The Ladder Polynomials

Two exact results, one honest boundary. Every figure below is reproducible from the code at the end.

This note addresses the open item in WP44 §2, Prediction 3 — the claim that the n-bonacci ladder is the exit sequence, “deferred to a companion paper” pending a connection between the ADE classification and the dm³ operator chain. It does not supply that connection. It supplies two exact results and one boundary.

xⁿ⁺¹ − 2xⁿ + 1

The n-bonacci characteristic polynomial is $q_n(x) = x^n - x^{n-1} - \cdots - x - 1$. Multiplying by $(x-1)$ telescopes the entire tail:

Verified symbolically for $n = 2,\ldots,8$; each $q_n$ is irreducible over $\mathbb{Q}$ over the same range. So the whole ladder — $\varphi$, tribonacci, tetranacci, pentanacci, hexanacci and upward — is a single one-parameter family, and the 2 in the middle coefficient is not put there. It is what the telescoping leaves. $\tau = 2$ is visible in the polynomial before any root is computed. VERIFIED

2 − r_n = r_n⁻ⁿ

Let $r_n$ be the root of $p_n$ in $(1,2)$ and write $r = 2 - \varepsilon$. Then $r^{n+1} - 2r^n + 1 = r^n(r-2) + 1 = -\varepsilon r^n + 1 = 0$, so

exactly — not asymptotically. The distance from the $n$-th rung to the limit is the reciprocal of that same rung raised to its own index. Verified to 50 significant digits for $n = 2,\ldots,20$. VERIFIED

n	rn	2 − rn	rn−n	2−n
2	1.6180339887498948	0.381966011250	0.381966011250	0.25
3	1.8392867552141611	0.160713244786	0.160713244786	0.125
4	1.9275619754829253	0.072438024517	0.072438024517	0.0625
5	1.9659482366454853	0.034051763354	0.034051763354	0.03125
6	1.9835828434243263	0.016417156576	0.016417156576	0.015625
12	1.9997555009373175	0.000244499063	0.000244499063	0.000244140625
20	1.9999990463165885	9.53683411e−7	9.53683411e−7	9.5367432e−7

Since $\varepsilon = r^{-n}$ and $r \to 2$, convergence is geometric with ratio $\to \tfrac12$: $\log_2(2-r_n)/(-n)$ runs 0.694, 0.879, 0.947, 0.975, 0.988, … → 1.

Consequence for WP44 Prediction 2. That paper states the maximum Lyapunov exponent “scales with the distance from $\tau = 2$ in the n-bonacci sequence.” That distance now has a closed form, so the prediction becomes quantitative: whatever the scaling law is, its argument is $r_n^{-n}$. MODEL

Every $r_n$ is also a Pisot number — all conjugates strictly inside the unit disc, checked for $n = 2,\ldots,12$; the largest conjugate modulus rises 0.618 → 0.980 and never reaches 1. VERIFIED

The ADE threshold

2 is not a constant chosen to make the ladder terminate. It is the threshold of a classification theorem with nothing to do with recurrences:

A connected simply-laced graph has adjacency spectral radius < 2 exactly when it is a finite ADE Dynkin diagram, and = 2 exactly when it is affine (extended) ADE.

Finite	ρ	Affine	ρ
A5	1.732050807569	Ã3 (3-cycle)	2.000000000000
A12	1.941883634852	Ã6 (6-cycle)	2.000000000000
E6	1.931851652578	D̃4 (4-leg star)	2.000000000000
E7	1.969615506024	Ẽ8	2.000000000000
E8	1.989043790737

This is the mechanism behind a line the corpus has carried as an assertion — “at E₈ the finite classification ends.” It ends at 2, and E₈ at 1.98904 is the closest finite diagram to the boundary. Two independent families accumulate at 2 from below: the Pisot side (dynamics) and the Coxeter side (classification, $\rho = 2\cos(\pi/h)$). VERIFIED

ϕ = 2cos(π/5) = ρ(A₄)

Verified to 30 digits, and 5 is the Coxeter number of $A_4$. The bottom rung of the ladder is an ADE spectral radius. Nothing above it is. Solving $h = \pi/\arccos(r_n/2)$:

n	h	integer?
3	7.783448	no
4	11.637159	no
5	17.000510	no — near miss, 5.1 × 10−4
6	24.502104	no
7	35.033998	no
8	49.859430	no

An exhaustive scan over $n < 40$ against $h < 60$ returns exactly one coincidence, $(n,h) = (2,5)$. That is expected, and the reason closes off a class of hoped-for identities: the ladder roots are Pisot, with conjugates inside the unit disc, while $2\cos(\pi/h)$ is totally real with conjugates in $[-2,2]$. The families are essentially disjoint, and $\varphi$ is the accident that manages to be both.

