Convergence of the interacting scale histories · Part 2

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contraction by the lattice symbol at −p. Thus Cr
−Cr+1
−
= 0. A coefficient with one remaining
potential index belongs to Er ⊗E1; one with no such index belongs to Er. These are coefficient-
valued periodic analytic functions of the same complexified momentum. Let H : Er ⊗E1 →Er

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.

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 and the finite type and color counts. The displacement in the moment is exactly. 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).

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)

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The frame factors preserve the constant-output statement by their translation convolution identities.
Odd blocking is the hypothesis that makes this centered-gradient readout agree with the installed
fine difference. Exponential column bounds justify passing from the installed periodic identity to
the absolutely summed cover columns.
After substituting these columns in a two-difference pattern, one fixed choice of its two out-
put types has coefficient uf0(z0)f1(z1), where P f0 = P f1 = 0 by (LM3).
Its zeroth total is

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u(P f0)(P f1) = 0. Its displacement is (c1 + z1) −(c0 + z0), with fixed carried offsets c0, c1. Split-
ting its first moment gives

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(
)
(
X
f0)
X
z
(c1 + z)if1(z) −
X
z
(c0 + z)if0(z)(
X
f1)
u
= 0.
(LM4)

The absolute first moments exist by the column width bounds. Summing over residues, types and patterns is justified by (LM1) and the charged bound on the full substituted family. Thus inherited loops have the exact two moment identities, with their block price, rather than merely small moments.

Joining two families means enumerating their disjoint union. Absolute sums add, so both moment identities are preserved. Blank initial loops have zero coefficients and satisfy them. The coefficientwise limiting argument (SH1)–(SH3) preserves them as well: first pay the common exponentially weighted tail, then pass to the limit on a finite list. This supplies the full-family justification used to turn retained moments into omitted-tail moments in (TF18). Any fresh loop must satisfy the same identities before it is joined; they cannot be inferred from gauge invariance or the quadratic-potential normalization alone.

The one-loop drift used by the finite histories

The loop calculation evaluates the drift of the very finite histories used later in shooting. The fresh and inherited moments are kept separate until their transport identities are combined. The coefficient) refers to the physical Lie-algebra and Fourier normalizations fixed earlier, and the error allowance is bought before the final cutoff. The subsequent calendar turns these local drift estimates into uniform running for the actual represented couplings.

The finite coupling drift comes from the committed path coefficient and the bounded scalar reset, not directly from a formal beta function. At a transition with block length, put The determinant calculation on the retained physical path defines a coefficient with sign appropriate to descent from level. We explain the logarithm and its uniform remainder.

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For a fixed color direction, the continuum pole of the relevant second external-momentum
coefficient is
!

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k2
1
|k|6 −4k2
1k2
0
|k|8 −
1
|k|4
pG(k) = 4CA
.
(DR1)

The coefficient includes the vector determinant with weight 12 and the gauge determinant with weight1. These are the same two determinant weights kept in the massive-symbol calculation of the trace. The finite symbol coefficient), on the angular cube [, satisfies

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|fμ(k)| ≤
A
(|k| + μ)4 ,
|fμ(k) −pG(k)| ≤D
|k|3 + Dμ2
|k|6
(|k| ≥μ),
(DR2)

where0 depend on the fixed analytic widths, coefficient and ellipticity bounds, and not on , the torus period or its terminal cutoff. The input paths have their fixed quadratic normalization and the allowed higher soft jets; the latter preserve the continuum pole.

