Nuclear Proton Lifetime Bound Rs

The proton lifetime bound

The proton, one of the building blocks of atomic nuclei, is very stable. The Super-Kamiokande experiment in Japan has set a lower limit on its lifetime: a proton, left alone, has not been observed to decay in at least 1.6e34 years, a number far larger than the age of the universe. This measurement does not say the proton is immortal, only that if it decays, it does so on a timescale that dwarfs any ordinary physical process.

Recognition Science offers a specific prediction for this lifetime. The framework's central cost function, which measures the price of a recognition event, forces a particular scaling law based on the golden ratio φ ≈ 1.618. The predicted proton lifetime is φ^163 years. To see how this number arises, take the measured lower limit of 1.6e34 years and ask what power of φ produces it: the exponent is log(1.6e34) / log(φ), which equals about 162.7. Rounding up to the nearest integer gives 163, and φ^163 years lands close to the experimental bound. This is a prediction, not a derivation: the framework identifies the lifetime as a specific power of φ, but the exact match to the measured value remains an open question.

The machine-checked library of formal theorems that accompanies the framework proves three general facts about the cost function, none of which are specific to the proton. It shows that the cost vanishes when the two inputs are equal, that it is never negative for positive inputs, and that φ − 3/2 is positive. These results are true for any pair of positive real numbers, not just for the proton's mass and the electron's mass. The module that carries the proton's name defines the cost as a function of the ratio of two masses, but it does not define what those masses are. The prediction itself lives in a research note, not in the formal theorems.

What the module does establish, in plain language, is a template. It shows that the framework's cost function behaves in a well-behaved way: zero at equality, nonnegative everywhere, and with a positive threshold value. The proton lifetime bound is the intended application of this template, but the formal content stops at the general properties. The gap between the template and the prediction is a definition of the proton mass and the electron mass in the framework's own terms. Until that definition exists, the φ^163 year prediction remains a research note, not a theorem.

This distinction matters for anyone reading the framework's claims. The cost function's properties are proved; the proton lifetime is not. The framework's library is honest about this gap, and the honest reading is that the proton prediction is a target, a place where the framework's ideas might connect to experiment, not a place where they already have.

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

noncomputable def cert : ProtonLifetimeCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

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 proton lifetime prediction is not a proved theorem in the framework's library. The module does not define the proton mass or electron mass in the framework's terms. The measured lower limit of 1.6e34 years is not a framework 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/Nuclear/Proton_Lifetime_Bound_RS.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: