14 The analytic stationary graph, loop moments and the cali- brated coupling history · Part 2

exponential weights are unchanged, whereas the real-coordinate maximum box is carried to its rotated image. The corridor is therefore transported as a set, with the boundspaint1+dim g, paint1 relating the two norms; it is not asserted to be the same coordinate box in a rotated basis. The status of Theorems 14.4 and 14.5 is on-shell recovery with an explicit off-shell defect bound, on the branch selected by the corridor. The uniqueness clause of Theorem 14.4 is what makes the branch a single-valued section rather than a choice.

14.2 The Fourier normalization and the kinetic readout

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The two-field determinant film has an internal momentum and two opposite external momenta. Its
R
[−π,π]4 F(k)d4k/(2π)4; its finite-
internal average is normalized Haar average, namely stock F =
period average is shopn F = n−4 P
k F(2πk/n). The curvature picture called profitable F is one
half of the internal average with the two external link and color indices in their specified order. The
factor one half is the quadratic Taylor factorial. Translation invariance makes the actual Hessian
entry zero unless its two Fourier momenta sum to zero, and otherwise that entry is the site count
times the finite-period average of the film.

To compare the two averages, expand the internal variable in its absolutely convergent Laurent series. Finite averaging retains exactly the coefficients whose four indices are divisible by the period; Haar averaging retains only the zero coefficient. Every remaining index has length at least the period. Spending half the common strip for summation and a quarter for the tail gives a fixed strip price times. Multiplication by the site count gives the recorded-Hessian error. Thus this internal integral represents the finite Hessian with a quantitative error and a fixed normalization.

Compose that picture with the momentum chart for real initial links. The readout leapTip()) is the divided coefficient Reality makes this diagonal coefficient real. Color invariance and simplicity make its color bilinear form a scalar multiple of the fixed invariant form, so its value is independent of the normalized color. The factor two in (DP1) is additional to the factor one half in the internal film average; both are retained in that definition.

Here is why this one transverse entry determines the leading kinetic tensor. Subtract its multiple of the normalized Wilson tensor, whose readout is one. Gauge invariance gives transversality. Signed covariance includes the phases from reversing initial links; at the first nonzero homogeneous order those phases have constant term one. The constant and linear homogeneous terms vanish by transversality and permutation covariance. Let) be the symmetric polarized degree-two term of the difference. Reflecting one axis shows that a component can survive only when its momentum occurrences and link occurrences have the same parity on every axis. Permuting axes

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moves the zero transverse diagonal readout to all transverse diagonal entries on single momentum
axes. Longitudinal entries vanish by transversality. Reflection parity then kills mixed momentum
entries in a diagonal link component and same momentum entries in an off-diagonal link component.
The only remaining possibility has its two distinct momentum axes equal to its two link axes.
Transversality at the sum of those two coordinate vectors makes twice that entry zero, since the
diagonal entries have already vanished. Bilinearity gives Q = 0. The degree-three term vanishes
under total inversion, whose momentum sign is odd and whose two link signs multiply to one. The
difference thus starts at degree four. This proves the one-axis kinetic identification used below,
including the Taylor normalization and reflected-edge convention.

Source–readout formulation. Two readouts are compared here, and each has an explicit fibre. The first is the pair of internal averages: shop reads a periodic analytic film on the finite dual

torus, stock reads its Haar average. They are two different readouts of the same film, and their difference is a recorded discrepancy, not an identification:

Proposition 14.6 (Aliasing readout). Let be periodic and analytic with sup on the strip of width. Then

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shopn F =
X
x∈nZ4
bF(x),
stock F = bF(0),
| shopn F −stock F| ≤A c(σ) e−σn/4,

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with a strip price c(σ) independent of n.

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Proof. The Laurent series converges absolutely, and the finite average of eik·x over the period-n
mesh is one if every component of x is divisible by n and zero otherwise. Shifting contours gives
| bF(x)| ≤Ae−σ|x|1. Every surviving nonzero index has |x|1 ≥n; write e−σ|x|1 ≤e−σn/4e−(σ/2)|x|1 ·
e−(σ/4)(|x|1−n) and sum the middle factor over Z4, giving c(σ) = [(1 + e−σ/2)/(1 −e−σ/2)]4.

The exactness class is controlled approximation in the period at fixed strip width, with the normalization (the factor one half of the quadratic Taylor factorial and the site count) retained on both sides. The second readout is leapTip, which reads a single transverse diagonal coefficient of a two-leg tensor. On arbitrary tensors its fibre is large; on the class of tensors that actually occurs it is as small as the kinetic identification requires:

Proposition 14.7 (One transverse entry determines the kinetic tensor). Let be a gauge-invariant, signed-covariant analytic two-leg tensor whose leapTip readout vanishes. Then its homogeneous parts of degree at most three vanish; starts at degree four.

Proof. Gauge invariance gives transversality, and signed covariance includes the phases from reversing initial links, which have constant term one at the first nonzero homogeneous order. Constant and linear terms vanish by transversality and permutation covariance. Let) be the symmetric polarized degree-two term. Reflecting one axis, a component can survive only if its momentum occurrences and link occurrences have the same parity on every axis. Axis permutations carry the vanishing transverse diagonal readout to all transverse diagonal entries on single momentum axes; longitudinal entries vanish by transversality. Reflection parity then kills mixed momentum entries in a diagonal link component and equal momentum entries in an off-diagonal one. The only remaining entry has its two distinct momentum axes equal to its two link axes, and transversality at the sum of those two coordinate vectors makes twice that entry zero, the diagonal entries having vanished. Bilinearity gives = 0. The degree-three term is odd under total inversion, whose two link signs multiply to one, so it vanishes.

Equivalently, on gauge-invariant signed-covariant analytic two-leg tensors the map leapTip() is injective modulo terms of degree at least four, and the normalized Wilson tensor is a section of it. Gauge invariance and signed covariance are properties of the target tensors; in the frame realization the tensors are those of the readout composed with, their Taylor coefficients at a flat raw configuration are taken with the raw directions conjugated into the stored frame (Theorem 9.2), and the invariant color form is unchanged by that conjugation, so the readout and its normalization are fibre constants.

14.3 Analytic exterior division with the origin jets retained

We give the analytic division used in issuing the loop coefficients. For one momentum coordinate set is a Laurent index, define

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(Pxi−1
a=0 eip·(x[i:=a]),
xi ≥0,
−P−xi−1
a=0
eip·(x[i:=−a−1]),
xi < 0.
Li,x(p) =

The empty sum is zero. Multiplication and telescoping give the entire identity

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qiLi,x = eip·x −eip[i:=0]·x.
(ED1)

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If F is periodic analytic and bounded by A on a strip of width σ, its Fourier coefficient bound on
an intermediate strip makes G(p) = P
x bF(x)Li,x(p) absolutely and locally uniformly convergent
on width σ/4. Indeed the finite sum has at most |x| terms and size at most |x|e(σ/4)|x|; the spare
width absorbs |x|. Summing the geometric coefficient bounds in each momentum coordinate gives
the explicit price

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21 + σ/4
v
,
(ED2)
∥G∥σ/4 ≤A 4
σ
σ/4

where is the number of momentum coordinates. Thus, including at every zero of; no singular quotient is defined on a mesh. If the numerator has zero jets below order comparison of its analytic Taylor series with ) gives zero jets of below order . Equivalently evaluate on nearby directions with nonzero-component, divide their one-variable series, and extend the resulting polynomial-jet identity to all directions by continuity.

For exterior coefficients let be contraction by the vector, and exterior multiplication by the-th coordinate vector with its orientation sign. The coordinate anticommutation rule gives id. Apply (ED1) componentwise to a cycle, and call the divided difference Substituting in this anticommutation rule shows that

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CIiDiu = u −combi u,

where comb evaluates at = 0 and deletes every exterior component containing. The identity is first algebraic where= 0, and extends to the zeros by analyticity. The residual is again a cycle. Indeed, after separating components containing, its contraction at momentum equals its contraction with the-th symbol set to zero, since every-containing component has been removed; the original cycle equation at = 0 supplies the remaining equations. Evaluation and deletion preserve the original strip, coefficient bound and Taylor vanishing order.