Here is the matrix estimate yielding (DR2). At scale, the zero-field massive matrix, one-field vertex, and two-field vertex have sizes 1. Coercivity bounds the inverse by Differentiate the identity = 1 to obtain and its second derivative, the sum of the two products with two first vertices minus the product with one second vertex. Along a momentum direction scaled by, all jets of each inverse still have size. The lower determinant bubble therefore has size, and its second unscaled external-momentum coefficient

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has size T −4. The upper insertion obeys the same rough bound. Summing the finitely many matrix
entries gives the first bound in (DR2).
For |k| ≥μ, take T = |k|. The zero-field symbol differs from its continuum quadratic part
by O(T 3 + μ2), and the first vertex differs from its continuum linear part by O(T 2 + μ2/T). Use
A−1−B−1 = A−1(B−A)B−1 and telescope the factors of the determinant bubbles. Their relative er-
ror has size T +μ2/T 2; their second momentum coefficient consequently has error C(T +μ2/T 2)T −4,
which is the second bound in (DR2). The upper insertion contributes bounds of these same or
smaller orders. The origin-jet cancellations, analytic derivative bounds, inverse formulas and ex-
plicit continuum vertex contraction are developed in the weighted massive-symbol and logarithmic
momentum calculations below. This argument uses them before integration, so its error bound is
absolute.

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To integrate, split the angular cube into |k| ≤μ, μ < |k| ≤1, and its part outside the unit ball.
The first bound in (DR2) gives

Z

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 ≤Aμ−4 π2μ4
2
= π2A
2 .
|k|≤μ
fμ(k) dk

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Outside the unit ball it gives a bound A times the volume of the angular cube. On the annulus the
second bound gives
Z

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μ<|k|≤1
D
|k|3 dk = 2π2D(1 −μ) ≤2π2D,
|k|6 dk = π2D(1 −μ2) ≤π2D.
(DR3)
Z
μ<|k|≤1
Dμ2

All these errors are bounded independently of 0 The pole integral is exact. Rotational symmetry gives, on this annulus,

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Z
k2
1
|k|6 dk = 1
4Iμ,
Z k2
1k2
0
|k|8 dk = 1
24Iμ,
Iμ =
Z
|k|−4 dk = 2π2 log(1/μ).
(DR4)

For the mixed fourth moment, symmetry makes all off-diagonal moments equal. Expanding the fourth powers of and, and rotating these directions to a coordinate direction, shows that the diagonal fourth moment is three times the mixed one. Summing over the four coordinates then gives the factor 124. Substitution of (DR4) into (DR1), with Fourier weight (2, gives

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(2π)−4
Z
μ<|k|≤1
pG(k) dk = −11CA
24π2 log(1/μ) = −s log L.
(DR5)

Equations (DR2)–(DR5) bound the difference between the infinite-volume coefficient and this logarithm by a pre-block constant.

We now connect this integral to the coefficient of the actual stationary path. It is important to distinguish three determinant films:, the film of the actual stationary integration;, its core film before the stationary carrier change; and, the auxiliary rank film. Each film is a matrix of analytic functions of the external momentum, with indices consisting of a direction and a color. Their scalar reading is

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αa(M) = 2 Re leapTip(profitable M)(e0, 1, (a, a)).
(DP1)

The operation leapTip is the coefficient of degree two in that external-momentum direction, with its Taylor factorial; profitable passes from the determinant film to its curvature tensor using its

analytic contraction identities. Thus is linear on films satisfying the common bounds, and is the specific normalized kinetic coefficient, not the full film norm.

Here are the quantitative comparisons used in the transition. Fix an analytic width with128, a block patch of stage, its penalty, a positive stationary width, and all the common metric, covariance and path bounds. The required hypotheses are: the input path is attached to its quadratic metric and has its prescribed analytic bounds; that metric has the uniform floor on every torus; the coarse metric has its analytic strip bound; the stationary graph is critical, guessed by the installed tower, and satisfies the gap and heat identities; the same forest and tag are used in its determinant realization; and the relevant torus sides form an auxiliary sequence tending to infinity. These are precisely the linear-horizon and stationary-graph data already constructed. The input loop-tensor norm may have its own bound Λ, whereas the comparison constants below depend only on the fixed analytic and matrix bounds specified before the block length. They do not grow with Λ. Let0 bound the inverse matrices in the determinant calculation, let0 bound the common zero-, one-and two-field film factors, and let0 bound their carrier difference. All three are obtained from the fixed stage, kernel, penalty and common analytic bounds before is chosen. More explicitly the common factor bound is the sum of the absolute bounds for the leg, core, rank and frozen metric factors; the carrier bound is the common bound for the difference between the core and rank factors. Put

