Separated and nested operator products · Part 2

remainder bound 4. Prune its insertion allowance to

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I′
v = z/(16x).

This is no larger than the available normalization. It satisfies the two elementary fusion inequalities

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

where, and2 is the integer block size at this step. This is a scalar multiplying every coefficient in this degree- fusion; it is not an arbitrary linear change of frame. Since

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|λ| = lq
m ≥1,
|λ−1| = |λ|−1 = l−q
m ≤1.

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For q = 0 both moduli equal one. 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. The target march allowances satisfy Wv ≤δmI′
v,
so their products meet the insertion substitution bound. Applying the local fusion estimate yields
a coefficient Q0 with
∥Q0∥coeff≤p′ = aq
m+1x 4/z

and the same remainder budget, representing the descended products simultaneously in every copy. All transformations of one copy into another are inherited from those of the original; the anchor is at zero. Restoring the polynomial coefficient, then applying the bounded-copy comparison to its actual local section, gives a physical label with quota at most 2 and root-test error 2 stakes where is the chosen final march amplitude. These are exactly the two outputs used in funding, with

The common cluster purchases used in this construction are made for every finite arity on the original family. They comprise the same positive activities, smallness inequalities

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(2Bi + 1)4 scurryi ≤1
2,
escurryi −1 ≤1,

the normalization lower bound, and the local fusion and analytic remainder estimates at every sufficiently late degree reference. The degree reference ensures the clock has reached at least degree plus three. These clauses are supplied by the already chosen scheduling and local analytic estimates; they do not entail replacing the Wilson family when the arity or output degree changes.

For the paid-product limit, a finite linear combination of screened words is encoded with its actual moving lattice labels. Its continuum compression and its lattice expression use the same scalar coefficients. Products of registers agree identically with the register of the concatenated list. Differences use the coefficients 11. Applying the paid-product inequality to this exact expression and passing to the selected continuum limit gives the compressed norm bound by the product of its insertion allowances. In particular it applies to the single mismatch times all exterior factors. 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. If is the original number of fields, is the number of remaining groups, 01, and define

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

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The sweep maintains K(i, j, s) ≤Qs
γ=1 Wγ.
At the finest level, s = H, so it is equality with
w = Q Wγ. When the groups stay separated at the next coarser level, their allowances become
Wγ/L4
i , where ai = Liai+1 and Li ≥2. 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.

This pays the entire one-step transport without a loss that accumulates separately in each tiling copy.

If separation fails, select a nontrivial grouping, preserving the distinguished anchor, whose representatives are separated at the coarser level. Let its size be, and let be the purchased cover bound for its groups. The local grouping estimate gives the exact product of new group allowances as the old product times license(; before transport each group also spends a factor 8. Since is no larger than this license and

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H(H −s′) ≥H(H −s) + H,
αH ≤αs,
8s′ ≤8H,

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we obtain
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 preceding transport inequality then pays the coarser step. This proves (OE8) by finite descending induction, in both cases. A regrouping strictly decreases the number of groups, so the exponent ) pays at most the fixed arity’s worth of grouping events, independently of the number of scale steps and tiling cells. Grouping the marker polynomial and then pushing it is identical to pushing first and grouping afterward, by the finite-dimensional integral identity already proved. Integrability is likewise retained under each such linear grouping.

Every step also retains the actual representatives, not just their count: their parent relation is inherited from the original positions, their copy transformations are inherited from the original collars, and their anchor representative is the original anchor. If a group is selected at a coarser level, its cover is bounded by the purchased. These are exactly the data needed for the local grouping estimate and the final normalization. Thus the induction does not require consistency choices beyond the existing finite marker grouping.

For clarity about the norm used throughout, a compressed word at noncoincident positions is its ground-space multiplication operator; it is assigned zero elsewhere only to make the notation total. For a character) is its phase correlation. 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 evaluations are contractive: the latter byΩ= 1, the former by the character norm. Conversely, the Gelfand norm is the supremum of the character values, so a common nonnegative scalar bound on all viewers is equivalent to the compressed operator norm bound. This identifies the all-phase norm in the OPE with the ground algebra already constructed.

