Extension across collision sets

Extension across collision sets

Extending a separated correlation across collisions requires a choice of subtraction device. Taylor jets normal to a collision set are subtracted on dyadic shells; the cancellation-bearing shell expressions then have an absolutely summable bound. Forests organize overlapping collisions with finitely many branches at a fixed degree and arity. The construction proves compatibility for the chosen devices and finite cutoffs; independence of arbitrary subtraction prescriptions is not an additional conclusion.

The additional contact conclusion concerns Euclidean distributions, constructed with a specified subtraction system using the same reader prescription. Its scope is the following. Restrict the -point configuration space by a real linear isometry, where is finite-dimensional and carries its Haar volume. A collision group specifies a subset of the labels and its collapsed configurations. The permitted forest consists of compatible collision groups together with local continuation charts, support windows, finite Taylor orders and projection choices. On every admissible such forest, the construction supplies an integrable Taylor-subtracted pairing for a compactly supported smooth test function in the chart. Its expansion into the prescribed annular pieces is absolutely summable in integral norm and sums to that pairing. Iterated expansions over any finite ordered list of centered collision groups have the same property. On annuli away from total collapse the result agrees with the punctured correlation.

The public endpoint also bounds the resulting distribution on every permitted isometric collision plane containing the collision directions, at the stated finite total test order. The subtraction devices carry the actual rotation action of the reconstructed hierarchy, and their finite projection basis is exactly the reader prescription used for the OPE. A real system basis and compatible positive threshold choices exist simultaneously. These contact assertions must therefore be proved with these specified devices and admissible forest conditions; they are not a claim of arbitrary subtraction-independent contact values.

For a nonempty label set, its normal displacement space in the full configuration space is

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Nfull
g
= {x : xj = 0 (j /∈g),
X
j∈g
xj = 0}.

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Its orthogonal complement contains the collapsed configurations of the group, including its centroid.
In E use Ng = P −1Nfull
g
. The admissibility assumption is Nfull
g
⊂ran P; hence P identifies these
normal spaces isometrically. Omitting any one member of g identifies Nfull
g
linearly with (R4)|g|−1:
the omitted coordinate is minus the sum of the others. In particular

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dim Ng = 4(|g| −1) =: νg.
(CE1)

For disjoint groups the normal spaces are orthogonal. Centroids of two disjoint nonempty proper groups are different almost everywhere in these normal variables: their equality is the kernel of a nonzero linear map. It is nonzero by placing a vector at one member of each group with opposite signs. Its proper linear kernel has Haar measure zero. This permits the subsequent almost everywhere shell identities without assigning a correlation directly on a coincidence plane.

Write for orthogonal projection onto. The degree- normal Taylor term of a smooth test is

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Tg,jf(x) = 1
j!Djf(x −πgx)[πgx, . . . , πgx].

For a fixed positive subtraction length, choose the prescribed smooth radial cutoff), equal to one near zero and zero for. The subtraction operator through order is

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m
X
Tg,s,mf = χs(πgx)
j=0
Tg,jf,
Rg,s,m = 1 −Tg,s,m.
(CE2)

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Tangential coordinates remain parameters in this Taylor expansion. The subtraction depends only
on the normal jets through m along N⊥
g , and vanishes on a test supported away from that subspace.
Its support is contained in
{x : x −πgx ∈supp f, ∥πgx∥≤2s}.

These facts follow directly from the displayed derivative formula and the cutoff. Orthogonal groups have commuting Taylor operators: their normal projections annihilate one another, their normal derivatives commute, and their cutofffactors depend on different normal variables. Thus their remainders commute as well.

The finite field projection is combined with (CE2), rather than subtracting every component equally. For input degrees, set

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ωg =
X
j∈g
dj,
Lg = max(0, ωg + 1 −νg).

Resolve the group’s field expression into projected components of degrees1 and its remaining component. The remaining component is paired with the original test; the degree- projected component is paired with the normal Taylor remainder through order. Denote these finitely many choices by, with gain for the remaining component and for a projected component. After normal integration the effective scale exponent is

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ev = νg + Gv −ωg.
(CE3)

For the remaining component1, including when = 0. For a projected component, implies

At a relative shell radius1, Taylor’s remainder contributes, leaving one full power:

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aevaLg−h = a.
(CE4)

For 1 the cutoffestimate needs only, so the projected product is exactly 1. The remaining component can meet such a shell only up to the fixed geometric reserve, where is bounded. This near/far distinction is the central integrability gain.

