Convergence of the interacting scale histories · Part 7

This supplies the input to the rate calculation below and hence (RB6); the retirement argument then uses the resulting bound with its existing delay order.

We supply the quantitative calculation behind (RB6). Put log, and . All derivative orders and stage parameters in this calculation are fixed before tends to infinity. A function of rate is eventually bounded in absolute value by for each Products add rates, a finite sum has the largest of its rates, and multiplication by any fixed constant preserves a rate. Thus a rate strictly smaller than20 gives the bound in (RB6), with coefficient one, after increasing the delay.

Write the actual reference logarithm, before centering, as

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g = log Ω+ L0 + Y + βf1 + W.
(RBG1)

Here Ωis the field-independent Gaussian normalization; is the logarithm of the normalized protected local integral after its Gaussian main part is removed; is the remaining Gaussian normalization term; is the difference between the prescribed coarse classical action and the fine classical action evaluated on the stationary graph; and is the difference between the prescribed coarse quantum action and the thermal determinant and Haar contribution on that graph. These are functions on the same real primary-coordinate space. In particular (RBG1) is an equality of the actual densities before any estimate is taken.

For a function, use the scaled derivative bound

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∥F∥n,A;z = max
0≤q≤n Aq∥DqF(z)∥.

The protected-integral comparison and interpolation (PL1)–(PL5) give, including at the origin. The derivative price of on the radial segment from zero to is

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d = X(x)gstep(β)2 + ε,
gstep(β) = β−1/2,
(RBG2)

with of rate zero. The integral itself is positive: the Gaussian measure is positive, the protected integrand is nonnegative, and on a nonempty open neighborhood of the zero fluctuation all protecting cutoffs are one and the exponential density is strictly positive. The Gaussian has positive density there. Its normalizing Gaussian determinant is positive by the vertical Hessian gap. Hence its real logarithm is defined on the chart. The estimates for its derivatives follow by differentiating its absolutely dominated integral and using the logarithm expansion established in (ASP1)–(ASP2). Let1 bound the reference coordinate and site counts, let be the stationary-response price, and let be the sum of the purchased bare-action jet price and the stationary-composition price. More explicitly, the coordinate conversion multiplies an order- multilinear price by the latter composition price is the fixed polynomial clump appearing in the ordered-partition derivative bound. Thus has rate zero. Classical matching removes the jets through order four, giving (RBG3)

The higher jets of, of orders three through, have a price of rate zero. This price is the bare jet bound multiplied by+Λ), together with the same stationary-composition polynomial and. Here is the bounded kinetic correction and Λ is the path allowance. The finite-period thermal comparison bounds its second jet at zero by

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∥D2W(0)∥≤A0,
A0 = M2{M4m(g)M} Θp e−γc.
(RBG4)

In this formulag) = (1 + dim g) is the fixed color-coordinate conversion factor. The reference side is Θ 0 are the purchased decay constants, and0 is bounded by the adjacent loop allowance. Only the displayed exponential dependence on is used below.

The first jet is smaller than a generic extensive-action bound would suggest. The signed symmetries give(0) = 0, while classical matching gives(0) = 0. Differentiating (RBG1) therefore yields (RBG5)

Taylor’s formula along segments in the chart, with these first and second jets and the third-and-higher jet prices, gives

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∥W −W(0)∥n,A;z ≤CnA3 + 3{(ε + d)A + A0A2}.
(RBG6)

For orders zero, one and two this follows by integrating the next derivative and using (RBG4)– (RBG5). For an order3, multiply the unscaled bound by. Taking the maximum and enlarging the three low-order contributions gives the factor three. Centering costs at most ; centering costs. Equations (RBG1), (RBG3), and (RBG6) now give

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∥g −g(0)∥n,A;z ≤dA + 2ε + βKnA5 + CnA3 + 3{(ε + d)A + A0A2}.
(RBG7)

Here. Consequently has rate1. The lower bound on the reference side makes Θ, and hence, have rate1. All remaining prices have rate zero. The respective rates on the right of (RBG7) are at most

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−23/16,
−2,
−19/16,
−21/16,
−23/16,
−15/8.

