Foundation Pair Kernel Green Fourier3 Outer Post Ibpamplitude Readout Im Eq Neg

A Fourier identity

The declaration outerPostIBPAmplitudeReadout_im_eq_neg_sin_integral is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It establishes a precise identity about the imaginary part of a complex-valued readout, which is a number extracted from a physical model. The identity states that this imaginary part equals the negative of a certain sine integral, an integral of the sine function over a specified range. In plain terms, it says that a particular oscillatory component of the readout is exactly the negative of another oscillatory component, a relationship that is not assumed but proved.

This identity is part of a larger effort to understand the Green function, which describes how a disturbance spreads through a discrete lattice. The framework models a three-dimensional cubic lattice where each point connects to its six nearest neighbors. The Green function is defined through a Fourier representation, which expresses the function as an integral over a cube of wavevectors. The declaration in question is a technical lemma that helps analyze the behavior of this Green function along a specific axis, a step toward proving that it decays like one over distance.

What the declaration does not claim is equally important. It does not establish the full asymptotic behavior of the Green function, nor does it identify the leading coefficient of that decay. The theorem is a single, exact identity about a particular readout, not a complete solution to the lattice Green function problem. It does not claim that the sine integral itself has a closed form, only that the imaginary part of the readout equals its negative. The framework's own documentation states that the two-sided decay rate and the full singularity extraction remain later rungs of the plan, and this declaration does not preempt those goals.

For a reader, the practical consequence is that a specific, checkable relationship holds exactly within the framework's model. It is a building block, not a finished edifice. The identity is one of many lemmas that, taken together, aim to show that the discrete lattice behaves like continuous space at large distances. This particular theorem confirms that a certain oscillatory piece of the readout has a simple, exact form, a small but solid step in a longer proof chain.

private def remPartialReadout (r : ℕ) : ℂ :=
  ∫ k : V3, 𝐞 (-⟪k, radialFreq r⟫) • remPartialCutoff k

/-- On-axis radial readout: `greenF r = Gint (r, 0, 0)`. Isotropy of the
leading term (later rungs) makes the on-axis limit the full answer. -/
def greenF (r : ℕ) : ℝ := Gint fun i => if i = 0 then (r : ℤ) else 0

What this page does not claim

The theorem does not establish the full asymptotic behavior of the Green function. The theorem does not identify the leading coefficient of the Green function's decay. The theorem does not claim that the sine integral has a closed form.

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/PairKernelGreenFourier3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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