Repeat on the residuals in the four axis directions, adding the four lifted differences. Every residual still has the original width, price, and order. Each new lift is therefore computed directly at width4, price at most times (ED2), and order at least1. There is no iteration of four strip shrinkages or four losses of Taylor order. Summing gives width4, price four times (ED2), and order1. The contraction of this sum telescopes to the original cycle minus the final residual. In positive exterior degree the latter is zero because every orientation contains a processed axis. In degree zero it is the original coefficient with all coordinates of this block set to zero; the required hypothesis is its vanishing on that entire coordinate plane. Positive order at the full origin implies this hypothesis only when this is the sole momentum block. This is why the exact harmonic-set check in (TEL2) is needed.

Suppose additionally that each coefficient vector lies in a prescribed linear subspace depending only on the other momentum blocks. Evaluation in a divided coordinate preserves that subspace.

The divided Fourier sum also does so: where= 0 it is the scalar multiple (, and at its zeros continuity and closedness of the finite-dimensional subspace give the same conclusion. Exterior multiplication is a signed selection of these coefficient vectors. Hence every residual and lift retains the other block constraints. For a dependent momentum equal to minus the sum of the independent ones, the linear shear exchanging that block with the dependent block is involutive. If there are + 1 independent blocks, its missing coordinate is the negative sum of their coordinates. A strip of width+2) is mapped inside width, since (. Apply the preceding coordinate construction on that narrowed strip and shear back with the same narrowing factor. This gives the dependent-block width and price used in (TEL3).

Source–readout formulation. Division is a section of contraction, and this is the form in which the loop coefficients are issued: a cycle is the readout, a primitive is a source datum over it, and the choice of primitive is the fibre. Write

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4
X
i=1
IiDi
combi−1 · · · comb1 u


Div u =

for the sum of the four lifted differences constructed above, so that the telescoping identity of the derivation reads Divcomb comb

For completeness we verify the single-axis identitycomb and the cycle property of the residual in one computation. Write for the interior product with the-th coordinate vector and for contraction with the symbol whose-th entry is set to zero, so that Scalar division commutes with, hence, using ), and = 0; therefore), first where= 0 and then everywhere by analyticity. Consequently

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CIiDiu = qiDiu −IiCDiu = u −u|pi=0 + Iiιi(u|pi=0),

and is the projection onto the components containing, which givescomb. Applying to this identity and using = 0 gives comb = 0, so the residual is again a cycle.

Proposition 14.8 (Division as a section of contraction). On analytic cycles of positive exterior degree (and on degree-zero coefficients vanishing on the block’s coordinate plane),Div = id. The fibre of over is Div + ker, and two primitives of the same differ by a-cycle of one higher degree. Div preserves the block constraints, loses one Taylor order and three quarters of the strip, and is bounded by four times (ED2).

Proof. The identity and the bounds were derived above; the description of the fibre is linearity of

The origin jets are retained on both sides: the proposition records the loss of exactly one Taylor order and of three quarters of the strip, and the degree-zero hypothesis is kept as a hypothesis (it is the reason for the exact harmonic-set check in (TEL2)). No singular quotient on a mesh is ever part of the realized record; where vanishes the primitive is defined by the entire identity (ED1).

14.4 The two-leg correction with both momentum blocks retained

Loop data and the running-coupling reader. The loop calculation reads local quadratic response from the same stationary integration used in the forward blocking step. Its internal momentum measure, external slot ordering and Taylor factorials determine the numerical coefficient.

The calculation supplies a drift estimate for the actual action coefficient; it does not declare that coefficient equal to a chosen reference line. Summing the drift will require the common calendar constructed next. Thus the coefficient and its controlled remainder, rather than only the formal logarithm, are the output consumed by calibration and uniform running.

