Marked integration and coupling reset · Part 3

Gauge invariance makes)] adjoint invariant. The proved simplealgebra result and evaluation at imply

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D2f(1)[Wa(x), Wb(y)] = cab(f, o)⟨x, y⟩,
D2f(1)[primitive(w), primitive(z)] =
X
a,b
cab(f, o)⟨wa, zb⟩.

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The latter uses the wedge-sum primitive and bilinearity, with the curl-reduction and invariant-
form hypotheses retained. The paid direction tWa(ξ) gives t2|caa| ≤cf. Also admissible is u =
t
2(Wa(ξ) + Wb(ξ)). At the critical identity repeated ordered left derivative equals the diagonal
Hessian and
4D2f(1)[u, u] = t2(caa + cbb + 2cab).

The paid bound on this derivative and both diagonal bounds yield. These estimates use only the paid directions at the identity, rather than a limiting periodic Fourier test. These are actual normalized paid directions. The projected matrix is on nonwrapping enlarged anchored support, zero otherwise. Bounds hold in either case; the nonwrapping case has the exact flux formula. Signed rotations and translations preserve this cutoffand radius-two support, so the flux pullback gives signed-plane covariance, including the zero case.

Paid calculus controls all ordered left derivatives up to order in the anchored direction class on the closed healthy set. Functions are smooth on full neighborhoods of tested points; these are derivatives of unmasked germs. Constant multiplication costs its absolute value, and sums or differences add bounds. Ordered Leibniz for an inner product has at most 2 splittings, yielding pairingCost(innerSL

Derivative and Hessian subtraction are linear on those neighborhoods; enlarging healthy support restricts tested configurations. These facts justify the residual and clover bills, including boundary points of the healthy sets. The lift retains the paid directions, smooth domains and nonwrapping cutoff. Recovery commutes with these finite-dimensional derivatives and wedge sums, so projected coefficients and their constants are recovered with the same scope rather than by differentiating masks.

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Coefficient recovery can be given entry by entry. At the identity section use the lifted wedge
jWa(ξ) and set
bcab = D2 bf(1)[jWa(ξ), jWb(ξ)].

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The critical-germ Hessian chain rule gives bcab = cab. Retaining the nonwrapping indicator then
recovers the actual projected matrix entry, including its zero branch. In particular the nonwrapping
cutoffin each projected entry has already been transported before the label sum is taken. The
transported direction norm on the section keeps tWa(ξ) and
t
2(Wa(ξ) + Wb(ξ)) in precisely the
original paid class. In particular the off-diagonal estimate recovers the explicit calculation

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2t2|cab| ≤4|D2f(1)[u, u]| + t2|caa| + t2|cbb| ≤6cf.

Thus the same matrix entries, flux pairing and factor 3 are recovered without introducing an ambient frame norm into the paid estimate. For higher ordered derivatives, the lifted readers are formed on the recorded smooth neighborhoods and restricted to the transported physical directions; section recovery applies before restriction to the healthy set. This preserves the stated Leibniz cost at boundary configurations as well.

13.15 Isotropy of the summed coupling projection

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For the actual countable activity listing with regularity, paid bounds and signed-motion label
permutations, define
To(a, b) =
X
i:anchor(i)=o
Πi(a, b).

The boundΠ and summable bills prove absolute convergence and allow finite plane sums to interchange. Reindexing by the exact motion permutation, including anchor, support, value and nonwrapping cutofftransformations, gives

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QTaffine(s,b)o(w, z) = QTo(reframes w, reframes z).

Test single-plane unit-color fluxes. Translations and plane permutations make every diagonal at every anchor equal to ∆= 1)). For distinct planes, reflect an axis in one but not the other. This fixes plane labels and reverses their mixed pairing, so its coefficient equals its negative and vanishes. Hence

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To(a, b) = ∆1a=b,
X
anchor(i)=o
QΠi(w, w) = ∆
X
a
∥wa∥2.

Use the actual clover flux and the same singleton healthy indicator at the anchor to recover the reset’s projected payout. Individual support indicators cannot be substituted without the retained faults accounting for that change. The diagonal paid estimate gives the sharper∆, where is the anchored listing budget: apply triangle inequality to the anchor-zero diagonal mass. No off-diagonal factor three is spent. Scalar and bound are obtained at the original finite period from the original label orbits. The lift retains those permutations, cutoffmasks and summable coefficient families. Recovery therefore commutes with absolute reindexing and plane tests, preserving both the scalar payout and the distinct fault terms required for indicator changes.

