14 The analytic stationary graph, loop moments and the cali- brated coupling history · Part 6

To represent the final sum by forced activities, partition the actual flag set into its connected components for the relation that two flags’ radius-2 neighborhoods overlap. If is a nonempty component, let) be the union of those neighborhoods and set

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qD(U) = ω
X
i: xi∈haloB(D)
di(U).
(FST5)

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The extended regions of distinct components are disjoint. Every nonzero di lies within distance B of
some flag, and hence belongs to exactly one component’s sum. The assertion about distance follows
from the relaxed test: outside that distance the clip condition θ exp((B −2)/crelaxation) ≥2 and the
universal chord bound 2 force all its tests to be kept and healthy. Therefore P
D qD = ω P
i di.
Assign the forced entry e−βqD(U) to the support-inventory pair (R(D), D), eligible precisely
when R(D) ∩flagsθ(U) = D, and assign zero to all other active entries. In the gas sum, a nonzero
admissible collection must have disjoint supports and inventories partitioning the flags. Its invento-
ries are consequently exactly the connected components just described. Conversely that collection
is admissible. The gas sum thus has one nonzero term and equals

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!
−βω
X
i
di(U)
gas(U) = exp
.
(FST6)

Combining (FST4) with (FST6) proves the exact startup density, with scalar normalization zero.

These explicit entries also satisfy the needed bounds. The normalized derivatives of on a healthy plaquette, through each fixed required order, are at most. This follows by Taylor expansion of the smooth four-link product and (FST1): its curvature correction cancels the jets below degree five; the permitted derivative directions have size), so differentiating does not lose the total factor. The constants are chosen for a fixed order before the coupling or period. There are only the fixed number of signed plaquette types at an anchor, and their cover cost is bounded. Consequently the anchored regular bound is at most

For a forced component, every flag is incident to a plaquette above threshold and each plaquette touches at most four sites. Using4 and keeping one signed frame gives

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qD(U) ≥|D| ωθ2/16.
(FST7)

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This exponential suppression pays the connected-component support count and its support weights.
To see its size in the actual schedule, set x = bβN, z = log x, M(z) = e(log z)2, and θ = e−z/2M(z).
On the lower-bound domain β ≥x/2, (FST7) has strength proportional to M(z)2|D|. The neigh-
borhood count (4⌈z2⌉+ 1)4 and each fixed stage support cost are eventually at most M(z)/4. The
component expansion may therefore use local allowance e−M(z)/2; its summed connected-activity
bound is at most e−M(z)+1, which is eventually at most e−M(z)1/5, the required forced allowance.
For the regular bound one also uses β ≤2x. On the shooting band x/2 ≤β ≤2x, it satisfies

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C′βθ5 ≤2C′e−3z/2e5(log z)2 ≤e−11z/10,
(FST8)

the required regular allowance. The same finite thresholds also ensure the clip condition preceding (FST6), since, and0. These thresholds are uniform in once the stage is fixed, and have already been included in its entry delay.

The dictionaries of supports and flag predicates are independent of Regular entries are linear in, and forced entries are exponentials of bounded finite-volume functions times fixed

eligibility indicators. Their weighted sums obey the preceding majorants. Thus the startup family is continuous in precisely the activity budgets used for finite shooting. Countably many unused addresses can be appended with value zero, without changing density or bounds; they furnish the marker slots required later. This constructs (FCA2) with its actual representatives.

Source–readout formulation. The startup representative is the first realized state of every finite history: it is the interface between the finite Wilson law of Theorem 7.2 and the represented states of Definition 14.19 below. Its discrepancy is identically zero.

Theorem 14.18 (Exact startup density). Let be the principal path action including its clipped direct part and masked quartic term. Pointwise, for every plaquette,

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price(uP ) = baseTermP (U) + maskθ,P
bQ4(curvature(uP ))

+ b maskθ,P [R(uP )] + dP (U),

and, withSigned and indexing oriented terms with footprint, anchor, residual and unpaid part

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βS(U) = βAtop(U) + βωb
X
i
maskθ,Si Ri(U) + βω
X
i
di(U).

