Convergence of the interacting scale histories · Part 4

The choices of these supplier functions use the original stage profiles and price lists. Raising the floor changes neither that profile nor either list. Apply (CAL1)–(CAL11) with this floor to select the block lengths once. They now determine the local patches at all later steps.

For each fixed requested finite horizon and profile, the local construction gives a positive width1. Its reciprocal is a countable numerical demand. More generally, let be any already selected demand array. Define the finite envelope

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j
X
j
X
HF (j) = 1 +
i=0
|F(n, i)|.
n=0

Increase only the coupling-entry delay at profile to at least exp(exp)). At every actual step , with outgoing profile + 1), the admissible clock then obeys

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HF (J(m + 1)) ≤log log bβm.
(CAL13)

For each fixed, the divergence and monotonicity of give a first step after which For all such steps and every + 1), the corresponding summand in the finite envelope yields

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|F(n, i)| ≤1 + log log bβm.
(CAL14)

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This buys every fixed demand eventually. It makes no assertion that infinitely many horizons have
been paid at one finite step. Apply it first to F(E, j) = ρ−1
E,j.
After the block lengths have been fixed, the local stationary and comparison estimates also have
finite coupling bills depending on the resulting patches. Let B1(j) and B2(j) be their numerical
bounds, and put B(j) = max(B1(j), B2(j)). Add this function to the entry delay by taking another
maximum. This gives, including at step zero,

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B(J(m + 1)) ≤bβm.
(CAL15)

If these bills also have a countable horizon parameter, apply the finite-envelope construction to that array as well, in addition to the all-step bound for its distinguished horizon. Every such enlargement is made with the lengths held fixed. The requirement is therefore a numerical delay condition, rather than a circular demand to choose a block length larger than a function of a subsequently reselected block length. Intersecting finitely many delay conditions is achieved by successive maxima. Each persists under all later compatible enlargements, so the earlier treasury, period, local geometry and observable conditions survive.

The additional local calendars use exactly the same finite-max construction. To spell out their quantitative admission, enumerate their finite requests by natural-number codes. Request specifies an integer precondition) and a finite real cost) for a legal adjacent pair where is the fixed stage profile. Before selecting lengths, include

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p∗(j) =
max
code(q)≤j pq(j)
(CAL16)

in their floor; an empty maximum is zero. After those lengths are fixed, there are only finitely many costs with code( and. Increase the entry delay until all are at most

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z( bβ) = 1 + log(2 + log bβ).
(CAL17)

This is possible because. For example a sufficiently large double exponential of their finite maximum suffices. The legal flight in each cost is the one formed from the already selected

, patch, kernel and stage; it is not a cost obtained from an independently chosen history. At an actual step, substitute) and. Monotonicity of implies that a request whose code is at most) has both its floor and its numerical cost paid at every . In particular every fixed request is paid on its admitted tail. All the post-length maxima are held fixed when increasing the delays again, so the statement persists under every subsequent compatible increase.

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For the stationary comparison, the finite-list assertion has a concrete index bound. A request
records an input degree d, an extended degree D > d, and a jet order k, and is admitted only when
2(D +k +1) is at most the incoming smoothing grade. At outgoing grade j, there are consequently
only finitely many such triples and adjacent predecessors.
The maximum of their local length
thresholds is the integer floor used in (CAL12). On the fixed patch, take a second finite maximum,
at least one, of their polynomial estimate constants, the local stationary constants, and all reciprocal
widths ρ−1
E,i with E, i ≤j. Call it Rj. Its post-length request is eRj; payment by bβm is exactly
Rj ≤log bβm. Every auxiliary derivative index in a fixed triple ranges over finite intervals, so it
introduces no supremum over arbitrarily high orders at one stage. This proves the finiteness of the
native stationary and comparison prices used above, at their actual chosen lengths.

This includes the finite jet, local coordinate and path readouts together, with one selected system of their dictionaries and bases. The inverse-map request has zero precondition. Its actual cost is the inverse estimate of the specified local patch map at its selected analytic width and axis forest. It can therefore be paid from step zero: at each, maximize its finitely many adjacent predecessor costs and use (CAL17). One does not need to assume a growth law for this inverse price. For a reporting window based at level, the affine running reference coefficient gives

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z( bβm) ≤z( bβk) + s|tm −tk| ≤(1 + s){2 + Zk + |tm −tk|},
(CAL18)

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where Zk includes z( bβk). To obtain the first inequality, differentiate z on v > 1: z′(v) = 1/[v(2 +
log v)] ≤1, and use | bβm −bβk| = s|tm −tk|. Thus the paid inverse cost has the same polynomial
reporting-age control required by the local comparison estimates, on the actual windows of the
actual history.

