Marked integration and the coupling reset · Part 5

The distinct connected-set tree estimate then bounds the sum of repeated connected terms whose union contains a fixed site by), provided the inflated vertex activities satisfy the readiness condition with budget and tree ratio 1. The actual union support weight is paid by the product of its vertex support weights: connected cover subadditivity applies to and each multiplicity1 only increases the available exponent sum. Nonnegative summation justifies all these successive sums.

For omitted-cluster restoration take two kinds of atoms. Forced atoms have nonnegative weights , nonempty supports, and sitewise budget at exponent Regular-log atoms are occurrence-indexed clusters, with their absolute cluster weights and union supports. The preceding logarithmic tail estimate gives their sitewise budget) at that exponent. Their combined repeated products bound the absolute value of every signed restoration product; forgetting original exclusions and retaining additional repeated forced atoms only enlarges the upper bound.

To obtain both a size tail and a compulsory-forced-atom gain, choose1 and0. Multiply every atom weight by, and every forced atom weight additionally by. After the geometric multiplicity allowance, the combined sitewise budget is at most

Searchable transcription

teμc
1 −σ +
tγ
(1 −σ)(1 −q).

Require this to be at most, with 0 0, and 4(2 Apply the preceding repeated connected estimate to these augmented activities. A term of total size acquires the factor. If it contains a forced atom, the exponent summed with multiplicities acquires at least, so its additional factor is at least. Dividing the augmented bound by these factors proves

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X
eAcB(∪i∈DXi)Wh(D, r) ≤t−me−μ
η
1 −q′ .

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connected (D,r), N(D,r)≥m
at least one forced atom
x∈∪i∈DXi

This is the omitted-cluster restoration bound used for the forced paid terms. Its applications use2, and = 0 the right side becomes 2. The compulsory atom is charged at least once, regardless of its multiplicity. The remaining input in this argument is the distinct connected-set tree estimate under the stated readiness condition, which belongs to the same rooted-tree bounds as the regular cluster estimate.

Switching real source formulas into the marker algebra

The nilpotent substitution in the marked expansion is defined through real switching values, not by a formal differentiation of an unbounded series. For a finite switch set, a real coefficient array , gives switch values

Searchable transcription

f(x, R) =
X
A⊆R
x|A|gx(A),
0 ≤x ≤1.

At 0 the array is uniquely determined by these values. Induct on (|R|): subtract all propersubset terms from (f(x,R)), then divide by 0. Equivalently the inverse is the finite Möbius formula

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gx(A) = x−|A| X
R⊆A
(−1)|A|−|R|f(x, R).

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A model for f is such an array with each gx(A) →g0(A) as x ↓0.
The switching jet is
Jf(A) = g0(A), independently of the model by uniqueness for x > 0.
A paid model includes
nonnegative bounds (b(A)) with |gx(A)| ≤b(A) for 0 ≤x ≤1 and P
A b(A) ≤c, uniformly over
any additional configurations. In particular |f(x, R)| ≤c on that interval.
For square-zero elements zi of the finite marker algebra, define

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Sz(f) =
X
A⊆I
Jf(A)
Y
i∈A
zi.

This substitution is additive on modeled families. To prove multiplicativity, multiply the two switch-value formulas. Their coefficient convolution is

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(g ∗x h)(A) =
X
B∪C=A
x|B∩C|g(B)h(C).

At = 0 only disjoint pairs survive. On substitution, every overlapping pair also vanishes because some occurs twice. Hence). Constants substitute to their scalar polynomial. Paid bounds multiply by the corresponding product bound, and addition adds their bounds.

Countable sums pass through this operation when their paid bounds are summable. For each subset, the corresponding coefficient arrays are dominated by summable bounds, so dominated convergence for series gives their limit at = 0 and commutes it with summation. There are only finitely many; applying the finite linear substitution proves

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X
!
=
X
n
Sz(fn).
Sz
n
fn

Apply this to the exponential series. The-th term has paid bound!, which is summable. Multiplicativity gives

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Sz(ef) = exp{Sz(f)}.

If the scalar mode of) is zero, the right side is its finite nilpotent exponential. This establishes the exponential identity without dividing by a density.

