Exact stress correction and local quantitative estimates

Exact stress correction and local quantitative estimates

Exact correction and quantitative near-kernel recovery. The three finite candidates share a filter and a common extraction. The correction inverse is applied to the actual detector and divergence coordinates, so the repaired stress obeys exact conservation and the specified scalar amplitude. A small detector reading is useful only after the quantitative near-kernel argument controls the other paid coordinate; it is not an exact kernel statement by itself.

The six local contributions below retain their separate estimates until their sum earns the required trace bound. Moving positive masses are chosen before the volume estimates, and the resolvent selectors keep both their finite windows and their errors. Determinant differentiation, vector/ghost poles and the angular factor eleven are evaluated with those same conventions. This construction recovers a Ward-normalized scalar coefficient from actual finite objects. It does not replace the finite calculation by the formal value of a continuum determinant.

25.1 The three finite candidates and their common extraction

Here are the actual finite functions behind the three columns used below. Write curv(Hol)), with the curvature coordinate and normalization of (SQ1)–(SQ12), and put . At a fixed issued ticket, the three tensors are

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J∅
ij(x) = Meld(J)ij(x),
(
0
i = j,
P
a Pija(x)
i ̸= j,
Jsh
ij (x) =
(TF11)
X
Jsc
ij (x) = δij
a,b
Paab(x).
4

The first is exactly the restored, localized principal tensor (TF0d), including its six antisymmetric differences and its scalar forward difference. The other two use elementary plaquette holonomies; there is no new stationary current in their definition. Their free coefficient vectors in the three-coordinate tensor basis are respectively the principal vector, (0 1), and (0 0). In the source’s three-coordinate convention the principal free vector isweyl. Its physical sign is fixed together with the Ward charge in (TR41)–(TR44); it cannot be changed independently of that normalization.

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All sixteen entries of each tensor, all sixteen individual stencil derivatives, and all four diver-
gences are extracted on the same filter. Explicitly these are the origin functions Jb
ij(0), elmi Jb
ij(0),
and P
i elmi Jb
ij(0), for b = ∅, sh, sc. The tensor entries have degree-four coefficient records and
the derivatives have degree-five records. These records include both the finite-cutofflabel and its

retained limit; extraction of the joint correlation book gives the corresponding fields Ω ∆ Thus extraction of a divergence is extraction of the sum of its actual finite derivatives, not an independently selected vector limit.

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The local bounds needed here have a direct source.
A plaquette has four links.
If
each matrix link has derivatives through order k bounded by the chart price d, its holon-
omy has price travelCost(G, k, 4, d).
Applying the curvature map gives price z(d)
=
The bilinear product has price 2k∥⟨·, ·⟩∥z(d)2.
curvatureCost(G) travelCost(G, k, 4, d).
Sum
the finitely many axes and multiply the scalar branch by 1/4. This gives a bound independent of
the torus and the ticket, supported in radius four. One stencil derivative enlarges the radius to
eight. The scheduled small-field extension makes these bounds into the coefficient and correlation
bounds used in extraction. The issued principal tensor has these same bounds by (TF0d) and its
localization estimates. The condition 20(high(x) + rim(x) + 10) < n makes all integer offsets used
in the coefficient tests nonwrapping.

Taylor projection commutes with the stencil divergence on these records: expand the finite translates and use the stencil moments from (TF0b). The degree-four stress coordinate therefore maps to its degree-five divergence coordinate. For the principal branch the antisymmetric difference corrections have zero relevant tensor coordinate; the scalar difference has zero trace coordinate. Consequently its free divergence scalar coordinate vanishes. This is the specific low-degree statement used later; it does not say that its entire finite divergence vanishes. The finite action, localized current, surgery polynomial and coefficient equations are issued by one ticket, so the six terms of

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(LCE6) concern this same first branch. The two elementary branches supply the correction columns
in (TR41).
With YiT = r4
i U−1
T and Pi = BiYi, their normalized limits and correction are precisely
i
(TR41)–(TR44): solve the two divergence coordinates for ci,0, ci,1, then set eTi = BiLi+ci,0Ωi+ci,1∆i.
This specifies the finite origin, normalization, and limiting meaning of all three candidates and both
correction objects.

