Convergence of the interacting scale histories · Part 8

two compared anchored expressions over a reference torus gives signed price Summing the unrestricted value bound over the comparison ball gives price). Both retain rate Let be the graft radius, letbound the cover number of its radius- support, and let be the fixed finite interpolation weights. If the derivative order and spatial derivative power of the budget are and, put

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Tz = 2j(1 + z)pj,
O(x) =
X
i
|λi|Tzpi(x).

For a signed measure with moment price at radius2, the finite interpolation derivative formula gives beacon envelope) at each site. There is one anchored beacon per site. Its support weight therefore gives anchored budget at most. The retired part has budget ). Since the regular graft is linear in these parts, the complete regular signed price is

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w(x) + AO(x)e(supportExponent +3)c∗.
(LCN10)

The cover boundis fixed before, and all radius factors have rate zero. Thus (LCN10) has rate8, and is eventually at most. This is the rough price at radius2 required in (LCN8). It applies to the deterministic source because the retirement index depends only on the layout and the matching identities identify the actual source with this graft.

The forced half uses separate all-configuration envelopes. Its primitive entries retain their exponential flag suppression. The attached local insight entries have a signed scalar price obtained from the comparison-ball and reference-torus value prices, multiplied by their ball count and support weight. Denote the resulting primitive and attached hit prices by and. The delay purchases 1, and

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γ = eκa + 2b,
(2B + 1)4γ ≤1,
e−κ(eγ −1) ≤forecastK .

The local contribution has rate8 and is eventually below the comparison ceiling. The forced forecast uses the primitive exponential suppression as well; it does not follow from comparing two power rates. To fuse the two families, inflate each old forced atom by, bound each attached exponential minus one by twice its logarithm envelope, and sum the connected components using the displayed support reserve. Each component contains an old forced atom, so undoing the inflation saves at least. The resulting hit price is at most1), as proved in the attached-log fusion argument. This gives the forced forecast at radius4, with one supportexponent unit reserved. Shrinking later to2 improves the moment inequalities and supplies the common forced envelope in (LCN3).

The sharper individual price in (LCN9) comes from universal retirement. Its remaining beacon envelope is a finite sum, with fixed coefficients, of

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gold(x),
Ni(x) gold(x),
2j(1 + z)pjNi(x) gold(x),

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where gold(x) = y−23/20. Every multiplier has rate zero. The fixed support exponential and turning
price preserve rate −23/20, proving the claimed rate of εj. Thus finite interpolation combines a
large-radius signed estimate and a sharper real individual estimate. For the primitive source, the
individual bound b is its regular price (FCC1) plus the beacon contribution. With x = bβi, it has
form
b = 3t Cjx−11/10 + εj(x),
εj has rate −23/20,
(LCN9)

where 0 is the freely chosen small primitive allowance. The three principal payments are the coefficient. The other terms are the turning fee times the retired tail, the tail itself, and the

beacon price times its fixed support exponential. The local estimates give them the stated rate at fixed block and stage. The larger signed radius is2, where fire( the rough signed price has rate4. Consequently the last term of (LCN8) has rate at most 20. Choose so that 6, and then the coupling-entry delay so that the two remaining terms cost at most (. Equations (LCN8)–(LCN9) give signed regular source price at most = chores. This is the numerator of the scalar price

The Hessian reading of this source yields the scalar correction price by linearity and its bound ∆Budget. Applying the regular reset price to that same signed source, including the committed coefficient bound log, gives regular output price once the small allowance and delay have been chosen. The forced reset entries use their common all-configuration envelopes; the forced and attached-log fusion estimates retain one unit of support-weight reserve and yield hit price at radius2. The expenses and the forced cutoffare the paid local reset inequalities, not a consequence of the scalar reading alone. Combining these two prices through

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(LCN1) proves the activity assertion in (LCN3). Thus the analytic recovery uses a signed estimate
obtained by a finite interpolation cancellation, with the exact exponent 301, rather than assuming
analyticity of arbitrarily selected primitive witnesses. In fact the output values are independent of
such incidental choices. To compare two representatives with identical coupling and activity data,
use the same constant affine input line but select the first representative at t = 0 and the second
at t ̸= 0. The bounds (LCN3) still hold. For q ≥4, the signed measure δt −δ0 has moment price at
most |t| at either radius. Therefore its two output differences are bounded by d|t| and fi|t|; letting
t →0 proves equality. This supplies the single-valued activity and scalar maps on the common
normed presentation.
Choose q ≥4 so large that