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The price estimate is coefficientwise. If |zi|w ≤1, define the envelope of mode A to be the
sum, over switch subsets D, of (b(D)) times the absolute coefficient of Q
i∈D zi in that mode. The
triangle inequality bounds the substituted coefficient by this envelope. Summing its weighted price
and using submultiplicativity gives at most P
D b(D) ≤c. Thus a paid scalar switching model
produces the local polynomial price required by restoration.

When every has zero scalar mode, the scalar mode of). Its singleton marker coefficient receives contributions only from switch subsets of size one; products of two or more zero-scalar factors cannot have singleton degree. These two facts prove the scalar and singleton coefficient readings used below.

Locality also passes at the coefficient level. If changing switch leaves a scalar switching family unchanged, its Möbius coefficients indexed by subsets containing vanish. If the output support omits a site in the parent support of leaf, the local silence identity gives exactly that switch independence. A nonzero coefficient can therefore involve only leaves whose required parent supports lie in the output support. Their scoped marker monomials then have all their pins there. The finite substitution preserves gauge invariance, locality and measurability. If every scalar switching value vanishes offa prescribed healthy region or inventory event, the Möbius coefficients and their limits

vanish there as well. These statements supply the support and permission properties of the marked corrections.

Reading the marked Mark-step output

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The reserve used by a retained step is constructed for the very finite leaf inventory just described.
Give each leaf its fee and normalized square-zero monomial zi. The parent support contains all pins
in the monomial’s scope, and |zi|w ≤1. The marked activity expansion produces a family K(t, A)
indexed by subsets of the finite leaf inventory. Its mixed jet in the leaves is evaluated on products
of the zi; because z2
i = 0, only distinct-leaf derivatives survive. Thus the marked polynomial is a
finite substitution in these jets. The coefficient bounds on the mixed jets yield local polynomial
bounds by the triangle inequality and ∥Q
izi∥w ≤1.

The output has an exponential correction and a forced activity correction. The latter is grouped by the coarse support-and-inventory barcode. Its scalar mode remains the unmarked background profile. For a singleton marker, only single-leaf jets can contribute; the single-leaf jet identity is

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∂iKj = feei Dj(Mi),

where) is the transported hinge of that same leaf. Multiplication by its normalized monomial cancels the fee, including the zero-charge case established above. Hence the singleton coefficient of the grouped forced correction is exactly the sum of the leaves’ quoted transported profiles. Absolute summability of the hinge families follows from the sound-door estimate: each hinge has a forced envelope whose total weighted mass is bounded by a fixed multiple of its leaf load. Summing over finitely many leaves and over the barcode fiber therefore commutes with coefficient extraction. This proves the singleton-profile identity used in the restoration step.

The coarse exact density identity is likewise an evaluation identity for the marked expansion: the Mark push of the priced ticket expression equals the exponential output times the coarse field polynomial. It is multiplied by the factored scalar displacement, which commutes with the Mark integral since it is a fixed finite-dimensional polynomial. The exponential output has zero scalar mode. The coefficient bounds and scoped locality give its served envelope; the forced mixed-jet bounds give the field envelope. These are the reserve’s two output budgets. The same construction is additive under summable grouped leaves because the underlying hinge map respects those sums. The switched marker expansion, primitive integral and connected-cluster arguments in this chapter construct the marked activity jets and prove their exact Mark identity.

The exact marked identity follows from the real switching identity once its paid bounds have been established. At a coarse configuration, let, let) be the sum of the regular log increments relative to the unmarked branch, and let) be the gas of the switched forced activities. The real identity is

Searchable transcription

a eL(x,R)G(x, R) = P
Buy(x, R)

(U),
0 ≤x ≤1,

with the actual Mark push. The paid regular increments have a summable total budget. The forced-activity bounds give a summable paid expansion of the gas. Apply to this identity, commute it with those summable series, and use its product and exponential laws. Since the regular log increment is zero at the unmarked switch, its substituted scalar mode is zero. The left side becomes

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X
!
a Exp
S
HS(U)
Field(K(U), FA(U)).

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For the right side, expand Buy into finitely many tickets. A ticket with leaf set D has switch
dependence x|D|1D⊆R times its scalar coefficient; its switching jet is concentrated at D. Its substi-
tution is therefore that coefficient times Q
i∈D zi. Each scalar ticket coefficient is integrable against
the Mark kernel. Finite linearity of integration proves that substitution commutes with the push
of this ticket sum. Its value is exactly the push of the priced polynomial inventory. This proves the
marked Mark identity from its real counterpart and paid envelopes, with all infinite summations
justified by their paid series bounds.

