A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition

Jeremy Rodgers

doi:10.5281/zenodo.24173681

Source Lift / The complete argument

Marked integration and the coupling reset · Part 4

A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition

Jeremy Rodgers · Independent Researcher · 2026-11-12
Shared publication DOI: 10.5281/zenodo.24173681

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the entry-cost rows with the actual fine curvature and prediction-mismatch curvatures. Add the background guide bound $A/L2 · t$ to a fine curvature row and A $·$ t to a noise row. The curvature comparison inequality then bounds the plaquette chord by

2/√cdirect(A/L2 + τ)t < t/4 — Source page 137.

and the prediction mismatch by — Source page 137.

2/√cpenalty(A + τ)t < Wt. — Source page 137.

Hence the fine cutoff, noise cutoff, site tag and pin tag all equal one. The forced-charge detector also equals one there: any putative flag would be witnessed by a nearby plaquette with chord greater than t, contradicting the t/4 bound. Each nonmandatory detector is its bounded auxiliary cutoff times (one of these quantities minus one), so it is zero under the radius-2B watch. A mandatory detector is defined to be zero already. The active-link cutoffcomplement likewise vanishes when the lifted link lies within its plateau, using the mean bound and $dξ ≤1.$

This proves the detector-to-warning implication from the local protected-mean, physical pathaccuracy, clean-patch tag and curvature-comparison estimates, with their three error contributions kept separate. Those local geometric estimates remain dependencies of this paragraph; the Gaussian probability argument does not supply them.

The joint Gaussian warning estimate

Let z be a centered finite-dimensional Gaussian vector of covariance $A ≥$ 0, and suppose $vTAv ≤$ $M∥v∥2$ for every real v. No independence of its coordinates is required. For a coordinate set D and a sign choice $εi ∈−1,$ 1, let v have those signs on D and zero elsewhere. Then $∥v∥2$ $= |D|$ and

E exp(sv · z) = exp(s2vTAv/2) ≤exp(Ms2|D|/2). — Source page 137.

If $εi zi ≥t$ at every $i ∈D$ , then $v · z ≥t|D|$ . Markov’s inequality, for $s ≥$ 0, bounds this orthant event by exp(( $−st + Ms2/2)|D|$ ). The union over its 2 $|D|$ sign choices gives

P(|zi| ≥t for all i ∈D)
≤2|D| exp((−st + Ms2/2)|D|). — Source page 137.

This also covers a degenerate covariance and an empty coordinate set.

Warnings are located at sites $oj$ , Gaussian coordinates at sites $gi$ . Suppose at most m warning addresses and at most p coordinate addresses lie over any one site. A warning at $oj$ is witnessed by some coordinate $gi$ within distance R whose absolute value is at least t. For a finite warning set W, choose one such coordinate for every warning. There are at most ( $VRp)|W|$ witness assignments, where $VR = (2R + 1)4$ bounds the number of sites in a native torus ball. For an assignment let D be its distinct coordinate image. Then $|D| ≤|W|$ . Each coordinate can witness at most $VR$ m warnings, so $|W| ≤VR m|D|.$

Choose $H ≥$ 0 with $M/2 + HVRm ≤t$ and apply the preceding rectangle estimate with $s = 1.$ Its exponential factor is at most exp( $−HVRm|D|) ≤$ exp( $−H|W|$ ), and 2 $|D| ≤2|W|$ . A union bound over witness assignments proves

P(every warning in W has a witness)
≤(2VRp exp(−H))|W|. — Source page 137.

Thus shared witnesses and overlapping warning neighborhoods have been counted explicitly. They are not counted as independent Gaussian events. For the actual warning family $R = 2B$ , where B is the support width. Its address census is the sum of the fine plaquette types, two coarse bond types, coarse and fine site types, and fine-link itinerary types. With block length L it is bounded by

m = L4|PlaquetteAxes| + |Axis| + |Axis| + 1 + L4
+itineraryDensity(L). — Source page 138.

The Gaussian coordinate census p is the tubular coordinate count of that same block. The
covariance upper bound M comes from its actual fluctuation covariance estimate. The warning
event uses threshold t = ξ tfield/g, where g = β−1/2. Band comparability gives tfield/g at least
one half of the fine monitor P. Put H = ε√P, ε = ξ/4. Eventually V2Bm ≤√P and M/2 ≤εP,
because the monitor dominates the fixed covariance cost and the polynomial width cost. Hence
M/2 + HV2Bm ≤2εP ≤t. The preceding bound gives warning probability at most E|W|, with — Source page 138.

E = 2V2Bp exp(−ε
√
P). — Source page 138.

The monitor’s growth makes this E smaller than every prescribed reference-coupling power, after the finite block and stage parameters are fixed. This is the rapidly decaying warning allowance used by the activity prices. Its scheduling uses the original monitor and covariance bound, with no reselection of a Gaussian for a given molecule.

The event is connected to the detector factors by a pointwise implication. A nonzero detector means the corresponding local quiet-watch condition fails. Unfolding that condition supplies a coordinate within distance 2B with $|zi|$ at least $ξ tfield/g$ . Thus a product with all its warning detectors nonzero lies in the joint event just bounded. The preceding plateau calculation proves this implication: under the quiet watch the curvature is at most one quarter of its threshold, the prediction mismatch is within its plateau, no nearby flag can occur, and every nonmandatory detector is a cutofftimes a vanishing plateau complement. A mandatory detector is identically zero. Taking the contrapositive therefore gives precisely the witness used in the probability bound.

The primitive integrand and its local extension

The primitive integrand is a function of actual group links, chart variables, masks and insertion factors. On the protected interior its logarithm has the stated relation to the group-valued readout. In particular, from exp( $l) = UB−1$ one Ordered group multiplication is retained throughout. obtains exp( $l)BU−1$ $= I$ ; there is no step that commutes $B$ with $U$ . The extension away from the protected interior retains its cutoffand discrepancy rather than claiming this identity on its full domain.

Here are the finite objects to which the integration estimates apply. Fix a coarse-to-fine step, its
fine tree F, a seed inventory, a coarse configuration V , local discrete choices k, and the off-tree group
variables Y . The active off-tree links are determined by the tree and seed inventory. For a color
array u, the fine configuration has link exp(ue)(replica V )e on an active link, the value supplied by
Y on an inactive off-tree link, and the fixed tree value on a tree link. The logarithmic signal equals
u on active links, zero on tree links, and the chosen cutofflogarithm of Ue(replica V )−1
on inactive
e
off-tree links. Thus the group configuration and the logarithmic signal are different objects. Their
identification by exponentiation is asserted only on the interior domain where the cutofflogarithm
agrees with the genuine chart inverse.

A letter is of one of four disjoint types: a block packet $A$ , an old forced index $q$ , a regular logarithmic index $λ$ , or a warning index $w$ . A molecule $p$ is a finite set of letters; repeated copies of a letter are not allowed. Its forced inventory is the set of its forced indices. Its seed inventory is the union of the seeds of its packet letters. The ground $H(p$ ) is the union of the enlarged read footprints of all its letters. In particular it contains every seed and every forced read. Admission requires all of the following: the overlap graph of letter footprints is connected, the packet letters form the permitted layout of their combined seed inventory, the forced inventory satisfies its sealing and incidence conditions with those seeds, and

flags(V ) ∩H(p) = flags(seed inventory of p).
(PRI1) — Source page 139.

We spell out the discrete tests in this assertion. A seed is a member of the disjoint union of fine plaquettes, coarse bonds and coarse sites. These types remain distinct even when their spatial positions coincide. A store $A$ is the triple of finite sets ( $Aplaq, Abond, Amflag$ ); its number of seeds is the sum of their cardinalities. The read set of a plaquette seed is its coarse projected start set, that of a coarse-bond seed its two pin endpoints, and that of a site seed its singleton. The place of $A$ is the union of these read sets; its dome $A[r]$ is their coarse radius- $r$ halo. Give the seed set the graph in which two seeds are adjacent when their radius- $lB$ halos intersect. A connected packet means a nonempty connected set in this graph.

