Separated and nested operator products · Part 3

component without any activity contributes zero under a selected old flag pattern: its required flag belongs to an old activity support, which would connect that activity into the component. Hence each contributing component is one of the two grouped types. Conversely selected grouped entries have disjoint group rosters and disjoint activity supports, so their expanded components have no further connections. Union and component partition are inverse operations, preserving coefficients and flagged unions. The separate flag condition holds for each grouped activity: a flag assigned to another would violate disjoint grouped supports. This proves OE11 at every configuration, independently of entry zeros. With no new markers no scalar entry can occur. Two empty-marker activities cannot connect, since their prescribed flagged supports are disjoint and there is no group marker. Thus empty connected collections are exactly single old empty-marker activity entries, preserving both value and original support-and-flag label. Each nonzero coefficient has its marker locations inside its support by locality, so every term required for either recovery is present. This proves preservation of the empty activity coefficient before any bound. Restoring the nonzero background makes OE11 precisely the served-polynomial identity.

All grouped supports contain their representatives. Entry dependence is confined to variables and flags in the extended support, and product dependence to its union. Gauge invariance and measurability persist, as do flag indicators. Nonzero ordinary groups are healthy on their whole support; activities have exactly their recorded flag intersection. Thus the same conditions hold for the next grouping. Copy covariance holds coefficientwise: a focused marker set uses its copy’s inventory; the copy permutation is a bijection taking extended supports and coefficients to the prescribed transformed ones. Reindex finite sums and products to obtain covariance of both polynomials, including empty markers and internal interactions, and hence their singleton functions and profiles after repeated regrouping.

Finally extract each fine served singleton. Restoration identifies its normalized local insertion, including the dominated activity correction. Anchor analytic transport gives a coarse function and dominated profile with unchanged leading counterterm and remainder multiplied by the driving constant. Define other copies by the prescribed reflection and translation and reindex activity labels by the actual motion bijection. Conditional integration intertwines these transformations; integrability permits coefficient extraction to commute with it. The marked transport theorem gives the full coarse representation at, and support substitution with target weights, occupancy and duty inequalities restores the served representation at. All estimates hold for the arbitrary finite occurrence set; no two-occurrence restriction is used. The lifted fusion source is this complete finite inventory, flag pattern and occurrence-fiber map. Its realization is the connected collection polynomial and its observable reader is full-fiber extraction followed by normalized singleton restoration. OE11 proves exact recovery of the grouped served polynomial; retaining the old inventory exposes the partial-fiber information discarded by the reader. OE10 pays the support transformation quantitatively, while empty-coefficient preservation recovers the same background and makes iteration compatible with the sweep.

Inventory bijection and exact served-polynomial recovery. For each retained scalar inventory value on extended support, its two routed entries recover pointwise as

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ah = 1Healthy(S)a,
au = (1 −1Healthy(S))a,
ah + au = a.

Configuration pullback preserves this split and every original support-and-flag label. Let L act coefficientwise on these finite polynomial inventories. It commutes with the full-fiber reader because neither operation changes the other’s coefficient indices. The source realization of (OE11) is therefore exactly its coefficientwise pullback, with the same finite nilpotent exponential and flagged gas.

More explicitly, retain an eligible old collection together with its unique connected-component partition. The forward reader replaces that collection by its ordinary or activity components; the inverse reader takes their union with their recorded old entries. On the selected flag pattern these operations are inverse, preserve the product of scalar coefficients, and preserve the union of flags. The zero contribution of an unhealthy scalar-only component is retained before summing. This gives recovery of every coefficient of (OE11), including the empty coefficient’s bijection with single old empty-marker activities. Individual zeros require no division.

Normalized singleton restoration thus uses the same nonzero background on both carriers. At the final one-step transport, coefficientwise pullback and extraction commute with the integrable conditional push; copy reindexing preserves the same activity-motion bijection. The retained marked transport at and support substitution at recover the entire served polynomial, with its dominated profiles and original duty bounds. The occurrence set remains the arbitrary finite set, so this recovery applies to every census used in the sweep.

20.9 Nested expansions on arbitrary finite trees

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Let t be a finite rooted tree, with fields of degrees wlat its leaves, and centers zv and radii
sv > 0 at internal vertices.
Zero and one child are allowed.
All subtree positions at v lie in
radius Msv; different immediate child subtrees have separation at least ηsv. Internal child radii
do not exceed parent radii.
The root radius is r ↓0; no positive lower bound is imposed on
child/parent ratios. Bounds on immediate centers imply these full subtree conditions after a small
uniform upper ratio choice: if child displacement constants are Ci, set C′ = P
i Ci and choose
0 < β ≤1, β ≤η/[4(C′ + 1)], below all previous child bounds. Descendant deviations are at most
C′βsv, so sibling separation loses at most 2C′βsv ≤ηsv/2 and the parent displacement is at most
(M + C′)sv. Induction gives the geometry. Put Wv = P
lbelow v wland qv = sv/sparent(v), with
virtual root-parent radius 1, so qroot = r. For fixed output degree m define

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Tm,ε(s) = rm−P
lwl−ε
Y
q−Wv−ε
v
.
(NE1)
v internal
v̸=root

Fix every internal truncation before taking the limit. The direct coefficient reads the full leaf cluster at the root; the iterated coefficient literally multiplies and sums the same homogeneous OPE channels up the tree through the assigned child horizons, ending in the fixed root channel ). Uniformly over admissible geometry and positive radii, for small enough

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|Cdirect| ≤Tm,ε,
|Cdirect −Citerated| ≤Tm,ε
X
qDv+1
v
.
(NE2)
v internal
v̸=root

With no nonroot internal vertices this sum is zero and equality is exact. The root horizon does not enter the sum, since its output is already fixed. With exterior positions separated from all tree positions and each other by 0, the common ground-algebra norm of the full leaf product minus its direct root expansion through is at most

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Tm+1,ε(s).
(NE3)

This serves both vacuum and every character. All finite expressions are smooth on their common open reader domain; for fixed requested degrees the tree geometry eventually belongs to it, by shrinking diameters and sibling distinctness recursively.

Suppose first the radii have rates, nondecreasing down the tree, and set ∆ 0, with virtual exponent zero. The engineering cost is

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E =
X
v internal
Wv∆v.
(NE4)

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Telescoping along each leaf path also writes this as its degree times its nearest internal parent’s
exponent, summed over leaves. Direct output rate is E −maroot, tested remainder rate E −(m +
1)aroot. NE1 adds slack ε P
v ∆v to this engineering rate. When P
v av = 1, this slack is positive:
some av > 0 and each av is bounded by the sum of all nonnegative advances. It absorbs fixed
constants even when individual gaps vanish.

Finite-tree channel and price recovery. Retain the complete finite rooted tree, all positive radii and centers, leaf labels, fixed internal horizons and the already chosen homogeneous channel bases. Lift each finite coefficient carrier by its fixed section. Every node coefficient is unchanged because the homogeneous coordinate reader recovers the same scalar. Thus each term in the finite iterated coefficient, indexed by the retained intermediate degrees and channels, has the identical product on both carriers. Summing those same indices gives

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CZ
direct = Cdirect,
CZ
iterated = Citerated,
T Z
m,ε(s) = Tm,ε(s).

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The tree reader keeps every child-to-parent ratio as an independent positive geometric datum. It
introduces no lower ratio bound, permits zero and one child, and retains the fixed root output
outside the nonroot error sum.
The tested root remainder is lifted by α, so its norm and all
vacuum/character evaluations recover (NE3).
The engineering identity behind this recovery is the exact finite path interchange
X
v
Wv∆v =
X
l
wl
X
v on the internal path to l
∆v =
X
l
wlaparent(l).

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Substitution in (NE1) gives its stated engineering rate plus ε P
v ∆v. If the tree has nint > 0 internal
vertices and P
v av = 1, then

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1 =
X
v
av ≤nint
X
v
∆v,
X
v
∆v ≥1/nint > 0.

This explicit retained slack absorbs fixed finite channel-count and map constants even when individual advances vanish. The same physical geometry and finite admissible reader domains are read at each node, so the resulting bounds preserve all of (NE1)–(NE4)’s stated finite-tree and finite-horizon scope.

The finite recursive proof retains leaves as actual labels of their actual degrees, with coefficient one; a bud is not a degree-zero projection, a convention that includes arbitrary input labels correctly. Child alternatives through are its homogeneous fields of degrees 0, weighted by direct coefficients. Parent alternatives multiply these coefficients by the OPE coefficient of those resulting fields. Inductively every raw subtree has (i) a finite comparable family at its parent’s exponent, to any prescribed tested accuracy, whose amplitudes pay engineering cost minus resulting degree times that exponent, and (ii) its direct coefficient and full tested remainder estimates at its own exponent. Leaves satisfy both. Products of child families add costs as in NE4 and sibling separation makes their positions admissible. If a child has the same exponent as its parent retain its comparable family. If it is strictly larger, use its second assertion to replace it through a sufficiently high

auxiliary finite horizon. For child exponent with cost Ωand later exterior bound replacing through auxiliary finite has error rate Ω. For requested, choose ), above all earlier requested horizons. Replace children in order, choosing each new horizon after the earlier finite degrees and their exterior costs are fixed. The exact telescoping sum has one error at a time, avoiding circular horizon choices. Finite readers transfer these tested errors to coefficients. Apply the single-center estimate to the resulting comparable family to complete both parent assertions. For prescribed truncations choose gain0 satisfying

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γ ≤(Dv + 1)∆v
at every nonroot internal v.
(NE5)

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The exact product difference
Y
j
xj −
Y
j
yj =
X
j
(xj −yj)
Y
i<j
yi
Y
i>j
xi

transfers one gain, not its multiple. For 0 all nontrivial immediate child gaps are positive. Replace through auxiliary to arbitrarily good accuracy. The auxiliary sum minus the desired sum is the finite omitted-channel sum; every term has a child degree at least gaining ( between its coefficient cost and the parent’s returned degree. Other degree-gap contributions are nonnegative. Recursive replacement of retained direct child coefficients gains the same by the product identity. The three exact differences therefore give

direct-to-iterated error rate (NE6)

For = 0, bound direct and iterated terms by the common raw rate; no high-accuracy substitution is needed at zero gap. With no nonroot internal vertices every child choice has coefficient one and permutation invariance gives exact equality. Take as the minimum of (; an attaining vertex contributes ratio power of rate. Positive NE1 slack absorbs constants; its one term is bounded by the NE2 sum. Direct/remainder rates give the remaining claims, and contractivity plus the character supremum gives NE3.

Recursive finite-family and truncation-gain recovery. A source alternative retains its actual ordered word, positions and scalar coefficient. Lift its field labels by the fixed coefficient sections and leave that scalar unchanged. Products and finite sums of alternatives then recover term by term under the same correlation reader. In particular a leaf retains its original degree and coefficient one, and a parent uses exactly the same intermediate channel indices. For a strict child exponent gap 0, one explicit allowed auxiliary choice is any integer

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H ≥max

Hprevious, 0,
Ω+ Wb −ρ

+ 1

.
a −k

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It recovers the required strict error rate and is chosen after the preceding finite alternatives and
exterior costs. Equal-gap children keep their comparable families and require no such division.
Let xj, yj be direct and recursively composed child scalar coefficients. The source product reader
preserves the exact finite identity
Y
j
xj −
Y
j
yj =
X
j
(xj −yj)
Y
i<j
yi
Y
i>j
xi.

Every term contains one changed coefficient, so it retains one gain under (NE5). For an omitted channel of degree + 1, the finer coefficient cost and parent return have net gain

. All other advances are nonnegative. Thus the same three finite differences recover (NE6), including its = 0 branch and the exact case with no nonroot internal vertices. Choosing an attaining minimum vertex and then the positive NE1 slack recovers the original sum of ratio powers in (NE2). The corresponding algebra remainder reader is isometric, giving the same all-phase (NE3) bound.