The shipping bound ensures that this finite inventory is admissible to the activity estimates. If a fine leaf has cover cost), its enlarged coarse birth support obeys

Searchable transcription

(q + 1)cB∗(birth(S)) ≤(q + 1)Cfoot + 5
4εv.

The tariffalso spends 2. With100, the selected relative width and exponent satisfy 13

4. Thus the original fine support weight pays both costs, leaving only the fixed multiplier . Charge each leaf to one of its scoped pins, use the parent-pin occupancy bound, and convert the rooted budgets to a sitewise bound. The remaining numerical demand is

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M siteFactor(B∗)e(q+1)Cfoot v′ ≤ribbon,

where is the inventory’s input budget. This is the shipping price. It pays the aggregate tariff after grouping equal activity labels, since those groups simply sum nonnegative leaf prices. Adding the unmarked activity reserve gives the admissible marked tariff. The geometric birth-support inequality and the scalar connected-activity estimates providing the ribbon, paid log increments and switched gas identity remain the lower-level inputs to this construction.

The numerical form of a retained insertion step

Write100 for fine and coarse support exponents, for their support widths, and for the occupancy bound. Choose 01. First replace fine marker weights by. Since every correction has degree at least two, its new monomial weight is at most times its old weight, so its unit price budget becomes at most. Reduce the support exponent to4. The finite transport inventory then has cost controlled by

Searchable transcription

T + eT T,
T = M siteFactor(B1)d + 4d.

Purchase a Mark-step reserve whose input budget is at least this quantity, and whose output field budget is. The response transport bounds must satisfy 21 and1, where are its two local envelope prices. After shrinking, these become 2 and, precisely the bounds required by the reserve. At the coarse level put siteFactor(. The numerical requirements are

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(2B0 + 1)4Q ≤1
2,
eQ −1 ≤1.

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The first pays the closure loss in support exponent, and the second bounds its final correction
price by one.
The matching condition on coarse weights Jv pays another half unit of support
exponent when passing from dIv to Jv, with price multiplier at most one. The reserve supplies
the coarse exponential exponent s0 + 1
2 and field exponent s0 + 1; these two half-unit expenditures
leave at least s0. The singleton feed identity and the restoration argument above identify the coarse
primary and forced insertions. This proves the retained unit-price served conclusion conditional on
the finite transport inventory and Mark-step reserve. The finite transport inventory and marked
reserve constructed in this chapter provide these inputs.

Separation pays for complex inflation

The large analytic disk in the omission argument is a geometric consequence of separated markers. Let8, let have at least two elements, and suppose distinct pins in have torus max-distance at least. If a support contains these pins, then

Searchable transcription

B|A| ≤64cB(S).

To prove this, choose a minimizing cover with) centers. Its halo admits a walk visiting the centers in at most (D(4B+2)) unit steps. For each pin choose a visit time whose center is within of it. Two chosen times differ by at least 0, by the distance triangle inequality and the unit-step walk bound. Thus their integer quotients by are distinct and ( where is the walk length. Since2 and

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|A|B2 ≤4(|A| −1)g ≤4N ≤4D(4B + 2) ≤64DB.

Division by 0 gives the assertion. Notice that the walk visits covering centers; visiting the pins themselves is unnecessary. Every nonzero higher correction has a nonzero mode with at least two markers, all pinned in its support. Hence64, and

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SB := eB/64 ≤ecB(S).

For, multiply a nonzero correction envelope by (|z|), and set the new envelope to zero for identically zero corrections. Then

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|z|aie(e−2)cB(Si) ≤aie(e−1)cB(Si) ≤aieecB(Si).

Thus multiplying every higher correction by preserves the price budget while reducing the support exponent by two. Locality, gauge invariance, measurability, scope and permission conditions survive scalar multiplication, and no scalar or singleton mode is introduced. This proves precisely the inflation estimate needed for holomorphy and retained transport in the omission comparison. It leaves the primary and forced singleton insertions at their literal values.

A positive denominator and an explicit vacuum bill

Let be normalized Haar measure and the real dimension of the ambient matrix space of the chosen representation. Normalize the matrix chord by the constant matrix norm chord(. For 0

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μG{g : chord(g) ≤r} ≥(r/4)dM .

