Separated and nested operator products · Part 3

Nested expansions

Nested expansions require an order of scales. The inner cluster is replaced first with the common coefficient reader, and its remainder is estimated before the outer replacement is summed. The scaleratio gain applies on the declared nested geometric domain. Terms with zero exponent are retained with their stated logarithmic or subpower qualification, rather than being assigned a nonexistent positive power gain.

Nesting keeps one reader At a nested node, the output of the inner expansion is an input to the outer one. Both therefore use the same homogeneous channel labels and finite projections on the physical filtration. The lower-degree terms are retained before the next substitution; otherwise the outer reader would see a different coefficient. The estimates distinguish a small ratio of nested radii from a fixed geometric separation, and keep the zero-exponent cases in their stated scope. Each nested tree and degree horizon is finite. Let be a finite rooted tree, with leaves carrying input fields and with every internal vertex

carrying a center and radius 0. A leaf also has its insertion position. Vertices may have zero or one child; all finite arities are allowed. At every internal vertex all positions in its subtree lie within of its center, and positions in distinct immediate child subtrees are at distance at least. Radii of internal children are at most their parent’s radius. The root radius is There is no lower bound on any child-to-parent ratio other than positivity.

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These conditions may equivalently be obtained from bounds on immediate child centers after
requiring a sufficiently small uniform upper bound on child-to-parent radii. To see this, induct over
the finite tree. Let Ci be the already obtained displacement constants of the child subtrees and
C′ = P
i Ci. Choose
β ≤η/[4(C′ + 1)]
0 < β ≤1,

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and below every child’s previously chosen ratio bound. The displacement of any descendant from
its immediate child center is at most C′βsv. Two such deviations reduce the immediate sibling
separation by at most 2C′βsv ≤ηsv/2.
The displacement from the parent center is at most
(M + C′)sv. This proves the stated full-subtree geometry recursively, with separation η/2.
Write wlfor each input degree, Wv = P
lbelow v wl, and

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qv = sv/sparent(v)

for a nonroot internal vertex. The virtual parent radius of the root is 1, so. For fixed output degree, define the common price

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Tm,ε(s) = r m−P
lwl−ε
Y
q−Wv−ε
v
.
(NE1)
v internal
v̸=root

Choose fixed truncation degrees at the internal vertices, before taking The direct coefficient is the coefficient of the entire leaf cluster at the root center. The iterated coefficient is the finite sum obtained by expanding each nonroot child cluster through its assigned degree, substituting its homogeneous basis fields at that child’s center, and repeating upward to the fixed root output. Multiplication of the successive coefficients and summation over all intermediate degrees and channels is meant literally. These are the same’s and homogeneous bases used in (OE1), with no new reader choice.

Uniformly in the admissible positions and positive radii, for sufficiently small

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|Cdirect| ≤Tm,ε(s),
|Cdirect −Citerated| ≤Tm,ε(s)
X
qDv+1
v
.
(NE2)
v internal
v̸=root

If there are no nonroot internal vertices, the iterated and direct coefficients agree exactly; the sum on the right is zero. The degree assigned to the root does not enter that sum, since the root output has already been fixed.

For exterior fields whose positions stay at least 0 from all tree positions and from one another, the full leaf product minus its direct root expansion through degree has ground-algebra norm at most (NE3)

The same estimate holds for the unselected law and every character, uniformly in their choice. The coefficients, the finite iterated sums and the tested expressions are smooth on the open set where all their finitely many reader configurations are admissible. For every fixed collection of requested degrees the geometry above eventually belongs to this open set. This follows recursively from admissibility of each child-center list and the direct leaf list; their diameters tend to zero and their required distinctness follows from the sibling separation.

We explain the scale accounting used to prove these assertions. If has logarithmic rate with nondecreasing down the tree and virtual parent exponent zero, set ∆ Then the engineering cost of the original leaves is

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E =
X
v internal
Wv∆v.
(NE4)

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This is also the sum over leaves of their degree times their nearest internal parent’s exponent: each
leaf’s path telescopes. Thus a direct output of degree m has rate E −maroot, and a tested remainder
through degree m has rate E −(m+1)aroot. The price (NE1) has exactly this engineering rate plus
ε P
v ∆v. The latter is strictly positive after normalizing P
v av = 1: at least one av > 0, and each
av is at most the sum of the nonnegative advances along all vertices. Therefore it absorbs every
fixed multiplicative constant, even when individual advances are zero.

