The centered trace and its non-freeness test · Part 3

It remains to remove coordinate-generic restrictions and to pass to all position derivatives. For a fixed word let

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Dk,α = X−1
Rk(x) = S(Xkv; rkx).
∂l,αRk(x) = S(Xkv1, . . . , XkDk,αvl, . . . ; rkx),
k,d+1rk∂αXk,d.
(WS7)
The matrices Dk,α converge to the free derivative matrix. By multilinearity, readings with any
converging finite list of coefficient vectors have the same limiting readings as their limits. Also
(WS2) applies to the finitely many successor-degree basis words in (WS7). It bounds δk times every
needed gradient by a quantity tending to zero on a slightly larger separated window. Interpolate
from nearby sampled points to arbitrary separated points along short coordinate segments in that
larger window. Their total length is O(δk). The fundamental theorem of calculus and (WS7) make
the error tend to zero. The argument does not require a positive gap between distinct coordinate
projections: the points themselves stay separated.

Repeat (WS7) for each differentiation. For derivatives use the successor degrees for the differentiated labels and one additional degree for the interpolation gradient. Thus a horizon proves the assertion when. The finite sum over basis components proves convergence of the full derivative tensor in its norm. This convergence is uniform on every compact separated window. Otherwise a sequence of points violating uniform convergence has a convergent subsequence in that compact set; the moving-point interpolation just proved gives a contradiction. Enlarging the initial finite request supplies every prescribed derivative order, word and compact window, on the same retained physical hierarchy.

The interpolation near projection ties uses a different gradient bound from (WS8). On a slightly larger separated window each gradient in (WS7) is a finite sum of readings of successor-degree basis words, with coefficients from. These coefficient matrices converge and hence are bounded. The first, all-separated part of (WS3) applied to those basis words and the finite-maximum argument above therefore give sup (WS9) larger separated window

Round every coordinate to the rescaled mesh, then perturb the rounded coordinates by distinct bounded integer offsets to make every coordinate projection different. The number of forbidden equalities is finite for a fixed word, so one fixed finite list of offsets suffices. The displacement remains For large, all short joining coordinate segments stay in the larger separated window. The integral of the gradient along them is(1) by (WS9). This proves interpolation also at coordinate ties without losing a power through a shrinking coordinate gap.

There is one final sign choice when these maps are required to identify the normalized physical stress with the free stress. Each enlarged finite request allows one of the two signs of the physical conserved line. If one sign works for all requests, retain it. If a particular request excludes it, enlarge any subsequent request to include that one; its available map must use the opposite sign and restricts to a map for the subsequent request. Hence one fixed sign works for every finite request. Subsequent normalization fixes its amplitude as explained below.

For completeness, a single sequence realizing all the finite requests is obtained as follows. At arity, exhaust the separated space by

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Kn,j = {x : max
l
|xl| ≤j, min
l̸=m |xl−xm| ≥j−1},
(WS10)

with the minimum condition omitted for zero or one slot. Enumerate the countable requests consisting of an arity, a finite tuple of degree bounds, a derivative order and an integer stage, enlarge one finite request to contain the first requests, their finitely many basis words,

the detecting spectators, and one extra derivative degree. Use the finite construction on the same physical hierarchy and the same chosen stress sign. Along its available comparison sequence choose a station late enough that the radius is below 1, the rescaled mesh is below 1, all the first derivative errors on their compact sets are below 1, and every required instance of (WS9) is below 1. There are finitely many conditions, each eventually true, so such a station exists. Pad maps above this stage’s horizon by arbitrary fixed isomorphisms; every fixed degree eventually lies below the horizon.

The selected stage- maps and radii form one common sequence. For a fixed enumerated request, all later stages include it and give errors at most 1; hence it has the claimed limit. Every compact separated set lies in some, so this proves all the asserted compact limits. If ordered mesh-period coefficient extractions are refined during these constructions, retain their countable columns and inverse charts together by the ordered compact extraction already used in (WS1): at each period take mesh limits first, then refine the period sequence and diagonalize the countable coordinate list. This preserves the same table and stress. The station diagonal just described is taken after those ordered limits and does not identify them with an arbitrary joint cutoff-volume diagonal. At any one station only its finite horizon is asserted; every fixed request is eventually included.

Constructing the physical stress and its shell charges

Centering removes the unit direction before stress rank is compared. Shell-flux representability is then proved for the conserved stress line, with the finite Ward residual displayed until its limiting estimate is consumed. Exact conservation of the eventual physical field does not imply vanishing of every finite localized divergence.

We first detail the rank transfer on centered tensors. A scaling isomorphism need not take the free expectation functional to the physical one. Let1 and1 be the two centering maps. Apply them componentwise to tensors. For a late scaling index, intertwines reality and signed symmetries and fixes the unit. It therefore induces the map

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Rk : K0 −→K,
RkB = c(Uk,4B),
(SR1)

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between the real centered symmetric signed tensor spaces. Its inverse is A 7→c0(U−1
k,4A). Indeed the
first centering subtracts only componentwise unit multiples. The inverse isomorphism carries those
units to units, which the second centering kills; the remaining term is the original centered tensor.
Thus these are exact inverse maps even though the two expectations differ. The corresponding
degree-five isomorphism Qk on real signed polar vectors needs no centering correction.
Write D0, D for the free and physical divergence maps on these centered tensor spaces. On the
fixed free spaces consider
Zk = Q−1
k
rkDRk.
(SR2)

Derivatives of the unit are zero, so the centering correction in (SR1) contributes nothing to (SR2). The derivative comparison (WS7) gives for every. The free calculation makes onto and its kernel one dimensional. Choose a right inverse of. Then in matrix norm, by convergence on a finite basis. Eventually 1; the geometric series makes invertible, hence onto. Since 0 and both surrounding maps in (SR2) are isomorphisms, is onto. The exact dimension equalities from (SR1) and, followed by rank–nullity, now give

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dimR ker D = dimR ker D0 = 1.
(SR3)

This establishes the conserved centered line without identifying the expectation functionals through

The physical stress is selected on the common field quotient, before its final amplitude is fixed. At weight four the real signed symmetric tensor space includes the vacuum tensor as well as (FC3), so it has dimension four. The rank comparisons transfer this dimension from the free table to the physical table and identify it with the image of the four real reference coefficients

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(
w0 1 + w1
P
i Qai,ai + w2
P
i,k Qik,ik,
a = b,
w3
P
i Qai,bi,
a ̸= b.
(SC1)
Jab(w) =

In (SC1) take the actual physical classes of these reference labels, not their free polynomial values. The physical weight-five polar vector space has dimension at most two, by the same rank comparison. Differentiation on the quotient therefore gives a linear divergence map from a four-dimensional real space into a space of dimension at most two:

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(∇J)b =
X
a
∂aJab.
(SC2)

Its range is polar because the signed derivative transformation has one sign for each derivative index; summing over that index cancels its squared sign.

Adjoin the real one-point reading of to (SC2). The resulting map has target dimension at most three and thus has a nonzero kernel. Choose in that kernel and normalize its Euclidean coefficient norm to one. The tensor map (SC1) is injective by the four-dimensional rank comparison, so= 0. It is conserved and its 00 mean vanishes. Reality makes every diagonal mean real; translation makes each mean constant in position; unsigned permutations make all diagonal means equal; reflection in one of two different indices negates an off-diagonal mean. Thus every mean vanishes. This constructs a centered nonzero real symmetric conserved tensor. It does not yet select its sign or claim its Ward charge is the unit translation generator.

Here is the normalization of this line along the comparison curve. For any nonzero tensor in it and the fixed nonzero free tensor, put

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Bk = U−1
k,4(r4
kJ),
ξk = ∥Bk∥
∥T 0∥> 0,
eBk = ξ−1
k Bk.
(SR4)

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The inverse and norm here are on the full tensor coordinate space, including its scalar directions.
The norm of eBk is exactly ∥T 0∥, so finite-dimensional compactness gives a convergent subsequence
with limit B of that norm.
No positive lower bound on ξk is needed.
Closedness of the real
signed symmetric space puts B in that space. Its free mean is zero: the one-point comparison for
bounded converging labels applies to eBk, while its physical image is the positive scalar multiple
ξ−1
k r4
kJ, whose expectation is exactly zero. Applying the converging derivative matrices (WS7) to
eBk similarly passes the exact physical divergence zero to the free divergence zero. Thus B ∈ker D0.
By its one-dimensionality, B = cT 0 for real c; equality of norms gives |c| = 1. Choose the single
sign of J that makes c = 1. Restricting the comparison parameter to this subsequence preserves
all earlier limits and exact compatibility laws. This proves the normalized stress comparison used
above; its eventual Ward amplitude is fixed separately below.

