Finite stress, scaling maps and shell charges

Finite stress, scaling maps and shell charges

Actual finite stress, followed by geometric station selection. The stress and current begin as functions on finite group configurations. Their Ward identity, Haar-Jacobian terms and quiet-loop localization are computed there before extraction. Rooted estimates, then the full-support phase argument, transfer these finite readings to the required characters. A single calibration matrix must work for the three panels and their joint higher-arity suppliers; centering removes the unit direction before the relevant rank assertion is used.

The station-selection equation is geometric:). It chooses a station at a requested physical radius. The actual interacting and Gaussian moments retain the IG3 comparison error at that station. Consequently the exact equality of radii and the limiting equality of moment readings have different logical roles. The physical conserved stress is then identified on the same quotient and with the same shell-charge normalization. Flux representability is proved with the actual geometry and faces; a coefficient rank count alone would not recover the ordinary face flux.

23.1 The finite trace candidate, its equation identity and its compensation

The local current and stress entering this chapter are finite configuration functions. For a base site , let ) = curv(

The curvature map is the real color projection specified in the startup construction, so this expression is defined on every group configuration. Gauge transformations act by the adjoint at a link-valued current and a scalar tensor by

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(Yb(o, U))(x,i) = 1{x=o}Cbi(o, U),
Jab(o, U) = 1
4δab
X
i,j
⟨Cij, Cij⟩−
X
i
⟨Cai, Cbi⟩.
(TF0a)

Color coordinates are converted back to the actual link-coordinate space by the fixed color isometry. At the identity every vanishes. Its derivative in a logarithmic link direction is precisely

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cab(o, A) = Aa(o) + Ab(o + ea) −Aa(o + eb) −Ab(o).

Thus(1) = 0 and(1) = 0. Its Hessian is obtained by replacing a product by. These formulas supply the actual quadratic jets, rather than an unspecified continuum stress approximation. Initially is symmetric and has identically zero trace.

The divergence used for the finite Taylor calculation is the fixed high-order antisymmetric stencil

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4
X
Daf(o) =
l=1
dl{f(o+lea)−f(o−lea)},
(d1, d2, d3, d4) = (4/5, −1/5, 4/105, −1/280). (TF0b)

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It has first moment one and its odd moments three, five and seven vanish. Substitution of the four
rational coefficients proves these identities. Taylor expansion therefore reads the ordinary derivative
exactly on polynomials of degree at most eight. Physical spacing factors are inserted only when
the finite tensor is rescaled.
During a stationary descent retain the full tuple (J, Y, E, Q, Ξ), with exact identities
X
a
Jaa =
X
a
∇aQa,
Jab −Jba =
X
i
∇iΞabi,

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X
a
DaJab −Eb ≡DYbP
through Taylor order four.
(TF0c)

Hereis the unit forward difference, and the last equivalence includes orders zero through four at the identity. Initially = Ξ = 0, and is defined as the difference on the last line, so that it is an equality. The Hessian of is the retained covariant quadratic form; differentiating, using (1) = 0, gives its two cross Hessian pairings. The local analytic change of variables preserves these four jets by the chain rule. The scalar rows Ξ remain analytic, gauge invariant, and zero through order one.

To see why the identities persist under blocking, denote the main averaging stencil by, the directional flux stencil by, and its correction by. Their finite coefficient identities are

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Fa = M + ∇aTa,
L5MDa = DaL4Fa.

Pull each fine scalar function back along the stationary section. Transport with with, and the difference potentials with their degree-three edge stencils. Add to the transported, and add 1 to the transported Ξ. Substitution proves all three identities in (TF0c); the two added terms cancel precisely the change from the main stencil to the flux stencils. The current is transported by the derivative of the stationary section, so the chain rule also preserves the last identity. No exact symmetry or zero trace of each blocked local representative is presumed.

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Localization is performed on the scalar and current functions themselves: gauge fix along the
prescribed finite axial paths, discard coordinates beyond the chosen radius, and retain the Taylor
polynomial through order four in those local coordinates. Denote the localized tensor by Jloc, and
localize Q, Ξ by the same operation. Restore its exact trace and its six antisymmetric divergences
by defining
X
X
eJ00 =
∇iQloc
Jloc
−
ii ,
i
i
i̸=0

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eJab = Jloc
ab
(a < b),
(TF0d)
eJba = Jloc
ab −
X
i
∇iΞloc
abi
(a < b).