A permitted layout $L$ for $A$ at parameters $l, B$ is a finite set of stores with the following three properties: each $A′ ∈L$ is a connected packet $,$

(A′)[lB] ∩(A′′)[lB] = ∅
(A′ ̸= A′′),
seeds(A) =
[
A′∈L
seeds(A′).
(PRI6) — Source page 139.

It exists by taking the connected components of the seed graph and converting each component back to its store. Conversely any permitted layout is this one: an edge cannot connect distinct members because their halos are disjoint, and each member is connected. Thus the packet layout is determined by the combined seed inventory. It does not introduce a multiplicity of choices or a choice of a graph witness.

For a finite set $I$ of old forced indices write $Pi$ for its actual fine support and $Ji ⊂Pi$ for its declared inventory. Sealing means $Pi ∩Pj = ∅$ for distinct $i, j ∈I$ . Incidence means

∀i ∈I, ∀y ∈Ji,
∃p ∈Aplaq with y ∈touched(p).
(PRI7) — Source page 139.

Here touched( $p$ ) is the fine set inspected for that plaquette seed in the original sector partition. This condition uses the inventory, whereas sealing uses the full supports. Their roles cannot be interchanged. The selector used in the packet integrand is defined at each fine site $y$ by

(
0,
y ∈flagsθ(U) \ S
i∈I Pi,
1,
otherwise.
OI,Z(U) =
Y
y∈Z
cI(U; y).
(PRI8)
cI(U; y) = — Source page 139.

For the packet, $Z$ consists of fine sites whose parents are inside its 300 $B$ dome in the prescribed sense. For the full gas take all fine sites. If a product of old forced factors is nonzero, their individual permission identities give $Ji = Pi ∩$ flags $θ(U$ ). On a sealed family, the full selector is one exactly when the union of those inventories is all flags. Otherwise it is zero. This proves the exact rewriting of the old gas occurrence as the sealed indicator times (PRI8) times its old forced product. If the product is zero, both expressions are zero without using a permission identity for a vanishing factor.

The footprint of a packet letter is its enlarged dome; that of every other letter is its prescribed inspected, projected halo. Define two letters to be adjacent when these coarse footprints intersect. The molecule ground is their union, as in (PRI1). Its admission is now completely specified by nonempty connectedness in this graph, the layout (PRI6) of its packet letters, sealing and incidence (PRI7) of its forced indices, and (PRI1). The empty seed inventory is allowed when there are no packet letters; its permitted layout is empty.

Compatible molecules have disjoint grounds. For such a finite family, admission of its union is equivalent to admission separately in every member, together with the global flag coverage. To prove the nontrivial localization of incidence, suppose a forced inventory site in one molecule is touched by a seed attributed to another. The parent of that site lies in the forced footprint of the first molecule and in the seed footprint of the second. Disjointness of grounds forces the two molecules to coincide. The same parent argument localizes the sealing test. Layout restrictions localize because the enlarged packet domes are contained in their grounds; joining the local layouts gives a global layout. Equation (PRI1) localizes by intersecting the global flag set with disjoint grounds. Finally every finite set of letters has a unique partition into connected components of its footprint graph. These statements prove that expansion into finite letter sets and regrouping into compatible connected molecules are inverse operations, with multiplicity one. Absolute summability proved below permits this finite combinatorial identity to pass to the countable alphabets. All these admission tests remain discrete masks and are not differentiated.

For the common lifted fluctuation $u = μ + β−1/2z$ , the four vertex values are, respectively,

vA = 1{packet connected}1{packet eligible}SA(u),
vq = Kq(U(u)) E2a,q(u),
vλ = eLλ(u,z) −1,
vw = Dw(u).
(PRI2) — Source page 140.

Here $E2a,q$ is the product of fluctuation cutoffs on the entire forced read footprint. Packet eligibility requires its place to lie in the chosen seed place and its declared flags to agree with the actual coarse flags throughout its 300 $B$ neighborhood. To define the profile factors, let $χ$ be the fixed smooth cutoffand put $hs(r) = χ((r/s)2$ ). For a tested fine plaquette use $ht/4$ (toll( $U, p)) if p /∈Aplaq,$ and its complement if $p ∈Aplaq$ . For a tested coarse bond use $hWt$ (wire(prediction( $U, k), V ; b)) if$ $b /∈Abond$ , and its complement otherwise. Multiply these factors once over every plaquette whose projected start set meets $A[300B]$ and every bond whose pin set meets that dome. These are the profiles in [0 $,$ 1] in the following formula. Let $nF , nP$ be the cardinalities of the finite frame-tag and predictor-point sets. At a site $x,$ the ticket factor is $nF$ times the actual frame-choice probability at $x$ . At a bond it is $nP$ times the actual predictor-choice probability for its connector. Multiply the site factors over $A[300B]$ and the bond factors over all pin sets meeting it. They are nonnegative and bounded respectively by $nF , nP$ , rather than by one. With the uniform average over the complete choice space,

Y
x
siteMass(U, k, x)
Y
b
pinMass(U, k, b).
(PRI9)
likelihood(U, k) =
1
|K| — Source page 140.

Indeed $|K| = n|Λc|$ $F$ $n| Bondc |$ $P$ , which cancels the inserted cardinality factors. This identity also includes zero-likelihood choices; their ticket product is zero. The frame probability is uniform over the admissible minimizing tags when that set is nonempty, with the prescribed uniform fallback; the predictor probability is its prescribed nonnegative normalized weight with its corresponding fallback. Thus the ticket retains the actual choice denominators, including the fallback branch.

The packet value is explicitly

SA(u) = e
−β(Qnear(A,u)+Qfar(A))+P
visit200B A ce+P
active304B A je(u)−NA log mβ
× OA(U) TA(U, k) PA(U, k, V ) Ea,A(u),
(PRI3) — Source page 140.

$√ 2π β−1/2)dim g, and NA counts the off-tree frozen links in the 10B visit where mβ = chart scale ( set. The terms ce are the link condensation terms; je are the local logarithmic Haar and cutoff corrections. The far term is the signed-orientation average of the extracted nonnegative row costs on the 200B posted set. The remaining four factors are the forced-order selector, the finite layout ticket, the product of sector profile factors, and the active cutoffproduct on the 304B neighborhood. The selector, profiles and cutoffs are in [0, 1]; the ticket has its finite combinatorial toll bound. The normalization deficit in (PRI3) is per frozen link, and is retained before that toll is estimated.$

Searchable transcription

√
2π β−1/2)dim g, and NA counts the off-tree frozen links in the 10B visit
where mβ = chart scale (
set. The terms ce are the link condensation terms; je are the local logarithmic Haar and cutoff
corrections. The far term is the signed-orientation average of the extracted nonnegative row costs
on the 200B posted set. The remaining four factors are the forced-order selector, the finite layout
ticket, the product of sector profile factors, and the active cutoffproduct on the 304B neighborhood.
The selector, profiles and cutoffs are in [0, 1]; the ticket has its finite combinatorial toll bound. The
normalization deficit in (PRI3) is per frozen link, and is retained before that toll is estimated.

$The regular logarithms Lλ comprise the old regular activity and cubic, quartic, loop and quadratic-tail energy pieces, the remote row and local chart terms, and the finite ledger terms from Gaussian completion and covariance reweighting. Their signs are the signs in the exponent of the original density: in particular an old regular occurrence contributes its negative value. A warn- ing vertex is zero when mandatory; otherwise it is its fluctuation cutofftimes the corresponding sector-profile complement. The Gaussian warnings are also retained in this warning alphabet. All factors in (PRI2) use this single configuration, signal and Gaussian variable. Put bp(Y, z) = Q α∈p vα(Y, z). The primitive value before the finite choice average is the literal integral$

