14 The analytic stationary graph, loop moments and the cali- brated coupling history · Part 7

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the drift inequality. No uniqueness of an inverse shooting map has been assumed. Determinism as-
serts only that equal finest states give identical forward trajectories; the later contraction argument
establishes the required uniqueness of their limiting calibrated data.
There is also a uniform interior margin. If the bottom value of any successful trajectory lies in
[3 ̄β/4, 5 ̄β/4], then
3
4
bβi ≤β(Ri) ≤5
4
bβi
(0 ≤i ≤N).
(FCA8)

At = 0 this is the given bottom interval. If it holds at, solve (FCA3) for the fine coupling and use (FCA4): its increment lies between (34) log and (54) log. Adding these bounds proves (FCA8) at + 1. This margin justifies the nearby-trajectory comparisons while retaining the original full bands.

Finally, assembling these states into the history preserves the common path at fixed, its period independence, the prescribed budget and clip, the finest bare Wilson law, and all exact push identities. Induction in the push integral identity gives the same finest Wilson partition function at every coarser level. It is this calibrated history, rather than a separately selected sequence of couplings, that enters the limits below.

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Source–readout formulation.
This subsection is where the scale index of the realized family
acquires its operations. The source objects are represented states; the readout is evaluation to a
density; the operation to be lifted is the block integration Pβ
i of densities; and the calibration is a
condition on a scalar readout of the whole history. The four groups of statements below correspond
to these four items.
States and the forward operation.

Definition 14.19 (Represented state and its readers). A represented state at level is

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Q =
path tensors, clip, budget, β, v, (Ej)j, (Kj, Pj)j, address layout

,

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with readers β(Q), v(Q), the layout L(Q) (address lists with supports, without coefficient functions)
and the density
Q. eval(U) = exp
v|Sitei| −βApath(U) −Exp(U)

 Gas(U).

It is admissible if it obeys the local regularity, symmetry and absolute activity bounds of the nonlinear step and the retained startup and transition conditions, including. A family is continuous if and are continuous, the layout is constant, and differences of activity values tend to zero in the absolute weighted derivative budgets (regular terms tested on the common healthy domain in the prescribed order and directions; forced terms on every configuration).

Theorem 14.20 (The forward recipe is an operation lift). At each there is one deterministic map on admissible level- states such that

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Readeval ◦TN,i,p = P β(·)
i
◦Readeval,

equal inputs give equal outputs (equality of represented moment states, including activities), continuous input curves give continuous output curves with the same layout, and 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|
,
Hb = P β(Q(b))
i
(Q(b). eval),

where by positivity of the push kernel and of the retained vacuum, and

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∆(b) = Averaging. delta
X. read(a(b))

is the source Hessian projection for the fixed listing and the parametrized source

Proof. Keep one primitive construction with its ordinal, tree choice and fixed support dictionaries; the output is the moment state it produces. The local integral comparison and the absolute activity estimates give continuity in the stated budgets, the reset identity gives the density equality and its coupling bound the drift inequality. In the exact normalized splitting the output trunk density times its attached-log exponential is(1); installing (FCA9) multiplies by exp(Site which proves the density identity at every configuration, including flag boundaries. (The site count, not the bond count, is what cancels.) The affine-curvature Hessian projection, rebates, faults and

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low- and high-path loan identities of the coupling-reset exponent prove the all-configuration repre-
sentation with the value (FCA10); in particular ∆is not defined by a two-probe contrast equation.
Equivalently, with τ(b) the primitive scalar before reset and lj(b, 1) its attached logarithms at the
identity,
v(b) = τ(b) −|Sitei|−1 X
lj(b, 1);
j

the resulting state has regular entries zero at the identity, no identity flags and vanishing kinetic and loop actions there, so its density at the identity is, which the density identity equates with(1). Thus (FCA9) and (FCA11) are the same normalization.

The probe and the dial.

Proposition 14.21 (The interval root is the Hessian projection) the affine function has a unique root in, depending continuously on the input; and , so that root is exactly ∆of (FCA10). Hence∆

Proof. The first inequality is (RSP4) with, and the second is (RSP2) with (RSP5). ) with 0 gives a unique real root, of modulus The projected state satisfies the full density identity; evaluating it at 1 and gives(∆) = 0, hence ). Continuity follows from continuity of the contrast, the coupling and the fixed-trial activity series, ) being fixed.

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Layout uniformity.

Proposition 14.22 (Uniformity over all presentations of a layout). For each layout carried by some admissible state there is one integer such that (FCC1) holds for every admissible curve with that layout and every allowed guide choice, with a threshold depending on the layout, stage, depth, period and fixed prices but not on the curve, its coefficients or its guides.