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Other diagonal entries are unchanged. The six pairs are 01, 02, 03, 12, 13, 23; their identities follow
individually by subtraction. Summing the four diagonal entries proves the trace identity. The same
construction has bounded support, enlarged by one difference step.
The localization error of a differentiated tensor is controlled at the actual stencil sites: (TF0b)
uses only shifts of length at most four, so its error is bounded by P
l|dl|(A+A) if the undifferentiated

error is at most. Summing the four directions gives four times this allowance. Support and anchor weights are enlarged by four at this step. The corresponding derivative and tail allowances are summed with the same absolute coefficients. Exponential path decay pays the discarded tails including their first moments; the stationary four-jet identity pays the local Taylor difference. These are the separate errors retained when the localized divergence is issued as a degree-five insertion.

Finally average the scalar tensor and its divergence over signed permutations and conjugation, with weight 1Signed, multiplying each component by its tensor or vector sign. Reindexing this finite average proves reality and the required signed covariance. The vector average kills the identity-configuration constants: the axis reflection reverses the vector sign and pairs identical constants. It does not increase a common rooted bound, because the sum of the absolute averaging weights is one, every transformed chamber has the same rooted seminorm, and the rooted triangle inequality applies. Transport commutes with the same transformations. Thus the issued vector and tensor have the correct exact finite symmetries before any coefficient limit is taken.

We first specify the finite objects whose limits enter the trace argument. This prevents a formal Taylor coefficient from being confused with a continuum insertion. At a fixed finite representative let be its rooted infinitesimal link-current vector field, the directional derivative of a configuration function along its left logarithmic link increments, and div its divergence for product Haar measure. In exponential coordinates this divergence includes both the ordinary coordinate divergence and the derivative of the logarithm of the Haar density. Write for a finite partial sum of the classical action, for a finite partial sum of the ordinary logarithmic corrections, and (TF1)

Here is the rooted lattice stress expression, with the same mean and curvature rows as in the finite stress construction. The finite sums are taken from the actual source dictionaries, rather than from new representatives of their limiting functions.

The protected configuration change along, with step 0, gives a finite difference identity. If denotes the resulting density-difference expression, its exact decomposition on the protected healthy region is

Searchable transcription

u
= Zb −βBb + (divν Yb)(1),
(TF2)
Zb = Rb + {divν Yb −(divν Yb)(1)} −DYbLQ + Ku,b +
X
γ∈F
Hu,b,γ.
(TF3)

The constant denoted by evaluation at 1 is evaluated at the identity configuration. The fourth term is the finite Taylor remainder of the changed action and Jacobian after subtracting their first directional terms, divided by. The last sum is the finite correction for the chosen finite shelves of activity terms: each summand is the corresponding changed activity difference with the coefficient and the factor dictated by that difference. Thus (TF3) defines a literal configuration function. To prove (TF2), expand the change of density, distribute), and use (TF1); the terms cancel. Splitting the activity change into the finite retained shelves and their remainder gives exactly the last sum. This is an algebraic equality before any limit or expectation is taken.

The correction shelves can be enlarged to approximate the full changed expression uniformly on configurations. The estimates for rooted activity sums above give nonnegative numbers with each summand regular on its actual support, and

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X
γ∈F
mass(Pγ)gγ ≤εcorr.
(TF4)

The finite four-term part of (TF3) is regular on the common neighborhood and has derivative allowance 2. The sum in (TF4) is included with its own supports, not paid as if every term had the largest common support.

The conversion of a regular function into a coefficient insertion is particularly explicit. Let Π be its Taylor polynomial along the local analytic section at the identity, projected to the degree-five frame, and weight its homogeneous components with the prescribed threshold powers. Call the resulting coefficient). Choose the primary local section)), and put the masked difference)) into one regular activity slot, with every other slot zero. The resulting formatted insertion equals on the protected region, has zero extra scalar term, and retains locality and symmetry. If is the norm of the finite Taylor reading map and the norm of this section construction, a bound for the original section gives the two respective bills

mass( mass( (TF5)

These are finite-dimensional operator bounds and the ordinary triangle inequality for Add this construction for the four terms and the finite correction sum. Taylor reading is linear, so the actual candidate coefficient is the sum of the four weighted Taylor readings plus the weighted readings of. Equation (TF4) bounds the size of the latter by (. This supplies the coefficient and its paid remainder together.

The equation identity for this candidate is also literal. Differentiate the finite change of variables in the normalized Haar integral, or integrate its density difference before dividing by. A spectator supported outside the changed links is unchanged. The resulting expectation is zero. Apply (TF2), subtract its identity-configuration constant, and then let the shelves increase and follow its admitted small-step schedule. Their uniform allowances justify these passages by dominated convergence. The same identity with a spectator derivative gives the finite lattice Ward contact term when that spectator meets the support. In particular equation directions have zero separated readings, precisely the equation cancellation already used in constructing the physical quotient.

There is a possible degree-five vector component in this procedure. The raw-to-free vector map has exactly the kernel proved in (FC9)–(FC10). Choose its real signed-equivariant section by the corrected polynomial lift and subtract the unique remaining multiple of. This is finite-dimensional subtraction before the coefficient limit. It explains why the trace candidate can be compared in fixed vector coordinates despite its original equation component.

The compensation by the quiet logarithmic part is exact at the coefficient level needed here. In logarithmic link coordinates the vector-current diagnostic is a sum, over, of the quadratic and cubic Taylor coefficients with two equal fixed-color variations in direction and opposite lattice directions. Its cubic part is the contraction denoted crest; denote the quadratic contraction by suite floft

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X
Hq(X) = crestq(X) −1
d̸=i
floftd,i,q(Xi).
(TF6)
2

The quiet logarithmic contribution is constructed with two moment cancellations; gauge invariance alone would not establish the required zero in (TF6). For each fixed ordered pair of plaquette types and color coordinates, let be the finite two-slot path coefficient and the displacement of the second slot from the first. Put

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C0 =
X
γ
cγ,
Ca =
X
γ
cγ(zγ)a.
(TF6a)

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Append a zero-displacement two-slot term with coefficient −C0+P
a Ca, and four terms of displace-
ments ea with coefficients −Ca. Repeat this for every plaquette and color pair. The augmented

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finite family then satisfies exactly
X
γ
cγ = 0,
X
γ
cγ(zγ)a = 0
(a = 0, 1, 2, 3).
(TF6b)

Its appended coefficients remain color invariant because that condition is a real linear subspace. The quiet logarithm is the signed and translated sum of this augmented family. The difference between it and the retained ordinary loop sum is precisely the finite appended part. Its coefficient sizes are bounded by the zero and first moments of the omitted path tail; exponential path decay bounds these moments, giving the small remainder already paid in (TF4).

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Here is why this construction cancels (TF6). Let T = Lquiet. Its Hessian at the identity vanishes
when one direction has bounded support and the curl of the other is affine on the enlarged support
neighborhood. In fact the Hessian of a two-slot term is the sum of the two products of its linearized
curl slots. With the first direction fixed, the other slot evaluated at displacement zγ is A + F(zγ),
with F additive. Summing over the family gives A P
γ cγ + P
a F(ea) P
γ cγ(zγ)a = 0 by (TF6b).
The same calculation applies to the reversed slot order and each signed translate. The nonwrapping
and support hypotheses ensure that these displacements are the actual integer displacements; in

the application 2(, where bounds the current support and the retained path length. For, differentiate twice at the identity. The current vanishes there, and the first derivative of a gauge-invariant scalar there is zero. The product rule consequently reduces the Hessian to (TF6c)

The directions have the bounded support of the current. Choose to be the polynomial link directions of degrees at most two that occur in the coefficient test. Their curls are affine on that neighborhood. The preceding Hessian cancellation makes both terms in (TF6c) zero. Terms with a constant pure-gauge direction vanish by the differentiated gauge identity. Expanding the cubic coefficient in the two position variables of the test uses just these Hessian pairings, so its centered-link reading is zero.

One must still convert the centered-link reading into the actual link reading. If the two position arguments are, the change is multiplication by

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e i((z0)i+(z1)i)/2.

The constant and first coefficients of the original reading vanish. The normalized third coefficient of this product is therefore its original third coefficient plus2 times its second coefficient. Apply the finite coefficient functional in the variables. The latter term is zero unless when, its value is floft. This follows by expanding the quadratic coefficient on ( and using 1. Summing over and gives exactly (TF6). Its centered-link version has just been shown zero, so) = 0. The factor 12 thus comes from this link-centering phase and the normalized Taylor coefficients. Subtracting the paid appended part gives the stated bound for the ordinary loop contribution. No quiet contribution is left undetermined.

The needed vanishing of the quadratic part can be derived without setting a mass regulator equal to zero inside an integral. For 01, the local resolvent estimates below give, for the massive comparison expression

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| cobd,i,q(Lμ)| ≤Cμ−1.
(TF7)

The proof differentiates the Fourier-resolvent expression twice; its pointwise integrand is bounded by. Counting lattice shells and multiplying by the momentum cell volume gives

in four dimensions. This applies to both the physical and comparison resolvents, hence to their sum. At each fixed positive, the resolvent identity identifies the unregularized current coefficient with cob), up to an error tending to zero in the retained large-volume limit. Given choose = min(1 + 1)]) first. The massive term has magnitude at most2; take the volume large enough to make the remaining error below2. The unregularized quadratic coefficient therefore tends to zero. The Taylor remainder bounds give the same limit for its finite-difference reading. Summing the finitely many shows that (TF6) has the same limit as its cubic part.

That cubic limit is, where is the one-loop slope constant in the coupling drift. Here is how the logarithmic slope fixes it. For the regulated determinant expression(log), the determinant estimates give (TF8)

The regulated response is), with = log. The real mean value theorem on an interval 0] supplies an intermediate regulator at which this response differs from by at most 2). First choose large, then use the local resolvent difference estimate to compare that response with the unregularized cubic current coefficient; this error tends to zero when the upper endpoint is sufficiently negative and the retained volume is sufficiently large. The intermediate regulators stay in the fixed positive interval [], so the large-volume estimates apply uniformly there. The triangle inequality proves the limit. Together with (TF7), it proves for every color. These are finite rooted Haar-Jacobian coefficient tests, not an assumption about a mollified continuum field.

Finally the passage through many adjacent coefficient transports retains a relative error. Put

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ωi = exp{−2
5 RLog( bβi)} −→0,
k = k(i) > i.
(TF9)

The chosen calendar ensures every adjacent transport error between and is at most times the next coefficient size. If denotes that coefficient, backward induction proves

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∥vj −vk∥≤ωi(k −j)
k −i
∥vk∥≤ωi∥vk∥,
i ≤j ≤k.
(TF10)

Indeed the induction hypothesis and1 imply; adding the next adjacent error gives the asserted bound at The calendar inequality follows from the positive reserve 10: logarithms and the number of intervening steps are eventually dominated by the exponential with exponent 110. Thus this accumulated relative error vanishes. The finite candidate, its equation identity, the quiet compensation and its scalar diagnostic consequently survive on the same retained field and enter the following trace comparison.

Source–readout formulation. The retained tuple ( Ξ) is the realized source record of the trace; its three identities (TF0c) are discrepancy columns kept at every level: is the recorded potential of the trace, Ξ of the antisymmetric part and of the equation term. The restored tensor (TF0d) is a section that makes two of the discrepancies exact again after localization. The slope constant enters the trace through, which reads the same number that was read in (DR5) from the coupling drift; the two readings are obtained from different readers (the kinetic reader and the cubic current reader) of the same determinant films, and their agreement is a derived consistency of the realized history and is not imposed. All objects are gauge-invariant cylinders or coefficient vectors, so the frame lift is the identity on values and the identities hold for pushed forward by the horizontal lift of Theorem 9.2, and so on.

The candidate itself is a literal configuration function at a fixed finite representative, and its defining relation (TF2) is an algebraic identity before any limit or expectation. This is the form

required by the method for an equation residual: the equation term is retained as a column of the realized record, and its vanishing in separated readings is proved (Proposition 18.2), not built into the definition. The possible degree-five vector component is removed by Proposition 16.5: the raw-to-free vector map has kernel exactly, and the unique remaining multiple of is subtracted before the coefficient limit. The quiet compensation is a motion inside the fibre of the moment readout of Proposition 14.10: it changes the presentation of the candidate by a family with vanishing zeroth and first moments and does not change the readings that define the trace.

23.2 The interacting-to-Gaussian comparison used for the trace

This comparison concerns the interacting Wilson family and its selected continuum phase. It is distinct from the calculation of a Gaussian constant coefficient in the preceding chapter. We derive the passage between them.

At a fixed retained comparison station take two physical lattice insertions, supported in its strict chamber, with their coefficient representatives converging along the common comparison filter. Let be the corresponding one-point Gaussian readings and let be the centered Gaussian two-point reading. The corrected singleton and pair polynomials are

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Dj = Aj −mj,
D01 = (A0 −m0)(A1 −m1) −z.
(IG1)

We next derive the finite interacting bound for these functions. This also specifies which Gaussian quantities are subtracted. For marked insertions let be the cumulant polynomial of the Gaussian profile at the rounded positions. The corrected cylinder is the coefficient polynomial obtained by multiplying the insertion generating polynomial by exp(), in the finite algebra in which each marker has square zero. Its singleton coefficients are, and its pair coefficient is exactly in (IG1). Thus the subtraction includes the Gaussian means as well as its connected pair coefficient. Let be a comparison station, its fixed maximum number of marks per chamber, and its monitor. Define

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Ik = log Hk
16M ,
Rk = e2Ik,
εk = e−Ik.
(IG4)

Here0. The initial marked density is the Wilson density times the product of the reflected copies of the corrected cylinder over all chambers of the tiling. Push this density along the same exact renormalization course as its unmarked Wilson density. In particular its vacuum integral is the original Wilson partition function. The comparison uses this density identity, not a replacement of the interacting law by a Gaussian law.

