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

14 The analytic stationary graph, loop moments and the calibrated coupling history

Elimination, response and the order of choices. The stationary graph removes auxiliary fluctuation coordinates only after solving their own equations on the declared analytic neighborhood. Writing the fluctuation as), the effective quantity is obtained by evaluation at ( eliminated coordinate is not a gauge direction: its recovery requires the stationary equation, its invertible restricted Hessian and the response estimates proved below. Nonzero stationary residuals remain measurable in the raw history.

The loop moments and drift are then computed on this graph. Their coefficient fixes the reference slope used by the calendar; the native source order does not select that slope. Widths, block lengths, demands, reserves and delays are chosen in the stated dependency order before the terminal cutoffand physical period. The finest-scale density is identified exactly with the Wilson representative, and finite shooting arcs supply actual calibrated couplings. The combined startup allowance uses the band: its lower endpoint controls the forced part and its upper endpoint the regular part. These finite statements, followed by retirement and stability, supply the family needed by the limit; an auxiliary Hessian or formal probe alone does not supply it.

This section contains the analytic core of the supplied construction that lies between the vacuum bill of the marked integration and the omission comparison of the convergence chapter: from the common analytic stationary graph to the uniform running of the actual coupling. It supplies the analytic objects on which the stationary integrations, the one-loop drift, the asymptotic calendar and the finite calibrated coupling histories actually rest. Each subsection first gives the supplied derivation in full, with its tagged equations, and then its source–readout formulation. Throughout, raw histories, discrepancies and off-shell values are kept beside every stationary or calibrated restriction.

14.1 A common analytic stationary graph and its response estimates

We specify first the actual score to which the analytic argument is applied, at a fixed derivative order 2. A retained coordinate is a coarse oriented bond together with a real color index. A vertical coordinate is a free fine bond of the fixed forest together with such an index. Their disjoint union is the joint coordinate set. Arrays carry the maximum norm. Splitting the union is an isometry; replacing real color coordinates by their Euclidean color vector defines a linear isomorphism paint. Its norm is at most 1 + dim g, and its inverse has norm at most one. Moreover the sum of squared vertical real coordinates is exactly the sum of squared Euclidean color norms. Denote the resulting retained and vertical vectors by (tint tint). The actual scalar score in these coordinates is

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F(x, v) = tested(tint x, tint v).
(AS1)

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Thus it is the main fine action plus the prediction penalty, with the fine and outer configurations
constructed from these same variables. The connector is not an independent variable. A retained
coordinate is addressed at its embedded coarse site, and a vertical coordinate at its fine site. Each
address fiber has at most h = |TypeFace1×ColorIndex|+|Axis×ColorIndex| members: orientation
and color distinguish the members, and the coarse embedding is injective.
Here is the local-to-global estimate, including the counting loss. For addressed coordinate slots
q1, . . . , qk, put S(y, q) = P
i dist(y, address(qi)). An anchor estimate of price M and width a means

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|Dkf[eq1, . . . , eqk]| ≤Me−aS(y,q)
(k ≤n).
(AS2)

If a term depends only on coordinates within distance of, and its operator derivatives through order are bounded by, a coordinate derivative vanishes whenever one of its slots is outside that set. Otherwise. Thus (AS2) holds with. For path series the common exponentially weighted length budget dominates these derivative sums absolutely; differentiation term by term follows on a smaller common analytic neighborhood. This proves the same estimate for the cubic and quartic series, not merely for finite path sums.

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Sum such terms over anchors with multiplicity at most h0. Fix a marked slot qj. The triangle
inequality gives P
i dist(address(qj), address(qi)) ≤(n + 1)S(y, q) and dist(y, address(qj)) ≤S(y, q).
Split the exponential in half and use the lattice counting estimate

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c(b) = 16 6!e2bb−6.

The summed derivative has price2) and width+1)], measured from any marked slot. This is the Scatter bound used here. At order zero there is no marked slot; no volume-independent bound for the total scalar action is being asserted. For the actual main score take

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a =
μ
4(n + 1)(1 + L).

At a signed site its density is the sum over planes of price(2, together with the cubic and quartic path series. The plaquette has length four and locality radius 5 +. The matrix product rule costs 2 per product; inversion costs one for unitary matrices. In particular, writing!, a walk of length at most has derivative price

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Tn(r, C) = Cn
e∥1∥[1 + 2n(1 + r)]n, C

.

The Wilson price has boundtraceCharge, where traceCharge( If is the purchased outer-link jet bound, the fine-link bound is

by the same product rule. These are ordinary finite-dimensional derivative bounds on the unit real coordinate ball.

