Quadratic kernels and their ultraviolet limit · Part 3

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V = B1 −T TB2T −b−1d1dT
1 ,
M = H1V H1.

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The stationary identities of the transition are TH1d1 = H2e1A0, T Te1 = d1Q and QA0 =
I −b−1dT
1 H1d1. Consequently,

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T TB2TH1d1 = d1[I −b−1dT
1 H1d1],

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so V H1d1 = 0 and Md1 = 0. Symmetry gives dT
1 M = 0. The two-sided exterior lift therefore
gives a three-form row F with dT
2 Fd2 = M on every torus. Replace F by (F +F T)/2; this preserves
the factorization and makes its real symbols Hermitian. Multiplying by B1 on both sides gives

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V = (d2B1)TF(d2B1).

This is the fund used in the positive correction. Analytic bounds are retained through the two exterior lifts, their symmetrization and the surrounding bounded rows. The exterior-division construction below supplies the analytic lift and its bounds.

The fine curvature charge annihilates, because and right exterior lift produces with. Multiplying by gives, the rebasing intertwining identity. The refined estimate takes a fine difference before lifting and keeps per residue; the residue sum then gives the bound on used in three-form absorption. The inverse-third-power argument below proves this quantitative bound.

The retained-support construction makes stagewise choices before the eventual block lengths. For a legal pair (), where4 and or, the linear transition estimate supplies a positive coarse-floor threshold). Choose

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l(i) = max(δ(i, i), δ(i + 1, i)).

This pays the coarse threshold for both possible incoming transitions at stage i. With fine and coarse parameters), choose the positive analytic width, positive coercivity parameter, cost, and integer block threshold supplied by each legal pair. At each stage also choose the positive clip cap and the bottom metric estimates. Take the minimum of the finitely many widths and coercivity parameters for the bottom and two outgoing possibilities, the maximum of their costs, and the maximum of their length thresholds. Weakening an analytic width or lower coercivity bound and increasing an upper cost preserves each estimate. Thus one table works for every allowed adjacent transition. This assembles the common price table from the metric, positive-margin and flight estimates established in this chapter.

For a fixed level schedule A, construct the linear trajectory backwards. Start with the top train at level N, and at each coarser level take the landing train of the flight determined by the block and adjacent smoothing indices. Denote the resulting train at i by. At every pair (N,i), the bottom estimate chooses a metric bed defined by that train and)). Choose as its transition tower the flight’s specified duo with the fixed penalty. The identity = land(destination()) aligns the bottom metric with this tower. The flight’s tangent equations give the primal and dual stationary identities on each scheduled finite torus; its rebased identity supplies the associated inverse and error data. The collected bottom bounds, transition bounds, and terminal normalization form the required linear horizon.

The pinning step retains these metric beds and the indicated tower components while arranging the additional horizon conditions for the given sheet profile. Since retention preserves the beds and the relevant components, their defined train metrics and destination response equalities survive.

This supplies Present for the pinned horizon as well as the admissibility statement H.Suffices. The bottom metric, flight and pinning estimates used in this assembly are proved in the harmonic, positive-margin and auxiliary-tower arguments in this chapter.

Harmonic values and finite-torus stationary equations

At = 0 every exterior signal vanishes; the regular factors bulb and bend equal one, and the principal patch multipliers are identities. Thus the filled potential response (pilot) and curvature response (wire) have principal value At a nonprincipal zero alias choose a coordinate with nonzero residue. Its line multiplier vanishes because(0) = 0 and= 0. The box mass filter vanishes, since its smoothing exponent is positive. Every nonprincipal pilot and wire value is zero. The pilot’s leading term contains = 0 and its correction contains the zero coarse longitudinal row. The wire’s first term contains the zero coarse adjoint derivative, and its second contains the vanishing flux. These explicit values establish the stationary symbol equations on the harmonic fibre; no harmonic projection is deleted.

For clarity, the finite Fourier transfer is as follows. Suppose a real matrix has representation , with Fourier matrices satisfying and. Taking adjoints and multiplying by yields]. Two real matrices with the same symbol and Fourier matrices agree, since their difference annihilates the invertible matrix. Products and differences of represented matrices become products and differences of symbols when the intervening Fourier matrices coincide.

The coarse Fourier Gram scalar is and the fine one is, with. The transpose of the block mean therefore has alias column): the mean carries and . The installed and columns are pilot and wire. The fine exterior derivative is on the scaled alias, and its precision is. Restoring these factors gives the exact finite real-matrix identities

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QfA0 = meanT Qc,
mean A0 = I −b−1Qc,
T Tdf = dc mean,
THcdc = HfdfA0.

The installed cross rows have these symbol representations because their residue Fourier sum is the alias Fourier matrix times the stacked alias column. Evaluation at every coarse discrete frequency then gives the represented identities above. Invertibility of the finite Fourier matrices proves the real matrix identities at every period. Together with the exact amended-metric preservation and pinning arguments, this supplies the finite linear horizons used in the flat entrance estimate. The cumulative-filter section gives the scalar covariance feedback formulas and the train identities used in this construction.

Scaled exterior identities for the stationary columns

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For scale S, set
qS
a (p) = S(eipa −1),
dS
r = qS∧,
δS
r (p) = dS
r (−p)T.

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On real momentum, δS
r is the Hermitian adjoint. The patch identities are

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Fr+1(ql)dL
r (ql) = dr(p)Fr(ql),
RL(ql,a)qL
a (ql) = qa(p).

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The second proves the first coordinatewise. Transpose at −p to obtain the dual transport identity.
Also dS
r+1dS
r = 0 identically.
At regular momentum let D = diag(S−2Da), ID = D−1 and α = S4ρ. The two-form metric is
α−1 mask2((ID)aa), with the pairwise-product convention above. The weighted exterior contraction
identity makes its potential precision

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K = ID −α−1IDdS
0 δS
0 ID.

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Indeed
expanding
the
two-form
wedge
product
yields
the
diagonal
weighted
scalar
P
a qS
a (−p)(ID)aaqS
a (p) = α; the off-diagonal terms are exactly the rank-one product subtracted
above. Thus, with P = α−1δS
0 ID and C = α−1IDdS
0 ,

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K = ID −IDdS
0 P,
KD = I −CδS
0 , DK = I −dS
0 P.

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Also KdS
0 = 0 and δS
0 K = 0 directly from the exterior chain identities. These are the rotor
identities used in matching pilot and wire.
Write Dl for the fine diagonal covariance at ql, Hl for its two-form pulse and Kl for its potential
precision. Use Dc, Hc, Kc for the coarse quantities, Fr, Vr for the patch multiplier and its dual, and
Pc for the coarse longitudinal row. Define

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El = m(−ql) tr(Dl)
Z
,
Z =
X
l
|m(ql)|2 tr(Dl).

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On the punctured real cube all summands in Z are nonnegative and the principal one is positive.
Hence Z ̸= 0 and P
l Elm(ql) = 1. Covariance recursion gives

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X
l
F1(ql)DlV1(ql) = Dc −b−1I.
The regular potential response is

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Al = DlV1(ql)Kc + EldL
0 (ql)Pc.

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Multiply by F1(ql) and sum over l. The patch identity turns the second term into P
l Elm(ql)d0(p)Pc.
The preceding identities and DcKc = I −d0Pc therefore give

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X
l
F1(ql)Al = I −b−1Kc.

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Similarly KlAl = V1(ql)Kc. The longitudinal term is killed by KldL
0 = 0, and the remaining
correction in KlDl is killed by δL
0 V1δ1 = 0. This proves the coarse averaging and primal stationary
equations on the punctured cube.
Set C = Hcd1Dcδ1. The weighted identities give δ1(I −C) = 0 and (I −C)Hcd1 = 0. The
regular curvature response is

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Tl = HldL
1 DlV1δ1 + ylV2(I −C),

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where yl is the original chain scalar and all fine factors are evaluated at ql. Left multiplication by δL
1
uses KlDlV1δ1 = V1δ1 and δL
1 V2 = V1δ1, giving δL
1 Tl = V1δ1. Right multiplication by Hcd1 kills the
complementary term. The potential response’s longitudinal term vanishes under dL
1 dL
0 = 0. Hence

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TlHcd1 = HldL
1 Al.

