The centered trace and its non-freeness test · Part 6

Searchable transcription

renormalized lift followed by U−1
i,5 . Equivariance under conjugation and signed symmetries makes
its image lie in the free divergence target. Applying the componentwise divergence comparison to
these exact formulas yields (TR46).
We give the finite-dimensional repair which supplies (TR47). This is needed because the limiting
degree-five map has a one-dimensional kernel. Let E, H be finite-dimensional real spaces, f : E →
H a surjection with ker f = Rx, x ̸= 0, and choose a right inverse U. Suppose si : E →H have
exact right inverses ui : H →E eventually, with ui →U. Suppose also that wi →x and siwi →0.
There is a linear functional A : E →R such that

Searchable transcription

y = Ufy + A(y)x.

Searchable transcription

Indeed y −Ufy lies in the one-dimensional kernel and its scalar coordinate is linear. Set Vi =
uif + A ⊗wi. Then Vi →IE, hence V −1
i
exists eventually and tends to IE. Also

Searchable transcription

siViy = fy + A(y)siwi −→fy.

Searchable transcription

Multiplication by V −1
i
gives si →f in operator norm. Pointwise convergence is sufficient in this
argument because all operator spaces are finite-dimensional: evaluate on a fixed finite basis and
reconstruct the operator continuously.
Let L : E →R be the bare scalar functional and ei the required precision scale. Assume the
stronger two estimates
eisiwi →0,
eiL(wi) →0.
(TR50)

Define. These functionals converge to, so they are uniformly bounded. Put ). On the vector, the right-inverse identity cancels the first part exactly and gives

Since , it follows that0 in operator norm. This proves (TR47) with also proves the convergence of needed for (TR46)–(TR48); no prior uniform convergence on its near-kernel direction was assumed.

In the present application is the nonzero free equation-of-motion vector, the kernel of the free degree-five map. The exact right inverses are obtained by lifting free target vectors at the same retained scale and undoing the renormalized coordinate map. The projection-lift identity makes exactly; the common scaling comparison gives. The near-kernel vectors are the issued equation-of-motion coordinates. Their two paid estimates are precisely (TR50). The bare scalar functional annihilates the limiting equation-of-motion vector. Thus this argument constructs the detector used in (TR42), with the precision scale = Age . For the correction candidates the sharper scalar estimate in (TR47) is compatible with their

Searchable transcription

qualitative coordinate limits for an additional reason.
Before taking the continuum limit their
issued coordinates differ from the ideal orbit coordinates by at most lossi = Age−1/4
. The ideal
i
scalar reading equals the target scalar reading exactly. Bounded linearity therefore bounds the
scalar discrepancy by a constant times lossi. Multiplication by ti = Age1/5
gives the decaying
i
power Age−1/20
. This proves the third error limit used after (TR47). The qualitative joint degree-
i
four and degree-five coordinate limits follow by taking the same finite-level orbit comparison, whose
coefficients approach the prescribed vector, and using the vanishing unweighted loss before passage
along the original extraction filter.

For the principal divergence coordinates, the explicit prelimit estimate which supplies the final limit in (TR48) is

Searchable transcription

∥vi∥≤2∥O∥Age1/6
i
,
|L(vi) + g| ≤∥M∥Age−1/4
i
+ 4∥M∥Age−1
i
+ 1
12H−30
i
+ 1
12H−100
i
.
(TR51)

Here is the fixed orbit map on degree-five coordinates and. To pass this bound to the continuum coordinates, first compare the issued representative with its projection to the retained coordinate core, with error Age . The retained representative has norm at most Age . Apply the bounded orbit map for the first bound and the bounded functional for its error contribution. The finite-level scalar calculation gives the remaining three terms and the value. Continuous linear maps and closed norm inequalities preserve these bounds along the extraction limit. Every term on the right of the second inequality tends to zero. The scalar finite-level calculation is developed below, and (TS1)–(TS5) supply the weighted near-kernel estimates required in (TR50).