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For explicit recovery of the anchor sum, let DN exhaust the labels at o and use the entrywise
recovered matrices bΠi = Πi. The tail estimate

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To(a, b) −
X
i∈DN
Πi(a, b)
≤3
X
t2
ci −→0
i/∈DN
anchor(i)=o

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shows that readout and section recovery preserve To entrywise. The recorded motion permutation
reindexes this same absolutely convergent family. Thus the diagonal plane test and the mixed
reflection test recover bTo = ∆I with the original ∆. Using the sharper diagonal estimate in that
one entry gives
t2|∆| ≤
X
anchor(i)=0
ci ≤c.

Finally the common singleton healthy indicator is part of the payout reader and is pulled back unchanged with the clover flux. Its recovery therefore preserves the scalar payout identity; individual support masks remain separate recorded readers whose differences supply the stated fault terms.

13.16 The local clover counterterm

Petal flux at is the signed average of four oriented plaquette curvatures per plane, all based at Each square touches and lies within radius two; singleton health bounds its chord by. Gauge

transformations act by the adjoint at, and signed motions by signed plane action. Curvature of a length- word is its matrix word minus identity followed by the color projector, of norm at most inverse color scale. Its paid derivative bill is colorBill, with

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colorBill(G, l, P, k) = G. colorScale−1 feeWalk(G, l, P, k).

Four coefficients of absolute value 14 give that same bill at = 4 for each petal. Ordered Leibniz gives petal-pair bill pairBill(, where

pairBill() = pairingCost( Color order) colorBill( power order)

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For real plane matrix q, define
2
X
Qq(U) = 1
a,b
qab⟨petals(U, o)a, petals(U, o)b⟩.

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Its bill is at most 1
2 pairBill t2 P
a,b |qab|. It is smooth, gauge invariant, radius-two supported and
linear in q.
On U(t) = exp(tv), differentiation of a based word at zero gives its signed link sum. For a
square this is
[vj(x + ei) −vj(x)] −[vi(x + ej) −vi(x)].

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For affine fields on the nonwrapping neighborhood it is their constant curl. Period n > 20 ensures
the requisite clover-coordinate increments; the signed average has derivative equal to positive-plane
flux w. Consequently
Qq(U(t))/t2 −→1
a,b
qab⟨wa, wb⟩.
2
X

Simple-group gauge invariance makes the identity a critical point. Taylor’s formula identifies the limit with half the diagonal logarithmic Hessian; thus the counterterm Hessian is exactly the quadratic form on curls. Hessian symmetry and polarization recover mixed affine pairs, establishing residual-decomposition normalization. The lift retains all four based square words, their signs, anchor and period hypothesis. Ordered word recovery and its differential reproduce the clover derivative before the Taylor limit, preserving the counterterm’s normalization and paid bill.

For each clover square based at, the ordered frame cancellation gives its physical holonomy reader

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Apply the original color projector to this recovered matrix word and then take the same four signed
averages. This defines P Z
o = petals(π(r, A), o), with P Z
o σ = petals(·, o). The counterterm reader is
therefore exactly
QZ
q = 1
a,b
qab⟨P Z
o,a, P Z
o,b⟩= Qq ◦π.
2
X

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The identity section recovers its word derivatives and radius-two support. At the critical germ the
Hessian recovery formula gives D2 bQq(0)[jv, jv] = D2Qq(0)[v, v]; the clover derivative then recovers
the stated quadratic form on the curl of v. Polarization gives the corresponding mixed pairings.
The norm and ordered derivative bounds are evaluated on these transported physical directions
with the original color metric, preserving the colorBill and pairBill constants.

13.17 Masked residual decomposition

For regular at anchor with support, set halo) and let be its projected-Hessian petal counterterm. Define

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(
0,
D+ healthy,
f(1) + 1{o} healthyqf(U) −1D healthyf(U),
otherwise.
rf = f −f(1) −qf,
df(U) =

Then exactly

On healthy all smaller tests hold and this is the definition of; on its complement the last term vanishes and this is the definition of. In particular the constant remains outside the original healthy set. For paid bound, projected entries satisfyΠ. With plane types, the counterterm bill is at most, where

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Mp = 1
2q2 3 pairBill(G, p).