Here is the precise two-leg use of the exterior division. There is one independent four-momentum ; the other is. Write , with its ordered plane basis, and let be contraction by the lattice symbol at. Thus = 0. A coefficient with one remaining potential index belongs to; one with no such index belongs to. These are coefficient-valued periodic analytic functions of the same complexified momentum. Let

contract the second index by the symbol at. Since these maps act on different tensor slots,

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HCr
−= Cr
−H.
(TEL1)

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There is no exterior sign in this convention. A coefficient has order a if all its real Taylor coefficients
of total degree below a at zero vanish. Each contraction increases this order by one.
The given two-potential tensor is u ∈E1 ⊗E1, of order s+2, satisfying both gauge contractions
C0
−u = 0 and Hu = 0. Apply the dependent-block division to obtain

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X ∈E2 ⊗E1,
C1
−X = u,
ord X ≥s + 1.

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The coefficient HX ∈E2 has order at least s+2, and is a dependent cycle because C1
−HX = Hu = 0.
A second dependent-block division gives

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Y ∈E3,
C2
−Y = HX,
ord Y ≥s + 1.

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Next lift the independent contraction: choose D ∈E3 ⊗E1 with HD = Y and order at least s. The
degree-zero obstruction for this last division must be checked on its actual harmonic set. There
is just one independent momentum block, so that set is exactly p = 0, rather than an unspecified
collision locus. Since s + 1 ≥1, every coefficient of Y vanishes there. This proves the required
obstruction is zero. In the general many-leg construction the other independent contractions de-
fine coefficient subspaces that remain constant when this block varies; the coordinate homotopy
preserves them. For two legs there are no other independent indices, so this preservation condition
is vacuous.
Now put
V = X −C2
−D ∈E2 ⊗E1.
(TEL2)

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It still has order at least s + 1. Its represented tensor is unchanged, since C1
−V = u −C1
−C2
−D = u.
Its remaining contraction is removed, since HV = HX −C2
−HD = HX −C2
−Y = 0. Apply the
independent-block division separately to the six E2 coefficients of V . This gives g ∈E2 ⊗E2, of
order at least s, whose two curl contractions equal u. All identities hold at zero momentum as
analytic identities, as well as at nonzero real momentum.
For clarity the width and norm losses can also be followed without any existence convention.
Put slimt(a) = a/(t + 2), and, for a coordinate set of size m,

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4
r + 1
!
2(1 + a/4)
m
,
P(a) = T4(a/2),
E(a) = a/16,
Br(a) =
Tm(a) = 16
4(ea + 1).
a
a/4

Here is four times the coordinate-division price in (ED2); bounds the finite sum defining a curl contraction on the strip. If has strip width 0 and coefficient bound0, set

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τ = E(σ),
w = E(τ),
μ = w/16,
b = P(σ),
c = bB0(τ),
d = cP(τ),
e = dT4(w).

The bounds for are respectively; their widths are4. Restricting to, the bound for is (TEL3)

For the final independent division there is exactly one remaining leg. Apply the coordinate-wall lift (ED1)–(ED2) with the four independent coordinates to each component of . The other coefficient indices are passive, so taking the maximum of their coefficient bounds introduces no extra factor six. The construction gives explicitly

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η = μ/4,
∥g∥η ≤A{b + eB2(w/8)}T4(μ),
ord g ≥s.
(TEL4)

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To see the contraction identity componentwise, denote the lift of the independent index by
Wa(b, p), where a is the dependent two-form index and b the independent two-form index. Then
P
b symbol1(p)b,iWa(b, p) = Va,i(p). Contracting this equality with symbol1(−p)a,h and summing
over a gives uh,i by (TEL2).
This is precisely the two-curl representation, with no omitted
polarization or identification of the two momentum slots. More generally repeated independent
divisions obey the recursion η0(σ) = σ, K0 = 1, ηt+1(σ) = ηt(σ/4), Kt+1(σ) = Tm(σ)Kt(σ/4),
where m is the ambient coordinate count.
Each step preserves the contractions in the other
independent blocks because their symbols do not vary with the divided block. For this application
only t = 1, m = 4 is needed. These prices depend on the input strip, not on the torus period or on
s. The tensor spaces, the harmonic-set check and the two identities in (TEL2) give the two-leg
construction used in both loop-moment applications below.