The scalar functional has an explicit invariant-coordinate formula. For the bare divergence vector parametrized by

Searchable transcription

ρ(bareVector w) = −4(w0 + 4w1).
(TR52)

Searchable transcription

The coordinate parametrization is an isomorphism for simple g. The equation-of-motion vector is
the image of (1, −1/4, 1), which (TR52) annihilates. More generally, if the vector belongs to the
equation ideal, its coordinates satisfy 2(w0 −w2) = 0 and 2w1 + 1
2w2 = 0; these force ρ = 0. The
field-space functional L is ρ after the degree-five coordinate isomorphism. This proves both its
equation-of-motion annihilation and the readings 0, −4 on the two columns of (TR45).
At finite level the same reading can be computed before quotienting. For a real regular local
vector V = (Va), the coefficient functional M on its degree-five cards satisfies

Searchable transcription

M

θ−2(altoD Va)a

= 1
12 Re Holt(V ).
(TR53)

Here is the actual cutoffthreshold. The local-card identity gives altoD DCard ; the factors cancel exactly. The card reading identifies the coefficient contraction with the local polynomial contraction Holt. Reality and nonwrapping of the support justify that identification without periodic alias terms. Thus (TR53) is a literal coefficient identity with the stated cutoff, rather than a remainder estimate.

Here are the local estimates underlying the coefficient calculation. Write for the large coupling parameter and put Ω) = exp(log log Ω

Searchable transcription

cx = θ2Ω−2(x),
tx = θ2Ω1/8(x),
σx = θ2x−1/100.
(LCE1)

All assertions in this paragraph concern sufficiently large 1. Products of the Ω add their subscripts; for every fixed and eventually. This follows by dividing(log log

by log, whose ratio tends to zero.

The regular local fields entering the exact split have the following derivative envelopes on the common healthy chart: the nonlinear principal discrepancy and the finite Taylor surgery error have envelope; the principal comparison field and its loop field have envelope; the finite-loop tail has envelope. These are envelopes through the prescribed chart order, on the stated local support, including measurability, gauge covariance and chart regularity. The large- threshold is common to the stated local estimates. At each fixed admitted, exhaust the finite loop and tail

shelves after specifying the required positive error allowance. Absolute convergence makes their errors arbitrarily small. These shelves may depend on; a single finite truncation uniform for all large couplings is not needed.

For clarity, the nonlinear estimate has a quantitative Taylor proof. Let(1 + high( , where is the fixed stage power. In exponential link coordinates the discrepancy between the nonlinear curvature expression and its linear comparison has derivatives through order four at zero bounded by. Its derivatives through the required higher order on are bounded by 2Ω1 is a fixed chart radius. The change from color coordinates to link directions has norm at most one, so these low-jet bounds persist under that change. Taylor’s formula with integral remainder, after rescaling, gives, uniformly for

Searchable transcription

∥minors(βK)∥Ck ≤4x 2Ω1/200(x) C5,Ds5 ≤σx,
k ≥5,
|β| ≤4x.
(LCE2)

Searchable transcription

The low-order terms fit this same estimate because x−30 ≤2Ω1/200(x)s5−j for 0 ≤j ≤4. The final
inequality follows from s5 = x−5/2 times a fixed subpower factor, whereas σx = x−1−1/100Ω2(x).
Thus the fifth-order Taylor remainder has a strictly decaying power to spare. The finite-difference
surgery uses h = θx−1/24.
Since h/s ≤x−1/24, choosing its order m ≥48 makes every fixed
subpower multiple of s(h/s)m+1 at most σxh eventually.
This controls the surgery error after
division by its difference step. The support margins and nonwrapping conditions used here are
the finite chart conditions already imposed when choosing the station; none is a new continuum
hypothesis.

We spell out how these local bounds survive projection and summation. Fix the degree-five frame and its local section, and denote their fixed reading prices by1 and1. If the section has growth exponent0, put. A field with derivative envelope has weighted coefficient size at most. This is the coefficient projection estimate: each coordinate is a bounded derivative reading, its fixed reading weight is included in, and summing the coordinate absolute values bounds the coefficient norm.

Searchable transcription

For the secondary surgery terms indexed by a finite set I, let Qi be their supports and gi their
derivative envelopes before multiplying by the difference coefficient.
The actual summable tail
bound is

 gi ≤σx.
X
teasi
es0 cover(Qi)
(LCE3)
h
i∈I

Here is the fixed support weight of the local budget, distinct from the chart radius. Multiplying each summand by its projection price at most and using the triangle inequality gives coefficient size at most ( for their sum. The same summation constructs its local representative and bounds its approximation remainder; this is why a bound for each individual term is not used in place of the required bound for their sum.