The representative selection in the grouping step is finite and explicit. For the positions in the anchor copy choose a subset of maximum cardinality containing the distinguished anchor and having pairwise lattice separation at least. Every omitted position is within separation

of one of the chosen positions, since otherwise it could be adjoined. Assign each position to such a representative, assigning representatives to themselves. This gives a surjection with a section fixing the anchor. When separation failed, the subset is proper, so its cardinality strictly decreases. Repeat this same relative selection in every copy. Reflections and translations preserve lattice

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separation, and the cross-copy separation was already proved; hence the entire selected configuration
is separated simultaneously.
The necessary finer-level two-point cover bound follows from the parent relation. If x, y have
parents at separation < B2
m, the distance from either point to its embedded parent is less than half
the block length Lm. Thus

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sep(x, y) ≤Lm(Bm + Bm+1 + 1)2 =: cm.

A pair at distance at most is contained in a halo of its first point. Covering this halo by -blocks costs at most

This is the purchased carpet bound. It applies separately to each point and its representative; the construction does not assume that the entire group lies in a single small halo.

We give the remaining moment argument for the compressed norm passage. Let be a fixed finite linear combination of screened words and let its labels vary along a refining regulator filter, converging in their fixed finite-dimensional label spaces. For each regulator pair, freeze the labels at that pair and use this ordinary static word in the finite even-moment expansion. For an even order , the expansion is a finite sum over choices of one word from at each of the 2 occurrences. Its coefficients are the products of the scalar coefficients and their conjugates, and its terms are the corresponding translated and reflected full lattice correlations. This is an exact expansion of the even-moment expression, including every subtraction in

Each such lattice correlation converges along the refining filter with the moving labels. Indeed expand each label in a fixed finite basis, expand the multilinear correlation as the finite sum over all basis choices, and use convergence of the finitely many coordinate products and the fixed-label point limits. For every sufficiently large replica displacement the limiting positions are generic, so all these fixed-label limits apply. Therefore the finite even-moment expression converges first along the regulator filter to its continuum replica expression. Sending the replica displacement to infinity gives the vacuum expectation of the corresponding even power of the ground compression, by the long-cluster moment formula already proved.

If the finite paid-product estimate bounds the whole even-moment expression in norm by its real part is at most. Both limits preserve that inequality, for every. The ground-algebra even-moment norm argument then gives. Equivalently, one may apply the same moment inequality to each character and use the Gelfand norm. Here is the product of the main and exterior insertion allowances; it is nonnegative because each paid allowance is nonnegative. Applying this construction to the exact register identity for the mismatch proves its norm bound. This proof uses finite-dimensional convergence of the moving label along the actual refinement filter; it requires no separate inner regulator limit for that label.

The local grouping allowances just used 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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Indeed each occurrence first receives δiWj, the allocation among at most H occurrences gives
the factor δi/H, and moving its support to the chosen representative costs exp((ei + 1
2)l). The
occurrence sets of distinct groups partition the old occurrences, so Q
γ W ′
γ = λi(H, l)s Q
j Wj exactly.
This proves the group-product formula used in (OE8).
To obtain the grouped local functions, apply the finite marker fusion estimate to those oc-
currence sets, using the cover bound l, the occupancy bound, and the two smallness inequalities

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displayed above. The original insertion allowance Wj controls its actual local analytic norm; multi-
plication by δi ≤1 preserves that bound. The positive background density permits normalization,
and the representation of each lattice stage identifies this normalized density with the stage’s ac-
tual background. Fusion produces one local function per group, together with dominated activity
profiles, representing the exactly grouped marker polynomial. Their normalized remainder bound
is 4/W ′
γ. Pruning each group allowance to W ′
γ/8 rewrites this as

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4/W ′
γ =
1
2(W ′γ/8),

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the bound required to transport the group to the next scale. The leading counterterm of a newly
grouped function is zero. The copied functions and profiles retain their common translations and
reflections, since the same groups and representatives are used in each copy.
For the terminal
grouping there is just one group in each copy; no factor 1/8 is needed before normalization, and the
common allowance is Q
j λi(H, l)Wj. Its remainder bound is 4 divided by this allowance, exactly
the input to the terminal step above.
Finally, the paid even-moment bound can be seen directly from its lattice layout. The two
groups of fields have separation scales ab and al. Choose buffer and insertion levels satisfying

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b ≤f ≤l≤n,
f ≤Q,

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and the finite packing inequalities

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|I| 10af < ab,
|J| 6an < al.