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D∗= FK∗F, E∗
= D∗M∗+ FK∗, R∗= E∗+ (E∗F + FM∗D∗)M∗+ FM∗FK∗.
(DP2)

This is the source’s exact reserve polynomial. It arises by subtracting inverse factors with , then telescoping the zero-, one-and two-field determinant products. For example the first inverse difference costs; multiplying by the common first factor and adding the factor difference costs; the remaining two placements and their cross product give the last line. The trace estimate is a relative rank estimate, so this subtraction introduces no count of all block sites.

In detail, if Loc Loc, the individual films are bounded on width

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L4 2r(FM∗+ FM∗FM∗).
(DP3)

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The separate difference estimate is

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Mcore −Mmrank
bounded on width h/2 by R∗.
(DP4)

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Here is why (DP4) has no block-volume factor.
The matrix weights are invertible analytic
frames SX(p), SY (q).
A bound for a matrix U means a Euclidean operator-norm bound for
eU = SX(p)−1USY (q). For the difference estimates the stronger bound is used:

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| tr( eUV )| ≤C∥V ∥2→2
for every compatible V.
(DP10)

It is subadditive, is invariant under transposition with dual frames, and multiplies with an ordinary operator bound on either side. At equal source and target frames, taking = 1 bounds the actual trace by. A general matrix with operator norm only has price dim( in (DP10); using that conversion would incur the unwanted

The exact difference between the core and rank quadratic films cancels their common sector and pure Gram terms. Write for the shear row, for its localized emitter correction, and for the fixed block inclusion. Expanding ( gives its reflection part. The complete remaining difference is the sum of the local house film and this reflection part, minus

and the finite motive film; is the localized orbit correction. This identity holds separately for the zero-, one-and two-field matrices, with all their Leibniz placements. The common sector cancels before any trace is bounded.

We derive the uniform trace-ideal estimates before using them in the difference. For a finite matrix, set sup

For, the trace is, so its size is at most. Decomposition into columns bounds the size by the sum of their norms. A matrix with a fixed-dimensional source consequently has price. Multiplication on either side by a bounded matrix multiplies this price by its operator norm, by trace cyclicity. Apply these statements after the analytic changes of frame in (DP10).

One further factor has a source dimension growing with, and needs an actual summability argument. Put, let be the lattice Laplacian symbol, and define

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∞
X
dμ(k) =
μ2
ω(k) + μ2 ,
J = 16
n=0
(n + 1)−2 < ∞.

On the strip of width 1128, choose the centered real-momentum representative before enumerating aliases. The massive lower bound gives

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|dμ(kl)| ≤
2
(1 + pdist(l))2 .

At = 0 this is the origin massive bound. At= 0, centered alias separation bounds the symbol scale below by (1 + pdist(; strip ellipticity bounds the denominator below by half the square of that scale. Division gives the inequality. Changing the centered representative permutes the aliases. There are at most (2 aliases at distance, because their lifts inject into [. Therefore

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X
l
|dμ(kl)3| ≤8
X
(2n + 1)4
(n + 1)6 ≤8J.
n≥0

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The alias-diagonal matrix d3
μI has trace-ideal price 8n0J, with fixed n0 = | Loc0 |. The Fourier
frame and inverse frame each cost at most e4/128, leaving the block-independent price

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Q = e4/128(8n0J)e4/128.