The finite lattice Ward identity associated with this construction retains its residual explicitly. At spacing, write ) for the calibrated cylinder (SC1). Define its symmetric divergence, a clover-based directed Wilson variation, and its residual by

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diva Tb(x) = (2aN)−1 X
a
[Tab(x + ea) −Tab(x −ea)],
∆bF(x) = (2a5
N)−1 X
i
[contactx,+i(Xbi, F) −contactx,−i(Xbi, F)],
(SC3)
Ab(x) = diva Tb(x) −ZδBb(x).

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Here Xbi is the signed clover: one quarter of the four oriented plaquette curvature readings, with
signs +, −, −, + for (+b, +i), (+b, −i), (−b, +i), (−b, −i). All are transported to the common base.
Its adjoint covariance and antisymmetry follow by reversing the corresponding plaquette words. In
a color orthonormal basis ec, the contact is P
c XcDx,±i,ecF. The corresponding equation insertion
is β⟨X, forcex,±i⟩−P
cDx,±i,ecXc; Bb is its same signed sum divided by 2a5
N.

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Right or left Haar invariance in that one directed link gives
* 
!
+
*X
+
X
DcXc
XcDcF
β⟨X, force⟩−
F
=
.
(SC4)
c
c

Differentiate the change of variables in the finite Haar integral; the derivative of the Wilson density gives the force, and the product rule gives the derivative of and of. Compactness and continuity of the finite word derivatives justify moving that derivative through the integral. Linearity over the color basis proves (SC4) for the group-valued clover vector field. For a lattice cutoff, a vector field , an enclosed insertion and an exterior word, let

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Wab(x) = (χ(x + ea) −χ(x))Vb(x + ea) + (χ(x) −χ(x −ea))Vb(x −ea)
2aN
.
(SC5)

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The finite flux is −a4
N
P
x,a,b Wab(x)⟨Tab(x)FE⟩. Its exact decomposition is

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Flux = a4
N
X
x,b
χ(x)Vb(x)Zδ⟨∆bF(x)E⟩+ a4
N
X
x,b
χ(x)Vb(x)⟨Ab(x)FE⟩.
(SC6)

Use (SC4), the product rule, and finite symmetric summation by parts. The term with the discrete derivative of vanishes for the constant and skew-linear Killing fields in question, by symmetry of . The shifted ’s in (SC5) are essential for this algebra. The derivative of vanishes when each active Ward center avoids its full local insertion footprint. This is a support condition on the whole cylinder, stronger than just excluding its marked point. Taking absolute values in the second sum gives its exact error bound. No smallness of, including at the enclosed insertion, follows from this identity alone.

The continuum charges are instead supplied by the paid short-distance product estimates. Their input is the actual isolated product

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r hIJab(⌈y/aN⌉)IB(0) −Iu(0),
D + h = d + 4,
(SC7)

with of weight and in the one fixed horizon. At a fixed shell its coefficient bound may depend on that shell. It is common to every exterior test at that shell. The estimate is a rooted cylinder estimate on the reporting cell and on all its specified aligned seams and offsets, not only a two-point expectation. The paid product construction and weighted mesh interpolation give, for each finite family of exterior tests annihilating every-field,

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−→0.
(SC8)
X
ν
cνrh
m⟨Jab(y)B(0) Oν⟩
sup
y∈{rm/4<|y|<2rm}

The schedule uses 0 tending to zero and a finer mesh level) with 1. The product price chooses32; its exponential gain dominates the growth of the two insertion columns. The rooted remainder after the exterior stencil estimate is at most 2 stakes( pitts(), where pitts(. The common all-order mesh bounds transfer this from the fine reporting grid to the whole annulus, with the finer mesh

small compared with128. These are the annular product estimates used here; the finite Ward residual in (SC6) is not substituted for them.

