Chemistry Ir5

IR5 module

Infrared (IR) spectroscopy measures how molecules absorb infrared light, which excites their bonds into vibration. A key rule of thumb is that absorption intensity scales with the square of the change in the molecule's dipole moment during the vibration, written |d(mu)/dQ|^2. Stronger dipole change means a stronger absorption band. This is a standard result in physical chemistry, independent of any framework.

The chemistry ir5 module in Recognition Science is a small formal file that attempts to connect this idea to the framework's central object, the cost function, a measure of the forced price of a recognition event. The module defines a domain cost as the cost function applied to a ratio of two real numbers, m and e, and it establishes three general facts about that cost: it equals zero when m equals e, it is never negative for positive inputs, and a certain threshold value, phi minus 3/2, is positive. These are all true statements about the cost function itself.

What the module does not do is establish anything specific about infrared absorption. The definition of domain cost uses m and e as abstract real numbers, not as dipole moment and charge. The docstring openly states that the file proves nothing specific to the subject, because the cost is defined without reference to it. The research note above the code records the intended direction: that for vibrations at the golden ratio amplitude, intensity might be about 1.4 percent of maximum, but this is an idea for future work, not a proved result.

In Recognition Science, the library's value here is as a template. The module shows how one would begin to attach the cost function to a chemistry concept, and it certifies that the cost has the basic properties any sensible measure should have. But the bridge from the abstract cost to a real dipole moment and charge, the step that would make this a theorem about IR absorption, remains open. The module is a placeholder with a clean foundation, not a finished derivation.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

This module proves any specific claim about infrared absorption intensities. The 1.4 percent figure is a proved result; it is a research note. The abstract cost function has been connected to any physical dipole moment or charge.

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/Chemistry/IR5.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: