The centered trace and its non-freeness test · Part 7

the volume large enough that the comparison and selector errors total less than2. This proves vanishing of that correction, including the actual second Taylor coefficient rather than merely its finite difference. Consequently

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

2
X
cresti −1
−→−12g.
(TR59)
midPayi = Re
d,a
floftd,a,i

The crest comparison identifies its limit with that of; each displayed correction tends to zero. This proves the scalar payment limit used to choose the finite-level accuracy in (TR55). The exact quiet-loop cancellation in (TR54) is local. For a finite roster of gauge-invariant loop functions, their periodized sum is

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P(U) = creset
X
X
seats s
pα(s · U).
(TR60)
α

Each summand is analytic on the permitted group charts, so the finite sum is analytic and gauge invariant. If is an analytic gauge-covariant vector field supported in a nonwrapping finite neighborhood, its directional derivative is again analytic, local and gauge invariant. At the identity it vanishes, since the vector field does. Gauge covariance follows by differentiating the identity for after conjugating the infinitesimal direction; locality follows by differentiating only links in the common support.

The quiet roster is chosen so that the Hessian of (TR60) pairs to zero between a direction supported in that neighborhood and the prescribed distant polynomial directions of degree at most two. The loop lengths are bounded by; the separation condition 2( ensures this statement uses actual separated links rather than wrapped copies. The derivative of at the identity maps every direction into the same supported neighborhood. Differentiating twice therefore expresses its Hessian pairing as two pairings of the Hessian of, each with one supported direction. Both vanish. In the third local difference quotient entering Holt, the remaining terms with constant-direction legs vanish by gauge invariance and the local Ward identity; the other terms are exactly these zero Hessian pairings. The finite difference quotient is consequently zero. The local coefficient identity expressing Holt as the finite sum of these third differences proves Holt((

To obtain the quiet roster from the original loop sum, the construction adds its finite boundary compensation roster. If denotes the original truncated loop sum and this compensation, the quiet sum is exactly. Differentiation gives, the split in (TR54). The compensation has the small boundary envelope calorie; the original truncated derivative has its larger local envelope. The compensated roster retains gauge invariance and the common loop-length bound, so the preceding cancellation applies. This accounts for the exact zero term and for its separate small boundary correction.

The derivative in (TR56) can also be read directly from its determinant definition. Its local primitive is a finite contraction of

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1
2 log det(A(U) + μ2I) −log det(B(U) + μ2I).
(TR61)

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The first matrix contains the physical quadratic term and gauge-fixing term; the second is its gauge
comparison matrix. For a fixed positive μ, both are invertible near the identity. The real matrix
derivative formula gives
d
dμ log det(F + μ2I) = 2μ tr(F + μ2I)−1.
(TR62)

Joint analyticity in the mass increment and the link chart permits the finite local coefficient derivatives to commute with this mass derivative. Apply (TR62) to (TR61), then the normalized local contraction yardsites cob, and sum over the ordered off-diagonal axis pairs. The resulting derivative is times the corresponding resolvent contraction. Substitution

supplies the second factor of, proving the derivative identity in (TR56). For its bound, the period-limit coefficient comparison identifies the normalized local contraction with the massive momentum integral up to an eventual error at most one. The latter obeys

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|α(μ) + g log(1/μ)| ≤C0
(0 < μ ≤1),

where is independent of and of the allowed regulator segment. The ordered-axis contraction is twelve times this coefficient. Hence

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| Re Ri(μ) + 12g log(1/μ)| ≤12(C0 + 1)

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eventually for each fixed μ, which is (TR56). The momentum-integral coefficient and its uniform
bound follow from the logarithmic momentum integral and weighted massive-symbol calculation
below.
The roster condition behind that Hessian zero has a simple exact form.
For every pair of
plaquette axes and color labels, its charges cα and relative offsets δα satisfy
X
α
cα = 0,
X
α
cα(δα)a = 0
for each axis a.
(TR63)

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These are zeroth- and first-moment cancellations. For any additive real function F on the offset
lattice, P
α cαF(δα) = 0, by expressing F in its four coordinate values. The Hessian of each two-leg
loop term is the sum of its two cross-leg curvature products. Shift the site summation so its first
leg is at the same site for every α. Where the supported direction has nonzero discrete curvature,
that site lies in its one-link enlargement. The other direction’s curvature, on this neighborhood
enlarged by the loop length, is affine:

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curl T(x + δα) = C + F(x) + F(δα).

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Consequently summing the charge-weighted products gives a fixed first-leg factor times (C +
F(x)) P
α cα + P
α cαF(δα) = 0.
Both leg orders vanish in this way.
The same argument ap-
plies to each signed rotation; the supported neighborhood grows by at most one link. Summing
over seats and rotations proves the Hessian cancellation used above. Thus the zero in (TR55) ulti-
mately rests on the explicit finite moment identities (TR63), affine curvature of the test directions
and the stated nonwrapping margins.

16.6.7 The logarithmic momentum integral

The massive resolvent estimates use a positive mass lower endpoint chosen after the error tolerance and before the volume limit. Shell counting separates the near-zero lattice momenta from the remaining uniformly controlled modes. The later moving-mass statement is obtained by combining those estimates with its scalar payment and selector bounds. It does not interchange the massless and volume limits without control.

We now derive the uniform bound used after (TR62). Use real angular coordinates , Euclidean radius, and normalized momentum weight The

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actual massive two-point symbol gives a continuous scalar integrand Fμ(k), its half second deriva-
tive in the chosen external momentum direction. The momentum coefficient has the exact integral
formula
α(μ) = c4
Z
B
Re Fμ(k) d4k.
(TR64)

To obtain this formula from the torus symbol, parametrize the unit-period torus by its half-open fundamental cube and change variables; the Jacobian is (2. The symbol is real analytic in external momentum and internal angular coordinates for fixed Its first and second external derivatives are bounded on the product of a compact external interval and the closed internal cube. Dominated differentiation therefore moves both derivatives under the integral. The half in the symbol’s second coefficient and the factor two in the definition of give exactly (TR64).

The massive symbol estimates have the following two forms, with constants0 independent of 0

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|Fμ(k)| ≤
A
(r + μ)4 ,
|Fμ(k) −PG(k)| ≤D
r3 + Dμ2
r6
(μ ≤r),
(TR65)

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where
 
k2
1
r6 −4k2
1k2
0
r8
−1
r4
!
PG(k) = 4Cadj
.
(TR66)

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The distinct axes 0, 1 specify the chosen external direction and polarization.
The comparison
construction establishes the first estimate at scale r + μ, and the second on the subfamily μ ≤r
at scale r; packaging all admissible symbols and masses into these two families gives common
constants. The estimates apply to the same mass μ, without shifting it to a preferred radius.
Let Aμ = {μ < r ≤1}. In four dimensions the unit ball has volume π2/2, so polar integration
gives
Z

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Aμ
r−4 d4k = 2π2 log(1/μ) =: Jμ,
Z
Aμ
r−3 d4k = 2π2(1 −μ),
(TR67)
Z
Aμ
r−6 d4k = π2(μ−2 −1).

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Rotational invariance determines the required angular moments without a sphere normalization
convention. Equal diagonal moments and P
a k2
a = r2 give
R
Aμ k2
a/r6 = Jμ/4. For a ̸= b, rotate a
unit vector to (ea + eb)/
√
2 and to (ea −eb)/
√
2. Add their fourth-power identities; the odd terms
cancel and give
R k4
a/r8 = 3
R k2
ak2
b/r8. Summing the mixed moments over b then gives

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Z
k2
ak2
b
r8
d4k = Jμ/24
(a ̸= b).
Aμ

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Substitution into (TR66) yields
Z
Aμ
PG(k) d4k = c4 4Cadj

1
4 −4
24 −1

2π2 log(1/μ) = −11Cadj
24π2 log(1/μ) = −g log(1/μ). (TR68)
c4

Split the entire cube into, and. The unit ball is contained in the cube. On the first region (TR65) bounds the integrand by, whose integral is at most2. On the

last region it bounds the integrand by, giving at most vol(). On the annulus the difference from the pole integrates, by (TR67), to at most

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2Dπ2(1 −μ) + Dπ2(1 −μ2) ≤3Dπ2.

All inequalities are absolute before summing the regions. Equations (TR64) and (TR68) prove

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|α(μ) + g log(1/μ)| ≤c4
h
Aπ2/2 + A vol(B) + 3Dπ2i
.
(TR69)

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This supplies the mass-independent constant in (TR56). It also displays the origin of the factor
eleven and the normalization g = 2b0.
The pole (TR66) comes from explicit vector and ghost traces. For a normalized adjoint-color
generator Tb, tr(T 2
b ) = −Cadj. This is the adjoint Casimir identity in an orthonormal color basis.
For the external polarization in axis 1 and momentum te0, the continuum vector vertex is
p(k)I4 + Q(t)

 ⊗Tb,
p(k) = −2ik1,
Q(t)ab = 2it(δb0δa1 −δa0δb1).