The filled pilot and wire equal these expressions away from the puncture. Their continuity extends the identities to = 0 once the Fourier and scale normalization factors are restored. The finite Fourier matrices use the coarse volume and fine volume separately when taking adjoints; their ratio is. Accordingly the installed columns use toll for and toll for, while the adjoint averaging column carries toll. With these factors the equations are exactly the primal, dual and transport equations on every compatible pair of tori. The Fourier transfer in “Harmonic values and finite-torus stationary equations” restores these factors and proves the asserted real matrix identities at every period.

Extending the three-form positive margin

Suppose the analytic Hermitian three-form symbol obeys

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⟨d2(p)a, Q(p)d2(p)a⟩≥v∥d2(p)a∥2

for every real, with 0. First this controls every vector at zero. For a complex three-form in four dimensions, the real-linear map from to the complex one-dimensional top-form space has a nonzero real kernel vector. The exterior contraction identity gives for some two-form. Since, the vectors) tend to. Apply the image bound and continuity of to obtain Let0 be its analytic coefficient bound and the fixed orientation-count constant. When all, the symbol departure estimate bounds every entry of. Increase to a positive value and choose). Then in that region.

Elsewhere choose with. Cartan decomposition gives, with

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y = d2(p)
qk(p)−1ιkx
,
z = qk(p)−1ιkd3(p)x.

Thus. Put. The quadratic-form and cross-term bounds imply

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⟨x, Qx⟩≥v∥y∥2 −2M∥y∥∥z∥−M∥z∥2
≥(v/2)∥y∥2 −(M + 2M2/v)∥z∥2.
Since ∥x∥2 ≤2∥y∥2 + 2∥z∥2, it follows that

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⟨x, Qx⟩+ [M + 2M2/v + v/2]Nδ−2∥d3(p)x∥2
≥(v/4)∥x∥2.

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Adding r d∗
3d3, where r = [M + 2M2/v + v/2]Nδ−2, makes Q uniformly positive. Near zero it
adds only a nonnegative form. This correction has finite-support Laurent coefficients and a common
analytic bound. It preserves the physical curvature contraction because d3d2 = 0. Enlarging the
upper cost therefore gives a uniformly positive full-space row with exactly the original dT
2 Qd2 at
every period. This is the positive extension used in pinning.

The auxiliary tower Gram identity

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Weighted convolution gives common analytic bounds at width γ = min{σf, σc, ν} > 0, where ν is
the three-form-root width. The full Gram row J stacks √clip.direct I with the bottom root row, and
is bounded by max{√clip.direct, rootCost}. Each exterior derivative has finite support: a bound

valid at all periods is the finite sum of times absolute stencil coefficients over orientations and sites. Transposition preserves that bound. Composing rows with an intermediate orientation space of size costs at most times their entrywise weighted bounds, by the distance triangle inequality and absolute convergence.

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Apply this first to JR, then to composition with the fine response T, then to multiplication
by J′T. The bound for T retains its factor L−2 throughout, so the mixed row costs CfL−2. The
normal row s−1d′TJ′T has an L-independent bound, as do I −J′R′J′T and PeR′J′T. Stacking takes
their maximum. Choose one upper price covering these and the original input prices. Keep A0, T
unchanged; every carried fine difference keeps its L−1−r or L−2−r bound at the reduced width and
enlarged price. This proves the estimates named F08; positivity and the Gram identity are separate
algebraic statements proved here.
On a fixed pair of tori put W = JTJ and W ′ = J′TJ′, the fine and coarse metrics from the full
Gram rows. Let R = W −1 and R′ = W ′−1. Retain the inverse response T; let d, d′ be the fine and
√
coarse first exterior derivatives, e the coarse second derivative, and s =
b. The factorized inverse
defect is

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W ′ZW ′ = eTP TPe,
Z = R′ −T TRT −b−1d′d′T.

Define and. Then and . For the mixed, normal and image rows

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M = JRTJ′T, N = s−1d′TJ′T,
O = stack(I −Q, X),

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direct multiplication gives

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MTM = J′T TRTJ′T,
NTN = J′b−1d′d′TJ′T,
OTO = I −Q + J′ZJ′T.

Substitution of makes their sum. This holds on the entire source of the stack, including harmonic coordinates. The transport identities

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TW ′d′ = WdA0,
T Td = d′ mean

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give
MJ′d′ = JdA0,
M TJd + NT(−s mean) = 0.

Also and = 0. The last follows from ( = 0 and = 0. These are the exact identities called F09 in the source. They use the same and defect, without an additional spectral estimate.

