19.20 Euclidean rotations and tested complex continuation

Let Euclidean generator Ω have vector field, label action and boost action Differentiating covariance on distinct configurations gives

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D(ΩaXi)iSv(X) = −
X
i
Sv[i←Lavi](X).
(BC1)

This follows by inverse rotation on finite-dimensional labels and smooth positional differentiation. Each label belongs to a finite-dimensional field space, so the multilinear scalar reading is differentiable by its finite basis expansion. Continue only after joint compact testing. Set . The parameters giving strict imaginary order for every support point form an open convex, containing the real axis, by compactness and affine gap inequalities. At , folded Euclidean points are). The divergence-free vector field

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Vw =
X
i
[xi,a∂ti −w(yi + wti)∂xi,a]

maps under the linear part of to times its rotation field, including = 0 without division. Chain rule, (BC1) and compact integration by parts give

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0 =
Z 
Fv(ζ(iw, X), x)
X
i
{xi,a∂ti + iwζi(iw, X)∂xi,a}φ
−iw
X
i
Fv[i←Kavi](ζ(iw, X), x)φ

dX.
(BC2)

Folded strict heights make the compactly supported product smooth; zero divergence makes its derivative integral vanish. Replace by. Pointwise holomorphy and joint continuity give uniform local Cauchy formulas on the compact support, so bounded integration is holomorphic in. Imaginary zeros accumulate at zero; the identity theorem makes the integral zero at one. The lift retains the compact test, folded map and both label generators. Recovery preserves the finite-dimensional chain rule and tested holomorphic integral, including signs and the nonsingular = 0 case.

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Absolute cone and tested rotation recovery.
Write τi = q0
i for the absolute time-frequency
coordinates. Every weakly increasing height vector has the unique description yi = c + P
j<i δj,
with c ∈R, δj ≥0. Consequently its physical pairing is

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n
X
n−1
X
n
X
⟨ey, q⟩= c
i=1
τi +
j=1
δj
i=j+1
τi.

Thus the cone on the right of (FT3) is exactly

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n
X
n
X
i=1
τi = 0,
i=j+1
τi ≥0
(1 ≤j < n).

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This calculation preserves the absolute coordinates and makes explicit both conclusions of the weak-
height reader. Scalar source recovery of c
Wv on every compact smooth test is equality with the
physical distribution, so the local vanishing argument and its distributional support are identical.
The same exponential tests gt and their e−t times polynomial seminorm bounds establish that
vanishing before any rotation or relativistic-spectrum step.

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For each finite label carrier let jd, pd be its section and recovery. The source label generators
are LZ
a = jdLapd and KZ
a = jdKapd = −iLZ
a on the horizontal image. For lifted labels,

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pdLZ
a jd = La,
pdKZ
a jd = Ka.

Finite label replacement and scalar table reading therefore commute with (BC1). Keep the physical folded configuration and compact test unchanged. Its linear derivative satisfies exactly

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DQw(X)Vw(X) = w(ΩaQw(X)i)i,
div Vw = 0.

Indeed its time component is and its spatial- component is), also at Hence the source tested integral in (BC2) equals the physical tested integral, with identical label signs. For arbitrary complex the compact support determines the same open convex parameter set. Equality holds pointwise in the integrand before bounded integration on that support; the local uniform Cauchy formula and identity theorem therefore recover its value zero at = 1. This keeps continuation at the tested level and introduces no spatial differentiability premise for strict waves.