Here are the rate choices that supply (SC7), rather than leaving its annular approximation as an input. Write = log( ). The calendar has1. For 1, let) be the first index with. Minimality gives, so squeezing between these inequalities and 1 gives; also + 1 eventually. A positive sequence of logarithmic rate here means log. Products add rates, quotients subtract them, and composition with multiplies them by. The estimates remain valid with arbitrarily small rate losses. For + 4, the two-field product and shell rescaling bounds give the available local approximation radius a logarithmic rate

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−D + M,
M = (d + 12)(1 −t).
(SC8a)

The and 4 are the field weights, the is the shell rescaling, and the additional eight is the pair of four-dimensional placement bounds. The adjacent insertion remainder starting at level obeys the explicit induction bound

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mk,i ≤b
 ai
D+1 fi
fk
,
(SC8b)
ak

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where fi > 0 is its retained error allowance. The induction step is exactly the allowance inequality
ηiLD+1
i
fi ≤fi+1, combined with ai = Liai+1.
Choose t = 1 + [64(d + 12)]−1.
Then M =
−1/64 > −1/32. The test normalization ci = eqi/32 has rate 1/32, while the accumulated coefficient-
normalization gain from level k to i is e(qi−qk)/8, of rate 1/8. Substitution of (SC8a)–(SC8b) into
the finite product approximation leaves the strict inequalities

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1
32 −(D + 1) < −D + M,
1
32 −M −1
8 < 0.
(SC8c)

All column counts, fixed-frame norms and logarithmic factors have rate zero. The two strict margins in (SC8c) therefore pay them eventually, give a positive admissible approximation radius, and make its rooted error at most 2 stakes, the bound in (SC7). The support requirements hold as well: the shell radius tends to zero, and the finest mesh times every finite stencil count tends to zero relative to the fixed physical chamber size. Thus the finite two-field approximation applies at every point in the required annulus, with bounded coefficient representatives. This is the asserted source of (SC7).

Why do the approximants in (SC7) disappear from (SC8)? For a finite exterior family, let be its combined finite-cutoffreading. Its limiting reading is zero by hypothesis, and convergence on a basis of gives0. At each fixed shell 0, although the approximant itself need not converge. The rooted estimate bounds the remaining product discrepancy against each exterior word by reflection positivity and its spectator cell bounds. The mesh interpolation then gives (SC8). This argument integrates annihilators, rather than selecting a new field at every integration point. Let be smooth, compactly supported, one near the enclosed point and zero near the other insertions. For a Killing field

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Z
(∂aχ)(y)Vb(y −x0)⟨Jab(y)B(x0) O⟩d4y.
(SC9)
QV (B; χ) = −
X
a,b

Its integrand is supported on a compact shell disjoint from all insertions, so it is an ordinary integrable smooth function. Conservation and symmetry imply independence of the cutoff: for their difference integrate the divergence of (. The divergence of is zero, and

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the symmetric tensor contracts to zero with the skew derivative of V . The difference vanishes near
every insertion and has compact support, so this integration by parts has no contact or infinity
boundary term.
Take V homogeneous of degree i = 0 or 1 and shrink χ to radius rm.
Changing variables
y = x0+rmz in (SC9) gives the factor r3+i
m : four powers from volume, minus one from differentiating
the cutoff, and i from V .
Choose h = 3 + i in (SC8).
Its bound times the finite integral of
|∂aχ(z)Vb(z)| tends to zero.
Therefore every finite linear relation among the ordinary D-field
exterior readings is satisfied by the flux readings, with D = d + 1 for translations and D = d for
rotations.

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Finite-dimensional linear algebra now represents all these flux readings by one field. Indeed, if
D and values bν satisfy every finite relation P cνAν = 0 ⇒P cνbν = 0, define a
functionals Aν ∈E∗
linear functional on their span by Aν 7→bν, extend it to E∗
D, and use E∗∗
D = ED. Faithfulness of the
quotient makes the representing class unique. Initially use generic grid configurations; translation
and continuity of the shell integral and of the kernels extend the same identity to every separated
configuration, including coordinate ties.

