Separated and nested operator products · Part 2

block length of that embedded parent. Distinct centers have parent separation at least + 1. Enlargement gives embedded separation + 1), while both deviations cost strictly less than and . The triangle inequality proves cross-copy separation at each intermediate level; the finest spacing handles within-copy separation. Use one formal marker per occurrence ). Descending integration preserves its polynomial and integrability. Grouping within each copy replaces the insertions by. Grouping in stages or once gives the same coefficients: each marker-set coefficient is the product over its full inverse image. This fixed linear map on a finite

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polynomial space commutes with conditional integrals. Thus descent of the products is exactly
grouped descent of the individual insertions. The sweep starts with wi = δjφjai/8. At each level
it joins exactly fibers acquiring a common parent, retaining one anchor per fiber, with at most
the original arity at termination. Its available normalization is bounded below by compensation of
Q
i wi, including the four-dimensional volume ratio and join licenses. The initial allowance is

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!
|I|, m + 1, j, 0, α, (δjφj/8)|I| Y
i
ai
α|I| compensation
.

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Here α ≤1 is no larger than every intermediate license; fewer surviving fibers only increase this
lower bound. This is xT . For rate-zero positive licenses ui, put αi = min(1, u0, . . . , ui). For t < 0,
eventually uj ≥et scalej; monotonicity gives αi ≥min(αi0, et scalei).
Divide logarithms by scale
and let t ↑0: αi retains rate zero and bounds all preceding licenses. The halo purchase bounds
the geometric license logarithm by 16
q
bβi, sublinear in scale; the other factors have rate zero too.
No power loss is introduced by the minimum.
At the last join choose z > 0 below surviving
normalization and purchased local constant x ≥1. The terminal analytic coefficient has remainder
bound 4/z. Prune the allowance to I′
v = z/(16x). It is available and obeys

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2I′
v 2(4/z) ≤1,
I′
v |λ−1| x(4/z) ≤1,

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where λ = (lm)q ∈C and lm ≥2 is the integer block size at this step. This λ is a scalar multiplying
every coefficient in the degree-q fusion; it is not an arbitrary linear change of frame. Since q ≥0,
|λ| = lq
m ≥1 and |λ−1| = l−q
m ≤1, with both moduli equal to one for q = 0; equivalently, scalar
multiplication by λ−1 has operator norm at most one on the coefficient space.
Thus the first
smallness expression is 1/x ≤1 and the second is |λ−1|/4 ≤1/4. A lower bound on the norm of
a general operator would not imply this inverse bound; the scalar identity is what proves it here.
In the lift the coefficient space is unchanged and λ acts by the same scalar on realized coefficients,
so the substitution bound is recovered with the same constants; the inverse bound depends on the
homogeneity of the fusion in one degree and would not survive mixing degrees. March allowances
Wv ≤δmI′
v meet substitution bounds. Local fusion gives

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∥Q0∥coeff≤p′ = aq
m+1x 4/z

with the same remainder budget simultaneously in every copy. Copy transformations are inherited from the original and the anchor is zero. Restore the polynomial coefficient and apply bounded-copy comparison to its actual local section: the physical label has quota at most 2 and root error 2 stakes, with final march amplitude . It yields a centered degree- label with quota 2 a mismatch supported in the prescribed central chamber, and root-test bound . These are exactly the outputs used above. We describe the geometry and algebra of that finite matching construction more precisely. The finite matching construction itself is the sweep-and-replica construction; its remaining formulas are addressed below. The input positions are required to remain in that chamber through level + 1, exactly the plot conditions already checked.

The lifted matching object retains the refining filter, replica geometry, occurrence polynomial, grouping map, licenses and local section. Grouping/descent commutation proves coefficient conservation; compact matching gives the same centered label to every exterior reader. The exact register identity preserves dependence on exterior variables. Amplitude and quota readers retain their paid errors, so the finite matching construction recovers OE6 without a viewer-specific replacement or an interchange of extraction limits.

