Quadratic kernels and their ultraviolet limit · Part 2

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(√cI, F)T(√cI, F) = cI + F TF = M.

The source chooses c’s allowed upper cap as half a uniform positive lower metric bound, intersects the finitely many positive analytic widths, and increases the common cost to cover the metric, inverse, diagonal, dual diagonal, and root rows. The resulting bed satisfies the required finite-torus floor identities and bounds for every permitted train. This reduces the remaining root step to the analytic square-root construction, rather than assuming a factorization with the desired decay.

For completeness, the analytic square root can be constructed with one branch on a collar uniform over the entire family. Suppose a real Laurent matrix row Q has strip width 0 and entrywise weighted coefficient bound A, and its real symbols obey with Increase A if necessary so. On the real axis, unitary diagonalization gives

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∥A−1Q(p) −I∥≤1 −v/A.

On the smaller strip/2, differentiating the absolutely convergent Fourier series gives the entrywise bound for symbol differences with constant2). Let C be one plus the norm of the fixed identification from entrywise matrices to the matrix operator norm, and choose

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τ = min(η/2, v/(2CD)) > 0.

For a complex momentum in this collar, comparison with its real part gives 2. Consequently 1 uniformly. The binomial series

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1/2
n
!
A−1Q(p) −I

n
P(p) =
√
A
X
n≥0

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defines a bounded holomorphic periodic matrix function on the collar. On the real axis, diag-
onalization reduces the series to the positive scalar square root, so P(p)∗= P(p), P(p)2 = Q(p)
√
and √v I ≤P(p) ≤
A I. The Laurent coefficients of Q are real, and conjugation at opposite
real momenta preserves the selected branch. Fourier inversion, taking real parts of the coefficients,
therefore recovers the same real-axis function.

Bounded holomorphy gives exponential Fourier decay by shifting each contour within the collar. Choose output width3, leaving another width for summing coefficient moduli. In four dimensions the weighted sum is bounded by, where bounds the entries of on the collar. The constants depend on and the fixed matrix size, not on the torus period. Discrete Fourier diagonalization gives and on every torus. The invertible discrete Fourier transform transfers the real-axis symbol identity to the real-coordinate matrices. Hence factorization and analytic decay hold simultaneously at all periods, as required for the bottom metric.

Smoothing histories and the actual block average

A block average is an actual map on the fine lattice, with its carry across residue seams included. Its Fourier multiplier therefore has both a principal alias and nonzero aliases. The latter cannot be discarded before the stationary problem is solved. Their lower denominator bounds, proved next, are what make the filled principal formula and all the subsequent contour estimates uniform on the prescribed complex collar.

Smoothing mass and physical mass The mass of a smoothing train below is a positive integer: the product of its block lengths. It fixes the rescaling in its Fourier multiplier. It is unrelated to the positive regulator masses in the later resolvent calculation, and to a spectral gap above the vacuum. Its useful property is multiplicative growth, which makes the inverse-square contributions of earlier smoothing stages summable independently of the depth of the history.

For a positive odd integer M define the line average and its centered version by

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RM(z) = M−1
M−1
X
u=0
exp(iuz),
PM(z) = e−i⌊M/2⌋zRM(z).

Both filters equal one at Summing the geometric series gives 1, including the zeros of1. Writing an integer in as yields ). For odd

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M⌊L/2⌋+ ⌊M/2⌋= ⌊LM/2⌋,

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so PL(Mz)PM(z) = PLM(z). Reflection of the centered sum gives PM(−z) = PM(z), and conjuga-
tion gives PM(z) = PM(−z). All these identities hold for entire functions.
Set AM(p) = Q4
a=1 PM(pa). A train is a finite chain of patches whose smoothing indices stay
fixed or decrease by one at each backward step. Its mass is the product of its odd block lengths.
A terminal train of index j has mass and smoothing function both equal to one. If v has index J
and a patch takes it to index j with j ≤J ≤j + 1 and block L, the new mass is L mass(v) and its
smoothing function is

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smoothnew(p) = AL(mass(v)p)J smoothv(p).

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Equivalently,

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smoothv = Aj
mass(v) extrav,
extranew = AJ−j
L mass(v) extrav .