We prove the recursive rate assertions, keeping all truncations finite. A leaf has just one alternative: its original field at its position, with coefficient 1. An internal child’s alternatives through horizon are the homogeneous fields of degrees 0, at its center, weighted by their direct coefficients. For any specified alternatives of the children, the parent coefficient is the product of their coefficients times the parent OPE coefficient for their resulting fields. A bud is never replaced by a degree-zero projection: its stage set is a singleton carrying its actual input degree. This convention includes arbitrary input labels correctly.

The main inductive assertion has two parts. Every raw subtree can be represented, up to any prescribed tested error rate, by a finite family whose positions all have separation at its parent’s exponent and whose coefficient amplitudes pay its total engineering cost minus the total resulting field degree times that parent’s exponent. The same raw subtree has direct coefficient and full tested remainder bounds at its own exponent, with its engineering cost. For a leaf both statements are immediate. At an internal vertex, combine the child families by finite products. Their costs add exactly to (NE4). Sibling separation makes the combined positions admissible.

If a child’s exponent equals its parent’s, retain the child’s comparable finite family obtained from its first inductive assertion. If it is strictly larger, use its second assertion to replace it through a sufficiently high auxiliary finite horizon. Specifically, if its engineering cost is Ω, the parent exponent is, its own exponent is, and later exterior costs are bounded by, the tested error rate is at most Ω

Given any requested rate, choose

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H > (Ω+ Wb −ρ)/(a −k),

and above any previously requested horizon. This makes the error smaller than requested. Perform the finitely many child replacements in order. At each step the preceding finite replacements and their field degrees have already been fixed, so choose the next auxiliary horizon to pay their actual exterior cost. The exact telescoping sum contains one replacement error at a time. This triangular choice avoids circular dependence among the horizons. Finite readouts of the tested errors then give the same comparison for coefficients. Applying the single-center estimate to the resulting comparable finite family establishes both assertions for the parent.

To compare with prescribed, rather than auxiliary, truncations, fix a gain0 such that

for every nonroot internal vertex (NE5)

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Inductively, each recursively composed child coefficient has its direct coefficient’s engineering rate,
and their difference improves that rate by γ. The product identity
Y
j
xj −
Y
j
yj =
X
j
(xj −yj)
Y
i<j
yi
Y
i>j
xi

therefore improves the product rate by, not by a multiple of. Multiplication by the parent coefficient preserves this improvement. For 0, every nontrivial immediate child has a strictly positive exponent gap, by (NE5). Replace all such children through auxiliary horizons with error of any desired rate, using the preceding triangular argument. Subtract the finite parent sum through the desired horizons. Its difference from the auxiliary sum is exactly the sum over omitted channel choices. In every omitted choice at least one child degree is at least + 1. That child’s coefficient costs its degree at the finer exponent and the parent coefficient returns it at the parent exponent; hence the choice gains at least (. All other degree-gap contributions are nonnegative. Thus the omitted finite sum gains. Replacing the remaining direct child coefficients by their recursive coefficients gains the same by the product identity. Adding these three differences proves that the direct-to-iterated coefficient error has rate at most

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E −maroot −γ.
(NE6)

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If γ = 0, direct and iterated coefficients already have the common raw rate, so their difference has
that rate by the triangle inequality; no high-accuracy replacement is needed at a zero exponent gap.
If there are no nonroot internal vertices, there is one choice at every child, its coefficient is 1, and
permutation invariance identifies the parent sum with the direct coefficient exactly.
Take γ to be the minimum of the finitely many (Dv + 1)∆v. Choose a vertex attaining this
minimum. Its ratio power qDv+1
has rate −γ, while the positive slack in (NE1) absorbs the fixed
v
constants in (NE6). Its single positive ratio term is bounded by the sum in (NE2), proving the
claimed sum of gains. The direct coefficient and exterior remainder rates similarly give (NE2)’s first
inequality and (NE3). Contractivity and the character supremum give the latter in the common
ground-algebra norm.

