Separated and nested operator products

Separated and nested operator products

A finite reader used consistently in every product. Operator-product coefficients are extracted with one homogeneous coefficient reader, chosen compatibly with the degree filtration, reality and rotations. A truncation retains finitely many channels and an explicitly bounded remainder. Recovery therefore means equality of the retained coefficients and of the tested replacement operation on its separated configurations, together with the same remainder bound. It does not invert a finite projection or assert convergence of an infinite series in the truncation degree.

Nested products must use that same reader at each stage. Lower-degree subtractions and the scale-ratio gains control the difference between the two actual conditional pushforwards. Their constants and radius thresholds keep the fixed word, degree and separation hypotheses. The common coefficients serve all permitted phases, but the correlation reading at each use still belongs to the chosen common hierarchy. The detailed derivations below establish these compatibility statements before any collision extension is attempted.

20.1 Separated products and fixed homogeneous readers

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Let Ed be the coherent finite-dimensional degree filtration and choose its reader once, compatibly
with reality and Euclidean rotations. For input degrees di write ∆= P
i di. Assume

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∥xi −o∥≤Mr,
∥xi −xi′∥≥ηr (i ̸= i′),
∥o −yj∥≥c,
∥yj −yj′∥≥c (j ̸= j′),
M ≥0,
η, c > 0.

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For every fixed truncation D and slack ε > 0, the expansion has, for all sufficiently small r > 0,

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

The threshold is before every allowed, where is the unselected hierarchy or any character. The same coefficients serve all, spectators and viewers. For each fixed channel, on its small-radius tail, (OE2)

These are finite-order asymptotics; noconvergence is asserted. Apply the real rotation-averaged finite reader at level to one field at the origin, obtaining . These respect inclusions in and reproduce included fields when

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

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For q ≤m this gives
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, include the second identity at m −1 into Em to see πm,q = 0. Inclusion compatibility
follows from L. Set Hm = T
i<m ker πi,m. Applying each lower projection to the recursion, using
induction on earlier homogeneous pieces and cancelling its matching term, proves Km(Em) ⊂Hm.
The same identities make the projections fix their own homogeneous space and annihilate the other
included pieces. Choose once a real compatible basis em,α of Hm, α ∈Cm. For cluster reader value
Am(x, a; o) define
hm = KmAm ∈Hm,
hm =
X
α
Cm,αem,α.

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The horizon-D packet is P
m≤D ιm,Dhm. Applying πj,D recovers exactly hj, independently of D.
For a single centered field OE3 recovers its inclusion. Spectator pairing of this packet is exactly
the finite sum in OE1, by linearity and component inclusion. These are exact identities before
the asymptotic bound. Define the error in the ground algebra as the compressed full cluster-and-
spectator word minus that finite sum of compressed centered outputs.
Vacuum and character
evaluation give precisely the scalar remainder. Contractivity of these evaluations and the Gelfand
norm identify the common algebra norm bound with the uniform bound over character viewers.
The averaged reader has the explicit finite-packet formula

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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)}.

Haar measure is normalized. Use this on the open set where every rotated readout is admissible and assign zero outside; the asymptotic uses eventually lie inside. Finite packets have integrands and all position derivatives continuous uniformly on compact(4) locally, so integration preserves smoothness and multilinearity. Translation and permutation follow termwise. Haar change gives orthogonal covariance; the second average gives conjugation compatibility. For one centered

degree- field,, coherence gives. Equivariance makes the first integrand constant, and real inclusions make the second average preserve it. This proves the single-field property used in OE3. Induction in the triangular recursion makes commute with rotations and conjugation, since each earlier projection does so and the inclusions intertwine both operations; thus is invariant and has the asserted laws. For the conjugate-linear involution on

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v = v + Cv
2
+ iv −Cv
2i

shows its fixed real space spans complex-linearly. A real basis there is complex independent: conjugate a complex relation and add and subtract to separate its real and imaginary coefficient relations. It is the required real basis, chosen once at each, before horizons and exterior words are specified.

In the source lift retain the whole packet, filtration, reader, projections and algebra-valued error together. Homogeneous readout recovers each exactly and is compatible with every finite horizon. Spectator evaluation then recovers OE1. A truncated reader loses higher channels and the residual; retaining the residual makes the original whole-word reading exactly recoverable. No separated expansion is promoted to a coincident-point operator identity.

Fixed homogeneous channel and residual recovery. To distinguish configuration readout from the homogeneous maps, write for the finite coefficient section and recovery, with . Define on horizontal images

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LZ
m,q = jmLm,qpq,
KZ
m = jmKmpm,
πZ
m,q = jmπm,qpq,
ιZ
q,m = jmιq,mpq.

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Their compositions recover the same triangular recursion and (OE3).
The lifted homogeneous
space is HZ
m = jmHm with fixed basis eZ
m,α = jmem,α. The normalized Haar and reality averages
commute with these fixed linear sections, on the same open admissible domain and with the same
zero convention outside it. Thus hZ
m = jmhm has exactly the original scalar coordinates Cm,α,
independent of any chosen horizon or spectator.
For P hom
D
= P
m≤D ιm,Dhm, its source packet and exact channel reader satisfy

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(P hom
D
)Z = jDP hom
D
,
πZ
j,D(P hom
D
)Z = jjhj
(j ≤D).

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In particular section recovery of each channel is exact at every finite horizon.
Let RD denote
the retained ground-algebra whole-word remainder, with all spectators still attached. Its source
remainder is RZ
D = α(RD), so

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∥RZ
D∥= ∥RD∥,
h(χ)(RZ
D) = χ(RD).

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The vacuum reader also agrees under the ground unitary. This recovers (OE1)–(OE2) with their
same uniform allowed-configuration/viewer tails and fixed-data dependence. Retaining RZ
D along-
side the finite packet recovers the whole reading; the construction asserts the same finite-order
asymptotics and no infinite-horizon convergence.

20.2 Uniform OPE rates and smooth interpolation

A rate at most means that for every positive slack the norm is eventually bounded by as . For a cluster of size and internal-separation exponent 0, set). The finite reader comparison gives

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

It holds for arbitrary varying rotations. Otherwise choosing a violating rotation at each bad parameter contradicts the comparison on that path. Compact Haar integration therefore preserves the uniform rate; conjugate comparison pays the real average. Apply fixed

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hn −πn,HZH
rate ≤(|d| −H −1)a.
(OE5)

For, the projection kills all old grades below; the grade- term has rate (, while the error has the better rate (. Finite summation gives rate ( for and each coordinate. If exterior clearance and internal exponents are 0, centered degree costs. A varying field of rate costs, by fixed finite-basis expansion. OE3 gives the exact remainder identity

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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 rate is. Each correction has rate ( . Finite summation preserves for arbitrary varying viewers, permitting the algebra norm bound. This last quantifier is what permits passage to the ground-algebra norm, rather than merely pointwise character bounds. Takinglog = 0 and spending arbitrary slack gives OE1–OE2. A failure of a uniform geometry threshold would select bad geometries at arbitrarily small radii, contradicting this pathwise rate. The coefficient family is consequently common to all horizons and phases. OE4 itself follows by testing the old scalar remainder at the finite detector configurations and fixed weights. Under varying rotations their distances stay positive and labels/output vectors have rate zero by compactness. Expand moving exterior labels in fixed finite bases. Multilinearity and finite sums preserve the rate; conjugation of the correlation gives the conjugate comparison, using the hierarchy’s reality identity. For completeness, the scalar remainder bookkeeping does not lose a grade.

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Recovery of uniform logarithmic remainder rates.
For fixed coefficient sections, hZ
n = jnhn
and the section norm preserve the rate of every finite-reader difference. Applying a fixed linear
map of norm C also preserves a rate: for a prescribed slack ε > 0, use the input estimate with
slack ε/2 and then choose T so large that C ≤eεT/2. The same argument absorbs any fixed finite
number of summands. Consequently the exact homogeneous map and finite remainder sum recover
(OE4)–(OE5) with unchanged exponents, rather than a shifted grade.
For the correction at degree j ≤D, the explicit exponent comparison in the retained geometric
regime is

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ξ −
(|d| −D −1)a + jb + |e|c

 = (D + 1 −j)b ≥0,
ξ = |d|a −(D + 1)(a −b) + |e|c.

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The source algebra remainder is RZ
D = α(RD), and thus

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∥RZ
D(T)∥= ∥RD(T)∥= sup
χ∈Θ
|χ(RD(T))|.

This uses the whole ground-algebra element at each. A violation of a common eventual bound selects one violating viewer at each violating parameter and contradicts the source estimate for arbitrary varying viewers. Varying rotations are retained in the same way before Haar integration;

the normalized integral cannot increase the uniform norm bound. Finite detector weights and compact rotation-dependent label coordinates have rate zero under the same reader.

Finally withlog and = 0, the rate bound is exactly The analogous channel rate is exactly A selected bad-geometry path is unchanged by section recovery, so the source’s uniform small-radius threshold argument applies to the same allowed configurations and fixed coefficient family at every finite horizon.