Pinning the auxiliary tower without changing the quadratic history

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The remaining assembly changes only auxiliary tower rows. Suppose a transition has metric factors
Hf, Hc with the stated analytic and positive lower bounds, response rows A0, T with their carried
fine-difference bounds, and inverse defect dT
2 Cd2, where C is analytic and positive on im d2. Extend
C to a uniformly positive full three-form row without changing dT
2 Cd2, take its analytic square root

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P, construct the auxiliary rows below, and apply their Gram identity. The preceding positive-
margin, root and Gram arguments justify each operation.
Let Ff, Fc be the full two-form Gram factors and Bf, Bc the inverse metrics.
Keep A0, T
unchanged. On the finite torus use mixed row FfBfTF T
c , normal row b−1/2dT
1 F T
c , and the image
row obtained by stacking

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I −FcBcF T
c
Pd2BcF T
c .

These formulas are supplied by common cover rows at all torus periods. The analytic auxiliary-row construction gives a positive width and common costs for them, including the required fine differences. Its inputs are fixed analytic estimates, clip profiles, and penalty; its constants are chosen before a particular block length. The exact Gram identity, using the bed floors and defect factorization, proves the required tower roll equations.

For each smoothing index first apply the positive extension and root operations, then choose the auxiliary-row prices for each of the two adjacent-stage possibilities. Intersect the finitely many widths with the original width and increase the cost to their maximum. Choosing at each transition the corresponding constructed tower gives a horizon with the identical beds and first two response rows and with the full horizon conditions at those new prices. Terminal normalization, crossing equations and bed bounds are inherited because these components were retained. The positive extension, auxiliary-row estimates and exact Gram identity therefore complete the pinning construction used in the retained horizon.

Completing the regular coefficient bounds and symmetries

All potential-and curvature-response bounds are made on one output momentum cube before choosing. The mass filter is bounded by decay) and its principal value has a positive real lower bound from the centered-average estimate. The line-filter factors are uniformly bounded, with positive principal lower bounds. At and have respective bounds grade(

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and C grade(l)−4, by the common-collar reciprocal estimate and the bulb and bend bounds. The
seed expression bounds S uniformly.
The sums H, Ta therefore have common analytic bounds. Their two smoothing factors con-
tribute decay2j, and each D contributes inverse squared grade. For j ≥4 these are bounded by
the uniformly summable decay2. Also U = |m(q0)|2 P
a fa + SH is bounded above. Every real
summand is nonnegative, and the principal filter and one fa have positive lower bounds. Thus U
has an L-independent positive lower bound. The reciprocal estimate bounds U−1 on a narrower
collar. These observations bound all factors in the regular correction coefficients of the potential
response; every matrix has a fixed face-orientation dimension.

For the curvature response, Λ is bounded because its only inverse factors are the principal mass filter and line filters. The other principal factors are bounded products and sums of controlled quantities. At= 0, the first branch has a signal of grade order one, of order grade(, and a smoothing flux with decay The second has two smoothing factors, the inverse-fourth-grade bound on, a bounded coarse factor and a further smoothing flux. Both products are bounded by decay), with constants fixed before

For real overlap identities, first use the regular stationary expressions. Every covariance and its scalar sum is periodic, and exterior signals and patch multipliers have the corresponding reciprocal periodicity. Shifting coarse momentum by 2 therefore permutes to + wrap both responses. If= 0 and both momenta lie in the enlarged real cube, neither is zero, so the

regular matching applies to both. For = 0 the identity is immediate. At nonzero momentum, scalar real symbols are even, Fourier rows conjugate at, and exterior matrices transform by entrywise conjugation. Substitution gives

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pilot(p, l) = pilot(−p, −l),
wire(p, l) = wire(−p, −l).

At zero, the principal formulas have their normalized values and all other branches vanish. Reflection therefore holds there too. These shift and reflection identities permit the common-collar Fourier export of the residue sums.

The inverse third power before residue summation

The response kernel called buckle is obtained from the symbol vault(wing) wire() with scale factor. Vault and its overlap identities preserve the common analytic estimates, so one actual fine difference of buckle has coefficient bound. Its fine curl sums the carried forward differences over four directions and the fixed two-form orientations, with incidence signs of modulus at most one. Hence its bound is 4Faces. Only orientation counts enter; no residue count is paid at this step.

