Extension across collision sets · Part 2

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on the permitted normalized supports, uniformly over the chart parameters. The subtraction length
may be fixed or varying, subject to the same support, halo and clearance conditions; it need not
equal R.
If f vanishes near the total diagonal N⊥
g , every term Cg in (CE16) annihilates it, by the normal
jet formula. Therefore the full score agrees with the punctured score on such tests, even when they
meet proper-subcluster collision sets. A test supported in any positive normal annulus has this
property. Normal rescaling preserves it, so (CE18) holds for the punctured pairing with the same
constants. No uniform positive lower bound on the annulus hole is required. These are the full,
output and remainder scaling assertions and their punctured counterparts.

A real basis can be chosen in each homogeneous contact grade simultaneously. 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 splitting’s stratum is invariant under conjugation and rotations, because its lower-grade map intertwines them. Every vector is a complex combination of conjugation-fixed vectors by the real-and-imaginary decomposition already used for. Extend a linearly independent subset of these real vectors to a maximal such subset; it spans the complex stratum, so finite-dimensionality makes it a finite real basis over. At degree zero start with the

nonzero unit, which is fixed by conjugation, and extend it. This gives a homogeneous system basis containing the unit. Expand a general input field into its finitely many homogeneous parts and then into this basis. Multilinearity of the score and reader gives the general contact expression as the finite sum of its basis expressions. Each term retains its actual assigned homogeneous degree in (CE16). Raising the ambient filtration horizon changes none of these parts, so the resulting linear extension is well defined.

The compatible radius-dependent prescription is also explicit. Given any 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(). Let) = 1 and, for 0, let ) be the prescribed smooth cutoffat. It is one for 0 and zero for. For every 0 these cutoffs are locally finite in. Moreover, a nonzero, implies the strict threshold spacing. Let) be the last nonzero cutoffindex. It exists, andas0. For every active clearance(

Indeed and the displayed threshold bound is stronger. Consequently for any fixed eventuallyclearance()). Define the finite smooth projection

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

Local finiteness makes this smooth for positive, even where the last active index changes. Biorthogonality of the homogeneous rows gives exactly

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rowx,n Ds(x)E = bn(s) rowx,n E.

For, all rows through equal one, and projecting (CE19) through degree is exactly the original degree- projection. Thus no limiting approximation is substituted for a finite reader in

the contact estimates. The construction may be slowed to meet all additional positive bounds while preserving these identities. It asserts no convergence as the growing cutoffhorizon itself varies.

This completes the contact endpoint’s real basis and threshold existence clauses. The affordable-expression rules, the uniform logarithmic compactness argument, and the mixed-label inclusion identities are proved in (UC1)–(UC3) below.

Uniform subpower bounds on contact layers

The layer estimates used in the contact construction are uniform in all radii. Their deduction from logarithmic-path estimates is the following compactness argument. Let be radii in (0 on a configuration set, let 0 be the bill without the small losses, and let be the normed quantity to estimate. Suppose that along every nontrivial filtered sequence with an increasing clock and log

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T
−→−sa,
0 ≤sa ≤1,

the bill has some exact exponential rate, and has upper rate at most. Then for every there is a single finite 0 such that

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

If this failed, choose violating the inequality with. Put

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pn,a = −log Ra(wn) ≥0,
Sn =
X
a
pn,a,
Tn = n + Sn.

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Take an ultrafilter refining the sequence tail. Compactness of [0, 1], for finitely many coordinates,
gives limits pn,a/Tn →sa ∈[0, 1] and n/Tn →k ∈[0, 1], with k + P
a sa = 1. The path hypothesis
applies. The supposed upper bound for F has rate c, while the violating comparison quantity

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enq(wn)
Y
a
Ra(wn)−ε

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has exact rate c + k + ε P
a sa > c. Positivity of the increment follows 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 term n in Tn handles sequences
whose radii remain away from zero.

Every bill used here has the required exact rate: constants, positive radii, products and fixed real powers have exact rates, and these operations generate the finite radius-power bills. Thus the argument does not require a common OPE threshold for all multiparameter paths or a common intermediate output horizon. It supplies the ordinary uniform layer inequality actually used in the contact integral.

Extending contact expressions from homogeneous to arbitrary labels

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Let M(d)
a
: E≤d →E≤a, 0 ≤a ≤d, be the fixed homogeneous splitting maps, with their inclusion
maps between grade caps. The contact route construction for homogeneous labels is multilinear in
each input. To see this before integration, fix all other inputs; its grade-transport identity expresses
the remaining dependence as a linear map. Hence addition and scalar multiplication hold in that
slot for the full expression-valued score.

For labels, define the mixed score by the finite sum

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

where is the homogeneous score for the fixed chart, partition and route parameters. If all inputs are homogeneous of their stated grades, every summand except for all vanishes, so (UC2) is the original homogeneous score.

Raising the grade caps leaves (UC2) unchanged. Embed every original tuple () in the enlarged finite index set. The splitting maps commute with inclusion, so those summands agree exactly. Any additional tuple has some; the corresponding split component of the included input is zero, and multilinearity makes that entire summand zero. This proves cap independence at the expression level.

On a test support containing only distinct configurations, compatibility of contact charts identifies each homogeneous local image with the ordinary field product times the test function. At each insertion point the splitting identity and coherent inclusion give

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

Expanding the finite product of these sums recovers the ordinary mixed-label field word. Evaluation by any allowed spectator then yields exactly the original separated correlation times the test function. Every sum is finite before integration; the previously proved per-grade integral bounds justify taking its integrals term by term. Thus the mixed extension has the prescribed separated restriction, cap independence and multilinearity on the full field space.