Searchable transcription

The regular logarithms Lλ comprise the old regular activity and cubic, quartic, loop and
quadratic-tail energy pieces, the remote row and local chart terms, and the finite ledger terms
from Gaussian completion and covariance reweighting. Their signs are the signs in the exponent of
the original density: in particular an old regular occurrence contributes its negative value. A warn-
ing vertex is zero when mandatory; otherwise it is its fluctuation cutofftimes the corresponding
sector-profile complement. The Gaussian warnings are also retained in this warning alphabet. All
factors in (PRI2) use this single configuration, signal and Gaussian variable.
Put bp(Y, z) = Q
α∈p vα(Y, z). The primitive value before the finite choice average is the literal
integral

ap =
Z
bp(Y, z) dνF (Y ) dγC(z),
(PRI4) — Source page 141.

with the product tree-gauge probability measure and the one covariance $C$ prepared for this sector. There is no normalization by the mass of an admissible event. Local discrete choices are subsequently averaged with weight 1 $/|K|$ ; their probabilities do not replace any of the profile or ticket factors.

For extension to coarse configurations outside the healthy domain, use the same finite formulas with the extended local row expressions and retain the admission masks at their actual enlarged output footprints. For a chosen ground $H$ , retain only the finite Gaussian coordinates $I(H$ ) read there. The submatrix $CI$ is positive definite at an actual admissible sector and has density

(2π)−|I|/2 exp

−1
2 log det CI −1
2zT C−1
I z

dz.
(PRI5) — Source page 141.

Integrate the extended product against this density and the tree-gauge measure, then take the uniform finite choice average. Set it to zero if the structural connectedness, layout or incidence tests fail. The marginal extension has its own fixed support and is not presumed positive or energetic at other coarse configurations.

On a healthy ground this agrees with (PRI4). The integrand is constant on fibers of the coordinate projection to $I(H$ ); its Gaussian pushforward has covariance $CI$ . The remaining local data agree on the ground. Integrating over these fibers and using (PRI5) proves the identity. The same finite-choice fiber argument proves independence of choices outside $H$ . For disjoint grounds the finite choice space is a product; writing its cardinality as the product of the cardinalities of the factors proves that the average of their product is the product of their averages. The tree variables have the analogous product-measure identity. No independence of overlapping Gaussian warning events is used.

The later continuity passage uses these same primitive integrals. Along an ultrafilter all finite seed stores and discrete tags can be fixed on one member of the ultrafilter. Keep the tree-gauge input $Y$ fixed, and compare the two histories on this fixed finite data. Their covariance densities and vertex factors have the asserted pointwise limits. The primitive bounds below give a common integrable majorant, so dominated convergence applies to each molecule. Since this holds along every ultrafilter tending to infinity, the difference of the primitive values tends to zero in the ordinary sequential sense. This explains why the fixed finite data in the continuity argument can be selected

after fixing $Y$ , without claiming that a continuously varying configuration has a globally constant sector.

Integrating the primitive vertex bounds

The primitive molecule body is a finite product of four kinds of vertices: connected block packets, old forced factors, regular exponential increments, and warning detectors. Fix a molecule p and let $B(p$ ) be its warning subset. The prepared estimates give nonnegative pointwise tariffs $fi$ for each vertex and integrated tariffs $ti$ , with

i∈p
ti = h|B(p)| Y
i∈p
fi.
Y — Source page 142.

Only warning tariffs receive the factor h. The packet tariffis zero unless the packet is connected
in the specified enlarged geometry. Otherwise it is its displayed block-packet income. Forced tariffs
are the old forced envelopes, regular tariffs bound both the local log and exp(log)-1, and warning
pointwise tariffs are the fixed storm bound. The source proves these four pointwise inequalities on
the same local data and sector.
Write Fp = Q
i∈p fi. The triangle inequality for a finite product gives | bodyp(Y, z)| ≤Fp for
every tree-gauge variable Y and Gaussian fluctuation z. If bodyp(Y, z) is nonzero, every warning
detector in B(p) is nonzero. The warning estimate then places z in the joint catastrophe event
CB(p), whose Gaussian probability is at most hˆ|B(p)|. The event and its covariance are common
to all the factors; no factorization of its probability is used. Therefore pointwise — Source page 142.

|bodyp(Y, z)| ≤Fp1CB(p)z. — Source page 142.

