The centered trace and its non-freeness test · Part 8

16.6.10 A normalized position coupling and its real remainder

A finite formal parity jet is distinguished from a real remainder estimate. Here the exact normalized coefficient identities give a bounded quotient at order four, and hence an actual ) error. The weaker frame class retains only the stated leading coefficient after its positive scalar branch and nonzero two-point normalization are imposed; the stronger remainder is not transferred to arbitrary frames.

The degree-four frame can be adjusted by two independent positive real factors while fixing the unit. Let be the centering projection and Π the rotation-invariant projection onto the scalar direction in the centered field space. They satisfy Π, and ΠΠ. The free scalar obeys Π, while the free Maxwell stress components obey = 0. Define ΠΠ (TR82)

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The three summands are complementary projections. Consequently A−1
x,y = A1/x,1/y. It fixes the
unit, preserves expectation, commutes with rotations and conjugation, multiplies O by x and
multiplies T 0 by y. These facts follow by applying the three projections successively; in particular
none of the centered components feeds the unit component.
Put P ′
s = PsAxs,ys. Multilinearity of the moments and their derivatives gives the exact trans-
formations
E′
s = ysxsEs,
B′
s = x2
sBs,
R′
s = ysx2
sRs,
Q′
s = x3
sQs.
(TR83)

For example the shell test contains one stress label and two scalar labels, accounting for its three factors. Let, which is real and nonzero by (TR79). Define, for sufficiently small positive

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s
b
Re Bs
,
ys =
b
x2s Re Rs
.
(TR84)
xs =

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The quotients tend to one by (TR78), so xs, ys > 0 eventually and both tend to one. Outside
that final interval take any total choice of frame; only the eventual identities are used. Equations
(TR83)–(TR84) impose
Re B′
s = b,
Re R′
s = b
(TR85)

exactly. This establishes existence of the normalized frame. Conversely any positive adjustment of the form (TR82) satisfying (TR85) tends to the identity. Indeed Re implies 1; positivity chooses1. Then Re gives Thus all such choices preserve the limiting frame and its reality, symmetry, unit and centering properties.

They also preserve every compact separated jet comparison. For any finite tuple of labels, expand in a fixed finite basis. Its coefficients converge to those of. Expand the moment multilinearly in that basis, commute its finite sum with the fixed position derivatives, and bound each difference by the sum of its coefficient error times the bounded free derivative and its bounded coefficient times the original comparison error. Compactness bounds the free derivatives uniformly on the separated compact set. Finitely many terms then give any prescribed error. This proves all the comparisons for, without a further extraction. Set

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c = (−d2)b ̸= 0,
λX(s) = Re E′
s
c
,
gX(s) =
q
λX(s).
(TR86)

This is a measured position-space charge, defined from a separated derivative test rather than from the physical stress coefficient. First apply (TR80) with the independently calibrated Wilson charge

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gW . Equations (TR83) and xs, ys →1 show Re E′
s/gW (s)2 →c. Hence λX/g2
W →1, so λX > 0
eventually,
gX/gW →1,
log(1/s) gX(s)2 →(2b0)−1.
(TR87)

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The positive square root fixes the branch. In particular this coupling is calibrated to the Wilson
charge before it is used in the Ward equations.
Transfer (TR76) to P ′, denoting its coefficients again by κ, d, e. The coefficient comparison
survives this asymptotically identity change, so d/g2
X →d2. The exact Ward equations (TR77)
still hold. Take their real parts, use (TR85), and write qs = Re Q′
s →q0 = Re Q0. They give

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κs −1 = −dsqs
b ,
ds = d2gX(s)2κs.
(TR88)

The second equality uses the definition (TR86), not a truncated expansion. Therefore

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κs −1
gX(s)2 →−d2q0
b
,
ds −d2gX(s)2
gX(s)4
→−d2
2q0
b
.
(TR89)

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For the first limit use d/g2
X →d2 in the first identity; it also implies κs →1. The second follows by
substituting the second identity into its numerator. Since a convergent real quotient is bounded,
(TR89) proves

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ds = d2gX(s)2 + O(gX(s)4),
4ds = −2b0gX(s)2 + O(gX(s)4).
(TR90)

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It also implies the signed cubic remainder is little o, by dividing the order-four bound by g3
X and
using gX →0. No full perturbative series is required for these assertions.
Finally take the trace and expectation of (TR76). Since the aligned scalar is centered, T 0 is
traceless and the unit has expectation one, one obtains es = s4⟨tr T⟩/4. Subtracting the selected-
vacuum expectation instead gives the exact centered-field comparison

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s4Θ = 4ds centerχ(P ′
sO).