Replica grouping and finite sweep recovery. Let send each occurrence ( its copy. On the finite square-free coefficient space define the source’s grouping reader by

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[zB]ΓgP = [zg−1(B)]P,
B ⊆V.

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For P = Q
(v,i)∈O(1 + zv,iFv,i), this gives ΓgP = Q
v(1 + zvqv) with qv = Q
i Fv,i. For a subsequent
grouping h the full inverse-image rule proves

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ΓhΓg = Γh◦g,
ΓgE[P | G] = E[ΓgP | G].

The second equality is coefficientwise linearity for the same finite integrable polynomial. Configuration pullback also commutes with Γ, so the source and physical descended polynomials have identical grouped coefficients at every step. The geometric reader retains physical positions, parent maps and copy motions. Its separation calculation is explicitly

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L(B2
m+1 + 1) −sum of the two deviations > LB2
m+1 ≥B2
i .

Thus the same fibers, anchors and joins occur at each scale, including the unchanged anchor copy. The source records the actual allowances, license minima and compensation before the final fusion; these scalar readers preserve positivity and the rate-zero minimum argument. The coefficient section at the terminal join recovers with the same, remainder budget and pruned allowance Restoring its grouped coefficient and using the same local section therefore recovers the physical label quota 2 and the root error 2 stakes on every copy, with . Neither copy grouping nor source recovery changes that final march amplitude.

20.5 Common purchases and the exact sweep compensation

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The common family supplies every finite arity with the same positive activities and smallness
conditions
(2Bi + 1)4 scurryi ≤1
2,
escurryi −1 ≤1,

the normalization lower bound, fusion and analytic remainder estimates at sufficiently late degree references. The clock then reaches at least degree plus three. These are scheduled estimates on the existing family; changing arity or output degree does not replace it. Encode a finite screened linear combination with its actual moving lattice labels and the same scalar coefficients in its lattice expression and continuum compression. Register multiplication is exactly concatenation; subtraction retains coefficients 11. Apply the paid-product inequality to that expression, then take the selected continuum limit, to bound its compression norm by the product of its allowances. This applies in particular to the intact mismatch times every exterior factor. This explains why the subtraction stays intact through the limit; estimating its two terms separately would discard the approximation accuracy.

The compensation inequality underlying the sweep can be written without any scale shorthand. For original arity, remaining groups1 and 0, define the exact compensation

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K(i, j, s) =
aj
4H α
H(H−s)
w.
(OE8)
ai
8

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The invariant is K(i, j, s) ≤Qs
γ=1 Wγ. Initially s = H and w = Q Wγ, giving equality at the finest
level. If separation persists, ai = Liai+1, Li ≥2, and each allowance becomes Wγ/L4
i . Since s ≤H,

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K(i, j, s) = L−4H
i
K(i + 1, j, s) ≤L−4s
i
Y
γ
Wγ.

If separation fails, choose a nontrivial grouping with separated coarser representatives and original anchor, of size, with purchased cover Local grouping multiplies the old product by license(, and each new group spends 8 before transport. Because is below this license,

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we have

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K(i + 1, j, s′) ≤K(i + 1, j, s)(α/8)H
≤8−s′ license(H, li, i + 1)s Y
γ
Wγ.

The transport inequality then pays the coarser step. Descending finite induction proves OE8 in both cases. Each regrouping decreases, so its fixed-arity exponent pays the grouping events independently of the number of scale steps or cells. Marker grouping commutes with push and preserves integrability by the finite linear integral identity. Retain the actual representatives: parents come from original positions, copy transformations from original collars, and the anchor is original. The purchased bounds every selected coarser cover. These are precisely the data used by local grouping and final normalization; no additional consistency choice is needed.

