Quadratic ultraviolet kernels and comparison

Quadratic ultraviolet kernels and comparison

The recovery problem at quadratic order. The finite Wilson measure is now fixed. The next task is to control its quadratic approximation under repeated blocking. A quadratic history retains the cochain spaces, block averages, precision forms, covariance symbols and stationary predictors at every intermediate scale. A precision is the quadratic form in the Gaussian exponent; its inverse is taken only on the specified gauge and harmonic complement. The source record must therefore retain that complement, rather than replacing a singular form by an unspecified inverse.

At the tangent level the frame readout is a surjection with section. An operator is realized by, whereas its quadratic form is pulled back by. These two constructions have different types. In particular the pulled-back form has the vertical frame directions in its radical and cannot be coercive there. The ensuing Fourier divisions, alias sums and positive-margin repairs are performed in the original cochain spaces and are then recovered through these maps. The principalpredictor amendment in Appendix I.1 is essential to cancellation before Fourier extraction; a bound on the separate singular pieces would not replace that identity.

For reporting level write scale(scale(. The spacings tend to zero. The supplied is the action Hessian on link potentials, with no additional factor one half, obtained from a curvature packet by; it does not specify the curvature metric away from curls. The row bound sums absolute entries over input sites at fixed input orientation, with weight, and takes the supremum over output links.

Retain the horizon hypotheses named H.Suffices P and Flat.meshed, including the actual destination response in each preceding history. In particularmin( and on the admitting tailmin(1))). Retain 2block(, one-step price at most block( for 0, and uniformly in

The terminal saddle difference is a curvature packet with cost topTab(Faces The exact crossing identity propagates this packet using the two actual histories’ responses, retaining neutrality and translation covariance. Width crossing licenses each propagation. In particular, this is an identity for the two actual histories rather than a comparison with a separately chosen limiting history. Its cost satisfies

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CN = topTab(N),
Ci ≤block(i)−θCi+1.

Before replacing the quote by its block bound, propagation multiplies the packet cost by quote(). The reciprocal Stair.weight constructs a nonnegative majorant whose recurrence is an equality with this actual quote. World.overlap then applies the polynomial-age decay lemma to that majorant, without assuming a continuum limit. The polynomial decay argument below then yields0, an integer0 and 0 such that

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Ryi(Qn,i −QN,i) ≤AT(k, i)d(aN/ai)c,
e ≤i ≤N ≤n.
(Q1)

To recover the cover quadratic kernel, let be the lattice coboundary and write interpreting through its two-point kernel. For the packet-to-kernel step, finite support of supplies width-one row and column bound. Weighted convolution givesFaces for1 and packet cost, independently of torus size. Equality of the packet form and kernel difference on all scheduled tori gives cover equality by the following uniqueness argument. In a periodized entry, the indicator selecting each fixed cover site converges to the indicator of the chosen site as torus side tends to infinity. Summable absolute entries dominate this sum; dominated convergence recovers the cover entry. Thus two summable kernels with identical periodizations agree entrywise. This is Flat.resolve, distinct from the matrix inverse named resolve elsewhere.

Q1 makes every entry Cauchy for fixed. Its limit obeys, by Fatou on weighted absolute rows,

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Uniqueness of entrywise limits gives the same family for every reporting level k; before e it may be
defined as zero, where no conclusion is imposed.
For completeness retain the scalar majorant proof, with nonnegative arrays, A ≥0, integer
d ≥0, Z ≥0 and θ > 0. All constants are uniform over the schedules and terminal indices under
consideration. Let ∆i = ti+1 −ti ≥log 3 and τi = 2 + Z + |ti −tk|. If fi ≤e−θ∆ifi+1 + gi, fN ≤gN
and 0 ≤gi ≤Aτ d
i , put K = 2(d + 1)/θ, q = e−θ log 3/2 and B = A/(1 −q). Here K > 0, 0 < q < 1,
τi ≥2, and τi+1 ≤τi + ∆i. The elementary inequality eu ≥1 + u gives

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K + τi+1 ≤(K + τi)e∆i/K.

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Since d/K ≤θ/2,
(K + τi+1)de−θ∆i ≤q(K + τi)d.

At the terminal index the bound follows from. If it holds at + 1, the recurrence gives . Thus backward induction gives , using is polynomially bounded, multiply the recurrence by = min( 0. The transformed forcing is polynomial because, and the new decay exponent is 0. The same argument gives , equal to a positive power of. A forcing supported only at automatically satisfies the weighted hypothesis, because its exponential factor is one on its support. For well-founded grades, finite polynomials in earlier-

grade bounds provide the forcing; next-scale ages are polynomially controlled when block lengths are polynomial in current age, since + log(block( + block(). Difference forcing retains polynomial absolute bounds of both histories, including the current grade, and difference bounds only at strictly earlier grades. Multiplication by polynomial absolute bounds preserves exponential decay. Finite sums take the minimum positive decay exponent.