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2e
(1/4)(q/4) ≤1,
2d ≤γ = 1/128.
(LCN4)

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At any center (λ, A) with Ω−1/32λ ∈[17/20, 23/20] and ∥A∥P ≤1/8, every neighboring center
within distance 1/8, followed by any direction of norm at most 1/4, stays in the enlarged scalar
window and activity ball used in (LCN3). Recover the line series at radius q/4. Polarization makes
its degree-n coefficient a symmetric multilinear map: the finite difference formula divides by n!, has
2n summands, and each sum of n unit vectors has norm at most n. Since nn/n! ≤en, the resulting
multilinear coefficient is bounded by

2e
(1/4)(q/4)
n
C,
C = 2d or 2fi.
(LCN5)

The formulas agree on overlaps because they recover the same scalar line coefficients. They extend through the isometric dense inclusion into the completion, preserving their operator norms. By (LCN4), the activity coefficients are bounded by, and the scalar coefficients by 2, at every order. For increments of norm at most 14, telescope each multilinear diagonal and sum:

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X
n≥1
n(1/4)n−1∥z −z′∥C = 16
9 C∥z −z′∥≤2C∥z −z′∥.
(LCN6)

Thus the local Lipschitz constants are 2 for activities and 4 for the scalar correction. To get the estimates for any two centers in the whole stated window and activity ball, split into distances at most 14, where (LCN6) applies, and distances greater than 14. In the latter case (LCN3) applied

to singleton measures bounds the two values separately by and; their differences are at most and 2. Both cases imply

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∥Φi(y) −Φi(x)∥≤8γ∥y −x∥,
|∆i(y) −∆i(x)| ≤8fi∥y −x∥.
(LCN7)

With, this is exactly the scalar constant 4 and activity constant 8 used in (FCB). All prices are for the fixed step of the common calendar and are uniform in the actual cutoffand period when its paid stage and support margins hold. The local analytic constants are chosen before the final history is selected; the completion introduces no new choice of primitive data. The common root envelope in this argument uses the energy gain proved before the expansion, in its actual order of quantifiers. For every sufficiently large odd block side, all sufficiently large fixed window widths admit a number 0 and a coupling threshold such that

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κ tsf(β, L)2 nseed ≤Eassessed
(ASP5)

for every admitted tube, sector, tree and packet store above that threshold. Here is its number of charged seeds, its small-field threshold and the sum of the actual fine, coarse and mismatch departure costs. The assignment of those costs to packet cores, the paid annuli and the three clipping ranges was proved in the energy chapter. Formula (ASP5) is a per-seed energy statement; the expansion additionally uses the support census, normalization expenses and reserves (ASP3)–(ASP4).

For adjacent scheduled profiles use the same angular covers for both copies of (ASP5). They provide one positive coefficient for each of the two requests; replace both by their smaller value and increase the common coupling threshold. The geometric block factor is fixed before these choices. The resulting calendar delays are chosen above both thresholds, and its actual coupling update identifies the next small-field threshold with the one in (ASP5). Every admitted step therefore has the claimed bound with the same charged seeds and the same angular read sets. The pointwise domination obtained this way gives one nonnegative envelope for all the signed entries, including when a parameter path changes which sector is active. Combining it with the tree and collar sums proves the all-root envelope used above.

Stability of finite calibration and its comparison weights

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The retained forward recipe also gives quantitative stability of the complete finite trajectory under
changes of its bottom coupling. This is the finite statement behind the auxiliary trajectories in the
uniqueness argument. Write xi = bβi and define

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p
βi/xi
wi = x−1/32
i
,
bi = xiwi = x31/32
i
,
λi =
wi
,
βi = xi(wiλi)2.
(FCB1)

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The positive square root is used. By (FCA8), wiλi ∈[17/20, 23/20]; this wider interval contains
[
p
3/4,
p
5/4]. The activity coordinate Xi is its image in the common completed activity chart.
At a fixed finite horizon all these coordinates live in one finite product of their scalar and activity
spaces.
Let γ = 1/128. The actual local analytic comparison, including the initial state at N, bounds
differences of activity outputs by 8γ times the maximum input-coordinate difference. Its scalar
correction output has Lipschitz bound 4si times that maximum. The weights are explicitly

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si = 2 choresi
θ2
i
,
choresi = v Cstage(i)x−11/10
i
,
(FCB2)

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where v > 0 is the fixed comparison cost and Cj > 0 the stage’s local analytic price. The common
small-field threshold is θi = x−1/2
i
exp((log log xi)2). Thus si ≤2vCstage(i)x−1/10
i
. These comparison
bounds apply to the same retained primitive operation at both points, as ensured by its fixed layout
and choices.
The calendar pays the cumulative weight inequality
X

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j<i
sj ≤γbi,
b0 ≤bi.
(FCB3)

Here is its derivation from delays. At a fixed stage transition the positive reference increment is log 0. The mean value theorem, for, gives

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(x + a)31/32 −x31/32 ≥a 31
32 2−1/32x−1/32.