The all-phase norm is the same ground-algebra norm throughout. At distinct positions the compressed whole word is; zero offthat domain only makes the notation total. Characters read phase correlations, and the vacuum functional reads the unselected correlation by common translation into the positive chamber and compression. For the unselected hierarchy its evaluation isΩΩ, because translating the entire finite word into a positive-time chamber leaves its correlation unchanged and the compression formula realizes that translated word. Both are contractive, while the Gelfand norm is the supremum over characters. Thus a common scalar viewer bound is equivalent to the compressed norm bound. The lifted sweep retains representatives, parent fibers, allowances and the exact marker polynomial; OE8 is its quantitative conserved budget. Its reader is the intact paid linear combination, followed by compression and character or vacuum evaluation. The same source therefore recovers both the compensation invariant and the common OPE norm without dropping the subtraction.

Exact sweep-budget and all-viewer norm recovery. At each finite sweep stage retain the tuple of actual representatives, parent fibers, anchors, group allowances and scalar licenses. Its readout preserves the physical coordinates and these finite numerical data. Thus and) are identical on the source and physical records. The two case updates recover the exact budget ratios

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K(i, j, s)
K(i + 1, j, s) = L−4H
i
,
K(i + 1, j, s′)
K(i + 1, j, s) = (α/8)H(s−s′) ≤(α/8)H.

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Because s ≤H and s′ < s, these are precisely the transport and join payments in (OE8). Finite
descending induction commutes with the readout: each grouping uses the same finite inverse-image
coefficient map and the same integrable conditional push. The number of strict regroupings is
bounded by the decrease in s, independently of the number of scale steps and replica cells.
For an intact paid linear combination E = P
lclM(wl), its source algebra element and viewers
are

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EZ = α(E) =
X
l
clMZ(wl),
h(χ)(EZ) = χ(E),
∥EZ∥= ∥E∥= sup
χ |χ(E)|.

The vacuum functional is likewise unchanged under the ground unitary. The original finite-lattice coefficient expression uses these same, including 11 for the discrepancy. Thus its paid-product reader retains subtraction through the selected limit before applying compression or any viewer. The recovered compensation and norm bills therefore concern one intact expression on the original common family.

20.6 Representatives, moving-label moments and grouping prices

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In the anchor copy choose a maximum-cardinality subset containing the distinguished anchor with
pairwise lattice separation at least B2. Each omitted position is at separation < B2 from a selected
one, or it could be adjoined. Assign each point to such a representative and each representative to
itself. This is a surjection with an anchor-fixing section. When separation failed the subset is proper,
so its size decreases. Repeat the relative selection in every copy; reflection and translation preserve
separation and the previously proved cross-copy estimate makes the whole selected configuration
separated. If two finer points have parents at separation < B2
m, the half-block deviations from
embedded parents give
sep(x, y) ≤Lm(Bm + Bm+1 + 1)2 =: cm.

The halo of the first point contains the pair and is covered with at most

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(2cm + 1)4 ≤16(1 + cm)4

-blocks. This purchased carpet bound applies to each point/representative pair; an entire group need not fit one small halo.

For a finite screened linear combination whose labels converge along an actual refining filter in fixed finite-dimensional spaces, freeze the moving labels at each regulator pair and use that static word in the order-2 even-moment expansion. It is a finite sum over one choice from at every occurrence, with products of scalar coefficients and conjugates and translated/reflected lattice correlations. All subtractions are exact. Expand labels in fixed bases. Finite coordinate products converge, and fixed-label point limits apply at the eventual generic replica positions. Thus first the regulator filter gives the continuum replica expression; then large replica displacement gives the vacuum even moment of its ground compression by GI3. A finite bound on the full expression passes through both limits for every. The ground-algebra even-moment norm argument gives , equivalently by character moments and the Gelfand norm. Here0 is the product of paid allowances. Applying this to the intact register mismatch proves its norm bound without a separate inner regulator limit for moving labels.

Anchor selection and moving-moment norm recovery. Retain the finite selected representative set, assignment and its section. The physical reader preserves positions and gives

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g ◦s = IR,
g(o∗) = o∗,
s(o∗) = o∗,
sep(x, s(g(x))) < B2
(x /∈s(R)).