Every admissible horizon has the same fixed terminal bare quadratic form. On each scheduled finite torus its stationary response satisfies

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Qcoarse = T TQfineT + b(I −PT)T(I −PT),
b > 0.

Here is the common probe and the actual extracted response. All spaces in this comparison are real with the stated physical metrics;below denotes their adjoint, represented by transpose in orthonormal coordinates. For two symmetric fine precisions, symmetric coarse precisions , and responses, the primal and dual equations give

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QT = P ∗A,
RW = P ∗B,
A = b(I −PT),
B = b(I −PW),

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W ∗(R −Q)T = BPT −(PW)∗A = B(I −A/b) −(I −B/b)A = B −A.

Explicitly, transposing gives. The ordered mixed products cancel without commutation of and, even when auxiliary factorizations differ. Common terminal forms therefore yield common evaluated quadratic forms backwards, and summable cover uniqueness yields common kernels. The source’s shadow clips change the numerical factors to 0 while retaining relaxation, radius, slow selector, schedule, patches, smoothing levels and the blocking penalty, so this comparison applies to those horizons. It asserts equality on link-potential quadratic forms, not equality of higher-order horizon data. The responses and their primal and dual equations are constructed in the harmonic and scaled-exterior arguments below.

Under the tangent readout, the packet is pulled back as a form, while responses use. The exact crossing and comparison identities therefore recover by the operator/form theorem. All row bounds apply on the transported section with the original color norms; the redundant vertical directions are not assigned a positive coercive floor. This identifies the source quadratic convergence mechanism on the realized branch without replacing its horizon assumptions.

Recovery of the ultraviolet limit and common output. At a fixed reporting lattice, choose the tangent readout and section once, independently of the terminal cutoff, with For the finite-cutoffforms and their limit put

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eQn,i = p∗
i Qn,ipi,
eFi = p∗
i Fipi,
j∗
i eQn,iji = Qn,i,
j∗
i eFiji = Fi.

The standard tangent readout is a fixed finite stencil at this level. Every entry of its pulled-back form is therefore a finite linear combination of entries of the physical form. Entrywise convergence commutes with this pullback and with section restriction. On the section, use the transported physical coordinate basis and the original distance and color norm. The weighted row bound of the restricted difference is then exactly), with the original polynomial and decay constants; this is not a bound in an arbitrarily chosen norm on the redundant coordinates. The same restriction recovers the Fatou bound and the family independent of reporting level. On each scheduled torus the readout stencil is its periodic version, so the finite-stencil pullback commutes with periodization of the summable kernel. Consequently the cover-uniqueness argument compares the same physical entries after readout. It requires no inverse of the redundant precision. When two horizon outputs agree at every cutoff, their section restrictions and their entrywise limits agree; this transfers the supplied common-output and entrance conclusions with their original admissibility conditions. It does not identify curvature metrics away from curls or nonlinear data of the two horizons.

11.1 The finite quadratic history contract and its terminal majorant

Before the horizon hypotheses are stated it must be said exactly what a finite quadratic history is. At each it consists of five real Laurent rows on lattice two-forms: an unamended metric , its inverse, an amended metric, its inverse, and a residual Gram root. Their weighted coefficient bounds have width 0 and price Ω()). At every torus period

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HH−1 = MM−1 = I,
F ∗F + cI = M,
Md1 = Hd1,
M(0) = I,
(1)

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where c > 0 is the direct clip coefficient, so the full Gram factor is the stack (√c I, F). Both
metrics are symmetric with quadratic forms between η(J(i))I and Ω(J(i))I, η > 0, and the terminal
unamended metric is I. Each transition carries a fine potential response A0, a curvature response
T, and mixed, normal and image rows satisfying the stationary and Gram identities derived in the
following subsections. The mixed row has bound Ω/L2; the normal and image rows have bound Ω.

A word of carried fine differences of or of has bound Ω or Ω respectively, through order2. A carried difference increments the fine residue, shifts the coarse displacement when the increment crosses its centred seam, and subtracts the original entry: it is the actual fine difference including the carry. These all-period conditions are constructed by the metric, positive-margin, auxiliary-tower and pinning arguments below; they are not additional hypotheses of the public theorem.