This separates the fixed-stage power from the extra smoothing introduced when the stage changes. The extra product is bounded independently of chain length. On Im, direct estimation of the finite exponential sum gives. Successive masses increase by factors of at least three, and is zero or one. Induction, equivalently summing the resulting geometric series, gives

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| extrav(p)| ≤e8ζ mass(v)/H,
mass(v) ≤H.

On the real principal region, gives

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| extrav(p/H)| ≥e−8c mass(v)/H.

Consequently smooth mass( has a positive lower bound depending only on. The normalized finite-sum proof is given in (QF1)–(QF4). At lift( mass(), the principal factor has inverse-linear decay in each centered integer alias coordinate, while the extra factor has the preceding upper bound. Hence

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| smoothv(ql)| ≤Cj
Y
a
(1 + | lift(l)a|)−j.

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The equivalent source normalization uses up(n) = 1 + |n|. This is the decay used in the high-alias
sums. The nonzero-alias reciprocal estimate supplies the required bound for RM.
These are the actual patch weights.
One centered box average of Fourier modes is AL; J
independent box rolls give AJ
L. Different rolls can have the same total displacement, and the patch
weight retains their multiplicities. A positive line path in direction a has L edges at displacements
0, . . . , L −1, with Fourier sum LRL(qa). Thus its directional patch mean has symbol

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LAL(q)JRL(qa).

Applying the defined mean to every discrete Fourier basis vector proves the source identity patch_service. The factor remains because the path integral counts edges. Wrapped paths on a small torus still count every edge occurrence; no distinct-visit assumption is used.

Uniform reciprocals at the nonzero aliases

A real lower bound is converted to a complex lower bound before inversion. Suppose is holomorphic with on a common complex cube of width, and on its real cube of width, where 0 and 0. Set

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r = min(η/4, θ, cη/(8C)).

For in the width- cube, put. The ball of radius2 about stays in the original cube. Cauchy’s estimate along the line from to, using on that ball, gives

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|fξ(p) −fξ(y)| ≤4Cwξ/η|p −y| ≤(c/2)wξ.

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Thus |fξ(p)| ≥(c/2)wξ. Its reciprocal is holomorphic on this common cube and bounded by
2/(cwξ). The width r is independent of ξ. This is the required reciprocal estimate; real positivity
alone would not prove complex nonvanishing.
In particular, define
λM(p) =
X
a
[Mχ(pa/M)][Mχ(−pa/M)].

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On real momenta this is P
a |Mχ(pa/M)|2. The bounds for χ give

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c2
−|p|2 ≤λM(p) ≤c2
+|p|2

on the real cube, uniformly in. For block and alias, put

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BM,L,l(p) = L2λM
(p + 2π lift(l))/L

.

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For l ̸= 0, choose a coordinate attaining grade(l) = 1 + maxa | lift(l)a|. The lower alias estimate
gives BM,L,l ≥(c−ζ)2 grade(l)2 on the real cube. The holomorphic upper bound is C grade(l)2.
Apply the reciprocal estimate with wl = grade(l)2 to obtain a fixed complex collar and |B−1
M,L,l| ≤
C′ grade(l)−2, uniformly in M, L, l ̸= 0.
The principal alias instead satisfies the exact identity
BM,L,0 = λLM and is not inverted on a collar containing zero.
Define
mix(l) =
Y
a
(1 + | lift(l)a|),
decayj(l) = mix(l)−j.

Since all factors are at least one, grade(mix(). Therefore grade( decaydecay Finally,

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X
l mod L
decay2(l)


4
X
n∈Z
(1 + |n|)−2

≤
< ∞,

uniformly in L. The centered representatives are injective, so each one-dimensional finite sum is bounded by the full integer series. The finite four-dimensional sum factors into the product of those four finite sums. This proves the decay summation used in the earlier feed and high-alias estimates, without paying a factor equal to the number of residues.

For the residue-column estimates, a word of fine differences contributes at most grade( The scale factor contributes The residue phase is uniformly bounded because residue coordinates are of size at most and Im. The symbol contributes decay). Absorption and summation therefore bound the residue sum by whenever 2. Reciprocal shifts permute aliases, giving periodicity; conjugation and reflection give real Fourier coefficients. Fourier estimates on a narrower collar yield the priced fine-difference rows. The coefficient and symmetry arguments below verify these symbol bounds for the actual potential and curvature responses, called pilot and wire in the source.