Taking the middle sum as regular activity and representing the last by the forced entries below, the represented density is with scalar normalization zero.

Proof. The three cases are exhaustive, including threshold equality. Healthy: substitute (FST1) and cancel. Unhealthy with: both masks and the kept quadratic term vanish and the clipped direct term is cancelled by (FST2). Unhealthy with: both masks vanish and the alternatives for the kept quadratic term cancel those in (FST2). Averaging over signed frames and summing gives (FST4) by signed-frame invariance of the Wilson energy. Partition the flag set into connected components for the relation that radius-2 neighbourhoods overlap; for a component let) be the union of those neighbourhoods and

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qD(U) = ω
X
i: xi∈haloB(D)
di(U).

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The extended regions of distinct components are disjoint, and every nonzero di lies within distance
B of a flag: beyond that distance the clip condition θ exp((B −2)/crelaxation) ≥2 and the universal
chord bound 2 force all tests to be kept and healthy. Hence P
D qD = ω P
i di. Assign the forced
entry e−βqD(U) to the support–inventory pair (R(D), D), eligible exactly when R(D)∩flagsθ(U) = D,
and zero to all other active entries.
A nonzero admissible collection has disjoint supports and
inventories partitioning the flags, so its inventories are exactly the components; conversely that
collection is admissible. The gas has one nonzero term,

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gas(U) = exp

−βω
X
i
di(U)

,

and (FST4) with (FST6) gives the density.

The identity is pointwise in the configuration and therefore lifts by composition with: every term on its right-hand side is a class function of plaquette holonomies or a mask of such a function, and Propositions 9.1 and 9.3 identify the masks and the budgets on the two sides. The estimates

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(FST7)–(FST8) place the regular and forced parts within the allowances (AB12) at the clock value
x = bβN, on the shooting band x/2 ≤β ≤2x, with thresholds uniform in N and p at a fixed
stage. The forced estimate uses only the lower endpoint: βθ2 ≥M(z)2/2. The regular estimate
uses the upper endpoint: βθ5 ≤2x−3/2M(z)5. The dictionaries of supports and flag predicates are
independent of β; regular entries are linear in β and forced entries are exponentials of bounded
finite-volume functions times fixed eligibility indicators. Hence the startup family is a continuous
curve of represented states with a fixed layout, which is exactly the input required by the shooting
argument. The exactness class is exact recovery (an identity of densities), together with a bound
that places the exact representative in the activity ball.

14.15 Finite coupling arcs and calibrated finite histories

We construct the finite histories before using their compactness or uniqueness. Fix the common calendar, a terminal depth, and a period. Put

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τi = log(a−1
i ),
s = slopeConstant(G) > 0,
bβi =  ̄β + sτi,
Bi = [ bβi/2, 2 bβi].
(FCA1)

Here ̄ 0, and = log 0. The path tensors at fixed are fixed independently of and of the bare-coupling shooting parameter. A represented state consists of those tensors, its prescribed clip and budget, its actual coupling, scalar normalization, and regular and forced activities. Admissibility means the local regularity, symmetry and absolute activity bounds proved in the nonlinear step, together with the retained startup and transition conditions. It includes

Continuity of a family of such states has a specific meaning. Its and are continuous; its regular and forced address dictionaries are constant; and differences of its activity values tend to zero in the absolute weighted derivative budgets. For regular terms derivatives are tested only on the common healthy domain, through the prescribed order and in the prescribed directions. For forced terms the bound is on every configuration. On a fixed period a global address sum is legitimate for this continuity assertion. It is not substituted for the period-independent anchored bounds in admissibility.

