Separated and nested operator products

Separated and nested operator products

An operator-product expansion is an approximation at a fixed finite degree horizon. One homogeneous coefficient reader is chosen on the physical filtration and used for every phase and truncation. Lower-degree terms are subtracted with that same reader, which is crucial when an expansion is inserted into a second expansion. The estimates keep the cluster separation and spectator separation parameters explicit.

The separated expansion and its fixed homogeneous channels

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Let Ed be the finite-dimensional field space of degree at most d, with the coherent inclusions already
constructed. A reader is chosen once for this filtration, its real structure, and its Euclidean rotation
action. For finite input degrees di, write ∆= P
i di. The expansion proved below has the following
precise form. If
∥xi −o∥≤Mr,
∥xi −xi′∥≥ηr
(i ̸= i′),

and the spectator points satisfy

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∥o −yj∥≥c,
∥yj −yj′∥≥c
(j ̸= j′),

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where M ≥0 and η, c > 0, then for each fixed truncation D and slack ε > 0, all sufficiently small
r > 0 satisfy

Ss(a(x)b(y)) −
α∈Cj
Cj,α(x, a; o) Ss(ej,α(o)b(y))

≤rD+1−∆−ε.
(OE1)
D
X
X

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j=0

Here may be the unselected hierarchy or any character component. The threshold precedes every allowed. The same coefficients serve every, spectator and viewer. For each fixed channel,

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|Cj,α(x, a; o)| ≤rj−∆−ε
(OE2)

on its corresponding small-radius tail. These are fixed finite-order asymptotic statements; no convergence asis asserted.

To define the homogeneous pieces, begin with the real, rotation-averaged finite reader at level Apply it to one field at the origin to obtain linear maps. They respect inclusions in the second index, and for reproduce that included field. Define recursively

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Km = 1 −
X
i<m
ιi,mKiLi,m,
πm,q = KmLm,q.

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Thus, whenever q ≤m,

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πm,qv = ιq,mv −
X
i<m
ιi,mπi,qv,
m
X
i=0
ιi,mπi,qv = ιq,mv.
(OE3)

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For q < m, the second identity at level m −1, included into Em, makes the first right side zero.
Hence πm,q annihilates all lower-degree fields. Inclusion compatibility follows directly from that of
Lm,q.
Set
Hm =
\
i<m
ker πi,m.

Induction in the recursive formula proves: apply each lower projection, use its inclusion compatibility, the induction hypothesis on the earlier homogeneous pieces, and cancellation of its one matching term. The same identities show that the projections fix their own homogeneous space and annihilate the others after inclusion. Choose the prescribed real compatible basis of

For an input cluster let be its finite averaged reader value, and define its new homogeneous output

The coefficients are exactly its coordinates in that fixed basis. Thus

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hm =
X
α
Cm,αem,α.

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The finite packet at horizon D is P
m≤D ιm,Dhm. Applying πj,D gives exactly hj for j ≤D, indepen-
dent of D. For a single field at its center the packet is precisely its inclusion, by (OE3). Pairing the
packet with a spectator correlation gives the finite sum in (OE1), by linearity and the component
inclusion identities. These are exact compatibility identities, not asymptotic approximations.

The ground-algebra error is defined before selecting a viewer: it is the compressed whole clusterand-spectator word minus the finite sum of compressed centered homogeneous outputs with those spectators. Evaluating it in the vacuum or any character gives exactly the scalar remainder in (OE1). Contractivity of all these evaluations and the Gelfand norm give equivalence between a common algebra norm bound and the uniform bound over character viewers. This is how the same error estimate will serve all phases.

The averaged reader used here has an explicit meaning. Write) for the old finite physical packet. For normalized Haar measure on the compact orthogonal group, put

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A0
m(x, a; o) =
Z
O(4)
Xm(k)P≤m(k−1x, X(k−1)a; k−1o) dk,
Am(x, a; o) = 1
2{A0
m(x, a; o) + A0m(x,  ̄a; o)}.