The scalar bounds use1. Choose 0 so that on 2. Such constants exist because extends continuously with nonzero value i at zero, and has no zero on that disk: = 1 requires a real multiple of 2. Compactness bounds this nonvanishing quotient above and below. If2, the geometric-sum identity gives for= 0, and Put = min. Centering leaves the modulus unchanged for real also2 and, put. Estimating each centered monomial minus one gives , where2). For we have and 1. Since

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y exp(kx/a0) ≥y + kx ≥1.

It follows that. The coefficient in front of this exponential is exactly one, which is essential when the estimate is composed along arbitrarily long smoothing trains.

For a complex alias with centered nonzero integer, the selected cube gives ) and Im, where4 and up(. These follow by separating from Re Also2 for odd The geometric-sum identity and ) give

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L|RL(q)|c−|q| ≤|χ(p)| ≤c+(3π/2),

so). For = 0, the finite-sum estimate gives. Centering changes this by at most, yielding the same inverse-alias bound for. Coordinate products give the four-dimensional estimate above. The remaining alias estimates concern the Laplacian denominator and the sum of the decay weights.

To install a response kernel, split a fine site into its centered residue modulo and a coarse displacement. Advancing by changes the residue to and adds a coarse displacement exactly when lift(. Denote that added displacement by). Then

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lift(r + ea) + Lca(r) = lift(r) + ea.

In the residue Fourier sum this identity gives a factor exp()) on the advanced residue. Subtracting the original residue value is exactly multiplication of each alias term by exp( Repeating this operation gives a product of those factors for a word of fine differences. Fourier inversion therefore identifies the actual carried differences of the cross row with the separately priced difference symbols. This identity follows from the real sums, rather than treating a residue wrap as an ordinary shift within the same copy.

For a fine-difference word, the symbol estimates give a kernel bound, with a positive width and uniform over all transitions at fixed smoothing indices, whenever2. There are finitely many such direction words. Intersect their widths and enlarge their costs to obtain one table for the row and all required differences. The coefficient estimates and the inverse-third-power estimate below supply the analytic symbol bound; the residue calculation identifies the actual carried differences to which it applies.

Laurent exterior division

Write1 in the lattice Fourier convention, and let) be exterior multiplication by on degree- forms. We divide a cycle without assigning a singular quotient at = 0. For a lattice monomial, let replace its-coordinate by zero. Define by summing monomials with-coordinates 01 when0, and minus the sum over1 when keeping other coordinates fixed. Telescoping gives

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qs(p)(Jsex)(p) = ex(p) −ex(p|ps=0).

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If P
x |ax|eη|x|1 ≤C, then Es preserves that bound. The sum defining Js has |xs| terms, each
of length at most |x|1. For θ = η/2,

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|xs| exp(θ|x|1) ≤θ−1 exp(η|x|1).

Therefore maps the width- bound C to the width-/2 bound 2C/. This proves absolute convergence of the transformed rows and licenses termwise telescoping. In particular as an identity of real-momentum symbols, including the hyperplane

Suppose maps degree-( + 1) forms to a fixed coordinate space and = 0. Let exterior multiplication by the-th coordinate vector. Define the relief row Obtain the residual by evaluating at = 0 and deleting columns whose orientation contains . Equivalently,

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Rs = (EsA)[I −Lr(s)Lr(s)T].

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The exterior contraction identity

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Lr+1(s)Tδr+1 = qsI −δrLr(s)T

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and δr(p) = δr(p|ps=0) + qsLr(s) imply

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Asδr+1 + Rs = A.

First prove this where= 0 by multiplying by, using, and cancelling that nonzero scalar. The symbols are continuous, so perturbing through nonzero real values proves the equality also at = 0. Furthermore = 0 because = 0. Thus the residual remains a cycle and one may repeat the operation.

