Convergence of the interacting scale histories · Part 11

and its complement. The finite difference sum tends to zero, and the two tails have an arbitrarily small uniform total. Hence the difference of the full series tends to zero. The same proof permits evaluation at arbitrary varying configurations. Forced comparisons are therefore preserved by summing a series, by collecting labels into finitely many barcodes, by a fixed injective reindexing with zero outside its image, and by finite sums and scalar multiples. They are also preserved by products: each fixed label is uniformly bounded, and ); a global bound on one absolute-budget family times the other’s tail budget controls the product tails.

Let Π denote the fixed weighted local section projection, with projection price. The core of the collected raw insertion is its seed plus the sum of Π applied to its bulk terms. For regular germs this projection is linear, including subtraction. Its paid bound gives

Searchable transcription

|Π(f −g)|1 ≤Cread paidBound(f −g).

At each label the right side tends to zero, and the projected rows have uniformly summable tails by that same price. Thus the projected core difference tends to zero whenever the seed difference does.

The residual and rebate operations used to remove this core are likewise linear on regular germs at the common threshold and support. Their fixed paid conversion price bounds the residual in the enlarged support and bounds the rebate everywhere. Apply these operations to the difference of the two bulk terms. Arbitrarily small paid difference bounds therefore give arbitrarily small residual paid bounds and arbitrarily small all-field rebate differences. Their uniform tail budgets are multiplied only by the fixed conversion price. Placing residuals in the common canonical support enumeration, with zeros at unused indices, preserves the regular comparison.

The new forced profile is the old forced profile plus the threshold hook of the collected rebates. The hook is a finite sum over support classes, multiplied by the actual compatible products of forced activities. These products agree in their discrete support and flag predicates because the two representations have the same forced address data. Their numerical factors have the forced comparison by the assumed comparison of the original forced activities, finite-product stability, and summable product budgets. Multiplying by the compared rebates and summing therefore preserves the forced comparison. Finally the common keeper barcode embedding collects these profiles and extends them by zero outside its image; this preserves both the convergence and tail bounds. Thus projection and collection pass the microscopic regular and forced comparisons to the output formatted insertion.

The common input decomposition for a shot

For fixed input capture and frame index, a shot is the merged sum of three kinds of contributions: the primary frame insertion, one contribution for each regular remainder label, and one for each forced barcode label. The input is the same book in both histories. The common incoming orientation fixes its threshold, forced addresses and barcode profiles. Stowing the forced input through the keeper therefore gives one common function of the fine configuration for every fixed forced label, independent of the history index and of which of the two histories is used.

Each of these contributions has a summable common tariff. For the regular inputs choose their paid regular bounds, and use tariffmass(). For forced inputs choose all-configuration bounds, and usemass(, with the same positive incoming forced scale. The input capture’s wealth bound makes both arrays summable. The finite primary contribution uses the larger of its heading tariffand its heading-plus-core tariff. The paid shot estimates bound not only each remainder but also its output core coefficient by this array. Thus it is a valid majorant for the countable comparison argument.

For a regular label the actual input ingredient is the real or imaginary part of its fixed masked regular function; the preparation construction has exactly that ingredient, its actual logarithmic key, and weight at most For a forced label the ingredient is the real or imaginary part of the common stowed function and has its actual gas key. If that function is zero, the output contribution is identically zero in both histories. Otherwise its input bounds and eligibility allow the same preparation construction, with weight at most. These are genuine fixed ingredients across the two histories, although their auxiliary preparation data may vary.

The nonconstant primary frame term likewise uses the fixed masked section insertion at the common incoming threshold, split into real and imaginary parts. For the constant frame index, preparation is unnecessary: the output is the same scaled unit insertion in both histories. Its forced profile is zero, and stowing that zero profile through any keeper leaves it zero; this proves equality even if the two keepers have different presentations.