The first formula is used on the open set where every rotated readout configuration is admissible; outside that set the source assigns zero. All subsequent asymptotic uses eventually lie in the open

admissible set. Each packet is a finite sum, and its integrands and all positional derivatives are continuous uniformly on the compact rotation group on a sufficiently small neighborhood. Thus integration preserves smoothness and multilinearity there. Translation and permutation invariance follow term by term. Haar invariance, with replaced by, gives orthogonal covariance. The second formula commutes with conjugation. For one field of degree placed at the center, coherence of the old readouts gives Equivariance makes the integrand constant after the exterior rotation; reality of the inclusions makes the second averaging leave this value unchanged. This proves the exact single-field property used in (OE3).

Induction in the triangular formula for now proves that commutes with rotations and conjugation: each earlier projection does so, and the inclusions intertwine both operations. Consequently is invariant under them and has the stated translation, permutation, reality and rotation laws. The real basis invoked above exists for a concrete elementary reason. If is the conjugate-linear involution on, then

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v = v + Cv
2
+ i v −Cv
2i

expresses every vector as a complex combination of fixed vectors. A real basis of the finite-dimensional fixed space is complex independent: apply to a complex linear relation and add and subtract to obtain separate real relations for its real and imaginary coefficients. It therefore is a complex basis of consisting of real fields. This choice is made once at each, before any truncation or exterior word is specified.

Here is the precise transfer of remainder rates through these choices. Use the source’s logarithmic parameter; a rate at most means that for every 0 the norm is eventually at most . Suppose the cluster is of size , is internally separated at that rate, and ), the reader comparison gives

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An(x, a; o) −An(o, ZH; o)
rate at most
(|d| −H −1)a.
(OE4)

This statement holds with every varying rotation, not just each fixed rotation. Integration therefore preserves the rate uniformly: otherwise choose a violating rotation at each violating parameter. The conjugate comparison supplies the second half of the real average. Applying the fixed linear map yields rate at most (OE5)

In particular, at, all old grades below are annihilated by. The remaining old grade has rate (; the error (OE5) has the better rate (. Finite summation proves the same rate ( for, and hence for every coordinate

For completeness, the scalar remainder bookkeeping does not lose a grade. Suppose cluster-to-exterior separation has rate, exterior internal separation rate, with 0 and centered field of degree, paired with the exterior labels of total weight, costs. For a varying field of rate, its cost is: expand in a fixed finite basis and use the fixed-field bounds on each coordinate. Reconstruct by (OE3). The exact identity for the new remainder is

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D
X
W(x, a; y, b) −
j=0
W(o, hj; y, b) = [W(x, a; y, b) −W(o, ZD; y, b)]
D
X
+
j=0
W(o, πj,DZD −hj; y, b).

The old remainder has rate

By (OE5), the-th correction has rate (. The finite sum thus has rate at most, uniformly for every varying viewer. This last quantifier is what permits passage to the ground-algebra norm, rather than merely pointwise character bounds. In the separated regime log = 0, these rates are precisely (OE1) and (OE2), after spending the arbitrary positive slack. To make the threshold uniform over all admissible geometries, a failure would select a bad geometry at arbitrarily small radii; restricting to those radii contradicts the just-proved rate along that selected path. This also explains why the coefficient family is common to every truncation and phase.

The finite readout comparison in (OE4) follows directly from testing the old scalar remainder. Each readout is a finite sum of fixed exterior configurations with fixed scalar weights. After a varying rotation, these exterior configurations remain a fixed positive distance from the shrinking center and from one another, and their rotated labels and loaded output vectors have rate zero, by compactness of the rotation group. The old scalar remainder at these moving tests has rate . To allow their moving labels, expand every exterior label in a fixed finite basis; the coordinates have rate zero and multilinearity preserves the rate in a finite sum. Summing the fixed detector weights and output vectors proves the rotated readout comparison. Conjugating the whole correlation and using the hierarchy’s reality identity proves its conjugate version with the same norm. This proves (OE4), including the necessary arbitrary varying rotations.

We detail the grading argument supplying the old estimates. At logarithmic scale, choose lattice levels with mesh exponents

where the actual cluster has exponent. Put

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τ = |g|d + 4 |I|(d −a).

