Convergence of the interacting scale histories · Part 3

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house price is consequently
CH = 4(4an1b)a.

Composition of the regularizer with the site-to-color row cancels = 1. Its trace-ideal price is and its ordinary price 4. The shear row has ordinary price 4, so the corrected row has price. The two reflection products have combined trace-ideal price

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CR = 4rt + 4(4p)r.

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For the orbit term let si bound the fixed stage row in direction i, and put S(z) = P3
i=0 4(4siz)si.
Its unweighted trace price is μ2S(B). The two changes to the dual massive frames cost at most
e4μ−1 each, yielding
CO = e4S(B)e4.

The motive film factors through the fixed-dimensional space, with inclusions of ordinary price one; its trace-ideal price is 4. All widths and derivative orders have been fixed before, and each cancellation of a block power has been displayed. These four estimates supply (DP10). Composing a difference factor with an ordinary bounded factor keeps (DP10). In particular the house film has price; the two reflection products have combined price has price 16, including its two film-composition factors; and the motive film has price 4 Loc. Thus one may use exactly

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K∗= CH + CR + 16CO + 4| Loc0 |.
(DP11)

Each price here is the absolute weighted column sum of its stated local factor with its zero-, one-and two-field placements, not a norm of a full trace. The analytic row and localized column estimates bound those sums by the fixed stage and strip prices. They use the same128, branch bounds and common stationary rows as (DP3), and are uniform in. The inverse identity gives difference price. For a two-field determinant frame, the direct term difference costs . Telescoping its three-factor bubble gives ( Taking its trace through (DP10) gives precisely (DP2) and (DP4). The factor 12 in the coreminus-rank determinant may be discarded when increasing this bound. This proves the relative projection estimate while retaining the factor in the unrelated individual estimates (DP3). The stationary carrier change has zero kinetic reading:

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αa(Mact −Mcore) = 0.
(DP5)

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To see this, write its diagonal film as 1
2(W(p) + W(−p) −V (p)/2). The stationary tax row has
zero jets below order two. Its mixed carrier row has zero jets below order one, by the stationary
graph identity. Hence each summand of W, a product of these two rows, vanishes to order at least
three. Every summand of V contains two tax rows and vanishes to order at least four. These films
are analytic on the common strip, so their external degree-two coefficient is zero. Linearity proves
(DP5).
For an analytic scalar function bounded by A on width r0, two successive Cauchy derivative
estimates give

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∥D2f(0)∥≤
A
r0/2
1
r0/4.

For the normalized reading (DP1), the source keeps the slightly weaker explicit price

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B(r0, A) =
A
r0/4
1
r0/8∥axes(e0)∥2.
(DP6)

The factor two in (DP1) cancels the Taylor factorial. Apply this estimate to (DP4), with and use (DP5). The exact projection bound is

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|αa(Mact) −αa(Mmrank)| ≤B(h/2, R∗).
(DP7)

No inverse power of the lattice period and no multiplier occurs in (DP7).

There is one more comparison, which is an equality rather than a bounded error. Let Π be the analytic film of the massive pole test on the fine lattice, with. Its finite-period analytic tensors approximate that test with error cents(, where0, and the rank film approximates the same test pulled back through the stationary uniform map with error cents(, where for fixed. The constants and analytic strip in these approximation statements may depend on. They are used for identification only, not as the uniform bound in (DP7).

Fourier evaluation of the stationary pullback gives the exact alias formula

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H(p) = L4 X
l
S
(p + 2πl)/L

[nl
−(p), nl
+(p)].
(DP8)

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Here S = Πμ, and nl
± are the incoming and outgoing stationary response columns. The analytic
contraction identities of S remove longitudinal columns. For l̸= 0, the remaining response columns
each vanish to order two at p = 0; their product therefore has no degree-two coefficient. For l= 0,
the primal and second-order stationary identities identify its contribution, through degree two, with
μ2S(μp). The discrepancy vanishes to order three. Taking two momentum derivatives introduces
another μ2, so the coefficient multiplier is L4μ2μ2 = 1. Consequently

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αa(Mmrank) = αa(Πμ).
(DP9)

The finite-period approximation just described proves that the analytic pullback in (DP8) is the rank film: let the auxiliary coarse and fine sides tend to infinity at fixed, keeping the same stationary graph and tags. The vanishing approximation errors identify their Fourier tensors and derivatives. The reality, gauge and signed symmetries of the actual pole test give its contraction identities and zero lower jets. No assumption about a momentum continuation between different lattice meshes is used.

