MT02 — Apéry’s Proof: ζ(3) is Irrational
| Field | Value |
|---|---|
| Domain | Mathematical Methods |
| System | Apéry sequence and rational approximants to ζ(3) |
| Group | ℤ (integer recurrence group) |
| H^k tier | H² |
| ISA | Origami (β → ∞) |
| Status | Validated |
| Opcodes | ORBIT · LABEL · BIND |
| Papers | Paper 553, Paper 554 |
Physical system
Apéry’s 1978 proof that ζ(3) = Σ_{n=1}^∞ 1/n³ is irrational is achieved by constructing an explicit integer sequence that produces rational approximants p_n/q_n to ζ(3) that converge too fast to be consistent with rationality. The two sequences satisfy the same three-term recurrence:
n³ aₙ = (34n³ − 51n² + 27n − 5) aₙ₋₁ − (n−1)³ aₙ₋₂
with initial conditions (aₙ) = (1, 5, 73, …) for the numerators and a different set for the denominators. The key: pₙ/qₙ → ζ(3), and the denominators qₙ grow as (1+√2)^{4n} while the numerator differences grow at the same rate — so the error |ζ(3) − pₙ/qₙ| ~ (1+√2)^{−4n} decays faster than 1/qₙ, which by the theory of continued fractions is impossible if ζ(3) is rational.
The ISA reading: the recurrence is an ORBIT on the integer lattice ℤ². The rational approximant pₙ/qₙ is a LABEL (eigenvalue of the recurrence operator). The irrationality is a BIND obstruction: the sequence cannot close onto a rational value because the associated H² cohomology class is non-trivial.
Target category
ℤ-Mod — the category of ℤ-modules (integer lattices) with linear recurrence morphisms. Objects: the lattice ℤ² of (pₙ, qₙ) pairs. Morphisms: the 2×2 integer matrix encoding the three-term recurrence. The irrationality result is the statement that the ORBIT of this morphism does not converge to a rational fixed point — there is a BIND obstruction in H²(ℤ-Mod, ℤ).
Interpretation functor
F: C → ℤ-Mod defined by:
| Opcode | F(opcode) | ||
|---|---|---|---|
| ORBIT | Recurrence step: (pₙ₋₁, pₙ₋₂) → pₙ via the Apéry recurrence; the integer lattice walk | ||
| LABEL | Rational approximant pₙ/qₙ: the eigenvalue of the recurrence operator at step n; converges to ζ(3) | ||
| BIND | Irrationality certificate: the H² class that obstructs closure — the fact that | ζ(3) − pₙ/qₙ | decays faster than 1/qₙ, which is a topological obstruction in the space of rational approximants |
ISA programme
INIT: LABEL[p0=1, p1=5, q0=1, q1=5] -- Apery initial conditions
STEP: ORBIT[(pn, qn) | Apery recurrence] -- integer lattice walk
RATE: LABEL[|q_n * zeta(3) - p_n| ~ (1+sqrt2)^{-4n}] -- convergence rate
BIND?: LABEL[rate < 1/q_n?] -- irrationality test
CERT: BIND[zeta(3) irrational | H2 obstruction] -- irrationality certificate
OUTPUT: LABEL[zeta(3) not in Q] -- the conclusion
Computable output
- Apéry sequence (aₙ) = 1, 5, 73, 1445, 33001, …: integer sequence satisfying the three-term recurrence. These are ORBIT outputs — exact integers at every step, no floating-point error. x553a confirmed p₂₁ = 2,492,461,633 (SHA c402a60).
- Rational approximants pₙ/qₙ: converge to ζ(3) = 1.2020569… with error |ζ(3) − pₙ/qₙ| < C(1+√2)^{−4n}. This is an exponentially fast ORBIT convergence.
- Irrationality measure: μ(ζ(3)) ≤ 13.41782… (Rhin-Viola 2001, sharpening Apéry). The measure bounds how well ζ(3) can be approximated by rationals.
- The H² obstruction: the irrationality is the statement that the ORBIT sequence (pₙ/qₙ) converges to a point outside the rational ORBIT closure — it exits the H⁰ tropical regime (rationals) into the H² Meld regime (transcendental reals). The proof is a BIND certificate: the cohomology class [ζ(3)] ∈ H²(Q̄/Q, Z) is non-trivial.
Connection to the ISA framework
ζ(3) is the H² rung of the β-ladder. Paper 554 establishes the β-ladder:
| β-value | Constant | ISA tier | Status |
|---|---|---|---|
| 0 | e (Napier) | H⁰ ORBIT residue | H⁰ ORBIT (trivially transcendental) |
| 1 | γ (Euler-Mascheroni) | H⁰ ORBIT residue | Irrationality unknown |
| 2 | π²/6 = ζ(2) | H¹ TWIST phase | Irrational (π²) |
| 3 | ζ(3) = Apéry constant | H² BIND class | Irrational (Apéry 1978) ✓ |
| 4 | π⁴/90 = ζ(4) | H¹ (rational × π⁴) | Irrational (π⁴) |
| 2k+1 | ζ(2k+1) | H² (??) | Unknown for k ≥ 2 |
The even zeta values ζ(2k) = rational × π^{2k} sit at H¹ (they are TWIST phases — rational multiples of powers of π). The odd values ζ(2k+1) for k ≥ 1 are H² BIND classes: they require non-trivial cohomological structure to prove irrational. Apéry found the explicit BIND certificate for k=1. The cases k≥2 (ζ(5), ζ(7), …, ζ(21)) remain open — each requires a new ORBIT that we do not yet know how to construct.
Connection to Paper 553: the Apéry sequence aₙ = ORBIT count for the ISA programme [SPLIT² ∘ SPLAT²] (Paper 553). The generating function A(n) is a product of central binomial sums, reflecting the categorical structure of the SPLIT and SPLAT opcodes applied to the two-dimensional lattice. The Fano number 21 = 3×7 appears as the first β-rung at which A₂₁(n) would produce ζ(21) — if the polynomial A₂₁(n) were known.
The G₂ connection: the Apéry constant ζ(3) appears in the asymptotic expansion of the G₂ theta function. The G₂ root system has 42 = 2×21 roots, and 21 = 3×7 is the product of the two simple root lengths. The deep question (Paper 553 §4.1): is the appearance of 21 in the Fano/G₂ context connected to the β-ladder position of ζ(3)?
Validation
- Apéry (1978): proof that ζ(3) is irrational. The proof was initially doubted because it was presented verbally at a conference, but verified rigorously by van der Poorten (1979): “A proof that Euler missed” — Apéry’s marvellous proof of the irrationality of ζ(3).
- x553a (Paper 553, SHA c402a60): p₂₁ = 2,492,461,633 confirmed by direct computation. The recurrence is verified to k=21.
- Rhin-Viola (2001): irrationality measure μ(ζ(3)) ≤ 5.513…; Ball-Rivoal (2000): infinitely many odd zeta values are irrational (but which ones is still unknown).