19.21 Tested boost identity and Schwartz motion

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The continuation at z = 1 gives
X
i
Ψv[i←Kavi](y; φ) = Ψv(y; Ba,yφ),
Ba,y =
X
i
[xi,a∂ti + (ti + iyi)∂xi,a].
(BC3)

Compact cutoffs converge in Schwartz topology. Continuity of this polynomial-coefficient differential operator and the tested functional extends (BC3) to every joint Schwartz test. Poincare test action is inverse placement (). For invertible, translation and weighted derivative seminorms, chain rule and the coordinate bound give

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pi,k(φ(A(· −b))) ≤(∥A−1∥+ ∥b∥)i∥A∥k2i
max
r≤i, s≤k pr,s(φ).
(BC4)

Compact-set derivative continuity as vary and higher weighted seminorm tail control prove strong Schwartz continuity. Boost inverse placement differentiates pointwise as

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d
ds(gs · φ)

s=0
= −Ba,0φ.
(BC5)

The lift retains inverse placement and polynomial differential test operators. Recovery preserves their finite coordinate actions, compact cutoffextension and seminorm bounds, including the inverse-action sign. The strong derivative justification continues with the source argument.

19.22 Strong test derivatives and finite boost covariance

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The boost derivative (BC5) holds in Schwartz topology. At parameter s its pointwise derivative is
the strongly continuous family D(s) = gs · (−Ba,0φ). Pointwise fundamental theorem writes the
quotient error as
R 1
0 [D(us)−D(0)](X) du. The same holds after every coordinate derivative. Taking
each weighted supremum bounds its seminorm by sup0≤u≤1 pi,k(D(us)−D(0)), which tends to zero
by continuity on the compact parameter interval.
This proves strong differentiation without a
Schwartz-valued integral. In (BC3), extra height terms i P
i yiΨv(y; ∂xi,aφ) vanish at the boundary
by each fixed-test limit. Hence
X

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i
Wv[i←Kavi](φ) = Wv(Ba,0φ).
(BC6)

Place all labels in one finite field space and set). Finite-basis expansion justifies the product rule; label derivatives in (BC6) cancel the negative test derivative. Repeating at moved labels/tests and using the group law gives

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W(ρ(gs)vi)i(gs · φ) = Wv(φ).
(BC7)

Translations and spatial rotations follow by strict-height analytic uniqueness and boundary change of variables with unit-modulus Jacobian. Covariant transformations form a subgroup by representation and test-action laws, including inverses. Coordinate boosts and spatial rotations then give full covariance with the stated group-generation input below. The lift retains finite label spaces, strong test actions and boundary pairings. Recovery preserves finite-basis differentiation and the cancelling height terms, transporting (BC6)–(BC7) with the same group-generation dependency.

Recovery of strong test motion and finite boosts. Use the same joint Schwartz test carrier on both sides of the readout. For)), the retained coordinate estimate is

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∥X∥≤(∥A−1∥+ ∥b∥)(1 + ∥A(X −b)∥).

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Together with the derivative chain rule it recovers (BC4) in every weighted seminorm. Thus test
recovery commutes with inverse placement, polynomial-coefficient differentiation and compact cut-
offapproximation. In particular the source tested boost operator is exactly BZ
a,yφ = Ba,yφ. For the
difference-quotient error Es in (BC5), its source seminorm reader is the same bound

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pi,k(EZ
s ) = pi,k(Es) ≤sup
0≤u≤1
pi,k(D(us) −D(0)) −→0.

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This recovers the strong derivative with its inverse-placement minus sign using the original pointwise
fundamental-theorem argument.
The finite-dimensional label section intertwines the boost action: ρZ(gs)jd = jdρ(gs), with
generator KZ
a
= jdKapd.
Every label in the fixed word lies in one common finite field space.
Writing bvi = jdvi, the scalar boundary reader obeys

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W Z
(ρZ(gs)bvi)i(gs · φ) = W(ρ(gs)vi)i(gs · φ).

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The recovered extra height term is i P
i yiΨv(y; ∂xi,aφ); each summand vanishes by its fixed
derivative-test boundary limit before finite-boost integration.
The label and test derivatives
of the displayed scalar pairing then cancel by (BC6) at every s, recovering (BC7).
The same
reader preserves the representation law, inverse test action and unit-modulus change-of-variables
Jacobian, so the covariance subgroup and its subsequent group-generation dependency remain
those of the retained construction.

