Extension across collision sets · Part 2

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component decomposition used by CE3. Each Cg depends only on the test germ at x −πgx by
CE2; singleton normals are zero. For partitions Q, R both separated on a test support, Q
i∈Q Si
expands into corrections indexed by partitions refining Q, since choices of partitions inside its
parts are exactly such refinements. A term not refining R has a group crossing two R parts. At
its collapse those labels coincide, whereas they remain separated on the test support. Its germ-
dependent factor annihilates the test, which is zero near that collapsed locus. All such crossing
terms vanish as scores after multiplication by the test mask. Both expansions therefore reduce to
common refinements, as masked scores before evaluation and integration. This proves the overlap
identity even for nonnested Q, R used in CE15.

Finite score recursion and global distribution recovery. For a test instruction its horizontal transport by . A score summand is transported to using the corresponding horizontal field section and the same tagged product. Denote this finite linear transport by. Its evaluation satisfies

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evZ(LA, fQE)(JEx) = UX ev(A, f)(x),
(D1D2)Z = DZ
1 DZ
2 .

Thus the order of test instructions is preserved. For disjoint groups the transported normal projections and Taylor operations commute, exactly as in the original score. The centroid field reader and its reshape are transported with the same fixed homogeneous output spaces and basis. Induction on then gives

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CZ
g = LCg,
PZ
g = LPg,
SZ
g = LSg.

The base is the actual singleton field and identity instruction. Every proper partition contains only smaller groups, and CE16’s finite sum uses the already intertwined centroid reader and normal Taylor instruction. This proves the inductive step, including the empty correction when also recovers the minus sign and normal jet order separately for every component.

For separated partitions, let mean that refines. On their common test support the exact masked-score identity is

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



Y
!
Y
i∈Q
Si



X
Y


[f] =
[f] =
j∈T
Cj
i∈R
Si
[f].
T⪯Q
T⪯R

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Here brackets denote action on the test before hierarchy evaluation. The bijection between choices
of partitions inside each part and refinements accounts for every term once. A crossing correction
annihilates the vanished germ at its normal collapse. Transport preserves that germ and the support
separation, so this same identity holds for L of each score. No nesting relation between Q and R
is required.
On a compact support K, transport the actual finite chart cover and its smooth masks by JE.
Let Aj be the local integral-norm constant for chart j, after fixing its frames and devices. The
common test order N = (2 + P
i di)|g| gives explicitly

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

∥Ks(f)∥L1 ≤
X
j
Aj∥θjf∥CN ≤
X
j
AjCK,j
∥f∥CN .

The first inequality uses linearity of the single root prescription; the overlap identity identifies each chart computation with it. Consequently, defining the horizontal root integrand by the lifted score,

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one has

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KZ
s (fQE)(JEx) = UXKs(f)(x),
SZ
s (fQE) = UXSs(f),
QXSZ
s (fQE) = Ss(f).

The same fixed-support seminorm bound proves continuity of the lifted functional on each test space and hence on the compactly supported smooth test-function space. Its order remains at most . Forest restriction commutes with these identities since it uses the same test score. This recovers the unselected reading and each character reading with the fixed length, projections and normal Taylor convention retained in the contact source.

For the initial continuation, retain a finite signed-expression proof tree. Leaves are bounded degree- labels priced and the unit priced one. Products require palette separation by a positive multiple of common radius and multiply prices. Carrying to is allowed for diameter) around the center and). Subtracting full projection through costs ( with that size condition and fixed reader clearance. Degree- projection costs (, collapses its palette to the center and requires no ratio bound. Palette enlargement is permitted. Along logarithmic paths the full field, separated product, coarsening and tested projection/remainder estimates prove each price by finite induction, retaining a remainder as its signed difference. Compact normalization of all log radii and the strict surcharge margin give the uniform bound

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|⟨E⟩χ| ≤Cεp
Y
j
R−ε
j .

Output testing at degree adds. Separated exterior fields have fixed clearance and bounded norms with radius one, so these bounds and hierarchy continuity supply the initial continuation. An additional root projection or remainder is allowed when its whole palette fits the root detector collar; its price is or respectively. Apply it after forming the root expression and before testing the lower forest; CE14 preserves that price under shell refinement.

The lifted global contact source retains its finite score recursion, test germ instructions and local equality partitions. Its overlap reader is equality of masked scores, so partition-of-unity assembly recovers one root prescription. CE15 proves continuity of its integral readout with the original finite derivative order. The signed proof tree and exact price update preserve previous cancellations and root operations; the contact distribution is recovered with these chosen devices, rather than from punctured values alone.

