dm³ 102 · Week 09 · Constant: Δ (n=4)

Δ — The Tetranacci Constant

Δ ≈ 1.928, the first supercritical rank, and what “supercritical” means in two different senses
dm³ 102 · Week 09 · Constant: Δ (n=4)
Δ — The Tetranacci Constant
Course: dm³ 102  ·  Constant: Δ (n=4)  ·  Source: Book 3, Chapter Δ

The Tetranacci sequence \(T(n)=T(n-1)+T(n-2)+T(n-3)+T(n-4)\) has characteristic polynomial \(x^4-x^3-x^2-x-1=0\), dominant root \(\Delta \approx 1.92756\). Theorem Δ.1 (proved, Book 3). Plugging \(c=\Delta\) into the fold potential \(V_\Delta(q)=q^3-\Delta q\) gives fold position \(q^*=\sqrt{{\Delta/3}}\approx0.802\) — overshooting \(\eta\)’s exact \(q^*=1\) but still short of the true critical value (\(q^*=1\) requires \(c=3\) exactly, and \(\Delta\approx1.928\) is still well below 3). Supercritical here shows up as richer transients: three decaying modes rather than two, higher entropic cost to enforce integer coherence, and secondary cycles that the pure Tribonacci case doesn’t exhibit.

Two different meanings of “supercritical” — don’t merge them
This is the same distinction Week 5 drew for \(\eta\), now on the other side: Book 3’s own text flags it directly — “the curvature coefficient \(c=\Delta\approx1.928\) is still less than \(c^*=3\)… the supercriticality here is in the recurrence depth (\(n>3\) summands), not in the fold-potential sense. The two uses of ‘supercritical’ are distinct and both appear in the literature.” So: \(\Delta\) is depth-supercritical (\(n=4\) summands, more than Tribonacci’s 3) while remaining potential-subcritical (\(c=\Delta<3\), the fold hasn’t reached the true threshold worked out in Weeks 7–8). Both are real, precisely defined, and not interchangeable.

The Tetranacci map \(T_\Delta: n \mapsto \lceil\Delta\rceil n+1\) does not converge universally to a single trivial cycle the way the true Collatz map (\(c=3\) exactly) does — it admits secondary cycles for large \(n\). This is the concrete behavioral signature of sitting past \(\eta\)’s exact fold without reaching the potential’s true \(c^*=3\) threshold.

This week’s content is grounded in Book 3, Chapter Δ — no material in this page depends on the external, unverified source removed from earlier drafts of this course.
-- dm³ 102 · Week 09 · Δ, Tetranacci

-- x⁴-x³-x²-x-1=0, dominant root Δ ≈ 1.92756
-- V_Δ(q) = q³ - Δq,  q* = √(Δ/3) ≈ 0.802  (past η's q*=1, short of true c*=3)

-- TWO senses of supercritical, both real, not the same:
--   rank-supercritical:      n=4 > 3          (Week 7-8 critDim sense)
--   potential-subcritical:   c=Δ≈1.928 < 3    (V_c fold-threshold sense)
example : (1.928:ℝ) < 3 := by norm_num
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