The common reference-period registry is fixed at the same pre-length stage. At outgoing grade its finite list consists of horizons and the allowed adjacent incoming grades. Their integer size requirements are included in (CAL16). With the resulting lengths fixed, its uniform finite-patch, chart and normalization prices give a real entry gate for that same list, which is included among the later delay maxima. This retains one registry for every eventual horizon and every reference torus used in the periodic-copy argument; a later numerical enlargement does not reselect it. The physical periods themselves are still chosen as in (CAL11).

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We explain the identification with the actual run, since it is needed for the order of quantifiers
in the theorem. The run’s stage profile is the original profile, its kernel is the original kernel, and its
factor at profile j is the length just selected. Hence (CAL12) at J(m + 1) is the actual step’s local
length inequality. Its stationary horizon satisfies the original primary price list. Projecting the two
maxima in (CAL15) gives each local patch payment, and therefore its stationary admissibility and
the delay required for the flat covariance limit. The preselected flat supplier may now be applied
with entrance index zero to this very horizon; the curvature and dual suppliers apply to it by the
other two floors. Their local results concern the same levels, patches and retained horizon as the
run. No new family is chosen when these numerical payments are consumed. The finite volume
and token comparisons are established afterwards on this fixed run, and only then is a requested
extraction order refined.

Paying the actual compulsory-emission forecast

The compulsory-emission estimate must retain its quadratic monitor exponent until it has been compared with the output forecast. Here is that comparison. Fix the block, the positive local emission parameter, and the finite output order. Put), with exp(log log. The integrated emission bound obtained from the packet and connected-cluster estimates is ) exp (EF1)

This is the bound before weakening its exponent in the dictionary estimate. The requested strong forecast has form

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C exp{−at2},
C ≥1,
t = y−1/2p
β−1/2 ,
β ∈[y/2, 2y].
(EF2)

In particular 0. After has been selected at this block and chart width, choose the positive forecast rate32, and, if necessary, decrease its requested rate to at most that its strict positivity margin is retained. This is a choice of the numerical forecast, made before choosing its entry delay. It gives

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at2 ≤4ap2 ≤μp2/8 ≤μp2/4.
(EF3)

For any prescribed 0, choose the entry delay so that 2. Equations (EF1)–(EF3) then put the actual emission bound below times (EF2), uniformly for every coupling in the band. The source uses the weaker sufficient condition16; the chosen32 leaves room for subsequent attachments. These choices are made separately for each of the finitely many legal adjacent stage pairs and combined by their maximum. Thus the forced forecast used by the step, and later by local grafting, is purchased with its actual quadratic decay and on its actual chart. A previously weakened subquadratic estimate is not needed for this payment.

Uniform reserve at every step, including startup

The numerical purchases also leave a squared reserve for errors that will later be divided by local normalization factors. We derive that reserve, since eventual smallness at each fixed stage alone would not supply a bound at every step of the chosen history. Put

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M(x) = e(log log x)2,
κ(x) = e−M(x)1/5,
x > e.
(CAL19)

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Thus κ( bβm) is exactly the forced allowance at step m. Fix a positive stage rate q. Eventually
qM(x) ≥2. Since M(y) ≥M(x) ≥1 for y ≥x > e,

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qM(y)2 ≥qM(x)2 ≥2M(x) ≥2M(x)1/5,
e−qM(y)2 ≤κ(x)2.
(CAL20)

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The stage rate in this formula is the positive rate of the actual already chosen post-length estimates.
Choose its threshold separately for each stage and include these thresholds among the entry-delay
maxima. At step m, use the outgoing stage J(m + 1), x = bβm, and y = bβm+1 ≥x. This gives
the squared-reserve bound simultaneously for every m, with the same post-length data. The same
delay can enforce log bβm ≥1.
Startup has a corresponding reserve without requiring a preceding block. In the startup formulas
write s = log x, so their large parameter is Ms = e(log s)2 = M(x) and their small remainder
exponent is γs = e−Ms/2. The startup estimates established above provide its local expansion with

packet-energy charge, extracted suppression charge, and remainder charge coupling2. The connected support budget for that remainder is

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e−Mse eγs−1 −1
 ≤e−Ms(e −1) ≤e−2M1/5
s
= κ(x)2
(CAL21)

for sufficiently large. The first inequality uses1, and the second follows from . In particular it pays the full connected budget, including its exponential conversion from the local remainder bound.