Source–readout formulation. The three candidates are three realized representatives read on one filter with one family of coefficient records. The correction is a motion along the two elementary columns, which span, together with the principal one, the three-dimensional tensor space of (FC3); by (FC14) the divergence reader separates exactly these directions modulo the equation direction so the corrected tensor is determined by its two divergence coordinates. Extraction of a divergence is extraction of the sum of the actual finite derivatives, on the same filter, and not an independently selected vector limit; this is what makes the corrected tensor a single realized object and not a combination of limits taken along different subsequences. The sign of the principal vector is not a free presentation choice: it is tied to the Ward normalization of the charge and cannot be changed independently. For the principal branch only the low-degree free divergence scalar coordinate is shown to vanish; the whole finite divergence is not asserted to vanish.

25.2 Exact stress correction and its scalar amplitude

The coefficient TR39 is obtained by finite-dimensional correction. The centered real symmetric covariant candidates Ω ∆ have free limits Maxwell stress, a traceless correction and a scalar tensor of trace 4. For their two-component divergence detector, putΩ∆ . The matrices converge to an invertible free map. In fixed bases factor a nearby matrix as); for2 the geometric inverse gives uniform bounds. Solve exactly (TR41)

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The detector estimate ∥Xi∥≤CRi gives ∥ci∥≤C′Ri, Ri/Bi →0. Since Ki has exactly the true
divergence kernel, eTi = BiLi + Sici is exactly conserved eventually. For ti ≥Ri ≥0, the scalar
detector has
li(Xi) →−g,
tili(KiΩi) →0,
ti(li(Ki∆i) + 4) →0.
(TR42)

Coefficient bounds kill both error products. Applying to TR41 and rearranging gives

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4ci,1 = li(Xi) + ci,0li(KiΩi) + ci,1(li(Ki∆i) + 4).

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Hence ci,1 →−g/4 = forceR, with sign and factor four fixed by the detector equation. The primitive
detector estimates are derived below. Write YiT = r4
i U−1
i
T componentwise and Pi = BiYi. The
candidate comparisons give PiLi →TMaxwell, PiΩi →TΩand Pi∆i →T∆. Thus

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Yi eTi = PiLi + ci,0
Bi
PiΩi + ci,1
Bi
Pi∆i −→TMaxwell.
(TR43)

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The stronger traceless estimates are Bi tr(PiLi) →0 and ci,0 tr(PiΩi) →0. The remaining scalar
contribution yields
Bi tr(Yi eTi) −→4 forceR Oscale.
(TR44)

Only the first coefficient needs a growth estimate; the second has the finite limit just established. Onto-ness of makes onto. Scaling and exact unit identifications equate the centered physical and free stress dimensions; the detector target has the free divergence-target dimension. Kernel equality and rank–nullity therefore give a one-dimensional physical conserved centered stress space, as in the free calculation. The actual centered Ward-normalized stress is nonzero. The free derivative map is nonzero because it attains a nonzero target coordinate; otherwise separating Choose an observer with derivative value observers would make that target zero. Its

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free translation-flux test has value y; the physical test of R, scaling positions, window and fields
together with the prescribed single derivative factor ri, tends to y. Vanishing R would contradict
this. Consequently eTi = viR with real vi. The shell test of both R and eTi tends to the same nonzero
y, for the latter by TR43 and tested derivative comparison. Linearity and division give vi →1,
Multiplying TR43–TR44 by v−1
hence eventual nonzero vi.
transfers both limits to R.
TR40
i
removes its unit shift and gives the selected trace limit used in TR2. The lifted correction source
retains the three candidates, exact kernel detector, inverse matrix and Ward shell test. TR41 is
its finite correction reader, and TR42 recovers its scalar coordinate with the numerical sign intact.
Rank–nullity and the nonzero charge reader identify the same physical stress line; normalization
by vi then recovers the actual Ward-normalized tensor rather than an arbitrarily scaled conserved
representative.