This counts all double-type entries and their petal pairing bills. Triangle inequality bounds the rebate’s paid bill and fault’s global absolute value by (2 +, while(1) costs at most. The fault vanishes on healthy and has a genuine failure witness. The rebate is regular there, vanishes at identity and has zero affine Hessian under the retained nonwrapping and period hypotheses: its counterterm has precisely the original affine Hessian. Gauge invariance and measurability pass to all three functions. Faults read enlarged-support data and final-threshold flags as allowed for forced terms. Signed motions transport petals, projected matrices and actual healthy sets covariantly,

commuting with decomposition. Projection and counterterms are linear, so differences of faults and rebates equal the fault and rebate of the difference. This permits the common parameterized source-series reset. The lift retains every distinct health predicate and the failure witness. Recovery preserves the two-case identity pointwise; derivative recovery acts on the regular unmasked rebate germ, preserving its zero affine Hessian and paid bound.

Write and for these three distinct healthy-set indicators. Their lifted readers are and, respectively. With and , the recovered pointwise decomposition is

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It follows by composition of the two cases, with the constant f(1) unchanged. The section recovers
all four terms and their exact predicates. If dZ
f (z) ̸= 0, then π(z) fails the enlarged-support healthy
test, so its recorded failure witness is the original forced-term witness. For affine physical directions
u, v, the smooth rebate germ instead satisfies

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rZ
f (σ(1)) = 0,
D2brf(0)[ju, jv] = D2f(1)[u, v] −D2qf(1)[u, v] = 0.

Here the critical-germ chain rule is applied to the unmasked rebate before any healthy-set restriction. Thus the measurable fault identity and the smooth residual cancellation are recovered with their separate domains; the same transported paid class retains the (2 + bill.

13.18 Low-path atoms and exact loan cancellation

A low index is a plaquette type and either a direct symbol or an ordered path-row pair (). Direct indices have mark direct, length zero and plain plaquette price. Pairs have

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m = cicj,
l = length(si) + length(sj),
P(U) = 1
2⟨si(U), sj(U)⟩.

Their flux analogues are respectively2 and2. Evaluate on anchor petals to obtain proxy. Home contains the anchor’s halo and both true path stencils for a pair, just the halo for a direct index. Let be home and singleton health indicators. Live is the clipped direct price or 1

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(
0,
hH = 1,
(α + ∆)J −∆mhoQ,
hH = 0.
F∆= ∆m(Q −P),
Lα,∆=

Healthy home implies kept selection, subthreshold direct plaquette, and = 1. Substitution, and the defining loan formula on its complement, prove in all cases

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(α + ∆)J + hHF∆= αhHmP + ∆mhoQ + Lα,∆.

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This includes a healthy anchor inside an unhealthy home.
The selector is local: outside the B aperture its exponential threshold exceeds universal chord
two by the decent-budget hypothesis; inside, plaquette corners lie in the aperture unless their test
is automatic. Slots read true route and plaquette stencils, proxies the radius-two clover contained
in H. Gauge conjugation preserves all chords, prices, predicates and transported inner products.
All these functions are measurable; a loan is a finite combination of continuous functions and
measurable healthy/kept indicators. Its extra flag read is precisely the final-threshold read allowed
for forced labels. Put t = p. threshold, ρ = c. relaxation. Kept slots obey ∥si∥≤teli/ρ. Chord
two and the direct clip kink inequality give globally |J| ≤|m|2tel/ρ. Healthy-anchor proxy bill is
1
2 pairBill(G, p)t2, so

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|Lα,∆| ≤|m|el/ρ 
2t|α + ∆| + 1
2|∆| pairBill(G, p)t2
.

A nonzero loan witnesses unhealthy, retaining length and relaxation for support charging. Loans are linear in ( ∆) and freshness in ∆, so parameter differences use these same estimates. Along an affine exponential line, differentiating each finite slot word gives the constant plaquette curl. This follows by differentiating the finite link product; the two nonconstant affine edge terms cancel to the constant curl. On nonwrapping homes the plain and proxy values divided by the line parameter squared have the same flux limit. Their difference has zero quadratic limit. Smooth healthy freshness vanishes at identity; its affine Hessian is zero by Taylor’s formula and polarization, under the retained period and nonwrapping hypotheses. The lift keeps ordered pairs, routes, masks and selector apertures. Literal recovery preserves the casewise cancellation and failure witness; differential recovery preserves the equal flux limits and zero affine Hessian.

For recovery of the parameterized loan family, define physical readers

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AH = (1 −hH)J,
BH = (1 −hH)(J −mhoQ).

Then its piecewise definition is equivalently. Lift and the two distinct masks by composition with. This gives

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LZ
α,∆= Lα,∆◦π,
F Z
∆= ∆m(Q −P) ◦π.

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Both are recovered by the section, and their parameter differences use exactly Lα−α′,∆−∆′ and
F∆−∆′. The full cancellation therefore pulls back term by term with the original mark m and
true length l. Nonzero LZ
α,∆implies (hH ◦π) = 0, recovering the original home failure witness,
while its bound retains el/ρ unchanged.
The freshness derivative is taken on its smooth germ;
section recovery of both plain and proxy flux limits gives its zero affine Hessian on the specified
nonwrapping domain.

13.19 Recovering the actual kinetic base

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Keep all ordered low pair labels. For fi = cisi(U) with P
i ∥fi∥< ∞, absolute domination by
1
2(P
i ∥fi∥)2 and continuity give
X
X

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2
.
1
2⟨fi, fj⟩= 1
i
fi
2
i,j

Thus live low sums are exactly the clipped direct price plus the kept shortening half-square. The selector is common to its entire row, so this holds on both branches. For flux analogues apply the same identity to zero-momentum symbol rows symbol0(). Their normalization

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c. direct δbi +
X
a
SabSai = δbi

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gives
X
2
X
low d
md analogued(w) = 1
a
∥wa∥2.

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Weighted path bounds ensure absolute convergence, allowing the common singleton mask and petal
substitution. The layer sums translated signed views divided by the number of signed orientations:
translations are summed.
Translation covariance moves the anchor-zero term to its translated
anchor; signed-site reindexing proves that live low plus masked cubic and quartic layers equal the
stipulated kinetic energy. The proxy layer equals P
o headsp(o, U), since signed reframe preserves
squared plane norms and the orientation count cancels its reciprocal. These are the reset’s actual
path identities. Only on everywhere healthy fields may all plain low masks be removed; the resulting
layer is the original unclipped second term, the orientation average of c. direct price +∥l∥2/2.

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High freshness retains the cubic and quartic atoms on their own footprints with coefficient
−(α+∆). Arity-r paid bills are startup coefficients times tr; 0 < t ≤1 permits a common t3 bound
for r = 3, 4. Each slot is first order along exponential link lines, so division of an arity-r atom by
the line parameter to power r has finite limit. Division by its square instead tends to zero. Taylor
expansion at the critical identity and polarization give zero diagonal and mixed Hessians in all
directions, retaining the existing masks and adding no boundary term. The lift retains ordered pair
indices, common selectors, symbol normalization and the orientation-normalized layer. Recovery
preserves absolute double sums and finite reindexing, so both kinetic and proxy identities, and the
high-term Hessian vanishing, recover with these exact normalizations.

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A direct tail estimate makes recovery of the ordered square explicit. For a finite label set D,
put M = P
i ∥fi∥and εD = P
i/∈D ∥fi∥. Then
X

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1
2|⟨fi, fj⟩| ≤MεD −→0.
(i,j)/∈D×D

Each finite pair sum is recovered through the original ordered labels and color inner product, so the same limit recovers the full row half-square. Its common selector can be multiplied before or after this limit. The zero-momentum symbol identity is retained as a matrix identity, and the same reasoning recovers its flux analogue after petal substitution. For a local path term, the kinetic layer has the precise reader

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X
X
Layer(g) =
1
| Signed |
b
gs,b.
s∈Signed

Composition with readout commutes with this finite sum, preserving the translation census and the orientation reciprocal separately. Consequently the kinetic and head-sum identities recover with

their original volume factors. The cubic/quartic freshness terms use their recorded footprints and smooth germ derivatives; their order greater than two recovers the zero Hessian before the existing masks are evaluated.

13.20 The unmarked local scalar

Use the same positive local pool ) as in the density expansion. Its paid, absolutely convergent trained connected-log series sums to log Incoming-to-current label equivalence preserves union support. Assign minus each stream as its fixed-support regular coefficient; its supportmask identity gives minus the trained log term. Absolute reindexing therefore gives regular sum log ). Adjoin original-support path scores, whose masked sum is

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−γR(V ) −β(billcubic(V ) + billquartic(V )) −billloop(V ).