Proposition 11.1 (Lift of the five-row contract). Let and be the potential readout and section of Theorem A.1. Two-form rows are read through, and . Then:

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1. the identities (1) hold verbatim for the realized rows, which act on the same two-form space;
in particular (√c I, F) is a Gram factor of M, so M ≥cI;

2. the exterior identity pulls back to on the whole redundant potential space, and both sides vanish on the vertical subspace

3. the realized potential response satisfies, and a word of carried fine differences of reads to the same word for, with the same residue increments and seam carries.

Proof. (1) The frame redundancy acts on potentials only, and = 0 shows that no two-form coordinate is changed; the rows and their products are literally the supplied ones. The quadratic form of is . (2) Compose with and use (3) The first identity is Theorem A.1. A carried fine difference is a fixed finite combination of lattice translations and index carries, which commute with and because is translation covariant; induction on the length of the word gives the claim.

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The loss in this lift is explicit: the amendment M −H is determined by (1) only offthe range
of d1, and the realization records which amendment the history actually chose. Two histories with
the same potential Hessian d∗
1Hd1 = d∗
1Md1 but different amended metrics are different realized
records with the same potential readout.
For the terminal-only recurrence of the flat convergence argument the majorant can be written
in closed form. With tj = log(a−1
j ),

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N−1
Y
bCi = topTab(N)
j=i
L−θ
j
= topTab(N) e−θ(tN−ti).

The empty product gives the terminal value and each backward step is exactly the required contraction. Put0. The triangle inequality gives, and so). The polynomial cap on the terminal price yields

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bCi ≤C T(k, i)l(1 + u)le−θu ≤C′ T(k, i)le−θu/2,
C′ = C sup
u≥0
(1 + u)le−θu/2 < ∞.

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Since e−u = aN/ai this is the asserted decay Ci ≤A T(k, i)l(aN/ai)c with c = θ/2. It uses no
continuum convergence. The general recurrence lemma, which also allows forcing at intermediate
levels, is proved with all its steps in the scalar majorant paragraph above (constants K = 2(d+1)/θ,
q = e−(θ/2) log 3, B = A/(1 −q) and the bound fi ≤B(K + 1)dτ d
i ). Because the majorant is a scalar
function of the realized scale clock and of the tagged terminal price, its lift is the identity on values:
bCZ
i = bCi, and the recovered kernel bound has the same constant C′ and exponent θ/2.

11.2 Width selection and neutral propagation

The entrance construction first fixes the retained-support linear choice, its positive clip curve and length demand. For its price choice, use) = max(tickets()) and require block(), including auxiliary steps before admission. Apply the linear horizon construction to the sheet with rearranged clips. The delay assumption at the admitting index supplies the widths checked below. The supplies construction on the shadow horizon gives one common family polynomial absolute bounds and decaying difference bounds. The same_shadow identity identifies every original finite-cutoffoutput with its shadow output; equality of these sequences transfers their limits and all bounds to every original horizon meeting its admissible price conditions. This proves the quadratic-history comparison. The responses themselves are constructed, and their primal and dual equations derived, in the harmonic and scaled-exterior subsections below (Subsections 11.9 and 11.10).

For a fixed price table choose) = min(1 0 before the lengths. If is the adjacent smoothing allotment, put

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B(j, u) = |Faces|2
2
w(j)h(j, u)2,
EP (j) =
l
2/w(j) + B(j, j)2 + B(j, j −1)2m
.

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The smoothing index stays fixed or increases by one at each step. Thus u = J(i) is j = J(i + 1) or
j −1. The requirement L ≥EP (j) gives both B(j, u)2 ≤L and L ≥2/w(j), and pays B(j, u)/L ≤
L−1/2 and 2w(u) ≤2 ≤Lw(j). These are the contraction and width-crossing inequalities with
θ = 1/2. Adjacency also makes w(u) no greater than either analytic width required at the crossing.
The thresholds depend on the already fixed prices, before L is chosen. With Z(k) = 3 + log bβk,
bβk > 1, and slope s > 0, the affine clock gives bβi ≤bβk + s|ti −tk|. Divide by bβk > 1 and use
log v ≤v −1 to obtain

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log bβi ≤log bβk + s|ti −tk| ≤(1 + s)[2 + Z(k) + |ti −tk|].

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Thus the admitting delay P.Ω(J(i)) ≤log bβi gives the required degree-one age bound. Together
with the previous inequalities this establishes every spacing condition used in the flat convergence
argument.
A nearest-neighbor coordinate path from x to z has length a = |z −x|1 and stays within that
distance of x. Such a path is obtained by changing each integer coordinate toward its target one unit
at a time. The Atlas one-step data supply a weighted forward directional difference row bounded
by c/Ld+1. At each intermediate site v, the triangle inequality gives

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eσδL(x,y) ≤eσa/LeσδL(v,y).