The seating lemma places every original insertion with the required interior margins in the appropriate translated reflection chamber. Translation copies are displaced by the prescribed replica itinerary. Within one replica the original spacing is unchanged; between different replicas all distances tend to infinity with the displacement. At sufficiently large displacement, and then sufficiently fine regulator, every required lattice separation exceeds its chamber scale. The same interior margins hold in all replicas because the itinerary preserves the chamber margins. At the 2 occurrences replace each field by its adjoint on the alternating even positions and keep it on the odd positions. Adjoining preserves its paid allowance. The mixed chessboard inequality on the two nested chambers now bounds the integral of the product of all these translated fields by the product of all their allowances, namely

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
! 


Y
2h
Y
j∈J
Dj



i∈I
Bi
.

The register translation identity and conjugation of finite products identify this integrand with the even-moment expression of the entire registered insertion. For a mismatch, is the mismatch times the exterior register, so the main group has exactly one occurrence per replica with allowance . Nothing expands that occurrence into two independently estimated fields. This proves the finite bound required by the compressed-norm limit.

We justify the nested-chamber inequality used in the last paragraph. A paid field has an interior support in its original central chamber and a bound for that chamber’s root test. Any permitted diagonal reflection and conjugation preserves both properties and its price. If its placement has a margin of three original chamber scales inside a larger chamber, the translated field is a copy, in the appropriate cell of the larger tiling, of another interior function with the same bound. This is the offset-and-reflection identity for the tiling copy map.

First collect all finer fields assigned to one larger cell. Copy this collected product to every cell of the larger tiling. Within the finer tiling all individual copied fields occupy distinct cells: distinct larger cells cannot meet in a finer cell, and two original finer placements in one larger cell are farther apart than the finer cell’s extent. Reflection preserves that separation. The finer chessboard inequality bounds the full copied product by the product of the finer root tests. Taking the root of order equal to the number of larger cells gives the collected product a larger-chamber root test at most the product of its individual prices. This root extraction is legitimate because the prices and root tests are nonnegative and the number of cells is positive.

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Now the coarser main placements occupy distinct larger cells, and no one of them occupies a
larger cell containing a finer exterior placement: otherwise the cell’s extent would contradict the
assumed main-main or main-exterior separation. Put each main field in its own cell, and each
collected exterior product in its occupied cell. One application of the larger chessboard inequality
bounds the full mixed product by the product of their root tests. Multiplying the collected bounds
gives exactly Q Bi
Q Dj. Thus the different insertion scales are retained; replacing them by a single
common scale is unnecessary.

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The common seating translation exists by two finite avoidance steps. First choose a translation
placing the main positions at distance five buffer scales from the coarse cell boundaries; the total
excluded intervals fit because |I|10af < ab. Then shift by at most one intermediate cell scale to
place the exterior positions at distance three fine insertion scales from their cell boundaries, using
|J|6an < al. The second shift consumes at most two of the five main buffer scales, leaving the
required three. The nested cell periods divide the larger periods. Replica displacements are integer
multiples of these periods, so they leave every boundary residue and hence all margins unchanged.
Actual torus separations follow from the Euclidean ones for sufficiently large periods: the physical
distance is at most twice the mesh times the wrapped lattice separation. Subsequent embedding
multiplies lattice separations by the mesh ratio, proving the required inequalities at every finer
cutoff.

The one-step sweep transport uses allowances . Its local driving estimate multiplies the analytic remainder bound by 6; this fits since 6 for2. Thus

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(6Li)/(2Wγ) ≤1/(2W ′
γ).

All marker tolls are finite sums of nonnegative products of allowances. Decreasing the allowances therefore preserves every toll estimate. The separated representatives give the support substitution bound, the occupancy hypotheses hold before and after taking parents, and the prescribed transition density gives the exact descended polynomial. The leading counterterms retain their copied scalar transformations, and 1 follows from At the initial fine scale, the solved activity representation gives remainder bound at most 4; taking 8 converts this ). These are precisely the numerical transport and initialization bounds used in the descending compensation induction.

The marker fusion has the following quantitative construction. For a surjection of old occurrence labels onto new group labels, with a chosen representative of each group, extend every old support by the fixed connecting cover from its occurrence to its representative. If this cover costs at most , the new allowance per old occurrence is

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f
Wi = (η/H)Wie−αl,
α = e + c.

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An old coefficient involving a set A of occurrences therefore gains (η/H)|A| in its weighted envelope,
after paying all connecting covers. The group allowance is the product of these f
Wi over its full
fiber. Distinct original occurrences retain distinct markers, so whenever a product is eligible its

marker sets are disjoint; the product of their tolls is the toll of their union. The latter is exactly the product of the full group allowances for the selected new markers. This is the algebraic reason fiber sizes can vary without changing the coefficient identity.