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Exact detector correction and Ward-line recovery.
Let Jst, Qst and Jdet, Qdet be the hori-
zontal maps on the finite stress and detector spaces. Transport KZ
i = JdetKiQst, SZ
i = JstSi, and
XZ
i = JdetXi. Then the exact inverse on the horizontal detector image gives

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cZ
i = −(KZ
i SZ
i )−1XZ
i = −(KiSi)−1Xi = ci,
eT Z
i = Jst eTi.

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In the first expression the output coefficient space is unchanged. For Ai = A + Ei = KiSi and
∥A−1Ei∥< 1/2, ∥A−1
i ∥≤2∥A−1∥. Thus the same detector bill bounds both correction coordinates
by C′Ri. Moreover KZ
i eT Z
i
= Jdet(Xi + Aici) = 0. Transport of the exact kernel equality makes
this genuine divergence zero, not merely zero at a selected detector.
Use lZ
i = liQdet. With εi = li(KiΩi) and δi = li(Ki∆i) + 4, the exact scalar identity yields

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|4ci,1 + g| ≤|li(Xi) + g| + C′Ri(|εi| + |δi|) −→0.

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Indeed Ri ≤ti and TR42 pays each error product, including when ti = 0. This recovers the actual
finite scalar limit forceR = −g/4 = −b0/2 with the original sign and factor four.
The transported scaling readers satisfy YZ
i Jst = Yi and PZ
i Jst = Pi, so TR43 is identical term
by term. Before the trace limit the exact decomposition is

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Bi tr(Yi eTi) = Bi tr(PiLi) + ci,0 tr(PiΩi) + ci,1 tr(Pi∆i).

The first two terms have their retained stronger vanishing estimates; the third uses the actual scalar coefficient limit and trace 4. This recovers TR44 without replacing the stronger trace estimates by ordinary candidate convergence.

Stress-space dimensions, detector rank and kernel equality are preserved by the horizontal isomorphisms. The nonzero Ward shell reading also remains identical. If is that common linear shell test, the exact ratio is

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vi = Fi( eTi)
Fi(R) −→y
y = 1,
y ̸= 0.

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The denominator is eventually nonzero; the same ratio is recovered from the lifted tensors. Thus the
physical stress line and its Ward normalization, including the subsequent v−1
i
transfer and TR40
unit removal, are retained. The primitive detector and stronger trace estimates remain explicit
dependencies developed next.

25.3 Scalar coordinates and quantitative detector transfer

The real invariant degree-four scalar basis 1 has an injective coordinate map and a continuous finite-dimensional left inverse. Apply it to the same running trace limit to obtain simultaneously

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U−1
i
(r4
i Θ) = ei1 + 4diOscale,
Bidi →d2,
Biei →0.

Exact scalar membership ensures these coefficients without unrelated subsequences. The free surjection has kernel Columns Ω= (0 1) and ∆= (0 0) are independent modulo that line: coordinate zero forces its coefficient to vanish. They span the quotient because (TR45)

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This proves the free invertibility used in TR41. Let si map bare degree-five coordinates to the free
divergence target, converging in operator norm. Candidate detector pi = KiAi and bare coordinates
ui satisfy
HN
i (pi −siui) →0
for every fixed N ≥0.
(TR46)

The clock dominates. For the correction columns, withrespectively Ω = 0 gives their limits. Uniformly bounded scalar detectors obey

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sup
∥u∥≤1
|ti(li(siu) −L(u))| →0.
(TR47)

The bare correction comparisons also give0. Split the full error as

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ti(li(pi) −L(u∗)) = tili(pi −siui)
+ ti(li(siui) −L(ui))
+ ti(L(ui) −L(u∗)).

