Three previews for 103. Σ (Pentanacci, \(n=5\), \(\approx1.966\)): quintic characteristic polynomial, four decaying non-dominant modes (two complex conjugate pairs), fold position \(q^*=\sqrt{{\Sigma/3}}\approx0.810\) — strong supercritical in the potential sense, still short of \(c^*=3\). Theorem Σ.1 (Book 3) ties five-fold biological symmetry (echinoderms, some flowers) to this curvature range.
Ω (Hexabonacci, \(n=6\), \(\approx1.984\)): the last individually-named ladder rung, within 1% of \(\tau=2\). Theorem Ω.1 (proved, Book 3): \(\lim_{{n\to\infty}}\rho_n=2\), derived from the n-bonacci polynomial’s asymptotic form \(x^n(1-1/(x-1))\to0\) as \(n\to\infty\), giving \(x=2\) — the same derivation style covered in 101 Week 6, now with the intervening rungs (\(\eta,\Delta\)) filled in from this course.
The through-line into 103: \(\Sigma\) and \(\Omega\) close out the individually-named ladder; \(\tau=2\) itself is the \(n\to\infty\) limit, proved in Theorem Ω.1 above. 103 opens with \(\rho\) (spectral radius) developed properly and the remaining unresolved threads from this course’s own audit — not with any claim that requires an external, unverified source to stand up.
-- dm³ 102 · Week 15 · Preview constants for dm³ 103
-- Σ ≈ 1.966 (n=5), q* ≈ 0.810 — strong supercritical (potential sense)
-- Ω ≈ 1.984 (n=6), within 1% of τ=2 — last named rung
-- Theorem Ω.1 (proved): lim_{n→∞} ρ_n = 2, from x^n(1-1/(x-1))→0
-- τ - Ω ≈ 0.0164 — "residual curvature", evocative framing per
-- source chapter, not yet a precisely independently-defined quantity.
-- ρ (spectral radius) previewed, developed fully in 103.
example : (2:ℝ) - 1.9836 > 0 := by norm_num