Statement of the public conclusion

Statement of the public conclusion

The assertion below is the supplied construction theorem in its public form. Its detailed proof occupies the subsequent sections. The Wilson family is fixed before a volume subsequence is requested, and the full mesh limit is obtained at each suitable fixed period before an outer volume subsequence is taken. An auxiliary compactness extraction is subsequently identified with that full inner limit. The concluding trace test concerns precisely the finite mass-shell class specified below. Let be a closed subgroup of) in its given faithful matrix representation. Its real Lie algebra is g : exp( for every

Assume that g is nonabelian and simple: every real linear subspaceg with [g is either zero or all of g. No connectedness hypothesis is added.

The common Wilson family. There is one Wilson family for this, with strictly increasing positive integer periods and denominators. Its mesh is and its periodic lattice has side. Group elements are assigned to its oriented links. With normalized Haar measure and the independent link prior, the probability law is

Searchable transcription

dμN,p(U) = Z−1
N,pe−βN,pS(U) Y
e
dν(Ue),

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


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

The family includes a token space and continuous, gauge-invariant local insertions for every token Each has a fixed finite set of integer offsets, independent of, controlling its insertion: if two configurations agree on the links whose endpoints lie in the translated offset set, their insertion values agree. There is a positive slope such that

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βN,p
log(h−1
N ) −s
−→0
(N →∞).
sup
p

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One hierarchy on a permitted refinement.
For every strictly increasing sequence p(k) there
is a strictly increasing subsequence index q(k) and one hierarchy T on p(q(k)) with the following
properties. Its fields have finite-dimensional complex normed label spaces Fd, d ≥0, with linear
injections into the label space of one Wightman theory that jointly cover that space. Tokens project
surjectively onto the disjoint union of these graded spaces. There is a compact metric character
space X, a full-support Borel probability measure μph on it, a selected character, and separated
Euclidean correlations Sχ
n for each χ ∈X and Sn for the unselected hierarchy. They are smooth
away from collisions, invariant under simultaneous permutations of slots and points, continuous in
χ on that separated domain, and obey

Searchable transcription

Sn =
Z
X
Sχ
n dμph(χ).

The subscript on the phase measure distinguishes it from the finite Wilson laws; it does not change the measure in the supplied statement. For a smooth compactly supported collision-free test and a finite token word the lattice smeared moment is

Searchable transcription

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

where is the centered physical placement. At each period whose centered open box contains supp, there is a full mesh limit

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rp,v(t) = lim
N→∞RN,p,v(t).

Along)) it converges to the integral of against the unselected correlation for the projected word. Every such fixed test is suitable at all sufficiently large periods of that subsequence. Thus the inner mesh limit is full at each suitable fixed volume; the outer volume limit uses the permitted refinement.

The selected Wightman theory. The selected theory has a normalized vacuum Ωand one common dense complex invariant domain, the span of finite smeared field words applied to Ω. Fields depend complex-linearly on labels and Schwartz tests, and their values on each domain vector depend continuously on the test. They satisfy the smeared adjoint identity, spacelike local commutativity and Poincaré covariance. Translations act by a strongly continuous unitary group whose joint energy-momentum projection measure is supported 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 every proper orthochronous affine Poincaré transformation. The selected Euclidean correlations are the values of a holomorphic continuation in the tube of strictly ordered imaginary times. Its smeared boundary limits on product Schwartz tests give the Wightman vacuum moments.

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Separated operator products.
For a cluster a = (ai) of graded fields, write |a| = P
i deg(ai).
There are finitely many channels at each degree j, with fields Oj,λ and coefficients Ca,j,λ(x; o), in
one system independent of phase and truncation. Given M ≥0, η > 0 and ε > 0, every fixed
channel satisfies, for all sufficiently small r > 0,

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|Ca,j,λ(x; o)| ≤rj−|a|−ε

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whenever |xi −o| ≤Mr and |xi −xi′| ≥ηr for i ̸= i′. For a fixed spectator word b, c > 0 and
D ≥0, the same cluster domain and the spectator separations |o −yl| ≥c and |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

This holds for all sufficiently small, uniformly in all phases and also for the unselected hierarchy. All terms belong to the same hierarchy. The threshold may depend on the fixed words and the stated parameters.

The centered trace and its comparison class. There is a centered degree-four field Θ. Let ) be its selected two-point correlation at spatial zero and Euclidean times2. Then

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

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 is of the form

Searchable transcription

Z
2
P(p)e−rEμ(p) d3p,
Eμ(p) =
q
R3
Eμ(p)t−1
μ2 + |p|2,

where and is continuous and positively homogeneous of a nonnegative integer degree. Here denotes the comparison mass called in the source, to distinguish it from the representation size. This is the public non-freeness test with its precise comparison class. No positive physical mass gap is asserted.

The public theorem is obtained by choosing blocking penalty = 1 in the detailed construction, taking the refinement for the requested volume sequence, and projecting its single set of witnesses. This preserves the common hierarchy across all conclusions. The detailed construction also supplies stress Ward identities, nested expansions and contact extensions. They are developed below on its stronger common witnesses; they are neither extra assumptions nor additional clauses silently added to the public theorem.