The finite-family calculus in that induction has explicit quantifiers. An alternative is a finite word at selected subtree positions, multiplied by a scalar amplitude of some finite upper rate. A family is eligible at exponent when each alternative’s positions are separated at. Two families agree to rate, up to exterior weight, when their complete correlations differ with rate at most against every fixed exterior label list of total degree at most, with arbitrary admissible position paths and arbitrary viewer paths. A supply at parent exponent and cost gives such an eligible family for every sufficiently large exterior exponent; for every it gives a comparable replacement agreeing in this sense. These choices concern exact continuum correlations throughout.

To change one branch while retaining all other branches, fix an alternative of the other branches. Let its field degree sum be and its amplitude have rate. Include its fields in the exterior test of the changed branch. Request agreement through weight and rate. Multiplying by that alternative’s amplitude returns rate. The finite sum over all other alternatives has the same rate. Since there are finitely many alternatives, one may take the maximum of their and first. Sibling separation and the old eligibility show that all these augmented exterior tests have the required separation. This proves the single-branch replacement formula. Induction over the finite branch set proves the triangular replacement assertion: at a given replacement only the previous choices and unchanged later branches enter its finite maxima. Transitivity follows from adding the complete correlation differences.

A tested agreement transfers to coefficient agreement by the actual finite reader. For a requested horizon, take at least the maximum degree sum of every exterior word in all detector supports through that horizon. These supports form a finite union. The rotating detector words retain their degree sums and have bounded label coordinates; their positions stay a fixed positive distance from the center and from one another. A shrinking eligible family is therefore admissible against every such moving detector eventually. Write each rotated reader as its finite sum of these tested correlations times its fixed weights and rotated loaded vectors. Apply the agreement estimate and then average over the compact rotation group. The arbitrary-path quantifier makes the bound uniform in the rotating group element, as proved for (OE4). Apply the real symmetrization and , then a basis coordinate; these fixed finite-dimensional linear maps preserve its rate. This proves the coefficient transfer with the same

There is a further debit when transferring a whole remainder, which is why agreement of products alone would not suffice. Let the desired remainder rate be

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ρ0 = M −(m + 1)(k −u),

before exterior cost, where the center-to-exterior exponent is 0. Choose a comparable replacement agreeing at rate, with enough exterior degree to cover both the actual exterior fields and the finite detectors through. Its full-product difference has at most this rate, hence at most. The coefficient differences through degree have rate; their centered exterior correlations cost. Each resulting term therefore has rate at most. Add the comparable replacement’s own complete remainder at that rate and subtract these coefficient differences. The exact resulting expression is the original family’s complete remainder. This proves the second inductive supply assertion with all coefficient subtractions retained.

The comparable-family estimate used above includes the boundary exponent cases. An alternative of field degree has amplitude of rate, and its positions have size and separation exponent. For 0, its degree- coefficient costs ( by the homogeneous reader bound. Multiplication gives = 0, test its amplitude times the word against the finite detector exteriors. The word has rate zero, the amplitude rate; the readout comparison with the empty family gives rate for its coefficient as well. Actual shrinking of its positions ensures eventual reader admissibility even though its exponent is zero. For the full remainder, when the exterior exponent, multiply the single-center remainder rate by the amplitude to obtain

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M −(m + 1)(k −u) + |e|b.

If, estimate the whole correlation and each finite centered term separately; the engineering degrees cancel and give exactly, the same expression at zero gain. Summation over the finite comparable family proves the estimate in every case.

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We now remove the auxiliary assumption that radii already possess logarithmic rates. Along any
approaching admissible family all internal radii tend to zero, since they are positive and bounded
above by the root radius. Put
T =
X
v internal
log(1/sv).