Every one is strictly below32, and20. The defining rate inequalities, with a sufficiently small positive slack and then a common delay, therefore make the whole finite sum at most. This derives the quantitative gain rather than assuming that a small integral automatically has a small logarithmic jet.

It remains to explain why the estimate applies to the five reference tori, whose sides need not be physical periods. Choose a physical coarse side divisible by the given reference side; the divisibility requirement was included in the block calendar. Copy the reference forest periodically to this larger torus. The uniform stationary construction gives a critical section on each torus. On the larger torus the newly constructed section and the retained physical section have the same zero value and solve the same critical equation in the same uniqueness corridor. They agree near zero. Their tested classical jets and their thermal second jets therefore agree there. The periodic-copy comparison transfers classical matching and the finite-period Hessian error to the reference section. Its decay exponent can be replaced by the minimum of the two positive exponents, and its price by the sum of the two prices. This is sufficient for (RBG4). No equality of the Gaussian integrals on different tori is required.

On a fixed reference torus, differentiation along the logarithmic link chart costs the factorial and direction factors already displayed in (RB4a)–(RB4b). The omitted-source allowance comes from (RB8), uniformly in the compact coupling band. The same normalized offers depend on the coupling value, so substitution of the actual coupling curve preserves these estimates and their continuity. Summing the finite-site logarithmic and omitted-source bounds gives the reference offer bound used in (FCC1). This completes the passage from the actual stationary integral to the retirement beacons.

The uniformity required in this last assembly can be made explicit. For a fixed transition between stage profiles, the local argument is purchased at derivative order + 6 for the stationary score and at the indicated higher orders for the bare-action and Haar jets. Its conclusion uses order2. The thermal comparison is first known for the retained critical section. A newly selected uniform analytic stationary graph is an extension of this section: the two solve the same

vertical equation, and uniqueness near zero identifies them. Therefore their classical matching and finite-period second-jet comparison are the same tests, with the enlarged common price and the smaller common positive width. This establishes the hypotheses of (RBG3)–(RBG5) for the graph used in (PL1), rather than replacing that graph by a different critical solution.

For each fixed incoming stage only finitely many outgoing transition tests are required. Take the maximum of their coupling delays. Enlarging the calendar’s delay preserves all the estimates, since they hold for every coupling above that delay. At an actual calendar step the identities for the block length and next coupling identify its local parameter with). The two adjacent loop caps are bounded by the same local loop allowance, and the kinetic correction satisfies

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|α| ≤drift + |sG log L| + 1.

The assumed stage bounds on paths and loops therefore place both sides of this actual step in the local comparison just proved. This accounts for the order of choices and for the fact that the estimates hold for all allowed coupling values in the compact band.

Finally put, let be the derivative order of the current budget, and let be its fixed conversion price for left derivatives. After retiring the source tail to budget, the centered retained source on any one reference torus of side has left derivatives bounded by

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Fi = q! 3qs + 2c4
i {T(s + s) + s}.
(RBG8)

The first term is the finite composition bound from the logarithmic coordinates to the prescribed left derivatives. The other terms pay the two source differences and the finite sum over sites; their factors are positive, so restriction to the omitted subfamily does not enlarge them. The retirement depth is common over the compact coupling band by (RB5)–(RB8), and common over the finite reference tori by taking its maximum. Each reference side is chosen above the read radius, and its finite joint-coordinate, fiber and site counts have the same preassigned size bound. No constant in this step is chosen from the later macroscopic volume subsequence.

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If the fixed interpolation stencil has signed weights ai, its support reach is e, and the budget
support power is p, transporting these local estimates to one insertion gives
X

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i
|ai| 2q(1 + e)pqFi.
(RBG9)

Indeed the directional substitution costs 2, the support enlargement costs (1 +, and the finite sum is bounded with absolute weights. Smooth dependence of each reference offer on its coupling allows substitution of the actual continuous coupling curve. Formula (RBG9) is the bound for the reference offer in (FCC1). It is on the same ordinary source and common finite stencil as the retirement construction, and supplies the needed passage from the local protected integral to its assembled source bound.

