dm³ 101 · Week 13 — G² — The Second Iteration

dm³ 101 · Week 13 · Synthesis

G² — The Second Iteration

Course: dm³ 101 · Synthesis · Source: Book 3, “The Recurrence Ladder — From π to Hexabonacci”, Zenodo 10.5281/zenodo.19117400 + Week 3

Week 3 established \(G = U \circ F \circ K \circ C\) as a well-defined composite map. This week asks: what changes on a second pass, \(G^2 = G \circ G\)? Near a fixed point \(x^*\) of \(G\) (Week 4’s Unfolding operator drives trajectories toward exactly such a point, by construction — UnfoldOp.decreases_Phi), the linearisation is governed by the same Lyapunov exponent from the correspondence table: \(\mu_{{\max}} = -2\). Two applications of a linear contraction with rate \(\mu_{{\max}}\) contract at rate \(2\mu_{{\max}}\) in the exponent — deviations shrink twice as fast under \(G^2\) as under \(G\), which is the formal content of “iterating gets you there faster,” stated precisely rather than as a slogan.

The gap τ − φ

\(\tau - \varphi = 2 - 1.618\ldots \approx 0.382\). Read against the criticality bridge (Week 6, Week 9): this is literally the distance, in curvature-coefficient units \(c\), between the subcritical Fibonacci rung (\(c=\varphi\)) and the embodiment threshold (\(c \to \tau = 2\), the \(n\to\infty\) limit of the ladder). It is not the distance to the actual fold threshold \(c^*=3\) — that gap, \(3-\varphi \approx 1.382\), is more than three times larger. Two different thresholds exist in this framework (the fold threshold \(c^*=3\) and the ladder’s limit \(\tau=2\)) and \(0.382\) is the distance to the second, not the first.

Interestingly, \(0.382 = 1/\varphi^2\) exactly (a standard golden-ratio identity: \(1/\varphi^2 = 2-\varphi\)), so the gap \(\tau-\varphi\) and \(\varphi\) itself are related by the same self-referential structure explored in Week 11 — another instance of the fixed-point pattern, not a new independent coincidence.

This week’s content is grounded in Principia Orthogona, Book 3 (“The Recurrence Ladder”) and AXLE_v6.lean — theorem names and Lean identifiers above point at the actual source files.

-- dm³ 101 · Week 13 · G² linearisation, μ_max from AXLE_v6.lean

-- Single pass: deviation ξ(t) ~ ξ₀ · e^{μ_max · t}, μ_max = -2
-- Two passes (G²): ξ(t) ~ ξ₀ · e^{2·μ_max · t}   -- twice the rate

-- τ - φ = 2 - 1.618... ≈ 0.382 = 1/φ²  (standard identity 2-φ=1/φ²)
-- Contrast: 3 - φ ≈ 1.382  (distance to the FOLD threshold c*=3,
-- not the LADDER limit τ=2 — two distinct thresholds, don't merge them)
example : (2:ℝ) - 3 ≠ 2 - ((1+Real.sqrt 5)/2) := by norm_num

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