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For each fixed stage the startup threshold is finite. Include it in the same entry-delay maximum.
At level zero the calendar identifies the initial stage with its first entry stage, so this threshold is paid
there too; at a successor level it follows from the previous entry payment and the positive coupling
increment. Consequently the chosen calendar gives (CAL21) at every possible finest cutoff. For
any actual coupling in [x/2, 2x], the startup expansion is still valid: its lower-coupling hypothesis
follows from the left endpoint, and its vacuum price is bounded using the right endpoint. The exact
startup density identity then turns (CAL21) into the required forced-activity bound. These choices
enlarge delays before the final calendar and history are selected, and preserve all earlier purchases.
They add no assumption to the theorem.

Inverse curvature coordinates and the startup fourth coefficient

The fourth-degree correction used in startup is determined by the Wilson plaquette price in curvature coordinates. It is separate from the earlier inequality bounding the difference between chord squared and curvature squared. Writeg for the projected curvature, and define

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k(a) = c(exp a).
(SQ1)

This real analytic map has(0) = 0 and. Indeed the exponential has derivative at zero, its inverse group factor has derivative, and taking the projected skew part of their difference divided by two returns. The finite-dimensional analytic inverse theorem gives open neighborhoods of zero and an analytic inverse with

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u(k(a)) = a,
k(u(z)) = z,
u(0) = 0,
Du(0) = I.

Set) = exp()). On a group neighborhood of the identity and a color neighborhood of zero,

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c(R(z)) = z,
R(c(U)) = U.
(SQ2)

For the second identity use the group logarithm chart constructed earlier: write = exp, shrink so that is in the inverse neighborhood, and substitute). Thus no global group logarithm or connectedness assumption is involved. Define the real analytic scalar function

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f(z) = price(R(z)).
(SQ3)

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Since R(0) = 1, the price is zero there. Its first derivative is zero because the Wilson price is a
squared chord. For R(tz), its matrix derivative at zero is the matrix of z, so the physical color
normalization gives
f(tz) = 1
2t2|z|2 + o(t2).

The second derivative is symmetric; polarization of this identity gives

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D2f(0)[z, w] = ⟨z, w⟩.
(SQ4)

Both inversion and conjugation preserve the price, while) and Apply (SQ2) to the nearby group elements and Uniqueness of the inverse curvature coordinates gives

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f(−z) = f(z),
f(Adh z) = f(z)
(SQ5)

as germs at zero, for every fixed. Differentiating these germ identities proves that every odd Taylor coefficient vanishes and every Taylor coefficient is adjoint invariant. The positive physical inner product is unchanged by the adjoint action. Use Taylor coefficients, including the factorial, to set

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Q4(z) = 1
4!D4f(0)[z, z, z, z].
(SQ6)

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This is the fourth-degree invariant polynomial called Q4 in (FST1). By (SQ4)–(SQ5), the Taylor
polynomial through order four is exactly 1
2|z|2 + Q4(z). Consequently

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f(z) −1
2|z|2 −Q4(z)
has all derivatives of orders 0, . . . , 4 zero at 0.
(SQ7)

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For a smooth family of actual group elements U(v) with U(0) = 1, (SQ2) identifies its Wilson
residual
R(U) = price(U) −1
2|c(U)|2 −Q4(c(U))
(SQ8)

with the composition of the remainder in (SQ7) and)), near zero. The chain rule shows that all derivatives below order five vanish: each term contains a derivative below order five of the outer remainder. This proves the exact flat-jet assertion underlying the startup estimate.

We next explain the uniform derivative bound, including arbitrary plaquette links and threshold equality. Let and . For a list of four-link directions , let) be the ordered left derivative of)), in the convention (AB3). It is jointly real analytic in the direction variables and the four matrix entries on their actual group locus. This follows by induction: differentiating the analytic expression along the analytic left increments preserves analyticity. It is homogeneous of degree one in every direction; in particular

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Jq(tA; U) = tqJq(A; U).
(SQ9)

Keep arbitrary and replace the first link by

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eU0(z) = exp(z)U3U2U−1
1 .
(SQ10)

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The resulting plaquette is exactly exp z. At z = 0, the residual has the vanishing property (SQ7).
Each ordered link derivative lowers this vanishing order by at most one: introduce one extra
real variable for its left increment, apply the vanishing composition to the joint variables, and
then differentiate in that variable at zero. Induction gives vanishing through order strictly below
max(5 −q, 0) for Jq(A; eU(z)).

For directions in a fixed ball, the external parameters consisting of these directions and the four group entries lie in a compact finite-dimensional set. On that set times the required derivatives of the analytic expression are bounded. Taylor’s estimate, obtained by integrating successive derivatives along the segment from zero to, therefore gives one such that if) = exp (SQ11)

This argument bounds actual left derivatives; it is not a differentiation of a healthy-set mask. The three unaltered links need not be small.

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The local logarithm estimate gives constants c0, b0 > 0 with | log V | ≤c0 chord(V ) for
chord(V ) ≤b0. Choose δ ≤min(b0, c−1
0 , 1). If chord(P(U)) ≤θ ≤δ, and |A| ≤rθ, apply (SQ11)
to A/θ, then (SQ9). It gives

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|Jq(A; U)| ≤Cq,rcmax(5−q,0)
0
θq+max(5−q,0) ≤C′
q,rθ5.
(SQ12)

For5 the last inequality uses; for5 its exponent is exactly five. All inequalities allow equality in the plaquette threshold. At a fixed derivative horizon take a maximum over its finitely many

At a plaquette anchored at, an allowed direction (AB4) restricted to its four links is at most, with depending only on the fixed power and the bounded plaquette distances from Pullback to these four links therefore converts (SQ12) into the derivative envelope used in (FST8). A plaquette wrapping around the torus still has these same four stored group entries and bounded periodic distances, so the estimate is unchanged. Summing over the fixed finite number of signed plaquette types and multiplying by gives the anchored bound. The constants are chosen before coupling, cutoffand torus period. This completes the startup Taylor and derivative input from WILSON and Residual.

The actual finest-scale Wilson representative

The Wilson startup is an exact identity for the whole density, although its activity dictionary is nonzero. The regular and forced entries cancel the added principal path terms configuration by configuration. For the allowance estimates, the forced suppression needs only2. The regular term also needs. Thus both allowances are asserted together on the shooting band ; the exact density identity itself does not acquire this restriction.

The startup used in finite shooting has nonzero activities; its total density is nevertheless exactly the bare Wilson density. We describe this cancellation. Fix a finest-level clip with threshold 0, and support width2. Put0. For a plaquette holonomy, let

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R(u) = price(u) −1
2| curvature(u)|2 −Q4(curvature(u)),
(FST1)

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where Q4 is the fixed fourth-degree invariant curvature correction. The finest path has row
√
b times
the elementary plaquette curvature, zero loop and cubic tensors, and quartic tensor bQ4 on the
same plaquette. Thus its quadratic-plus-quartic part is the corresponding local Taylor polynomial.
It uses the entire plaquette footprint, including a wrapping plaquette.
For each plaquette P, write uP for its holonomy, rP = chord(uP ), and kP = b| curvature(uP )|2/2.
Define its unpaid nonnegative part by

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price(uP ) −clinearθrP ,
rP > θ,
kP if the footprint is healthy, else b price(uP )

dP (U) =




(FST2)

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


−
kP if its anchor is kept, else 0
,
rP ≤θ.

The kept predicate is the exponentially relaxed plaquette test of the clip. On a kept anchor its own plaquette satisfies. The identities price() = chord( curvature(chord(), and 4 show 0, they give the stronger Let) denote the actual principal path action, including its clipped direct part and its masked quartic term. Checking separately the healthy case, the unhealthy case with, and the unhealthy case with, gives the pointwise identity

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price(uP ) = baseTermP (U) + maskθ,P
bQ4(curvature(uP ))
 + b maskθ,P [R(uP )] + dP (U). (FST3)

The first two terms here are precisely the plaquette terms in. In the healthy case substitute (FST1) and cancel. In the large-plaquette case both masks and the kept quadratic term vanish, and the clipped direct term is , canceled by (FST2). In the remaining case both masks vanish and the two alternatives for the kept quadratic term cancel those in (FST2). These are exhaustive cases, including threshold equality.

Average (FST3) over all signed coordinate frames with weight Signed, and sum over plaquettes. Index these oriented terms by, with footprint, anchor, residual and unpaid part. Signed-frame invariance of the Wilson energy gives

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βS(U) = βAtop(U) + βωb
X
i
maskθ,Si Ri(U) + βω
X
i
di(U).
(FST4)

Take the middle sum as the regular activity action. 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).
For the forced estimate, the lower bound β ≥x/2 is sufficient.
On the actual shooting band

, (FST7) has strength proportional to. The neighborhood count (4

and each fixed stage support cost are eventually at most4. The component expansion may therefore use local allowance; its summed connected-activity bound is at most which is eventually at most, the required forced allowance. The regular bound 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.

Finite coupling arcs and calibrated finite histories

Finite shooting selects a coupling on a continuous finite arc with Wilson startup. The estimates control that arc in the specified band, and the common calibration condition fixes the retained endpoint. Retirement removes only terms whose exact density identity and purchased error have already been recorded. The limiting history is unique within this fixed recipe and calendar; uniqueness across arbitrary choices of recipe is not part of the construction.

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