The centered trace and its non-freeness test · Part 9

this fixed stage. Its defining expression is a fixed finite sum of products of the coupling, threshold powers, path costs, signed-motion count, and the local site census. The coupling is at most 4 the other growing factors have subpower growth. The expression therefore has a rate bound of exponent 2, and is eventually at most In the expense inequalities take the truncation, the separation count, and

Searchable transcription

c′ = eP/2,
r = ex,
1 ≤w ≤eP/32.

The fixed factor satisfies 4 eventually. Thus 4 . Likewise (TS9) gives , and hence

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Writing u = e−MTlease, we have
8eP u ≤8x3eP−M ≤ex.

The forced allowance is the specified rapidly decreasing function smoke() = exp[Wave(1 Its decay dominates , giving smoke(

eventually. Consequently the total expense is at most 3. The site factor (2 has subpower growth, so (21 eventually. Finally1, which is the required output allowance here. Also 41 follows from 81. The wing tariffis the already imposed local surgery tariff. This proves all the displayed expense conditions simultaneously for every weight in the stated interval.

For each fixed stage, the preceding bounds impose only one eventual lower bound on. The stagewise threshold selection enlarges the original coupling-entry delay to exceed these bounds at every permitted stage transition. Taking the maximum with the other construction thresholds preserves them all; later delay increases preserve this choice. Thus the physical calendar used in (TS6)–(TS8) satisfies the expense bounds before the finer cutoffor volume is chosen.

16.7.1 Finite density surgery and the equation insertion

The finite lens changes one bond by ordered group multiplication. Health and chord inequalities preserve the exact flag set, so exponent differences only involve supports containing the modified source. The resulting finite subset expansion is exact. Only the explicitly stated finite-shelf approximation is an error, and its allowance is selected after division by the node step has been included. Haar invariance gives the zero-integral identity for the same stock density used throughout the marker construction.

Why the marker is a finite representative The equation insertion is constructed from a local change of density before the continuum quotient is taken. Its flags and replicated marks retain where that change is supported, and its expense calendar gives the root bounds needed by spectators. Thus the eventual equation-null class has an actual finite local representative on the chosen Wilson family. The surgery identity and the weighted estimates below establish this assertion together; a formally vanishing coefficient symbol would not supply the same conclusion.

The lens itself has a finite, exact definition. For an oriented bond issuing from, let change its link by left multiplication by); for a negatively read bond the equivalent stored positive link is changed by the corresponding right inverse multiplication. Let) denote the scalar stock of a support. Truncate the represented exponent by support-cover number:

Searchable transcription

HM(U) = v| Site | +
X
covB(Q)≤M
SQ(U).

The lattice is finite, so this sum is finite. For the bounded local function), the lens is

Searchable transcription

L(U) =
Z
h
J(Tν(j)U, j)eHM(Tν(j)U)−HM(U) −J(U, j)
i
dνG(j).
(TS10)
G

The admissibility conditions on these data are concrete: ) reads the fixed support ); a nonzero value requires health at with threshold is measurable; and its chord is at most, with 8. These conditions are preserved under the signed chamber copies used above.

To justify expansion of (TS10), put and suppose= 0. Every link changes in chord distance by at most. The chord of a plaquette changes by at most 4, by telescoping its four unitary factors. A plaquette whose value changes touches, so its chord at is at most Its chord at is therefore at most4. Both are below. Unchanged plaquettes have identical threshold comparisons. Hence the flag sets at agree exactly. The same reasoning after any signed motion shows that their relaxed kept-plaquette tests agree, since each relaxed threshold is at least

A stock whose support omits is unchanged: the modified bond has an endpoint at, all internal links of that support agree, and its flag data agree. Thus

Searchable transcription

HM(V ) −HM(U) =
X
SQ(V ) −SQ(U)
.
z∈Q
covB(Q)≤M

For this finite set of supports, set

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(
eSQ(V )−SQ(U) −1,
J(V, j) ̸= 0,
0,
J(V, j) = 0.
tQ(U, j) =

The distributive law and exponential addition give the exact identity

Searchable transcription

J(V, j)eHM(V )−HM(U) =
X
A⊆{Q:z∈Q, covB(Q)≤M}
J(V, j)
Y
Q∈A
tQ(U, j).

If) = 0, both sides are zero; this also explains the zero branch in the definition of avoids any need to estimate exponent differences on configurations where the insertion is absent. This finite subset expansion is the local surgery expansion priced in the expense inequalities. Every term reads only the inspected union of and the supports in that formula. The cover-extent inequality bounds its distance from by

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pierSize = (l + M)(4B + 2) + 10B

when cov and 0. The resulting lens reads the origin-centered window of radius 2 pierSize. Its gauge invariance follows from the displayed covariance of, invariance of the stocks, and the change of Haar variable Measurability and boundedness follow from the finite stock expansion and its envelope. These facts provide the locality, gauge invariance and integrability used in the copied-lens and root estimates.

For the equation marker used above, specialize (TS10) as follows. On a positive source link let be a unit color direction, let) = exp(), and take

For a reversed link use the equivalent reversed multiplication in (TS10). The current is gauge covariant at the source, so the covariance of and is exactly the one required there. The step

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h = θx−1/24, with its fixed node multiples, eventually satisfies the stronger chart condition 64D ≤θ;
thus it also satisfies the 8D ≤θ flag condition of (TS10).
Let Le,tih(U) be this lens with the finite stock exponent, and sum over the active links, color
basis indices and nodes:
M(U) = 1
X
X
i<m
wi Le,tih(U).
(TS11)
h

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e,c

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Haar integration is already part of each lens. The coefficient wi/h is its actual normalization. For
every node its lens integral is zero for the stock density, by the Haar change of variable; the matched
course transports this identity to the corresponding finite Wilson equation expression. Expanding
by (TS10), its empty subset supplies the main density variation and its nonempty subsets are
exactly the secondary terms F in (LCE6). Taylor expansion with the node moments above gives
the current divergence and action derivative, with the remainder E.
The exact healthy-region
identity is therefore
M = Z −β
X
a
Da eJab + (divν Yb)(1),
(TS12)

up to the explicitly retained finite-shelf approximation, uniformly at most after division by. The shelves are exhausted only after choosing this error allowance, so division by is paid in their selection. The normalization and support costs in the rooted estimate can be read directly from (TS11):

Searchable transcription

X
e,c,i
|wi|/h ≤cardB(x)
X
i<m
|wi|

/h ≤x2

eventually at each fixed. The active-link count and fixed node sum have rate zero, while

has rate 124, strictly below two. Each lens is supported in the window of radius 2 obtained above. At the actual middle stop its copied root bound is at most T1900000); the preceding total coefficient expense is at most T(100000). The rooted triangle inequality and T) give

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∥M∥root ≤Ti(−1800000).
(TS13)

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Here Ti(a) = exp{a(log bβi)48/10000}; all finite local radii and prices are absorbed before choosing
the sufficiently late stop. The rescaling from that stop to the reporting scale has degree-five factor at
most Ti(100000). The healthy error is exponentially smaller than any fixed negative Ti power, and
the global polynomial bound and exceptional-event estimate pay its complement. Signed averaging
removes the constant in (TS12) and preserves the root bound. This is the normalized equation
marker used in (TS1)–(TS5), including the costs of the full joint replicated array; it is not a formal
integration-by-parts symbol without a local representative.