Apply the four axes successively to the preceding residual and add the relief rows. Once an orientation contains a processed axis its column is zero and stays zero. In positive degree every orientation contains an axis. After four operations the residual is zero and the sum satisfies The explicit finite width losses preserve a positive analytic width, so retains exponential coefficient decay uniformly for a common input bound. Fourier evaluation yields at every period. Conversely, = 0 at all periods implies the symbol cycle identity by Fourier uniqueness. That reverse implication uses all periods.

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Transposing gives the corresponding left lift. A two-sided lift requires a correction because an
arbitrary left lift need not remain a right cycle. Suppose A is a two-form row with Ad1 = 0 and
dT
1 A = 0 on all periods. Choose W by a left lift so dT
2 W = A. Put F = Wd1; then dT
2 F = 0. Lift
again on the left to obtain a degree-four row T with dT
3 T = F. Now Td0 = 0: its left derivative
vanishes since dT
3 Td0 = Wd1d0 = 0, and a left cycle in top degree four vanishes by the same lifting
theorem, whose next form space in dimension four is zero. Right-lift T to obtain X with Xd1 = T.
Define K = W −dT
3 X. Then Kd1 = 0 and dT
2 K = A, using d3d2 = 0. Right-lift K to obtain B
with Bd2 = K. Thus dT
2 Bd2 = A. Every step is a finite composition of the preceding analytic
lifts, exterior derivatives, sums and transposes, so the same familywise exponential decay statement
holds. This proves the exterior factorization used for the fund; the refined block-length power in
the sale estimate is additional quantitative information.

The regular two-form symbols

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Fix a smoothing train v with scale parameter M. Set za(p) = seedQM(p, a) seedQM(−p, a) and
λM(p) = P
a za(p). These za are nonnegative at real momentum. The positive smoothing factors
are h0 = halo(v) and la = lid(v, a). Let ra = drain(b, v, a) be the regular remainder and put

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ga = la + λMra,
z(1)
a ra
laga
.
h = h−1
0
−
X
a

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Here z(1)
a
is the analogous factor for M = 1. The exact identity z(1)
a h0 = zala, after taking
common denominators, gives

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λMh =
X
a
z(1)
a /ga.
(2)

The two regular symbols are diagonal on the oriented two-forms. At orientation, the symbol called vault has entry and the symbol called pulse has its reciprocal; off-diagonal entries vanish. At = 0, every and equals one. These are the values of the same analytic expressions at zero, with no separately substituted harmonic value.

The positivity proof uses 0 and0 on the real cube, so 0. For gives0. More quantitatively, if and the comparison constants for and with

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are c2
−and c2
+, then (2) gives

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h ≥c2
−
Cc2
+
.

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At p = 0, h = 1, giving the joint lower bound min(1, c2
−/(Cc2
+)). Hence h never vanishes. Pulse
and vault are therefore positive, real diagonal mutual inverses throughout the real cube. Uniform
bounds and positive lower bounds for ga, h give analytic narrower-collar bounds for their reciprocals

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and their finite products. Taking Fourier coefficients gives the exponentially decaying metric and
inverse rows used above.
Away from zero, set Da = ga/λM and ρ = P
a z(1)
a /Da. Equation (2) gives ρ = λ2
Mh. Thus

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vault = ρ mask2(Da),
pulse = ρ−1 mask2(D−1
a ),

where mask takes the pairwise products of diagonal entries on two-form orientations. Periodicity of and proves agreement under reciprocal-lattice shifts on overlapping real cubes. The regular expressions give the values through the apparent pole. At the terminal train, and 1 , so both two-form symbols are the identity, as required for the bare Wilson quadratic form.

The centered-average, nonzero-alias and cumulative-filter estimates in this chapter provide the smoothing bounds, regular remainder bounds and scale-uniform comparison of used in (2).

The depth-independent analytic estimate works as follows. Each earlier train contributes with weight mass. A hop of block length multiplies its mass by, dividing the sum of old weights by; the new list adds one train of mass one. Hence, if) is the sum of inverse-square masses over that list,

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t(top) = 0,
t(hop(v)) = 1 + L−2t(v).

Since3, induction gives2, uniformly in depth. Each train’s actual contribution has analytic bound mass, so the finite list has bound 2. This controls the function called levelNoise; an unweighted contribution is not accumulated at every renormalization step.