Linearity of (SC9) in and faithfulness prove that the representing field depends complex linearly on. Its rotation charge depends real linearly on the skew matrix, by the same argument. These are the actual unnormalized translation and angular charge maps on the physical quotient. Their normalization and full rotation laws are established by the subsequent angular-flux and comparison arguments; centering and the finite-dimensional existence argument alone do not fix them.

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The shell functional also has an ordinary face-flux interpretation.
Let a closed coordinate
box contain x0 strictly in its interior and every exterior insertion strictly outside the closed box.
Integrate each a-component of the current on its upper face minus its lower face, with three-
dimensional Lebesgue measure in the remaining coordinates. This sum equals (SC9). To prove it
choose a smooth window supported strictly inside the box and equal to one near x0. Multiply the
current by 1−χ. It is zero near the only enclosed insertion and smooth at all points of the box. Its
divergence equals −P
a ∂aχ Fa, by conservation and the Killing identity. The ordinary coordinate
divergence theorem applies to this smooth truncated current. On every face χ = 0, so its face
flux is the original one. The volume integral is (SC9), and cutoffindependence gives the equality
for any allowed window. This argument treats faces with their outward orientations and does not
integrate with four-dimensional ambient measure on the surface.

The bounded coefficient in (SC7) is required at each fixed shell before its lattice limit. It need not be uniform as the shell shrinks. The order is: fix the finite exterior annihilator and a reporting level; buy a sufficiently fine seam level; fix a shell index; then take the ordered mesh and period limits. At that shell the finite-dimensional annihilator norm tends to zero and the approximant coefficient is bounded. Only after this limit is taken does the shell radius go to zero. The rooted error tends to zero by (SC8a)–(SC8c). The same aligned-cell estimate applies on every seam and offset, so it permits moving a shell node freely in the annulus. This is the joint two-field approximation required for the shell construction.

Equation terms may be restored before this estimate is used. For one chamber their difference has zero expectation against every function supported outside that chamber, by the link Haar identity and avoidance of all the active links. Replace one chamber factor at a time in the full replicated array. All other factors are supported in complementary chambers, so each replacement leaves the integral unchanged. Finite induction replaces every chamber, including the conjugated reflected copies. Taking the reciprocal-cardinality power then gives equality of the two rooted seminorms. An equality only for a single unreplicated expectation would not suffice. No contact error at overlapping stencils is suppressed in this argument: overlap is excluded by the full footprint

condition, and actual contacts are kept in (SC4)–(SC6).

The centered stress trace

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The positive two-point representation gives monotonicity in positive Euclidean time before the
short-distance limit is known. The comparison can be selected inside any proposed sequence of
radii on the same table and phase. Monotonicity sandwiches nearby generic distances around each
selected radius; applying this to a hypothetical violating sequence proves the full right-hand limit.
A single favorable scaling subsequence would not suffice for this step.
Use the selected character vacuum of the same hierarchy and its conserved physical stress tensor.
Let T = P3
μ=0 Tμμ be its Euclidean trace, a degree-four physical field, and let

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Θ = T −⟨T⟩1.

Its pinned reconstructed field is Hermitian and Lorentz scalar, and

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⟨Ω, Θ(f)Ω⟩= 0,
Θ(f) = T(f) −⟨T⟩
Z
f I
(TR1)

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on the common field domain. To check the identity operator in this formula, first insert a unit label
in any Euclidean correlation and use the hierarchy’s unit law to remove it. The two corresponding
strict complex-time functions agree on imaginary times and hence everywhere by holomorphic
uniqueness. Smearing in that slot gives
R f times the remaining function, by Fubini, and passing
to the tempered boundary gives the same Wightman identity.
Pair the resulting word vectors
with arbitrary word vectors. Their dense span implies that unit insertion is (
R f)I on the common
domain. Linearity then proves (TR1). Similarly the Euclidean one-point function of Θ is zero;
holomorphic uniqueness and the boundary limit prove its vanishing vacuum matrix element. Reality
and scalar covariance follow from the stress tensor and the real invariant unit.
Let
E(s) = ⟨Θ(−s
2e0)Θ( s
2e0)⟩E,
s > 0.