For the old graded estimates choose mesh exponents 0, with actual cluster exponent, and put). Physical matching supplies a degree- centered label an amplitude and a survival filter for their actual continuum limits. Its label budget rate is its common mismatch against all permitted exterior tests has rate

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τ −(q + 1
16)(a −b).
(OE6)

The same is used at every detector and phase. Its degree- readout,, has rate. For = 0 use the full budget. For 0, lower-degree approximation gives whose distance from is bounded by the degree-(1) fare times the original budget. The degree- reader kills by coherence; its value on, bounded by the fare of exponent (, has the required rate. This subtraction is essential: using just the norm of would lose all of this graded improvement. Choose with (). Testing OE6 at the finite detector list and adding this readout estimate bounds the actual degree- coefficient by on the lattice screen. For truncation, choose with (). Let exterior clearance exponent be, internal exponent0, and weight. Reconstruct from all its readers. For subtract the true coefficients; OE6 pays each difference and its exterior cost. For , the graded budget gives

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

The product-to- mismatch has that upper exponent too. Their exact sum is the product minus its degree- truncation, so the screen remainder has rate

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τ −(N + 1)(a −c) + |r|u.
(OE7)

Every statement retains arbitrary varying viewers. Smooth interpolation applies to the actual coefficient or remainder, using its rough bound, screen bound and positional derivatives. The rough coefficient rate is since detector exteriors stay distant; the rough remainder rate is at most by separate bounds on the original and finite truncation. Derivatives are those of the same hierarchy and finite detector sums. Interpolation gives the screen rates at arbitrary configurations. Choose and strictly between exterior clearance and. The supplied clock has the required mesh levels. As0, the coefficient exponent tends to ( and the remainder to. For any strict requested exponent first fix small enough, then levels and auxiliary horizon. A moving center changes no distances or readers. The empty cluster has only the degree-zero unit coefficient, zero positive-degree coefficients and exactly identity truncation, requiring no anchor approximation. This covers the cases excluded when an anchor in the cluster was chosen above. We detail the grading argument supplying the old estimates.

Here is the interpolation detail. 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. Let) have rough rate in a corridor of size/separation exponent, screen rate at mesh exponent, and every

coordinate derivative replace a label by a degree-one-higher derivative label. Choose integer with . Descend the total degree threshold: the rough bound already proves the improved estimate for. If it holds at one higher degree, move to a screen point within rate The small ball stays in the corridor. The derivative has rate; its operator norm is bounded by the finite sum of its coordinate norms. The mean-value inequality gives

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

Add the screen value and iterate down to every original degree. Uniform derivative bounds and eventual differentiability throughout the moving ball follow by selecting a violating point if they failed, then applying the pathwise hypothesis. Finite-dimensional rounding with finitely many forbidden coincidences finds within 2 mesh units, depending only on the number of cluster positions. Derivative replacement holds for full correlations, finite detector sums and the finite-product-rule remainder; the centered exterior factors do not move with cluster positions. Finally, the derivative replacement identity holds for the actual functions in question. No derivative of a chosen approximating label is required.

The lifted rate source retains the common matched, survival filter, detectors, finite-basis coordinates and all slack/level choices. Its triangular reader OE5 recovers the new channels from the old packet with the paid residual. The exact remainder identity preserves the degree and exterior costs; interpolation recovers arbitrary geometries from the same screened functions. Retaining the arbitrary varying-viewer quantifier is essential for recovering the compression algebra norm rather than merely separate character limits.

Graded screen and interpolation recovery. Retain the same centered matching label amplitude and refining filter for every detector. For a degree- readout and the retained lower-degree approximation, its section lift satisfies

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RZ
h jqQ = jhRhQ = jhRh(Q −s),
Rhs = 0.

Thus its fare and whole-label budget combine with exponent ( on both carriers. This is the same lower-degree subtraction that yields the screen coefficient bound. The chosen satisfies the displayed mismatch inequalities before testing any viewer, so section recovery retains the finite low/high grade split giving (OE7).

For interpolation, scalar correlation, detector and fixed-truncation remainder readers agree at every actual configuration. Their positional derivative-label identities therefore agree as well. The lifted screen point is the same physical rounded point, with unchanged excluded coincidences and distance bill. Applying the mean-value reader to the actual function gives the same exponent cancellation

The selected-path argument for uniform bounds on the moving ball is preserved, so this recovers the descending induction without taking a derivative of the matched label The final slack choice is quantitative. Setting and yields exactly

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τ −ha = (|g| −h)x + (|g| + 8|I| + h)δ,
τ −(N + 1)(a −c) + |r|u = |g|x −(N + 1)(x −c) + |r|u
+ (|g| + 8|I| + N + 1)δ.

For any strict requested upper exponent choose to fit its positive gap first, retaining the stated positive mesh ordering; then choose the levels and auxiliary horizon. This explicit readout preserves the source’s quantifier order, moving-center invariance, and exact degree-zero unit behavior of the empty cluster.

