Position coupling, real remainders and equation markers · Part 2

On its common chamber support the rooted seminorm gives

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X
!
|wm,α|
max
α
∥back(Lα)∥root ≤Tau−1800000
i
.
∥back(Mm)∥root ≤
ρ(x)
α

The exact exponent is 1000001800000. Horizontal realization retains every lens, beacon, support and scalar coefficient, so this same finite readout also preserves the pointwise

bound. The surgery order is fixed before the eventual threshold; that threshold remains uniform in finer cutoffand volume. The local lens estimate below remains the explicit dependency for the individual marker terms. For local films and billed labels, set Billing and physical transport give

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

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Each original error is at most Tau−430
i
; the triangle inequality costs 1 + β ≤Tau230
i
, using 0 <
β ≤2 bβk. The signed finite divergence orbit Dd commutes with insertion, transport and pullback
on the common window; its seminorm estimate gives the same error for back(DdJ) −I(a5
i DdV ).
Real orbit projection places the label in the invariant subspace. Billed coefficients at most two
give Ve size at most 2 + 2β; fixed orbit cost c ≥1 gives c(2 + 2β).
Eventually 6c ≤t = bβk,
t ≥1, so this is at most 6ct ≤t2 ≤t4. Multiplication by a5
i gives the physical budget. The
crown bound, common support and TS7 then supply the corrected-divergence probe estimate with
its coupling and spacing factor retained. The lifted surgery source retains every address, weight,
lens and common chamber. Its finite marker reader TS6 preserves the paid sum; its billed reader
TS7 retains the actual coupling difference rather than two independent normalizations. Weighted
spectator readout and the invariant detector recover the target vector on the same refinement.

26.6 The local lens root estimate and its funded calendar

Copy the Haar-averaged surgery difference into the reporting- chambers with signed motions and conjugation. A copied base stays at dist(0) from its center. Choose the-to- block ratio above both 2 pierSize +2+2 dist(0) and 64+2 dist(0). Inspected supports and changed bonds are disjoint with the required marked separation. Motions preserve bounds and gauge conditions; one component of product Haar integration is its ordinary Haar integral, so each chamber coefficient is exactly its lens copy. Use two slots per chamber with equal positive weight). For support exponent, cover, function bound and truncation, choose0 satisfying

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4ce(s−3)l ≤eb,
Fe(s−3)l+2w(k)2/c ≤r,
4e−mTlease ≤1,
Twing ≤1,
t = r + 8ebe−mTlease + 2ebbforced,
(2B + 1)4t ≤1,
et −1 ≤uallowed.

The finite local expansion and census give support exponent4, budget one, including the factor. Tariffs sum separately at each disjoint surgery base without a chamber-count factor. Transport to with+1). Distinct anchors make doubled-marker occupancy at most two, allowed by2. Retained insertion and positive scalar normalization give, for chambers and Wilson lens-product moment

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|Q|w(m0)2v ≤e| Sitem0 | treasury(m0)+4v.

Restored integration uses the same positive Wilson partition on both sides, which cancels. Census gives Site; thus

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|Q|1/v ≤exp[(aJ/am0)4 treasury(m0) + 4]
w(m0)2
.
(TS8)

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At J = i and its prescribed basin m0, the subpower surgery and separation radii fit the calendar
block ratio uniformly in volume. Treasury and weights make TS8 at most Tau−1900000
i
in the same
reporting chamber.

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Billed film and rooted lens recovery.
The source retains both films, their actual coupling
and spacings. Pullback and physical insertion are linear on the same chamber, so the transferred
error is the difference of the two individual errors, with coefficients 1, −β. Its rooted seminorm is
therefore at most (1+β) Tau−430
i
≤Tau−200
i
. The same signed finite orbit commutes with all three
maps, recovering TS7 for the divergence. Its coefficient payment is explicitly

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c(2 + 2β) ≤c(2 + 4t) ≤6ct ≤t2 ≤t4,
t = bβk ≥1,

with 6 eventually. Multiplication by the unchanged recovers the precise physical degree-five budget and coupling label.

