Convergence of the interacting scale histories · Part 12

exceed every label in that set. Their panels therefore agree eventually. Removing the selected terms from the regular source preserves its sparse envelopes and addresswise convergence. Likewise the selected site contribution is a finite sum over this common set with the same fixed selection weights. Its comparison follows term by term. There is no passage through a discontinuous moving selection rule.

A trial on the fixed small torus is pulled back through its prescribed padded shape map. Its support lies within the fixed radius, and a healthy neighborhood of that radius maps into the all-healthy trial torus. The derivative pullback costs at most

Searchable transcription

M = 2B.order(1 + z)B.power B.order.

Using tolerance on the trial torus proves comparison after pullback. The ornament is a finite sum over the trial types and sites, with fixed ray weights, so it has a regular comparison. The corresponding failed-trial contributions have a forced comparison by the same finite-sum argument. The site correction is exactly the selected regular contribution minus its value at the identity configuration minus the failed-trial contribution. The identity is healthy; hence evaluating there preserves the regular comparison. Subtraction proves the uniform site comparison, and the finite set of sites gives the forced comparison required for the correction.

These facts assemble the common source: its regular part is the unselected regular source together with the trial ornament; its hard part contains the original hard source and the negative frame average of the site correction. Every sum uses a common disjoint label set. This proves comparison of the complete source, rather than merely of its total evaluated function.

For clarity, the marked-product step can also be written without the source notation. A marked family ) carries sparse bounds for

Searchable transcription

|un,i(t, V )| + |∂tun,i(t, V )|

and convergence of both quantities along arbitrary configuration sequences, at the prescribed parameter values. An unmarked compared logarithm ) becomes a constant marked family. Its exponential factor is1; on a common bounded interval the mean-value theorem transfers the logarithmic comparison to this factor and preserves the sparse envelopes with the fixed exponential multiplier.

Extend both families by zero through the same label embeddings. A resulting forced atom is a finite product of the embedded marked factors and these unmarked exponential factors, multiplied by its eligibility indicator. The two histories have exactly the same eligibility predicate on the common configuration. Product differentiation,

Searchable transcription

∂t
Y
i∈A
ui =
X
i∈A
(∂tui)
Y
k∈A\{i}
uk,

therefore proves comparison for both the atom and its mark derivative. No continuity of the eligibility indicator is needed: it multiplies both differences by the same number in. The sparse product bound established earlier controls the sum over these finite subsets. Widening the source is another common zero extension, so the same argument applies after widening. Finally evaluate at = 0. Its absolute value is bounded by the marked envelope, and its comparison is the derivative part just proved. This supplies the forced marked reset comparison used in the two-history output limit.

Analytic control of omitted insertion terms

One estimate needed in the continuum comparison is obtained by scaling all restored insertion errors with a single complex parameter. The elementary analytic fact is this: if is holomorphic on 2, and there, then

Searchable transcription

|f(1) −f(0)| ≤2B
S .
(IC1)

For every on the segment from zero to one, its closed radius-2 disk lies strictly inside the radius- disk. Cauchy’s derivative estimate gives2). Integrate along that segment to obtain (IC1). Here the normalized insertion moment at level is

Searchable transcription

Z
(P(U))t d priorj(U)
Stampj,t(P) =
ZN
,

where is an insertion-polynomial coefficient and 0 is the original Wilson partition function. The denominator is the same at every intermediate level, by the exact transport identity. Suppose is served by the insertion family at level, and its restoration formula is restore(

Searchable transcription

P(z) = restore(zP0, zP1),
f(z) = Stampj,t(P(z)).

Then, while(0) is the restoration with the two error inputs removed. The retained insertion estimates give holomorphy on the disk of radius = brisk( They also give integrability for every in that disk. The scalar complex parameter uses the available supportexponent margin of two; no geometric support is changed.

Searchable transcription

Transport P(z) to a fixed earlier level k < j. The retained transport theorem keeps its served
insertion bound there.
Integration is unchanged by transport, including after taking the finite
polynomial coefficient t; coefficient extraction is a continuous linear map and commutes with this
integrable finite-dimensional integral. The root small-field estimate at k therefore gives the uniform
disk bound
| Sitek | treasuryk +n(x + y + 2)


|f(z)| ≤exp
=: Bk,t.
toll(wk, t)

Here is the number of insertion slots, are their two root norm allowances, and the toll uses their positive weights. All volume dependence is at the fixed reporting level; there is no exponential cost involving the increasingly fine level Applying (IC1) gives

Searchable transcription

Stampj,t(P) −Stampj,t(restore(0, 0))
 ≤
Bk,t
brisk(Bj)/2.
(IC2)

This is the uniform omission estimate. It follows from the exact insertion transport and the common analytic disk, and does not require convergence of the intermediate representations of the mixed pin terms.