Proof. Choose a state with layout and form the canonical curve on the closed incoming band by retaining its path, cutoffand address lists, setting every coefficient function and the vacuum scalar to zero and assigning coupling, with the address symmetry permutations kept. Zero functions satisfy every bound, so the curve is admissible and continuous. At a fixed coupling the weighted connected series has a summable majorant; for a target 0 take a finite set of terms outside which the sum is below2. Each of these is either kept or eventually in the finite retirement roster, so one depth removes them all; continuity in the deviation budget bounds the change of the whole weighted series by2 in a neighbourhood, and the retired part is a subfamily of the same majorant, so the neighbourhood works for every larger depth. Finitely many neighbourhoods cover

the band; their maximal depth gives a common tail bound, intersected with the beacon bounds = gold 0). For another admissible curve) with the same layout compare its guide at with the canonical guide at: paths, lists, cutoffs, budgets and couplings agree, the matching map is the identity on the lists and preserves the finite retirement roster and debris predicate, and 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). Hence

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1debris(d,i)Wi(Q(t)) = 1debris(d,i)Wi(Uβ)

as local functions, which preserves every derivative envelope; adding the principal unretired contributions gives (FCC1). Apply this with the standard allowances 116 and with those needed for the contraction, intersect the eventual sets with and choose

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Shooting.

Lemma 14.23 (Interval restriction). Let be continuous on with. Let be the first point with and the last point of with. Then and

Proof. The points exist by the intermediate value theorem and compactness of the level sets. A value below] would force a later crossing ofbefore, and a value above earlier crossing of; both contradict the definitions. The case is included.

Theorem 14.24 (Calibrated finite histories). For every there is a nonempty closed bare- parameter interval and complete forward trajectories

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b 7−→(R b
N,p,i)0≤i≤N

of the deterministic recipe such that the finest coupling is, each level is continuous and admissible, every adjacent pair satisfies (FCA3), and the bottom couplings take the endpoint values 3 ̄ and

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5 ̄β/4. In particular there is b∗
N,p with bottom coupling exactly  ̄β; put RN,p,i = R
b∗
N,p
N,p,i. Moreover any
successful trajectory with bottom value in [3 ̄β/4, 5 ̄β/4] satisfies

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3
4 bβi ≤β(Ri) ≤5
4 bβi
(0 ≤i ≤N).

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Proof. Start with [lN, uN] in (FCA2) and descend. If an interval carries trajectories through levels
i+1, . . . , N with endpoint level-(i+1) couplings li+1, ui+1, apply (FCA3) to the level-(i+1) curve;
the crossings (FCA6) and Lemma 14.23 give a closed subinterval on which the level-i coupling stays
in [li, ui] and attains both endpoints. Retain the higher states and insert the new coarse state; every
higher density identity, startup condition and continuity estimate survives restriction. At level zero
l0 = 3 ̄β/4, u0 = 5 ̄β/4, and the intermediate value theorem gives b∗. For (FCA8) induct upward:
solving (FCA3) for the fine coupling and using (FCA4), the increment lies between 3
4s log Li and
5
4s log Li, while bβi+1 −bβi = s log Li.

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The calibration fibre.
The bottom-coupling map BN,p : [aN,p, bN,p] →[3 ̄β/4, 5 ̄β/4], b 7→
β(R b
N,p,0), is a continuous surjective readout of the bare parameter. The calibrated history is a
point of the fibre B−1
N,p( ̄β). At this point of the argument nothing makes this fibre a single point:
no uniqueness of an inverse shooting map is assumed, and existence used only the intermediate
value theorem.
Uniqueness within the retained recipe is proved afterwards by the comparison

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weights (Theorem 14.29); the uniqueness of the limiting calibrated data is a further statement
obtained later by contraction. Determinism says only that equal finest states give equal forward
trajectories. The coupling normalization is thus imposed at the bottom (value  ̄β) and propagated
upward by (FCA8); it is a calibration condition on the realized history and not an output of ad-
mission. All objects are gauge invariant or numerical, so the frame lift of a represented state is
QZ = (. . . , (Ej ◦π)j, (Kj ◦π, Pj ◦π)j, . . . ) with the same β, v and layout; by Theorems 9.4 and 7.2
it is admissible exactly when Q is, QZ. eval = Q. eval ◦π, the push identity (FCA3) holds for the
lifted recipe T Z(QZ) = (TQ)Z, and BZ
N,p = BN,p.

