The previous post computed ζ(2) — natively the construction hit the recurring $z = 0$ obstruction, but ζ(2) = π²/6 gave a composition fallback through the Machin direct-π PIVP. ζ(5) has no such fallback. This post records the three routes a GPAC / polynomial-PIVP construction of ζ(5) can take, and the obstruction that each one hits. Under the limit-equals-exact-target rule, ζ(5) currently has no PIVP of any kind. This is the headline open question for the holonomic side of the project.

Where ζ(5) sits in the Apéry ladder

The even zeta values $\zeta(2), \zeta(4), \zeta(6), \ldots$ are all rational multiples of powers of $\pi$ — Euler’s $\zeta(2k) = (-1)^{k+1} \frac{B_{2k} (2\pi)^{2k}}{2 (2k)!}$. So once we have a polynomial PIVP for $\pi$, every even zeta value drops out as a composition: square $P$, take a rational multiple, done. The ζ(2) post shows this in eight states.

The odd zeta values are different. $\zeta(3)$ has an Apéry recurrence and Beukers’ triple-integral residual, both of which yield a clean holonomic ODE in a single variable $z$ — this is the textbook example of the residual generating-function pipeline (and also the textbook example of the $z = 0$ obstruction when one tries to lift the ODE to a polynomial PIVP without a fixed point at the initial value).

$\zeta(5)$ does not have an analogous Apéry construction. The strongest known result is Rivoal–Ball (2001): infinitely many of $\zeta(3), \zeta(5), \zeta(7), \ldots$ are irrational, without isolating any single odd $\zeta(2k+1)$ beyond $\zeta(3)$. Zudilin (2001) refined this: at least one of $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational. Whether $\zeta(5)$ specifically is irrational is open. Whether $\zeta(5)$ admits a polynomial PIVP whose continuous-time limit is exactly $\zeta(5)$ is also open, and the two questions are entangled — both come down to producing a sufficiently sharp Diophantine / analytic structure for $\zeta(5)$ that the literature has not yet found.

Route A — direct polylogarithm

The most elementary route mirrors the dilogarithm route to ζ(2). Starting from $\zeta(5) = \mathrm{Li}_5(1)$ and the polylogarithm recurrence

we get a chain $\mathrm{Li}_{1}, \mathrm{Li}_{2}, \mathrm{Li}_{3}, \mathrm{Li}_{4}, \mathrm{Li}_{5}$, each related to the next by a single ODE step with a $1/z$ factor. The chain encodes ζ(5) as the final state’s value at $z = 1$.

The obstruction is the same one that killed Route A for ζ(2), only now iterated five times instead of once. Each step has a $1/z$ in the right-hand side. The natural drive that clears all five $1/z$ factors is $\dot z = z(1-z)$, which has $z = 0$ as a fixed point — so the trajectory cannot leave the IV. Replacing the drive with $\dot z = (1-z)$ avoids the fixed point but leaves $1/z$ factors in the lifted right-hand sides; absorbing each into auxiliary states (a $u_k := 1/z^{k}$ chain) introduces fresh $1/z$ factors in the auxiliary dynamics. The obstruction iterates rather than resolves.

This is the same recurring $z = 0$ structural feature recorded in posts 010, 011, 012. For ζ(5) the polylog chain inherits it five times over — nothing new is learned.

Route B — Zudilin’s third-order recurrence

The closest analogue to Apéry’s ζ(3) recurrence is Zudilin’s 2002 recurrence for ζ(5) (arXiv math.NT/0206178). It is a third-order linear difference equation with degree-9 polynomial coefficients:

with explicit polynomials $a_{0}(n), a_{1}(n), a_{2}(n) \in \mathbb{Z}[n]$ of degrees $3, 9, 8$ respectively. Three independent solution sequences exist: denominators $q_{n}$, numerators $p_{n}$, and a “parasite” sequence $\tilde p_{n}$. The Diophantine approximations $p_{n}/q_{n} \to \zeta(5)$ converge geometrically.

The asymptotic rate is determined by the leading-coefficient ratios. The indicial polynomial is

with three real roots

The denominator sequence $q_{n}$ grows like $|\mu_{3}|^{n}$, and the residual $p_{n} - \zeta(5) q_{n}$ decays at the second-largest-modulus rate $|\mu_{2}|^{n}$. The per-step convergence ratio for $p_{n}/q_{n}$ is therefore

about $-3.85$ digits of accuracy per increment in $n$. This is fast in the discrete sense — comparable to a Chudnovsky-class series for $1/\pi$ — but it is not enough for a Diophantine criterion of the Apéry type, which is why ζ(5) irrationality remains open even with the recurrence in hand.

For the GPAC question the recurrence introduces three obstructions of its own.

