Geometric Shadows in Apéry's Polynomial: $G_2$ Factorisation, p-Adic Valuation Bounds, and the Transcendental Obstruction to Ab Initio Derivation

We identify three structural shadows of $G_2$ geometry in Apéry's 1978 proof that $\zeta(3)$ is irrational, and report three computational experiments that strengthen the connection.

The three geometric shadows (carried over from v1.0)

First, the Apéry polynomial $A(n) = 34n^3 + 51n^2 + 27n + 5$ factors as $(2n+1)(34T(n)+5)$, where $T(n) = \frac{1}{2}n(n+1)$ is the $n$-th triangular number. The factor $(2n+1)$ is the spin-orbit boundary of the $n$-th octonionic weight shell, and the constant $5$ counts the Mac Lane pentagon identities governing associativity coherence in the $G_2$ root system. The coefficient $34$, which also appears as the middle coefficient of the Apéry characteristic polynomial $x^2 - 34x + 1$, encodes the self-similar structure of the recurrence via the Pisot unit $(1+\sqrt{2})^4$.

Second, we conjecture that the Beukers kernel $1/(1-uvw)$ is generated by the iterated $G_2$ associator trilinear form with Fano coefficient $|[e_i,e_j,e_k]| = 2$, and that the prefactor $5/2$ in Apéry's series $\zeta(3) = \frac{5}{2}\sum_{n\geq 1} \frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$ equals the ratio $\pi_\mathrm{ML}/\kappa = 5/2$ of the Mac Lane pentagon constant to the Fano coefficient.

Third, we conjecture a Fano-Weighted Volume theorem: the non-associative correction to the octonionic Fundamental Theorem of Calculus accumulates as a 2-cocycle on triangulated surfaces (not a 1-cocycle on edges), and the Beukers triple integral $\zeta(3) = -\int_{[0,1]^3} \frac{\log(xyz)}{1-xyz},dx,dy,dz$ is the finite integrated form of the Associator Penalty $\mathcal{A}$ introduced in the companion paper on Non-Associative Calculus (doi:10.5281/zenodo.20025384).

New in v2.0: three computational experiments

§4 — $p$-Adic Valuation Audit. A numerical audit of the first 100 terms of the Apéry sequence $q_n$ across primes $p \leq 11$ shows that the denominators of $q_n$ sit with zero slack against the $G_2$-equivariant integrality bound $v_p(\mathrm{denom}(q_n)) \leq 3\lfloor \log_p n \rfloor$. Shifting any coefficient by 1 (e.g., $34 \to 33$) reduces integrity from 100% to 8%, demonstrating that the constants $34, 51, 27, 5$ occupy the unique ground state of the $G_2$ vacuum.

§5 — Ore Algebra Recovery. The Apéry recurrence $(n+1)^3 u_{n+1} = A(n),u_n - n^3,u_{n-1}$ is recovered ab initio in 1.67 seconds from 30 exact integer terms using \texttt{ore_algebra.guess} in SageMath, confirming that $34, 51, 27, 5$ are the unique minimal recurrence coefficients for this sequence.

§6 — The Null-Space Flip (Transcendental Obstruction). A parametric compatibility analysis over $\mathbb{Q}(n)$ reveals a fundamental obstruction to polynomial derivation of the recurrence from the Beukers geometry. At certificate degree $d = 3$ the system is over-constrained (67 non-trivial compatibility equations, inconsistent). At $d = 5$ (2.3-hour computation, 648 certificate columns, 688 monomial rows) the certificate space becomes a \emph{topological sponge}: its left kernel has dimension 141 but produces zero non-trivial compatibility equations, absorbing the recurrence vectors entirely. No polynomial degree pins $(a_0 : a_1 : a_2)$ uniquely. The root cause is that the true WZ certificate contains $D^{n+2}$ with $n$ in the exponent — transcendental in the algebraic sense, outside the polynomial ring $\mathbb{Q}[x,y,z]$ for any finite degree.

Together the three shadows and three experiments suggest that $\zeta(3)$ is the canonical irrationality invariant of the $G_2$ exceptional structure, and that the $\zeta(21)$ conjecture of doi:10.5281/zenodo.20029647 is its 7-dimensional counterpart.