The local activity norm and analytic comparison estimates

The norms in finite calibration are the paid activity norms, not the uniform norm of the resulting density. Fix the current budget and a common address layout. Let 0 be its regular and forced normalization allowances. For an activity pair

= inf has anchored regular derivative budget has forced hit budget (LCN1) The regular budget tests every left derivative through’s order, in its permitted directions, on the healthy domain of that address, with its exponential support weight. The forced hit budget takes the

absolute bound on every configuration and sums over addresses hitting each site. Both include the fixed locality and invariance conditions. The infimum is attained: take the least supremum envelope for each regular derivative family and each forced coefficient, and pass every finite nonnegative partial-sum inequality to the infimum. Hence is equivalent to the two displayed budgets with that exact. Triangle inequalities for derivatives and coefficients give subadditivity; scaling gives

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Fix, for each signed translation, one permutation implementing that symmetry on the layout.
The activity pairs obeying all these fixed permutations form a linear subspace HP . Its seminorm is
(LCN1). Use its separated completion bHP ; the map from the original presentation is an isometry
after identifying seminorm-zero differences. This retains the off-mask regular germs required by
the derivative budget, rather than identifying them merely by their evaluated values outside the
healthy domain. The scalar-plus-activity space has norm ∥(λ, A)∥= max(|λ|, ∥A∥P ). The actual
retained activities have norm at most 1/8 in these coordinates.

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We explain how the local primitive estimates give an analytic estimate in this norm. For a
finitely supported signed measure w = P
j cjδtj on [−1, 1], say it has moment price r at radius
p ≥1 if


X
≤r
for every n ≥0.
(LCN2)

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j
cj(tj/p)n

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A family f(t) has comparison price C at radius p if every such measure satisfies ∥P
j cjf(tj)∥≤rC.
This is the precise signed-combination bound used here. If f(t) = P
n(t/p)nan with P
n ∥an∥≤C,
the bound follows by interchanging a finite sum and an absolutely convergent series. Conversely,
define the moment vector ((t/p)n)n in l∞. The signed-combination bound defines a bounded linear
map on their span and then its closure. Coordinate vectors belong to that closure: subtract the
already recovered lower coordinates from the moment vector at a small nonzero t, divide by (t/p)n,
and let t →0. Its image supplies the coefficients. Restricting from radius p to p′ < p sums the
geometric coefficient majorant and gives an absolutely convergent series. This is the recovery step
used to turn signed-combination bounds into analytic line estimates.

On an affine activity line, regular occurrences and forced coefficients are affine in on the same configuration: their masks and inventories are fixed. The finite calibrated coupling along the line is, with1. The comparison center has Ω 40], and the climate bounds ensure the whole line stays in the allowed coupling band. Assume1 for1. The primitive’s signed regular and forced estimates, followed by the Hessian projection and the fixed dictionary fusions, give prices

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output activity: d at radius q/2,
scalar correction: fi at radius q,
fi = choresi /θ2
i . (LCN3)

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The regular signed bound is tested in the anchored derivative budget. The forced signed bound
has one nonnegative summable hit-envelope common to all line parameters and configurations. For
any weights satisfying (LCN2), these two envelopes give budgets rd bP ′, rd kP ′ on the actual output
layout, so (LCN1) gives the activity price in (LCN3). The scalar price follows by applying the
linear Hessian reading to the same signed regular source estimate and dividing its paid bound by
θ2
i . This explains both the norm and the threshold denominator in (LCN3). The primitive locality,
derivative and signed-sum bounds used in this step are the marked integral and source projection
estimates of the preceding chapters; no assertion about analytic dependence of a chosen forest or
local guide is required.