20.3 Compact matching limits and exterior-price passage

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A finite plot at k ≤m < m + 1 ≤j obeys

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

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On sufficiently large periods the wrapped positions retain their locations, are separated by B2
j
lattice units and share central parent zero through m + 1. Matching along the original extraction
order gives degree-q labels with a paid local mismatch of amplitude A and quota at m at most p.
Quota comparison bounds their norm in the fixed finite reference space by Kp. Bounded finite-
dimensional compactness gives a nontrivial refining filter and a limit Q, on which the quota and
paid mismatch survive. The survival record retains the actual filter, geometry, insertion, reference
horizon and quota.
Fix positive slack ρ for exterior fields. Their bounded-copy envelopes are

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μj = (1 + Yj)a−ej−ρ
l
+ 1.

At mesh exponent their product has rate (. Choose auxiliary exponents strictly between exterior clearance and matching support, and a further pair strictly above exterior internal exponent. These gaps pay every finite width and separation requirement; a fixed number of fields adds no exponent. On the same survival filter the mismatch still uses the same. Its compressed exterior rate is at most, with amplitude rate. Choose and then0 before any eventual bound, giving. Contractive evaluation permits arbitrary varying phase viewers. Using both cluster and exterior envelopes gives the full-correlation engineering rate. These

arguments justify all exterior costs used in the grading and interpolation proof. For approaching plots the strict exponent inequalities eventually give cluster size below16, spacing above and ordered level indices. The subpower consumes no strict gap. One compact matching limit can therefore be selected pointwise at each parameter, with the common amplitude and quota rates required by OE6.

For matching accuracy take block exponent, input exponent and exterior-support exponent. The input budget product has rate and the volume factor rate so the positive initial amplitude has rate, where). The terminal cost is

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

Columns, chart factors and copy controls have rate zero; adjacent + 1 have the same mesh exponent. Thus the cost rate is and output quota is 2 . Choose

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zT = exp

[(q + 1
16)(a −b) −τ]T

,
AT = 2 stakes /zT .

The stock-opening level has exponent and stakes have upper rate zero, giving the OE6 error rate. March affordability offers gain (), chart affordability (); expenditure uses only (). Both leave strict positive margins absorbing all rate-zero constants. Room also follows from: the clock has + 1 and scale increments at least log 3, hence

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Jm + 5 ≤J0 + 5 + scalem / log 3,
log(Jm + 5)
scalem
−→0.

A fixed readout radius and this subpower width fit inside the strict mesh gap. Make these finite matching choices before the continuum limit; they give the centered label, quota and local root

error in the same central chamber through+1. The two affordability conditions in the matching construction hold eventually. No exterior independence is assumed. On the common finite lattice,

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

The left side is one paid main insertion with the given exterior insertions. The original extraction along the survival filter yields the two right-hand terms with the actual limiting, proving the screened compressed-difference bound. Approximate arbitrary exterior positions by screened positions avoiding the center and cluster, coordinate by coordinate; separation persists and both limiting configurations are noncoincident. Smoothness passes scalar inequalities at every character to the limit; their supremum gives the algebra norm. The same argument without replacement bounds 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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Survival-filter and exterior-price recovery.
A matched source record retains the actual re-
fining filter F, finite reference carrier, labels Qλ, paid mismatch, quota and geometry. Its coefficient
section gives
Qλ −→
F Q,
jqQλ −→
F jqQ,
pq lim
F jqQλ = Q.

The section norm retains the same compact bound. Scalar quota and mismatch inequalities are part of this same record on that filter; their recovery does not select a replacement label after choosing an exterior word. At each common finite lattice configuration, pullback preserves the exact register identity:

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L
(registerx −O(Q, 0)) registery

 = L
registerx⊔y −O(Q, 0) registery

.

The normalized integral and paid insertion readers therefore recover the same correlated discrepancy. Passing along F and then through the fixed screened-position approximations preserves the chosen . The resulting algebra element is recovered isometrically by, so taking all character readings gives the same norm bound for arbitrary exterior points. For clarity, the two strict affordability margins read exactly as

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(q + 1)(a −b) −(q + 1
16)(a −b) = 15
16(a −b),

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(q + 1
8)(a −b) −(q + 1
16)(a −b) = 1
16(a −b).

They absorb the same rate-zero prices before the continuum limit. The source keeps adjacent-level exponents and central chamber in that same finite matching choice. The exterior envelope reader likewise retains until the auxiliary exponents and have been fixed to meet the requested strict slack. Thus its recovered rate is with the source’s choice order and arbitrary-viewer scope.

20.4 Finite sweep, replicas and coefficient conservation

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Copy the input list into every cell of a fixed coarse tiling, with its specified reflection and translation,
writing Fv,i for copied insertions and qv = Q
i Fv,i. The anchor copy is unchanged. At m + 1 all
original members have parent zero, so copied parents are exactly the tiling centers.
At every
finer level a copied position has its center’s coarser parent and lies within half the intervening