Nuclear Neutron Lifetime Structure Neutron Decay Phase Space Positive

A small but solid step

When a free neutron decays into a proton, an electron, and an antineutrino, the process releases a small amount of energy, about 0.782 MeV. This energy is called the Q value of the decay. The Q value matters because it sets the stage for everything else: if the Q value were zero or negative, the decay could not happen at all. In the language of particle physics, the decay is kinematically allowed only when Q is positive.

The phase-space factor is a measure of how much room there is for the decay products to arrange themselves, given that energy. For beta decay, this factor scales as the fifth power of the Q value, written Q^5. A larger Q value means far more available configurations for the final particles, which translates into a faster decay rate. The exact relationship is a standard result in nuclear physics, derived from the density of final states in quantum mechanics.

In the Recognition Science framework, a machine-checked library of formal theorems, this simple physical fact is captured in a formal declaration. The theorem neutron_decay_phase_space_positive proves that if the Q value is positive, then Q^5 is also positive. This is a direct consequence of the positivity of the Q value itself, and it is established by a short, verified argument. The declaration also connects this to the broader structural placeholder for the neutron lifetime, which states that the decay is allowed and the mean lifetime is positive.

This theorem is a necessary condition, not a sufficient one. It confirms that the phase-space factor is not zero and not negative, which is a prerequisite for any meaningful lifetime calculation. However, it says nothing about the actual value of the lifetime. The full derivation of the neutron's mean lifetime, which is measured to be about 881 seconds, remains an open problem in the framework. The experimental discrepancy between bottle and beam measurements is also not addressed by this theorem.

What this declaration establishes is a small but solid logical step. It ensures that the formal structure is consistent with a basic kinematic fact. It does not, by itself, predict the lifetime or resolve any experimental puzzle. It is a building block, not the finished building.

/-- Positive decay Q-value forces positive fifth-power phase-space factor (`Q^5`). -/
theorem neutron_decay_phase_space_positive : neutronDecayQ ^ (5 : ℕ) > 0 := by
  exact pow_pos neutron_decay_allowed 5

/-- Free-neutron beta-decay Q value in MeV (structural placeholder). -/
def neutronDecayQ : ℝ := 0.782

/-- Free-neutron mean lifetime in seconds (experimental reference scale). -/
def freeNeutronMeanLife : ℝ := 881

What this page does not claim

This theorem does not derive the numerical value of the neutron lifetime. This theorem does not address the experimental bottle-beam discrepancy. This theorem does not claim that the phase-space factor is the only factor determining the decay rate.

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

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Derived articles

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