Signed-expression price recovery. Retain the finite expression tree together with each radius register, center, palette and price. The horizontal field section intertwines the actual fixed projections: for a degree reader Π and its finite sum Π

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ΠZ
h J = JΠh,
(1 −ΠZ
≤h)JE = J(E −Π≤hE).

Products use the recorded separated palettes; carrying and palette enlargement retain the same physical positions and centers. Thus induction over the finite tree recovers its expression and price at every node. In particular the price updates remain exactly

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p(E1E2) = p(E1)p(E2),
p(rem≤h E) = p(E)(R/R′)h+1,
p(ΠhE) = p(E)(R/R′)h.

These are the assigned prices on the permitted operations, not norm identities. The remaining component is the displayed signed difference; its two pieces are not priced separately. The remainder operation retains) and clearance, whereas the degree projection retains its original absence of a ratio bound and its collapsed center palette. The viewer map is transported by

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the same ground-algebra isomorphism, so its scalar evaluation is unchanged.
On any allowed
logarithmic path, both the quantity and assigned bill therefore have the same rates before and
after lift. Normalized logarithmic compactness and the same full surcharge Q
j R−ε
j
recover the
stated uniform all-viewer bound. Exterior words at radius one introduce no new radius loss. A
root output or remainder is retained in its original position in the tree, after forming the root field
and before testing the lower forest; its Rh or Rh+1 price is preserved by CE14.

21.5 Normalized contact scaling and compatible finite cutoffs

For normal scaling retain the normalized test

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fR(x) = R−νgf(x −πgx + R−1πgx).
(CE17)

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The frame multiplying Ng by R and fixing its complement has determinant Rνg, while fR composed
with it is R−νgf. These cancel exactly. The one-root budget becomes Ru−ωg−ε, where u = 0 for
the full reading, u = h for degree-h output and u = h + 1 for remainder through h. Thus
Z
∥Ks(fR)(x)∥dx ≤C∥f∥CN{g}Ru−ωg−ε.
(CE18)

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It is uniform on permitted normalized supports and chart parameters. The subtraction length
can be fixed or vary under the same halo, support and clearance conditions; it need not equal
R. If f vanishes near the total diagonal N⊥
g , its normal jets annihilate Cg in CE16. Hence full
and punctured scores agree on these tests even if proper-subcluster collision sets are met. Every
positive normal annulus qualifies, also after scaling, with no uniform lower bound on its hole. CE18
therefore bounds full, output and remainder punctured pairings with the same constants.

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Normalized normal-test recovery.
Set SR = (1 −πg) + Rπg, so S−1
R = (1 −πg) + R−1πg and
fR = R−νgf ◦S−1
R . On the horizontal copy use SZ
R = JESRQE. Then

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(fZ)R = fRQE,
(fZ)R ◦SZ
R = R−νgfZ,
| det SZ
R| ∥(fZ)R ◦SZ
R∥CN = ∥f∥CN .

The determinant is taken in the transported orthonormal frame; it is Thus the one-root volume, normalized test and scalar continuation price combine as

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Rνg R−νg RuR−ωgR−ε = Ru−ωg−ε.

This recovers CE18 for each of its three root readers with the same finite derivative count, permissible support and subtraction length. Moreover preserves and is invertible for each A test vanishing in a neighborhood of that subspace retains this property under normalization. Its normal jets vanish, so CE16’s root correction vanishes exactly and the punctured and full pairings coincide. Proper-subcluster collisions may remain in the support; they are treated by the retained lower corrections. The recovery uses only a positive annulus hole at each test, with no uniform lower bound on it and no requirement

Choose real bases simultaneously in homogeneous contact grades. This is a coordinate basis for the strata of the already fixed splitting; it does not replace the OPE prescription or its projection functionals. The strata are invariant under conjugation and rotations since their lower-grade maps intertwine them. Conjugation-fixed vectors span each stratum by real/imaginary decomposition; a maximal independent real subset is a finite complex basis. In degree zero start with the nonzero real unit and extend it. Expand any field into its finite homogeneous parts and basis coordinates.

