Completion of the proof

Completion of the proof

The common witnesses assembled here establish the public theorem’s quantifiers. All degree horizons, derivatives, label coordinates, operator products and trace comparisons are consumed on one Wilson family and one composite refinement. The inner mesh limit remains full at every suitable period; only the outer volume limit is subsequential. The detailed construction supplies additional stress and contact information on these common witnesses without changing the list of clauses in the public theorem.

We assemble the choices to verify the quantifiers of the theorem. Fix the given group and representation. Choose blocking penalty 1. The linear kernel construction gives its positive widths and stagewise lower bounds on block ratios. The nonlinear geometry, Gaussian and connected-activity estimates give further stagewise bounds, with block ratio chosen first, then the remaining fixed parameters, and finally the coupling-entry delay. The coefficient, detector, approximation and trace estimates impose bounds in this same order. At each stage only finitely many demands of its prescribed finite degree and derivative order occur. Their maximum is a finite demand. Increase the stage parameters to meet it. Increasing later block lower bounds and delays preserves the earlier requirements. Taking the increasing degree and derivative orders in this construction meets every fixed finite request eventually.

The exact one-step identities and the banked-history convergence then construct the calibrated Wilson family, including its common insertion tokens, before any volume subsequence is specified. Its represented action at the finest cutoffis the Wilson action in the theorem. Positivity of its partition function and the local insertion construction give the stated probability laws, continuity, gauge invariance and fixed finite stencil locality. The finite arcs (FCA7), with their exact Wilson startup and density identities, provide these calibrated histories. The summed drift argument (RUN1)–(RUN5), using the realized calendar (CAL1)–(CAL11) and the actual update (DR7), gives the positive logarithmic coupling slope uniformly in cutoffand period; its finest-cutoffspecialization is the bare-coupling statement of the theorem.

Now fix the requested strictly increasing sequence of periods. Retain the coupling limits and perform the common hierarchy and coefficient extraction. Compose its successive refinement maps. The fixed-volume Cauchy argument and the smearing passage give the full mesh limit at every suitable fixed period; the refined period sequence gives its outer limit. Thus the auxiliary compact selections do not weaken the inner limit in the statement. The field quotient and the physical detector identify one finite-dimensional field filtration on this table.

The compression algebra constructs its compact metric phase space with full-support probability measure. Its character hierarchies, including the selected one, have the stated smooth correlations and barycenter identity. Apply the heat, tube, boundary and Gram constructions to these same hierarchies. They give the Hilbert fields on the common cyclic domain, the adjoint and locality identities, the strongly continuous spacetime action and its forward-cone momentum measure. The heat-ground argument gives the unique vacuum in every character. The label-separation argument gives field faithfulness and the nonzero unit field. The strict imaginary-time and boundary identities identify the reconstructed moments with the corresponding Euclidean hierarchy.

Choose the equivariant operator-product reader on this field filtration once. The separated expansion, its coefficient bounds and its tested remainder hold for every character and for the unselected hierarchy. The nested replacement and integration arguments retain this reader, and the collision-extension construction retains its basis and finite projections. Their outputs therefore refer to the same fields and correlations, including when several truncation degrees or collision patterns are requested together.

Finally, the Gaussian comparison and position normalization determine the physical stress tensor on that table. The finite equation-marker bounds and weighted near-kernel estimates identify its centered trace and fix its amplitude. The spectral and position-space calculation proves

Searchable transcription

lim
r↓0 r8 log2(1/r) Sχ
2 (Θ(−r/2, 0), Θ(r/2, 0)) = 3 dimR g
π4
> 0

in the selected phase, with the same scalar diagnostic under the phase identifications established above. The mass-shell asymptotic argument excludes the finite comparison class specified in the

theorem; the coherent-state argument gives the additional Maxwell countertests described in the trace chapter. Every witness in these conclusions has been constructed on the one chosen Wilson family and retained on the one composite refinement. Passing to the public fields, operators, correlations and channel coefficients is therefore their identification on that common table. This proves the theorem.