The ghost vertex is. The free propagators are the scalar inverses, and the shifted denominator is . Spatial and color traces factor. Direct multiplication gives . Thus the two bubble traces, near = 0, are

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Hgh(t) = p(k)2(−Cadj)
r2D(t)
,
Hvec(t) = (4p(k)2 −8t2)(−Cadj)
r2D(t)
.

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Their denominators are nonzero for sufficiently small t when r > 0. Differentiating twice, using
D′(0) = 2k0, D′′(0) = 2, and p(k)2 = −4k2
1, gives

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H′′
gh(0) −1
2H′′
vec(0) = 2PG(k).
(TR70)

This proves the pole and its sign, including the half weight of the vector determinant. For the actual massive symbol the same logarithmic determinant differentiation separates each second coefficient into an upper term and a lower bubble term. Denote their second derivatives by for the vector matrix and for the ghost matrix. The exact scalar combination is

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h
1
2(Uv −Lv) −(Ug −Lg)
i
.
(TR71)
Fμ(k) = 1
2

This equality is before any bound. The upper terms and bubble terms each have common bounds of order in the admissible scaled family. Since multiplication by 12 decreases the absolute value, (TR71) yields the first bound in (TR65) upon taking On the family, take. The more precise estimates are

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|Uv| ≤u/T 2 + vμ2/T 6,
|Ug| ≤x/T 2 + yμ2/T 6,
|Lv −H′′
vec(0)| ≤A(T + μ2/T 2)/T 4,
|Lg −H′′
gh(0)| ≤B(T + μ2/T 2)/T 4.
(TR72)
Combining (TR70)–(TR72) bounds |Fμ −PG| by
uT + xT + A + B
T 3
+ (A + B + v + y)μ2
T 6
.
The angular cube has T ≤R∗= 1 + 2π, so

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D = max(uR∗+ xR∗+ A + B, A + B + v + y)

is a valid common constant in the second bound of (TR65). All constants are chosen on the family of all admissible regulator symbols, masses and internal momenta; restricting afterward to preserves those choices. This proves the scalar assembly of the two bounds. Their upper-term and bubble estimates are the matrix estimates (TR72), whose propagator and derivative bounds are developed next.

16.6.8 Weighted matrix derivatives for the massive symbols

Weighted matrix differentiation retains the order of factors in the finite determinant and symbol calculations. The vector and ghost contributions are combined only with their prescribed multiplicities and Fourier measures; the angular factor eleven belongs to this normalization. The subsequent position Ward tests determine the measured coupling rather than assuming that a bare coupling has already been measured.

Uniformity for moving positive masses The regulator mass can vary along the retained sequence, but it remains strictly positive when the finite inverse and determinant are differentiated. The weighted symbol estimates state how their constants depend on that mass and the annular momentum scale. Only after the corresponding uniform bounds and volume errors have been proved is the mass sent toward zero. This order is what permits assembling the vector and ghost contributions without exchanging a singular limit with an uncontrolled inverse.

We describe the matrix calculus behind (TR72). All matrices here have fixed finite index sets. Use a submultiplicative matrix normfor which tr. An analytic one-variable matrix family), at scale 0 and size0, has the required weighted bound if

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T j∥A(j)(0)∥∗≤CL
(j = 0, 1, 2),
(TR73)

with one common for the whole indexed family. Product differentiation gives sizes, with constant at most 4: the second derivative has the four terms. Sums add constants, and scalar multiplication multiplies the size by a bound on the scalar. If(0) is invertible, has size 0, and, then the inverse has size at the same scale. Indeed,

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(A−1)′ = −A−1A′A−1,
(A−1)′′ = 2A−1A′A−1A′A−1 −A−1A′′A−1.

The three weighted bounds are at most, and (2 . Their sum is a valid common constant. Analyticity of the inverse near zero follows either from its determinant formula or from a local geometric inverse. For two invertible families the exact identity

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A−1 −B−1 = −A−1(A −B)B−1
(TR74)

transfers a relative error size into inverse error size. Products of several differing factors are compared by replacing one factor at a time; (TR73) controls every term. These statements retain the raw derivatives, including their binomial coefficients.

