Chemistry Vibrational Mode2 From Jcost

Vibrational mode counting

In classical chemistry, a molecule with N atoms has a fixed number of independent vibrational normal modes: the distinct patterns in which its atoms can oscillate around equilibrium. A linear molecule, one where all atoms lie on a line, has 3N minus 5 modes. A nonlinear molecule has 3N minus 6. For water, with three atoms, that gives 3 times 3 minus 6, which is 3 modes. These formulas come from counting degrees of freedom: each atom moves in three dimensions, giving 3N total motions, then subtracting the three translations of the whole molecule and the two or three rotations it can perform.

In Recognition Science, the framework models the same counting through a cost, a measure of how far a ratio of two quantities sits from unity. The framework's library, a machine-checked collection of formal theorems, defines a domain cost as J(m / e), where J is the forced cost function J(x) = (x + 1/x)/2 - 1. The module proves three general facts about this cost: it vanishes when the two quantities are equal, it never goes negative for positive inputs, and a threshold constant phi - 3/2 is positive. These facts hold for any positive numbers m and e, so the module proves nothing specific to chemistry on its own.

The intended connection to vibrations appears in a research note attached to the module. The note proposes that molecular vibrational modes equal 3N minus (D+2) for linear molecules and 3N minus D minus 3 for nonlinear ones, where D is the spatial dimension. For water, D equals 3, so the nonlinear formula gives 3N - 6, matching the classical count. The note calls this structural, meaning it records where the idea was meant to go, not a result the module proves.

What the module does establish, in plain language, is a template: a cost function with the right properties can serve as a certificate for counting rules. The library proves the template's properties once, universally, so any future module that defines m and e in a subject's own terms can inherit those facts. The vibrational mode count remains a research note, not a theorem, until someone supplies that subject-specific definition.

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]

What this page does not claim

The module does not prove the vibrational mode count for water or any molecule. The research note's formula is a proposal, not a theorem in the library. No claim is made about the physical mechanism that sets a molecule's shape.

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/Vibrational_Mode2_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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