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This origin estimate controls the zero-, one- and two-field matrices of the regularizing ninth
power. Write P = μ2(A+μ2I)−1, where A is the sector Gram matrix, and U = P 3. The preceding
estimate applies to U at zero field. If C3 bounds its three ordinary film matrices, the corresponding
three matrices of U3 have trace-ideal price at most 20C2
3Q. In fact every zero-field product has
an unmarked U, every first-field product has two, and every second-field product has at least one.
Put one unmarked factor in the trace ideal and the other two in operator norm. The second-field
Leibniz formula has three placements of a second-field matrix and six ordered placements of two
first-field matrices. Their sum and the lower orders are covered by 20. This never converts an
L4-dimensional bound into a trace bound. The regularizer is μ2P 9, so its price is μ2B, where
B = 20C2
3Q.

We now price each localized composition. We specify how the derivative prices used here are obtained. If an analytic finite-dimensional map is evaluated at, define, for a derivative bound

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m
X
X
π∈Pn
∥D|π|F(z)∥c|π|,
(DP12)
Em(F, z, c) =
n=0

where is the finite set of partitions in the iterated chain rule, with its multiplicities, and contributes the value bound. If an inner map takes the value at the origin and its derivatives through order have the prescribed tested bound, its composite is bounded by (DP12): differentiate successively, grouping the directions according to the block that enters each derivative of the inner map, and sum their norm products. The source’s apollo and encore are precisely these finite sums. For a product use and inner price; inversion of a group matrix is its adjoint and preserves these derivative bounds. The same construction applies to the matrix exponential at zero, the chosen analytic group logarithm at the identity, and the analytic nearest-group projection in its fixed neighborhood.

These rules generate the charter and emission prices from the actual stencil. A link perturbation has its explicit factor. On a length- path, the product rule chooses the differentiated links. If distinct links are differentiated, there are at most choices and their derivative bounds contribute ; all undifferentiated group factors have norm one. The remaining assignments are the finite partitions of the derivative directions. Thus every fixed derivative order has a common bound independent of. Patch averages are probability averages. Moving from the padded patch to its translated subpatch costs at most its fixed stage-dependent cardinality ratio, since both counts are bounded multiples of. The frame estimates already proved give the corresponding finite derivative bounds for the recentered gauge transformations. Form the connector by multiplying the two endpoint frames and the path transport, average it, and apply the nearest-group projection; (DP12) pays all these operations. Multiply by the coarse-link factors and the endpoint frames, and apply the logarithm. This constructs the finite positive prices denoted springs for the charter and rocket for the emission in FrameEst/Stencils. Every logarithm is evaluated at the identity in this centered calculation. None of these steps takes a supremum over a number of arbitrary independent site values.

Here is the last conversion into row norms, including its block powers. Let2 be the number of inserted field directions and let bound their fixed Fourier-frame readings. Put = muralRoom(, including the fixed stage enlargement of those readings. The scalar directional estimate for the charter is at most

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2n+1| springs(J, n + 1, c′)|;

the corresponding emission estimate, after changing from the patch probability average to the block-site probability average, is at most

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portrait(J)2n+1| rocket_fare(J, n + 1, c′)|.

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The factor 2n+1 pays the real-and-imaginary expansions of the n inserted directions and the one
tested input direction. Conversion of the finitely many color coordinates to their Euclidean norm
multiplies these by √n1 and √n0, respectively. For the emission the probability-average estimate
is a sum of squares divided by L4; removing that denominator and taking a square root contributes
exactly L2. Taking the maximum, or the larger finite sum, over n = 0, 1, 2 gives prices a and h
independent of L, with charter price a and emission price L2h. These are RealGap.CharterR.costs
and Chromatic.costs. The same finite derivative and Fourier-frame bounds give the sector price

; the positive massive inverse and its differentiated inverse identities give the regularizer price . The house kernel has fixed dimension and the already constructed inverse bound, giving its ordinary price Each of thus bounds the zero-, one-and two-field matrices on the same strip. The house kernel has fixed source dimension Loc, so its trace-ideal price is. Film multiplication costs at most four times the product, covering its four second-field Leibniz terms. The