Foundation Proton Radius3 From Jcost

Proton radius and the cost function

The proton's charge radius is a measured quantity: about 0.841 femtometers, where one femtometer is a millionth of a billionth of a meter. In Recognition Science, the framework's central object is a cost function, a rule that assigns a nonnegative number to a ratio of two quantities, meant to measure how far a recognition event is from a perfect match. The library named foundation-proton-radius3-from-jcost defines that cost function as J(x) = (x + 1/x)/2 - 1, the same J that the framework's core theorem forces from five plain conditions.

The library proves three facts about this cost function. First, when the two inputs are equal, the cost is exactly zero: J(1) = 0. Second, for any two positive inputs, the cost is never negative. Third, the number phi - 3/2, where phi is the golden ratio, is positive. These are general properties of the cost function, not statements about protons. The library's own docstring says so plainly: it proves nothing specific to the proton, because the cost is defined on a ratio without connecting that ratio to any physical quantity.

What would make this library a theorem about the proton radius is a definition of the two quantities m and e in the proton's own terms. The pack includes a research note that tries such a definition, but the arithmetic does not work: one attempt gives 572 femtometers, far too large, and another gives 0.016 femtometers, far too small. The note ends with the word "Structural," meaning the intended link is not yet established.

In Recognition Science, the framework models the proton radius as an open target, not a derived result. The library establishes only the general cost-function facts, and the physical bridge from those facts to a proton radius remains to be built. A reader should take the library as a small, correct piece of the framework's foundation, and as a clear example of what the framework does not yet claim.

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]

What this page does not claim

The library derives the proton charge radius from the cost function. The library proves any statement specific to the proton or to any physical system. The research note's arithmetic is a valid derivation.

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/ProtonRadius3FromJCost.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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