The Pentanacci sequence \(P(n)=P(n-1)+\cdots+P(n-5)\) has characteristic polynomial \(x^5-x^4-x^3-x^2-x-1=0\), dominant root \(\Sigma\approx1.96595\). Fold position at \(c=\Sigma\): \(q^*=\sqrt{{\Sigma/3}}\approx0.810\) — strong supercritical in the potential sense, still short of \(c^*=3\). Four non-dominant roots form two complex-conjugate pairs, all with \(|\lambda_i|<1\) relative to the dominant root — the spectral gap holds.
Theorem Σ.1 (Pentanacci biological instantiation)
Five-fold symmetric biological systems — echinoderms (starfish, sea urchins), some flowers — are proposed as dm³ transitions with curvature parameter in the range \([\Delta,\Sigma]\). The fold operator activates with 5 branches rather than 3, at higher entropic cost than the Tribonacci/Collatz case. Read this the way 102 asked you to read every biological application: the five-fold anatomy is real and well documented; the dm³ curvature-parameter framing is a proposed interpretive layer on top of it, not an independent structural finding.
Gap to \(\tau=2\) is now \(2-\Sigma\approx0.034\) — less than a tenth of \(\varphi\)’s original gap (\(\approx0.382\)) four rungs ago in 101.
This week’s content is grounded directly in the AXLE/Book 3/5 sources cited above — no material in this page depends on the external, unverified source removed from dm³ 102.
-- dm³ 103 · Week 02 · Σ, Pentanacci
-- x⁵-x⁴-x³-x²-x-1=0, dominant root Σ ≈ 1.96595
-- q* = √(Σ/3) ≈ 0.810. Gap to τ: 2-Σ ≈ 0.034
example : (2:ℝ) - 1.966 < 0.035 := by norm_num