We make explicit the gain that pays (LCN3), since an individual bound on each line value would not imply the signed-combination bound. Use the 301 distinct interpolation nodes

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0 ≤j ≤300, in [−1, 1], and their Lagrange polynomials Lj. If w has moment price η at radius q,
let cj = P
t wtLj(t) and w0 = P
j cjδrj. The coefficient bounds of these finitely many polynomials
give P
j |cj| ≤Cqη for one fixed Cq ≥0. Every polynomial of degree at most 300 is interpolated
exactly, so w −w0 annihilates its moments through that degree. If V ≥q, its remaining moment
prices at radius V are bounded by η(1 + Cq)(q/V )301: factor (q/V )n from the degree-n moment
and use n ≥301. Thus a family with signed price B at radius V and individual value bound b has
signed price at radius q at most

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Cqb + (1 + Cq)(q/V )301B.
(LCN8)

This is valid for the regular derivative budget as well as a scalar norm, because finite sums, absolute scalar multiplication and envelope sums preserve that budget.

We derive the preliminary large-radius signed price used in (LCN8). Fix the stage, block ratio and address layout. Write), and let 0 be the fixed turning price. We first explain the parameter estimate on each vertex. A function polynomial in the line parameter, of degree at most two, is reconstructed exactly from its values at

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f(t) = 1
2t(t −1)f(−1) + (1 −t2)f(0) + 1
2t(t + 1)f(1).
(LCN11)

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The sum of the absolute coefficients of these three Lagrange polynomials is 4. If each nodal function
has all required spatial derivatives bounded by vdm at derivative order m, (LCN2), applied to each
parameter power, bounds its signed interpolation by 4p2vdm at radius p ≥1. The same identity
holds after spatial differentiation: its parameter coefficients are fixed scalars. Thus this is a signed
derivative bound, with no factor P
t |wt|.
Logarithmic vertices enter as ef(t) −1, rather than f(t). If their nodal derivative budget is
at most v ≤1/2, first recover the budget of f using f = log(1 + (ef −1)).
The scalar value

bound is; all derivatives of log(1 +) through the fixed order are bounded on [ Its finite chain-rule sums therefore give a bound with the same spatial derivative scale, for a constant depending only on that order. Apply (LCN11), and then the absolutely convergent exponential series in the signed moment norm. Multiplication in that norm multiplies prices, so a log price1 gives; the differentiated exponential is controlled by the finite chain-rule sums at the same scale. Consequently there are constants1 such that the signed price of this vertex is at most, provided1. These are the estimates FullVolume/Ray.Badge.starpath and White.faded, applied at the three displayed nodes.

They apply to the actual vertices because the line dependence is placed in the couplings and occurrence amplitudes, not in a newly chosen forest. With the spatial configuration held fixed, an old vacuum logarithm is constant or is multiplied by the square of the affine coupling-amplitude line, hence is quadratic in. A regular occurrence has the prescribed degree-at-most-two amplitude representation on this line; multiplication by its fixed local enforcement factor preserves that degree. The forced occurrence has the same representation. Warning vertices, including their common steering and course variables, have the retained parameter-independent derivative envelope

and therefore satisfy the constant-function signed estimate. In the differentiated formula the background, steering and Gaussian-coordinate variables are varied together as specified in the common chart. The spatial chain-rule prices are included in the nodal bounds. These are the four cases in FullVolume/Sector/Grant.mural, lured, loglift and alerts. In particular polynomial interpolation is applied to the log before exponentiation, not to the exponential.

The preliminary signed price now follows by the connected expansion with these vertex prices. Three nodal log envelopes sum to at most 3, where the old target has rate at most , their enlarged price has rate at most

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−7/16 + 2/256 = −55/128 < −2/5.

Thus both 32 and1 hold eventually, and the reweighted regular and log prices fit the target of rate5. This is the explicit target enlargement in FullVolume/Sector/Reign.shows. The finite-order product rules multiply these prices at each connected tree. The support exponent exceeds the final exponent by more than three; split this excess into three positive reserves to pay the tree-degree factors, collar enlargement and log attachments. The positive tree sums have ratios at most 12, hence cost at most 2 each. All remaining fixed-order spatial and collar prices have rate zero. Multiplying them by the enlarged forecast of rate5 is eventually at most Restricting the moment radius to2 improves the bound. The forced entries retain a common nonnegative hit envelope by the same positive majorants. This establishes the preliminary signed estimate for every finite combination satisfying (LCN2); it does not infer it from individual activity bounds.