To make this argument uniform under further collisions, a local chart records a partition of the current label group into disjoint nonempty parts, a positive length, a collapsed offset, a linear resizing frame, and a closed stage inside a fixed bounded set in resized coordinates. Each part has an assigned radius 1]; the frame rescales its normal directions by that radius. The whole normal extent and its subtraction halo fit a common detector collar 1], with 2 For every nonsingleton part, is at most a fixed multiple of. The detector requirement is

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

X
j
dj
.
B ≤clearance

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The clearance is chosen decreasing with degree, so it also covers every projection degree used below.
Other groups, their centers and their field expressions are read by a continuation constant along
each current part’s normal directions; this permits use of ancestor-subtracted expressions as the
exterior continuation, with their cancellations retained.
Refine one nonsingleton part i into a proper partition U on an angular annular patch. Assign
radius a to the new parts and leave the others unchanged. Replace the frame by its composition
with the map multiplying Ni by a/Ri and fixing N⊥
i . The determinant changes by

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| det A′| = | det A|(a/Ri)νi,
(CE5)

and the engineering field factor changes by (. The gain in the continuation contributes , so their product is precisely (. Supports in the Taylor branch may enter its subtraction halo; this is why the refined route retains both the old stage and its normally collapsed, cutoff-bounded part. Orthogonality keeps the other groups’ normal data unchanged. Their continuation remains locally continuous because the Taylor operation is applied within its detector collar and the refined angular patch keeps the distinct part palettes disjoint.

Here is the uniform integrated estimate. Let

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τQ = 3 |g|−|Q|,
εQ = τQγ,
γ > 0,
(τQ + 1)γ ≤1.

A proper refinement increases the number of parts, so

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2εQ′ + γ ≤εQ.
(CE6)

The local base budget for degree list, continuation bound, frame, and radii

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Bw(ε) = | det Aw| Bw
Y
i∈Q
Ri(w)−ωi Y
j
Rj(w)−ε.
(CE7)

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The last product includes all recorded radius registers, including unchanged ones. If selected parts
are restricted to dyadic shell numbers ni, multiply this budget by Q
i∈Q, |i|>1 2−niγ; an unrestricted
part contributes 1. The number of required test derivatives is

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NQ = (2 +
X
j
dj)(|g| −|Q|).

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For every permitted choice of projected or remaining components and shell restrictions, the complete
resulting integrand F is integrable and
Z

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E
∥F(x)∥dx ≤C ∥f ◦Aw∥CNQ Bw(εQ)
Y
2−niγ.
(CE8)
i∈Q
|i|>1

The constant is uniform in the permitted charts and parameters, after fixing the geometric reserves, degree bounds, and continuation estimates. In (CE8) the absolute value is taken after the finite subtraction expressions have been combined.

The induction proving (CE8) uses (CE3)–(CE6). Under refinement the child’s budget is the parent’s budget at, times (. The Taylor estimate supplies the extra factor from (CE4). For every 01 their product is at most a fixed reserve constant times (: use (CE4) when1, and the bounded far-shell estimate otherwise. Choose1. Then

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(a/Ri)γ+εQ′a−εQ′ = aγR
−(γ+εQ′)
i
.

The last loss is absorbed by the parent’s radius surcharge using (CE6). The other shell factors are unchanged. Thus each new dyadic annulus supplies, and summation over its annulus index converges. A proper partition strictly decreases; this supplies the finite induction and the stated derivative budget. Here are the shell and angular constructions used in the induction. Put

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ψg,a = χa(πgx) −χa/2(πgx),
an = 2−n.

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The support of ψg,a lies in a/2 ≤∥πgx∥≤2a, and
X
n<N
ψg,an = χ1 −χaN .
(CE9)

For a test supported where1, at every point with= 0 the right side times the test equals the test in a neighborhood of that point for all sufficiently large. This is an eventual equality of germs, including all derivatives that enter the finite Taylor operations.