The same relative assignment is used in every copy. Hence its parent-based two-point carpet bill remains (2 for each point/representative pair. It is not replaced by a single cover assumption for an entire group.

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For each regulator parameter, retain the actual finite-basis coordinates of every moving label.
The source section fixes these coordinates, so every finite even-moment expansion, with all scalar
coefficients and conjugates, is identical before taking limits. Coefficient products and fixed-label
correlations converge on the same refining filter. Its continuum replica expression and subsequent
large-displacement limit therefore recover the same ground-algebra moments. In the commutative
algebra the resulting norm test is
Z

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Θ
|χ(ME)|2h dν(χ) = ⟨Ω0, (M∗
EME)hΩ0⟩≤c2h,
h ≥1.

For 0, if a character value exceeded, continuity would give an open set of positive measure on which it exceeded some. The inequalities contradict. For = 0, already the second moment and full support force zero. Thus the Gelfand reader gives. The algebra isomorphism preserves these moments and norms exactly. This recovers the paid bound for the intact mismatch with its moving labels and original filter, without inserting a separate inner limit for those labels.

The actual local grouping allowances are

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λi(H, l) =
(δi/H)δi
exp((ei + 1
2)l),
W ′
γ =
Y
j∈γ
λi(H, l)Wj,
ei = supportExponenti .
(OE9)

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Each occurrence receives δiWj, allocation among at most H costs δi/H, and moving support costs
the displayed exponential. The fibers partition the old occurrences, so their product is exactly
λi(H, l)s Q
j Wj, as used by OE8.
Apply finite marker fusion with these fibers, the cover and
occupancy bounds and both smallness inequalities. The original allowances bound the actual local
analytic norms and δi ≤1 preserves them. Positive background normalization is the actual stage
density by the stage representation.
Fusion gives local group functions and dominated profiles
representing the exactly grouped polynomial, with remainder 4/W ′
γ. Pruning to W ′
γ/8 gives 4/W ′
γ =
1/[2(W ′
γ/8)], the next-step bound. Newly grouped leading counterterms are zero. Translations and
reflections of functions/profiles persist because all copies use the same grouping. At the terminal join
there is one group per copy and no 1/8 pruning before normalization: its allowance is Q
j λi(H, l)Wj,
with remainder four divided by it.

20.7 Nested chessboards and the intact paid even moment

For two field groups of separation scales, choose

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b ≤f ≤l≤n,
f ≤Q,
|I|10af < ab,
|J|6an < al.

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The seating lemma gives the required interior margins in translated reflection chambers. Replica
itineraries preserve those margins; within a replica original spacing persists and between replicas
distances diverge. At sufficiently large displacement and then fine regulator all lattice separations
exceed their chamber scales.
At the 2h occurrences use adjoints on alternating even positions
and unchanged fields on odd positions. Adjoints preserve allowances. The mixed nested-chamber
chessboard inequality bounds the integral by

! 
 Y



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2h
Y
j∈J
Dj



i∈I
Bi
.

Register translation and finite-product conjugation identify this with the even moment of the whole registered insertion. Finally, the paid even-moment bound can be seen directly from its lattice layout. For a mismatch, is that mismatch times the exterior register; the main group has one occurrence per replica with price. It is never split into two independently estimated fields.

To prove the nested inequality, a paid interior field with chamber root bound retains both price and support under permitted diagonal reflection/conjugation. A margin of three original chamber scales inside a larger chamber makes its placement a tiling copy of an interior function with the same bound, by the offset/reflection identity. Collect finer fields in each larger cell and copy that product to all larger cells. Their finer cells are distinct: different larger cells cannot meet in one finer cell, and finer placements in a common larger cell are farther apart than its finer cell extent. Reflection preserves this. Apply finer chessboard, then take the root of order the positive

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number of larger cells. This root extraction is legitimate because the prices and root tests are
nonnegative and the number of cells is positive. The collected larger root bound is the product of
its nonnegative individual prices. Coarser main fields occupy distinct larger cells, none occupied by
finer exterior fields, since otherwise the cell extent contradicts separation. Apply larger chessboard
to those main fields and collected products. This gives Q Bi
Q Dj without replacing the two scales
by one. The common seating translation exists by two finite avoidance steps. First translate main
points five buffer scales from coarse boundaries, possible by |I|10af < ab. Then shift at most one

intermediate scale to put exterior points three fine scales from boundaries, possible by This consumes at most two main buffer scales and leaves three. Nested cell periods divide the larger ones and replica displacements are their integer multiples, so residues and margins persist. For large periods Euclidean distances are at most twice mesh times wrapped lattice separations; subsequent embedding multiplies lattice separation by mesh ratio. Thus all finer cutoffinequalities follow.

One-step transport uses . Its analytic remainder cost 6 for

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(6Li)/(2Wγ) ≤1/(2W ′
γ).

Marker tolls are finite sums of nonnegative allowance products, hence remain bounded after decreasing allowances. Separated representatives give support substitution, occupancy holds before and after parents, and transition density gives the exact descended polynomial. Counterterms preserve their copied scalar transformations and 1 since . Initially the solved representation gives 4; choosing 8 converts it to 1 ). These are the initialization and transport constants of OE8.

The lifted source retains the anchor-fixing surjection and section, individual connecting covers, convergent label coordinates, regulator filter and replica itinerary. Its moment reader preserves every signed coefficient before both limits, and its nested root reader preserves the two chamber scales and actual mismatch. OE9 and the pruning identity recover the exact grouped polynomial and its next-step bound. These retained data prevent loss of approximation accuracy through representative selection, grouping, or compressed-norm recovery.

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Grouping allowances and nested-root recovery.
Retain the actual local allowance W ′
γ, cover,
occupancy, normalized stage density and dominated profiles from (OE9). Their configuration reader
is the same local grouped function and its coefficient polynomial. Conditional-integration recovery
preserves that positive normalized density before the remainder is read.
Pruning is the scalar
operation c
Wγ = W ′
γ/8, so the source recovers exactly

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4
W ′γ
=
1
2c
Wγ
.

The terminal group retains its stated unpruned allowance instead. Zero leading counterterms and the common copied transformations are part of this same record. For a finite paid expression let) denote the actual alternating-adjoint replica integrand. Configuration pullback commutes with every finite product and conjugation, and normalized Wilson recovery gives

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

2h
Y
i∈I
Bi
Y
j∈J
Dj

⟨L E2h(Z)⟩Z = ⟨E2h(Z)⟩,
|⟨E2h(Z)⟩| ≤
.

The source root reader keeps the two chamber scales separately: first the finer copied-product inequality, then its positive larger-cell root, then the coarser mixed inequality. Each support, reflection, price and seating residue is the same physical datum under readout. For the mismatch its main occurrence is retained as one paid expression in every replica.

Finally the next-step allowance is unchanged under source recovery. The original estimates 6 and 8 for the initial allowance recover the transport and initialization bounds respectively. Monotonicity of the finite nonnegative toll sums preserves all subsequent decreases of those allowances. Thus the grouped remainder, nested even moment and paid sweep use one normalized stage and the original numerical constants.

20.8 Quantitative marker fusion and exact restoration

Let ∆map old occurrences onto groups, with an injective representative section. Extend old supports by the chosen connecting covers, each of cost at most, and set

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f
Wi = (η/H)Wie−αl,
α = e + c.

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An old coefficient on occurrence set A gains (η/H)|A| after its cover costs are paid. A group’s
allowance is the product of f
Wi over its full fiber.
Distinct occurrences keep distinct nilpotent
markers. Eligible products have disjoint old marker sets, so their toll product is exactly the toll
of their union and, for complete fibers, the corresponding product of group allowances.
With
occupancy at most s, scalar weighted bound a, activity bound b and Wi|ci| ≤T, the extended
inventory envelope is
γ = 2(1 + s)η(sTeα + a) + (1 + s)b.
(OE10)

For nonempty. At most old siblings charge a representative, cancelling The injective section places representatives at old locations, retaining occupancy. Reallocation yields the incidence factor 1+; two scalar inventories yield the factor 2, and the activity inventory is charged once. At an original site singleton scalars contribute at most, because singleton cover number is at most one; ordinary nonempty coefficients contribute, activities. This proves OE10 without dropping the unit contribution. Finite eligible products are bounded by products of entry envelopes, using the disjoint-marker toll identity. The connected-set forest estimate at excess

exponent applies when (2. At exponent it bounds both grouped ordinary and activity coefficients by1. This is the same connected-set estimate already proved for the general insertion calculus, now with the extended supports and group tolls. Choose. Singleton restoration with the normalized nonzero background gives the new function and dominated activity profile, with remainder 2 . The actual empty background activity coefficient is retained, so the profile is admissibly dominated, not merely numerically bounded. Higher coefficients give the simultaneous served representation. The sweep uses = 1, recovering OE9.

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Weighted fusion and singleton recovery.
Retain the occurrence-to-group surjection ∆, its
representative section and the actual connecting covers. For an old roster A write t(A) = Q
i∈A f
Wi.
The source toll reader has the exact identities

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t(A)t(B) = t(A ∪B)
(A ∩B = ∅),
t(∆−1(J)) =
Y
γ∈J
W ′
γ.

Thus eligible full-fiber products retain their weighted coefficient price. Configuration pullback fixes the cover costs, occupancy and all scalar allowances; it preserves coefficientwise sup bounds and the same sitewise nonnegative envelope (OE10). The forest reader therefore has identical inputs and on both carriers. Let denote the two grouped polynomials. Coefficientwise pullback commutes with their singleton readers:

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[ζγ]L P ord = L [ζγ]P ord,
[ζγ]L P act = L [ζγ]P act.

The empty-marker activity and actual normalized background are retained by this same map. Consequently restoration recovers both the local function and its admissibly dominated profile, with remainder 4 , and all higher served coefficients. A prescribed copy motion is a bijection of the retained inventories and covers; reindexing the same finite coefficient products commutes with these singleton and empty-marker readers. Thus copy covariance and the hypotheses for the next grouping survive exactly, including the sweep values2 and

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Here is the full inventory identity. For scalar markers ci, ordinary polynomials AS and flagged
activity polynomials KS,F , take one singleton entry for each ci and one entry for every nonempty
coefficient of AS. Extend each support by representatives of all mentioned groups. Split each scalar
value into its value on healthy extended support and its value on unhealthy extended support,
zero on the complementary configurations. Their sum is exactly the value. Retain every allowed
activity coefficient, extended support and original flagged data. These finite inventories recover
P
i cizi + P
S AS and each KS,F coefficientwise; locality places all mentioned markers inside their
support. Connected eligible collections receive union supports and markers, with their prescribed
flagged intersections. Expansion of the finite nilpotent exponential and gas, followed by the unique
connected component partition, yields

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"
X
!
#
X
!
i
cizi +
X
S
AS
eAS
Gasζ(f
K).
(OE11)
R∆
expz
Gasz(K)
= expζ
S

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Here the full-fiber reader is
(
ζ∆(A)a,
A = ∆−1(∆(A)),
0,
otherwise.
R∆(zAa) =

For retained monomials, new rosters are disjoint exactly when old rosters are: a shared group would place its entire nonempty fiber in both. Thus the reader commutes with eligible products, while it is not asserted to preserve arbitrary products involving partial fibers.

Two entries connect if their group rosters meet, or their extended supports meet and both are activity-collecting types (unhealthy scalar or old activity). Ordinary grouped entries are nonempty connected eligible collections with no such type. Grouped activity entries are connected eligible collections containing an old activity, with their flagged intersection equal to the union of the old flagged subsets; the coefficient is the product if this condition holds, zero otherwise. Old activity supports inside each collection must be pairwise disjoint, as in the original gas. An unhealthy scalar