14.16 Exact density identity after retirement

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The retirement used above is an exact decomposition before it is an estimate. Keep one anchored
listing for the whole input curve. For its full regular source write Afull(V ) = P
j aj(V ). Partition
its indices by the chosen retirement predicate and write aj = akeep
j
+ atail
j
, including their original
healthy masks. The same predicate is used throughout the compatible curve. The local absolute
envelopes make both subseries summable, hence

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Afull = Akeep + Atail.
(RET1)

The forced gas is retained in the kept source exactly; it is not split into two multiplicative copies. If the density of the full source is

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ev|Λ|−H(V )−Afull(V ) G(V ),

then the identical integrated density after retirement is

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e−Atail(V )ev|Λ|−H(V )−Akeep(V ) G(V ).
(RET2)

This follows by substituting (RET1) in the exponential; in particular the tail has a minus sign. Summability is needed here to identify the sums in (RET1), rather than merely cancelling finitely many formal letters. Replacing the old listing by the anchored common listing preserves each masked occurrence, so it preserves this equality on all configurations.

The full-source density identity follows from the connected expansion already established. Its empty-flag pool is strictly positive and equals the exponential of its absolutely convergent connected logarithm. The scalar normalization contains the positive Gaussian and coordinate prefactors, the incoming scalar volume term, and the reference-forest normalization. Taking their logarithm per coarse site defines. The cubic, quartic, loop and local correction scores have the exact totals displayed above. Their signs cancel those terms in, leaving precisely the logarithm of the positive pool. Multiplying by the forced pool then recovers the original conditional integral, including its prefactors. Thus (RET2) holds before either the kept source or the tail has its sharper output budget.

The kept-source estimate and the tail estimate are purchased separately on this same listing. The former gives the desired small regular price together with the original forced forecast; the latter is the remainder used by the finite reference beacons. Reattaching that remainder restores (RET2). Consequently neither continuity nor projection replaces the exact density by a merely close one.

Source–readout formulation. Retirement is a change of presentation within one fibre of Read: the pair () is a finer record than, the sum map (RET1) is the readout, and (RET2) is the statement that the density read from the finer record is identical to the density

read from the coarser one. Neither continuity nor projection replaces the exact density by a merely nearby one. The kept-source estimate and the tail estimate are purchased separately on the same anchored listing, and reattaching the tail restores the identity. The forced gas is kept in the kept source exactly and is not split into two multiplicative copies. The exactness class is exact recovery; the estimates on the two parts are additional information about the finer record and do not enter the identity.

14.17 Finite reference beacons and weighted retirement continuity

Reference views as coefficient readers. A beacon is a prescribed finite reference volume used to read a local coefficient from a represented density. Its comparisons must keep the folding maps and boundary-face contributions. The selected finite difference removes the stated face terms and retains the interior coefficient. That is deliberate information removal with a proved annihilation identity, not recovery of the full boundary-dependent density. The coefficient reader and its error bounds are then used in the common reference chart and the stability estimates.

The finite beacons used in (FCC1) have an explicit five-volume construction. For integer half-width, use reference torus sides 2( 4, and coefficients

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ρj = (1, −4, 6, −4, 1)j
24 4!
.
(RB1)

Fold each coordinate of a reference torus by reflection into its interval of length. To pull a configuration to this folded torus, a positively traversed original edge is read directly, a negatively traversed edge is inverted at its lower endpoint, and a collapsed edge is the identity. Translate this folded view to a chosen coarse anchor. Denote the resulting group configuration by . This finite word map defines the views even at folded faces.

Fix the local read radius, with 2. A position in a folded coordinate has a type: within of the left face, within of the right face, or in the interior. A boundary type records the face and its distance 01. Its number of preimages under folding is independent of; an interior type has 2( preimages. In four coordinates a full type therefore has a multiplicity which is a product of four constant or affine functions of. The fourth difference (RB1) kills every such product having a constant factor, and gives one on the product of four interior factors: its fourth difference is 4! 2. Hence for any function of the four-coordinate type,

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4
X
j=0
ρj
X
x∈torus2(w+j)
f(typej(x)) = f(all-interior type).
(RB2)

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This is a finite counting identity, including faces and edges of every codimension.
Let fj(x, V ) be the compatible anchored local reference densities. Center them by f0
j (x, V ) =
fj(x, V ) −fj(x, 1). Compatibility means that on healthy configurations two such readings agree
whenever their radius-r cover inputs agree, and they obey the exact signed and translation laws.
Define the beacon at o by

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4
X
j=0
ρj
X
x
f0
j (x, Sj,oV ).
(RB3)
bo(V ) =

Choose its support radius + 4. Introduce also the shifted expression which uses anchor stage) for the-summand and masks that summand on its radius- neighborhood. Summing this expression over gives exactly the sum of the masked beacons, by the finite translation permutation of the torus. On a healthy radius-2 neighborhood all these masks are one. Their folded local

readings depend only on their four-coordinate type; compatibility and the signed laws identify equal types. Equation (RB2) reduces the shifted expression to the original centered local density at. The all-interior type reads its unchanged cover neighborhood. Thus the difference between the assigned retained density, its identity value and this shifted expression vanishes on that healthy neighborhood. This is the local remainder used in the forced part of the graft. Its global identity follows by summing over; there is no discarded face correction.