The n = 5 near-miss is flagged deliberately. $h = 17.000510$ is exactly the kind of number that becomes a false claim if nobody computes the next three digits. It is not 17.

S_n, all the way up

No discriminant is a perfect square in $\mathbb{Q}$, so no rung lies in $A_n$: every Galois group on the ladder contains an odd permutation.

n	Gal(qn)	|G|	n!	status
2	C2	2	2	proved
3–6	Sn	6–720	6–720	proved
7	S7	5040	5040	proved — Jordan
8–10	Sn	—	—	NOT PROVED HERE

n = 7 is rigorous. $q_7$ is irreducible, so the group is transitive; 7 is prime, so transitive implies primitive; Frobenius sampling returns a transposition; and a primitive group containing a transposition is $S_n$ (Jordan). One transposition suffices — no sampling assumption enters.

n = 8 is evidence, and the honest word is evidence. The transposition type first occurs at p = 17921, after 2,052 usable primes, against an expected density $\binom{8}{2}/8! \approx 1$ in 1,440. An earlier scan to $p < 4000$ found none — that was under-sampling, and reporting it as absence would have been a false negative wearing the grammar of a result. 21 of the 22 cycle types of $S_8$ have appeared. But 8 is not prime, transitivity does not give primitivity, and Jordan does not apply. OPEN

Maximal Galois group means no hidden structure: no proper subfield, no partial symmetry, no relation among conjugate roots beyond the ones every polynomial has. Which isolates n = 2 twice over, by two independent measures — the only rung with a small Galois group ($C_2$, the field $\mathbb{Q}(\sqrt5)$), and the only rung whose root is an ADE spectral radius. Two unrelated tests, one exception, the same rung. OPEN — whether that is one fact or two is not settled here, and it is the sharpest question this note produced.

The boundary, stated

VERIFIED The single-polynomial collapse and its irreducibility over the tested range; the exact gap identity; the Pisot property; the ADE/affine spectral threshold at 2; and $\varphi = 2\cos(\pi/5) = \rho(A_4)$.

MODEL That the shared accumulation at 2 is structurally meaningful rather than a coincidence of two families that both happen to live in $(1,2)$.

OPEN The McKay correspondence connecting ADE to the dm³ operator chain. Nothing here supplies it. What this note does is make the target precise: any such correspondence must explain why one family is Pisot and the other is not, and why they meet only at $\varphi$. That is a sharper question than WP44 was able to ask, and a correspondence that cannot answer it is not the right one.

Two references this note should have carried

The results above were computed rather than looked up, which is the right order and also the reason a note like this can end up sitting beside a literature instead of in it. Two papers address exactly these objects.

The discriminant is known in closed form

Luca, On the discriminant of the k-generalized Fibonacci polynomial, II (The Fibonacci Quarterly 62:3, 2024), gives

$$\mathrm{Disc}(q_k) \;=\; (-1)^{\binom{k+1}{2}-1}\, \frac{2^{\,k+1}k^{\,k} - (k+1)^{\,k+1}}{(k-1)^{2}}$$

VERIFIED That formula reproduces every entry of the discriminant table in § 5, for $k = 2$ through $10$ — nine agreements, checked by ladder-polynomials-verify.py block [6]. The table here was computed before the formula was known to this note, so the agreement is a cross-check in both directions and not a restatement of one source by the other. Luca’s paper proves something further that this note does not need: $|\mathrm{Disc}(q_k)|$ is itself a $k$-generalized Fibonacci number only for $k = 2, 3$.

The Galois group has its own paper

Luca cites Martin, The Galois group of $x^n - x^{n-1} - \cdots - x - 1$ (2004), as the reference for the Galois group of precisely this family. This note has not read it, and says so rather than guessing at its contents.