The projection gain underlying (UC1) can now be made precise. Along a logarithmic clock , an upper rate means a bound by eventually for every 0. A finite expression is affordable at layer and cost when it is a finite sum of separated field words whose coefficient for a word of total field weight has rate at most. This class is closed under finite sums, scalar coefficients with their added rate, and eventual equality. Its supports follow its finite placement palette.

A tested error of precision at a palette means that after multiplication by any separated affordable spectator word, at terminal layer one and of cost, its expectation has upper rate This quantifies over the unselected hierarchy and every character, including varying choices of character. An expression has funds at layer, cost, if it has some finite terminal affordability bound and, for every precision, differs by such a tested error from an affordable expression at layer, cost

When a palette lies at radius rate about a center and 01, the one-scale OPE replaces an affordable word by a finite sum of fields at that center to arbitrary tested precision at layer. Indeed choose the output horizon so large that the gain) dominates the prescribed precision deficit. The coefficient estimate assigns every retained output its rate, where is its output weight. The tested remainder estimate applies to each allowed spectator geometry on this path. The choice of horizon is allowed to depend on that path and precision; it is not asserted to be uniform before (UC1). Finite sums and scalar coefficients give the same replacement for every affordable expression, and adding its existing tested error gives the same replacement for every expression with funds.

For a centered finite sum of fields with coefficient bounds, the retained coefficient row of weight has rate at most. 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. To transfer this

statement from a finite approximation to the original expression, test the error against the finite detector family for that row, including its rotations and conjugates. All these tests lie outside the fixed row aperture. Their placement distances remain bounded below independently of, so their mutual separation has layer zero. If probeFee() is the sum of their fixed weights, a precision- error gives row rate at most +probeFee(). Choose sufficiently small to absorb this cost. Hence an expression with funds () has row rate and its projection through any finite horizon still has funds (

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This row estimate includes the full rotation average. For any varying rotation, finite-dimensional
covariance and compactness of the orthogonal group preserve the rate of a coefficient vector. The
rate bound for every varying choice is uniform over rotations eventually: otherwise selecting a vi-
olating rotation at each bad index contradicts that bound. The normalized Haar integral is then
bounded by the same exponential bound. Conjugation preserves norms, and the fixed finite decod-
ing maps preserve rates. This proves the detector-row estimate without exchanging a nonuniform
family of limits and an integral.
Let Π≤w be the prescribed finite projection. Subtracting it and passing to a coarser layer u ≤s
gives
u, c −(w + 1)(s −u); e −Π≤we

.
Funds(s, c; e) =⇒Funds
(UC3)

For a centered finite field sum this is immediate: all outputs of weight at most cancel, and any surviving weight + 1 satisfies

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c −ds ≤c −(w + 1)(s −u) −du.

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For a general expression choose its centered approximation at precision min(q, q−maxFee(w)−wu).
Projection of its error costs at most maxFee(w)+wu, where maxFee(w) = Pw
j=0 probeFee(j). Both
the original error and its projection consequently fit precision q. Subtract them and use the finite-
sum result for the approximation. Since q was arbitrary, (UC3) follows. At u = s, simply subtract
the projected affordable expression, with no gain. Repetition of (UC3) supplies the full w + 1 gain
at every signed ancestor move of the contact forest.

Funds multiply across separated palettes with addition of costs. Suppose has funds ( has funds (), their palettes are separated at layer, and1. Fix terminal affordability costs for. For any desired tested precision, first approximate by an affordable with error precision. After choosing, enlarge its layer to one and record its finite terminal cost. Then approximate by an affordable with error precision. The identity

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ef −gh = (e −g)f + (f −h)g

places both terms at precision, by the definition of tested precision and the two recorded terminal costs. The product is affordable at layer, cost: concatenate its separated words and multiply their coefficients. This proves the multiplication rule for funds. Induction proves it for any finite family of pairwise separated palettes.

The order of these approximation choices matters. Although has a controlled outgoing layer- cost, its terminal cost at layer one can be much larger. Choosing the second approximation only after recording that finite cost pays for it without imposing a false uniform bound on all OPE horizons. The arbitrary-precision definition is exactly what permits this successive choice.

Funds also imply the stated expectation bounds. To test against an affordable spectator of layer, cost, with1, first enlarge to terminal layer one and record its finite cost . Choose an affordable layer- approximation to whose error precision is. The direct separated correlation estimate bounds the approximation times at rate, and the definition

of precision bounds its error times at that same rate. Addition proves the desired tested bound. Taking the spectator to be the empty word gives expectation upper rate, uniformly over all character choices. This closes the expectation and multiplication rules used with (UC3).

The avoidance geometry used for the equation lift can also be given explicitly. A lattice step that would traverse the forbidden positive edge in either direction can be replaced by three steps: move one unit in a different coordinate, make the intended step, and move back in that coordinate. The replacement has the same endpoints and avoids the forbidden stored edge. Replacing each offending step of any fixed coordinate path constructs a path between any two cover sites avoiding that edge.

For a plaquette of axes, its boundary can use a fixed forbidden edge based at only if its own base belongs to

This follows by listing its four stored positive edges, including the bases of its two reversed steps. Outside this finite set, take an avoiding transport path to the plaquette base as above; both the transport and the boundary then avoid the edge. These are the safe sampling probes used in the finite-dimensional span argument.

Avoidance on the integer cover yields exact independence on all sufficiently large tori, uniformly over translations of the stencil. For each of the finitely many used cover edges choose the side larger than twice the largest absolute coordinate of its base and of the forbidden base, plus one. Their centered wrapped lifts then recover those bases exactly. Distinct stored cover edges stay distinct after wrapping. The link variation therefore changes none of the path factors, so their holonomies, curvatures, adjoint transports and all finite products are unchanged. Taking the maximum over the finite used-edge set gives the common side threshold in the marking identity.