= 1 and kill the first, TR47 and bounded the second, bare comparison the third. Exact readings4 give TR42. For the principal renormalized bare divergence, (TR48)

Even if grows, TR47 gives error at most0. Bounded handles proving. Bounded and eventual1 also yield, the same norm bound used by the correction solver. The elementary absorption principle used throughout this step is: if, and0 in a normed space, then0. The lifted detector source retains the bare coordinates, moving map, clock rates and scalar functionals. TR46–TR48 commute the divergence reader with the scaling identification with paid growth, while TR45 recovers the exact two-column quotient inverse. The continuous scalar left inverse then recovers both trace coordinates on the same refinement.

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Scalar coordinates and paid detector-map recovery.
On the invariant scalar space let
C(e, d) = e1 + 4dOscale and choose its continuous left inverse LC.
Horizontal transport gives
CZ = JC and LZ
C = LCQ, so LZ
CCZ = 1. The inverse scaling reader satisfies (UZ
i )−1Ji = U−1
i
,
hence the same simultaneous pair (ei, di) is recovered at every finite i. Applying the fixed linear
map to the running trace limit recovers Biei →0 and Bidi →d2 on the same refinement.
In the free correction quotient, the coefficient reader is explicitly

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[w] 7−→(w2 −w0, 4w1 + w0),
(x, y) 7−→[xΩ+ y∆].

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It is well defined because adding any multiple of (1, −1/4, 1) changes neither coordinate. TR45
proves that these maps are inverse. Transporting the detector target therefore recovers the same
free two-column inverse, not just its rank.
Equip the horizontal finite spaces with the transported norms. The bare coordinates can be
retained unchanged, with sZ
i
= Jdetsi, pZ
i
= Jdetpi and lZ
i
= liQdet. Their composition obeys
lZ
i sZ
i = lisi, and all TR46 remainder norms agree. Write M = supi ∥li∥and εi = ∥ti(lisi−L)∥→0.
The three-term correction error then has the explicit bound

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|ti(li(pi) −L(u∗))| ≤MHi∥pi −siui∥+ εi∥ui∥
+ |ti(L(ui) −L(u∗))| −→0.

This is an identical bound after horizontal readout. It retains the exact free scalar values zero and 4 for the two correction columns. For the growing principal vector, 0 and with give

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|li(Xi) −L(vi)| ≤M∥Xi −sivi∥+ εi∥vi∥/ti ≤M∥Xi −sivi∥+ Cεi −→0.

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Thus the same controls recover both the scalar limit and the solver’s norm bill. Finally if Bi ≤2Hi
and HN
i ei →0 for every fixed N, then ∥Biei∥≤2Hi∥ei∥→0. Applying this to both preexisting
detector errors recovers the paid running-factor transfer used in TR42–TR44.
No unbounded
coordinate error is suppressed by merely taking an unweighted limit.

25.4 Near-kernel detector repair and paid coordinate estimates

The absorption rule is and0, by norms. Both preexisting bare/physical detector and renormalized-divergence errors decay faster than every inverse power of. Since, absorption permits their multiplication by in TR42–TR48. For a real

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signed-symmetry invariant scalar u, the tensor Jab = δabu belongs to the classified free stress space.
Its coordinate trace is 4w01+4(w1+4w2)Oscale = 4u. Thus scalar membership follows. Expectation
kills Oscale and fixes the unit; its nonzero free two-point norm then proves the scalar coordinate map
injective. Inverse scaling preserves those real symmetries and membership. The physical detector
is exactly
(KiT)a = Bir5
i U−1
i,5 ((div T)a).
(TR49)

Positive prefactor and invertibility give equality of detector and divergence kernels. The map is the componentwise renormalized bare lift followed by ; symmetry places its image in the free target. Componentwise comparison gives TR46.