Multilinearity extends the score from these basis inputs, retaining each actual homogeneous degree in CE16. Higher ambient horizons change none of these parts, so this linear extension is well defined. For positive requested bounds choose

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gn = min

fn, 2−n, clearance(n)

,
t0 = g0,
tn+1 = min(tn/8, gn+1).
2(n + 1)

Then0 and 2(clearance(). Take) = 1, and the prescribed smooth 0, equal one for and zero for. They are locally finite for = 0 with, strict threshold spacing forces) = 1. The last active index) exists and tends to infinity as0. Active 0 obeyclearance(), since. For fixed this gives eventuallyclearance(

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N(s)
X
Ds(x)E =
j=0
bj(s) insertRowx,j E.
(CE19)

Local finiteness gives smoothness for positive, also when the last index changes. Biorthogonality gives exactly row) row

For all rows through have weight one, so projection through is precisely the original finite reader. Slowing thresholds to meet any further positive preserves these identities. No convergence in the growing horizon is asserted. The affordable-expression rules, logarithmic compactness and mixed-label inclusions used here are the UC1–UC3 constructions below.

The lifted normalized-test source includes the normal volume and scaling frame; its integral reader cancels the Jacobian against CE17 exactly. Restriction away from total collapse recovers the punctured score while retaining proper-subcluster corrections. The basis and threshold source retains every finite row and its biorthogonal insertion. CE19 recovers each fixed finite reader exactly below its threshold, so a variable cutoffintroduces no replacement of the finite reader assumed in the contact estimates.

Real contact bases and fixed-row cutoffrecovery. Write for the conjugation on a homogeneous stratum. Its horizontal transport is . For every in that stratum,

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v = vR + ivI,
vR = v + κv
2
,
vI = v −κv
2i
,
κvR = vR,
κvI = vI.

The same formulas on the horizontal copy give and. A maximal complex-linearly independent subset of these fixed vectors spans, since otherwise the displayed decomposition supplies another independent fixed vector. Transport therefore gives the identical finite real system basis, including the unit in degree zero. The rotation representation transports on the whole stratum; individual basis vectors may mix. For a finite homogeneous expansion in each input, multilinearity gives the exact finite basis sum of scores. Applying to that sum recovers the original contact score term by term, with the assigned homogeneous degrees unchanged when its ambient horizon is raised.

The positive sequence is part of the retained source data. Inductively 0 and8, proving all its stated spacing, clearance and decay conditions. If 0 and = 0, then

so) = 1; for = 0 this follows from = 1. On a compact positive interval 0, all indices with 2 vanish identically, which proves local finiteness and smoothness without differentiating the last-active-index function. For each fixed activates every index through; hence . An active 0 satisfies more precisely

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ks < 2ktk ≤
k
k + 1 clearance(k) ≤clearance(k).

For any fixed positive, eventually andclearance( Let denote the fixed degree- row and its biorthogonal row insertion used in CE19. Transport both maps with the same horizontal sections of their domain and output. Their composition identity remains = 0 for and the prescribed degree- row for. Because the cutoffsum is locally finite, exact map transport gives

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DZ
s (x)J = JDs(x),
rZ
x,nDZ
s (x)J = Jnbn(s)rx,n,
ΠZ
≤nDZ
s (x)J = JΠ≤n
(0 < s ≤tn).

Here is the degree- output section and the last identity uses the fixed finite projection on its indicated domain. Thus every finite reader is recovered at an actual threshold, with no limit in the growing horizon required. Any additional positive requested bounds are imposed in before the same recursion; they retain all these exact row identities.

21.6 Uniform subpower bounds and arbitrary-label contact scores

Let 1] be the finite radius registers on a configuration set, with positive bill and normed reading. Assume that on every nontrivial filtered sequence with increasing log1, the bill has an exact rate and has upper rate at most. Then for every 0 one finite constant gives

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m
Y
a=1
Ra(w)−ε
(w ∈U).
(UC1)
∥F(w)∥≤Cq(w)

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If not, select violations with C = en. Put pn,a = −log Ra(wn), Sn = P
a pn,a and Tn = n + Sn. On
an ultrafilter refining the sequence tail, compactness of finitely many coordinates gives pn,a/Tn →sa
and n/Tn →k, with k + P
a sa = 1. The path hypothesis gives upper rate c for F, whereas the
violating comparison quantity enq(wn) Q
a Ra(wn)−ε has exact rate c+k+ε P
a sa > c; the increment
is positive because k and all sa are nonnegative and their sum is one. A quantity of strictly larger
exact rate eventually exceeds one of smaller upper rate, contradicting every chosen violation. The
added n covers radii bounded away from zero. All bills here are generated by constants, positive

radii, products and fixed real powers and thus have exact rates. No uniform intermediate OPE horizon or threshold across paths is required to obtain this ordinary uniform layer bound. Let be the fixed homogeneous splittings. At fixed chart, partition and route the homogeneous contact score is multilinear before integration: its grade-transport identity, with other inputs fixed, is a linear map. Hence addition and scalar multiplication hold in that slot for the full expression-valued score. For arbitrary labels set