The derivative allowance of (RB3) is equally concrete. If the centered full reference sum on side ) has left-derivative bound on the small reference chart through order, and the ambient budget has direction power, then

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4
X
j=0
|ρj| 2k(1 + z)pk qj.
(RB4)
∥bo∥budget ≤

On a healthy neighborhood the folded reference configuration can be gauge transformed into that small chart, by the local axial chart construction; its displacement leaves the fixed margin required by the reference estimate. Gauge invariance transfers that estimate back. Each pulled-back direction reads one original edge or its reversed adjoint transform. Its norm is bounded by (1 + reversal preserves the color norm. Repeated left derivatives of the finite word substitution produce at most the 2 factor in (RB4). Finite summation over the five centered reference sums proves the displayed allowance. These are the actual finite offers used for the beacon, with their common threshold and derivative order. The’s in (RB4) are supplied by the finite reference construction. Putorder, turnFee( = gold( 0, andSite(2(. Its actual allowance is

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qj = k! h 3k + 2Vj{s(h + h) + h}.
(RB4a)

The centered chart expression has boundsD, at every admitted point of the chart of radius = eaves(), with margin 2 furl(). The logarithm of each varying link recovers its local coordinate. If bounds that logarithm and is the size of a permitted left variation, the scheduled inequalities are

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(1 + royalty(G))C ≤1,
k!J(1 + royalty(G))C ≤p.

The chain rule bounds the coordinate map through order by, with slope one. Composition with the chart expression costs at most! !

On the strict healthy chart the exact decomposition expresses this chart function as a constant minus the retained full reference sum minus the summable activity source. The source budget bounds the latter by, through the same left-derivative order. Centering at the identity costs at most twice this number; derivatives of positive order kill the centering constant. The triangle inequality proves (RB4a). To include a healthy endpoint on the boundary of the strict chart, approach it along1. This path keeps the prescribed chart margin and is eventually strictly healthy. Regularity of the retained reference function gives continuity of each left derivative at the endpoint, and hence the same bound there. No threshold indicator is differentiated.

The reference estimates hold eventually in their approximation depth uniformly on the compact coupling band. The five reference volumes therefore have a common depth. The actual result after folding is

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4
X
j=0
|ρj| 2k(1 + z)B.power k qj.
(RB4b)
gleam =

The radii and reference cardinalities have rate zero, and has rate20; so does this allowance. The small chart required here is supplied on any healthy neighborhood by axial gauge: its link distances are bounded by a fixed multiple of (2 times the threshold. The interior extension and its inverse chart place it inside radius2 furl. The radius has rate zero and the threshold rate2, whereas decays more slowly; the scheduling thus gives the strict margin eventually.

The reference offers are indexed by the coupling itself: at a curve parameter, the selected offer is the one at). It is therefore the same offer for two presentations with the same coupling, independently of how that point was reached. The fixed-reference finite-volume estimates hold eventually in their approximation depth simultaneously on the compact coupling band; (RB4b) gives the number denoted by gleam in (FCC1). The compatibility needed above comes from the same local cylinder formulas in the different reference views.

Here is also the weighted-tail step behind the retirement argument. At one finite torus all supports have cover number bounded by its site count. Thus multiplying any summable envelope by its support exponential, even the early exponent + 3, costs only a finite constant depending on that torus. Changing its derivative anchor also costs a finite constant:

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K =
(1 + D)p
k,
(RB5)

where is the maximal torus separation; the direction triangle inequality gives this bound. Consequently the local source estimate supplies one summable envelope in the common anchored listing. If the debris indicator of each fixed address is eventually zero, dominated convergence for series makes its weighted debris sum tend to zero at that parameter. This eventual removal is literal: a nonkept address eventually belongs to the finite retirement roster, whereas a kept address never belongs to the debris.