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Source–readout formulation.
The source object is the two-curvature coefficient g ∈E2 ⊗E2;
the readout is the double curl Curl2(g)h,i = P
a,b symbol1(−p)a,h symbol1(p)b,i ga,b, valued in doubly
transverse two-potential tensors.
Theorem 14.9 constructs a section s2 with Curl2 ◦s2 = id on
analytic doubly transverse tensors of order at least two, bounded by (TEL4) and lowering the
order by two.
The readout is not injective: its kernel contains C2
−(·) in the first slot and the
analogous image in the second.
The correction −C2
−D in (TEL2) is precisely a motion inside
that kernel fibre, chosen to make the intermediate primitive transverse in the remaining slot. The
realized history stores (X, Y, D, V, g); the readout recovers u and nothing more.

Theorem 14.9 (Two-curl primitive). There exists of order at least whose two curl contractions equal, constructed as follows.

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1. Dependent-block division gives X ∈E2 ⊗E1 with C1
−X = u, ord X ≥s + 1.

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2. HX ∈E2 has order at least s+2 and C1
−HX = HC1
−X = Hu = 0; a second dependent-block
division gives Y ∈E3 with C2
−Y = HX, ord Y ≥s + 1.

3. Independent division gives with and order at least; its degree-zero obstruction lives on the harmonic set, which for a single independent block is exactly where every coefficient of vanishes since

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4. Put
V = X −C2
−D ∈E2 ⊗E1.

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Then ord V ≥s + 1, C1
−V = u −C1
−C2
−D = u and HV = HX −C2
−HD = HX −C2
−Y = 0.

5. Independent-block division of the six coefficients of gives

All identities hold as analytic identities at zero momentum as well as at nonzero real momentum.

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Proof. Each step is an application of Proposition 14.8 together with (TEL1); the cycle con-
ditions needed for each division are the displayed ones.
For (5), let Wa(b, p) denote the lift
of the independent index, a the dependent and b the independent two-form index.
Then
P
b symbol1(p)b,iWa(b, p) = Va,i(p), and contracting with symbol1(−p)a,h and summing over a gives
uh,i by (TEL2). No polarization is omitted and the two momentum slots are not identified. In the
many-leg construction the other independent contractions define coefficient subspaces constant
in the divided block, which the coordinate homotopy preserves; for two legs that condition is
vacuous.

14.5 Production of the fresh loop moments

The moment normalization of the fresh determinant loop is obtained before its path representation is issued. Let be the common analytic two-field picture of the determinant and Haar term, the normalized coarse metric row with(0) = 1, and the inherited-loop picture. The latter vanishes to order four in real external momentum. Set equal to the kinetic readout of and subtract times the normalized metric picture. The one-axis identification proved above shows that this difference, and therefore slots(

vanishes to order four. It retains its two gauge-contraction identities, reality and color invariance. Its analytic norm is bounded by bedCharge(, where bound and bounds the metric row. Thus this normalization does not discard the inherited loop price.

Apply (TEL1)–(TEL3) with = 2: the input has zero jets below order four, and its two-curvature coefficient has zero jets below order two. In particular the second-contraction correction uses the harmonic set = 0 checked there; the coefficient norm and positive strip width have the period-independent prices in (TEL3).