For a single lens retain every chamber motion, conjugation, copied base and the stronger separation inequality. Product Haar readout of one chamber coordinate is exactly its original Haar integral; thus the marked coefficient is the actual copied lens. Horizontal configuration realization preserves all these integrals and disjoint supports. The two markers per chamber retain weight ) and occupancy at most two at every scale, so the original multiplicity and weight inequalities remain the admissibility conditions.

Write 0 for the common finite Wilson partition factor. The restored marked integral has the exact fully marked coefficient, while the served insertion estimate has the same factor multiplying its bound. Cancelling it before any root gives

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

For1, take theth root and use the actual tiling census Site to recover TS8. Both the numerator and denominator of this normalized coefficient are identical in the horizontal Wilson readout. Thus the rooted estimate retains no extra chamber-count factor.

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The expense sum remains the stated r + 8ebe−mTlease + 2ebbforced at each surgery base, with
the w(k)2/c coefficient payment included.
Disjoint supports allow the retained basewise tariff
combination; they do not replace this by a new tariffmultiplied by the chamber count. At the
summit calendar J = i, the same block ratio, basin treasury and weights recover the individual lens
bound Tau−1900000
uniformly on the volume tail. This supplies precisely the local lens dependency
i
of the finite marker sum TS6; its explicit large-coupling expense allocation follows next.
The
expense conditions in this argument are consequences of the same large-coupling choice; they are
not additional assumptions on the final theory. At fixed stage and cover l put B = ⌈(log x)2⌉,
P = B40 + 1100, M = B60 + 1100, ex = e−P/8. Eventually M ≥4P and for any fixed C, j,

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Cxj ≤eP/32.
(TS9)

Taking logs proves this from log and degree forty of. Lease tariffis at most: its finite product/sum formula contains coupling at most 4 and subpower threshold, path, motion and site factors, hence rate at most two. Use and 1. For fixed

, eventually 4 and. Therefore 4 and . With; the forced smoke(

obeys 2 smoke(. Thus1 and 41 eventually. The wing tariffis already imposed. All hold for every weight in the interval. At each fixed stage they require one eventual lower bound; enlarge the original coupling-entry delay above it and all other thresholds. Taking maxima and later delays preserves them before finer cutoffor volume is chosen. The lifted lens source retains chamber copies, both markers, the common partition factor and exact cell census. TS8 is its rooted reader independent of chamber count. TS9 pays every expense within the original calendar; retaining that threshold order recovers the uniform local estimate used by the finite marker sum TS6.

Expense-calendar recovery. The scalar source retains the stage, fixed cover, actual coupling entry, and numerical choices. Horizontal readout is identical on each of these entries. For fixed, the logarithm of is at most a fixed constant plus eventually; therefore recovers TS9. The same finite tariffformula and subpower factors retain the eventual

lease bound uniformly over its admissible representations at that stage. For the chosen weight interval and , the coefficient payment is explicitly

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xc0e2w2/c′ ≤e3P/32−P/2 = e−13P/32 ≤e−P/8 = ex.

The other fixed-support inequality follows from 4 and. The retained truncation and polynomial lease bound give

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8eP e−MTlease ≤8x3eP−M ≤ex

eventually. The forced allowance retains its actual rapidly decaying function, giving 2 smoke( . Thus all three contributions to the same expense satisfy simultaneously for every allowed weight. In particular its polynomial site factor times tends to zero, and eventuallylog 2, so1. The same lease payment implies 41; the wing tariffremains the already imposed local condition.

These are scalar inequalities of the retained calendar, so they recover exactly after source realization. For each fixed stage the finite list of eventual lower bounds on is combined by taking its maximum with the existing entry thresholds. Increasing the entry delay preserves every bound. The choice remains before finer cutoffand volume, and introduces no additional condition on the continuum law. This supplies the expense allocation dependency for the rooted lens and marker readers with its original quantifier order.