Combining (DP7), (DP9), and (DR2)–(DR5) now gives the exact allowance used in the transition: log (DR6)

where0 is the bound for the massive-symbol integral remainder. The stage construction fixes, and hence, before the block length. Afterwards it imposes one lower bound on odd, and post-block bounds on the analytic norm and loop allowance of the output path. In the transition notation these latter bounds have the forms Θ, and ΛΛΛ They may depend on; they do not enter The stationary tensor is produced using arbitrary growing torus sequences satisfying the stated floor and stationary identities; after coefficient identification it is one cover path and works on every realized torus satisfying the construction’s patch and support margins. Thus the auxiliary large-volume limit imposes no extra period-dependent error on (DR6), and the estimate is uniform over the terminal cutoffs whose paths have those same pre-block bounds. The scalar reset has∆1. Since its actual coarse coupling is + ∆, we obtain

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|βN,p,i+1 −βN,p,i −s log Li| = |αN,i + ∆+ s log Li|
≤|αN,i + s log Li| + 1 ≤di+1.
(DR7)

Thus the one-step drift used in the shooting and running arguments includes both the calculated path coefficient and the actual nonlinear reset. The choices of large block lengths and late coupling entry below pay this drift simultaneously with all the local activity and insertion bounds.

Massive auxiliary symbols and the uniform alias trace bound

The massive comparison used in the preceding determinant argument is fixed as follows. With the lattice exterior symbol1, put

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s(p) =
X
a
qa(−p)qa(p),
dμ(p) =
μ2
s(p) + μ2 ,
cμ(p) = μ2dμ(p)9,
μ = L−1.
(MA1)

On real momentum) is the nonnegative squared curl frequency. The scalar massive covariantsite operator at a group field is ; its normalized inverse is . The auxiliary regulator is times its ninth power, and the comparison site operator is plus that regulator. These are ordinary finite positive matrices when 0. Their symbols at the identity are respectively. The ninth power is part of the construction, not a variable exponent sent to infinity.

For link fields, copy the site regulator into each of the four link directions at the link’s initial site. Add it to the physical quadratic Hessian and to . This defines the positive auxiliary link matrix. The associated scalar functional is one half of its logarithmic determinant minus the logarithmic determinant of the site comparison operator. The copying operation respects gauges and unsigned permutations of axes. Reversed links move the initial site, so this auxiliary functional is not assigned a signed symmetry by that copying argument. The physical signed laws are established separately.

The symbol at the identity is the physical transverse quadratic symbol plus the longitudinal symbol plus) times the identity. On real momentum the gap estimate bounds its quadratic form below by min(

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210
(s(p) + μ2) ∥v∥2
2,
(MA2)

where 0 is the physical transverse floor. The scalar companion has the same bound without the transverse factor. Indeed the physical and longitudinal pieces bound their respective curl and gradient components; in the small-frequency region; in the other region the frequency term already pays a fixed multiple of. This proves the required positive floor including zero momentum.

To see the uniform trace summation, aliases are

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pl = L−1p + 2πL−1el,
l ∈(Z/LZ)4,
(MA3)

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using the fixed centered integer lift el. Write r(l) = maxa |ela|. There are at most (2n + 1)4 indices
with r(l) = n, by injection into the integer box. Consequently, uniformly in L,
X
l
(1 + r(l))−6 ≤J0,
J0 = 16
X
n≥0
(n + 1)−2 < ∞.
(MA4)

This bound uses a box estimate for each shell; its excess over the actual shell size is harmless. For odd, on the strip of imaginary width 1128, first choose the real part of the coarse momentum in its fixed reciprocal fundamental cell. The lattice sine lower estimate on every nonzero alias and the complex perturbation estimate for give

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|dμ(pl)| ≤
2
(1 + r(l))2 .
(MA5)

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At the zero alias the massive denominator directly gives the bound two. For a nonzero alias, its
real frequency is at least (r(l) + 1)μ in the norm used by the sine estimate, and 1
2((r(l) + 1)μ)2 ≤
|s(pl) + μ2|; divide μ2 by this lower bound. A general coarse momentum differs by a reciprocal
integer translate; this permutes the alias set. Thus summing (MA5) cubed gives, on that whole
strip,
X

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l
|dμ(pl)3| ≤8J0.
(MA6)