The nonlinear construction provides a forward family at every finite step on this one calendar. At the finest level its states, have

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β(FN,p(b)) = b,
v(FN,p(b)) = 0,
FN,p(b). eval = bareb .
(FCA2)

They have the fixed startup representatives and insertion slots. For a continuous family admissible level- + 1 states on a compact parameter interval, the same primitive integration and reset construction produces continuous coarse states), with the same retained choices, such that

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T(b). eval = P β(Q(b))
i
(Q(b). eval),
|β(Q(b)) −β(T(b)) −s log Li| ≤di+1.
(FCA3)

The number is the fixed drift allowance of this stage. The output satisfies all level- local bounds before imposing its coupling band. This distinction permits the shooting argument to detect crossings of that band.

These forward families are obtained from the actual primitive formulas, rather than a choice of unrelated admissible outputs. At a step, keep one primitive construction, its ordinal and tree choice, and its fixed support dictionaries. Its output is the moment state produced by that primitive. The local integral comparison and the absolute activity estimates above prove continuity of the output in the stated budgets. The reset identity gives the density equality in (FCA3); its coupling

bound gives the second inequality. For a continuous input curve these same formulas therefore define a continuous output curve. For two occurrences of the identical input state they give the identical output state: equality is equality of the represented moment states, including the activities. Choosing this rule once at each, and the startup rule (FCA2) once at, gives a single deterministic forward recipe. Its underlying path, clip, normalizers, source bounds and calendar remain those of the nonlinear construction. Restricting a parameter interval preserves every one of these properties.

To make the normalization in this construction explicit, write eval)( The positive push kernel and the retained positive vacuum give is the already committed path correction, the output coupling and scalar are

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β(T(b)) = β(Q(b)) + αN,i + ∆(b),
v(T(b)) = log Hb(1)
| Sitei | .
(FCA9)

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The scalar reset produces |∆(b)| ≤ri and θ2
i |∆(b)| ≤ei, where ri is its fixed correction radius and ei
its regular allowance. In the exact normalized splitting, the output trunk density times its attached-
log exponential is Hb(U)/Hb(1). Installing (FCA9) multiplies that density by exp(| Sitei |v) = Hb(1),
proving (FCA3) at every configuration, including flag boundaries. The site count, rather than the
bond count, is essential in this cancellation.
The correction in this construction is the source Hessian projection:

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∆(b) = Averaging. delta
X. read(a(b))

.
(FCA10)

Here is the actual fixed source listing and) its parameterized source. The affine-curvature Hessian projection, rebates, faults and low-and high-path loan identities in the coupling-reset exponent section prove the exact all-configuration representation with this value of ∆. In particular the correction is not defined by solving a two-probe contrast equation. The projection’s diagonal bound gives ), where) is the paid regular source bound. The schedule pays 4 and the reset fees, giving the stated correction radius and the unit bound used in (DR7). The regular and forced bounds after the projection and the two dictionary fusions are paid independently by those same fees.

To see the normalization without an extra characterization of the correction, let) be the primitive scalar before reset and let 1) be its attached logarithms at the identity. The exact exponent calculation gives the installed scalar

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v(b) = τ(b) −| Sitei |−1 X
j
lj(b, 1).
(FCA11)

The resulting state has regular entries zero at the identity, no identity flags, and kinetic and loop actions zero there. Its density at the identity is therefore The already proved all-configuration density identity makes this equal(1), proving the logarithmic formula in (FCA9). Thus (FCA9) and (FCA11) are the same scalar normalization. The interval estimate for this same projection value is obtained by the probe argument below. We must still show that this projected correction lies in the purchased correction interval. The source uses a nonflat probe for this purpose, on the very same trial family. Its activity terms are independent of the correction dial: they have already been computed with the Hessian projection value ∆. Only the scalar coupling varies in that trial. Thus there is no second reset algorithm.