We now establish the microscopic comparison of these preparations. Fix the scale, its block ratio , the degree, and the incoming dictionary. Both histories have that same dictionary and budget at this scale. Their couplings approach each other, their regular and forced activity coefficients have the weighted comparison already proved, and their stationary kernels and path responses have the common limits. Thus their local Gaussian covariances, means, saved fields and vertex factors have precisely the comparisons established earlier in this chapter.

A preparation of a fixed real ingredient retains its wing type and a common bound on its input weight. Converting the preparation to the common local chart preserves its ingredient, wing and weight. Consequently the two converted real ingredients agree exactly, as do the two imaginary ingredients. This is stronger than convergence of a possibly changing representative. The support address of a local integrated term depends on the wing and its fixed combinatorial fragrance address. It therefore agrees in the two histories and throughout the comparison sequence. Denote this support by. The insertion format ensures 0

Apply the mixed marked-integral comparison to each such address. Its hypotheses are now the identical ingredient just identified, the compared Gaussian data and vertices, and their common summable derivative envelopes. The change to a fixed Gaussian integration variable, the two dominated-convergence arguments, and the finite-order derivative estimate give: for every address and every 0, the difference of its two real bulk terms eventually has paid bound the common output threshold and derivative order. The same statement holds for the imaginary bulk terms. Their separate envelope arrays have the uniform tails of the marked cluster majorant. Multiplying by the fixed block factor and recombining real and imaginary parts preserves these conclusions. Extending the address enumeration by zero preserves them as well. We have therefore proved the bulk comparison, including both regularity on the common support and the summable tails needed for projection.

The forced part also retains its actual address data. Its construction first restores the missing ordinary logarithms, then restores the marked slots, and finally applies the coupling reset to the common shelves. For the ordinary logarithms, the finite-set/tail comparison of the reset families proves convergence and common tails. The marked slots have the mixed-integral comparison just obtained; inserting the prescribed finite subsets and extending by zero preserves it. Products of the restored ordinary exponentials with the marked factors preserve convergence and their summable envelopes. Finally the reset comparison applies to the same path family, threshold, shelves and rebate operations on both sides. The forced output profiles consequently have uniformly summable bounds and satisfy

Searchable transcription

Fn,α(Un) −F ′
n,α(Un) −→0
for every sequence of configurations Un.
(HL1)

This is the strong all-configuration comparison used here; no continuity of the masks is involved.

For clarity, (HL1) implies the corresponding uniform comparison. Otherwise there are subsequence, and configurations where the difference exceeds. Complete these choices to a sequence by arbitrary configurations at the other indices. It contradicts (HL1). Thus the supremum of the difference tends to zero. Combined with the uniform tails, the finite-set/tail argument also proves convergence of each summed forced profile.

The resulting real and imaginary forced comparisons recombine as

Searchable transcription

Ld(Fn + iGn) −Ld(F ′
n + iG′
n) = Ld(Fn −F ′
n) + i(Gn −G′
n)
,

with the same fixed and. This proves the required forced comparison for the raw complex preparation. Its primary seed has the already compared coefficient. The linear projection estimate, residual and rebate bounds, and common keeper extension from the preceding section now give convergence of the projected core and arbitrarily small bound on the collected remainder. The countable ingredient summation and finite-frame assembly therefore use proved local comparisons, all on the original common input book.

Summing local insertion comparisons

The local comparison feeding the transfer metric preserves the same coefficient supports in the two histories. It consists of convergence of their finite core coefficients to each other and, for every 0, an eventual remainder bound for their difference in the common output orientation. The difference is formed by appending the negative of the second labelled remainder to the first, so it remains a legitimate formatted insertion on that common support.