The local coefficient inventory consists of two scalar-entry contributions and the activity-entry contributions. If the occurrence occupancy is at most, the scalar coefficients have weighted bound, the activity coefficients bound, and, extending their supports and charging representatives gives total sitewise envelope

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γ = 2(1 + s)η
sTeα + a

 + (1 + s)b.
(OE10)

For nonempty marker sets, ( When a chosen representative is charged, there are at most old siblings in its fiber; the factor cancels precisely that multiplicity. The representatives themselves satisfy the same occupancy bound, because the section is injective and their locations are old occurrence locations. These observations give the 1 + incidence factor in each contribution. The scalar-entry inventory has two copies, producing the first factor 2; the activity inventory is charged once. Summing their nonnegative envelopes proves (OE10).

Expand the grouped coefficients as finite eligible products of these inventory entries. Give a product the product of its entry envelopes. The disjoint-marker identity in the preceding paragraph proves that this is a valid weighted coefficient bound. The connected-set forest estimate, at excess support exponent, applies provided

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(2B + 1)4γ ≤c.

It bounds both the grouped ordinary coefficients and grouped activity coefficients by support exponent. This is the same connected-set estimate already proved for the general insertion calculus, now with the extended supports and group tolls. Selecting gives the required two coefficient budgets

Read the singleton coefficient of the grouped ordinary polynomial as the new local function and the singleton coefficient of the grouped activity polynomial as its activity profile. The singleton restoration identity and the normalized nonvanishing background give their exact restored polynomial and a local remainder bound

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2Z/W ′
γ + 2Z/W ′
γ = 4Z/W ′
γ.

The domination construction for the activity singleton retains the actual background activity coefficient at the empty marker. Hence it yields an admissible dominated activity profile, not only a numerical estimate on one coefficient. The higher coefficients give the simultaneous served-insertion representation. In the sweep specialization = 1, these are exactly the bounds and hypotheses used in (OE9).

Copy covariance of these new singleton functions follows at the level of every grouped marker coefficient. A marker set focused in a given copy involves only inventory entries in that copy; its transformed set is focused in the transformed copy. The copy permutation is a bijection on these inventories, takes extended supports to their transformed supports, and conjugates each coefficient by the prescribed scalar transformation. Reindexing the finite sums and products proves covariance for the grouped ordinary and activity polynomials. Reading their singleton coefficients then proves covariance of the local functions and profiles. The same argument includes the empty marker and all interactions internal to a copy, so repeated regrouping preserves the hypotheses required at the next scale.

We spell out the inventory underlying this restoration. Write for the scalar marker entry, for the ordinary polynomial on a support, and for the activity polynomial on a support and

its flagged subset. For every scalar, take a singleton-marker entry on its occurrence location; for every nonempty marker coefficient of, take that coefficient on. Extend each of these supports by the representative locations of the groups mentioned by its markers. Split its scalar value into two entries: one equal to the value when this extended support is healthy and zero otherwise, and the other equal to the value when it is unhealthy and zero otherwise. Their sum is exactly the original value, at every configuration. Keep every allowed coefficient of , with the same extended support and its original flagged data. These are the two scalar-entry inventories and the

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single activity inventory in (OE10). They are finite on each lattice.
Summing the two scalar entries recovers P
i cizi + P
S AS coefficient by coefficient. Collecting
activity entries by their original support-and-flag label recovers KS,F , also coefficient by coefficient.
Each nonzero coefficient has its marker locations inside its support by locality, so every term
required for either recovery is present. The grouped construction forms eligible connected collections
of these entries and assigns their union support, union markers, and flag intersection to the resulting
coefficient.
Expanding the marker exponential and activity product, then partitioning a finite
collection into its connected components, gives

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

Herereads the old coefficient on the full inverse image of each new marker set, and the exponentials are the finite exponentials in the nilpotent marker algebras. The eligibility condition requires disjoint old markers and complete group fibers. It is exactly what identifies the left coefficient with the right one; the connected-component partition is unique. The healthy/unhealthy split routes a component to the ordinary polynomial or the flagged activity polynomial without changing its scalar value. This is a finite polynomial identity, so it holds regardless of zeros of any individual entry.