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The physical approximation supplies, for an auxiliary horizon q, a centered label Q of degree q,
an approximation amplitude, and a survival statement for their actual continuum limits. Its label
budget has exponent τ −qa, and its mismatch against all allowed exterior tests has exponent at
most
τ −(q + 1
16)(a −b).
(OE6)

The same mismatch is used at every detector; no refitting of the centered field to a detector or phase is allowed.

For a mode of degree, applying its physical readout to costs only. For = 0 this follows from the whole-label budget. For 0, the approximation-to-lower-degree estimate gives a label belonging to filtration degree1, with distance from bounded by its degree-( fare times the original budget. The degree- readout vanishes on, by coherence and inclusion compatibility. Its value on therefore equals its value on; the bounded linear readout and the fare’s exponent ( give the asserted exponent. This subtraction is essential: using just the norm of would lose all of this graded improvement.

Choose large enough that

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(h + 1)a ≤(q + 1
16)(a −b).

Testing (OE6) at the finite detector list and adding the preceding readout bound proves that the actual degree- coefficient has exponent at most on the lattice screen. For a truncation at

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degree N, choose instead q ≥N with

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(N + 1)a ≤(q + 1
16)(a −b).

Let the exterior clearance exponent be, its internal exponent be0, and its total weight be. The centered degree- correlation costs. Reconstructing the centered through all its readouts, split its finite expansion into degrees and. For the first group subtract the true coefficients; (OE6) pays each difference and the centered exterior correlation. For the second group use the graded budget

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τ −ha + hc + |r|u ≤τ −(N + 1)(a −c) + |r|u.

The original product-to- mismatch has this same upper exponent by the choice of. The exact sum of these three differences is the desired product minus its degree- truncation. Thus the lattice remainder has rate at most

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τ −(N + 1)(a −c) + |r|u.
(OE7)

All statements retain arbitrary varying viewers throughout.

The passage from these lattice screens to arbitrary points uses the smooth interpolation estimate, applied to the coefficient, or to the product minus its fixed truncation. Its hypotheses consist of the rough correlation bound, the screen bound just proved, and the positional derivative bounds for these same functions. For coefficients the rough rate is, since the detector exteriors stay at positive distance. For remainders it is at most, obtained by separately bounding the original product and the finitely many truncated terms. The derivative bounds are those of the same hierarchy and finite detector sums. Interpolation gives (OE7) with the displayed at arbitrary approaching configurations. Finally choose, and an intermediate strictly between the exterior exponent and. Mesh levels with each of these positive exponents exist on the supplied clock. As0, the coefficient exponent tends to (, and (OE7) tends to

For any strict requested upper exponent choose 0 small enough first, then the mesh levels and auxiliary horizon. This proves the old rate estimates with their exact quantifier order.

Translation to a moving center changes no distances or readouts. The empty cluster requires no approximation: its coefficient is the readout of the degree-zero unit, its positive-degree coefficients vanish, and its truncation is exactly the identity insertion in every exterior word. This covers the cases excluded when an anchor in the cluster was chosen above.

Here are the details of the smooth interpolation step. Consider a family) of scalar functions, whose label list has total degree, and suppose every positional coordinate derivative replaces one label by its prescribed derivative, raising total degree by one. Assume that in a corridor of configurations of size and separation exponent

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fT (p, z) has rate ≤a|p| + c,

while on screened points of mesh exponent it has rate. Choose an integer such that (. We prove the better rate for every degree by descending induction with the statement that it holds whenever = 0 the rough bound suffices. For the induction step move to a screened mesh configuration at distance of rate. Every point in the ball of that radius remains in the same corridor because. Each derivative there is a function with

label degree + 1, so the preceding induction statement bounds its rate by. The finite coordinate basis bounds the operator norm of the derivative by the sum of its coordinate norms. The mean-value inequality on the ball gives

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fT (p, z′) −fT (p, z) rate ≤b(|p| + 1) + d −b = b|p| + d.

Adding the screened value proves the induction step. Taking includes every original degree.