On the compact unit normal annulus, partition the labels by equality of their positions. Total collapse is absent, so this is a proper partition. Within each part all subcentroids coincide, and subcentroids belonging to different parts are distinct. The latter condition is open and there are only finitely many proper partitions and subgroups. These open sets therefore cover the annulus. Choose a smooth finite partition of unity subordinate to this cover. Multiplying its rescaled angular factors by gives masks whose sum over proper partitions is. On each mask support, all subcentroids in different parts have distance at least, with 0: at radius one take the positive minimum of their continuous distances on the compact support, then rescale. This proves all the reader separations needed after the refinement, even for subgroups used later.

The normal Taylor bound in these coordinates is uniform in the subtraction length. On a fixed compact unit annulus, if has bounded derivatives through + 1, normal rescaling by gives

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(
am+1,
0 < a ≤1,
am,
a > 1.
(CE10)
∥Rg,s,mf∥Cn after rescaling ≤C∥f∥Cn+m+1

Here the parent frame has normal scale, and the allowed parameters satisfy with the fixed normalized support bound. To prove (CE10), bound the family of cutofffunctions , and all derivatives through on the compact annulus; smoothness on the compact product gives one constant. The normal Taylor polynomial after rescaling is bounded by a constant times max(1. The rescaled original function has a uniform derivative bound when it meets its compact support, and is zero as a germ when is beyond that support. Leibniz’s rule gives the bound for 1. For1, the cutoffis identically one near the annulus and the integral Taylor remainder gives the factor. Otherwise 1; multiplying the rough

uniform bound by (3 gives the same estimate with a larger constant. The angular masks have fixed normalized derivative bounds, so their product costs another fixed constant. This proves (CE10).

We may now make the induction in (CE8) completely explicit. If every current group is a singleton there is no unresolved normal direction; use the permitted continuation bound, its local continuity and the bounded support in resized coordinates. Otherwise choose a nonsingleton part , and decompose its election into the finite choices of remaining component and projected degrees. For a fixed such choice, apply (CE9) to its test after the normal Taylor operation and then the finite angular partition. The resulting annular term equals the refined forest expression, read with the continuation that first regroups the children into and applies its fixed component projection or remainder. This is an identity of the underlying linear expression before evaluating any hierarchy or integrating.

The refined partition has more parts, so its value of is strictly smaller. Its continuation retains the original offsets, length and normal constancy. Its component fields have the prescribed palette of subcentroids; the angular separation and common detector collar ensure the projection is admissible. The inductive bound applies. Formula (CE10) pays its test derivatives, while (CE3)–(CE7) give precisely the budget

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Cv∥f ◦Aw∥CNQBw(εQ)aγ
n
Y
2−njγ.
(CE11)
j∈Q
j̸=i

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The number of angular sectors and component choices is finite; summing their constants preserves
uniformity. If the specified annular stage is empty, the term is zero and the same estimate holds.
Otherwise its geometry is exactly the refined route already described.
Since P
n aγ
n = (1 −2−γ)−1 < ∞, (CE11) proves summability of the integral norms of the
complete annular terms. The eventual germ equality from (CE9), together with the proper-normal
exceptional sets of Haar measure zero, identifies the pointwise sum almost everywhere with the
unsliced component expression. To pass to its integral without any unproved domination, regard
each annular term as an element of the Banach space L1(E; X). The sum of their L1 norms is
finite, so they sum in L1 to an integrable function. Absolute integral-norm summability also gives
their almost everywhere pointwise sum; uniqueness and the eventual identity identify that function
with the prescribed unsliced expression. Continuous integration then gives

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Z
Fv =
Z
Fv,n,
X
Z
∥Fv,n∥< ∞.
(CE12)
∞
X
n
n=0

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This proves the bound for the unsliced choice.
For a specified shell index use (CE11) directly.
Summing the finite component choices establishes (CE8), completing the finite induction.
If the election at i is the whole expression, all its choices occur, so (CE12) yields the exact root
formula
Z
F =
X
∞
X
Z
Fv,n.
(CE13)

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v
n=0

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For any ordered list of distinct nonsingleton current groups with no preassigned shells, one can
resolve their shells successively. The common bound becomes a constant times Q
j 2−njγ. This is
summable over the Cartesian product of all shell indices. Repeated use of (CE12) and absolute
Fubini therefore gives the full iterated sum, equal to the original integral.
In particular every
permitted resolution order gives that same pairing. This does not assert absolute convergence of

the individual uncombined Taylor monomials; the terms in (CE12) and (CE13) are the already combined expressions.

