The irrationality measure of {\pi} is at most 7.101862832357

Technical Summary: The Irrationality Measure of $\pi$ is at Most 7.101862832357

1. Problem Statement
The paper addresses the determination of the irrationality measure $\mu(\pi)$, defined as the infimum of real numbers $\mu$ such that the inequality $|\pi - p/q| > q^{-\mu-\varepsilon}$ holds for all integers $p$ and sufficiently large positive integers $q$. While the exact value remains unknown, the goal is to establish the tightest possible upper bound. The current state-of-the-art prior to this work was established by Zeilberger and Zudilin [5], who proved $\mu(\pi) \le 7.103205334137\dots$ using a specific contour integral construction with fixed parameters.

2. Methodology
The author introduces a refinement of the Zeilberger–Zudilin integral by incorporating two independent numerator exponents, $A_1$ and $A_2$, into the integrand:
$$I_{A_1, A_2}(N) = -i \int_{4-2i}^{4+2i} \frac{(x-5)^{A_1 N} Q_0(x)^{A_2 N}}{x^{N+1}(x-10)^{N+1}} dx$$
where $Q_0(x) = (x-4+2i)(x-4-2i)(x-6+2i)(x-6-2i)$.

The methodology proceeds through four main stages:

• Arithmetic Construction: The integral is transformed via a change of variables ($t = x-5$) into a rational function $R_n(t)$ integrated over a vertical segment. The paper derives a partial-fraction decomposition of $R_n(t)$, identifying coefficients $c_{j,n}$. By analyzing the $p$-adic valuations of these coefficients (specifically at primes 2 and 5), the author constructs integer linear forms $M_n J_n = U_n + V_n \pi$. A critical component is the "removable-prime" saving, where a product $\Phi_n$ of specific primes is shown to divide the denominators of the coefficients, thereby reducing the growth rate of the clearing denominator $M_n$.
• Analytic Estimation: The paper estimates the asymptotic growth of the coefficients $V_n$ and the decay of the integral $J_n$.
  • The coefficient growth rate $r$ is determined using a positive-coefficient saddle-point method (Lemma 4.1) applied to the generating function of the coefficients.
  • The integral decay rate $s$ is determined by deforming the integration contour to a path passing through a complex saddle point $y_c$, maximizing the phase function $F(y)$.
• Optimization of Parameters: The author defines an explicit auxiliary bound function $B(\alpha, \beta)$ (where $\alpha, \beta$ are normalized exponents) representing the resulting irrationality measure estimate. The paper demonstrates that the original Zeilberger–Zudilin point is not a local minimizer for this two-parameter function. A search within the "diagonal family" ($A_1 = A_2$) identifies a specific rational point $(A_1, A_2) = (1857/2785, 1857/2785)$ that yields a lower bound.
• Local Optimality Proof: The paper proves that this specific rational point is a strict two-dimensional local minimizer of the bound function $B(\alpha, \beta)$ within the admissible arithmetic chamber. This is achieved by deriving an exact logarithmic variation formula for the prime-saving term and analyzing the directional derivatives across twelve sectors in the parameter space.

3. Key Contributions and Results

• New Upper Bound: The primary result (Theorem 1.1) establishes that:
  $$\mu(\pi) < 7.101862832357$$
  This improves upon the Zeilberger–Zudilin bound by more than $0.001342501780$, a reduction of approximately $0.0189\%$.
• Parameter Selection: The specific parameters used are $(a, b, c) = (1857, 3714, 5570)$, corresponding to normalized exponents $(\alpha^*, \beta^*) = (1857/5570, 1857/2785)$.
• Local Minimality: Theorem 1.2 and Theorem 6.3 prove that the chosen parameter point is a strict local minimizer of the auxiliary bound function $B(\alpha, \beta)$. The proof involves a rigorous analysis of the directional derivatives of the prime-saving term $\phi(\alpha, \beta)$, showing that any small perturbation in the parameters increases the bound.
• Rigorous Numerical Enclosure: The paper provides explicit, rigorous rational enclosures for all intermediate constants, including the prime-saving rate $\omega$, the coefficient growth rate $r$, and the integral decay rate $s$. These calculations avoid floating-point assumptions, utilizing exact rational arithmetic and interval bounds for logarithms and Bernoulli numbers.

4. Significance and Scope
The paper claims significance in two specific areas:

1. Quantitative Improvement: It provides a new upper bound for $\mu(\pi)$ that improves upon the Zeilberger–Zudilin bound, demonstrating that the fixed-parameter construction can be refined by introducing a second independent exponent.
2. Structural Insight: It establishes that the previously optimal diagonal point is not a local optimum for the two-parameter deformation. The work provides a detailed geometric and analytic understanding of the "arithmetic chamber" where these bounds are defined, proving that the new point is a local minimizer in a two-dimensional sense.

Limitations and Claims
The author explicitly states the modest scope of the results:

• The result is an upper bound, not the exact value of $\mu(\pi)$.
• The local minimality (Theorem 1.2) is a local statement about the specific auxiliary function $B(\alpha, \beta)$ within the admissible chamber. It does not claim global optimality over all possible constructions, all possible denominator savings, or all positive parameters.
• The improvement is specific to the family of integrals defined by the two-exponent deformation; it does not imply that the true irrationality measure of $\pi$ is close to this bound, nor does it rule out better bounds from entirely different constructions.

Disclosure
The paper notes that Generative AI (OpenAI's GPT-5.6 Sol and GPT-6 Astra) assisted in parameter exploration, proof development, computation, and manuscript drafting. However, the author asserts full responsibility for all mathematical claims and the final text.