26.7 The exact local lens and its flags

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For an oriented bond from z, Tν(j) left-multiplies its link; a negative reading uses the corresponding
right inverse on the stored positive link. With scalar stocks define the finite truncation HM(U) =
v| Site | + P
covB(Q)≤M SQ(U). For bounded local J the exact lens is

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L(U) =
Z
G
[J(Tν(j)U, j)eHM(Tν(j)U)−HM(U) −J(U, j)]dνG(j).
(TS10)

It requires fixed read support), nonzero only at health threshold measurable equivariant, and chord at most with 8 Signed copies preserve these assumptions. Put. When= 0, link changes are at most and plaquette changes at most 4 by telescoping unitary factors. Changed plaquettes touch, have

chord at most and at most 3; unchanged plaquettes have identical tests. Thus flags agree exactly, also under signed motions and relaxed thresholds at least. Stocks omitting are unchanged, since internal links and flags agree. Therefore

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HM(V ) −HM(U) =
X
[SQ(V ) −SQ(U)].
z∈Q
covB(Q)≤M

Define= 0, zero otherwise. Finite distributivity gives

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J(V, j)eHM(V )−HM(U) =
X
A⊆{Q:z∈Q, covB(Q)≤M}
J(V, j)
Y
Q∈A
tQ(U, j).

On the absent-insertion branch both sides vanish; no exponent bound there is needed. This is the expansion paid by the surgery expenses. For cov, its inspected union lies within pierSize = ( of, and the lens reads the origin-centered radius-2 pierSize window. Gauge covariance of, stock invariance and Haar change prove gauge invariance. Finite stock envelopes prove measurability, boundedness and integrability. The lifted lens source retains the oriented stored-link operation, exact flags and finite stock roster. Its subset reader recovers TS10 on both presence branches without differentiating a hard mask; support and Haar readers recover the same local invariant lens used in TS8.

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Exact lens, flags and finite-subset recovery.
The local source retains the oriented stored
bond, scalar stocks, finite truncation, insertion and measurable group displacement. For configu-
ration realization π and section σ, use JZ(z, j) = J(πz, j), HZ
M = HM ◦π, and on the horizontal
section define T Z
ν(j)σU = σ(Tν(j)U). Then the entire integrand, including the subtraction, satisfies

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JZ(T Z
ν(j)σU, j)eHZ
M(T Z
ν(j)σU)−HZ
M(σU) −JZ(σU, j)
= J(Tν(j)U, j)eHM(Tν(j)U)−HM(U) −J(U, j).

The same normalized Haar reader therefore gives). The negative-bond convention is retained as right inverse multiplication on its actual stored positive link, so this identity also covers reversed readings.

On the presence branch the four-factor plaquette telescope retains its 4 bound, and 8 gives at the original configuration. Thus both ordinary and relaxed flag comparisons recover exactly. Every stock omitting has the same internal links and flags, so its exponent difference is zero. The constant volume term cancels as well. Let be the retained finite support list containing. Its exact scalar reader is

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J(V, j)
Y
Q∈Q
(1 + tQ(U, j)) =
X
A⊆Q
J(V, j)
Y
Q∈A
tQ(U, j).

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The presence branch is the original exponential by addition of the stock differences. On the absence
branch its retained zero factors and J(V, j) = 0 make both sides zero. Horizontal readout preserves
this branchwise identity and its finite support list, without an estimate on an unused exponent
difference.
The common inspected union, cover bound and actual 0, z ∈P therefore retain the same radius-
2 pierSize window. The gauge action transports the changed configuration using ν(gzj) = gzν(j)g−1
z ;
insertion covariance and stock invariance identify the transformed integrand. The normalized Haar

change recovers its unchanged integral. The finite stock envelopes provide the same measurable bounded integrand, so this is an ordinary integrable reader. This recovers locality, gauge invariance and the exact lens used by the copied-marker estimate, with all admissibility conditions retained.

