The $\zeta(21)$ Apéry Generalization: Cubic Pisot Units, $G_2$ Geometry, and a Conjectured 4-Term Recurrence

CORRECTION NOTICE (2026-08). The premises here are correct: (1+√2)⁴ = 17+12√2 really is the root of x²−34x+1 governing Apéry's recurrence; no Apéry-like proof is known for any odd zeta beyond ζ(3); and Hurwitz's theorem really does restrict continuous cross products to dimensions 3 and 7. The inference does not follow from them. Hurwitz constrains dimensions 3 and 7, not zeta arguments 3 and 21, and the step from "7-dimensional cross product" to "ζ(21)" is a construction internal to this framework rather than a consequence of the cited theorem. The word "consequently" in the abstract should be read as "we propose".

One stated ingredient also needs checking. The polynomial x³−4x²+3x+1, described as the G₂ minimal polynomial associated with ℚ(cos(π/7)), is not the minimal polynomial of 2cos(π/7) — that is x³−x²−2x+1. Its roots (2.8019, 1.4450, −0.2470) are a different quantity. Since the derived characteristic polynomial is obtained from it by Newton power sums, the derivation should be rechecked before the ζ(21) irrationality prediction is relied upon. A revised version is in preparation.

Roger Apéry's proof of the irrationality of $\zeta(3)$ relies on a 3-term recurrence governed by the quadratic Pisot unit $(1+\sqrt{2})^4$, a root of $x^2 - 34x + 1 = 0$. Decades of subsequent searches for analogous recurrences for $\zeta(5)$, $\zeta(7)$, and higher odd zeta values have failed.

Using the Adelic Simplicial Architecture (ASA), we propose that this failure is topological. Hurwitz's theorem dictates that stable continuous cross products exist exclusively in dimensions 3 and 7. Consequently, $\zeta(3)$ (the 3D Pachner fold) and $\zeta(21)$ (the 7D Pachner fold) are geometrically privileged — the only odd zeta values whose internal boundaries close without shattering.

Attempting to force $\zeta(21)$ into Apéry's 3-term template is a structural incompatibility: the 7D cross product is governed by the 7-fold rotational symmetry of the octonions, whose geometric expansion base resides in the totally real cubic field $\mathbb{Q}(\cos(\pi/7))$. This cubic field natively generates a 4-term recurrence. We derive the exact characteristic polynomial $x^3 - 2{,}492{,}461{,}633,x^2 + 5{,}676{,}976{,}825{,}495,x + 1 = 0$ using Newton power sums on the $G_2$ minimal polynomial $x^3 - 4x^2 + 3x + 1$, and establish that the dominant coefficient clears the Diophantine horizon $e^{21}$, predicting that $\zeta(21)$ is irrational.

New in v2.0: We prove that $A_{21}(n)$ must satisfy the anti-palindromic identity $A_{21}(-n-1) = -A_{21}(n)$, a consequence of the odd corner exponent $s = 7$ in the recurrence. This forces $(2n+1) \mid A_{21}(n)$ over $\mathbb{Q}$. Since the leading coefficient $p_{21} = 2{,}492{,}461{,}633$ is odd, the integer-clean factorisation takes the form $2A_{21}(n) = (2n+1)\cdot\mathrm{IntBulk}_{21}(n)$ with $\mathrm{IntBulk}_{21} \in \mathbb{Z}[n]$ of degree 20. The anti-palindromic constraint reduces the free parameters in $A_{21}(n)$ from 22 to 11, giving the tightest known structural constraint on the unknown recurrence polynomial.