Nuclear Nuclear Magic Numbers2 From Jcost

Nuclear magic numbers

Nuclear magic numbers are the counts of protons or neutrons that make an atomic nucleus unusually stable: 2, 8, 20, 28, 50, 82, and 126. A nucleus with one of these counts sits lower in energy than its neighbors, which is why it resists decay and why nuclear physicists treat these numbers as special. The pattern was established experimentally through the mid-twentieth century, with Maria Goeppert Mayer and J. Hans D. Jensen sharing the 1963 Nobel Prize in Physics for explaining it with a shell model, where nucleons fill discrete energy levels much like electrons in an atom.

Recognition Science offers a different route to these numbers. The framework starts from a single forced cost function, J(x) = (x + 1/x)/2 - 1, which it proves is the only cost function satisfying five plain conditions. The idea is that nature keeps a discrete record of recognition events, and the cost of recognition is forced, not chosen. From that cost function, the framework derives a phi-ladder: a sequence where each step multiplies by the golden ratio φ ≈ 1.618. The research note in the module applies this ladder to nuclear magic numbers with the formula N_magic_k = φ^k × 2, producing the sequence 2, 3.2, 5.2, 8.5, 13.7, 22.2, 35.8, 58. The note compares this to the measured magic numbers 2, 8, 20, 28, 50, 82, 126 and calls the match "structural," but it is a research note, not a theorem.

In Recognition Science, the module named NuclearMagicNumbers2FromJCost establishes three general facts about the cost function, and nothing specific to nuclear physics. It proves that the cost vanishes when the two inputs are equal, that the cost is nonnegative for positive inputs, and that φ - 3/2 is positive. The module defines a domain cost as J(m/e), where m and e are real numbers, but it never defines what m and e mean for a nucleus. The docstring says plainly: the Lean code proves nothing specific to this subject, because domainCost is defined without reference to one. The module is a template, shared verbatim with 2383 sibling modules, and what would make it a theorem about nuclear magic numbers is a definition of m and e in nuclear terms.

The practical upshot is a clean separation. The framework has a proved cost function and a proved phi-ladder, but the step from those to nuclear magic numbers remains a research hypothesis with a named falsifier: if a measured magic number deviates from φ^k × 2 beyond experimental error, the identification fails. The module itself does not close that gap; it only certifies that the cost function has the three basic properties. A reader should treat the magic-number sequence as an intriguing numerical coincidence under active investigation, not as a derived result.

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)

def canonicalThreshold : ℝ := phi - 3 / 2

What this page does not claim

This module derives the nuclear magic numbers from the cost function; it proves only general properties of J(m/e). The phi-ladder sequence is a measured match to nuclear magic numbers; it is a research note with no experimental receipt. The framework's cost function is the same as the nuclear shell model's potential; the two are unrelated formalisms.

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/NuclearMagicNumbers2FromJCost.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.

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