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Here is the summation principle needed for countably many local comparisons. Suppose the i-th
pair of insertions each has a common summable bound fi ≥0, and its difference has a remainder
bound tending to zero with the history index. The coarse bound on that difference is 2fi. Given
r > 0, choose a finite set E with P
i/∈E 2fi ≤r/2. On the finitely many indices in E, eventually
choose the common bound v = r/[2(|E|+1)]. Use v on E and 2fi outside. This is a valid per-index
bound and its total is at most r. The paid merge bound therefore gives an eventual remainder
bound r for the full merged difference.
Absolute convergence and the same label enumeration
identify this merged difference with the difference of the two merged insertions. The finite core
sums converge by the same finite-set/tail split. Thus countably many local comparisons assemble
into the full comparison.

The marks of a fixed coefficient input are obtained by a finite sum over the frame indices and the prescribed motion chips. Their shot inputs are the same captured coordinate-axis data in both histories. The local comparison for each such shot passes through its fixed motion and through this finite sum by the formatted-insertion operations. Hence the resulting site marks have converging core difference and arbitrarily small common remainder bound.

Finally each actual coefficient readout equals that site’s core, and its recorded capture equals capture of that site mark under the common output keeper. Each site mark has a finite paid remainder bound. The identity collecting a difference under the keeper identifies the difference of their recorded captures with capture of the formatted difference mark. Its arbitrarily small remainder bound bounds the capture norm by the same small amount. This proves convergence to zero both of the coefficient-isomorphism outputs and of their recorded capture outputs. These are precisely the fixed-input comparisons used in the transfer argument; the preceding microscopic comparison supplies their local analytic input.

The transfer comparison metric

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Let |x|1 = P
ν |xν| be the coefficient size in the fixed finite frame. A Gram datum P contains a
complex-linear capture map gP into the insertion-data seminormed space, with ∥gP x∥≤a|x|1. The
comparison relation P ∼ε Q means ε ≥0 and

Searchable transcription

∥gP x −gQx∥≤εa|x|1
for every x.

It need not separate distinct Gram data. It is symmetric, satisfies the triangle inequality in and every pair is comparable with = 2. For coefficient maps, the corresponding comparison is . Both uniform comparisons follow from bounds on the finitely many coordinate axes: expand in that basis and sum absolute values.

An adjacent transfer consists of a flowΦ) on Gram data and a coefficient isomorphism ). Its established paid inequalities are

Searchable transcription

|x|1 ≤b|Sn(P)x|1,
|(Sn(P) −Sn(Q))x|1 ≤cε|x|1,
Φn(P) ∼ε/2 Φn(Q)

whenever At fixed input, both the flow outputs and the maps) are Cauchy in their respective comparisons. These statements concern the actual integrated transfer maps; outside the eventual admitted schedule they may be completed by the zero capture map and the identity coefficient map. This harmless completion supplies total sequences while preserving all eventual statements.

To recover flow comparison from coefficient-normalized quantities, suppose where are coefficient isomorphisms, their difference has comparison cost, the maps differ by at most, and. Put. Then

Searchable transcription

gXi −gZi = (Tk −Uk) + gZ(Wk −i).

The two terms are bounded by and. Thus whenever (1 +. This proves the normalized comparison without an inverse estimate on an untested ambient space.

Pointwise convergence of the normalized maps along every pair of escaping history subsequences implies the uniform Cauchy comparisons. First apply it to each of the finitely many coefficient axes, obtaining a common bound and hence a uniform map bound. Next use the preceding inverse-normalization estimate for the flow. Finally, if an eventual pairwise tail bound failed, one could choose bad indices. Both escape to infinity and contradict the assumed subsequence convergence. This turns the subsequence statement into an actual double-tail Cauchy estimate.

If inputs themselves are Cauchy, the flow outputs are Cauchy too: insert one fixed late input between Φ) and Φ). The two outer errors are halved by contraction and the middle error is small by fixed-input convergence. Similarly insert in the coefficient maps; their two outer errors cost times the input error. Taking the input tolerance to be + 1)] and the middle tolerance2 proves the uniform coefficient-map comparison. These establish the precise transfer properties used in the finite-depth approximation and coefficient Cauchy argument below.