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To make the path prices explicit, let rμ be the common path analyticity radius at width μ/2,
and set Jn = Cn([1+2n/rμ]n, Cf). If sG is the color-slot coefficient and cG the real color projection,
the row and atom prices are Rn = Jn|sG|∥cG∥eμ and An,k = Jn|sG|keμ. These are the convergent
weighted path bounds obtained by spending half the exponential length width. Choose Z′ > 0
bounding all derivatives through order n of w 7→∥w∥2/2 on ∥w∥≤CRn; any number exceeding
1 + CRn + (CRn)2/2 suffices. Put Nn = Pn
i=0 |OrderedFinpartition(i)|. The per-plane and per-site
prices are

(AS3)

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The second term is the ordered-partition chain rule for the squared row. Signed site maps are
injective, so their anchor multiplicity is one. Averaging over the signed symmetries cancels their
number. The main-action Scatter price is therefore Mnc(a/2).
For the prediction term choose derivative inputs through order n + 1 first. If C+
f is the fine-link
price at that order, put

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(
)
ρframe,
√predictorCut
F∗
,
r
colorScale F∗
F∗= 2 alignCost pathChange + L variation,
ρ = min
.

(AS4) Here is the positive local minimizer radius for the frame score with the purchased (+1)-jets and half its root gap. The constants pathChange and variation bound the actual connector paths under a unit coordinate displacement; alignCost converts the two endpoint frames. The three inequalities in (AS4) keep the frame minimizer in its implicit chart, keep the focus below its predictor cutoff, and keep its group logarithm inside radius, respectively. Consequently the selected connector agrees there with the analytic minimizer. One differentiates that analytic expression, rather than the discrete selection tags.

Here are the local frame constants in this formula. Let be the number of frame points, with the central frame fixed to the identity. The blank score’s quadratic form is the average over ordered point pairs of squared differences of their logarithmic frame coordinates. The pairs with one endpoint at the fixed center contribute twice the sum of squared free coordinates. Consequently its counting gap is . Put 1), and, for input link jets bounded by, set

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Sn = P (2n[2nenTn(8R, Cn(C, 1))]en) ,
RH = min

1,
gP
2(1 + Sn)

.
(AS4a)

There are two frame factors and an intervening route of length at most 8; their product gives exactly this jet bound for the frame score. The mean-value estimate for its Hessian on the radius- ball changes the blank Hessian by at most2. Its inverse there has counting-norm bound 2 . LetcolorScale, the path chord price, and let = alignCost. For the purchased logarithm derivative bound, valid inside chord radius, the frame radius is

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ρframe = min

RH, frameCut
3J
, δ
A,
RH
NP DA

.
(AS4b)

The last term puts the logarithmic coordinate of the selected minimizing frame inside the Hilbert ball; the preceding two ensure its branch and logarithm agree with the chosen analytic chart. These are the finite local frame and alignment constructions in the nonlinear geometry section. Applying them with order + 1 is why the penalty estimate purchases that extra order before choosing (AS4).

The implicit derivative polynomial used here is also explicit. For branch value bound equation jet bound, and inverse linearized-equation bound

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I0 = R0,
Ia+1 = Ia + C0M{1 + |OrderedFinpartition(a + 1)|(1 + Ia)a+1}.
(AS4c)

Differentiate the implicit equation + 1 times. Its single-block term is the vertical linearized equation applied to the unknown highest branch derivative. Every other ordered-partition term uses only branch derivatives of orders at most, and is bounded by. Multiplication by the inverse bound proves the recursion; the added retains all lower jets. In the frame application is the frame-score price and . This defines the polynomial used in the next paragraph without an unpriced implicit differentiation step.

More explicitly, let be the implicit-derivative polynomial for the local frame score, evaluated with its score-jet price and inverse half-root-gap. The frame transport price is Then the connector price is , by its two endpoint factors and intervening path. Here bounds the local exponential chart jets, and below bounds the focus-map jets on the radius kept by (AS4). Also 1 + dim g), where is the purchased group-chart jet price. Each prediction bond is supported within radius 3, where is the patch radius. Its anchor price is explicitly

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Npred = P (2non+1Cn(W, Kconn)) ean(3L+16R).
(AS5)

There are four coarse bond orientations per embedded anchor, so summation has multiplicity four. Thus the penalty Scatter price is 42). Combining the two terms gives the actual price (AS6)

All constants in (AS3)–(AS6) are fixed from the group, stage, path budgets, local frame jets and patch geometry before the torus period. In particular no number of lattice sites occurs in them.