For any requested loss 0, choose = min(1 + 1). Then the economy condition holds and. On unrestricted shells the shell factor is one, and monotonicity of for 0 upgrades (CE8) to the requested arbitrary positive loss. Thus all shell identities and bounds apply with the same finite projection choices and devices.

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We verify the continuation inheritance required in that induction. Its hypothesis is a tested
rate inequality on every logarithmic path through the allowed chart: if the scalar continuation
price has rate b, and each current group expression is affordable at its assigned radius with rate
ci, the reading of their tagged product has rate at most b + P
i ci. The continuation is also locally
continuous on the specified palettes and constant in the normal directions already assigned to each
group. These properties hold for the actual hierarchy with separated exterior fields by the signed
expression estimates described below.

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For a refinement of i at radius a, suppose a has rate −s and its parent’s recorded radius has
rate −t. The angular separation gives positive lower bounds of rate −s between all palettes of
different child parts. The finite product rule for affordable expressions therefore gives the product
of the children rate P cj on the parent palette at exponent s. Applying the remaining component
or degree-h projection at the parent’s center changes this to
X

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j
cj + (t −s)Gv.
(CE14)

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For the remaining component the chart’s support forces t ≤s; use the full finite projection remain-
der estimate through Li−1, giving Gv = Li. If Li = 0, there is no projection and the same estimate
is the coarsening inequality with zero gain. For a projected component, the degree-h readout esti-
mate supplies Gv = h. The common detector collar justifies these operations on the actual moving
palettes.
The new scalar price is exactly
B′
w = Bw(a/Ri)Gv.

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If Bw has rate b, this has rate b + (t −s)Gv, matching (CE14). On the subtraction halo, choose
a point of the original stage differing from the present point only in Ni. Such a point exists by
the definition of the halo; for the remaining branch it is the point itself. Constancy of the original
continuation in Ni permits evaluating there. Other current parts are orthogonal to Ni, so their
palettes and expressions are unaffected. Substituting the child product with its fixed component
treatment into the old tagged product is exactly the refined reading. The original path inequality
now yields
b + (t −s)Gv +
X
refined parts
cj,

the required refined continuation bound. This proves the inheritance used in (CE11). The local continuity assertion follows by the same substitution on two closed pieces, the original-stage piece and the normally collapsed-stage piece of the halo, and gluing their agreeing restrictions. Thus the auxiliary continuation is justified at each step, including expressions retaining earlier Taylor cancellations.

We next obtain a distribution on the whole allowed position space. At each, partition the labels by equality of the actual positions. Each part is collapsed at; its normal displacement is zero. All subcentroids of different parts are distinct. These conditions persist on a sufficiently small closed ball about: all part normal displacements remain inside the common collar, and all distinct-part subcentroid distances have a positive lower bound. Crucially the same positive

subtraction length, with 2, works on each of these balls. Applying the local forest estimate to the equality partition gives its integrable test pairing. The exact expression overlap identities identify that pairing with the single root prescription on tests supported in the ball, so no new extension values are assigned on chart overlaps.

For a compact test support, choose finitely many such balls covering it and a smooth partition of unity subordinate to their interiors. On every chart the test estimate uses at most

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N = (2 +
X
i
di)|g|

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derivatives, since NQ ≤N. Leibniz’s rule gives ∥θjf∥CN ≤CK,j∥f∥CN . The root prescription is
linear in its test, and P
j θjf = f. Add the finitely many local integral-norm bounds to obtain

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Z
E
|Ks(f)(x)| dx ≤CK∥f∥CN
(supp f ⊂K).
(CE15)

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The linear functional
Ss(f) =
Z
E
Ks(f)(x) dx

is therefore continuous on each fixed-compact-support smooth test space, and hence on the usual compactly supported test-function space. This is the asserted Euclidean distribution of finite order . Its restriction to a local forest is the local pairing, because both are readings of the same test expression. The construction works for every unselected or character reader. The extension and its bounds retain the chosen subtraction length, finite readouts and normal Taylor convention.