Fixed-level convergence of transported insertions

We explain the compactness mechanism for insertion coefficients before taking the continuum limit. At a fixed physical period, let be the integrated Gram data at level in the-th retained history. Adjacent levels satisfy, eventually in

Searchable transcription

Wi,n = Ti,n.flow(Wi+1,n).

The paid transfer bounds make each flow contract the comparison error by one half and carry the two-part Cauchy property of its input to its output. All admitted Gram data at one level have mutual comparison error at most two. Iterating levels and replacing the last input by fixed admissible data therefore approximates, eventually, with error at most 2 2. For fixed the approximating sequence has the two-part Cauchy property by the finite transfer bounds. Choose so the approximation error is below a quarter of the desired tolerance, and then use this Cauchy property. The triangle inequality proves that itself has that property at every fixed level. Only finitely many eventual identities are used for any such approximation.

Here is the inverse-normalization argument for coefficient transport. In the finite-dimensional coefficient norm, suppose eventually, is Cauchy, the maps are Cauchy uniformly on bounded inputs, and

with fixed 0. First is bounded, so is eventually bounded by the same inequality. Next

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Sn(yn −ym) = xn −xm + (Sm −Sn)ym.

The first term tends uniformly to zero for large; the second does so by boundedness of and the uniform map comparison. The inverse bound proves Cauchy. Equivalence of the finite-dimensional coefficient norm and the ambient norm converts this to the usual Cauchy statement.

Apply this successively to the actual adjacent transfer relations for the coefficients normalized by the reference inverse. Their lowest retained coefficient is eventually the fixed terminal coefficient , by the based transfer condition, and is therefore Cauchy. The Gram comparison just proved supplies the Cauchy comparison of each transfer map; its paid inverse bound supplies the uniform Induction upwards gives Cauchy coefficient sequences at every fixed level. This uses inverse bounds on the actual graph input at that level, without assuming a bounded product through arbitrarily many future levels.

The corresponding captured insertion data are Cauchy as well. If is the integrated capture map at that level and its Cauchy coefficient, write

Searchable transcription

gnxn −gmxm = gn(xn −xm) + (gn −gm)xm.

The uniform map bound controls the first term, and the second part of the Gram comparison controls the second on the bounded coefficient sequence. This constructs the Cauchy sequence of captured data used for the fixed-level density limit.

In the retained-history construction the terminal horizons tend to infinity. Thus every fixed level eventually lies below the horizon, all coefficients and captured data above refer to the original common history choices, and the path at that level is the prescribed stationary path. The history conditions provide the corresponding uniform lawful, scalar, regular, forced and path budgets. At a fixed level the Cauchy coefficient sequence is bounded; multiplying its bound by the fixed normalization scalar gives the uniform insertion payment bound. The exponential insertion and each forced profile are measurable: they are sums of masked measurable regular terms, followed by continuous pullback and either complex conjugation or identity according to the chosen motion. These are the measurability inputs for the fixed-level integral passage.

Uniform convergence of fixed-level insertion integrands

At a fixed level the comparison required for integration is uniform over all configurations; no continuity across changing masks is assumed. More precisely, a sequence of functions has the needed property if it is eventually uniformly bounded and is uniformly Cauchy over its entire

argument set. This property is preserved by eventual equality, fixed pullback, finite sums and products, and continuous functions on the finite-dimensional bounded target ranges. For the last assertion the common bounded range lies in a compact ball, on which the continuous function is uniformly continuous.

For captured insertion data, let) be its summed masked regular remainder and let ) be its forced profile with barcode. The capture norm and paid local mass bounds give absolute convergence and

uniformly in. Masks have absolute value at most one; for the regular terms apply the healthy paid bound when the mask is one and use zero otherwise. For the forced profile restrict the summable forced budget to the specified barcode. Both maps are linear in: absolute convergence permits termwise subtraction, including subtraction under the fixed masks. Thus the same inequalities hold for differences. A Cauchy sequence of captured data consequently gives uniformly Cauchy regular remainders and every fixed forced profile, with eventual uniform bounds.