26.8 Finite density surgery and the equation insertion

For the equation marker used above, specialize (TS10) as follows. On a positive source link let be a unit color direction, let) = exp(), and take

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For a reversed link use the equivalent reversed multiplication in (TS10). The current is gauge
covariant at the source, so the covariance of J and ν is exactly the one required there. The step
h = θx−1/24, with its fixed node multiples, eventually satisfies the stronger chart condition 64D ≤θ;
thus it also satisfies the 8D ≤θ flag condition of (TS10).
Let Le,tih(U) be this lens with the finite stock exponent, and sum over the active links, color
basis indices and nodes:
M(U) = 1
X
X
i<m
wi Le,tih(U).
(TS11)
h

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e,c

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Haar integration is already part of each lens. The coefficient wi/h is its actual normalization. For
every node its lens integral is zero for the stock density, by the Haar change of variable; the matched
course transports this identity to the corresponding finite Wilson equation expression. Expanding
by (TS10), its empty subset supplies the main density variation and its nonempty subsets are
exactly the secondary terms F in (LCE6). Taylor expansion with the node moments above gives
the current divergence and action derivative, with the remainder E.
The exact healthy-region
identity is therefore
M = Z −β
X
a
Da eJab + (divν Yb)(1),
(TS12)

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up to the explicitly retained finite-shelf approximation, uniformly at most e−prism(x) after division
by h. The shelves are exhausted only after choosing this error allowance, so division by h is paid
in their selection.
The normalization and support costs in the rooted estimate can be read directly from (TS11):
X
e,c,i
|wi|/h ≤cardB(x)
X
i<m
|wi|

/h ≤x2

eventually at each fixed. The active-link count and fixed node sum have rate zero, while

has rate 124, strictly below two. Each lens is supported in the window of radius 2 obtained above. At the actual middle stop its copied root bound is at most T1900000); the preceding total coefficient expense is at most T(100000). The rooted triangle inequality and T) give

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∥M∥root ≤Ti(−1800000).
(TS13)

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Here Ti(a) = exp{a(log bβi)48/10000}; all finite local radii and prices are absorbed before choosing
the sufficiently late stop. The rescaling from that stop to the reporting scale has degree-five factor at
most Ti(100000). The healthy error is exponentially smaller than any fixed negative Ti power, and
the global polynomial bound and exceptional-event estimate pay its complement. Signed averaging
removes the constant in (TS12) and preserves the root bound. This is the normalized equation
marker used in (TS1)–(TS5), including the costs of the full joint replicated array; it is not a formal
integration-by-parts symbol without a local representative.

Source–readout formulation. The equation marker is a literal local representative and not a formal integrationby-parts symbol. In the terms of the method it is the equation-residual column of the realized record at the reporting scale: each lens integral vanishes for the stock density by the Haar change of variable, the matched course transports this to the finite Wilson equation expression, and the identity (TS12) holds on the healthy region with the finite-shelf approximation retained. The normalization cost, the support window and the root bound (TS13) are properties of the realized representative with the costs of the full joint replicated array. Signed averaging removes the constant in (TS12) and preserves the root bound.

Completion. The closing assembly of the supplied construction names the witnesses constructed above. Blocking penalty 1 is chosen. The calibrated Wilson family and its common insertion tokens are constructed before any volume subsequence, by the exact one-step identities, the finite arcs (FCA7) with their exact Wilson startup (Theorem 14.18) and density identities (Theorem 14.20), and the banked-history convergence. The summed drift argument (RUN1)–(RUN5), on the realized calendar (CAL1)–(CAL11) with the actual update (DR7), gives the positive logarithmic coupling slope uniformly in cutoffand period, whose finest-cutoffspecialization is the bare-coupling clause. For a requested increasing period sequence the refinements are composed; the fixed-volume Cauchy argument and smearing give the full mesh limit at each suitable period and the refined period sequence the outer limit (Section 18.9); the field quotient and detector give one finite-dimensional filtration (Theorem 18.3); the compression algebra gives the phase space; heat, tube, boundary and Gram constructions give the fields, with Euclidean covariance from Theorem 19.3; one equivariant operator-product reader serves every character; and the Gaussian comparison, position normalization, equation-marker bounds and near-kernel estimates determine the stress, its centred trace and its amplitude, with the full limit of Theorem 24.1. Every witness lives on the one chosen Wilson family and the one composite refinement.