The underlying symbol has a zero-field matrix), one-field vertices) and their incoming counterparts), and the two-field vertex). Along1, the symbol estimates give sizes 1, respectively, in (TR73), for both shifts. The shift-independent zero-field matrix has the same bound. Products of symbol families preserve this collection of bounds: a one-field vertex of a product has its two Leibniz terms, and its two-field vertex has the four terms corresponding to both placements and both cross placements. Inversion preserves it with overall size inverted, by (TR74) and the differentiated inverse formulas. This accounts for all placements of the external legs.

These sizes follow from precise Taylor cancellations at the origin. The zero-field matrix differs from by a function whose jets of orders below three vanish. The one-field vertex differs from its continuum linear vertex by a function whose jets of orders below two vanish. Their derivatives through orders three and two, respectively, are uniformly bounded on the fixed angular ball. Taylor’s estimate therefore makes their remainders of orders and, including their

weighted derivatives through order two along the admissible rays. The continuum one-field vertex itself is linear, so has size. The two-field vertex has uniform derivatives through order two at the unscaled size one. The symbol’s fixed analytic-strip bounds give all these derivative constants by Cauchy estimates, with chain-map norms included. For instance, bound the highest required derivative on the ball, use the vanishing lower jets, and integrate successively along the segment from zero to to obtain for. This proves the stated scaled Taylor bounds.

The admissible scale obeys

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|k| ≤T ≤|k| + μ,
μ ≤T,
0 < μ ≤1.

On the angular cube the lattice momentum magnitude) satisfies). Hence

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s(k)2 + μ2 ≥T 2/64.
(TR75)

The ghost massive matrix is this scalar times the identity. The physical vector quadratic form, including its gauge-fixing term, is bounded below by min( , where 0 is the common physical ellipticity constant. The elementary coercive inverse estimate gives both inverses norm at most a fixed constant times. Thus the inverse calculus above applies uniformly, even as both momentum and mass tend to zero. Let) denote the massive inverse and the incoming and outgoing one-field vertices. The lower bubble matrix is

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G(k)V−(−te, k + te)G(k + te)V+(te, k).

Its size is. Taking its second derivative and trace by (TR73) gives used in (TR71). The upper matrix is). Split into its massless vertex and mass-regulator vertex. In the massless part the first inverse is independent of; its norm is while the raw second derivative of the massless two-field vertex is bounded without a scale loss. This part therefore contributes. The regulator’s two-field vertex has size at derivative scale, and multiplication by) gives size. Its second derivative costs another yielding. This proves the upper estimates in (TR72), and the rough estimate follows from the full sizes and the bounded range of

For the bubble comparison take, and compare with the continuum matrices. The zero-field matrix discrepancy has size . Write

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E = T + μ2/T 2,
T 3 + μ2 = ET 2.

Equation (TR74) gives inverse discrepancy size. The vertex discrepancy has size ; the continuum vertex has size. Telescoping the four bubble factors therefore bounds their matrix difference at size. Its second derivative and trace are at most, exactly the last two estimates in (TR72). This completes the matrix power counting underlying (TR65); the origin-jet identities and common strip bounds invoked here are the corresponding symbol inputs.

The origin cancellations used in this calculus can be stated independently of the family bounds. For a real analytic function, the statement) on every real ray implies that all its jets of degrees below vanish. Indeed its first nonzero homogeneous Taylor polynomial would be nonzero on some real diagonal, by polarization, contradicting the ray estimate. Conversely, vanishing of these jets and a bounded order- derivative give the uniform Taylor estimate used in (TR73). Thus ray constants may depend on the ray; only the final derivative bounds must be common.

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For the lattice difference symbol qa(k) = eika −1, qa(k) −ika vanishes to order two and qa(k)
vanishes to order one. Therefore
X

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a
qa(−k)qa(k) −|k|2

vanishes to order three: write each product discrepancy as the sum of one second-order remainder times a first-order factor and one first-order linear factor times a second-order remainder. The ghost zero-field matrix is this lattice scalar times the identity, proving its required third-order cancellation. For the vector zero-field matrix, its exact gauge-fixed form is

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Avec(k) = d1(k)∗M(k)d1(k) + d0(k)d0(k)∗.

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The metric coefficient satisfies M(0) = I. Subtract the same expression with M = I. Its discrepancy
is the product of two first-order exterior differences and the first-order metric discrepancy, hence
vanishes to order three. The free exterior identity says that the expression with M = I is the lattice
scalar times the identity. Combining with the preceding ghost calculation proves the vector third-
order cancellation. This uses the actual quadratic symbol normalization, including its longitudinal
gauge-fixing term.
The one-field ghost vertex has the exact component expression
h
−e−i(ka+pa)qa(k) + qa(−k −p)eikai
(Tb)xy.