The finite core insertion is also controlled uniformly in the configuration. For each coefficientaxis vector, its placed core function is continuous on the compact finite lattice configuration space and hence bounded; its healthy mask cannot increase that bound. Expanding a general coefficient in these finitely many axes gives a uniform Lipschitz bound in its coefficient norm. The constant coefficient is handled in the same way. Therefore a Cauchy sequence of core coefficients gives uniformly Cauchy core insertion functions, including their masks.

The format and capture identities identify the actual exponential insertion entry with the sum of this core function and the captured regular remainder, eventually in the history index. They identify the actual forced profile with the captured barcode profile as well. The preceding uniform bounds therefore prove the fixed-level convergence assertions for both actual entries and all profiles. A fixed translation, signed motion, complex conjugation when required, and fixed normalization scalar preserve these assertions. This proves the translated insertion convergence at any fixed anchor from the origin construction, without choosing a new normalization there.

For later use, integration of such a sequence against a probability measure is Cauchy whenever its terms are eventually measurable: the uniform bound gives integrability, and the norm of the difference of two integrals is at most the uniform norm of their difference. Applied to the finite polynomial restore expression, finite-dimensional continuity and finite sums/products preserve the same convergence. The common positive lower bound on its unmarked denominator then permits the normalized quotient limit. This connects the captured-data convergence to the fixed-level normalized expectations.

Exact unmarked density and component regrouping

Here is the unmarked version of the molecule expansion, including its coverage condition. Fix one forest throughout, one probability Haar measure in its gauge coordinates, and the common output configuration. A packed address consists of a store, a finite warning-label set, and the remaining labelled lists. The map sending this data to its molecule uses the prescribed block layout of that store. Reading the roster, warning ownership and lists from the molecule is a left inverse. Conversely, every molecule satisfying that layout is recovered from the read data. Thus this map is injective and misses only addresses whose activity is zero.

For each store and boundary choice, the local expansion has summable terms. Its absolute majorant is the scalar tribute times a product of nonnegative vertex tariffs. The tarifffamily is summable; hence its finite-subset products are summable, since their total is bounded by the exponential of the tariffsum. Restricting to the injectively encoded addresses preserves summability.

Integrating the expansion and then summing the finite boundary choices gives the store’s exact cut entry. The scalar = levy deducted

is unchanged by replacing the store and assigning a boundary choice: the neutrality identity cancels their apparent effect on the scalar correction, with the forest held fixed. If there are boundary choices, the tribute for each is exactly Their sum is therefore times the averaged molecule receipt. Summing over stores, and using the full discrete sector coverage, yields the absolutely convergent identity

Searchable transcription

push(R; V ) = OX
X
p
receiptX(p).

This is a density relative to probability Haar in the output variable; no unit partition normalization has been introduced.

To factor this sum into connected activities, partition each molecule into the connected components of its actual enlarged supports. These components have pairwise disjoint grounds and their union reconstructs the molecule. A compatible family contributes only if its roster flags cover exactly the output flag set, and each component satisfies its own layout, eligibility and flag-read conditions. These conditions are equivalent to the union molecule’s layout and approval. For example, intersecting the union flag set with one component’s ground leaves precisely that component’s flags: flags from another component cannot enter, by disjointness and the fact that a component’s flags lie in its ground.

Define a component activity by planting all output flags in its local data and then applying its own eligibility test. On its ground, this planted data agrees with the data from the union roster. The same holds for the selected regular keys: a key belonging to another component cannot meet this ground. Locality of the integrated activity therefore identifies the planted component activity with the union-data activity. Independence on disjoint grounds factors the latter product, including the independent finite boundary averages. Hence

Searchable transcription

receiptX
 [
p∈A
p

= 1{compatibility and exact flag coverage}
Y
p∈A
aX(p).