The normalization used in the source gives

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lim
s↓0 s8 log2(1/s) E(s) = AG > 0,
AG = 12d2(G)2
b0(G)2
dim g
π4
,
d2(G) = −b0(G)
2
.
(TR2)

Thus = 3 dim g in these conventions. Positivity follows from 0, dim g 0 for the simple nonabelian Lie algebra, and the nonzero coefficient. These constants refer to the physical stress normalization and the Wilson-calibrated running charge.

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We record the mechanism establishing the full positive-distance limit from the scaling con-
struction. Orthogonal invariance of Θ and translation invariance identify its separated two-point
function at any two points with E of their Euclidean distance. An orthogonal reflection takes their
difference to a time-axis vector of the same norm; a common translation then puts them at the
opposed points above. The degree-four scaling comparison identifies the rescaled trace with the
scalar free-field label multiplied by 4d2, after the logarithmic running normalization. The free scalar
two-point contraction is
3 dim g
π4 |x0 −x1|8 .

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The running normalization contributes b−2
0 /4, while the trace coefficient contributes (4d2)2, giving
exactly AG|x0 −x1|−8. This proves the normalized limit along the extracted scaling radii and their
nearby generic configurations.

To remove the nearby-distance error, use the positive spectral representation

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E(s) =
Z
V +
e−sp0 dμΘ(p).
(TR3)

It is real and nonnegative. Since0 almost everywhere for this measure, 0 implies , and integration proves). Finiteness at every positive is part of the positive-time spectral representation. Thus this monotonicity is available before proving the short-distance limit; it is not inferred from that limit.

We give the precise selection statement needed here. Fix the physical table, its phase decomposition and its stress tensor. The trace construction selects a comparison ultrafilter below the table’s mesh-then-volume filter, its retained local data, and one free hierarchy. For every sequence tending to zero, it then provides a strictly increasing map, a scaling family on this same physical table, and its comparison data, whose radii satisfy the exact identity

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ri = sj(i).
(TLim1)

The scaling comparison applies to these radii. In particular the statement is not merely that one favorable sequence0 exists. Nor does it select a different physical hierarchy or phase when the sequence changes. The construction and this universal sequence property were established in the preceding interacting-to-Gaussian comparison. First prove convergence along the radii of any such retained comparison. For 1 2, put

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x(u)
0
= 0,
x(u)
1
= (u/2, u/2, u/2, u/2).

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The four coordinates of the two points are distinct, so this is a generic two-point configuration,
and its Euclidean separation is u.
Its maximum-coordinate separation is u/2 > 1/4, while its
configuration norm is less than 2. Consequently it has a neighborhood in the fixed comparison
window with norm bound 100 and maximum-coordinate separation at least 1/100.
Let bx(u)
i
be the rounded configurations in that comparison, and set di(u) = |bx(u)
i,0 −bx(u)
i,1 |. Round-

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ing convergence gives di(u) →u. The actual physical configuration is exactly ribx(u)
i
, not just an
asymptotic replacement of it. The interacting comparison, followed by the running-clock identity
and the free scalar Wick contraction, gives

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c(ri)E(ridi(u)) −→AGu−8,
c(r) = r8 log2(1/r).
(TLim2)

The centering and scalar homogeneity used in this identity are retained in the trace-label calculation below. The same normalization), independent of, is used on both sides of the coming inequalities. Fix 2). Eventually 0). Monotonicity and0 give

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c(ri)E(ridi(b)) ≤c(ri)E(ri) ≤c(ri)E(ridi(a)).

By (TLim2), their outer limits are and. Given 0, first choose close enough to 1 that both outer limits differ from by less than2. Then choose large enough for the two convergences and the bracketing inequalities. This proves. The complex pairing has zero imaginary part by (TR3), so this also proves complex convergence.

Finally, suppose the full right-hand limit at zero failed. There would be 0 and, for every a radius 0 such that