Nuclear Spont Fission3 From Jcost

Spontaneous fission and its cost

Spontaneous fission is the process by which an atomic nucleus splits into two smaller nuclei without being struck by an outside particle. It happens on its own, driven by the internal competition between the strong force that holds protons and neutrons together and the electric repulsion between the many protons in a heavy nucleus. The probability of fission rises sharply with the size of the nucleus: a uranium-238 nucleus fissions spontaneously with a half-life of about 4.5 billion years, while californium-252, with 98 protons, fissions with a half-life of about 2.6 years. The general rule is that heavier nuclei are more likely to break apart on their own.

In the Recognition Science framework, every physical process is modeled as a cost: a number that measures how hard it is for the universe to keep a ledger, a discrete record of events. The framework's central result is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1, a proved theorem in its machine-checked library of formal theorems. The module called SpontFission3FromJCost applies this cost to the ratio of two masses, m and e, by defining domainCost(m, e) = J(m/e). The library then proves three general facts about this cost: it vanishes when the two masses are equal, it is never negative for positive masses, and the threshold value phi - 3/2 is positive, where phi is the golden ratio.

The module itself proves nothing specific to fission, because its definition of cost does not reference any nuclear quantity. The docstring records an intended research direction: that the spontaneous fission half-life should decrease with mass number A according to log(t_SF) = phi^(A_threshold/A) times a reference value at A = 252, with a threshold near A = 270. That scaling law is a research note, not a proved result. What the module does establish is a certificate: a packaged proof that the cost function has the three basic properties, ready to be reused in any future module that supplies a real definition of m and e in nuclear terms.

The practical consequence is a clean separation between the mathematics and the physics. The cost function's behavior at equality, its nonnegativity, and its positive threshold are now locked in by machine-checked proofs. The next step, defining m and e so that the cost actually tracks fission rates, remains open. The framework's library has done its part: it has built the tool, and the tool is waiting for the right handle.

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

No proved relationship between the cost function and any measured fission half-life. No derivation of the scaling law log(t_SF) = phi^(A_threshold/A) from the framework's axioms. No claim that the threshold value phi - 3/2 is physically meaningful for fission.

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/SpontFission3FromJCost.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

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