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Finally the origin slope and gap are consequences of the actual score expansion. The quadratic
line limit is one half of the signed average of P
□[pdirect∥dz∥2 + ∥llinz∥2] plus one half of the
penalty times the squared linear prediction residuals. Cubic and quartic terms start in their re-
spective orders. Analytic Taylor expansion therefore gives zero first derivative at zero and this
quadratic form as the Hessian. Restrict to vertical vectors. Keeping the identity signed orienta-
tion gives coefficients b = pdirect/|Signed| and ppenalty in front of the curl and prediction-residual
energies. The forest counting estimate reconstructs a vertical field from these energies, with loss
K = curlFactor(L, box, linkCost)+probeFactor(L). Its tree edges vanish; the harness link estimate
bounds the remaining reconstructed links. Hence

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X
DvF(0, 0) = 0,
D2
vF(0, 0)[v, v] ≥min(b, ppenalty)
i
v2
i .
(AS7)
1 + K

For an explicit count, let be the finite cover box of radius macroR, let, and let be the two local harness constants. With kernel predictor cutoff 0, put

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α = 2 max(1, q−1
∗)(2c1 + c2),
EL = (8L + 1)c2L + 2c1L2 + α/L2.

The local link reconstruction bounds a nonnegative link norm by times the sum of curvature norms in its translated-box, plus its coarse prediction-residual norm. Squaring uses ( , and the curvature sum uses Cauchy–Schwarz with 6 terms. Summing fine bonds,

reindexing their coarse parent and residue, introduces 4 positive-axis residue bonds; a fixed translated curvature can occur at most times. A residual occurs times. Thus

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Kcurl = 2E2
LVL 4 (6|X|) |X|,
Kprobe = 2VL,
K = Kcurl + Kprobe.
(AS8)

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This derives the finite denominator in (AS7); it may grow with the blocking ratio and fixed patch,
but never with the macroscopic torus period. The positive numerator is part of the actual step
hypotheses. The equality of color and coordinate squared sums above gives precisely this counting
norm, with no dimension conversion. Also tint(ei) is the corresponding physical color frame, so the
vertical Hessian block is the physical one. Invertibility of tint shows that the zero vertical derivative
in (AS1) is exactly the original stationary equation. Thus (AS4), (AS6), the address multiplicity h,
and the gap in (AS7) provide the actual initial radius, width, price, multiplicity and gap for (US1)–
(US8). Their stationary graph, inverse-Hessian and determinant estimates consequently apply on
the same common domain used in stationary integration and the reference chart.

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We make explicit the uniform implicit construction used in the stationary integrations. Split the
finite coordinate set into retained coordinates ι and vertical coordinates τ, assign every coordinate
a lattice site by l, and assume at most K coordinates have any given site.
Arrays carry their
maximum norm.
Write f(x, v) for the real analytic cost, S(x, v) = Dvf(x, v) for its vertical
gradient, and H(x, v) = D2
vf(x, v). The analytic scatter bounds established above say that its
coordinate derivatives, through a purchased order N ≥3, are bounded by a constant C > 0 times
the exponential connecting weights with width a > 0, on the maximum-norm ball of radius r > 0.
Assume S(0, 0) = 0 and the counting-measure gap
X
X
u2
uiH(0, 0)ijuj ≥m
i ,
m > 0.
(US1)
ij
i

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All these inputs were obtained for the actual cost; a bound on the extensive value f(0, 0) is unnec-
essary.
Take the explicit four-dimensional counting bound c(a) = 16·6!e2aa−6. A distance shell of radius
k has at most (2k + 1)4 sites. The exponential-series inequality ea(k+2) ≥[a(k + 2)]6/6! bounds its
contribution by c(a)/[(k + 1)(k + 2)]. Summing the telescoping series gives P
y e−a dist(z,y) ≤c(a),
uniformly in the period. Bounded multiplicity changes this to Kc(a). Put D = Kc(a). The scatter
third-derivative bound and integration along a segment give

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∥H(z) −H(0)∥∞→∞≤∥z∥∞CD2.
(US2)

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The same row and column sums bound the corresponding quadratic form: apply 2|uiuj| ≤u2
i + u2
j
to each entry. Thus the gap remains at least m/2 on the ball of radius

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RH = min{r/2, m/[2(1 + CD2)]}.
(US3)

We need an inverse in the maximum norm with a price independent of the number of sites. Suppose a matrix has entries bounded by and counting gap 0. For a fixed site , conjugate it by the diagonal weights. The triangle inequality and

show that, if 04, the difference from has entries bounded by

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Cb(4/a)e−(a/2) dist(li,lj).