This identifies the scalar in the normalized frame with the actual physical trace used in the spectral and exclusion arguments.

The projection in (TR82) is constructed by compact-group averaging. Average the real orthogonal action on degree-four fields against normalized Haar measure, on the left and on the right as in the aligned frame construction, and compose the two averages. Haar invariance shows that its image is fixed by every orthogonal transformation and that it is unchanged when its argument is rotated. It fixes every invariant vector; hence it is idempotent. Since the action commutes with conjugation, the average does also. Invariance of expectation shows that averaging preserves expectation, so it commutes with centering in the required identities. The scalar is fixed. Each off-diagonal component of changes sign under reflection of just one of its indices; its invariant projection is therefore its own negative and is zero. The projected diagonal components are equal by rotation covariance. Their sum is zero because is traceless; all diagonal projections are consequently zero. This proves every projection property used in (TR82).

The leading scalar coefficient also transfers to the weaker position normalizations asserted by the theorem. Such a frame need only be real, rotation-aligned, unit-preserving, have its centered scalar in the prescribed positive branch, and have the centered scalar two-point norm at the fixed separated test tend to the free nonzero value. The branch classification places its centered scalar on the same positive ray as that of a reference plotted frame:

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center(LsO) = asPsO,
as > 0.

The ratio of the two centered two-point norms equals . Both norms tend to the same nonzero free value, so 1, and positivity gives1. Taking the trace and centering in the tensor decomposition yields center(tr center(), and the analogous identity for. The centered scalar is nonzero eventually by its nonzero two-point norm. Cancelling it gives , hence for every equally normalized coupling. Unit mixing in disappears under centering. This proves precisely the leading transfer, without transferring the stronger order-four bound to arbitrary frames.

All spectral pairings used in this discussion are ordinary integrals at each fixed positive scale. A dilated compact continuous test remains compact continuous and is integrable against a cone-supported measure with finite future Laplace damping, by the compact-test argument following (TR12). A dilated Schwartz test remains Schwartz and is integrable by (TR10)–(TR11). On the limiting cone measure the same bounds apply to its polynomial density ( p. Thus the compact and Schwartz versions of the spectral limit, and their relative and joint position versions, carry their integrability assertions as well as their limiting identities.

16.6.11 The same scalar diagnostic in the selected vacuum

The scalar diagnostic is first normalized in the unselected hierarchy and then interpreted in the selected vacuum through an exact centered label identity and positivity. Convexity and secants control the selected scalar function. This argument keeps the phase fixed; it does not obtain a desired result by replacing the selected vacuum as the radius changes.

The frame was positioned using the unselected hierarchy. Its interpretation in the selected vacuum follows from the exact centered label identity and positivity. Let, let and write1. The coefficient comparison gives

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s4Θ = 4dsws,
log(1/s) 4ds −→−1.
(TR91)

For any fixed separated two-point shape, rotation and translation reduce the trace moment to ). The short-distance limit (TR2), with the fixed dilation of the logarithm accounted for, gives log

Substitute (TR91) and divide by the square of its nonzero limiting coefficient. It follows that tends to the free scalar two-point function. The exact unit remainder in the uncentered label comparison gives), hence0, since log(10 and the second limit in (TR91) is nonzero. The unit insertion rule gives

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⟨vs(sx1)vs(sx2)⟩χ = ⟨ws(sx1)ws(sx2)⟩χ + m2
s.
(TR92)

Thus the uncentered selected scalar moment has the same pointwise free limit at every separated shape. The derivative diagnostic requires a further argument; differentiating this pointwise limit would be unjustified. For fixed 0, the real function

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Us(t) = Re⟨vs(ste0)vs(0)⟩χ,
t > 0,

is convex. In fact (TR91) and (TR92) express it as

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Us(t) =
s8
(4ds)2 E(st) + Re(m2
s),
(TR93)

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for all sufficiently small s. The first prefactor is nonnegative, and E(st) =
R e−stp0 dμ(p) is convex
in t, since each real exponential is convex and its integrals are finite. The additive constant does
not affect convexity.
We use the following elementary consequence. If differentiable convex functions Us on (0, ∞)
converge pointwise to V , and V is differentiable at 1, then U′
s(1) →V ′(1). For fixed a < 1 < b,
convexity bounds U′
s(1) between the secant slopes from a to 1 and from 1 to b. Pass to the pointwise
limit in these two fixed slopes, then let a ↑1, b ↓1. This proves the assertion with the correct order
of limits. Here (TR92) gives V (t) = cGt−8, so

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U′
s(1) −→−8cG = b.
(TR94)

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Smoothness of the selected separated moments identifies this derivative with the real first-position
scalar derivative Bsel
s .
Apply physical conservation to (TR76), now pairing the resulting derivative identity with the
selected scalar field. Derivatives of the unit vanish, exactly as in (TR77). Therefore

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κsEsel
s
+ dsBsel
s
= 0.