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Bd(f1, . . . , fk) =
X
0≤ai≤di
A(ai)(M(d1)
a1 f1, . . . , M(dk)
ak
fk).
(UC2)

For homogeneous inputs only their own grade tuple survives, recovering the original score. Raising caps embeds every original tuple () in the enlarged finite index set and preserves it by splitting/

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inclusion compatibility; each added tuple has some ai > di, whose component is zero. Multilinearity
kills that term. Thus cap independence holds at expression level. On distinct-configuration test
supports, each homogeneous contact image is its ordinary field product times the test. The exact
identity
d
X

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a=0
inserta(M(d)
a f) = insertd(f)

and finite product expansion recover the mixed-label word. Allowed spectator evaluation gives the separated correlation times the test. Per-grade integral bounds permit termwise integration of the finite sum, giving the full field-space extension with its prescribed separated restriction. The lifted uniform-bound source retains the actual radius-power bill, clock and all path hypotheses; the compact violation argument recovers one finite UC1 constant without selecting a different reader for each radius. Its mixed-label source retains the coherent splitting and the finite score tuple. UC2 commutes exactly with cap inclusion and separated readout; finite integration preserves these identities. Consequently extension to arbitrary labels loses no homogeneous component or assigned degree.

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Uniform-layer and mixed-label recovery identities.
For a radius-power bill q = C0
Q
a Rβa
a
with C0 > 0, its exact path rate is

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log q
T
−→−
X
a
βasa = c.

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The section retains every named radius and this positive bill. The horizontal norm of the lifted
reading is the original norm, so both sides of UC1, as well as any selected violation, recover exactly.
For the normalized violating array of the preceding argument the strict increment satisfies the
explicit lower bound
k + ε
X
a
sa = ε + (1 −ε)k ≥min(1, ε) > 0.

Thus the contradiction retains a strictly positive margin even when all radius rates vanish, in which case = 1. Taking the viewer as part of the selected configuration retains the same argument for varying characters and the unselected reading, under the original path hypotheses. This transports the common constant without assuming that intermediate path-dependent horizons or thresholds are uniform.

For clarity write for the horizontal maps on a capped label space and on its homogeneous output. Use the transported splitting and inclusion maps

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(M(d)
a )Z = JaM(d)
a Q≤d,
(ιd′
d )Z = J≤d′ιd′
d Q≤d.

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Their coherent composition identities follow by QJ = 1. For fixed chart, partition and route, let
the homogeneous score lift satisfy AZ
a (Ja1v1, . . . , Jakvk) = LAa(v1, . . . , vk), where L is the score
transport already defined. UC2 then gives exactly

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BZ
d (J≤d1f1, . . . , J≤dkfk) = LBd(f1, . . . , fk).

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Every sum here is finite before score evaluation or integration.
For larger caps d′
i ≥di, split-
ting/inclusion coherence makes each old summand identical and each new summand zero in a slot
with ai > di. Consequently both sides are independent of that cap enlargement as expression-valued
scores, not only after testing. For homogeneous inputs only their own tuple remains.

On distinct configurations the recovered field expression is

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

k
Y

di
X
ai=0
insertai(M(di)
ai fi)
k
Y
=
i=1
insertdi(fi).
i=1

The product retains its original order. The same finite expansion holds for transported insertions, scores and any allowed spectator. The finite sum of the per-grade integral bounds supplies integrability of the mixed score and permits termwise integration. Applying its horizontal readout therefore recovers the separated mixed-label correlation with the original test function, while retaining every actual homogeneous degree used in its contact correction.