We describe the bounds on this push. At the middle stop its regular marked sources have values ), with specified fired marker sets and support cubes. They occur together with the forced sources and the unmarked background in the finite density expansion described in the marked-integration chapter. If is the corresponding coefficient of the negative Gaussian cumulant, then on the healthy local chart

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|Wα(U) + Cα| ≤b−31/32(t),
|Cα| ≤b1/4(t).
(IG5)

Here is the station coupling,) = exp( boil()), and boil(. To obtain the first bound, expand the actual insertion product by its finite cumulant expansion. The local Gaussian comparison of its calibrated chart coefficients gives an error at most) for each coefficient; the finite exponential map from cumulants to moments is continuous on this fixed finite coefficient space. Its chosen neighborhood makes the cumulant-polynomial discrepancy plus that error at

most). The exact negative cumulant coefficient cancels the corresponding free coefficient. The second bound follows because the rounded Gaussian profile stays in a compact subset of the generic position domain, where its finitely many coefficients are bounded, while

Here is the polynomial estimate behind the local Gaussian comparison just used. Denote coefficient size by coin, and let bound the degree of each insertion polynomial. After the actual linear substitution into the common key and residual coordinates, write these polynomials as write for their calibrated free polynomials, included in the same coordinates. The calibration is the exact identity that their weighted truncations through the assigned degree agree. If the substitution has weighted size at most, the original polynomial has size at most, and the weight radius is1, every remaining monomial has weight at least. Hence

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coin(Pi −Fi) ≤aibk0ρ−(di+1) ≤ξ.
(IG8)

This is the high-degree tail estimate; it uses the actual solved calibration maps. If coin( 1, telescoping a product over any subset of the insertions gives

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Y
!
i∈J
Pi −
Y
i∈J
Fi
≤|J|ξ(B0 + ξ)|J| ≤nξ(B0 + ξ)n.
(IG9)
coin

Substitution of the Gaussian variables commutes exactly with their Wick functional: the substituted covariance is the pullback of the original covariance and its mean is the pullback of the original mean. These are equalities of finite Gaussian moments, including the residual coordinates, not limiting approximations.

Let all means and covariances have absolute value at most1, and let bound their discrepancies from the zero-mean continuum free data in the key coordinates. Put and . Wick’s recursion bounds the functional on a degree- polynomial by coin(), and its change under these mean and covariance discrepancies by coin( One proves both assertions by induction on the monomial degree: remove one variable, pair it with each remaining variable or replace it by its mean, and, for the difference, telescope the resulting products. The at most + 1 choices at each step and the bound fit the displayed constant.

Combining this bound with (IG9), and then comparing the free-key polynomials under the two Gaussian data, gives total error at most

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CB,N
nξ(B0 + ξ)n + Bn
0 E
.
(IG10)

The common covariance estimates and small-mean estimates in the Gaussian comparison chapter give precisely these bounds on the pulled-back data, uniformly over the allowed small middle-stop means. The station choices make (IG10) at most). Thus the estimate feeding (IG5) follows from the actual covariance comparison, exact calibration and a finite polynomial calculation. It is not an additional interacting-to-free hypothesis.

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These coefficient estimates are weighted in the density, as follows. A regular source in a support
cube of radius QB has support charge at most exp(e(2Q + 1)4), since that cube occupies at most
(2Q + 1)4 budget blocks. Its marker weight at radius Rk is at most RM
k = b1/8(t), by the identity
between the monitor and the station coupling. Consequently multiplication of (IG5) by the support
charge and marker weight fits the prescribed regular-source bound. The same calculation with the
additional unit of support charge fits the Gaussian compensation source bound. The forced sources
already satisfy their full-configuration bounds. The one-step marked integration estimate therefore
propagates this compensated density from the middle stop down to the width stop, preserving the
unmarked background and bounding the total source weight per marker and the total forced weight