The body is measurable, and the uniform finite bound makes it integrable against the product of
the probability tree-gauge measure and the reference Gaussian measure. Integrating the displayed
indicator majorant and then integrating Y proves
Z
|bodyp(Y, z)|dνY dγz — Source page 142.

≤FpγCB(p) ≤
Y
i∈p
ti. — Source page 142.

This proves the primitive integrated tarifffrom the four vertex bounds and the joint catastrophe estimate. In particular an empty warning set costs no probability factor. Correlated warnings are paid by their actual joint event.

For a marked team, remove the team’s selected keys from p and multiply that remaining body
by its paid ingredient product. Each paid leaf ingredient has absolute value bounded by its assigned
leaf weight, so their product is bounded by the team weight WM. If the team keys are not all in p,
define the term to be zero. Otherwise the preceding argument gives the marked bound
Z
Y
|markedBodyM,p| ≤WM
ti.
i∈p\keys(M)

The corresponding uniform pointwise bound uses the fi instead of ti. Taking these products over
all finite molecule subsets is summable whenever the primitive tarifffamily is summable: the sum is
bounded by WM times the product of 1+ti, hence by WM exp(P ti), with finitely many compulsory
keys held present. This supplies the summable integral-norm bound used in the primitive Mark
expansion. Its remaining analytic inputs are the block-packet bound, old forced and regular-factor
envelopes, warning storm bound and joint Gaussian catastrophe estimate. — Source page 142.

Factoring the regular gas and restoring forced exclusions

At one Mark step split primitive activity labels into regular labels with empty flag inventory and forced labels with nonempty flag inventory. Fix a compatible forced set $A$ , and let $YA$ be the union of its enlarged supports. The allowed regular gas in the presence of $A$ removes the regular labels meeting the forced exclusion region. Denote the full regular gas by $Z0$ , and its convergent cluster logarithm by $L0$ . The removed logarithmic terms are precisely regular occurrence clusters whose union meets that region. If their sum is $MA$ , the restricted gas is

Z0,A = exp(L0 −MA) = Z0e−MA. — Source page 143.

This is an equality of absolutely convergent series: cutting the regular label family decreases its absolute budgets, and subtracting its cluster sum from the full cluster sum leaves exactly the indicated missing clusters. It retains their original occurrence indices and factorials.

Multiply by the forced product and sum the compatible forced sets with the required total inventory. Expand $e−MA$ by positive multiplicities of missing-cluster addresses. The resulting term consists of the forced set and a finite multiplicity family of negative regular clusters; its weight is the product of the forced activities and the cluster factors ( $−h)r/r$ !. Each selected missing cluster intersects the forced exclusion region. Absolute summability follows by the restoration bound, or at fixed volume by the summable exponential of the absolute missing-cluster series. Therefore grouping these terms does not change their sum.

Decompose every such term into overlap-connected components of its atom supports. Every component containing a missing cluster also contains a forced atom: that cluster meets the union of the selected forced supports. A component consisting only of forced atoms is already allowed. The resulting components have pairwise disjoint supports; their forced inventories partition the original inventory. Conversely, union of any permitted disjoint collection of these components recovers the original forced set and each missing cluster with its multiplicity. The multiplicity factorials factor over components because no atom address belongs to two distinct components. Thus the map is bijective and preserves the product weights.

The original gas, multiplied by its external scalar prefactor, is consequently $Z0$ times the gas of these connected restored forced components. This proves the regular-log factorization and forced regrouping identity once the primitive Mark expansion has been supplied. It uses exact inventories and actual overlap exclusions; the more permissive repetition count used for the absolute bound does not alter this equality.