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Using κs →1, ds/g2
W →d2, and (TR94) yields
Re Esel
s
gW (s)2 −→(−d2)b = c.
(TR95)

This proves the selected-vacuum diagnostic claimed for the same measured fields. The argument needs only pointwise scalar asymptotics, convexity and the exact conservation identity; it claims no higher-rate comparison between selected and unselected diagnostics.

The quantitative near-kernel vector for the trace sensor

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The near-equation vector is an explicit linear combination of transported degree-five insertions. Its
weighted sensor estimate is stronger than bare convergence to the equation subspace. The proof
therefore constructs a funded local marker with a genuine finite density variation and replicated
root bound. A formal integration by parts symbol, without its local representative and expense
estimates, would not supply this weighted conclusion.
The near-kernel vector required in (TR50) is a specific combination of two transported degree-
five insertions. Write bβk for the prescribed clock and βk for its extracted coupling; the common
calibration gives bβk/2 ≤βk. Let Ak(c) = AgeK(c, k). For every fixed c,

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Ak(c)/βk −→0.
(TS1)

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Indeed the clock estimate gives Ak(c) ≤bβ1/2
k
eventually, while β−1
k
≤2 bβ−1
k . Their product is at
most 2 bβ−1/2
k
→0.
Let Pk be the real orbit projection of the principal degree-five logarithmic insertion, and Ek
the corresponding orbit projection of the degree-five divergence insertion associated with the un-
corrected stress. They are expressed in the same transported coordinate chart and use the same
extracted coupling. Define
vk = β−1
k Pk −Ek.
(TS2)

The principal bounds proved in the finite scalar calculation give6) and Lease( . The divergence insertion satisfies

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Ek −→−Stem5,
Ak(1/5) Lease(Ek) −→0.

Here Lease(Stem5) = 0, because its bare vector is an equation-of-motion vector and the scalar sensor annihilates that equation subspace. The limit of is obtained by projecting the same finite three-coordinate stress card; its coordinates tend to the negative equation vector. The scalar equality on these cards is exact, so its transported error has the displayed weighted bound. Equations (TS1)–(TS2) consequently give

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vk −→Stem5,
Ak(1/5) Lease(vk) −→0.
(TS3)

For the second assertion expand the linear functional in (TS2). The principal contribution is ) Lease(0 by (TS1) and its finite limit; the divergence contribution is the preceding weighted error.

The other required estimate concerns the measured vector), where is the common retained scaling sequence. Before transport, (TS2) is times the orbit of the coefficient difference

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Xk −βkZk.

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The exact finite local divergence relation makes this difference a served equation insertion. Its
coefficient charts and actual lattice functions have common limits by the already established loga-
rithmic insertion comparison. The degree-five core projection commutes with the orbit operation,
and the coordinate change is linear, so (TS2) is exactly its transformed, β−1
k -normalized limit; this
is an identity, not an asymptotic replacement.
The retained equation estimate on these coefficients gives, for every integer p ≥0,

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bβ p
ki si(vki) −→0.
(TS4)

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Only the first component needs to be estimated before the finite real orbit projection; the remaining
components are its prescribed symmetry transforms. In particular take p = 1. The clock estimate
Aki(1/5) ≤bβki eventually and positivity give

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Aki(1/5)si(vki) −→0

by comparison of norms. Together with (TS3), this supplies both weighted leaks required by the finite-dimensional repair argument (TR50). That argument produces uniformly bounded functionals, convergence, and

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Aki(1/5)∥μi ◦si −Lease ∥op −→0.

Thus the quantitative scalar sensor is obtained from an actual near-equation insertion, with its growing weight paid before the stress correction is solved.