19.23 Dual cone and smooth frequency cutoff

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For absolute momenta with total zero and forward-cone proper suffix sums Pj = Pn
i=j pi, summation
by parts gives, for closed dual-cone successive height differences,

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n
X
⟨p, y⟩=
j=2
⟨Pj, yj −yj−1⟩≥0.
(DC1)

Endpoints vanish; cone self-duality gives each inequality, including the empty tuple. For interior height choose) in the open cone. If2, the point) remains inside it for nonzero cone. Applying (DC1) gives

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⟨p, u⟩≥(r/2)∥p∥,
(DC2)

with zero momentum immediate. This is a common neighborhood damping margin. Convolve the indicator of the cone’s open distance-two neighborhood with a nonnegative unit-integral smooth bump supported within radius 12. The cutoff is one within distance one, supported within distance 52, with closed support inside distance three. All derivatives are globally bounded, since derivatives can be put on the compactly supported bump and the indicator has absolute value at most one:. These use full absolute-frequency Euclidean distance, including dimension zero. For closed height and in closed support, choose cone within three. Then

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−⟨y, x⟩≤⟨y, z −x⟩≤3∥y∥.

The lift retains suffix momenta, dual heights and this explicit convolution. Recovery preserves summation by parts and distance margins, transporting the damping and derivative bounds in the same absolute-frequency metric.

19.24 Closed-height multipliers and Fourier conventions

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On cutoffsupport and ∥y∥≤R, the preceding exponential bound gives ∥Dl
xe−⟨y,x⟩∥≤e3RRl.
Leibniz and cutoffbounds yield, in the source’s polynomial-growth form,

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∥Dk
x(ηe−⟨y,·⟩)(x)∥≤Ck,R(1 + ∥x∥)Nk.
(DC3)

Derivatives vanish offclosed support. Thus multiplication acts continuously on Schwartz tests uniformly for bounded closed heights; interior heights also retain (DC2) damping. These are the cutoffand damping estimates needed for the subsequent tube construction. For fix inverse radian Schwartz maps

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H g(q) =
Z
e−i⟨x,q⟩g(x) dx,
Pf(q) = (2π)−N
Z
ei⟨q,x⟩f(x) dx.

For closed height define). (DC3) and Leibniz control its seminorms by finitely many test seminorms, uniformly on bounded height sets. Joint multiplier smoothness gives jet convergence on compact frequencies, and one extra polynomial weight controls tails. Hence is Schwartz-continuous even at boundary heights. The lift retains these inverse transforms and multiplier seminorm bounds. Recovery preserves finite derivatives and weighted-tail control, transporting closed-height continuity with the exact Fourier constants.

Dual-height and multiplier readout. Retain the full absolute-frequency Euclidean space and its closed cone as physical reader coordinates. The source record for a cutoffis () with

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η(q) =
Z
1{dist(q−h,C)<2}b(h) dh,
b ≥0,
Z
b = 1,
supp b ⊂{∥h∥≤1/2}.

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Its readout is this actual convolution, preserving the distance-one unit region, distance-three closed-
support bound and derivative bills.
For the suffix-sum linear map S(p) = (Pj)n
j=2 and height-
difference map ∂y = (yj −yj−1)n
j=2, the dual reader is precisely

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⟨p, y⟩= ⟨S(p), ∂y⟩
when
X
i
pi = 0.

This retains (DC1), its interior ball margin (DC2), and the empty tuple without changing coordinates or pairings.

Keep the Schwartz test space unchanged under recovery. Writing, its retained derivative bounds and the multiindex Leibniz formula give explicitly

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α
β
!
∂α(Dyg) =
X
(∂βmy)∂α−βg.
β≤α

If) = max sup, (DC3) therefore yields

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si,k(Dyg) ≤C′
k,R si+N,k(g),
N = max
j≤k Nj,
∥y∥≤R.