Here are the finite-family quantifiers used above. An alternative is a finite word at selected subtree positions times a scalar of finite upper rate. Eligibility at means positions separated at that exponent. Agreement to rate up to exterior weight means the full correlation difference has that rate against every fixed exterior list of weight at most, arbitrary admissible position paths and viewer paths. A supply at parent exponent and cost provides such comparable families for sufficiently large exterior exponent and every, in the exact continuum hierarchy. To replace one branch, fix the other branches’ alternative, of degree and amplitude rate. Request weight and accuracy for the changed branch; multiplication restores rate. Finite maxima over all other alternatives suffice before the replacement. Sibling separation ensures admissible augmented exteriors. Successive replacements use only previous choices and unchanged later branches; transitivity adds complete differences. For coefficient transfer choose covering every finite detector word through. Rotating detector words retain degree and bounded coordinates and remain a fixed positive distance apart and from the shrinking cluster. Finite detector sums, compact rotation averaging, real averaging, and a coordinate reader preserve agreement rate . Uniformity under arbitrary rotation paths follows as in OE4. For complete remainder transfer an extra debit is needed, which is why agreement of products alone would not suffice: if desired pre-exterior rate is, request replacement agreement covering actual exterior and detector weights. The product difference has rate at most. Degree- coefficient differences cost at most. Add the replacement remainder and subtract all these coefficient differences to recover exactly the original full remainder. For a comparable alternative of degree, amplitude rate and geometry exponent 0, its degree- coefficient costs (, giving. Include its fields in the exterior test of the changed branch. At = 0, finite detector words cost zero, so its amplitude times their correlations has rate; comparison with the empty family gives the same coefficient bound. Write each rotated reader as its finite sum of these tested correlations times its fixed weights and rotated loaded vectors. Actual shrinking ensures eventual admissibility even at zero exponent. For multiply the single-center remainder by the amplitude, giving, separately bound the full correlation and each finite centered term; engineering degrees cancel to Finite summation covers both boundary cases.

Finite-family agreement and whole-remainder recovery. Retain each alternative as its ordered word, scalar amplitude and geometric path. Its lift preserves every complete correlation against the same exterior labels and viewer. Hence agreement to rate through weight is preserved with all its original universal path quantifiers. For a fixed other-branch alternative of degree and amplitude rate, requesting () before the replacement gives exactly rate after multiplication. Taking the finite maxima first recovers the triangular branch procedure. The coefficient reader uses the same finite detector words, normalized rotation average, real average and homogeneous coordinate maps; its recovery therefore preserves that agreement rate as well.

Write) for the complete tested remainder of a family, and) for its centered degree- coefficient field, with the fixed homogeneous basis sum understood. For a comparable

replacement the exact readout identity is

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m
X
Rm(F) −Rm(G) = Corr(F −G) −
j=0
Corr(Cj(F) −Cj(G); exterior).

Both source and physical readers retain this subtraction before any estimate. With requested agreement, itsth centered term has upper rate

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ρ0 −mu + ju + |e|b = ρ0 −(m −j)u + |e|b ≤ρ0 + |e|b.

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This recovers the precise debit needed for the full remainder, not only the product. For the boundary
cases the same scalar rate identities give M −mk = M at k = 0 and M −(m + 1)(k −u) + |e|b =
M +|e|b at u = k. The source keeps actual shrinking and eventual admissibility in the zero-exponent
case, so no positive gap or degree loss is inserted in these cases.
No prior existence of logarithmic rates is required. For any approaching admissible family, every
internal radius tends to zero; set
T =
X
v internal
log(1/sv).

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Eventually T > 0 and tends to infinity. The normalized log-radius vector lies in a finite compact
cube and has a limit a on a nontrivial refining filter, with P
v av = 1. Radius monotonicity forces
av ≥aparent(v) by exponential-rate comparison. Geometry gives the size rates and actual shrinking;
sibling separation gives separation at each strictly larger exponent. The proof applies even when
some av = 0, with positive total slack. If a uniform small-radius threshold failed, select violating
arrays at arbitrarily small radii. Extracting a refining filter of those same arrays yields eventual
validity, a contradiction. Include exterior geometry and a violating viewer in the selection when

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necessary. Thresholds thus depend only on the fixed tree, labels, horizons, M, η, ε, δ, without lower
ratio bounds.
R |u|dμ ≤B, take an almost everywhere
For any measure space (K, μ) and complex u ∈L1 with
measurable admissible plot map at one fixed radius array. Smooth coefficients on the common open
domain, zero extended outside, are measurable. If |g| ≤C where u ̸= 0, then

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|ug| ≤C|u|,
Z
ug dμ
 ≤BC.
(NE7)

Apply to the direct coefficient and its direct-minus-iterated difference with NE2’s two bounds. Both are integrable, hence so is the iterated coefficient, and subtraction proves the weighted NE2. No product measure or absolute continuity assumption is used. Jointly varying exterior positions and characters are allowed for the tested remainder: smooth position dependence and continuous character evaluation on the compact metrizable character space give joint measurability on the guard domain. Apply NE7 with. The raw correlation has bound by adding its degree-zero terms and remainder with their engineering rates, so it too is integrable. The truncation is a difference of integrable expressions, its finite terms can be integrated, and the integrated full discrepancy is bounded by. Probability phase mixtures use = 1, with identical coefficients outside the phase integral. Covariance concerns the full homogeneous vectors, not invariant scalar basis channels. Leaves emit their actual labels. Internal vertices include the child emissions into their fixed horizon spaces, apply the multilinear at their centers, and sum through their own horizon. Multilinearity gives exactly the finite channel formula. Induction using the laws of and inclusions gives translation, conjugation and orthogonal covariance; applying only at the root gives the degree- law. The basis can transform nontrivially within; no assertion that

its vectors are individually rotation invariant is needed. The full tested discrepancy inherits these laws.

The lifted nested object retains the finite rooted tree, all radii, centers, actual leaf degrees, fixed horizons and full homogeneous vectors. Its iterated reader is their finite multilinear substitution; its direct reader retains the full cluster with the same fixed basis. The exact telescoping comparison retains the residual between readers, with NE2 quantifying recovery at arbitrary positive ratios. Keeping that residual recovers the original coefficient exactly; NE3 similarly retains the tested whole-word remainder. Filter normalization proves uniform reader bounds without a scale-gap assumption. NE7 transports these same readers under joint weighted integration and phase mixture.

Normalized radii, weighted readers and vector recovery. For the nested source record retain its radii unchanged under the physical readout and section. In particular the finite vector

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av(T) = log(1/sv)
T
,
X
v
av(T) = 1,
avirtual parent = 0

is literally the same function of and. Its compact filter limit therefore needs no new choice of physical scales. For below its parent, gives) eventually, and hence the same inequality in the limit. Each radius has rate relative to; even at its actual convergence to zero remains part of the retained geometry. The geometric size estimate gives the required envelope rate at vertex, and also actual convergence of every descendant displacement to zero. The bounds and retain, respectively, the descendant envelope and separation at every exponent strictly above. Induction over the tree thus gives all the regularity

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hypotheses of the preceding rate proof, including vertices with av = 0. With ∆v = av−aparent(v), the
preceding finite-tree argument retains ε P
v ∆v > 0. Selecting a violation before compact extraction
retains that very position, exterior and viewer record; its refined eventual bound contradicts the
selected violation. This is the recovery of the common threshold, including all-viewer quantifiers
and arbitrarily small positive ratios.
m for the direct and iterated readers. Their lifted versions satisfy CZ = C ◦p
Write Cdir
m and Cit
and CZ ◦j = C. For any measurable plot z(i) the exact identities
Z
Z

Z
 ≤BTm,ε
X
u(i)CZ(jz(i)) dμ(i) =
u(Cdir
m −Cit
qDv+1
u(i)C(z(i)) dμ(i),
m) dμ
v
K
K
K
v̸=root

use the same fixed radii and the same zero extension on the common open domain. Domination by supplies integrability before subtraction. Joint position and character readouts similarly give for the complete tested discrepancy; the raw bound then makes its finite truncation integrable. Thus finite coefficient sums and probability phase integrals retain their order and coefficients.

For a subtree rooted at, denote its emitted vector by and its fixed horizon by. With the prescribed inclusions understood, the recursive reader is exactly

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Dv
X
j=0
ιj hj
(xw)w child of v; (ιwew)w child of v

,
el= the actual leaf label.
ev =

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The lifted recursion uses hZ
j = jjhj(p ·) and the same inclusions, so induction gives peZ
v = ev and
eZ
v = jev for section inputs. Multilinearity recovers every finite channel term. If ρj(R) is the actual
orthogonal action on Hj, use ρZ
j (R) = jjρj(R)pj on its horizontal image. The same induction
intertwines these actions, translations and conjugation. At the root retain only hm to recover the
specified degree-m output. This transports the full homogeneous vector and its tested discrepancy
while allowing basis channels to mix under ρm(R).