The principal local support has bounded weighted cover, so its support weight is at most a fixed = exp( TileNo). Its derivative envelope is 2. Combining it with (LCE3) bounds both the total projected coefficient and its local representative error by

Searchable transcription

K0(4Atx + σx) ≤(4A + 1)K0θ2Ω1/8(x) ≤θ2Ω1/6(x).
(LCE4)

The last inequality follows from ΩΩ0. Similarly,

Searchable transcription

K0σx ≤cx,
rcx ≤θ2Ω−1(x).
(LCE5)

Indeed their respective ratios areΩ) andΩ), both tending to zero. These inequalities prove the total envelope and all four small coefficient envelopes used next. They also show explicitly why the local errors still vanish after division by

Searchable transcription

We identify those six contributions in terms of the finite functions, to avoid ambiguity between
this split and the six antisymmetric tensor differences in (TF0d). At the current scale let Yb be the
localized current, eJ the restored tensor, PA the finite principal action, and LQ the finite ordinary
loop sum. Write QQ for its augmented quiet sum, with the zero and first path moments removed
as in (TF6a)–(TF6c). Define DY f by differentiating f along the left logarithmic link current, and
write divν Y for its Haar divergence. The six objects are

Searchable transcription

Wb = altoD{β(P
a Da eJab −DYbPA)},
Bb = divν Yb −(divν Yb)(1),
Kb = DYbQQ,
Db = DYb(QQ −LQ),
Eb = altoD{h−1Tm,h(Yb, −βPA −LQ)},
Fb =
X
γ∈F
weightedPγ Hγ.
(LCE6)

In this display already denote coefficient vectors; are scalar configuration functions until altoD is applied. The sign identity is, so the loop contribution is The operator altoD means the degree-five Taylor reader with its actual threshold weights, on the common local support; the secondary readers retain their own supports. The current here is the localized version of (TF0a), transported through the stationary descent, not an independently chosen continuum vector.

The term is a concrete finite-difference remainder. If 0 are the distinct nodes and their Lagrange polynomials on nodes, use

Searchable transcription

wi = li(0)/ti,
Dm,hf =
X
i<m
wi{f(tih) −f(0)}.

Searchable transcription

Their exact moment identities are P
i<m witj+1
= 1{j=0} for 0 ≤j < m, by interpolation of tj at
i
zero. Therefore Dm,hf −hf′(0) starts in order hm+1, with its integral Taylor remainder. Apply this
to the changed current pairing times the exponential action ratio on each active link, and average
the color directions over the Haar adjoint orbit of a unit color basis vector. Sum over the finitely
many active links. This is precisely Tm,h, including its current derivative and density derivative.
Its bounds after division by h are (LCE2).

The low-jet estimate for the first term in (LCE6) uses the quantitative stationary error as well as the identity (TF0c). Here is that step in detail. Write = boom , and for the stationary error. Equality of the analytic jets of and through order four gives

Searchable transcription

Dj(F −R′)(0) = DjE(0),
0 ≤j ≤4.
(TF12)

The issued stationary chart bounds the analytic coefficient norm of by shock( on radius = east(). Converting the coefficient bound to ordinary derivatives through order four costs the previously defined analytic derivative price 4), called gable in the source. Localizing the tensor to Meld costs at most 4), where

There are two further replacements: the original current by its localized current, and the full principal action by its finite retained action. Use the exact identity

(TF13)

The current difference has weighted price; the localized current has price at most trophy(. The full action derivative has bound

Searchable transcription

U(x) = bankCost(B(x))K momentPriceP (D/2)[GeoSize +2 paths],
T(x) = bankCost(B(x))Kx−100 + e−D rim(x)/4U(x).
(TF14)

Here 0 is the issued path width; depend on and the fixed derivative order four, and the geometric and path prices are fixed by the stage. Cofinal finite path cuts give the tail derivative bound. The weighted product rule for contracting a current with an action derivative bounds (TF13) by ) = vocation) + vocation (TF15)