If the union approval fails, either flag coverage fails, making the displayed contribution zero, or at least one component fails its eligibility condition and its factor vanishes. Thus this equality includes all zero cases.

The connected-component map is injective because union is its left inverse. Every family outside its image has zero compatible contribution: a nonzero compatible family consists exactly of the components of its union. Reindexing the absolutely convergent sum proves

Searchable transcription

push(R; V ) = OX
X
A finite
1{compatibility and exact flag coverage}
Y
p∈A
aX(p).

Finally 0: its Gaussian chart mass and kernel normalization factors are positive, and its other factors are exponentials or positive powers of those masses. The extensive scalar stays outside the connected gas. The argument has not asserted locality of that scalar or performed the subsequent coupling reset.

The frozen data in this identity are the store and boundary choice after fixing the gauge slice; the threshold, width and selector belong to the fixed paid chart. The comparison of these expressions uses the common drafts, principal covariance matrices and Gaussian densities constructed earlier in

this chapter, together with the bill and protected-mean comparison and the four vertex cases. For a marked product it uses the same comparison with its finite marker coefficient retained. The next subsection states how these addresswise comparisons pass through the common source, its products and the reset; the signed line estimate (LCN11) gives the corresponding parameter control. No varying discrete approval predicate is differentiated.

Comparison of the common source and its marked products

Fix a period, a derivative order, and a budget of order at most, with threshold equal to the common temperature. All addresses and all finite trial tori in this paragraph are fixed before either history is varied. A regular comparison means common sparse summable envelopes in the -seminorms and convergence, for each address, in those seminorms. A forced comparison means common sparse envelopes in the uniform norm and addresswise uniform convergence. These are stronger than pointwise comparison of the total action; the distinction permits the subsequent products and reset operations.

For the connected logarithmic stream, the local Gaussian comparison already proved gives convergence of every derivative through order, along every sequence of healthy configurations and admissible test maps. This implies uniform convergence in the healthy-configuration derivative seminorm: otherwise choose a violating configuration and test map for each member of a bad subsequence, contradicting that sequential statement. On a fixed torus choose1 + dist( for every pair of sites. Conversion from admissible derivatives to anchored-derivatives costs at most

Apply the uniform derivative result with tolerance. The common stream envelopes, multiplied by, remain sparse and give the required regular comparison. The forced connected stream is handled by the same bad-sequence argument with order zero; its separate sparse envelopes were obtained from the forced integral estimate. In particular, the ultrafilter statements used to express local Gaussian limits here imply ordinary sequential convergence, since every free ultrafilter on every bad subsequence would give the same contradiction.

The path contribution is the sum of four already defined terms: the low path term with coefficient, the cubic and quartic terms with coefficient, and the loop quadratic term with coefficient1. Their coefficient arrays converge in the common weighted path norms. The scalar sequences in the two histories converge to the same respective limits and are uniformly bounded. For any such scalar pair and regular pair , use

Searchable transcription

anFn −a′
nF ′
n = an(Fn −F ′
n) + (an −a′
n)F ′
n.

The uniform scalar bound preserves sparse envelopes, while the two terms on the right tend to zero at each address. Thus all four path terms compare. Finite frame averaging preserves this conclusion: transport by a frame costs the fixed factor turnFee(), and multiplication by the averaging weight costs Rim(). For a target tolerance, use the original comparison with tolerance (Rim()turnFee()). The finite frame index multiplies each sparse envelope only finitely many times.

Consequently the full regular source, the negative connected logarithmic stream plus these path terms and their prescribed frame average, has a regular comparison. The hard source has a forced comparison, including its zero branch and its canonical label embedding. The enumeration used to place these terms in the common dictionary is the same for both histories.

The finite trial correction requires one additional observation. At the fixed period both selection panels are eventually contained in a single finite label set, and their cutoffparameters eventually