Foundation Rs Quantum Tunneling Rate

The tunneling rate

Quantum tunneling is the quantum mechanical phenomenon where a particle passes through a potential energy barrier higher than its own kinetic energy. The probability of this happening decays exponentially with the barrier's width and height. The standard form for the tunneling rate is rate = ν exp(−G), where ν is an attempt frequency and G is the Gamow factor, a dimensionless measure of the barrier's impenetrability.

In Recognition Science, the framework models the Gamow factor as G = 2·J(φ)·Q_barrier/Q_kinetic. Here J(φ) is the framework's cost function evaluated at the golden ratio φ, and Q_barrier and Q_kinetic are quantities representing the barrier height and the particle's kinetic energy. The framework's library of formal theorems proves general facts about the cost function J(x) = (x + 1/x)/2 − 1: it equals zero when its input is 1, it is never negative for positive inputs, and φ − 3/2 is positive. These are properties of the cost function itself, not specific to tunneling.

The module named RS_Quantum_Tunneling_Rate defines domainCost(m, e) = Jcost(m/e), a cost based on the ratio of two quantities. Its machine-checked theorems establish that this cost is zero when m equals e, that it is nonnegative for positive inputs, and that the threshold φ − 3/2 is positive. The module also bundles these facts into a certificate structure. However, the module does not define what m and e mean in tunneling terms; they are left as arbitrary real numbers. The research note in the source states the intended interpretation, but the formal proofs do not depend on it.

In Recognition Science, this is how the framework's central cost function connects to a physical rate. The framework proves the cost function's uniqueness from five plain conditions, and here it appears as the exponent prefactor in a tunneling rate. The framework's contribution is the derivation of J(φ), not the tunneling formula itself, which is standard physics. The module demonstrates that the cost function has the properties a barrier factor needs: it vanishes at equality, stays nonnegative, and has a positive threshold.

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 the tunneling rate formula itself, which is standard physics. This module defines m and e in terms of a specific physical barrier. This module shows that the framework's tunneling rate matches any measured value.

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/RS_Quantum_Tunneling_Rate.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: