Foundation Pair Kernel Newtonian3 Dirichlet Integral Eq Pi Div Two Sub Error

The Dirichlet integral

The Dirichlet integral is the improper integral of sin(t)/t from 0 to infinity. Its value is π/2, a result known since Leonhard Euler in the 18th century. The integrand, sometimes called the sinc function, oscillates with decaying amplitude, so the integral converges conditionally, not absolutely. This means the order of integration matters; the standard definition takes the limit of the integral from 0 to R as R grows without bound.

The framework's machine-checked library of formal theorems proves this convergence with an explicit error bound. The declaration dirichlet_integral_tendsto states that the integral from 0 to R of sin(t)/t tends to π/2, and the proof includes a bound on the remainder: the difference between the finite integral and π/2 is at most 2/R. This is a sharp, quantitative statement. It does not merely say the limit exists; it says how quickly the partial integrals approach it.

This result is part of a larger campaign within the framework to trace the constant 4π in three-dimensional geometry. The Dirichlet integral appears when computing the Fourier transform of the Newtonian potential 1/||x||. The framework's library proves the one-dimensional leaf of this calculation, and the error bound is the tool that controls the limit. The broader goal is to show that the 4π appearing in the inverse-square law is a fact of pure three-dimensional geometry, independent of any physical model.

In Recognition Science, this theorem is one rung on a ladder. It establishes a classical analytic fact with machine precision. It does not, by itself, derive any physical constant or prove the existence of three spatial dimensions. It is a lemma, a carefully isolated piece of mathematics that later steps may use. The declaration does not claim to have proved the full Newtonian Fourier identity, which remains a stated target, not a proved theorem.

The practical consequence is a verified building block. Anyone working within the framework can rely on the Dirichlet integral and its remainder bound without rechecking the analysis. The theorem is a small but solid stone in a larger edifice, and its proof is checked by a computer, leaving no room for a hidden gap.

private abbrev E2 := Fin 2 → ℝ

What this page does not claim

The declaration does not prove the full Newtonian Fourier identity lim_N (2π)⁻³ ∫_{||k||<N} cos(k·x)/||k||² dk = 1/(4π||x||). The declaration does not derive any physical constant or prove the existence of three spatial dimensions. The declaration does not claim the Dirichlet integral converges absolutely.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PairKernelNewtonian3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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