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Manuscript VI@1 · VI.tex · figure · b5b60afb

Finiteness and completeness

Finite-renormalization spaces and certificate requirements › Derivative counting alone is insufficient · lines 2149–2167

VI@1#fig:VI-catalogue-bounds

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metric / coordinatescalarswitching defectconditional quantumsupport / uncompleted
Finite
z1,…,zn,∑iai≤pz_1,\ldots,z_n,\qquad \sum_i a_i\leq p
finitely many monomials, then linear constraints and quotients
Not finite from a derivative bound alone
Cn=κ2m4∫gnφ2,n=1,2,…C_n=\kappa^2 m^4\int g^n\varphi^2,\qquad n=1,2,\ldotslinearly independent as functionals of (g,φ)(g,\varphi)

not asserted to be allowed in a fixed two-insertion renormalization class

Complete
action
composite
switch
boundary
Ward
lower-order normalization
finiteness does not prove coverage
a finite sample still has a finite-dimensional image

Finiteness and completeness are different requirements. Explicit bounds on jet variables, polynomial degree, and coefficient dependence yield a finite-dimensional local class. A derivative bound alone does not control arbitrary powers of a dimensionless switch. Even a finite list needs a separate proof that it covers the allowed renormalization responses of the specified composite.

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