Here is an ambient-space proof, requiring no coordinates on the group. A collection of group matrices with mutual distances greater than gives disjoint ambient balls of radius2, all inside the ball of radius 2. Comparing Euclidean volumes bounds its cardinality by finite maximal such collection exists: otherwise successive insertion would violate this cardinality bound. Its radius- closed balls cover the group. By unitary invariance of the matrix norm, their intersections with the group are left translates of the chord shell. Haar invariance and the union bound give 1(shell()). Combining the inequalities proves the assertion.

On a finite four-dimensional torus Λ, take the shelter of configurations with every positive oriented link in the chord shell of radius4, where 01. Product Haar measure gives this shelter measure at least

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(s/16)dM|E|,
|E| = 4|Λ|.

The chord of a product is at most the sum of the chords, and inversion preserves the chord. Every plaquette in the shelter consequently has chord at most, the field threshold, there are no flags there; its activity gas is exactly one. Signed translations preserve the shelter as they only permute links and invert some of them.

Write for the path bound, Λ for the loop cap, for the regular-activity allowance, for the forced allowance,0 for the coupling, and for the direct quadratic clipping coefficient. Define

Searchable transcription

 
Cdirs2
2
+ C2s2
2
!
l= β(Cθ3 + Cθ4) + Λ0θ2,
b(s) = |P|
,
e0 = δ + l,

where is the finite set of plaquette-axis pairs. Each healthy atom of degree is bounded by its coefficient magnitude times: every transported curvature leg has norm at most the threshold, since its footprint is healthy and parallel transport preserves the norm. Absolute summation with the atom budgets bounds the cubic, quartic and loop remainders per site by Λ respectively. The quadratic base is nonnegative. Thus the channel-averaged action is bounded below byΛ

On the shelter, every curvature leg has norm at most. Summing the absolute path coefficients gives norm at most (Cs) for the linear path field, so the quadratic base per site is bounded by (b(s)). The same remainder estimate bounds the channel action above byΛ)). The regular activity exponent has absolute value at mostΛ: charge each occurrence’s weighted budget to its designated root site and sum those root budgets. If is the scalar vacuum exponent, the full exponent therefore satisfies

Λ everywhere Λ on the shelter

The represented density is nonnegative. At the finest scale this follows from the positive bare Wilson density; downward induction preserves it because each Mark kernel is positive. Integrate over the shelter, where the gas is one, and use nonnegativity outside it. The resulting denominator bound is

Searchable transcription

ZR :=
Z
R(U) dμ(U) ≥exp{v0 −|Λ|(e0 + βb(s)) −4|Λ|dM log(16/s)} > 0.

Combining this with the insertion norm bound proves the explicit normalized estimate

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tw(A)|
R [X(U)]A dμ(U)|
ZR
≤exp
|Λ|
2e0 + κ + βb(s) + 4dM log(16/s)
 + |V |(x + y + 2f)
	 .

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The scalar v0 cancels. This bound applies to the full denominator, with no change of law or
restriction of the normalized numerator to the shelter.
For a selected history use s = bgk = bβ−1/2
k
, with β ≤2 bβk.
The loop price is monotone in
nonnegative β, as is βb(s).
Hence the coefficient of |Λk| in this bound is at most the explicit
treasury

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τk = 2{δk + lk(2 bβk)} + κk + 2 bβkbk(bgk) + 4dM log(16/bgk).

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This establishes the root estimate used in the omission argument. Its extensivity is explicit; no
volume factor is silently absorbed in an error tending to zero.
The treasury also has a useful stagewise bound.
Set P(s) = exp{(log log s)2} and Θ(s) =
s−1/2P(s). For every fixed b and every p > 0, eventually P(s)b ≤sp, because b(log log s)2/ log s →0.
Consequently sΘ(s)3 ≤1, sΘ(s)4 ≤1, and 2P(s)vΘ(s)2 ≤1 uniformly for v ≤1, eventually.
Substitution into the actual thresholds and loop caps gives lk(2s) ≤4Ck +1, while δk, κk ≤1. Also
sbg2
k = 1. Therefore, eventually,

Searchable transcription

τk ≤2{1 + (4Ck + 1)} + 1 + |P|(Cdir,k + C2
k) + 4dM{log 16 + 1
2 log bβk}.

Stage parameters are kept visible in this inequality; it asserts logarithmic growth in the reference coupling only after those parameters are fixed.