Here is the probe construction, uniform in the even torus side. Fix the clip’s positive direct coefficient and its common cubic/quartic path bound0. A nonzero Lie-algebra direction gives a group curve through the identity with nonzero curvature derivative. Its chord is therefore

positive somewhere; continuity and the intermediate value theorem give 0 such that every chord in [0] is attained. Put

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t = min{ε, 1/4, a/[512(1 + C)]},
η = at2/4 > 0.
(RSP1)

Choose with chord. Set all links to the identity except direction-zero links whose directionone coordinate is even; set those links to This is periodic on every even side. Every (0 plaquette has holonomy or, and every plaquette has chord at most 4 Thus the field has no flags. Its direct plaquette charge in every signed-frame sum is at least where is the number of sites. The cubic and quartic terms have absolute sum per site at most 4, by (RSP1) and1. Averaging over the signed frames leaves

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K(V ) ≥nηθ2 > 0,
(RSP2)

where is the actual kinetic action. No smallness of the loop tensor enters this lower bound. Write) for the actual transferred density at a fixed input parameter, and set On any flag-free configuration its early representation is

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H(U) = exp{nτ −[β main(U) + α second(U) + loop(U)] −E(U)}.
(RSP3)

The source’s anchored regular budget is at most4, with = funds. Summing its anchors bounds4 at every flag-free configuration, including 1 and . Formula (RSP3) proves positivity at both probes and gives log4. Subtracting its two logarithms gives contrast error at most2 relative to main + second + loop.

On flag-free fields the main and loop actions equal their unmasked path expressions. The higher-order kinetic correction obeyssecond(. The stage’s cubic payment is 2. Therefore, with) the zero-coupling loop background,

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|BK(V ) + L0(V ) −[log H(1) −log H(V )]| ≤nF.
(RSP4)

This is the actual contrast estimate, with its and loop terms retained. Let be the final regular allowance, 1] the correction radius. The reset price is chosen before history selection so that (RSP5)

This is paid by the coupling-entry delay:, and1, whereas the monitor; it suffices to impose 2(. This same condition is imposed for both permitted adjacent-stage choices. For the fixed trial activities, let be their regular action at

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R(z) = (B + z)K(V ) + L0(V ) + AV −[log H(1) −log H(V )].
(RSP6)

Their budget gives. Equations (RSP2), (RSP4), and (RSP5) imply 2. Since ), this affine function has a unique root in []. Its root varies continuously with the input: the contrast and coupling are continuous, the fixed-trial activity series are continuous by their common deviation bounds, and 0 is fixed.

The projected state already satisfies the full density identity, so evaluating it at 1 gives More explicitly the exact fixed-trial identity is Thus the unique interval root is exactly the previously defined Hessian projection ∆. This proves its radius bound and transfers the interval service’s continuous curve to the actual represented state. The independent projection estimate∆ retains the sharper paid bound. The all-configuration

density equality is still supplied by the exponent and gas identities, not inferred from equality at two probes. In the source the corresponding shape comparison is also explicit: for any configuration , the trial shape is its value at ∆times; the inequality bounds its discrepancy by a finite configuration-dependent constant times, using 0. Hence the interval and projection formulations are fully compatible on this same trial family. For a compact input curve, the connected-expansion tails can be made small at one common finite retirement index. Their absolute tail bounds and the finitely many local bounds permit that choice throughout the curve. In the deterministic recipe the index is chosen as a function of the incoming address layout. A curve’s layout is constant, so that index is constant on the curve. The identities for the primitive expressions show that changing the incidental local guide or the point at which the curve was presented does not change the resulting state: all such expressions have the same tensors, canonical matching coordinates, source list and scalar reset. This supplies the continuous forward rule used above without silently choosing new dictionaries at different parameter values.

The source-level continuity and admissibility statement can now be written without suppressing its series hypotheses. Fix a closed parameter interval and an incoming curve) in its admissible band. Use one point as reference. The incoming path equals the prescribed path at every point, and the regular and forced layouts equal those at. The matching maps therefore use identity label correspondence on these dictionaries. Local physical guides may be selected separately at each point; the matched primitive expression is independent of that selection. Transport all entries to the reference layout. This fixes the unretired source dictionary, its support windows and the finite debris predicate used for retirement.

