dm³ 101 · Week 15 — Preview: μ, η, Δ — The Middle Operators

dm³ 101 · Week 15 · Preview

Preview: μ, η, Δ — The Middle Operators

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

dm³ 102 opens exactly where the criticality bridge (referenced since Week 6) becomes the main subject. Three previews, stated precisely so 102 can move fast:

μ — the exact Lyapunov exponent (Theorem μ.1, proved)

At the fold \(q=1\), \(c=3\): \(V(q)+2=(q-1)^2(q+2)\), a double root. Linearising the transverse dynamics at this point gives \(\mu_{{\max}} = -2\) exactly — not fit to data, not a round-number approximation, but forced by the double-root structure itself. The Gronwall stability radius follows directly: \(\varepsilon_0 = |\mu_{{\max}}| / (2(1+\sup\|\text{{Hess }}V\|)) = 2/(2\cdot3) = 1/3\) exactly.

η — Tribonacci at criticality (Theorem η.1, proved)

\(\eta\), the \(n=3\) ladder rung (\(\approx 1.839\)), is the unique n-bonacci constant whose associated curvature coefficient sits exactly at the fold threshold \(c^*=3\) — recall from Week 9 that \(\varphi\) (\(n=2\)) does not. This is also, non-coincidentally, the constant tied to the Collatz map \(T:n \mapsto 3n+1\), where the odd-branch coefficient is exactly \(c=3\): Theorem C.1 establishes the Collatz map as the unique integer map at criticality.

Δ — Tetranacci, the first supercritical step

\(\Delta \approx 1.928\) (\(n=4\)) overshoots the fold: \(q^* = \sqrt{{\Delta/3}} \approx 0.802\), landing past \(\eta\)’s exact \(q^*=1\). Supercritical systems (Theorem \(\Delta\).1) show richer transients, multiple stable branches, and higher entropic cost before settling — the crystal law \(g^6=33\) still closes, but at higher dissipation.

The through-line for 102: subcritical (\(\varphi\), no transition) \(\to\) critical (\(\eta\), the unique integer-coherent fold, Collatz) \(\to\) supercritical (\(\Delta, \Sigma, \Omega\), richer but costlier transitions) \(\to\) the \(n\to\infty\) limit \(\tau=2\). This 101 course has given you the constants and the fixed-point machinery; 102 gives you the Collatz application in full and the Gronwall-basin picture in detail (\(\varepsilon_0=1/3\) symmetric analytical bound vs.\ the numerically-certified, asymmetric inner boundary \(r^*\approx0.776\) — AXLE Issue #13, value certified, general asymmetry theorem still resting on an axiom pending full ODE formalisation).

Update: r* is now a certified number, not just an estimate

An earlier draft of this page cited \(r^* \approx 0.80\), which was a rough figure. The actual value is \(r^* = 0.77594059\) (\(\approx 0.776\)), reproducibly certified to 8 decimal places by certify_rstar.py in the AXLE repo — deterministic bisection search plus high-precision (rtol \(10^{{-12}}\)) DOP853 integration of the toy-model ODE \(\dot r = r(1-r^2)+2(r-1)e^{{-z}}\), \(\dot z = r^2 - 2(r-1)^2 e^{{-z}}\). The value is also recorded directly in Lean, Dm3Arithmetic.lean: noncomputable def r_star : ℝ := 0.77594059, with r_star_pos and r_star_lt_one proved. What is not yet proved: Chain_updated.lean still states inner_basin_is_asymmetric as an axiom, “pending dm³ ODE formalisation” — so the number is certified, but the general theorem that the basin really is asymmetric (rather than just this one numerically-verified instance) is not yet derived from first principles in Lean. Hold both facts at once, the way this course has asked you to all along.

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 15 · Preview constants for dm³ 102

-- μ_max = -2 exactly (Theorem μ.1, proved — double root at q=1,c=3)
-- ε₀ = |μ_max| / (2·(1+sup‖Hess V‖)) = 2/6 = 1/3 exactly
-- η ≈ 1.839, critical rung, tied to Collatz c=3 (Theorem η.1, C.1)
-- Δ ≈ 1.928, first supercritical rung, q* ≈ 0.802 (Theorem Δ.1)

-- r* = 0.77594059 (≈0.776) — numerically certified, 8 decimal
-- places, via certify_rstar.py (bisection + DOP853 ODE integration,
-- rtol=1e-12). Also in Lean: Dm3Arithmetic.lean, r_star := 0.77594059,
-- with r_star_pos / r_star_lt_one proved. AXLE Issue #13.
-- Still an AXIOM (Chain_updated.lean): inner_basin_is_asymmetric —
-- the general asymmetry theorem, not just this numeric instance,
-- is pending full dm³ ODE formalisation in Lean.
example : (1:ℝ)/3 = 2/(2*3) := by norm_num

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