For the finite repair let be onto with ker= 0, and right inverse. Suppose has exact right inverses, and0. A linear gives by the kernel coordinate. Set. Eventually its inverse exists and tends to identity. Since, multiplication by proves in operator norm; finite-basis reconstruction makes pointwise convergence sufficient. For a bare functional and precision, assume (TR50)

Define, uniformly bounded, and). Exact right-inverse cancellation gives

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QiViy = A(y)[Lui(eisiwi) −eiL(wi)] →0.

Thus0 in norm, proving TR47 for and the needed convergence without assuming prior uniformity near its kernel. Here is the nonzero free equation-of-motion vector. Lifting target vectors at the same retained scale and undoing coordinates gives exactly and Issued equation coordinates give and the two paid TR50 estimates. The bare functional annihilates. The resulting precision is = Age . Correction coordinates differ prelimit from ideal orbit coordinates by at most Age ; ideal scalar readings are exact. Bounded linearity therefore bounds the scalar discrepancy by a constant times loss Bounded linearity and multiplication by give Age 0, proving the third error in TR47. The same joint degree-four/five orbit comparison and unweighted loss give qualitative limits on the original extraction filter.

True-divergence and near-kernel repair recovery. The scalar membership argument is transported by the same real signed-symmetry maps: the diagonal tensor of and its classified trace recover. Expectation and the nonzero free scalar two-point norm retain the injectivity used by the scalar coordinate reader. For the detector let denote the horizontal tensor maps. Its exact formula gives

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KZ
i J4T = Bir5
i U−1
i,5 Q5J5 div T = KiT.

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Here the detector output is in the unchanged free coordinates. Since Bir5
i > 0 and Ui,5 is invertible,
zero detector value recovers zero true divergence in both realizations. Equivariance keeps the real
target restriction and componentwise TR46 comparison.
To expose the repair maps, the source retains f, U, A, si, ui, wi with siui = I, and the actual
decomposition y = Ufy+A(y)x. Their finite-dimensional horizontal transports satisfy the identical
composition laws. In the original coordinates put Vi = uif + A ⊗wi and bi = siwi. Then exactly

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siVi = f + A ⊗bi,
si = (f + A ⊗bi)V −1
i
.

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As Vi →I, eventually ∥V −1
i
∥≤2, and hence

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∥si −f∥≤∥f∥∥V −1
i
−I∥+ ∥A∥∥bi∥∥V −1
i
∥−→0.

This recovers operator convergence from the actual right inverses and near-kernel vector, including its previously uncontrolled direction. For the precision scale put). Right-inverse cancellation gives the exact rank-one map

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Its norm is at most
∥Qi∥≤∥A∥∥V −1
i
∥
∥Lui∥∥eisiwi∥+ |eiL(wi)|

 −→0.

Thus both weighted near-kernel payments in TR50 are retained. Transported norms and maps give the same estimate, so recovers TR47 without assuming prior uniform convergence on the near-kernel direction. In this application its exact right inverse uses the same retained scale and projection-lift identity. The issued/ideal scalar error is paid explicitly by Age Age = Age 0, on the original extraction filter and joint degree-four/five coordinates. The subsequent TS1–TS5 estimates remain the dependencies supplying TR50 itself.

For principal coordinates the prelimit bounds are

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∥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 degree-five orbit map and. Compare the issued representative to its retained-core projection at error Age; the latter has norm at most Age. Apply the bounded orbit map and scalar functional. The finite scalar calculation below supplies the other terms and. Continuous linear maps and closed norm inequalities pass through extraction; every error tends to zero. TS1–TS5 below supply the weighted near-kernel hypotheses. The lifted repair source retains exact right inverses, near-kernel vectors, precision scales and closed prelimit inequalities. Its coordinate reader reconstructs the missing kernel direction; the scalar reader then recovers the required weighted accuracy. Keeping the issued and projected coordinates together pays both qualitative and weighted limits on the same extraction filter.

25.5 Local estimates for the coefficient calculation and the six contributions

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

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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

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

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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.

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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

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

The last inequality follows from ΩΩ0. Similarly,

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K0σx ≤cx,
rcx ≤θ2Ω−1(x).
(LCE5)