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Continuity in the deviation budget extends that small-tail bound to a parameter neighborhood.
More explicitly, spend half a requested allowance on the tail at the center and half on the full source
difference to a neighboring parameter, converting the latter to the common anchored weighted norm
by the finite constant (RB5) and its support-weight counterpart. Restricting the difference to the
debris subfamily cannot increase its positive majorant. Thus the neighborhood works for every
sufficiently large depth.
A finite subcover of the compact band gives a single depth.
Combin-
ing it with the uniform beacon depth proves the simultaneous retirement used in (FCC1). The
canonical fixed-layout curve argument already given then transfers this depth to every compatible
presentation; the construction here supplies its weighted continuity and actual finite beacons.

Here are the reference and continuity inputs with their finite-volume dependence exposed. In this section alone is a fixed reference side, is the fixed budget,order, power, andsupportExponent. The five signed fourth-difference coefficients in (RB1) are denoted throughout by; the formerly used for them in (RB4b) mean the same coefficients.

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They are unrelated to the parameter λ in the local fusion argument.
The representation constant used here is royalty(G) = √m, where m is the dimension of the
fixed faithful unitary representation. In particular it is at least one. It is distinct from the Fourier
aliasing price, which depends on a strip width and an analytic norm.
The reference-chart radius and margin have the explicit values

log log log (RB6)

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The logarithm of the normalized finite Gaussian integral is centered by subtracting its value at
the identity. On the reference chart ∥P∥+ 2d0 < p0, with no flags, its derivatives through the
purchased order satisfy ∥Dlf(P)∥≤hp−l
0 . The lower bound on its unnormalized integral is strictly
positive. These statements are obtained on the actual reference forest and its Gaussian marginal,

with the same stationary section and coupling, by the Gaussian differentiation and connected estimates (ASP2a)–(ASP2c), the local integral normalization (PRI2)–(PRI5), and the logarithm series (ASP1)–(ASP2). In particular the bound concerns the centered logarithm, not the integral itself. The reference side must exceed the prescribed patch and covariance read radius. The path width, path norm, loop allowance, correction parameter and derivative horizon are fixed first; the odd block length is then above their finite lower bounds; finally is above their common delay. Bounds for the two adjacent loop allowances and the correction parameter put both sides in these same estimates.

The exact logarithmic density identity on the strict healthy chart has the form

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f(P) = c −
X
y∈Site(n)
Fy(actual P) −
X
α
Aα(actual P).
(RB7)

Here is the retained finite ordinary reference density and the emitted regular source. Their actual occurrence indices and supports are retained in the sum. The nonretained source consists of the omitted ordinary occurrences and omitted short stationary paths, not of a difference between Gaussian integrals on different tori. Its anchored early allowance is at most once its approximation depth is large. The retained-source part has allowance at most turnFee(). Summing these two bounds gives turnFee( per site. Summation over sites and centering (RB7) gives the second term of (RB4a); the reference logarithm gives its first term after the logarithmic chart chain rule already proved there. This explains every summand in

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We give the tail argument that supplies the omitted-source allowance used in (RB7). A regular
address is a pair (oα, Xα), its anchor and finite support. For nonnegative numbers aα, its early
bound u means
X
sup
aα exp{(e + 3) covB(Xα)} ≤u.
(RB8)
o
α:oα=o

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Sum over the V anchors. Since each exponential is at least one, P
α aα ≤V u. This proves absolute
summability of the derivative envelopes, including derivatives in any fixed finite list of directions:
divide those directions by one common scalar until they satisfy the threshold bound, apply (RB8),
then multiply by that scalar to the derivative order. The same scalar works for every address.
Termwise differentiation on the strict healthy open set therefore follows from uniform absolute
convergence of the derivative series through order k.
Changing every anchor to the one in the common listing multiplies its envelope by at most
Kanc = ((1 + D)p)k, as in (RB5). On this fixed torus,

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covB(X) ≤V,
Ksup = exp{(e + 3)V }

bounds every early support weight. Thus an unweighted summable envelope gives the summable weighted envelope. For the debris predicate

is not kept and does not belong to the depth- finite roster

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one has 1Pm(α) →0 at every fixed address: ordinary occurrences exhaust their finite shelves and
short paths exhaust their length cutoffs. Dominated convergence for the nonnegative series yields
P
α1Pm(α)gα →0. This proves a small early bound, rather than merely a small unweighted sum.

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For clarity, source deviation has the following meaning. Two sources have identical regular and
forced address layouts. There are nonnegative envelopes cα, dα such that on each regular healthy
domain every left derivative through order k of their difference is bounded by cα, and on every
configuration every forced coefficient difference is bounded by dα. Both total weighted sums
X
X
ee covB(Xα)cα,
ee covB(Yα)dα
α
α