Foundation Neutron Proton Diff Rs5

The neutron-proton gap

The neutron and proton are the two building blocks of atomic nuclei. They are nearly identical in every way except one: the neutron is heavier. The difference is small, about 1.293 mega-electronvolts (MeV), yet it shapes the universe. It decides which atomic nuclei are stable, whether stars can fuse hydrogen into helium, and why the world is made of matter rather than just radiation. This mass gap is a fixed, measured fact of nature, and any theory of fundamental physics must account for it.

In the Recognition Science framework, the gap is approached through a cost function, a mathematical rule that assigns a price to any ratio of two quantities. The framework's central theorem proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The module Neutron_Proton_Diff_RS5 defines a specific cost for the neutron-proton system as J(m/e), where m is the neutron mass and e is the proton mass. The machine-checked library of formal theorems proves three general facts about this cost: it is zero when the two masses are equal, it is never negative for positive masses, and the golden ratio minus 3/2 is a positive threshold.

These three facts are true, but they are also generic. The module proves nothing specific to neutrons or protons because the definition of the cost does not reference their actual masses. The research note attached to the module records the intended goal: J(phi) times 10.96 MeV equals 1.293 MeV, matching the measured gap. But this numerical agreement is not a theorem. It is a hypothesis, because the module never defines what m and e are in physical terms. The note itself says the module proves nothing specific to this subject.

What the module does establish is a template. It shows that the cost function vanishes at equality, stays nonnegative, and has a canonical positive threshold. These are the properties any candidate definition of m and e would need to satisfy. The module is a placeholder, a scaffold on which a real physical derivation could be built. The work that would turn it into a theorem about the neutron-proton gap remains open.

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

The module does not derive the 1.293 MeV neutron-proton mass difference as a theorem. The module does not define the neutron mass or proton mass in physical terms. The numerical agreement with the measured gap is a hypothesis, 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/Foundation/Neutron_Proton_Diff_RS5.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: