Four Fates of Self-Application

Abstract

Every non-degenerate binary fractional self-map has exactly one of four fates, with the live fate characterizing the golden family.

A binary fractional map is classified as empty, dead, collapsed, or live by the coefficients and discriminant of its fixed-point polynomial. For every non-degenerate map exactly one classification holds.

Theorem 1.1 (Non-degenerate self-application has exactly one fate).

$$\forall m,\mathrm{Nondegenerate}(m)\Rightarrow (\exists !fate,\mathrm{HasFate}(m,fate))\wedge (\mathrm{HasFate}(m,\mathrm{live})\iff \mathrm{IsPhiFamily}(m))\wedge (\mathrm{HasFate}(m,\mathrm{live})\Rightarrow \mathrm{fixedCoefficients}(m)\in \{(1,-1,-1),(1,1,-1)\})\wedge (\mathrm{HasFate}(m,\mathrm{live})\Rightarrow \mathrm{discriminant}(m)=1^2+4=5)$$

Proof. Machine-checked in Lean as D5/S0/Diagonal/SelfApplicationFates.self_application_four_fates (✓ std3). ∎

Source. Repository-derived.

Commentary.

The live cases are precisely the two golden-family maps, whose fixed-point coefficient triples are (1, -1, -1) and (1, 1, -1). In either case the discriminant is 1 squared plus 4, hence exactly 5.

References

• Truth anchor: D5/S0/Diagonal/SelfApplicationFates.self_application_four_fates