At high aliases the smoothing factor has decay indexed by, while the inverse nonzero-alias divisor costs an inverse squared grade. Their product is dominated by the summable reference decay of order two when4. Its sum is independent of the alias side length. The same argument gives the inverse-square-mass bound for each contribution. Combining the high-alias sum with levelNoise bounds drain. The alias lower bounds, inverse-divisor bound and reference decay sum used here were proved above; their scalar assembly is also displayed in the cumulative-filter construction.

The factor lid is the product of its smoothing expression at and. A uniform positive smoothing lower bound, together with the positive lower line-filter bound, makes lid uniformly positive. The same holds for halo. The bound on drain and the bound on bound every above. Equation (2) then bounds below. Only halo, lid and occur in the rational denominators defining. Their positive lower bounds permit a common smaller holomorphic collar and analytic bounds for. No bound for the singular quotient 1 at zero is required.

Amendment and preservation of the potential precision

Let be the unamended two-form metric and. Both are symmetric positive matrices on each finite torus. Put and, for

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A = B + lXTX,
G = A−1,
R = I + ld2BdT
2 ,
D = R−1.

The source names these rows pledge, amend, bail and treble, respectively. Positivity gives and, so their analytic inverses exist on uniform narrower strips. Since = 0, we have = 0. Hence and

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dT
1 Gd1 = dT
1 Hd1.

The amendment changes the two-form extension while preserving the potential precision exactly. Also; multiply by on the left and on the right to obtain. These are the rebasing identities used above. At zero momentum = 0, and gives

Uniform inversion uses the fixed orientation dimension, rather than the torus site count. If the real symbols satisfy, then diagonalization gives det in dimension and. On a common narrower collar the determinant stays separated from zero: analytic coefficient bounds control its change from the real slice, and the collar can be reduced to lose at most half the positive bound. Adjugate divided by determinant gives uniformly bounded holomorphic inverse symbols. Periodicity and real-coefficient symmetry survive. Fourier coefficients decay exponentially, and discrete Fourier evaluation gives the inverse matrices at every period. This proves the inversion needed for and

The same Fourier calculation transfers all real-axis spectral inequalities to finite tori. The discrete Fourier matrix has Gram matrix; that positive scalar multiplies both the quadratic form and the squared norm and cancels. Thus a bound uniform in momentum gives the identical bound on every finite torus, including side one.

Constant-curvature transport

For a translation-invariant Laurent row, its total spatial sum is its zero-momentum symbol and is independent of the starting site. For an installed cross row, the starting site also has a residue modulo the block length. The required independence is therefore a separate assertion. For the specified duo response, its zero-momentum entry at each residue z is toll( 2) times the identity in face orientations. In the alias formula all nonzero aliases vanish at the origin; the zero alias equals the normalized identity. Consequently the spatial sum of each orientation column is independent of that residue as well.

The property survives convolution. If the spatial sums of are the orientation matrices , absolute convergence gives total sum for their convolution. The fine inverse and coarse metric are translation-invariant rows, and the intermediate curvature response has the constantresidue property just proved. Their sandwich therefore has constant spatial sums. This is the condition called travels_route, required to preserve packet neutrality. The regular curvature-response formulas and harmonic values establish the alias vanishing and normalization used here.

Exterior factorization of the unamended defect

Exterior factorization separates the part of a two-form kernel seen by curls from its complementary directions. The calculation is performed in Laurent symbols before evaluating on a torus, so the same algebraic identity holds at all permitted periods. The zero Fourier mode is then restored by its explicit harmonic equation. This order avoids claiming that division on the punctured momentum domain alone determines the harmonic fibre.

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From symbols to a finite history
The exterior factorization supplies the auxiliary rows needed
by the adjacent stationary problem. Those rows are then evaluated at every discrete torus momen-
tum. Invertibility of the finite Fourier transform turns the symbol identities into actual matrix
identities, including the separately checked zero mode. The subsequent assembly labels such as
“Present” and “H.Suffices” record precisely this collection of identities and purchased bounds for a
scheduled horizon; they are not invocations of an additional theorem in an unavailable file.
On a coarse torus let H1 be the original metric and B1 = H−1
1
its inverse, with fine counterparts
H2, B2. For the installed response T and penalty b > 0, define