The required uniform derivative bound on this moving ball follows from the pathwise hypotheses without assuming uniformity: if it failed arbitrarily late, select a violating point in each such ball, and use that selected path in the derivative hypothesis. The same selection argument supplies eventual differentiability at every point of the ball. Screened approximation itself is elementary finite-dimensional rounding with finitely many forbidden coincidences: a fixed number of mesh shifts suffices to avoid them, with distance bounded by 2 times the mesh size, where depends only on the number of cluster positions. Thus the displacement rate is indeed. Finally, the derivative replacement identity holds for the actual functions in question. It holds for full correlations by the hierarchy’s derivative law, for a readout by its finite sum of full correlations, and for the remainder by subtraction and the finite product rule (the centered exterior factors do not vary with the cluster positions). No differentiability of a chosen approximating label is used.

We also specify the compactness and exterior-price passage in the physical matching step. A finite plot at levels satisfies

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8∥xi∥< am+1,
∥xi −xi′∥> 2ajB2
j
(i ̸= i′).

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At sufficiently large periods its wrapped positions retain their actual locations and are separated by
B2
j lattice units, while every position belongs to the same central block through level m + 1. The
matching estimate therefore produces, along the original extraction order, degree-q labels whose
products differ from the original cluster by a paid local insertion of amplitude A, and whose quota
at level m is at most p. The quota comparison bounds their norm in the fixed finite-dimensional
reference space by Kp. Finite-dimensional bounded compactness gives a nontrivial refining filter
and a limit label Q. Both the paid mismatch and the quota bound persist on that filter. This is
the precise survival statement used above: it retains the actual refinement filter, geometry, paid
insertion, reference horizon and quota, in addition to convergence of the labels.

To test that same mismatch against an exterior word, first fix arbitrarily small positive slack The bounded-copy construction for an exterior field of degree gives a per-field envelope

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μj = (1 + Yj)a−ej−ρ
l
+ 1.

At a mesh exponent, its product has rate (. Choose auxiliary mesh exponents strictly between the exterior clearance exponent and the matching support exponent, then a further pair strictly above the exterior internal exponent. Their strict gaps ensure all finite support widths and separation requirements for the paid-insertion estimate; any fixed number of fields costs no exponent. The common survival filter makes the paid insertion estimate apply to the very same After the continuum limit the compressed exterior mismatch has rate at most

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δ + (|e| + ρ|J|)ξ,

where is the amplitude’s rate. Choose these auxiliary exponents with, then0, before requiring any eventual bound. This gives exactly. Contractive evaluation gives the identical bound for arbitrary varying phase viewers. The two-group full-correlation bound follows by the

same argument with the product of the cluster and exterior envelopes, giving engineering rate . These arguments justify all exterior costs used in the grading and interpolation proof.

For the approaching plots, the strict exponent inequalities ensure their two geometric conditions eventually: the cluster size is smaller than16, its internal spacing exceeds 2 , and the selected level indices are ordered. Here is subpower relative to the mesh, so it consumes none of the strict gaps. Consequently the compact matching limit exists at each approaching parameter. Selecting one such limit label pointwise gives the family used in (OE6), with the same amplitude and quota rates.

The matching accuracy is funded as follows. Let the chosen block level have exponent, the input mesh exponent, and the exterior support level exponent The individual input budgets have product rate The many-copy initial amplitude includes the volume rescaling factor of rate), hence is a positive quantity of rate, with the same ) as above. Its degree- terminal cost is explicitly

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eT = 4(1 + columnsm) 2aq
m+1 blingm
4
xT
.

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The column count, the displayed chart factor, and the copy controls have rate zero; the adjacent
level m + 1 has the same mesh exponent as m. Thus eT has rate τ −qa. The output quota is 2eT .
Choose
zT = exp
(q + 1
16)(a −b) −τ
T

.

The stock-opening level has the same exponent as its starting level, and its stakes have upper rate zero. The error amplitude is

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AT = 2 stakes
zT
,

which has the upper rate in (OE6). The two affordability conditions in the matching construction hold eventually. For the march condition, the available exponent is the full gain (); for the chart condition it is ( ). The chosen expenditure uses only ( ). Both leave strict positive margins. All numerical constants, stocks, columns and chart prices have rate zero, so they are absorbed by those margins. The remaining room condition also follows from : the clock satisfies + 1, while each block scale increment is at least log 3. Hence + 5 + scale log 3 and log( scale0. A fixed readout radius and this subpower extra width are dominated by the strict mesh gap.

