\(\varphi\) satisfies \(\varphi^2 = \varphi + 1\), equivalently \(\varphi = 1 + 1/\varphi\): it is the fixed point of the map \(x \mapsto 1 + 1/x\). Iterating from any positive starting value converges to \(\varphi\) regardless of the start — \(x_0=1 \Rightarrow x_1=2.000 \Rightarrow x_2=1.500 \Rightarrow x_3=1.667 \Rightarrow x_4=1.600 \Rightarrow x_5=1.625 \Rightarrow \cdots \to \varphi\). This is a genuinely elegant, self-contained piece of mathematics: a number defined entirely in terms of itself, stable under its own defining map.
gronwall_contraction_below_stability_radius). It is not a claim that \(G\) converges to \(\varphi\), or that \(\varphi\) is a fixed point of \(G\) in any literal sense — \(G\) acts on trajectories in a state manifold \(X\); \(\varphi\) is a fixed point of a one-dimensional real map. The resemblance is in the logical form (self-reference stabilizing to a unique value under iteration), not in the objects themselves.
This is a genuinely useful pattern to be able to name precisely, because dm³ leans on fixed-point arguments repeatedly — the Gronwall stability radius \(\varepsilon_0=1/3\) is itself a contraction-mapping bound. Being able to say “this is the same proof pattern, applied to a different space, under different hypotheses” rather than “these are the same thing” is precisely the discipline Week 3’s falsifiability conditions were trying to instill.
AXLE_v6.lean — theorem names and Lean identifiers above point at the actual source files.-- dm³ 101 · Week 11 · φ as fixed point, structural analogy to G -- φ = 1 + 1/φ ⟺ φ² - φ - 1 = 0 example : (1 + Real.sqrt 5)/2 = 1 + 1/((1 + Real.sqrt 5)/2) := by sorry -- standard algebraic identity; illustrative, not the point -- The point of this week is NOT a new Lean fact — it's recognizing -- that G's well-definedness (Theorem A, Week 3) and φ's fixed-point -- equation are both contraction-mapping instances, without conflating -- the two distinct mathematical objects. example : True := trivial