Multiplying the curl by the bounded coarse metric gives the charge kernel with the same

scale. Apply the analytic right exterior lift on the fixed coarse orientation columns; it retains the prefactor, so the lifted kernel called sale is bounded by per fine residue. Its Gram row sums over residues and the fixed fine orientations. The bound is therefore

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C′′L4(L−3)2 = C′′L−2.

This is the inverse-square scale used in three-form absorption. Estimating before taking the fine difference would lose the power; the actual proof takes the fine curl before lifting.

Filling the stationary predictor at the principal alias

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Predictor convention and amendment A1.
The regular principal coefficient uses the paren-
thesized expression below. The extraction placed m(q0)−1 on its first term only. To see the required
scope of that multiplier, retain the directional rail in the covariance identity. Put m+ = m(q0),
m−= m(−q0) and, for this calculation only, a = | pullLinea(q0)|2fa/(m+m−), F = P
c fc, g =
m+m−a + ST and U = m+m−F + SH. Thus a is a reduced principal coefficient, rather than
an unannounced replacement of the source’s directional multiplier by m+m−. The corresponding
weights are E0 = m−F/U, r0 = m−a/g, and

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T
g −H
U

.
E0 −r0 = m−S(FT −aH)
Ug
=
S
m+

In the real even reduced convention this is the shorter calculation; on the complex collar the two factors retain the source’s opposite-momentum convention. This is the cancellation needed before extracting Fourier coefficients. The amendment changes the parentheses, and retains all nonprincipal branches, the collar hypotheses and the harmonic restoration.

For one transition, use

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ql = p + 2π lift(l)
L
,
q0 = p/L,
S = λmass(land)(p).

Put = bulb(land = bend(land) = massFilter() and = bulb(wing). The land and wing are the coarse and fine smoothing trains of this transition. For

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Dl,a =
bulb(wing, ql, a)
blow(mass(wing), p, l).

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Set Ta = P
l̸=0 | pullLinea(ql)|2Dl,a+b−1. On real momentum the notation is an absolute square;
on complex momentum it means the product at ql and −ql. The alias covariance recursion gives

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ga = | pullLinea(q0)|2fa + STa.
(3)
Put H = P
l̸=0 |m(ql)|2 P
a Dl,a and
U = |m(q0)|2 X
a
fa + SH.

Away from = 0 this equals S times the original sum of weighted covariance entries. The first term has a positive real lower bound; the other term is nonnegative, so U has a positive lower bound. The earlier alias estimates bound and U on a common collar, and the reciprocal lemma bounds there. Thus U is the regular denominator used in place of the original singular covariance sum.

Define the regular correction coefficients

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G0,a = m(q0)−1(Ta/ga −H/U),
Gl,a = m(−ql)[(
X
c
Dl,c)/U
−Dl,a|RL(ql,a)|2/ga], l ̸= 0.

Every denominator is regular: and the nonzero-alias divisor. The estimates above bound uniformly and by decay) grade( for= 0. The principal formula follows from (3). For the original normalized covariance weights and directional rebates, the identity is . The original longitudinal factor is times the regular row

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h−1d0(p)∗diag(g−1
a ).

Thus S cancels before Fourier coefficients are extracted. The pilot is its regular transverse leading term plus

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diag(Gl,a)dL
0 (ql)
h−1d0(p)∗diag(g−1
a ).

The transverse leading terms are

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l = 0 :
diag(fa) flux1(q0) diag(g−1
a ),
l ̸= 0 :
S diag(Dl,a) flux1(ql) diag(g−1
a ).

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Here flux1 is the transpose of the directional one-form patch multiplier, and dL
0 has the specified
scaled normalization. The scalar cancellation and these regular formulas supply the value at zero.
This is the filled potential response; the original expression with an isolated longitudinal denomi-
nator is not separately asserted to extend holomorphically.
The potential-response bound follows term by term. Its leading term has decayj(l). Its cor-
rection has decayj(l) grade(l)−2 times a signal bounded by a constant times grade(l); every other
factor is bounded. Since the grade is at least one, the same decay bounds the correction. Sec-
tion 5.24 proves the covariance recursion; Section 5.16 gives the stationary equations; Section 5.15