The prices in (TF15) have the following explicit meanings. Set

Searchable transcription

cG(a) = # Loc(G, 1) e2a6! 24
a6
,
gG(a, k) = (1 + cG(a))k,
vocationG(a, C, D) = 16 cG(a/5)C gG(a/5, 3)D.
(TF20)

Searchable transcription

Here Loc(G, 1) is the fixed finite set of first-order link direction types (axis and color); its cardi-
nality has no volume dependence. The lattice shell count bounds the sum of e−a distance over these
directions by cG(a). Expanding a multilinear derivative in its coordinate directions and summing
these decays gives gG(a, k). For a current-action contraction, write DY F = P
lYl∂lF. The lo-
calized current has its first three derivatives bounded by Ce−(a/5) distance(l,0)gG(a/5, 3); the action
derivative has four derivatives bounded by D and vanishes at the identity. In its fourth product
derivative the term requiring a fourth derivative of the current is multiplied by that zero. The

remaining binomial coefficients sum to 2= 16. Summing ingives exactly (TF20). The vanishing action derivative follows because its one-color linear coefficient is invariant, and simplicity excludes an invariant color vector; it holds for both the full action and its invariant finite tail. This proves the contraction bound in (TF15) using only three current derivatives and four action derivatives. The scheduled radii and cuts give), and

Searchable transcription

x−100gG(σ/5, 4) + x−70gG(v/5, 4) + V (x) = O(x−64) ≤x−30
(TF16)

eventually. These bounds are uniform in the admitted finite torus and hold for every sufficiently large finite cut at that fixed stage. Equations (TF12)–(TF16) supply the second, third, and fourth derivatives as well as the zeroth and first ones. They are the quantitative hypothesis used in the rescaled Taylor remainder (LCE2). In that application the higher derivative bound and 2 thinM(4, give the order-five rescaled remainder, with(1 + high) Summing the Taylor terms gives the stated envelope.

The quiet derivative has zero scalar contraction by the two path-moment cancellations (TF6a)–(TF6c), whose full-family justification is given immediately below. The derivative of the appended quiet correction is. The term keeps the Haar Jacobian, and its centered constant is explicitly removed. The last term is obtained by expanding each finite surgery exponential into support factors, with coefficient; its entire weighted sum is bounded in (LCE3). Thus the four small terms are exactly altoD, and the two non-small terms have their actual scalar calculations.

For completeness, the retained quiet moments are small because the full issued loop family has zero moments. For each ordered pair of plaquette types and pair of color indices, define

Searchable transcription

ca,b
γ
= 1{types of γ=a} coeffγ(b),
zγ = offsetγ,1 −offsetγ,0 .
(TF17)

Searchable transcription

The path bounds include absolute summability of these coefficients and of each ca,b
γ (zγ)l, together
with their normalization
X
γ
ca,b
γ
= 0,
X
γ
ca,b
γ (zγ)l = 0
(l = 1, . . . , 4).
(TF18)

Searchable transcription

These are curvature loop value and first-jet normalizations, distinct from the potential jet normal-
ization. They are carried in the issued path data as the summable loop terms and their zero sums;
joining path families preserves them by absolute summation. The path-data construction retains
these identities along with the quadratic metric normalization, so (TF18) applies to the family
actually used in (LCE6).
Let L be the retained finite set. Subtracting its sum from (TF18) gives
X
γ∈L
ca,b
γ
= −
X
γ /∈L
ca,b
γ ,
X
γ∈L
ca,b
γ (zγ)l = −
X
γ /∈L
ca,b
γ (zγ)l.
(TF19)