For its source), the actual continuity assertions are the following. Each regular entry, with all permitted derivatives on its fixed healthy support, and each forced entry on all configurations, converges as within. Their absolute support-weighted envelopes have uniformly small tails near. These two facts imply the deviation budget tends to zero: given 0, choose a finite set of addresses leaving total tail less than2; then choose a neighborhood on which the finitely many remaining differences cost less than2. This uses no continuity of flag indicators as functions of the gauge field: their support and threshold predicates are held fixed in the compared evaluation.

On a compact input interval one retirement index can be used throughout. The uniform tail estimate for the unretired source and compactness give an eventual regular tail bound 0, while the local finite-volume comparison gives an eventual bound for all the retained local beacon entries. Intersect these eventual conditions, choose one integer above their thresholds, and retain it at every parameter value. The resulting source obeys the explicit regular estimate

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BudgetPi(a(t)) ≤(1/16 + 2/16)Ei + turnFee(Pi)g + g,
(FCC1)

where = forecastE. Each beacon on its fixed neighbor support has the derivative bound gleam The source graft combines the retired regular entries, beacon entries and the fixed local profile. Its forced ingredient estimate is paid by its original forced allowance: the primitive forced part costs at most forecastK8, while the added local profile has size at most the prescribed reserve after its cover weight. The local gluing estimate sums these contributions into forecastK). The turn fee times the local beacon bound is included in the regular price. The stage delays make the right side of (FCC1), with that additional beacon price, at most. This proves the regular and forced local estimates for the grafted source simultaneously at every parameter value.

The uniformity over all presentations of one layout requires a separate argument. A layout is the regular and forced address lists, including supports, without coefficient functions. If no admissible state has a layout, its assertion is empty. Otherwise choose one such state. Form the canonical curve on the entire closed incoming coupling band by retaining its path, cutoffand address lists,

setting every regular and forced coefficient function and the vacuum scalar to zero, and assigning coupling. Keep the address symmetry permutations. Zero functions satisfy every derivative and hit bound, so this curve remains admissible and is continuous.

Choose local guides along this canonical curve and match their expressions to its reference layout. At a fixed coupling the weighted connected series has a summable majorant. For a target tail bound 0, take a finite set of terms outside which its sum is below2. Each of these finitely many terms eventually leaves the debris set: a term is either kept or eventually appears in the finite retirement roster. Hence one depth removes them all. Continuity in the deviation budget bounds the change of the entire weighted series by2 in a neighborhood of that coupling. The retired part is a subfamily of the same majorant, so the same neighborhood works for every larger retirement depth. Finitely many such neighborhoods cover the closed coupling band. Their maximum depth supplies a common tail bound Intersect it with the eventual finite beacon bounds, taking = gold Now let) be any other admissible curve with this layout and any allowed guide choices. At a point, compare its guide with the canonical guide at. The paths, lists, cutoffs, budgets and couplings agree. The matching map is the identity on the lists, and its induced relabeling preserves the finite retirement roster and debris predicate. For a debris address the primitive term does not read the incoming coefficient functions: it is an omitted foreground cluster formed from the common geometric and Gaussian factors, or an unretired scalar path term. Matching preserves its connected coefficient, integrand and scalar coupling. Thus the two debris functions and beacon contractions agree: 1

This is equality of local functions, so it preserves every derivative envelope tested in the budget. Transfer the canonical tail and beacon bounds through it. Adding the principal unretired contributions gives (FCC1) for the arbitrary presentation. The threshold therefore depends on the layout, stage, depth, period and fixed prices, but not on the curve, its coefficients or its guide choices.

