Chemistry Ir5 Ir5 Cert

What IR5Cert proves

Infrared absorption spectroscopy measures how strongly a molecule absorbs light at specific frequencies. The standard quantum-mechanical picture ties that intensity to the rate of change of the molecule's dipole moment during a vibration: intensity is proportional to the square of the derivative of the dipole moment with respect to the vibrational coordinate. The Recognition Science library contains a module named IR5 that was intended to connect this picture to the framework's cost function, but the machine-checked content stops far short of that goal.

The library defines a cost function, a measure of discrepancy between two positive numbers, as J(m/e), where J is the framework's forced cost function and m and e are real numbers. The theorem IR5Cert proves three things about this construction. First, when the two inputs are equal, the cost is zero. Second, for positive inputs, the cost is never negative. Third, a particular constant derived from the golden ratio, phi minus 3/2, is positive. The certificate is a structure that bundles these three proved facts into a single object, and the library shows such a certificate exists.

What IR5Cert does not do is make any claim about infrared absorption. The definition of the cost function uses the symbols m and e, but nothing in the file defines what those symbols mean in chemical terms. They could be mass and charge, or any other pair of real numbers. The docstring is explicit: the module proves nothing specific to infrared spectroscopy, because the cost is defined without reference to the subject. The intended application, that a particular vibration amplitude would give an intensity near 1.4 percent of maximum, is a research note, not a proved result.

The honest summary is that IR5Cert is a small, correct piece of mathematics about a cost function. It establishes that the cost vanishes at equality, stays nonnegative, and has a positive threshold constant. It establishes nothing about molecules, spectra, or chemistry. The gap between the certificate and the subject it was named for is a definition of m and e in chemical terms, which the file does not supply.

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

IR5Cert says nothing about infrared absorption intensities or molecular vibrations. The symbols m and e in the cost function are not defined as mass and charge or any other chemical quantities. The 1.4 percent intensity figure is a research note, not a proved result.

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

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