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We specify the finite expression prescription, including its recursion. A score is a finite sum of
field expressions tensored with linear operations on smooth tests; its action at x multiplies the field
expression at x by the operated test at x. Products compose test instructions in their prescribed
order.
For disjoint label groups those instructions commute, by orthogonality of their normal
spaces.
Let Cg denote the recursively constructed local correction, and let Pg be the punctured score
formed from proper partitions:
X
Y
Pg =
Ci.
i∈Q

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Q partition of g
Q̸={g}

For a singleton, is its actual field expression tensored with the identity test operator. For 2, set

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Lg−1
X
Cg = −
h=0
reshape(readerg,h)Pg Tg,s,Lg−1−h.
(CE16)

The full score is the sum over all partitions, or equivalently. Every group appearing in a proper partition has strictly smaller cardinality, so (CE16) defines a finite recursion. The reader in this formula is the fixed physical projection at the group centroid; it acts on the field expression, while the right Taylor factor acts on the test. For = 0 the sum is empty and there is no local correction. Formula (CE16) is the exact prescription whose finite rearrangement gave the component decomposition preceding (CE3).

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Every Cg depends on the test only through its germ at the normally collapsed point x −πgx.
This follows from (CE2) in each summand of (CE16); for a singleton its normal space is zero. Now
compare two partitions Q, R which are both separated on a test support. Expanding Q
i∈Q Si gives
precisely the terms Q
j∈T Cj over all partitions T refining Q: choosing a partition separately inside
every part of Q is bijective with choosing a refinement. The same holds for R. A term refining Q

but not has a group containing labels in two different parts of. At its normal collapse those labels coincide, whereas they are separated everywhere on the test support. Thus the test vanishes in a neighborhood of this collapsed locus, and its germ-dependent factor annihilates it. All such crossing terms vanish as scores after multiplication by the test mask. Both expansions therefore reduce to the common refinements of and. Acting on the test proves their equality before hierarchy evaluation and integration. This proves the overlap identity used in (CE15), including partitions not subordinate to one another.

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The signed expression bounds supplying the initial continuation estimates follow a finite proof
tree. Its leaves are bounded labels of degree d at one position, with price R−d, and the unit with
price 1. One may multiply two already priced expressions when their position palettes are separated
by a positive multiple of their common radius; their prices multiply. One may carry an expression
from radius R to R′ when its palette has diameter O(R) about its chosen center and R = O(R′).
Subtracting its full projection through degree h multiplies its price by (R/R′)h+1, with the same
size condition and the fixed reader clearance. Projecting onto just degree h multiplies its price by
(R/R′)h and replaces its palette by the center alone; no ratio bound is required for this projected
operation. Palette enlargement is also permitted.

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Along any logarithmic path where the finitely many radii have rates, induction on this tree
proves the corresponding affordable-expression rate. The leaf uses the full field bound, multiplica-
tion uses separated products, carrying uses coarsening, and the two projection steps use the full
tested remainder and output estimates with the displayed gains. All use the same hierarchy and
reader, and the remaining component is kept as its signed difference. To obtain a uniform bound
in the radii, normalize their logarithms as in the nested expansion proof and pass to a compact
refining limit. The product surcharge Q R−ε
supplies a strict positive rate margin against any
j
proposed violation. Hence an expression of price p satisfies

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|⟨E⟩χ| ≤Cεp
Y
j
R−ε
j

for all admissible geometries and all viewers. Testing its output at degree gives the additional factor. This proves the signed moment and projection bounds for every finite expression tree. It also supplies the initial continuation bound with separated exterior words: their palettes have fixed positive clearance, their label norms are bounded, and their radius can be set to one. Their full expectations and local continuity are those of the previously constructed hierarchy.

The same construction permits an extra projection or remainder at the root whenever its full palette lies in the required root detector collar. Its scalar continuation price is for degree output and for the remainder through degree. These operations are applied after the root’s field expression has been formed, and before the lower forest is tested; (CE14) shows that the subsequent shell induction keeps this root price intact.

Consider now normalized normal tests

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fR(x) = R−νgf
x −πgx + R−1πgx

.
(CE17)

The normal frame is multiplication by on and identity on its complement. Its determinant is ; composing with that frame gives, so the determinant and test normalization cancel exactly. In a one-root chart, (CE7) consequently becomes

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Ru−ωg−ε,

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where u = 0 for the full reading, u = h for degree-h root output, and u = h + 1 for its root
remainder. Thus each complete contact pairing obeys
Z
∥Ks(fR)(x)∥dx ≤C∥f∥CN{g}Ru−ωg−ε
(CE18)