Physics Neutron Lifetime3 From Jcost

Neutron lifetime from cost

The neutron is an unstable particle that lives, on average, about 879.4 seconds before decaying, roughly 15 minutes. A free neutron's lifetime is a measured quantity, not a derived one in standard physics. In the Recognition Science framework, the idea was to connect this number to the golden ratio φ ≈ 1.618, which appears throughout the framework's cost structure. The research note in the module records an attempt: τ_n = φ^17 × 0.246 s ≈ 879 s, where 0.246 is a fitted factor and φ^17 ≈ 3571.

The machine-checked library of formal theorems in the framework does not prove that formula. The module defines a cost (a penalty assigned to a recognition event) as a function of mass divided by charge, Jcost(m/e). It proves three general facts about this cost: it equals zero when the two inputs are equal, it is never negative for positive inputs, and a certain threshold value, φ − 3/2, is positive. These are properties of the cost function itself, not of the neutron.

The reason is a matter of definition. The module defines domainCost as Jcost(m/e) without specifying what m and e mean for a neutron. The docstring states this plainly: the paragraph about the neutron lifetime is a research note recording where the idea was meant to go, not a result. The same body of text is shared verbatim with 2383 sibling modules, each one a template waiting for a subject-specific definition of its variables.

In Recognition Science, the framework models physical quantities by forcing them through a cost structure. The golden ratio emerges from the framework's central theorem about cost functions. But for the neutron, the connection stops at a numerical coincidence. The measured lifetime is 879.4 s; the formula φ^17 × 0.246 gives about 879 s, close but with a free factor. The framework proves the cost function's general properties, and it proves the threshold is positive, but it does not derive the neutron's lifetime from first principles.

What the module does establish is a template. It shows what a certified claim about the neutron would need: a definition of m and e in the neutron's own terms, so that the cost function actually describes the particle. Until that definition exists, the module remains a general statement about costs, not a statement about neutrons. The reader can see exactly where the framework's proof ends and where the physics would have to begin.

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]

structure NeutronLifetime3Cert 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

What this page does not claim

The neutron lifetime formula τ_n = φ^17 × 0.246 s is not proved by the module. The framework does not derive the measured neutron lifetime from its cost function. The factor 0.246 in the research note is fitted, not derived.

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/Physics/Neutron_Lifetime3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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