21.7 Projection gain, tested funds and safe probes

At clock, an expression is affordable at layer, cost, if it is a finite sum of separated words whose coefficient of total weight has upper rate. Along a logarithmic clock, an upper rate means a bound by eventually for every 0. This is closed under finite sums, scalar rate addition and eventual equality, with its finite position palette. A precision- tested error, multiplied by any separated affordable spectator of terminal layer one and cost, has expectation upper rate, for the unselected and every possibly varying character. An expression has funds ) if it has a finite terminal affordability bound and, at every requested precision, agrees to that precision with an affordable expression of layer, cost. For palette radius rate and 1, one-scale OPE gives a centered finite approximation at layer to arbitrary precision: choose horizon so) pays the precision deficit. Retained output of weight has coefficient rate. This horizon may depend on path and precision; uniformity is obtained later by UC1. Finite sums and an existing tested error give this replacement for all expressions with funds. A centered finite sum with these rates has weight- row rate: lower weights vanish and higher-weight splitting uses fixed linear maps. For a general funded expression, test its error against the row’s finite detectors, rotations and conjugates, outside its fixed aperture. A field of weight below gives zero, while splitting a field of weight at least applies a fixed linear map and cannot worsen its coefficient rate. Their distances are bounded below and their layer is zero. If their fixed weight sum is probeFee(), the row error rate is at most + probeFee(); choose precision to pay that fee. The row bound and finite-projection funds follow. Arbitrary varying rotations preserve rates by finite-dimensional covariance and compactness; selecting a violating rotation would contradict this pathwise bound. Hence normalized Haar averaging, conjugation and finite decoding preserve the same rate without interchanging nonuniform limits; this proves the detector-row estimate. The projection gain underlying (UC1) can now be made precise.

For the prescribed finite projection the exact gain rule is

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Funds(s, c; e) =⇒Funds
u, c −(w + 1)(s −u); e −Π≤we

,
u ≤s.
(UC3)

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For centered sums all weights through w cancel and every remaining d ≥w + 1 obeys c −ds ≤
c −(w + 1)(s −u) −du. For general e, approximate at precision min(q, q −maxFee(w) −wu), with
maxFee(w) = Pw
j=0 probeFee(j). Projection of the error costs at most maxFee(w) + wu, so both
errors fit precision q. Subtract them and apply the finite sum result. At u = s simply subtract
the projected affordable expression with zero gain. Repeated UC3 supplies each signed ancestor
move’s full w +1 gain in the contact forest. Funds multiply across separated layer-s palettes, s ≤1,
by adding costs. For e, f with funds (s, c), (s, d) and finite terminal costs a, b, first approximate e
by g at precision q −b. Then enlarge g to terminal layer one and record its actual finite cost c′.

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Approximate f by h at precision q −c′. The identity

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ef −gh = (e −g)f + (f −h)g

puts both errors at precision, while concatenation makes affordable at cost. Induct over finite separated families. This ordered choice is essential: terminal cost can greatly exceed the outgoing layer cost, so it is recorded before requesting the next approximation; no uniform bound over horizons is imposed. The arbitrary-precision definition is exactly what permits this successive choice. For expectation against a spectator of layer, cost, first record its terminal cost then approximate at precision. Direct separated bounds and error testing both give rate . The empty spectator gives expectation rate, uniformly over characters. This closes the multiplication and expectation rules.

Tested-funds and finite-gain recovery. Lift a finite word using the same ordered labels and positions, and retain its scalar coefficient and total weight. Its coefficient rate is unchanged. For any allowed terminal spectator and transported viewer, the exact scalar identity is

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⟨eZvZ⟩χZ = ⟨ev⟩χ.

The same identity holds for the unselected hierarchy. The horizontal section and readout therefore transport affordability, precision- tested errors and finite terminal bounds in both directions on the retained expression class. In particular the funds quantifier remains “for every precision, there exists a finite approximation,” with that approximation allowed to depend on the path and common horizon is inserted in this transport.

For a fixed row of weight, transport its actual detector family, aperture, rotations, conjugates and finite decoding maps. Positions are unchanged, so the fixed positive placement distances and layer zero are unchanged. Its error estimate retains the exact fee +probeFee(). Given any extra rate tolerance 0, choose the precision to satisfy +probeFee(. This recovers the row rate from the affordable approximation. Varying rotations are retained as part of the tested path. The resulting eventual uniform bound over rotations bounds their normalized Haar integral by that same bound; horizontal transport commutes with this finite-dimensional integral. Thus the full averaged reader, including conjugation and decoding, has the recovered row estimate. For UC3 put = maxFee( and = min( If is the chosen finite approximation, the exact residual identity is

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(e −Π≤we) −(a −Π≤wa) = (e −a) −Π≤w(e −a),
q0 ≤q,
q0 + Kw ≤q.

Both tested errors therefore meet precision before and after horizontal transport. For a surviving weight

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[c −(w + 1)(s −u) −du] −(c −ds) = (d −w −1)(s −u) ≥0.

This recovers precisely the cost gain and its zero-gain case.

For separated products the successive precision requests remain for, followed only after recording’s finite terminal cost by for. The two errors in then have rates at most; their finite sum has the same upper rate. The product approximation retains cost and the original separated palette. For a spectator of layer, outgoing cost and terminal cost, the request gives error rate, equal to the direct approximation bound. These identities transport the multiplication and expectation rules, including the empty