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The same row bound therefore applies to forward and backward steps, with the factor eσa/L. Tele-
scoping yields
X

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y
|U((z, i), (y, j)) −U((x, i), (y, j))|eσδL(x,y) ≤aeσa/Lc/Ld+1.

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The centered estimate uses d = 2 and spends half the input exponential width on a. The actual
response and its one-step bound are constructed in the atlas, response and exterior-lifting arguments
that follow.
For a packet K, neutrality means P
z K(x, (z, a)) = 0 for each held x and each
orientation a separately. Its propagated kernel is

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K′(y, z) =
X
x,v
T(x, y)K(x, v)U(v, z).

Weighted row/column absolute majorants justify every regrouping: first sum the nonnegative absolute majorant in the output variables, then in the input variables. The resulting finite product of bounds dominates the full multiple series, including regrouping over the finite orientation sets. Before taking absolute values, subtract from the right factor the value at the site of with the orientation of; neutrality cancels the removed sum orientation by orientation. Explicitly the centered factor issiteorientation)). For 0 and 2 use2 and, which follows from for0. The degree-two centered right row therefore costs; the left column costs. A packet of cost consequently gives folded

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The L−1 gain requires centering before the norm. Site-independent orientation-wise output sums
of U preserve neutrality. Summing K′(y, (z, b)) in z then gives a linear combination of the zero
orientation-wise sums of K, so it vanishes. Shifting fine summation variables by L times a coarse
shift proves covariance. Splitting absolutely convergent sums into residue classes proves exact torus
periodization T ∗
torusKtorusUtorus. Applying the mixed stationary identity identifies this periodized
packet with the coarse quadratic difference. These operations lift with the original orientations and
absolute majorants. The predictor spatial estimates and constant-curvature transport property
remain the stated dependencies; the secured pilot and wire bounds supply the former.

Quadratic comparison and neutral-cost recovery. The scalar recursion source retains each nonnegative forcing value, age and scale increment. Its readout is the identity on these entries, so the majorant, terminal condition and exponentially weighted recurrence recover literally, with the same and well-founded grade order. The price table and widths are fixed before the chosen block length; neither the lift nor the shadow quadratic comparison reorders these choices.

For finite physical tangent spaces use sections and readouts. Transport as and a precision as the pulled-back form . Restriction to the sections then recovers every stationary and mixed comparison identity, including, in the original metrics. On curvature packet coordinates transport each response and packet with their corresponding section maps. The orientation-wise cancellation of the constant right predictor is retained before forming the absolute majorant. Its scalar cost reader is exactly

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(|α|qL2) (|β|(2/τ)cL−3) p = |α|q|β|(2/τ)cL−1p.

Thus the width crossing, neutral gain and periodized quadratic difference recover with the same actual histories, and no positive form is imposed on redundant vertical directions. Absolute convergence licenses the same finite-orientation regrouping and residue-class periodization used in the physical comparison.

11.3 Full-space correction and analytic roots

On the whole three-form coordinate space the source symbol is, with and Choosing + 2 before requiring gives . Here is the fine amendment, the coarse amendment, and a fixed orientation count; 0 depend only on the fixed smoothing pair and penalty. At real momentum the symbols are Hermitian, and for every full three-form vector

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⟨x, Ax⟩≥[−a2C + l −ma2B/L2]∥x∥2 ≥∥x∥2.

Thus the coarse threshold is chosen before the length, which then absorbs the already fixed fine amendment. For, a uniformly bounded right inverse of the actual square finite matrix is also a left inverse. Consequently is invertible and has, hence An independent analytic row estimate supplies the upper bound. This is a full three-form bound, not merely a curl-image bound. The rebased identity is

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B[Y −T ∗XT −b−1d1d∗
1]B = d∗
2Rd2.

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Indeed substitute Y = B0+lR∗
0R0, X = b0+mF ∗
0 F0, the unamended defect R∗
0V R0 and SR0 = F0T,
so the bracket is R∗
0(V + lI −mS∗S)R0. Then use R0B = Fd2 and symmetry of B. The exterior
lifts, regular symbols and analytic inverse bounds of this section construct B0, b0, S and F with
the estimates used here. These are the source’s amended inverse metrics and installed response; no
arbitrary extension is substituted.
For M ≥b0I and 0 < c ≤b0/2, the residual M −cI ≥b0I/2 has a real analytic root row F.
Subtracting the constant row cI supplies its analytic strip bound. On each finite torus F ∗F =
M −cI, so the augmented row has the exact Gram identity

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(√cI, F)∗(√cI, F) = cI + F ∗F = M.