Searchable transcription

Since |zγ,l| ≤length γ and the coefficient is bounded by its atom budget, an omitted bound
P
γ /∈L budgeta(γ) (1 + length γ) ≤A bounds every retained moment in (TF19) by A. Thus the
coefficients appended at displacement zero and the four unit displacements in (TF6a) have total
coefficient magnitude at most 5 #{color pairs}#{type pairs}(A + 4A). Indeed there are five dis-
placement choices for each type pair, and each coefficient has magnitude at most A + 4A; the
unit-displacement coefficients individually have the sharper bound A. Their paths have length at
most ten, so a weighted atom budget at width a costs at most an additional e10a. Fixed-order local

function and derivative bounds follow by the finite plaquette product estimate used for (TF11). The exponential path estimate gives momentPrice after a sufficiently large cut and radius, for width 0 and full loop price. This proves the small appended-tail bound used for. The augmented finite family, meanwhile, has exactly zero constant and first moments by its explicit coefficients. Its Hessian against a supported direction and a direction with affine curl therefore vanishes by (TF6b), proving the scalar cancellation for. Smallness of the correction and exact cancellation of the augmented family are separate consequences of (TF18) and (TF6a), respectively.

The issuance is constructive: for each of these regular scalar functions subtract the local section of its weighted Taylor card and put the masked difference in one regular activity slot, as in (TF5). Add the four principal formatted insertions and the finitely many secondary ones. Their coefficient is exactly the sum of their Taylor cards by linearity, and their support-weighted prices add. Equations (LCE3)–(LCE5) then prove both the bound on the issued coefficient and the paid remainder of its representative. This proves existence of the candidate with its coefficient identity and its budgets simultaneously. The principal issued divergence decomposes into six vector contributions:

Searchable transcription

V = W + altoD B −altoD K + altoD D + E + F.
(TR54)

Searchable transcription

The decomposition is exact because the local split satisfies y + D = K, and the regular-card map
is additive on this regular split. The central term K has zero Holt contraction by the finite-level
cancellation identity, so its scalar reading is zero by (TR53). The term B gives Re Holt(B)/12.
Its difference from the finite-level scalar payment midPay /12 is at most calorie/12. Each of the
four remaining terms W, altoD D, E, F, after multiplication by θ−2, has coefficient norm at most
Wave(−1). Applying the operator norm of M and summing gives
M(θ−2V ) −midPay
 ≤4∥M∥Wave(−1) + calorie
12
.
(TR55)
12

The total vector’s coefficient-size envelope isWave(1Wave(16). These estimates use the coefficient-size bound, which dominates its norm, and the actual positive threshold to cancel. The scalar payment comparison midPaylandPayshock, with landPay gives the last two terms and limiting value in (TR51). This reduces that limit to the local scalar

cancellation, the four explicit local envelopes and the payment comparison; it introduces no independent continuum assumption. The finite scalar payment itself is obtained by differentiating a massive comparison, with the mass parameter kept strictly positive until after the large-volume estimates. Let) be its scalar resolvent primitive and) its differentiated shell quantity. The source proves

Searchable transcription

d
dx Re Ri(ex) = −Re Si(ex),
|Re Ri(ex) + Lx| ≤C
(x ≤0),
(TR56)

eventually for every fixed, where and the same works for every such The derivative is a derivative of a convergent finite-volume resolvent expression; strict positivity of its massive matrix gives its inverse throughout this calculation. For any fixed0, the real mean value theorem gives a point), hence ], with

Searchable transcription

| Re Si(μi) −L| ≤
2C
b −a.
(TR57)

Searchable transcription

Only the two endpoint estimates in (TR56) are used, so their separate eventual assertions can be
intersected. No uniform convergence in the mass parameter is needed. The mean-value choice may
depend on the finite-volume index.
Let Xi be the local payment before its final second-derivative correction.
The comparison
supplies
Xi −Si(μi) −μ2
i Ki(μi) →0,
|μ2
i Ki(μi)| ≤C1μi,
(TR58)

for every moving mass in any fixed interval [ 0. To prove Re, first choose negative that4. Then choose large enough that the right side of (TR57) is at most 4. The interval lower endpoint is now fixed and positive, so (TR58) applies to the mean-value masses. Its vanishing error and the two preceding bounds prove the desired convergence. The choice order is essential: the volume limit is taken with this fixed positive lower mass, after choosing the error tolerance.

We derive the local resolvent bound in (TR58). Use a periodic lattice of side, angular momentum spacing, and normalized momentum weight. Write lift() for its centered momenta. If, any subset of momenta with has at most

Searchable transcription

(3R/h)4
(LR1)

members. Indeed their centered integer coordinates have absolute value at most. The lift is injective, and the integer coordinate box has at most (2 points. This upper bound remains true when the coordinate box wraps on the torus. For1, group momenta according to

Searchable transcription

2jμ ≤|kx| + μ ≤2j+1μ,
j ≥0.