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Since x−1/10/x−1/32 →0, eventually the upper bound in (FCB2) is at most γ times this increment.
Include this one eventual threshold in the coupling-entry delay for each of the two allowed adjacent-
stage possibilities. It follows at every realized step that si ≤γ(bi+1−bi). Summing gives P
j<i sj ≤
γ(bi −b0) ≤γbi. Positivity and monotonicity of xi give b0 ≤bi. Thus (FCB3) is a consequence of
the same schedule, not a supplementary assumption on a chosen pair of trajectories.
Compare two successful trajectories of the deterministic recipe, with bottom-coupling difference
δ. Let
U = max
0≤i≤N max{|λ′
i −λi|, ∥X′
i −Xi∥}.

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The local comparisons give ∥X′
i −Xi∥≤8γU, including the finest-state comparison, and correction
differences at most 4siU. Subtracting the finite coupling recursions cancels their common path
coefficients:
β′
i −βi = δ −
X
j<i
(∆′
j −∆j).
(FCB4)

On the other hand (FCB1) factors their difference exactly as

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β′
i −βi = bi(λ′
i −λi)(wiλ′
i + wiλi).

The last factor is at least 17 1. Equations (FCB3)–(FCB4) imply

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|λ′
i −λi| ≤|δ|
bi
+ 4γU ≤|δ|
b0
+ 4γU.

Taking the maximum over all levels and both coordinates proves

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U ≤|δ|
b0
+ 8γU,
U ≤2|δ|/b0.
(FCB5)

Only finite products and finite sums occur at this stage. If = 0, (FCB5) gives identical finest normalized couplings, hence identical finest actual couplings. Their startup states are equal by their fixed startup rule. Descending determinism of the primitive recipe then gives equality of all represented states. Thus the successful finite trajectory with a prescribed bottom value is unique within this retained recipe, although monotonicity of a scalar coarse readout was unnecessary for existence.

The arcs prove existence for every bottom value in the nearby interval. Combining them with (FCB5) gives continuous dependence of its normalized scalar and activity coordinates on that value,

on the same finite calendar. For example, choose a bottom perturbationb, with the fixed small comparison radius 0, and small enough to stay in [3 ̄ ̄4]. Then2, placing the entire perturbed trajectory inside the common analytic comparison neighborhood. This constructs the nearby finite trajectories used in passing local comparison estimates to banked limits. Equations (FCB1)–(FCB4) also identify explicitly the positive scalar weight and summable correction weights in that argument.

Uniform running of the actual coupling

The drift estimate also proves the uniform running bound needed in the public theorem. Define

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D0 = 0,
Di+1 = Di + di+1.
(RUN1)

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For the calibrated finite history, (FCA3), βN,p,0 =  ̄β, and the exact recursion for bβi imply

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|βN,p,i −bβi| ≤Di
(i ≤N).
(RUN2)

Indeed subtract the two recursions and apply the triangle inequality at each step. The right side depends only on the calendar, so this bound is uniform in both and The stronger asymptotic drift condition of that calendar is

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di+1
log Li
≤J(i + 1)−2,
J(i) −→∞.
(RUN3)

Here) is the stage index, not a cutoffor period. Given 0, choose so that for every. Summing (RUN3) gives, for

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Dm ≤Dj + ε
2(τm −τj).

Since, eventually. Together with (RUN2) this proves

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∀ε > 0 ∃m0 ∀m ≥m0 ∀N ≥m ∀p,
|βN,p,m −bβm| ≤ετm.
(RUN4)

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This is uniform control of the actual finite-cutoffcoupling, including m = N. It is stronger than
merely writing down the reference line bβm =  ̄β + sτm.
Dividing by τm > 0, and then absorbing the fixed intercept  ̄β/τm, yields

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βN,p,m
τm
−s
 −→0.
(RUN5)
sup
N≥m, p

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Explicitly apply (RUN4) with error ε/4, and take τm ≥4(| ̄β| + 1)/ε; the triangle inequality gives
an error smaller than ε. Taking m = N proves the bare coupling slope in the theorem, uniformly
in the physical period.
Set b0 = 11CA/(48π2), so 2b0 = s. For arbitrary cutoffand period sequences Ni ≥i eventually
and pi, let gi = β−1/2
Ni,pi,i. Positivity of the coupling and (RUN5), followed by continuity of the square
root and reciprocal at s > 0, give

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gi
q
log(a−1
i ) −→(2b0)−1/2.
(RUN6)

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The equality used here is β−1/2√τ = (β/τ)−1/2 for β, τ > 0. This regulator coupling is defined
from the inverse action coefficient. Its numerical leading logarithm does not by itself identify it
with a coupling defined by a physical position-space observable. That separate identification, and
its normalization, are proved in the trace and Ward calculation.

Comparing omitted interactions at a common normalization

The fine-to-intermediate comparison uses one fixed denominator throughout. If is the finest level and is the period, define, for a polynomial-valued integrable insertion at level

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R [P(U)]t dμm(U)
Im(P; t) =
ZN,p
,

where is a polynomial mode, is the product Haar prior, and 0 is the original finest-level Wilson partition function. In particular the denominator is not replaced by a new partition function when changes. The root estimate, for insertion weights

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|Ik(P; t)| ≤Bk(t) := exp{|Λk| τk + |V |(x + y + 2)}
toll(w, t)
.

Here is the vacuum treasury, (x,y) bound the two insertion envelopes, and the positive toll is the coefficient-extraction weight. This bound follows by applying the small-field integral bound to the represented density, bounding its vacuum bill by the treasury, and using the exact identity between the integrated represented density and

At an intermediate level, separate the two omitted interaction polynomials from the retained insertion data. Multiply both omitted parts by a complex parameter, restore the resulting density, and call it (P(z)). Then (P(1)) is the full insertion and (P(0)) is the insertion with those parts omitted. The paid analytic estimate makes

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Y (z) = Ij(P(z); t)

holomorphic for, where = brisk( 2. For every such, transport (P(z)) through the actual retained Mark steps from to. The retention estimate preserves the served-insertion bounds with root weights; integration of the transported polynomial equals integration of the original polynomial. Coefficient extraction commutes with this integral because it is a continuous linear map on the finite-dimensional polynomial space. The denominator remains, so the root estimate gives) uniformly throughout this disk.

For any holomorphic bounded by on 2, every disk of radius (S/2) centered on the segment ([0,1]) lies inside that disk. Cauchy’s derivative estimate gives on the segment. Integrating along the segment proves

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|Y (1) −Y (0)| ≤2B
S .

Consequently the omitted interactions change the normalized coefficient by at most 2 The root cost depends on, not on the intermediate volume at The analytic paid bound, the retained transport estimate, and the root small-field bound are the substantive inputs to this comparison and must be established below.

The holomorphy under the integral in this comparison can be justified directly. The polynomial algebra has finitely many modes; its truncated exponential is a finite sum of powers. For a fixed configuration the restored polynomial is a finite sum of products of this truncated exponential and the local field factors. Its masks and flags depend on the configuration, not on. Replacing the omitted terms by therefore makes each mode an entire function of. All masks are measurable: a prescribed flag pattern is a measurable event, and restoration is a finite sum of products on these events.

The weighted polynomial norm is equivalent to the ordinary finite-dimensional coefficient norm when all insertion weights are positive. The served-insertion estimate for the scaled omitted terms,

together with the envelope bounds and nonnegative coupling, gives a single finite constant such that

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∥P(z, U)∥≤Y
(|z| < Sj, U ∈Cj).

Since the Haar prior is a probability measure, this is an integrable majorant. It proves integrability at every. Dominated integration of Cauchy’s formula on small circles, or equivalently Morera’s theorem with Fubini, proves holomorphy of each integrated coefficient. Thus no differentiability of the configuration masks is needed. The served-insertion envelope proved in the marked-integration chapter gives this uniform majorant.