These are the same test seminorms on the source carrier. On frequency tails one additional weight gives a factor (1 + outside; on the remaining compact set joint smoothness gives jet convergence as varies. This explicitly recovers closed-height Schwartz continuity. The Fourier readers retain the same inverse maps H P and the factor (2, so composition with recovers exactly the multiplier used in the subsequent tube construction.

19.25 Schwartz-valued tube lens

For open-tube define. Interior damping and the supportdistance bound giveon support, 0, with) sufficient. Frequency derivatives cost powers of the linear functional norm. Leibniz and (1+! prove

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sup
q (1 + ∥q∥)ke(c/2)∥q∥∥Dj
qλZ(q)∥< ∞.
(TL1)

Thus the lens is a Schwartz test with spare exponential margin. For small real-linear complex functionalmin( 1), exponential remainders satisfy

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∥Dj(eL(q) −1 −L(q))∥≤∥L∥2(1 + ∥q∥)2ea∥q∥.
(TL2)

Order zero uses the series remainder, order one uses1, and higher orders use with 1. Multiply by the lens and use (TL1) to bound every seminorm of by . For2, derivatives of the affine terms vanish and the bound follows from using1. This proves the joint complex Frechet derivative

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dλZ(H)(q) = i⟨H, q⟩λZ(q).
(TL3)

For the tempered frequency functional set

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Fv(Z) = (2π)−N c
Wv(λZ).
(TL4)

The lift retains the actual Schwartz lens, spare margin and Fourier factor. Recovery commutes with its continuous distribution pairing and seminorm Taylor estimate, preserving the joint derivative and tube definition.

19.26 Holomorphic emission and damped-distribution identity

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Continuity of c
Wv transfers (TL3) to complex differentiability of Fv, hence full-tube holomorphy.
At this point no spectral support assertion is needed for holomorphy itself; the support condition
will make the result independent of the cutoff. Fix interior Y and Schwartz g(q) = η(q)e−⟨Y,q⟩,
gX = ei⟨q,X⟩g. Its transform is (H g)(· −X) and

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pi,j(h(· −X)) ≤2i[pi,j(h) + ∥X∥ip0,j(h)].
(TL5)

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This follows from ∥q∥≤∥q −X∥+ ∥X∥, splitting according to which summand is larger. Finite-
seminorm continuity of the tempered functional gives continuous polynomial growth of c
Wv(gX),
hence absolute integrability after joint Schwartz testing. For compact test φ, uniform jet control
permits passing the functional through the compact integral; Fourier inversion yields

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c
Wv(DY (Pφ)) =
Z
Fv(X + iY )φ(X) dX.
(TL6)

Compact cutoffs extend this to all Schwartz tests: left continuity and right polynomial-growth dominated convergence justify both passages. Thus the emission is the actual damped boundary distribution at every interior height, without a fictitious norm on Schwartz space or an exchange of unbounded integrals. The lift retains the translated Schwartz tests and finite-seminorm estimates. Recovery preserves their continuous pairing and compact-cutofflimit, transporting (TL5)–(TL6) with exact Fourier constants and absolute integral convergence.

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Schwartz lens and emission recovery.
Retain (η, Z, λZ) as a source test record on the same
absolute frequency space. Its test readout is the actual λZ, and its scalar distribution readout is
c
W Z
v (g) = c
Wv(g). The damping margin and (TL2) therefore give in each Schwartz seminorm si,j
the recovered second-order remainder

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si,j
λZ+H −λZ −i⟨H, ·⟩λZ

 ≤Ci,j,Z∥H∥2

for sufficiently small in the same open tube. This is a family of seminorm bounds, rather than one norm on the test space. Continuous distribution reading recovers the derivative

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dFZ
v (Z)(H) = (2π)−N c
Wv(i⟨H, ·⟩λZ) = dFv(Z)(H).