Assign boundary points to one of their adjacent groups. A group has at most (6 points by (LR1); its summand ( is at most (2. Therefore

Searchable transcription

n−4 X
x
(|kx| + μ)−5 ≤
64
X
j≥0
2−j ≤2 · 64
(2π)4μ.
(LR2)
(2π)4μ

Only finitely many groups are nonempty at a given; extending their sum by zeros justifies the infinite geometric majorant.

Here is the integrand estimate to which (LR2) applies. At 0, scale an external momentum direction by The zero-field matrix, one-field vertex and two-field vertex have sizes 1 with bounded scaled derivatives through the finite orders in use. These follow from their vanishing origin jets and the analytic symbol bounds proved in the massive-symbol section. Ellipticity gives the inverse size. Differentiating gives

Searchable transcription

D(A−1) = −A−1(DA)A−1,

Searchable transcription

D2(A−1)[u, v] = A−1(DA[v])A−1(DA[u])A−1 + A−1(DA[u])A−1(DA[v])A−1 −A−1(D2A[u, v])A−1.

These formulas, and one further differentiation when needed, retain the same inverse size under scaled derivatives. Hence the contracted resolvent-insertion integrands have scaled analytic-jet bounds of size, with constants depending on the fixed directions and finite matrix dimensions but not on

For such an analytic integrand , let [! be its homogeneous Taylor coefficient. The scaled bound gives

Searchable transcription

z3|[V ]3(p)| ≤Cz−2/3!,
hence
|[V ]3(p)| ≤C′z−5.

For the second coefficient it gives. The angular cube and1 give a fixed upper bound, so this too is bounded by. Summing the finite matrix entries and applying (LR2) bounds both contracted coefficients by a constant times

The local readings are explicit finite coefficient selectors. For a scalar function on integer arguments set

Searchable transcription

Pan f = 1
2(f(1) −2f(0) + f(−1)),
Flop f = 1
12[8(f(1) −f(−1)) −(f(2) −f(−2))].

Searchable transcription

On powers of degree at most two, Pan selects the quadratic coefficient; on powers of degree at
most three, Flop selects the linear coefficient. The mixed selector is i Pana Flopb, evaluated on the
specified two-direction momentum contour, with the corresponding powers of h−1. Its fixed finite
coefficients bound it by a finite sum of the directional third coefficients just estimated. The local
second selector is Pan on the opposed-momentum contour, with h−2. Analytic Taylor remainder
bounds show that each selector differs from its indicated derivative coefficient by a term tending to
zero as h →0, with mass restricted to a fixed [δ, 1]. For δ > 0, the massive matrices have a common
positive lower bound, so their analytic inverse neighborhoods and the higher derivatives needed for
these remainders are bounded throughout this interval. This also allows a mass μn depending on
n. Absorb a final selector error at most 1 into μ−1
n, since μn ≤1.

The physical local lens is the prescribed half-weighted physical resolvent contraction minus its gauge comparison contraction. Both have the preceding bounds; their matrix tickets identify the actual finite matrices with the Fourier inverse matrices. Linearity and the triangle inequality therefore prove, with constants fixed before the positive lower mass cutoffis requested,

Searchable transcription

∃Cd,a,b ≥0 ∀δ > 0 ∀(μn)n,
δ ≤μn ≤1 ∀n
 =⇒|Kd,a,b,n(μn)| ≤Cd,a,bμ−1
n
eventually. (LR3)

Searchable transcription

The eventual threshold may depend on δ and the moving mass family. Once δ is fixed, h ≤δ ≤μn
eventually, as required in (LR2). Sum (LR3) over the finitely many tensor indices. Multiplication
by μ2
n yields C1μn, the remainder bound in (TR58).
For the local second-derivative correction, its exact massive comparison has error tending to
zero at each fixed 0 < μ ≤1, and its massive term is at most Cμ2μ−1 = Cμ by the second-selector
version of the same argument. Given ε > 0, choose 0 < μ ≤1 with Cμ < ε/2, and only then take