These choices are inserted in the finite matching construction before the continuum limit. It yields a centered degree- label with quota 2 , a mismatch supported in the prescribed central chamber, and root-test bound . The input positions are required to remain in that chamber through level + 1, exactly the plot conditions already checked. The finite matching construction itself is the sweep-and-replica construction; its remaining formulas are addressed below.

There is also no hidden independence assumption in the exterior test passage. On a common finite lattice the original cluster register and its centered replacement are multiplied by precisely the same exterior register. The algebraic identity is

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(registerx −O(Q, 0)) registery = registerx⊔y −O(Q, 0) registery .

The paid insertion bound applies to its left side as one main insertion and the given number of exterior insertions. Taking the original extraction limit along the survival filter gives the two terms on the right with the actual limiting label. On screened configurations this proves the norm bound for their compressed difference. For arbitrary exterior positions, approximate by screened lattice

configurations avoiding the finite set consisting of the center and all cluster positions, coordinate by coordinate. Their separation inequalities persist eventually, and the hierarchy is smooth at both noncoincident limiting configurations. For each character the scalar inequalities therefore pass to the limit. Taking the supremum over characters gives the compressed algebra norm bound. The same approximation, without a replacement, proves the corresponding bound for full products. Thus the extension to arbitrary exterior points uses actual continuity of the already constructed hierarchy and one fixed replacement, not a fresh replacement chosen after the test.

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We describe the geometry and algebra of that finite matching construction more precisely. Make
one copy of the input list in every cell of a fixed coarse tiling, with the prescribed reflection and
translation of that cell. Write Fv,i for these copied insertions and qv = Q
i Fv,i. The anchor copy is
unchanged. At level m + 1 all members of the original list have parent zero, so the parents of the
copied list are exactly the tiling centers. At every finer level the copied position has the same coarser
parent as its center and lies within half the intervening block length of the embedded parent. Two
distinct centers are separated at the parent level by at least B2
m+1+1. If the intervening enlargement
factor is L, their embedded separation is at least L(B2
m+1 + 1); the two deviations cost strictly less
than L. The propagated budget satisfies B2
i ≤LB2
m+1. The triangle inequality therefore gives
the required cross-copy separation at every intermediate level. Within a copy the given finest-level
spacing supplies the remaining separation condition.

Represent the copied insertions by a polynomial with one formal marker per occurrence ( Descending integration preserves this polynomial and its integrability. Grouping markers within a copy into one marker replaces by. Grouping in stages or all at once has identical coefficients: the coefficient of a specified marker set is the product over its full inverse image. Since grouping is a fixed linear map on a finite-dimensional polynomial space, it commutes with each conditional integral. This proves the conservation identity between the descended polynomial of the products and the grouped descended polynomial of the individual The descending sweep starts with positive single-field allowances

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wi = δjφjai/8.

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At each level it joins precisely the fibers that have acquired a common parent, retaining one anchor in
each fiber; thus the final number of fibers is at most the original arity. Its available normalization is
bounded below by the compensation of Q
i wi, including the four-dimensional volume ratio between
levels m + 1 and j, and the licenses used in these joins. The initial allowance is

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!
|I|, m + 1, j, 0, α, (δjφj/8)|I| Y
i
ai
α|I| compensation
.

The chosen1 is no larger than any intermediate license, so fewer surviving fibers only increase this lower bound. This is the finite quantity whose rate was denoted above.

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The licenses can be chosen uniformly down to every preceding level while preserving rate zero.
If ui > 0 is the license and log ui/ scalei →0, set αi = min(1, u0, . . . , ui).
Then 0 < αi ≤1
and αi ≤uj for all j ≤i. For any t < 0, the rate-zero hypothesis gives uj ≥et scalej for large
j. Monotonicity of the scale implies αi ≥min(αi0, et scalei). Dividing logarithms by the scale and
letting t ↑0 proves that αi still has rate zero. The halo purchase bounds the logarithm of the
q
bβi, whose ratio to the scale tends to zero. The other license
geometric part of each license by 16
factors have the same rate-zero property. Thus this minimum introduces no power loss.
At the last join, let z > 0 be at most the surviving normalization and let x ≥1 be the purchased
local coefficient constant. The terminal normalization estimate gives a local analytic coefficient with