The passage from the retained equation insertion to (TS4) is detected by degree-four spectators. At each retained horizon choose the finite local coefficient family for on the same actual region. Its locality window contains both the equation surgery and the divergence film: the first is contained in the inspected radius-canvasR tower, enlarged by eight; the second in the radius-townD tower. Use Sliver = max(canvasR +8 townD +1)

This window is subpower in the clock. Its site count and the total uniform norm allowance are eventually at most the fourth power of the clock. Thus both local representations fit in the common chamber before applying the retained insertion comparison. On the healthy region, the exact finite equation identity expresses the difference of the two films as the surgery marker plus its centered normalization. The surgery approximation error is

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bounded by its small normalization times e−prism; after division by that normalization it is at most
e−prism. Outside the healthy region use the sum of the uniform film bounds, covered by the same
fourth-power allowance. These are the local and global bounds used to establish the equation probe
estimate; no continuity of a hard healthy-set indicator is asserted.
For each fixed degree-four free spectator w, the resulting two-point estimate after taking the
coefficient limit is
S

xi; a5
i project(Xki −βkiZki), U(4)
i
w
 ≤Tau(ji, −100).
(TS5)

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The left-hand side is an actual fixed-placement hierarchy limit of its lattice two-point moment.
The two lattice positions are the prescribed near point and the origin.
They are distinct and
have a common nonwrapping placement on the eventual volume tail. The spectator is its retained
corrected lift, so the coefficient and correlation limits use the same refinement.
The coordinate identity in (TS2) converts the first insertion of bβ p
kisi(vki) under the scaling frame
into
bβ p
kib5
i a5
i project(Xki −βkiZki),

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where the physical scale is aibi. It follows from linearity of the transported chart, the exact cancel-
lation of β−1
ki βki, and the definition of the degree-five scaling factor. Consequently (TS5) bounds
each tested response by
bβ p
ki tiltCap(ji)5 Tau(ji, −100) −→0.

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The clock and tilt factors have rate zero in the horizon variable, whereas the last factor has strictly
negative rate. This proves the limit for every fixed p. Finally the real invariant-vector detector,
obtained from these degree-four tests and the same frame comparison, converts their vanishing
responses into convergence of the measured invariant vector to zero. This is (TS4); the estimate is
proved on the exact degree-five near-kernel combination rather than inferred from its unweighted
limit.
The marker estimate in this comparison is a finite local sum. At the summit scale k = k(i), put
x = bβk and let ρ(x) > 0 be the surgery parameter. For fixed surgery order m, the marker is

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Mm(U) =
X
wm,α
ρ(x) Lα(U).
(TS6)
α∈Am,x

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Here Am,x consists of the surgery addresses in the prescribed finite box and their finite orbit indices;
Lα is the local lens obtained from that surgery. The orbit census gives
X

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α∈Am,x
|wm,α| ≤cardB(x) wheat(m).

The box census has subpower growth in, and the surgery parameter bound consequently gives

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cardB(x) wheat(m)/ρ(x) ≤x2

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eventually. On the chosen scale calendar this is at most T100000
i
. Each lens has support in the same
window of radius 2 coast(x), that window fits the comparison chamber, and its pullback to the finest
lattice has rooted test norm at most T−1900000
i
. The local lens estimate applies at every address in
the sum because its beacon is in the common enlarged box and within the allowed distance from
the origin. The rooted test norm is a seminorm on these chamber-supported functions. Pullback
commutes with this finite sum and its scalar coefficients. Thus (TS6) has rooted test norm at most

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T100000
i
T−1900000
i
= T−1800000
i
.

The surgery’s pointwise bound also gives for every configuration. All these assertions hold eventually in, uniformly in the finer cutoffand volume; is fixed before that eventual threshold.

We spell out the label transfer as well. Let be the two local films used in the surgery, and their billed degree-five labels. With fine comparison spacing, summit spacing, and actual coupling, set

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q = (ai/ak)5,
Ve = Xe −βYe,
Je = q(xe −βye).

The billing approximation, physical normalization, and transport give

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∥back(Je) −I(a5
i Ve)∥root ≤T−200
i
.
(TS7)

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Indeed the two individual errors are at most T−430
i
; the triangle inequality multiplies their sum by
at most 1 + β. Since 0 < β ≤2 bβk, the calendar bound 1 + β ≤T230
i
pays this loss.
Let Dd denote the signed finite orbit sum defining the relevant divergence component.
It
commutes with the physical insertion map, with coefficient transport, and with pullback on the
common window. Applying its rooted seminorm estimate to (TS7) gives the same stated error for

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back(DdJ) −I(a5
i DdV ).