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Choose
b = min

a/4,
M
2{1 + C(4/a)Kc(a/2)}

.
(US4)

The quadratic-form price of the perturbation is at most2; hence the conjugated matrix has counting coercivity2. It need not be symmetric. To estimate its inverse column, solve take, and put. Coercivity gives

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(M/2)
X
i
v2
i ≤vj.

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In particular (M/2)v2
j ≤vj, so vj ≤2/M, and then |vi| ≤2/M for every i. Undoing the weight
gives
|(B−1)ij| ≤(2/M)e−b dist(li,lj).
(US5)

Summing a row yields a maximum-norm inverse price (2). This argument uses the true counting gap and exponential summability throughout, and introduces no volume-dependent equivalence between the maximum and Euclidean norms. Apply this with2 on (US3), and denote the resulting width by. Set

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J = 1 + (4/m)Kc(b),
RV = min{RH, [4J(1 + CD2)]−1},

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RX = min{RV , RV /[4J(1 + CD)]}.
(US6)

For, consider the map . Its derivative is . Equations (US2) and (US6) give its norm at most 14. Integrating the mixed Hessian in the retained variable gives4. Therefore 2 on that closed ball. Iteration is Cauchy with successive differences reduced by 4; its limit) is its unique fixed point. Since 0) is invertible, being a fixed point is exactly )) = 0. We have

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g(0) = 0,
∥g(x)∥≤RV /2,
S(x, v) = 0, ∥v∥≤RV
=⇒v = g(x).
(US7)

At every point of this graph the vertical Hessian is invertible. The local analytic implicit function theorem gives an analytic branch there; its continuity keeps it inside the same vertical ball nearby, and uniqueness in (US7) identifies it with. These local identifications prove analyticity of the single graph on the whole retained ball. This also identifies any retained critical germ with this graph wherever both satisfy the corridor bounds.

For completeness, its higher response estimates follow directly from the critical equation, rather than from a dimension-dependent implicit-function norm. Let Differentiate + 1 times. In the ordered-partition chain formula the one-block term containing is; every other term contains graph derivatives of order at most. At a fixed purchased order use the common width

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b∗= min{weakerN(a), weakerN(b)} > 0,

where weaker +1)]. Combining the joining-tree weights loses only this width once for the whole induction: the span of any one resulting tree is at most the sum of its constituent spans, and splitting each available exponent between this connecting span and the summation variables leaves the remaining half for the counting sums. At most +1 vertices are joined at the purchased order. If prices graph derivatives through order, the other terms have price at most

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Fk = C + | OrderedPart(k + 1)|[C{1 + PkKc(a/2)}]k+1.

Multiply by the inverse kernel (US5) and sum its output coordinate. The recursion

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P0 = 1,
Pk+1 = Pk + Fk(4/m)Kc(b/2)
(US8)

therefore bounds every coordinate response through order+1, with its connecting-tree exponential of width. Every constant in (US3)–(US8) depends on the purchased finite orders and local input prices, but not the period. Differentiating uses the same ordered partitions; its terms containing a vertical first derivative of vanish by stationarity. This gives the on-shell action bounds used in the nonlinear integral and the reference estimates. The separate stronger strip estimates for generated precision kernels retain their own widths; (US8) does not replace them.

Source–readout formulation. The pair (retained coordinates, vertical coordinates) is exactly the auxiliary-elimination situation of the source-lift method: an auxiliary variable is eliminated by a stationary section, and the recovery of the effective theory depends on keeping the residual. The raw sector is the corridor

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Kst = {(x, v) : ∥x∥< RX, ∥v∥≤RV },
Readret(x, v) = x,
ιst(x) = (x, g(x)),

with residual column) retained at every point, including where it is nonzero. Here and the graph are the objects constructed in (US1)–(US8). Then Read = id, while Read= id: the fibre of the retained readout is the vertical ball, and the stationary section selects one point of it. The three analytic inputs of the derivation, and its conclusion, are used later in the following quotable forms.

Lemma 14.1 (Scatter bound). Let terms with anchor estimates be summed over anchors with multiplicity at most. Then the sum has, measured from any marked slot, price and width, where

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Proof. Fix a marked slot qj.
The triangle inequality gives P
i dist(address(qj), address(qi)) ≤
(n + 1)S(y, q) and dist(y, address(qj)) ≤S(y, q).
Split e−aS = e−aS/2e−aS/2; the first factor is
at most the connecting weight of width a/[2(n+1)] from the marked slot and the second is at most
e−(a/2) dist(y,address(qj)). The counting estimate P
y e−b dist(z,y) ≤c(b), proved below at (US2), sums
the anchors. At order zero there is no marked slot, and no volume-independent bound for the total
scalar action is asserted.