Rescale by a real, divide by, and let0. The exponentials tend to one and giving. Analyticity then makes the discrepancy from this linear expression vanish to order two. The gauge-fixing vector contribution has corresponding linear expression

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[−ikjδia −i(ki + pi)δja](Tb)xy.

Adding the Wilson curvature vertex gives

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δij[−i(2ka + pa)] + 2i(pjδia −piδja),

times, exactly the continuum vertex used in (TR70). The Wilson curvature contribution is obtained by differentiating its group commutator to second order and then applying the third-order chain rule; the Hessian of a commutator at identity is the sum of the two products of first derivatives with their reversed products subtracted. Terms from a group factor with zero first jet vanish. This fixes the curvature-vertex sign.

The permitted perturbing symbol has its one-field part vanishing to order two by its prescribed soft order three. Adding it therefore preserves the same continuum linear vertex. The common strip estimates on the base symbol and the perturbation give the derivative bounds required after these jet identities. This explains how the actual, possibly perturbed, symbol supplies the cancellations in (TR73), while retaining one continuum pole for the entire permitted family.

16.6.9 The determining position-space Ward tests

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The coefficient can be read in the position-space frame by a finite determining system. This step
distinguishes the formal first even coefficient from the later real remainder estimate. Let O = Oscale,
let T 0 be the normalized free Maxwell stress tensor, and write δv for the tensor δabv. For a real
invertible degree-four frame Ps preserving the stated conjugation and symmetries, the physical
tensor decomposition is
P −1
s
(s4Tab) = κsT 0
ab + dsδabO + esδab1.
(TR76)

Here all three coefficients are real and chosen simultaneously from the exact tensor identity. The physical coefficient result above gives for any coupling with the same leading normalization, in particular for the Wilson charge. Since0, it also gives

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Use the separated two-point configuration x = (e0, 0), with one exterior field O. At the scaled
configuration sx, pull both labels through Ps. Define Es as the first-point divergence of the moment
with the stress label PsT 0; define Bs as the same divergence with Ps(δO). Thus Bs is just the time
derivative in the first position of the two-scalar moment. These derivatives are with respect to the
unscaled configuration x, so the chain rule supplies a factor s multiplying the physical differentiated
field. For a smooth shell cutoffχ around the first point, define Rs as the corresponding translation
shell-flux test with stress PsT 0, inserted field PsO and exterior field PsO; define Qs with stress
Ps(δO) in its place.
Conservation and the translation Ward identity give exactly

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κsEs + dsBs = 0,
κsRs + dsQs = Bs.
(TR77)

To check the scale factors, the shell-flux integral changes by: its volume contributes and its cutoffderivative contributes. Multiplying (TR76) by this flux functional and using the chain-rule factor on the translated field yields the same factor on both sides, which cancels. The unit term has zero divergence and zero translation shell flux; these are exact unit identities. No coincident insertion or contact identity is used in (TR77).

The position-frame jet comparison on compact separated configurations gives

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Bs →B0,
Rs →B0,
Qs →Q0(χ).
(TR78)

The second limit uses the free translation Ward identity. The first is explicit. Put = 3 dim g The free scalar moment on is

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B0 = −8cG ̸= 0.
(TR79)

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Consequently the second equation in (TR77), together with ds →0, implies κsRs →B0. Dividing
by Rs →B0 ̸= 0 gives κs →1. Divide the first equation by u2
s and by κs; then

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Es
u2s
= −ds/u2
s
κs
Bs −→−d2B0.
(TR80)

Thus the tested conservation column is known through its first possible even coefficient; the other three columns are needed only at order zero. For clarity, the determining formal system uses an unknown order-zero stress amplitude and a two-entry invariant jet (), the latter corresponding to powers 1. Its verified columns are

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B = B0,
E0 = 0,
E1 = −d2B0,
R = B0,
Q = Q0(χ).

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The equations are

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κE0 + d0B = 0,
κR + d0Q = B,
κE1 + κ′E0 + d1B + d0B′ = 0
for every κ′, B′.
(TR81)

Since= 0, the first forces = 0, the second forces = 1, and the third forces Conversely those values satisfy every equation. In the signed parity convention this gives the trace jet (0 0), because the trace multiplies the scalar coefficient by four and 4. This formal parity statement alone asserts no real order-four remainder; that stronger estimate requires the normalized frame construction below.