B1: the eval point is algebraic. The natural generating function is $Q(z) = \sum_{n \ge 0} q_{n}\, z^{n}$, holomorphic on the disk $|z| < 1/|\mu_{3}|$. The “natural” evaluation point at which $Q(z)$ encodes the limit $\zeta(5) = \lim p_{n}/q_{n}$ is at $z^{*} = 1/\mu_{3}$ — a reciprocal root of an irreducible cubic over $\mathbb{Q}$, so a degree-3 algebraic number. A polynomial PIVP with rational coefficients cannot drive $z$ from a rational starting value to $z^{*}$ in finite continuous time and reach $z^{*}$ exactly; the trajectory asymptotes to $z^{*}$ only if $z^{*}$ is itself a fixed point of a rational-coefficient drive, which forces the drive’s defining polynomial to have $z^{*}$ as a root — impossible for an irrational algebraic number over $\mathbb{Q}$.

This is the same flavor of algebraic-eval-point obstruction that killed BBP at base $1/\sqrt{2}$ in post 011. For BBP the obstruction was at degree 2 ($\sqrt{2}$); for ζ(5) it is at degree 3.

B2: the readout is a ratio. Even granting B1 (suppose we lift the holonomic ODE for $Q(z)$ to a polynomial PIVP and evaluate at some rational $z_{*} \neq z^{*}$), the readout for $\zeta(5)$ is the asymptotic ratio $\zeta(5) = \lim p_{n}/q_{n}$, which corresponds to a residue-or-asymptotic operation on $(P(z), Q(z))$, not a finite-state polynomial readout. To extract the ratio inside the PIVP one must compute $1/Q(z_{*})$, which requires an inverter — exactly the construction the residual generating-function approach was meant to eliminate.

The ζ(3) case avoids B2 because Beukers gives an isolated residual integral $A_{n} - \zeta(3) B_{n} = \iiint (\cdots)^{n}$ — the residual is a single non-negative integrand, and the constant term of its generating function is exactly $\zeta(3)$ as a definite integral. No ratio is needed.

B3: no Beukers-style residual integral is known for ζ(5). The multi-integral approach of Vasilyev (2001) and Krattenthaler–Rivoal (2007) produces five-fold integrals over $[0,1]^{5}$ that yield linear combinations of $1, \zeta(3), \zeta(5)$ — not a clean residual isolating $\zeta(5)$. Producing an explicit family

with $F_{n}$ a non-negative rational function whose Picard–Fuchs operator has order at most 4 and rational-in-$z$ coefficients — that would simultaneously give the missing GPAC route and a candidate irrationality proof for $\zeta(5)$. As of 2026 it is an open problem in transcendence theory.

Route C — composition

ζ(2) had ζ(2) = π²/6, and the existence of a polynomial PIVP for π made the composition route immediate. ζ(5) has no closed-form reduction to constants the project currently has PIVPs for. ζ(5) is not a rational multiple of any $\pi^{k}$ — that would put it in the even-zeta family, contradicting Rivoal–Ball-style results. ζ(5) is not known to be expressible in terms of Catalan’s constant, Apéry’s constant ζ(3), or any other “compound” constant for which a PIVP exists. Conjecturally the family $\{1, \pi^{2}, \zeta(3), \zeta(5), \zeta(7), \ldots\}$ is $\mathbb{Q}$-linearly independent — a strong form of the irrationality conjecture for odd zetas. If so, no rational composition of currently-PIVP-computable constants can reach ζ(5).

So Route C, which saved us for ζ(2), does not exist for ζ(5).

What this post is recording

• Three routes, all blocked.
  • Route A (Li₅ chain): $z=0$ obstruction iterated, same as posts 010, 011, 012.
  • Route B (Zudilin recurrence): algebraic eval point + ratio readout + missing residual integral, three independent obstructions any one of which is fatal.
  • Route C (composition): no known reduction.
• Polynomial PIVP whose continuous-time limit is exactly ζ(5): none currently known. This is the headline open question for the holonomic side of the GPAC project.

What would unblock things

The cleanest single advance would be a Beukers-style residual integral for ζ(5): explicit rational sequences $A_{n}, B_{n}$ with

for some polynomials $P, Q, R$ with $Q, R > 0$ on $(0,1)^{k}$, and a Picard–Fuchs operator of order $\le 4$ in the auxiliary variable arising from the family. This would:

1. Convert Route B’s ratio readout into a difference readout, removing the inverter.
2. Provide a generating function $R(z) = \sum I_{n} z^{n}$ with $R(0) = \zeta(5)$ as a clean definite integral, mirroring the ζ(2) integral form in post 012.
3. As a side effect, give the analytic structure that has so far eluded irrationality proofs for ζ(5) directly.

The $z = 0$ obstruction for the lifted PIVP would still need to be resolved, but at least the problem would shift from “construct the analytic object” to “lift a holonomic ODE without a fixed-point drive” — a problem we can name precisely and track across cases.

For now, ζ(5) sits in our taxonomy as a constant that GPAC visibly cannot yet compute under the limit-equals-exact-target rule. That is the honest entry; the post is here to keep the open question on the page.

行有不得，反求諸己。其身正而天下歸之。 When action does not achieve what we sought, look inward. When the standard is straight, what was sought returns of its own accord. — 《孟子 · 離婁上》