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The real orbit projection places a5
i DdV in the required invariant label subspace.
To check its
coefficient budget, each billed input has transported coefficient size at most 2, hence Ve has size at
most 2 + 2β. If c ≥1 is the fixed degree-five orbit cost, its divergence has size at most c(2 + 2β).
Eventually 6c ≤bβk = t, with t ≥1, so

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c(2 + 2β) ≤6ct ≤t2 ≤t4.

Multiplication by gives exactly the physical coefficient budget. The previously obtained crown bound for, its chamber support, and (TS7) now give the required physical probe estimate for the corrected divergence label. In particular this transfer retains the coupling multiplying , its degree-five spacing factor, and the actual common readout.

Here is how the local lens estimate entering (TS6) is obtained. A single lens is the Haar average of a local surgery difference. Copy it into every chamber of a tiling based at a coarser reporting level, applying the chamber’s signed motion and conjugation. If the original surgery is based at , each copied base lies at distance dist(0) from its chamber center. The embedded centers are separated by at least the block ratio between levels and. Choose that ratio larger than both

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2 pierSize +2 + 2 dist(0, z),
64B b + 2B + 2 dist(0, z).

The inspected surgery supports are then disjoint, the changed bonds are distinct, and the stronger separation needed for the marked expense bound holds. All rotations and optional conjugations preserve the local boundedness and gauge conditions. Integration in one component of a product Haar variable is its ordinary one-component Haar integral, so the marked coefficient at each chamber is exactly that chamber’s copy of the original lens.

Introduce two marker slots per chamber and give each the same positive weight). The surgery expense inequalities have the following explicit form. Put for the support exponent, for the support-cover bound, for the bound on the inserted local function, and for the surgery truncation. With the ordinary and tail tariffs denoted by and, choose1 and such that

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4ce(s−3)l ≤eb,
Fe(s−3)l+2 w(k)2
c
≤r,
4e−mTlease ≤1,
Twing ≤1.

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The remaining expense is

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t = r + 8ebe−mTlease + 2ebbforced,
(2B + 1)4t ≤1,
et −1 ≤uallowed.

The local surgery expansion and its support census then give a served marked insertion with support exponent4 and budget 1. The factor is included in its coefficient bound. The tariff summation is performed separately at each surgery base; disjointness of the inspected supports permits their combination with no additional factor equal to the number of chambers.

Transport this insertion from down to an intermediate level, retaining the two markers per chamber and weights satisfying

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w(j) ≤δjw(j + 1),
m0 ≤j < k.

At each scale the embedded chamber anchors are distinct. Hence the occupancy of the doubled marker set is at most two, which is within the available multiplicity This proves the admissibility of the marker transport at every step. The retained insertion estimate and positive scalar normalization bound its final, fully marked coefficient. If is the number of reporting chambers and the physical Wilson moment of the product of the copied lenses, the result is

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|Q| w(m0)2v ≤exp[| Sitem0 | treasury(m0) + 4v] .

This coefficient identity follows also directly by integrating the restored marked insertion: the partition identity transports the intermediate integral back to the finest Wilson integral, and the same positive Wilson partition factor occurs on both sides and cancels. The chamber-count identity is Site

Taking the-th root therefore proves the local rooted estimate

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|Q|1/v ≤exp
(aJ/am0)4 treasury(m0) + 4

w(m0)2
.
(TS8)

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Thus the bound is independent of the number of copied chambers.
For the summit calendar take J = i and m0 its prescribed basin. The surgery window radius and
the necessary separation radius are subpower functions of bβk; the calendar’s block ratio eventually
exceeds them uniformly in the volume. Its treasury and weight choices make the right side of (TS8)
at most T−1900000
. The same window fits the reporting chamber. This establishes the local estimate
i
used in (TS6), with the finite surgery expense conditions displayed above.

The expense conditions in this argument are consequences of the same large-coupling choice; they are not additional assumptions on the final theory. To see the numerical allocation, fix a stage and the integer. Set

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B = ⌈(log x)2⌉,
P = B40 + 1100,
M = B60 + 1100,
ex = e−P/8.

These are respectively the prism and surgery truncation used above, and eventually. For every fixed constant and integer

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Cxj ≤eP/32
eventually.
(TS9)

Indeed log eventually, while the exponent has degree forty in; taking logarithms proves the claim. The lease tariffhas an upper bound, uniformly over the admissible representations at