Convergence of the interacting scale histories

Convergence of the interacting scale histories

Convergence requires one common numerical schedule for all fixed finite demands. The order is: choose the finite stage profile, then its block length, then its post-length constants and coupling-entry delay, and only afterwards the terminal cutoffand physical period. Increasing a delay does not reselect a previously fixed block. This prevents circular parameter choices in the stationary estimates, shooting construction and actual coupling running.

A common analytic stationary graph and its response estimates

The stationary graph solves the vertical critical-point equation in a common analytic neighborhood. Positivity of its vertical Hessian gives the inverse used in the response estimates. Retained coordinates and vertical coordinates have different norms; the explicit color-coordinate maps and their bounds below justify each passage between them. No extensive scalar action is assigned a volume-independent zeroth-order norm.

We specify first the actual score to which the analytic argument is applied, at a fixed derivative order2. 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 ) = tested(tint tint (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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DvF(0, 0) = 0,
D2
vF(0, 0)[v, v] ≥min(b, ppenalty)
X
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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Kmcurl = 2E2
LVL 4 (6|X|) |X|,
Kmprobe = 2VL,
K = Kmcurl + Kmprobe.
(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.

We make explicit the uniform implicit construction used in the stationary integrations. Split the finite coordinate set into retained coordinates and vertical coordinates, assign every coordinate a lattice site by, and assume at most coordinates have any given site. Arrays carry their maximum norm. Write) for the real analytic cost,) for its vertical gradient, and ). The analytic scatter bounds established above say that its coordinate derivatives, through a purchased order3, are bounded by a constant 0 times the exponential connecting weights with width 0, on the maximum-norm ball of radius

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Assume S(0, 0) = 0 and the counting-measure gap
X
ij
uiH(0, 0)ijuj ≥m
X
i
u2
i ,
m > 0.
(US1)

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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.

The Fourier normalization and the kinetic readout

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The two-field determinant film has an internal momentum and two opposite external momenta. Its
R
internal average is normalized Haar average, namely stock F =
[−π,π]4 F(k)d4k/(2π)4; its finite-
period average is shopn F = n−4 P
k F(2πk/n). The curvature picture called profitable F is one
half of the internal average with the two external link and color indices in their specified order. The
factor one half is the quadratic Taylor factorial. Translation invariance makes the actual Hessian
entry zero unless its two Fourier momenta sum to zero, and otherwise that entry is the site count
times the finite-period average of the film.

To compare the two averages, expand the internal variable in its absolutely convergent Laurent series. Finite averaging retains exactly the coefficients whose four indices are divisible by the

period; Haar averaging retains only the zero coefficient. Every remaining index has length at least the period. Spending half the common strip for summation and a quarter for the tail gives a fixed strip price times. Multiplication by the site count gives the recorded-Hessian error. Thus this internal integral represents the finite Hessian with a quantitative error and a fixed normalization.

Compose that picture with the momentum chart for real initial links. The readout leapTip()) is the divided coefficient Reality makes this diagonal coefficient real. Color invariance and simplicity make its color bilinear form a scalar multiple of the fixed invariant form, so its value is independent of the normalized color. The factor two in (DP1) is additional to the factor one half in the internal film average; both are retained in that definition.

Here is why this one transverse entry determines the leading kinetic tensor. Subtract its multiple of the normalized Wilson tensor, whose readout is one. Gauge invariance gives transversality. Signed covariance includes the phases from reversing initial links; at the first nonzero homogeneous order those phases have constant term one. The constant and linear homogeneous terms vanish by transversality and permutation covariance. Let) be the symmetric polarized degree-two term of the difference. Reflecting one axis shows that a component can survive only when its momentum occurrences and link occurrences have the same parity on every axis. Permuting axes moves the zero transverse diagonal readout to all transverse diagonal entries on single momentum axes. Longitudinal entries vanish by transversality. Reflection parity then kills mixed momentum entries in a diagonal link component and same momentum entries in an off-diagonal link component. The only remaining possibility has its two distinct momentum axes equal to its two link axes. Transversality at the sum of those two coordinate vectors makes twice that entry zero, since the diagonal entries have already vanished. Bilinearity gives = 0. The degree-three term vanishes under total inversion, whose momentum sign is odd and whose two link signs multiply to one. The difference thus starts at degree four. This proves the one-axis kinetic identification used below, including the Taylor normalization and reflected-edge convention.

Analytic exterior division with the origin jets retained

Division retains the harmonic obstruction Exterior division first solves a contraction equation on the regular momentum set. Analytic extension requires the stated vanishing along the relevant harmonic coordinate planes, not merely a value zero at the single full origin. The division formula below keeps the origin jets and the constraints from the other momentum blocks. These data are needed by the two-leg correction that follows; deleting a harmonic component instead would change the supplied stationary identity.

We give the analytic division used in issuing the loop coefficients. For one momentum coordinate set is a Laurent index, define

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(Pxi−1
a=0 eip·(x[i:=a]),
xi ≥0,
−P−xi−1
a=0
eip·(x[i:=−a−1]),
xi < 0.
Li,x(p) =

The empty sum is zero. Multiplication and telescoping give the entire identity

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qiLi,x = eip·x −eip[i:=0]·x.
(ED1)

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If F is periodic analytic and bounded by A on a strip of width σ, its Fourier coefficient bound on
an intermediate strip makes G(p) = P
x bF(x)Li,x(p) absolutely and locally uniformly convergent
on width σ/4. Indeed the finite sum has at most |x| terms and size at most |x|e(σ/4)|x|; the spare

width absorbs. Summing the geometric coefficient bounds in each momentum coordinate gives the explicit price

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21 + σ/4
v
,
(ED2)
∥G∥σ/4 ≤A 4
σ
σ/4

where is the number of momentum coordinates. Thus, including at every zero of; no singular quotient is defined on a mesh. If the numerator has zero jets below order comparison of its analytic Taylor series with ) gives zero jets of below order . Equivalently evaluate on nearby directions with nonzero-component, divide their one-variable series, and extend the resulting polynomial-jet identity to all directions by continuity.

For exterior coefficients let be contraction by the vector, and exterior multiplication by the-th coordinate vector with its orientation sign. The coordinate anticommutation rule gives id. Apply (ED1) componentwise to a cycle, and call the divided difference Substituting in this anticommutation rule shows that

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CIiDiu = u −combi u,

where comb evaluates at = 0 and deletes every exterior component containing. The identity is first algebraic where= 0, and extends to the zeros by analyticity. The residual is again a cycle. Indeed, after separating components containing, its contraction at momentum equals its contraction with the-th symbol set to zero, since every-containing component has been removed; the original cycle equation at = 0 supplies the remaining equations. Evaluation and deletion preserve the original strip, coefficient bound and Taylor vanishing order.

Repeat on the residuals in the four axis directions, adding the four lifted differences. Every residual still has the original width, price, and order. Each new lift is therefore computed directly at width4, price at most times (ED2), and order at least1. There is no iteration of four strip shrinkages or four losses of Taylor order. Summing gives width4, price four times (ED2), and order1. The contraction of this sum telescopes to the original cycle minus the final residual. In positive exterior degree the latter is zero because every orientation contains a processed axis. In degree zero it is the original coefficient with all coordinates of this block set to zero; the required hypothesis is its vanishing on that entire coordinate plane. Positive order at the full origin implies this hypothesis only when this is the sole momentum block. This is why the exact harmonic-set check in (TEL2) is needed.

Suppose additionally that each coefficient vector lies in a prescribed linear subspace depending only on the other momentum blocks. Evaluation in a divided coordinate preserves that subspace. The divided Fourier sum also does so: where= 0 it is the scalar multiple (, and at its zeros continuity and closedness of the finite-dimensional subspace give the same conclusion. Exterior multiplication is a signed selection of these coefficient vectors. Hence every residual and lift retains the other block constraints. For a dependent momentum equal to minus the sum of the independent ones, the linear shear exchanging that block with the dependent block is involutive. If there are + 1 independent blocks, its missing coordinate is the negative sum of their coordinates. A strip of width+2) is mapped inside width, since (. Apply the preceding coordinate construction on that narrowed strip and shear back with the same narrowing factor. This gives the dependent-block width and price used in (TEL3).

The two-leg correction with both momentum blocks retained

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Here is the precise two-leg use of the exterior division. There is one independent four-momentum
p; the other is −p. Write Er = ΛrC4, with its ordered plane basis, and let Cr
−: Er+1 →Er be