Finite stress, scaling maps and shell charges · Part 3

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These same readings control the inclusions and derivatives without requiring a bound on the
inverse in the physical coefficient norm. In a rescaled position variable, integration by parts is
exactly
Z
t(x)S(rk∂αB, . . . ; rkx) dx = −
Z
(∂l,αt)(x)S(B, . . . ; rkx) dx.
(WS4)

The derivative of a compactly supported test has the same support constraint. Apply the converging injective reading matrix at the receiving degree to the two sides. A left inverse close to the fixed free reading matrix then proves

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U−1
k,d+1Iphys
d,d+1Uk,d −→Ifree
d,d+1,
U−1
k,d+1rk∂phys
α
Uk,d −→∂free
α
.
(WS5)

Every inverse used in this deduction acts on a finite detector matrix with a fixed injective limit. It does not presume that the physical charts are uniformly conditioned.

One can make the first line of (WS5) exact. Choose left inverses of the free adjacent inclusions. Average them over the finite signed permutation group and then average with their complex conjugates. The averaged maps are still left inverses and now respect both symmetries. Define recursively

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Xk,0 = Uk,0,
Xk,d+1 = Iphys
d,d+1Xk,dPd + Uk,d+1(1 −Ifree
d,d+1Pd).
(WS6)

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Substitution of Ifree
d,d+1v proves exact compatibility with the adjacent inclusion, hence with every
inclusion by composition. Induction in d, using (WS5), gives U−1
k,dXk,d →1. Consequently these
maps are eventually isomorphisms, preserve the unit and symmetries, and still satisfy the second line
of (WS5). On the finitely many exceptional indices choose any isomorphisms; these choices do not
affect a limit. Conjugation by matrices tending to the identity preserves all the finite-dimensional
limits just obtained.
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 larger separated window (WS9)

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.

Source–readout formulation. The scaling maps are the interface between the two realized sectors over the common label algebra. They are not canonical: each is selected by compactness and diagonalization and corrected by (WS6), and the statement proved is about the limit of readings, not about the maps. What is recovered is the short-distance germ of the interacting hierarchy at

scale, read in free coordinates, to all derivative orders. What is not asserted: uniformity in the degree, a bound onor its inverse as, or that the interacting theory is free. The lift transports these maps as linear maps between label spaces, on which the frame realization acts trivially.

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Theorem 23.2 (Scaling comparison). There are radii rk ↓0 and isomorphisms Xk,d : F free
d
→Fd,
real, signed-covariant, unital and exactly compatible with inclusions, such that for every word, every
derivative order and every compact subset of the separated configuration space,

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∂γ
x S(Xkv; rkx) −→∂γ
x Mn(x; v)

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uniformly, where Mn is the free hierarchy (FM3), and X−1
k,d+1rk∂αXk,d →∂free
α
.

Proof. This assembles (WS1)–(WS17), Theorem 23.1 (for the two-point comparison and moving labels) and the detector argument of Theorem 16.7.

One calibration matrix serves the three panels (trace, stress and rotation): the same finite reader list and the same matrix of readings are used for all three, so the three conclusions are statements about one realized family and can be combined. This is the common-witness requirement of the method in the form in which it is needed here.

23.5 Constructing the physical stress and its shell charges

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).

Source–readout formulation. Theorem 23.3 is a representability statement: the flux reader, a priori only a family of numbers indexed by exterior words, is the reading of a field because it respects every linear relation among field readings. This is exactly the method’s criterion for a function on the readout to descend from the source (here: to lift to a label class), and uniqueness is faithfulness of the quotient. The amplitude and sign of the stress are fixed in two separate steps ((SR4) for the sign relative to, and the Ward normalization of Theorem 19.3); the finite lattice identity (SC6) keeps its residual explicitly and is never used as if that residual were small.

Theorem 23.3 (The charges are fields). For each of weight and each Killing field there is a unique class in whose exterior readings equal the flux readings for all exterior words; it depends complex linearly on and real linearly on the skew matrix. The same number is the ordinary face flux through any closed coordinate box containing in its interior and all exterior insertions outside.

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Proof. If functionals Aν ∈E∗
D and values bν satisfy P cνAν = 0 ⇒P cνbν = 0, then Aν 7→bν is
a well-defined linear functional on the span, extends to E∗
D and is represented by an element of
E∗∗
D = ED; faithfulness of the quotient (Theorem 18.3) makes the class unique and gives linearity.
Generic grid configurations are used first and translation and continuity extend to all separated
configurations. For the face flux multiply the current by 1 −χ with χ supported strictly inside the
box: it is smooth on the box with divergence −P
a ∂aχFa, the divergence theorem applies, and
χ = 0 on every face.