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Manuscript II@2 · II.tex · figure · 50d64def

The remaining function

Meaning of the certificate › The remaining global function · lines 1717–1734

II@2#fig:remaining-function

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H(β)=m(β)−23nℓa(β)−23nℓb(β)H(\beta)=m(\beta)-\tfrac23 n_{\ell_a}(\beta)-\tfrac23 n_{\ell_b}(\beta)
Domain X\mathcal X
β0\beta_0: H=0H=0
β+\beta_+: H=0H=0
β−\beta_-: H=0H=0
elsewhere: not decided here
θℓa=θℓb=−23q(d)⟹rS(d,θ)=0\theta_{\ell_a}=\theta_{\ell_b}=-\tfrac23 q(d)\Longrightarrow r_S(d,\theta)=0

One choice for all three labels.

global identity or injective evaluation → membership
complete separator on a new family → nonmembership

Failure of this two-contact candidate would not exclude every other fit.

The remaining-function test and the larger affine fitting problem have different quantifiers.

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