Added 2026-09-17 — the comparison that was missing. The question this section asks (what is the Galois group of the ladder?) has a classical twin it was never set beside: the cyclotomic family xⁿ − 1, whose group is (ℤ/nℤ)*, abelian of order φ(n) < n, solvable for every n, and soluble in radicals for every n. That is the case where reducing a geometric problem to a polynomial pays — it is why Gauss gets the 17-gon and why Euler's reduction of the division of the circle to xⁿ − 1 = 0 was worth making.

The ladder is the same manoeuvre with the opposite outcome. book4/cyclotomy-ladder-verify.py computes the discriminants here by resultant over ℚ and confirms that none is a perfect square for n = 2…8, so no group sits inside A_n; the group is generically S_n, of order n!, non-solvable from n = 5. At n = 8 the two group orders differ by a factor of 10,080. The historical source for the cyclotomic side is O. Neumann, Cyclotomy: From Euler through Vandermonde to Gauss, in Bradley & Sandifer (eds), Leonhard Euler: Life, Work and Legacy (Elsevier 2007), pp. 323–362; the reading is discussed in Vol VII · Euler. Martin (2004) is still unread here — that gap is unchanged.

The rest of the standing ground

• Irreducibility of $q_k$ over $\mathbb{Q}$ is classical for this family; the note verifies it on $k = 2\ldots10$ and claims nothing beyond the tested range.
• That the ladder roots are Pisot is classical — they are the multinacci numbers, and the check here (conjugate moduli rising $0.618 \to 0.980$, never reaching 1) is a reproduction, not a result.
• The ADE / affine spectral threshold at 2 is the Smith–Coxeter classification of simply-laced graphs by spectral radius. § 3 uses it; it does not extend it.

None of this weakens § 1 or § 2. The collapse to $x^{n+1} - 2x^n + 1$ and the exact identity $2 - r_n = r_n^{-n}$ are elementary and stand as written; what § 7 supplies is where to check them against, which is the difference between a note that can be cited and one that cannot.

import sympy as sp
from mpmath import mp, mpf, findroot
x = sp.symbols('x'); mp.dps = 50

# §1 collapse + irreducibility
for n in range(2, 9):
    q = x**n - sum(x**k for k in range(n))
    assert sp.simplify(sp.expand((x-1)*q) - (x**(n+1) - 2*x**n + 1)) == 0
    assert q.as_poly().is_irreducible

# §2 exact gap identity
for n in range(2, 21):
    r = findroot(lambda z: z**(n+1) - 2*z**n + 1, mpf('1.9') if n > 4 else mpf('1.6'))
    assert abs((2 - r) - r**(-n)) < mpf('1e-40')

# §3-4 ADE threshold and the phi coincidence
import numpy as np
rho = lambda A: max(abs(np.linalg.eigvalsh(A)))
def path(n):
    A = np.zeros((n, n))
    for i in range(n-1): A[i, i+1] = A[i+1, i] = 1
    return A
assert abs(rho(path(4)) - (1 + 5**0.5)/2) < 1e-12          # rho(A_4) = phi
for n in range(3, 9):
    r = max(np.roots([1.0] + [-1.0]*n).real)
    h = np.pi / np.arccos(r/2)
    assert abs(h - round(h)) > 1e-6                          # no further coincidences

# §5 the n=8 transposition
n, target = 8, sorted([2] + [1]*6)
q = sp.Poly(x**n - sum(x**k for k in range(n)), x)
D = sp.discriminant(q.as_expr(), x)
for p in sp.primerange(3, 60000):
    if D % p == 0: continue
    t = sorted(int(sp.Poly(f, x).degree())
               for f, _ in sp.factor_list(q.as_expr(), modulus=p)[1])
    if t == target:
        print("transposition at p =", p); break     # -> 17921

Companion files: book4/ladder-polynomials-verify.py (regenerates every number on this page, including the § 7 cross-check; all checks pass), book4/ladder-polynomials.md (source of this page), Chapter 11 (the log-ζ coordinate repair, audit item M1, closed 24 August 2026), and WP44 — Disaster Theory and the Climate Catastrophe Manifold.