Here are the product and summation steps in that signed estimate. A radius- series certificate means an actual representation on [

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f(t) =
X
n≥0
(t/R)nbn,
X
n
∥bn∥≤A,
R ≥1.
(ASP1)

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For a finite signed combination w = (wt), put Λwf = P
t wtf(t). If |P
t wt(t/R)n| ≤η for every
n, absolute convergence gives ∥Λwf∥≤ηA. This estimate uses cancellation of the combination;
replacing wt by |wt| would lose it.
Products preserve (ASP1) with the product allowance, by
the absolutely summable Cauchy product.
Summable families preserve it by the double-series
theorem. Exponentiation gives allowance eA−1 for ef −1, by summing the positive scalar majorant
P
q≥1Aq/q!.

For the simultaneous spatial derivative estimates, fix a base coarse field. Suppose each diagonal derivative through order, in a unit spatial direction, has (ASP1) allowance at order another factor has allowance, the derivative of their product is

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q
r
!
q
X
Dr
vf Dq−r
v
g.
r=0

The product rule for the series certificates bounds its allowance by

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q
r
!
q
X
AbrCdq−r = AC(b + d)q.
(ASP2)
r=0

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Induction gives, for a finite product of vertices, allowance Q Ai and slope P bi. Thus the signed
cancellation is retained before multiplying the vertices. After applying Λw, polarization bounds the
full symmetric q-linear derivative by polarCost(q)ηAbq. Taking P
q≤j polarCost(q) as one common
constant controls all orders through j. Finite linearity commutes Λw with these derivatives.
Here the polarization constant used by the source is explicitly

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j
X
Pq = 2q(q + 1)q,
P≤j =
q=0
Pq.
(ASP2a)

For a symmetric-linear form and vectors1, expand

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X
!
X
i∈S
vi, . . . ,
X
i∈S
vi
S⊆{1,...,q}
(−1)q−|S|A
= q! A(v1, . . . , vq).

Every term whose list omits an index cancels by the alternating subset sum; a surviving list contains every index exactly once, giving! identical terms. If the unit diagonal bound is, each subset term is bounded by (, by homogeneity. There are 2 terms and!1. This proves the stated constant, including = 0, and normalization of the nonzero vectors proves the operator norm bound. Applying it after the signed parameter combination gives the allowance at all orders through. It uses no bound depending on the dimension of the spatial chart.

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The Gaussian integration prices entering these allowances also have a direct finite-dimensional
proof. Let γΣ be the centered Gaussian actually used for a packet, with Σ ≥0 and |Σii| ≤B. The
one-coordinate marginal gives
Z
etzi dγΣ = et2Σii/2 ≤eBt2/2.

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Thus (1 + |zi|)q ≤eqzi + e−qzi implies
R (1 + |zi|)q dγΣ ≤2eBq2/2. Independence of the coordinates
is unnecessary. For a local edge set I, let J be its colored coordinate set, a = |I|, b = |J|, and

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FI(z) = 1 + a + b +
X
i∈J
|zi|,
Y = 1 + a + 2b.

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Choose a coordinate attaining the maximum when J ̸= ∅; then FI(z) ≤Y (1 + |zi|). The empty
set case follows directly. Consequently
Z
FI(z)q dγΣ ≤Y q{1 + 2beBq2/2}.
(ASP2b)

This proves integrability as well as the bound, by domination by a finite sum of the two exponential moments. A tilt changes only the marginal on. Its covariance Σ is positive definite. With1, its exact density multiplier is

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1 −c2
!
Mc(z) = cb exp
2
zT
J Σ−1
J zJ
,
0 ≤Mc(z) ≤cb,
FI(cz) ≤cFI(z).
(ASP2c)

This is obtained by the substitution in the Gaussian density; the determinant contributes. The quadratic form is nonnegative, proving the bound. The remaining scalp factor and this multiplier combine exactly into the original scalp times times the exponential of the tilt’s quadratic cost. Applying the product and exponential estimates to that equality gives derivative price charge(, with spatial slope bounded by. Formula (ASP2b) pays its moments.