For switched leaves the same argument holds at each real switch value 0 $≤x ≤$ 1, with the same supports, threshold masks and inventory tests. The paid bounds established above apply uniformly there. Two subsequent dictionary fusions put the restored components into the final coarse dictionary. Each fusion preserves the gas up to the exponential of its recorded unmarked regular log contribution. These extra logarithms do not depend on the switch selection. The unmarked normalization identity says that the final coarse exponent, its fusion logarithms, and $L0(0, ∅$ ) reproduce the original scalar prefactor. Substitution into the factored switched gas identity yields

eEA(U)eL0(x,R)−L0(0,∅) GasA(K(x, R); U) = P(Buy(x, R))(U). — Source page 143.

All divisions in the regular-gas ratio are legitimate because the real gas/log identity proves $Z0 >$ 0, including the unmarked branch. The primitive Mark expansion, dictionary fusion and unmarked scalar normalization proved in this chapter establish the scalar equality, with the stated exclusion restoration and component regrouping.

Paid regular and forced cluster terms

A switched primitive activity is a finite ticket sum. Each ticket carries switch dependence $x|D|1D⊆R$ and a scalar factor bounded by the product of its marked leaf weights and all unmarked activity weights not selected by the ticket. Its switching model is concentrated at $D$ . Summing tickets gives a paid bound equal to the product of the augmented activity tariffs. The selected mixed-activity estimate enlarges this bound by its displayed support and polynomial-moment weights; it sets it to zero when the activity’s eligibility conditions fail. These are pointwise paid bounds uniform in configurations.

A regular cluster term is the product of its $n$ switched primitive activities, multiplied by $φ(z)/n!.$ The product rule for paid models gives the bound

n
Y
|φ(z)|
j=1
uij,
n! — Source page 144.

where $ui$ are the selected primitive paid bounds. The rooted cluster estimate just proved gives a support-weighted sitewise sum at most $P0/(1 −ρ$ ) for their regular cluster envelopes, when the selected primitive budgets are $P0$ and the tree ratio is $ρ <$ 1. Subtracting each cluster’s unmarked value doubles its paid bound. These differences are the regular log increments in the switched identity; they are not independently selected regular activities.

The forced restoration terms combine forced primitive activities and negatives of regular cluster terms, each with its explicit occurrence multiplicity. For an atom occurring $r ≥$ 1 times the factor is $hr/r$ !; different atom factors are multiplied. The paid product and power bounds therefore give exactly the corresponding product of envelope powers divided by factorials. Restrict to connected terms with at least one forced atom and the required flag inventory. These restrictions depend on configuration and on the fixed combinatorial term, not on the switching parameter, so multiplying by their indicators preserves the paid bounds. The omitted-cluster restoration and primitive-integral arguments in this chapter control their support-weighted sum.

Restoring omitted clusters with positive multiplicities

The restoration exponential uses a different indexing from the occurrence tuples in the logarithm. Its index is a finite set $D$ of atom addresses together with a positive multiplicity $ri ≥$ 1 at every $i ∈D$ . Its weight and total size are

hri
i
ri! ,
N(D, r) =
X
i∈D
ri.
Wh(D, r) =
Y
i∈D — Source page 144.

An absent atom is absent from $D$ , rather than being included with multiplicity zero. For nonnegative $hi ≤σ <$ 1, summing each positive multiplicity gives

X
hri
i
ri! ≤
hi
1 −σ.
ri≥1 — Source page 144.

Indeed $ri$ ! $≥$ 1 and $hri$ $i$ $≤hiσri−1$ . Multiplying these inequalities over a fixed distinct set $D$ reduces the repeated-atom bound to a distinct-set bound with inflated vertex weights $hi/(1 −σ).$ The sitewise smallness hypothesis guarantees each required $hi ≤σ$ : choose a site in its nonempty support and retain its single summand in that site’s budget. All extra support factors and (1 $−σ)−1$

Source: A Wilson Yang–Mills Construction: A Natural-Language Mathematical Exposition. Both papers belong to one Zenodo record. No external funding.