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In particular the diagonal alias matrix d3
μ ⊗1Ch has nuclear norm at most 8hJ0.
This is the
three-massive-arc trace bound used inside the ninth-power regulator.
Here are the frames in which it is used. Let TL(p) have entries L−2eipl·exδαβ, with row x, α and
column l, β. The finite character sum shows that it is unitary at real p; its explicit inverse is the
opposite-phase character matrix. On the fixed complex strip the two matrices have operator norms
at most e4/128, by separating their real unitary Fourier factor from the diagonal complex phases.
Furnishing an alias symbol means conjugating its diagonal alias blocks by these two matrices. Thus
its three-arc nuclear bound is at most

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e4/128 (8hJ0) e4/128.
(MA7)

The weighted frames additionally use

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hl(p) =
q
pebWeight(pl, μ),
AL(p) = TL(p) diag(hl(p)−1)TL(0)−1.
(MA8)

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Here pebWeight(p, μ) = P
a |ei Re pa −1|2 + μ2.
Thus the positive weight uses the real part of
momentum even when p is complex. Such a weight is a norm-comparison device; it is not asserted
holomorphic. For complex differentiation, freeze the weights at the real anchor and use the analytic
TL(p) factors.
Comparing the original and frozen weights at nearby momenta gives the fixed
height allowance. The link-plus-site frame is the direct sum of (MA8) on link components and the
unweighted site frame.

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This separation justifies extending the real floor to a complex strip without a hidden dimension
factor. Test the conjugated matrix against two fixed vectors and apply the one-variable Cauchy
estimate to the resulting scalar analytic function. If its operator norm is bounded by M on a ball of
radius R, its derivative is bounded by 2M/R on the half ball. Integrate along the segment from the
real anchor to a nearby complex momentum. Choose that distance so that the perturbation norm
is less than half the real lower bound in (MA2). The Neumann series then inverts the perturbed
matrix with at most twice the anchored inverse bound. Apply this in the frozen frames, then restore
the compared positive weights. All constants are fixed before L; none comes from summing matrix
entries to estimate an operator norm.

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The derivative films used above are the actual zeroth, first and second coefficient matrices of
the local analytic rows. A row’s real directional derivative bound is first converted componentwise,
p
and then its color-coordinate square sum is taken. For the link row this costs exactly
| Loc(G, 1)|;
p
for the site row it costs
| Loc(G, 0)| and the displayed local sampling-density factor. Pulling the
site row into the block adds L2, since the normalized site square sum has L4 terms. These are the
link and site row allowances used in the weighted operator bounds. The rows are differentiated
through third order when constructing a second-order film of their first derivatives; that shift of
order is retained.

For the regulator film, form three consecutive cubes of the normalized inverse film in (MA1), giving , and then multiply by. At the common center this is exactly. Its trace-ideal bound uses one unsplit cube as (MA7), while the other factors receive operator bounds. In

a first or second product derivative, choose the corresponding three-arc occurrence after applying the exact inverse and product rules; the column placements in (DP10)–(DP12) give the listed alternatives. The resulting finite sum is bounded by the positive sum of their tariffs.

Finally the house term is the physical Hessian film sandwiched between its two row films; the orbit term is the sum over four initial-site copies of the regulator film. Sandwiching a trace-ideal factor by operator-bounded factors preserves its nuclear bound, with the product of the two operator allowances. Sum the four copied terms and conjugate to the common frame; the fixed Fourier norms cost at most on each side at the wider intermediate allowance. The reflection correction is the sum of its two cross products with the regulated site row, and the fixed extra block contributes 4 Loc(. Addition and the same two-sided multiplication rule give the full trace toll in (DP10)–(DP12). Thus the cancellation there is backed by the massive symbol (MA1), the alias sum (MA6), the real and complex floors, and the actual local row derivative bounds. It does not require treating an arbitrary large block matrix as uniformly trace class.

The complex denominator estimate used in (MA5) can be proved before alias summation. Put , and1. Integrating the derivative of the exponential gives , and the same estimate at. Expanding their scalar product yields

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|s(p) −s(x)| ≤4t(u + 4t) + 4tu = 8tu + 16t2.