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Eventually every summand is nonnegative, T > 0, and T →∞.
The vector with coordinates
log(1/sv)/T belongs to the finite compact cube [0, 1]Inner(t). On a nontrivial refining filter it con-
verges to a; summing the coordinates proves P
v av = 1. Hence each sv has rate −av on that
filter. Monotonicity sv ≤sparent(v) implies av ≥aparent(v), since the opposite strict inequality would
eventually force the smaller radius to exceed the larger one by the exponential-rate comparison.
Take the virtual parent exponent to be zero.

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The geometric size estimate Msv gives the required envelope rate at vertex v, and also actual
convergence of every descendant displacement to zero. The sibling lower bound ηsv gives separation
at every exponent strictly larger than av.
Induction over the tree thus gives all the regularity
hypotheses of the preceding rate proof, including vertices with av = 0. Since P av = 1, the slack
ε P ∆v is strictly positive, as required in (NE4). Applying the rate estimates gives (NE2) and
(NE3) eventually on this refining filter.

If a single common small-radius threshold for all admissible scales and positions failed, choose a violating scale and position array at arbitrarily small radii and restrict to those radii. The compact layer extraction just proved gives a nontrivial refining filter of this very array where the desired inequality holds eventually, contradicting its selected violations. The same argument handles the exterior geometry and all viewers; for viewers one first selects a violating viewer at each parameter. Thus the final thresholds depend only on the fixed tree, field labels, horizons,, and exterior clearance, with no lower bounds on the ratios. This proves the uniform quantifiers in (NE2)– (NE3).

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The bounds admit arbitrary joint weighted integration.
Let (K, μ) be any measure space,
R |u| dμ ≤B. Let z : K →Plot(t) be almost everywhere measurable, with
u ∈L1(μ; C), and
admissible geometry wherever u ̸= 0, at one fixed array of radii s. The direct coefficient and the
finite iterated coefficient are smooth on their common open reader domain. Extending them by
zero outside that domain gives measurable functions. If |g(z(i))| ≤C on the support of u, then

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|u(i)g(z(i))| ≤C|u(i)|

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almost everywhere, so this product is integrable and

Z
u(i)g(z(i)) dμ(i)
 ≤BC.
(NE7)

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Apply this first to the direct coefficient with C = Tm,ε, then to its difference from the iterated
P
v̸=root qDv+1
coefficient with C = Tm,ε
. The iterated coefficient is integrable because it is the
v
difference of those two integrable quantities. Subtracting their integrals proves the weighted version
of (NE2). No product structure or absolute continuity assumption on μ is used.

For jointly varying exterior positions) and phase characters), the same conclusion applies to the complete tested remainder with. Its measurability follows on the common open guard domain from smoothness in positions and continuity of the character evaluation of each fixed ground-algebra element. The character space is compact metrizable, and these separate properties give joint measurability. Almost everywhere measurable may therefore be composed with it. The full raw correlation is integrable too: its bound is, obtained by adding the finite degree-zero terms to the degree-zero remainder and applying the engineering rates. The truncated expression

is then the difference of two integrable expressions. Consequently the finite sum may be integrated term by term and the norm of the full integrated discrepancy is at most. A probability mixture of characters is the special case = 1; its expansion uses the identical coefficients outside the phase integral.

Finally, covariance concerns the full homogeneous vectors, rather than individual scalar basis channels. At a leaf the emitted vector is its actual label. At an internal vertex first form the emitted vectors of the children, include each in its fixed horizon space, and apply the multilinear to their centers for each through the parent’s horizon. The finite sum of these included outputs is the emitted vector of that subtree. Expanding multilinearity shows that its coordinates are exactly the finite iterated channel formula. Translation invariance, conjugation compatibility and orthogonal covariance now follow by induction from the corresponding laws of and the inclusions. Applying just at the root proves the same laws for the degree- iterated output. The basis can transform nontrivially within; no assertion that its vectors are individually rotation invariant is needed. The full tested discrepancy inherits these laws from the hierarchy and these homogeneous outputs.

The separated and nested expansions, their compatible coefficients, all-viewer estimates, joint weighted versions and covariance have now been translated.