Thus the source emission is the same jointly holomorphic full-tube function, with the specified Fourier normalization. For fixed interior , the compact-support test integral identity behind (TL6) is, as an equality of Schwartz functions,

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(2π)−N
Z
gX(q)φ(X) dX = DY (Pφ)(q).

Uniform jet control on the compact support permits the continuous frequency reader to pass through this integral. The two resulting scalar readings are exactly the two sides of (TL6). For a general joint Schwartz test, the same compact cutoffs converge on the left in the Schwartz topology and on the right under the polynomial-growth bound from (TL5). Hence recovery preserves the absolutely convergent smeared emission on every interior height, before cutoffindependence is used.

19.27 Tube identification and cutoff-independent covariance

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A cone-supported distribution annihilates Schwartz tests vanishing near the cone. For compact
smooth g, the tests ηe−⟨Y,q⟩g and e−⟨Y,q⟩g agree near it, hence have equal readings. Compact support
is essential to the second test. At strict temporal y, bounded continuous fv,y defines tempered
uy. Apply (FT2) to e−⟨ey,q⟩g; exponentials cancel and cutoffremoval gives c
Wv(Deyg) = buy(g).
Schwartz continuity and compact cutoffs extend equality to all tests. Fourier inversion and (TL6)
identify their smeared functions; continuity and full-support Lebesgue upgrade almost-everywhere
to pointwise equality:
Fv(X + iey) = fv,y(X).
(TI1)

Thus this full tube continues the actual strict waves. For real Lorentz, letsatisfy . The height and frequency cones are invariant. Lenses) and) have identical exponentials, and both cutoffs are one on a neighborhood of each cone point. Their cutofffactors need not agree globally, but both equal one near every cone point: use the intersection of the unit ball about and the inverse image of the unit ball about. Their difference has closed support disjoint from it; hence the supported distribution gives them the same value. Fourier boundary covariance therefore gives (TI2)

The same local support comparison proves independence of permitted cutoffchoices. The lift retains the cone-supported distribution, dual action and test domains. Recovery preserves these local lens identities and Fourier inversion, transporting (TI1)–(TI2) without an invariant-cutoffassumption or unrestricted exponential multiplication.

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Cutoffremoval and covariance recovery.
Let C be the retained closed frequency cone. The
source frequency reader is the same tempered functional c
Wv with support in C. If h ∈S vanishes
on a neighborhood of C, choose smooth compact cutoffs κR with κRh →h in S. Each compact
test κRh is supported away from C, so

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c
W Z
v (h) = lim
R
c
Wv(κRh) = 0.

This makes the support-based Schwartz readout explicit. For compact smooth it applies to ; compact support ensures the test belongs to. At strict temporal, the Fourier test and damping reader thus recover

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c
W Z
v (Deyg) = buy(g),
Z
FZ
v (X + iey)φ(X) dX =
Z
fv,y(X)φ(X) dX.

Continuity on both sides then gives exactly (TI1) pointwise, on the original strict-wave configurations. For a retained Lorentz motion, the dual frequency map is for the stated Euclidean pairing. For each the open set

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{q : ∥q −p∥< 1} ∩{q : ∥r∨q −r∨p∥< 1}

contains and makes both cutofffactors equal to one. Its union over is the neighborhood on which the two compared lenses agree. Their difference is Schwartz and the preceding support reader therefore annihilates it. Together with the recovered boundary covariance this recovers (TI2) with the same finite label action and dual map. Replacing the two cutofffactors by any two permitted choices gives the identical argument for cutoff-independent emission. The readout retains each individual cutoffand does not require it to be invariant.

19.28 Adjacent spacelike analytic germs

Fix adjacent slots + 1 and their transposition. Choose + 1) with except. Exactly the selected gap is flat; every other ordered gap is strict. For two arguments this profile is zero. Both orders belong to their closed height cones. At a real spacelike difference , putv v, and

v + sinh + (cosh