For128, the right side is at most ( 2: use 2 . Since, the reverse triangle inequality proves

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|s(p) + μ2| ≥1
2(u2 + μ2) > 0,
|dμ(p)| ≤2,
|cμ(p)| ≤29μ2.
(MA9)

The coarse strip has width 1128; each alias divides its imaginary part by, so its fine momentum has exactly the width128 required here. In particular the inverse denominator is analytic throughout the strip. For completeness the weights in (MA8) obey

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hl(p) ≥μ,
∥q(pl)∥2 ≤5hl(p)

on the unit coarse strip. At coarse points of distance at most one, the preceding exponential estimate gives. Using shows that the two positive weights have ratio bounded by a fixed constant in both directions. Freeze at a real anchor while retaining ) and its explicit analytic inverse. These are the analytic frames used in the Cauchy estimate. In ordinary unweighted norm their bounds are

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∥AL(p)∥≤e4σμ−1,
∥AL(p)−1∥≤e4σ sup
l
hl(p).
(MA10)

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For a real anchor, h2
l ≤16 + μ2 ≤17. These estimates display the mass powers; they must not be
silently replaced by two mass-independent unweighted bounds.
Here is an explicit form of the strip-inverse argument. Suppose a matrix K, in the two specified
weighted frames, has norm at most A, its real lower singular bound is a > 0, and the changes to
frozen analytic frames cost at most C on a ball of radius σ. Test the analytic matrix on two unit
vectors and apply the scalar Cauchy estimate. This gives derivative bound k = CAC/(σ/2) on the
half ball. On the strip
η = min{σ/2, (a/2)/(1 + k)}

the displacement from its real anchor changes the matrix by at most The lower singular bound is therefore2, so its weighted inverse has norm at most 2. This also proves invertibility

and analyticity of the inverse by the determinant formula. The constants are independent of the dimension of the block. In the site application the real lower form bound is proportional to , and the two inverse weights cancel it; it is that weighted matrix to which the assertion applies.

The remaining mass bookkeeping can now be stated exactly. The normalized inverse film has bounded ordinary operator coefficients because its factor cancels the two inverse-weight losses. The regulator film has both operator and trace-ideal price times a fixed price, by its definition . Converting it back to the weighted form costs at most on each side, hence cancels as this. This accounts for the mass-independent regulator toll in the link determinant comparison. It does not remove the explicit in its ordinary frame.

Finally the actual matrices behind the last paragraph of this chapter are as follows. Let be the fixed inclusion of the link block and let be the complementary site projection; let read the initial-site slot of direction. Write for the regulator film, for the differentiated row, and for the local second variation of the scalar correction. With the rear operation denoting the transpose at reversed momenta, including the reversed external legs, the relevant products are

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3
X
House = RrearHR,
Orbit =
i=0
Srear
i
CSi,
Motive = P rearP.
(MA11)

These are exact products of films, so their first and second coefficients are their ordinary ordered product derivatives. The motive is constant and its finite site rank is the reason its trace allowance Loc(. The orbit allowance is the regulator trace allowance multiplied on each side by the Fourier-frame bound. The house allowance follows by first assigning the finite local trace allowance to, then multiplying by the two row bounds. The two reflection cross products are bounded in the same order. The total toll is the positive sum

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payLedger + reflectLedger + 4(4 orbitLedger) + popLedger.
(MA12)

Each factor in this expression has thus been paid either by a local finite rank, the alias trace sum, or a weighted operator norm; no trace norm of a general extensive Hessian is assumed.

Constructing the common asymptotic calendar

Every finite stage contains only finitely many requests of its admitted degree and derivative order. Their maximum is finite. A countable family is met by letting the stage order increase, so that each fixed request is eventually present. This is weaker than a uniform estimate at all orders at one stage, and is exactly what the diagonal construction needs.

We give the numerical schedule construction explicitly, since both finite shooting and uniform running use it. For each stage4, all pre-block parameters have already been chosen: its derivative order, support exponent, analytic width, path bound, column bound and drift allowance ). Let

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A(j) = log

1 + paths(j) + width(j)−1 + columns(j) + relaxation(j) + supportExp(j)

.

Choose an odd integer block length L3, at least every previously required lower bound, such that log L min( log L (CAL1)

This is possible because the numerators are fixed before this choice and 1 log0. To make the block lengths nondecreasing, take for each the maximum of the finitely many lower bounds

through, then an odd integer larger than that maximum. These choices prove the drift and logarithmic-cost inequalities without making the drift allowance depend on the selected terminal cutoff.

After fixing the block lengths choose the positive protection exponents, decreasing with, small enough for all post-block requests and at most 1. This can be done by taking successive finite minima of positive admissible upper bounds. Next choose the stage’s coupling-entry and scaleentry thresholds; every analytic estimate proved with these parameters has one eventual threshold, so a finite maximum handles all its requests. Later increases of these thresholds preserve their inequalities.