The retained transport statement has a descending induction with no limiting step. At the top, scaling the omitted parts spends two units of support exponent. Suppose the transported polynomial has the required served bound at (m+1). Its first-degree coefficient is the prescribed insertion manifest there, by the relay identity. Integration through the Mark kernel gives the prescribed manifest at, by the insertion feed identity. The parent map sends the chosen pins at (m+1) to those at. The matching assumptions supply their occupancy bound and the allowed reweighting of the insertion weights; the wake data supply the local finite transport tales and their estimates. The one-step duty estimate now gives the served bound at with the full support exponent at that level. This proves the assertion down to. Integrability at each stage follows from integrability of the Mark push. This induction explains why the two-unit expenditure occurs only at the top; it does not accumulate once per step. The finite inventory and one-step duty estimates in the marked-integration chapter establish the inputs to this induction.

Comparing the varying Gaussian integrals

A change of protected variables also changes the Gaussian law. If ) has Jacobian the general density ratio is

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JV (z) ρCV (z′(z))
ρC0(z)
.

For positive covariances on the same-dimensional real space this is

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JV (z)
 det C0
1/2
exp

−1
2z′TC−1
V z′ + 1
2zTC−1
0 z

.
det CV

The fixed-covariance expression follows only when. The determinant and inverse derivatives in the protected-variable estimates below are precisely what control the genuinely varying case.

The fluctuation space at a fixed step is one finite-dimensional Euclidean space of dimension A prepared datum has Gaussian density), mean displacement in physical coordinates, and scale 0. Use the common physical variable. Then

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z = g−1(w −tX),
cX(w) = g−DρC(g−1(w −tX)),

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and for every integrable vector-valued function A,
Z
A(z) dνC(z) =
Z
cX(w)A(g−1(w −tX)) dw.

This is ordinary translation invariance of Lebesgue measure followed by scalar change of variables, including its determinant. It also proves integrability of the transformed integrand.

The common local covariance bounds give with 0 independent of the history index and gauge-slice variable. The protected mean gives: sum its finitely many squared coordinate bounds and take the square root. The coupling window gives fixed . For these imply

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1 + ∥r(w −tX)∥≤(1 + v + vB0)(1 + ∥w∥),
e−τ∥r(w−tX)∥2 ≤eτu2B2
0e−τu2∥w∥2/2.

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The second inequality uses ∥w∥2 ≤2∥w−tX∥2+2B2
0. For any fixed set of tested bonds its face-scale
function is at least one and at most K0(1+∥z∥), since each coordinate is bounded by the Euclidean
norm. Consequently every transformed integrand whose original norm is bounded by a fixed power
of that face scale has a common majorant

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C1e−τu2∥w∥2/2(1 + ∥w∥)k.

This is Lebesgue integrable: in finite dimension Gaussian decay dominates any polynomial. Its constant is independent of the gauge-slice variable.

The integral convergence argument is applied twice. At each gauge-slice point, the common majorant and pointwise convergence of the difference of the two transformed integrands give convergence of their Lebesgue integrals by dominated convergence. Fubini identifies these with their original Gaussian integrals for almost every gauge-slice point, simultaneously for the countably many history indices. Their norms are bounded by the integrals of the two majorants. The outer gauge-slice measure is a probability measure, so this constant is integrable. A second dominated-convergence application proves convergence of the full Gaussian–gauge integrals.

For ordinary primitive derivatives, differentiation under the integral identifies the derivative of the local activity with the integral of its differentiated integrand. Its paid bound is

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C
1 + (p facescaleK + 1)

j ≤C(p + 2)j facescaleKj,

because the face scale is at least one. Thus it has the required polynomial majorant. Marked primitive derivatives have the same form of estimate from their common marked budget. The undifferentiated marked amplitude has a uniform constant bound, which is the case

Pointwise comparison of the transformed integrands is taken with one common amended draft. At each fixed gauge-slice point, the draft data have only finitely many discrete possibilities; an ultrafilter selects one possibility eventually. The same viability, threshold and choice data are then used in both histories. Their density factors and local vertex products compare in these common physical coordinates; products retain both real and imaginary marked ingredients. The preceding two-stage integral argument converts this comparison into convergence of ordinary derivatives, marked derivatives and marked amplitudes. The finite reserve products and boundary-choice averages in the actual primitive definitions preserve it. Their healthy-support local identities identify these formulas with the actual leaflet and marked-grove germs near the tested base configuration. This is the Gaussian integral passage used in the single-stream comparison, with all discrete masks retained inside the common draft.

Common drafts and Gaussian density factors

The finite choice used in the preceding integral argument is precise. A draft records a gauge-slice variable, a store and a boundary choice, together with the fixed selector, scale and width. After fixing the gauge-slice integration variable, the paid conditions fix the latter three parameters. Only