Write the resulting coefficients, for fixed plaquette and color types, as a real Laurent family ). Taking real parts replaces an analytic coefficient on real momenta by the average of it and its conjugate at opposite momentum. This preserves the represented real tensor and its vanishing order. Project its finite color coefficient vector onto invariant tensors; this is a fixed real linear projection, of finite absolute matrix-sum price, which leaves the already invariant represented tensor unchanged. It commutes with reading the absolutely convergent Laurent series and with its derivatives. Thus

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bP(p) =
X
z∈Z4
P(z)eip·z,
bP(0) = 0,
D bP(0) = 0,
X
z
|P(z)|eη|z| < ∞
(LM0)

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for a fixed positive output width η.
Exponential summability also gives P |ziP(z)| < ∞, by spending part of η. The series may
therefore be differentiated on the real momentum axes. Its value and first derivative at zero are
P P(z) and i P ziP(z). Equation (LM0) proves both sums zero, separately for each plaquette and
color choice. The actual path issue enumerates these types and offsets. Its first leg is based at
zero, its second at z, and each leg includes the four plaquette edges. Its length is |z| + 8, so the
path price differs from the Laurent price only by the fixed factor e8η and the finite type and color
counts. The displacement in the moment is exactly z. Reindexing by this enumeration proves the
summable zeroth and first unwrapped displacement moments of the issued atoms. These are the
fresh-loop identities needed in the next subsection and in (TF17)–(TF19).

Source–readout formulation. The fresh loop is produced by composing three maps whose source–readout roles are different. (a) The kinetic readout of the two-field picture is subtracted;

by Proposition 14.7 the remainder lies in the fibre of that readout over zero and therefore starts at degree four. (b) The two-curl section of Theorem 14.9 is applied to the remainder with this chooses one primitive in the kernel fibre of the double curl, with the bound (TEL4). (c) The real part and the invariant-color projection are applied; both are fixed real linear maps of finite price which fix the already real, invariant represented tensor and commute with the Laurent series, so they move the coefficient only inside the fibre of the represented tensor. The issued path atoms are then a relabelling of the Laurent family: first leg at zero, second at, length + 8, displacement exactly. The two vanishing statements of (LM0) are thus statements about the chosen source coefficient, not about the represented tensor alone, and they are the input of Proposition 14.10 below. The normalization step does not discard the inherited loop price: the norm bound bedCharge( keeps all three contributions.

14.6 Production and transport of the inherited loop moments

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The full-family identities (TF17)–(TF19) require absolute summation before truncation. We give
the construction that transports these identities across a block. For a two-curvature atom a, let qa
be its coefficient with its plaquette and color types fixed, and let δa ∈Z4 be the displacement from
its first leg to its second. Its length bounds |δa|. A charged exponential bound implies
X
a
|qa| < ∞,
X
a
|qaδi
a| < ∞,
(LM1)

because. Thus all changes of summation below are absolute, including those with a displacement factor.

For one atom translate its first leg to the origin in the finite torus sum. Along a fixed route from zero to, telescope the second leg. Subtract its value and its first differences at the origin. The remaining discrete second-order Taylor difference can be written as a finite sum of products of differences, one on each curvature factor, by summation by parts along successive route edges. Explicitly the removed terms are

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qa
X
x
Fa(x)Ga(x),
qaδi
a
X
x
Fa(x)∇iGa(x).
(LM2)

The discrete Taylor identity is exact on the torus: each link telescope is a translated finite difference and the sum of a translated product can be shifted by the same lattice translation. Repeating this along the two route indices generates at most the square of the route length in remainder terms. The exponential budget absorbs this square after any fixed positive width discount. If the full family has zero zeroth and first displacement moments, summing (LM2) gives zero separately for each type and color. Its total action therefore equals the sum of the two-difference remainder patterns. This proves the representation used to inherit a quiet loop; the subtraction is performed

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on the whole summable family, not on an arbitrary finite prefix.
Let ψi(r, b; z) be the installed fine-gradient column for a coarse residue r, curvature type b,
direction i, and unwrapped displacement z.
These columns are the product of the fine frame
inverse, the differentiated linear tower, and the coarse frame. Whenever the stationary-column
construction uses two curvature legs, its dependent-momentum join is precisely (TEL1)–(TEL3),
with the retained Taylor orders stated there. On harmonic input the unmodified tower has residue-
independent total columns. Install it on a one-site coarse torus. Its output is then constant on
the fine torus; a fine difference of that output is zero.
The intertwining identity between the
differentiated column and the fine difference consequently gives
X

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z
ψi(r, b; z) = 0.
(LM3)