14 The analytic stationary graph, loop moments and the cali- brated coupling history · Part 5

stationary chart and its comparison construction are available. Raise the original length floor to their pointwise maximum:

) = max (CAL12)

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.

Source–readout formulation. The countably many local demands are readers of the calendar and of the fixed finite patches, and the enlargements act on the calendar alone. The compatibility statement is therefore that each enlargement commutes with every reader already purchased: a demand array is a function of c which is monotone under the two permitted enlargements, so a later enlargement cannot invalidate an earlier payment. This is what (CAL12)–(CAL18) establish, in the order: stage data and length floors, then lengths, then entry delays.

Identification with the actual run. The order of quantifiers of the public theorem needs the following identification. The run’s stage profile and kernel are the original ones and its factor at profile is the selected length; hence (CAL12) at+1) is the actual step’s local length inequality, its stationary horizon satisfies the original primary price list, and projecting the two maxima in (CAL15) gives each local patch payment, its stationary admissibility and the delay required for the flat covariance limit. The preselected flat supplier applies with entrance index zero to this very horizon, and the curvature and dual suppliers by the other floors. No new family is chosen when the 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. In the source–readout record this is the statement that the demand arrays are read from c and the fixed patches; the enlargement operations act on c alone and commute with every reader.

14.11 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.

Source–readout formulation. In the lift and are scalar functions of the tagged level and coupling. The comparison of (EF1) with times (EF2) is an inequality between two readouts of the same realized record, uniform over the fibre of couplings] above one clock value. The forecast is thus purchased with its actual quadratic decay on its actual chart, and not on a reference chart from which it would have to be transported. The finitely many legal adjacent stage pairs are part of the calendar record c of Theorem 14.16, and the maximum over them is one of its permitted enlargements.

14.12 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 = log, so their large parameter is ) and their small remainder exponent is. 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.

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Source–readout formulation.
In terms of Theorem 9.4, the reserve is the statement that the
forced component of the realized activity has norm at most κ( bβm) in units of kPm, at every step

and at startup, with the same address layout; (CAL19) is the same function as the normalization (AB12). The squared form is retained because later errors are divided by local normalization factors: the reserve is a property of the realized record, not of a limit. The displayed inequality (CAL21) is typeset from the supplied text with one typographical restoration: the text prints the exponent of its first bracket with a stray comma, “”, which is the thin-space command of the original with its backslash lost; the reading is the one consistent with the first inequality stated after it.

14.13 Inverse curvature coordinates and the startup fourth coefficient

Exact startup and permitted shooting. The first output is an exact finite Wilson density in the represented activity form, including its normalization; an estimate alone would not identify the starting physical measure. The second output is a finite coupling history whose bottom value is the prescribed calibration. The shooting argument uses its two-sided band, while a forced-only estimate can use a lower-bound-only domain. These domains are kept separate below. Retirement changes the presentation of a state through the exact forward identity. Uniqueness is within this recipe and calendar, not across independently chosen constructions.

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 Extra/WILSON_STARTUP/Curvature and Residual.

Source–readout formulation. The startup fourth coefficient is a Taylor datum of a class function of the plaquette holonomy, and the only question for the lift is whether its jet bounds and its domain are fibre constants. They are.

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Proposition 14.17 (Uniform flat-jet bound). Let U
=
(U0, U1, U2, U3)
∈
G4, P(U)
=
U0U1U−1
2 U−1
3 , and let Jq(A; U) be the ordered left derivative (AB3) of R(P(U)) in four-link
directions A1, . . . , Aq. Then Jq is jointly real analytic on the group locus, multilinear,

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Jq(tA; U) = tqJq(A; U),

and there are constants such that

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|Jq(A; U)| ≤Cq,r|z|max(5−q,0)
if |A| ≤r, |z| ≤1, P(U) = exp z,

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|Jq(A; U)| ≤Cq,rcmax(5−q,0)
0
θq+max(5−q,0) ≤C′
q,rθ5
if chord(P(U)) ≤θ ≤δ, |A| ≤rθ,

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where | log V | ≤c0 chord(V ) for chord(V ) ≤b0 and δ ≤min(b0, c−1
0 , 1). The three links U1, U2, U3
are arbitrary, and equality in the plaquette threshold is allowed.

Proof. Analyticity follows by induction, differentiating an analytic expression along analytic left increments. Keep and replace the first link by

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eU0(z) = exp(z) U3U2U−1
1 ,

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so that the plaquette is exactly exp z. At z = 0 the residual has the vanishing property (SQ7). Each
ordered link derivative lowers the vanishing order by at most one (introduce an extra real variable
for the left increment, apply the vanishing composition to the joint variables and differentiate at
zero); by induction Jq(A; eU(z)) vanishes to order max(5 −q, 0) in z. For |A| ≤r the directions and
the group entries range over a compact set, so on that set times |z| ≤1 the relevant derivatives are
bounded and Taylor’s estimate gives (SQ11). For (SQ12) apply (SQ11) to A/θ with |z| ≤c0θ ≤1
and use (SQ9); for q ≥5 use θq ≤θ5.

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By Theorem 9.2 the same envelope holds for R ◦P ◦π on Z in raw directions of the same
norm, and by Proposition 9.1 the hypothesis chord(P(U)) ≤θ is the same event for raw links. The
curvature coordinate is covariant, c(π(r, A)p) = Adr−1
x c(Ap), and f and Q4 are adjoint invariant by
(SQ5), so R(π(r, A)p) = R(Ap): the startup residual is a frame-blind function and its Taylor data
are recovered exactly. The inverse curvature coordinate itself is not frame-blind; only the invariant
functions built from it are read out. The quartic form Q4 and its coefficient are determined by the
Wilson price through (SQ1)–(SQ4); they are computed, not chosen.

14.14 The actual finest-scale Wilson representative

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.