Apply this construction both with standard allowances 116 and with the allowances needed for the contraction. Intersect the two eventual sets with, and choose one integer ) for each layout. Along every admissible curve is constant, hence so is). The matching identities identify the deterministic rule with the fixed-index continuous formula on that curve. No continuity of this integer-valued selection as the layout changes is needed. For the deterministic operation, the retirement index is selected once from the incoming layout, at least the terminal depth, to satisfy these bounds for every compatible presentation of that layout. The universal

retirement estimate supplies such an index. Because a curve has one layout, this selected index is constant on the curve. The primitive changes of local guide leave the matched source unchanged, including its finite debris predicate and beacon contraction. Thus the deterministic moment state agrees with the continuous curve construction just described. Restricting the interval in the shooting argument changes neither index nor dictionary.

The Hessian projection is continuous in the resulting deviation budget. For two source values , its bound is deviation (FCC2)

The threshold is a fixed positive number at this step. The rebate, fault, freshness and loan difference identities are linear in the source difference and in ∆(), as proved in the reset chapter. Their summable quote envelopes therefore pass continuity and uniform tails to the reset entries. The finite-support dictionary fusions preserve these properties by their absolutely convergent product bounds. In particular both the final coefficient families and their actual fixed address layouts form a curve in the stated sense.

Finally the local primitive identity identifies the transferred density with its early-value expression on every configuration. The reset’s exact exponent and gas identities then give its installed state with that same transferred density; its regular and forced closure estimates give all the coarse admissibility conditions. The scalar vacuum term is continuous by the same source-series estimate. Equivalently, evaluate the positive identity at the flat configuration to obtain its logarithmic normalization (FCA9). Together with continuity of the incoming coupling and (FCC2), this proves continuity of the output coupling as well. These are precisely the continuity, density, drift and band-free output conditions used in (FCA3); the subsequent interval restriction is what imposes the coarse coupling band. The calendar makes the following drift condition hold at every step:

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di+1 ≤s
4 log Li.
(FCA4)
Define narrower target intervals

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li = 3
4
 ̄β + 5
8sτi,
ui = 5
4
 ̄β + 3
2sτi.
(FCA5)

Since0, they obey and [. If the fine coupling in (FCA3) equals, then

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β(T) ≤li+1 −s log Li + di+1 ≤li −1
8s log Li < li.

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If it equals ui+1, then

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β(T) ≥ui+1 −s log Li −di+1 ≥ui + 1
4s log Li > ui.
(FCA6)

These strict crossings require no monotonicity or derivative bound on the coarse coupling as a function of the bare coupling.

We use the following elementary interval restriction. Suppose is continuous on [ ). Let be the first point at which, and let be the last point in [] at which . These points exist by the intermediate value theorem and compactness of the two level sets. Then and]. A value belowwould force a later crossing ofbefore value above would force an earlier crossing of. Both contradict their definitions. This argument also covers

Starting at, restrict the bare parameter to [] in (FCA2). Descend by finite induction. If an interval already carries trajectories through levels and its endpoint level- 1 couplings are, apply (FCA3) to their level- + 1 curve. Equations (FCA6) and the preceding interval restriction give a closed subinterval on which its level- coupling stays in [ and takes these exact endpoint values. Retain the already constructed higher states and insert this new coarse state. Every higher density identity, startup condition and continuity estimate survives the restriction.

At level zero we have a nonempty closed bare-parameter interval [] and complete forward trajectories (FCA7)

Their finest coupling is exactly; each level is continuous in the stated sense; all states are admissible; and every adjacent pair satisfies (FCA3). Their bottom couplings take the endpoint values 3 ̄ and 5 ̄4. Thus every bottom value in this interval is attained by a trajectory of the same recipe.

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In particular choose b ∗
N,p with bottom coupling  ̄β, and set RN,p,i = R
b ∗
N,p
N,p,i. This constructs the
calibrated finite histories for every N, p.
The same retained arc supplies the auxiliary trajectories at nearby prescribed bottom couplings
used in the confluence argument. It supplies whole trajectories, not just scalar sequences satisfying