The allowed upper cap of is half a uniform positive metric lower bound. To retain the root construction, let have strip, weighted coefficient bound 0 and real symbols . With2) and fixed entrywise-to-operator norm factor 1, choose min()). On the real axis, unitary diagonalization gives Differentiating the absolutely convergent Fourier series on the smaller strip2 gives the entrywise symbol-difference bound with constant. Take to be one plus the norm of the fixed identification from entrywise matrices to operator norm. Comparison with Re in the width- collar gives 2, and therefore 1 uniformly. Thus

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1/2
n
!
P =
√
A
X
(A−1Q −I)n
n≥0

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is a uniformly bounded holomorphic periodic branch, equal to the positive square root on real
In particular P(p)∗= P(p), P(p)2 = Q(p) and
momentum by diagonalization of the series.
√
√vI ≤P(p) ≤
AI there.
Reality of the original Laurent coefficients preserves this branch
under conjugation at opposite real momenta. Fourier inversion, taking real parts of the coefficients,
thus recovers the same real-axis function. Shift each integration contour within the collar. With
output width e = τ/3, a further width e remains to sum absolute coefficients. In four dimensions
this gives a weighted coefficient sum at most K[2(1 + e)/e]4, where K bounds the entries of P on

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the collar. These constants depend only on v, A, η and the fixed matrix size, independently of the
finite period. Discrete Fourier diagonalization proves P ∗
torus = Ptorus and P ∗
torusPtorus = Qtorus for
every period: the real-axis identity holds at every discrete frequency, and the invertible discrete
Fourier matrix transfers it to real coordinate matrices. Thus the factorization and analytic decay
hold simultaneously for every period, as required in the bottom construction. This reduces the
remaining root step to the analytic square-root construction, rather than assuming a factorization
with the desired decay. Intersecting finitely many positive widths and enlarging the common cost
pays 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. The lift transports these
full-space bounds on their declared section, retaining each original inverse and norm.

Full-space correction and root recovery. For each physical exterior degree retain a section and readout with. The full three-form bound is recovered on, where is the entire original three-form space:

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⟨j3x, p∗
3Ap3j3x⟩= ⟨x, Ax⟩≥∥x∥2
(x ∈H3).

Here the pairing denotes the pulled-back form evaluated on the section. Thus every original three-form vector is included, while the redundant vertical kernel is allowed. Transport exterior maps as and the installed response by its corresponding operator lift. Restrict pulled-back metric forms to their sections before composing the rebased identity. Every intermediate cancels, and the result is exactly in the physical metrics. The original inverse bound for is used on its physical full space; no inverse of on redundant coordinates is introduced.

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For the residual metric root, retain the physical output coordinates of the row F and realize it
as F Z = Fp. Its augmented row is HZ = (√c p, Fp), so

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(HZ)∗HZ = p∗(cI + F ∗F)p = p∗Mp,
HZj = (√cI, F).

This is a factorization of the pulled-back metric form. Separately, if is the positive analytic square-root operator of, its operator realization is . It satisfies ( and In the transported section metric it recovers the original positive self-adjoint root; no choice of an ambient redundant-coordinate inner product is implicit. The root branch, collar and Laurent coefficients are the original retained analytic data. At each period the same row factorization and section identities recover the torus Gram identity, with precisely the original widths and costs on the physical section.

11.4 Smoothing histories and actual block averages

For positive odd, retain the entire functions

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RM(z) = M−1
M−1
X
u=0
eiuz,
PM(z) = e−i⌊M/2⌋zRM(z),
AM(p) =
4
Y
a=1
PM(pa).

Both line functions equal one at zero. Geometric summation, including the zero set by the polynomial identity, gives1. Splitting the integer index as gives ). For odd, the centers obey, giving Center reflection and conjugation give) and ̄

A train descends through smoothing indices which stay fixed or decrease by one backwards, and its mass is the product of its odd block lengths. A terminal train has mass and smoothing function one. At a hop of block from index to + 1, the new mass is mass() and its smoothing function becomes(mass( smooth). Equivalently,

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

On Im, finite sums give This latter representation separates the fixed stage power from the extra smoothing inserted at stage changes. Masses grow by factors at least three, and the extra exponent is zero or one, so summing the geometric mass series gives extra for mass at most. The real principal estimate yields the companion extra lower bound and hence a positive lower bound