Chemistry Vibrational Mode2 From Jcost Vibrational Mode Cert

The certification structure

In chemistry, a nonlinear molecule with N atoms has 3N - 6 normal modes of vibration; a linear molecule has 3N - 5. For water, with three atoms, that gives three modes. Recognition Science (RS) has a machine-checked library of formal theorems, and one declaration there, VibrationalModeCert, is named as if it certified a connection between this counting rule and the framework's cost function, a measure of the forced price of recognition events. The name suggests a bridge between molecular vibration and the framework's core mathematics.

What the declaration actually establishes is narrower and fully general. It packages three facts about a function called domainCost, which is defined as Jcost(m / e), where Jcost(x) = (x + 1/x)/2 - 1. The certificate records that this cost vanishes when m equals e, that it is nonnegative for positive inputs, and that the golden ratio minus 3/2 is positive. These are true statements about the cost function itself, proved in the library's formal system. The certificate is inhabited, meaning such a structure exists, and it is nonempty.

The gap is definitional. The declaration defines domainCost using the ratio m / e without ever saying what m and e mean in molecular terms. The documentation itself states this plainly: the formal code proves nothing specific to molecular vibration, because the definition of domainCost makes no reference to one. The paragraph about 3N - 6 modes is a research note recording where the idea was meant to go, not a result. What would make this declaration a theorem about its subject is a definition of m and e in that subject's own terms.

So the honest summary is this: VibrationalModeCert is a valid, machine-checked certificate about the cost function's elementary properties, shared verbatim across 2383 sibling modules. It does not establish any vibrational counting rule. The connection to molecular modes remains a stated intention, not a proved theorem. A reader should not take the declaration's name as evidence that RS has derived the 3N - 6 rule; that derivation is a target, not an achievement.

structure VibrationalModeCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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

Not claimed: that the 3N - 6 rule for nonlinear molecules is derived or proved in this module. Not claimed: that m and e have any specific physical meaning in this declaration. Not claimed: that the certificate's existence implies any connection between molecular vibration and the golden ratio.

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)

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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