Choose the initial coupling ̄ 1 above the first-stage entry threshold, and sufficiently large that exp((log log ̄ (CAL2)

The ratio (log log log0 proves this. For fixed, the required lattice-size bound at logarithmic scale has the explicit form

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Vj(u) = Sj
1 + c(u) + c(u + log Lj)

qj100(c(u) + 1)
+ 100
(j + 3)Lj + 1

,
(CAL3)
c(u) =
l
log2( ̄β + su)
m
.

Here and are the fixed shell-count and growth requests from the local estimates. The last two terms include the small-field and patch buffers. For each fixed stage,) and its fixed translate are); hence ). Choose an integer entrance bound such that and log 3 imply simultaneously

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 ̄β + su ≥betaEntry(j),
u ≥scaleEntry(j),
u > 0,
log Lj ≤u/j,
Vj(u) ≤eu.
(CAL4)

Each condition holds eventually as log 3gives one common integer There are also stagewise comparison-window requests) satisfying0, and finite readiness thresholds. Choose integers so that for every and log 3 log L

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rj ≤u,
wj(u) ≤k/2.
(CAL5)

Indeed eventually(2 log L; multiplication by the upper bound on proves the second inequality, and the lower bound on supplies readiness. Integer rounding only strengthens the resulting bound by

We now build the stage clock, rather than presupposing one meeting all these requests. Use an unshifted stage number0, corresponding to, and write for the chosen integer requests after this shift. Define plateau starting indices recursively by

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a0 = 0,
kh = Ch + 2ah + 1,
ah+1 = ah + Bh+1 + 2kh + 1.
(CAL6)

Set when. The sequence is strictly increasing, so is defined everywhere, nondecreasing, increases by at most one in a step, and tends to infinity. Also(1) = 0. Its recursion gives two useful properties:

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E(i + 1) > 0 =⇒BE(i+1) ≤i,
(CAL7)

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and, for every h, kh ≥Ch and

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E(i) = h
whenever
kh −⌊kh/2⌋≤i ≤kh + ⌊kh/2⌋.
(CAL8)

For (CAL7), the first index of plateau For (CAL8), + 1 places its lower endpoint above, while places its upper endpoint strictly before Thus every stage receives a sufficiently long central comparison window, however large its requests. Define L + log (CAL9)

Then log 3 log L, by monotonicity of the blocks. The first inequality proves For every noninitial plateau (CAL7) lets us apply (CAL4) at. In particular

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0 ≤log Li
τi
≤J(i + 1)−1 −→0.
(CAL10)

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This is the gentle-mesh condition. Equations (CAL1) and (CAL9) give (FCA4) and (RUN3). The
stage order, power and support exponent stay nondecreasing, the protection exponent decreases,
and (CAL2) gives the small-field threshold at every scale. These properties all belong to the same
calendar.
For volume, let di = Q
r<i Lr = eτi. On every noninitial plateau (CAL4) gives VJ(i+1)(τi) ≤di,
so any positive physical period pays its lattice-size requirements. Only finitely many steps belong
to the initial plateau, since E →∞. Choose an integer M larger than their finitely many size
requirements and the required minimum period, and set the physical periods to

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Pp = 4(M + p + 1),
lattice side at (i, p) = Ppdi.
(CAL11)

These periods are strictly increasing and divisible by four. They pay the initial exceptions directly and the later requirements by the preceding denominator bound. They also provide arbitrarily large periods divisible by any prescribed positive integer: take + 1), which is at least, and then. The selected complete construction may subsequently retain a period subsequence; it keeps all the size inequalities.

Finally use (CAL5) at, which lies between the two bounds preceding (CAL10). The requested window has width at most2, and (CAL8) keeps that entire window on stage Thus the comparison windows, entry delays, local volume margins, gentle mesh and asymptotically small drift are realized together. No selection of a new calendar for a particular history, cutoff, period or observable is required.

Countable local demands on the same calendar

Why countably many demands can be met At any one stage only finitely many derivative, degree and support requirements have been activated. Their maximum determines a finite threshold at that stage. The increasing stage sequence eventually activates every fixed finite request, while later delays preserve the choices already made. This is a diagonal schedule of finite demands, not one estimate uniform in every degree and derivative order at once. The order of choices below is the mechanism that makes these quantifiers compatible.

The calendar construction must also accommodate estimates whose numerical values use the chosen block lengths. We describe this ordering explicitly. First choose the stage profiles, kernel, primary analytic price list, and the secondary price list for the flat covariance and exterior constructions. The three local constructions then supply finite integer lower bounds for block lengths at profile. Two further integer lower bounds) ensure the local stationary chart and its comparison construction are available. Raise the original length floor to their pointwise maximum:

) = max (CAL12)