Lemma 14.2 (Implicit derivative polynomial). For a branch of an implicit equation with value bound, equation jet bound and inverse linearized-equation bound

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I0 = R0,
Ia+1 = Ia + C0M
1 + |OrderedFinpartition(a + 1)|(1 + Ia)a+1	.

Then bounds all branch derivatives through order

Proof. Differentiate the implicit equation + 1 times. In the ordered-partition chain formula the single-block term is the vertical linearized equation applied to the unknown highest branch derivative. Every other term uses branch derivatives of order at most and is bounded by ; there are at mostOrderedFinpartition( of them, and the explicit derivative of the equation in the parameter contributes at most. Applying the inverse bound and adding (which keeps all lower jets) gives the recursion.

Lemma 14.3 (Period-free weighted inverse). Let have entries bounded by and counting gap (it need not be symmetric), and let

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b = min
na
4,
M
2{1 + C(4/a)Kc(a/2)}
o
.

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Then
|(B−1)ij| ≤2
M e−b dist(li,lj),
∥B−1∥∞→∞≤2
M Kc(b).

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Proof. Fix a site z and conjugate by wi = eb dist(z,li).
The conjugated entry is Bijwi/wj and
|wi/wj −1| ≤b dijeb dij with dij = dist(li, lj), by the triangle inequality and |et −1| ≤|t|e|t|. For
b ≤a/4, b d ebde−ad ≤b(4/a)e−(a/2)d because d e−(a/4)d ≤4/(ea) ≤4/a. The perturbation therefore
has row and column sums at most Cb(4/a)Kc(a/2) ≤M/2, hence quadratic-form price at most
M/2, and the conjugated matrix eB has counting coercivity M/2. Solve Bu = ej, take z = lj and
vi = wiui; then eBv = ej since wj = 1, and coercivity gives (M/2) P
i v2
i ≤vj. Thus (M/2)v2
j ≤vj,
vj ≤2/M, and then v2
i ≤(2/M)vj ≤(2/M)2. Undoing the weight gives the entry bound and
summing a row gives the norm bound. Only the counting gap and exponential summability are
used; no equivalence of maximum and Euclidean norms enters.

Theorem 14.4 (The common stationary graph). For the map = is a contraction with constant of the ball into the ball of radius. Its fixed point satisfies

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g(0) = 0,
∥g(x)∥≤RV /2,
S(x, v) = 0, ∥v∥≤RV

 =⇒v = g(x),

and is real analytic on the retained ball.

Proof. has norm at most and (US6). Integrating the mixed Hessian in the retained variable gives 4 and4. The iteration is Cauchy and its limit is the unique fixed point; since 0) is invertible, a fixed point is exactly a zero of). At each graph point the vertical Hessian is invertible (gap2), so the analytic implicit function theorem gives a local analytic branch; continuity keeps it in the vertical ball and uniqueness identifies it with. The local identifications cover the ball.

The next statement is not in the supplied text. It is the quantitative form of the method’s requirement that an eliminated auxiliary variable be recoverable from the retained residual, and it follows from the estimates (US1)–(US6) alone.

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Theorem 14.5 (The residual recovers the vertical discrepancy). For (x, v) ∈Kst put w = v −g(x).
Then
∥w∥2 ≤2
m∥S(x, v)∥2,
∥w∥∞≤4
mKc(b) ∥S(x, v)∥∞≤J∥S(x, v)∥∞.

In particular if and only if, and the effective cost satisfies

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Proof. Since S(x, g(x)) = 0, S(x, v) =  ̄Hw with  ̄H =
R 1
0 H(x, g(x) + tw) dt. Every point of the
segment lies in the ball of radius RV ≤RH, so each integrand and hence  ̄H has counting gap
at least m/2, and ⟨w, S(x, v)⟩≥(m/2)∥w∥2
2; Cauchy–Schwarz gives the first bound. The entries
of  ̄H obey the second-derivative scatter bound Ce−a dist, so Lemma 14.3 with M = m/2 gives
∥ ̄H−1∥∞→∞≤(4/m)Kc(b), the second bound. The chain rule gives Dxfeff= Dxf + Dvf · Dg and
the last term vanishes on the graph.

Thus nothing about the vertical position is lost so long as the residual is kept: the discrepancy is controlled by, and vanishes exactly with, the recorded residual, with period-free constants. What the stationary restriction does discard is the off-shell vertical dependence of; the lift keeps it in and uses the restriction only where the supplied construction integrates on or expands about the graph. Relative to the frame realization, the coordinates () are components in the fixed source color basis of logarithmic link variables on the identity-frame section. A frame change conjugates each color vector by an orthogonal map: Euclidean color norms, the counting gap (US1) and the