The supplied public conclusion and its recovery target

The supplied public conclusion and its recovery target

The following records the opening theorem’s content (input lines 7–61) in typeset notation. Its mathematical assertions are supplied results. They specify the target of the translation, rather than consequences selected by native admission. Let) be closed in its given faithful representation. Its real Lie algebra is exactly the matrices with exp( for every real, and is assumed nonabelian and simple: every real linear subspace contained in it and stable under commutators with it is either zero or the whole

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algebra. No connectedness assumption is added. There is one calibrated Wilson family with strictly
increasing positive integer periods Lp and denominators dN, mesh hN = d−1
N and lattice side LpdN,
with law
dμN,p = Z−1
N,p exp(−βN,pS(U))
Y
e
dν(Ue),

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1 −1
m Re tr(Ux,iUx+ei,jU−1
x+ej,iU−1
x,j )

.
S(U) =
X
x,i<j

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Its token insertions IN,p,v,x are continuous, gauge invariant and local on a fixed finite offset stencil
for each token v, uniformly in N, p, x.
A finite set of integer offsets depending only on v con-
trols each insertion at every N, p, x: agreement of configurations on edges whose endpoints belong
to the translated offset set implies equality of the insertions. Its positive coupling slope obeys
βN,p/ log(h−1
N ) →s > 0 uniformly in p.
For every strictly increasing period index sequence p(k) there is a strictly increasing refinement
q(k) and a single hierarchy on p(q(k)). Its graded finite-dimensional complex normed spaces Fd
inject into, and jointly cover, the field-label space of one Wightman theory; tokens project onto their
disjoint union. A compact metric character space carries a full-support Borel probability measure
and a selected character. The separated correlations Sχ
n and unselected Sn are smooth away from
R Sχ
collisions, permutation invariant, continuous in the character and satisfy Sn =
ndμ(χ).

For every smooth compactly supported collision-free test and finite token word, the lattice smeared moment is the source’s -weighted site sum at centered physical positions. Explicitly, with the original centered placement, this is

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h4n
N
X
x1,...,xn
t(xphys
1
, . . . , xphys
n
)
Z
n
Y
i=1
IN,p,vi,xi dμN,p.

At each suitable fixed period it has a fulllimit. Along the refined periods, converges to the unselected tested correlation. Suitability means that the centered open period box contains the test support, and holds eventually along the refined sequence. The full inner limit is retained before the outer subsequence limit.

The selected theory has normalized vacuum Ωand a common dense invariant domain equal to the span of finite smeared field words on Ω. Fields are complex linear in labels and Schwartz tests and continuous in the test on each domain vector. They satisfy the smeared adjoint identity, spacelike commutativity and proper orthochronous Poincare covariance. Translations are strongly continuous and unitary, with joint momentum projection measure in the closed forward cone. The invariant vacuum is unique up to scalar, every nonzero label gives a nonzero smeared operator, and at least one field acts nontrivially. The public symmetry coordinates cover all proper orthochronous

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affine Poincaré transformations. Strictly ordered imaginary-time tube values recover selected Eu-
clidean moments; product-Schwartz smeared boundary limits recover vacuum Wightman moments.
For a graded cluster a, write |a| = P
i deg(ai). One phase- and truncation-independent channel
system Oj,λ, Ca,j,λ(x; o) has finitely many channels at each degree. For fixed M ≥0, η > 0, ε > 0,
each fixed channel eventually satisfies

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|Ca,j,λ(x; o)| ≤rj−|a|−ε

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when |xi −o| ≤Mr and |xi −xi′| ≥ηr. For a fixed spectator word b, c > 0 and D ≥0, the same
cluster domain and spectator separations |o −yl| ≥c, |yl −yl′| ≥c give

Sχ(a(x), b(y)) −
λ
Ca,j,λ(x; o)Sχ(Oj,λ(o), b(y))

≤rD+1−|a|−ε.
D
X
X

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j=0

The assertion is uniform in all phases and includes the unselected hierarchy; its threshold may depend on the fixed words and stated parameters. Finally a centered degree-four field Θ has selected two-point function) with

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lim
r↓0 r8 log2(1/r)GΘ(r) = 3 dimR g
π4
> 0.

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It cannot agree at every sufficiently small radius with a finite complex linear combination of finite
products of the specified mass-shell integrals.
Each factor integrates E(p)t−1P(p)e−rE(p)/2 on
R3, where E(p) =
p
mass2 + |p|2, mass ≥0, t is a nonnegative integer and P is continuous and
positively homogeneous of nonnegative integer degree. This is the stated comparison class, not an
exclusion of every generalized free theory. The supplied proof chooses blocking penalty b = 1 and
projects the single compatible witness family through its common refinement.

Scope of the stated conclusion. The supplied construction proves more than the public statement records: it also supplies stress Ward identities, nested expansions and contact extensions, developed on its stronger common witnesses. These are not additional assumptions and not additional clauses of the public theorem. The reconstruction follows this division. The recovery target of the whole paper is the conclusion stated in this section; the further results are lifted where they are constructed and are recorded as such in the claim-status register (Section H). No positive mass gap is asserted in the supplied conclusion, and none is asserted anywhere in this paper.