Finally a warning event with probability at most and a polynomial factor with second moment at most contribute at most, by Cauchy–Schwarz. Haar variables have total mass one and do not change this bound. All polynomial moment prices are included by taking twice the required derivative degree. If the colored coordinate count is bounded by

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2b ≤2 covB(Y ) ballCost(B) density(L) dim g

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and the tilt margin satisfies 2δ ballCost(B)density(L) dim g ≤α, then log(1+δ) ≤δ gives (1+δ)2b ≤
eαcovB(Y ). This is precisely the reserved cover weight; the Gaussian determinant factor is paid before
the final summation.
The interpolation step preceding this argument supplies those certificates at the real nodes.
Explicitly, at k + 1 distinct nodes ti, let li be the Lagrange basis and Bk = P
i
Pk
r=0 |[tr]li|. A

degree- family with nodal derivative allowance has (ASP1) allowance: expand and bound. For optional exponential vertices the nodal hypothesis is on1. When its allowance is at most 12, log(1 +) has bounded derivatives on [2] and log(1 + . The finite chain rule therefore bounds the nodal logarithm through the same spatial order. Interpolate this logarithm, and only then exponentiate by (ASP1). The condition with its fixed derivative constants included in, pays this last exponential step. This explains why the source can use polynomial dependence of the log without asserting polynomial dependence of the exponential.

The remaining tree summation is positive. If vertex prices have hit allowance at support exponent 0, the rooted forest estimate of the connected expansion gives for the terms with nonroot vertices the bound

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eδ covB(Y )
4 ballCost(B)γ
m
.
(ASP3)
δ

This estimate is proved by summing the leaf label at its first overlap with its parent: a radius- cover of the parent has at most its cover number times ballCost() possible hit sites. The hit bound pays that sum; the reserve pays the parent cover factors and the enumeration of parent assignments. The rooted occurrence enumeration and factorial weights give the factor 4

already included in (ASP3), as in the absolute tree estimate used earlier. A designated marked root instead pays its own weighted allowance, multiplied by the same geometric factor. For ratio at most 12, summing0 costs at most two. Polynomial factors from the spatial direction growth are paid by the collar reserve:

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collarj,p = (shift(x))pj sup
r≥0
rpje−δr.
(ASP4)

The supremum is finite (with the usual value one when = 0); it is just the elementary absorption of a fixed polynomial by a positive exponential. The shift is bounded by a fixed multiple of the local support width plus one.

The packet’s additional factor (1 +1, is also paid on the whole packet. The number of relevant off-tree color coordinates satisfies shares() Delay(, by the link census and the 10 visit bound. Since 1 +, and any fixed positive derivative constant is at most its exponential, these factors are absorbed by increasing the coefficient of Delay( shares( in the packet income. They do not change its gain exponent or replace a whole-packet cost by a single-site hit.

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In the final scheduling set α = (Qexp −e)/4 > 0, where e < Qexp is the requested output
exponent. Then e + 3α ≤Qexp. Spend the three α’s separately in the tree bound, collar bound
and attached-log bound. The enlarged root price has rate −2/5; the degree ratio in (ASP3) has
that rate times a width factor of rate zero, so it is eventually at most 1/2. The fixed derivative and
collar factors also have rate zero. Their product with the root price is eventually below bβ−3/8, since
−2/5 < −3/8. The parameter radius is R = bβ1/256, eventually at least two; restrict to R/2 for the
final bound. These are the simultaneous choices giving the common nonnegative hit envelopes for
the signed forced entries and the series certificates for the attached logs.

Retirement selects a fixed subset of this same nonnegative majorant. Adding the boundary correction and turning operation gives the common signed price

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w(x) = A{y−3/8 + 4y−7/16}.

Every reference torus and comparison ball in the graft has size polynomial in) at a fixed stage. Their site counts) and), and their derivative factors, have rate zero. Summing the