The empty activity coefficient is preserved in (OE11). A connected eligible collection with no new marker can contain no scalar entry, since each scalar entry has a nonempty old marker. Starting from any one of its activity entries, a connection to a second entry would have to use a group marker or intersect a flag support. There is no group marker, and distinct old empty-marker activity entries have disjoint prescribed flagged supports. Thus the collection consists of a single old empty-marker activity entry. Conversely each such entry is eligible and contributes its original value and its original support-and-flag label. This bijection proves preservation of the empty coefficient before any envelope estimate. After restoring the nonzero background, (OE11) is the served-polynomial identity used to construct the new local functions.

The sitewise scalar-entry estimate in (OE10) can also be read directly from this inventory. At a site there are at most singleton scalar entries, each of weighted size at most, and a singleton support has cover number at most one. Their contribution is at most The nonempty coefficients of the ordinary polynomials contribute at most. The two healthy/unhealthy copies are each bounded by this same envelope. Activity coefficients contribute at most. Extending the supports and reallocating to representatives gives the 1 + factors already described and the occurrence factor, proving (OE10) with no omitted unit term.

For precision, the eligibility and connectedness in (OE11) are as follows. An old marker set is retained byexactly when it is a union of complete fibers. Its new marker set is ∆(). For any old monomial,

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(
ζ∆(A)a,
A = ∆−1(∆(A)),
0,
otherwise.
R∆(zAa) =

For a product of retained monomials the new marker sets are disjoint exactly when the old ones are disjoint: an overlap in a new group would force its entire nonempty fiber into both old sets. Thus reading the full-fiber coefficients commutes with such eligible products.

Connect two inventory entries when their group rosters meet, or when their extended supports meet and both entries are of the type that must be collected into an activity. The latter types are the unhealthy scalar entries and all old activity entries. A grouped ordinary entry is a nonempty connected eligible collection containing no such type. A grouped activity entry is a connected eligible collection containing at least one old activity entry, with its total flagged intersection prescribed to be the union of the old flagged subsets. Its coefficient is the product of the inventory coefficients when this flag condition holds, and zero otherwise. The old activity supports inside a collection are required to be pairwise disjoint, as they were in the original gas expansion.

Partition an eligible old collection into components for this relation. A component containing an unhealthy scalar entry but no activity entry has zero contribution whenever the old flag pattern is selected: the unhealthy entry requires a flag in its extended support, and that flag must belong to an old activity support, which would connect that activity to this same component. Thus every contributing component is either an ordinary entry or a grouped activity entry as defined above. Conversely, a selected collection of grouped entries has disjoint group rosters, and its activity supports obey the selected disjoint-support pattern. Its expanded components therefore have no additional connections to each other. Taking their union and then taking components are inverse operations. The union of the old flagged subsets equals the union of the grouped flagged subsets. Finally, under the selected pattern, a grouped activity’s own flag condition holds separately: a flag in its support assigned to another grouped activity would contradict disjointness of the grouped activity supports. The coefficient product is unchanged by this bijection. Summing proves (OE11).

Locality and measurability also survive this construction. Every grouped support contains its mentioned representatives. Each entry depends only on the original variables and flag pattern in its extended support; the product depends only on their union. Its flag indicator is measurable and gauge invariant, and every underlying coefficient is measurable and gauge invariant. A nonzero ordinary grouped entry is healthy on its entire support, since all its scalar entries are healthy there. A nonzero activity entry has exactly its recorded flag intersection. Thus the new ordinary and activity polynomials satisfy the same support and flag conditions as the old ones, enabling the next grouping step.

For the final identification of one-step transport, extract each singleton coefficient of the fine served polynomial. The singleton restoration law identifies it with the normalized local insertion, including its dominated activity correction. The local analytic transport estimate gives at the anchor one coarse local function and a dominated coarse activity profile, with the same leading counterterm and remainder multiplied by the local driving constant. Define the functions in every other copy by the prescribed reflection and translation of this anchor function; reindex the activity labels by the actual activity-motion bijection. The conditional integration intertwines these transformations, so their normalized insertions agree with the corresponding singleton coefficients of the pushed polynomial. Coefficient extraction commutes with this integration, since the fine polynomial is integrable. The marked transport theorem then gives the full coarse activity representation at weights, not just singleton identities. Finally the support substitution estimate with target weights, the occupancy bound, and the one-step duty inequalities restores the served representation at. These are exactly the analytic transport, singleton cancellation and served restoration estimates developed above, with occurrence set. No two-element property of the earlier occurrence set is used: all coefficient and support estimates are stated for an arbitrary finite marker set. This completes the transport identification required in the sweep.