Thus the combined regular dictionary has additive sum equal to this expression minus log Anchors are chosen once from nonempty supports before field evaluation. Signed translated views preserve complete sums by finite path-seat and absolute connected-label reindexing, transporting supports and eligibility. Here the dictionary uses full-seat averaging, including translations, with pulled healthy masks; this preserves each complete sum exactly and retains its distinct normalization from kinetic orientation layers.

The positive scalar token is sector prefactor times incoming vacuum exponential times fixed deducted chart factor. PutSite log(token). Separating this from the same-threshold clipped-base exponential gives overhead

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(
)
τ|Sitec| −β averageorientations
X
i
baseTerm(V, i)
O(V ) = exp
.

Insert the regular sum into the frozen channel exponent. Cubic, quartic and loop bills cancel their path scores; the receipt correction cancels the regular receipt term, leaving log + log. Hence

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exp{Efrozen(V ) −γR(V )} = O(V )C(V ).

A prior fusion adds its consumed attached-log sum on the left, recovering the earlier normalization formula. Positivity from the real log theorem justifies exponentiating log. The same forced canopy-pool gas obeys ) gas( ) = push( ). Multiplying yields the early evaluator identity for every field, including zero gas, without division by gas or evaluation at a selected field. The lift retains the positive token, fixed anchors, exact pool and both averaging conventions. Recovery preserves their scalar equalities and the absolute stream sum, transporting the unnormalized push even at gas zeros. Let be the full finite seat set used for the dictionary, including translations. For its complete pulled sum, invariance gives, and therefore

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X
1
|S|
s∈S
Gs = G.

The realized record retains separately from Signed, so readout preserves this normalization and the kinetic layer’s distinct translation sum. The field-independent token 0 has the same scalar reader on the source object, giving with the original site count. Pull back the exact pool, overhead, receipt and canopy gas together. Their recovery identities are

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e(Efrozen−γR)◦π = (OC) ◦π,
(OC gas) ◦π = push(R; ·) ◦π.

Section recovery returns the original scalar and evaluator identities. These are multiplicative identities on every admitted field, so zeros of the canopy gas retain the same unnormalized push without any cancellation of that gas.

13.21 Pool averaging and literal score series

Probability averaging of single-forest activities preserves each common absolute tariffby triangle inequality. The thin and thick tariffs thus bound averaged entries and deposits. Renewed reserve supplies log-series readiness with unchanged supports and compatibility. Hence

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log C(V ) =
X
q connected
Gas. term(entriesPool(V ), q),
X
!
C(V ) = exp
q
Gas. term(entriesPool(V ), q)
> 0.

The scalar proof uses this gas of averaged entries. Split each nonzero molecule family into nonemptyflag and empty-flag components, retaining nonempty ground. Their subtype union is injective and recovers every nonzero family, reindexing the integrated gas into thick and thin families. At fixed thick family, excluding intersecting thin supports sets their entries to zero and decreases absolute budgets. The same log theorem and omitted-cluster restoration give excluded gas

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!
−
X
q
missingA(q, V )
C(V ) exp
.

Multiply by the banked thick factor and sum to obtain the pool density identity. Throughout this step the forest averaging is performed at the level justified by disjoint-ground factorization, rather than assuming arbitrary products commute with a common forest average. Forest averages factor at disjoint grounds as proved earlier.

A low score label is a signed translated seat and low atom, with coefficientSigned times its marked plain value and support equal to the viewed full home. Pullback of the actual healthy mask gives exactly a summand of. Cubic, quartic and loop score labels carry their transformed footprints and coefficients respectivelySignedSigned andSigned times the unmasked atom. Footprint masking gives the placed atom. Absolute atom budgets justify the sums. Reindexing seat translations by signed action identifies them with orientation averages of all anchors, preserving masks. This proves the exact score total used above, with no change of masks. Definition expansion gives exactly

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inChannels(β) = β average
X
baseTerm +β(billcubic + billquartic) + billloop .

These are convergent equalities before smallness estimates or cubic coefficient cancellation. The lift retains pool entries, thin/thick subtype maps, omitted-cluster labels, seats and transformed masks. Recovery commutes with the justified probability averages and absolute series, preserving the exact score and channel identities in the reset.

For the pool reader, denote a single-choice integrated entry by) and its normalized finite averaging weights by. Then

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 ̄ap =
X
X
λXaX(p),
X
X
λX = 1,
|aX(p)| ≤bp =⇒| ̄ap| ≤bp.

The local integral recovery already proved gives the same) on readout, hence the same averaged entry and tariff. Form the gas and its connected logarithm from these recovered ̄ At fixed