Physics Axion Mass3 From Jcost

Axion mass from J-cost

The axion is a hypothetical particle proposed to solve a puzzle in particle physics: why the strong force does not seem to violate a symmetry called CP. The axion mass is not fixed by the theory that proposes it, so physicists look for ways to predict it from deeper principles. Recognition Science offers one such route, using its cost function, a measure of how expensive a recognition event is, forced by five plain conditions to equal J(x) = (x + 1/x)/2 - 1.

The module called Axion_Mass3_FromJCost does not, in fact, establish the axion mass. It proves three general facts about the cost function J applied to a ratio m/e: it vanishes when m equals e, it is nonnegative for positive inputs, and the number phi - 3/2 is positive. These facts hold for any positive numbers m and e. The module defines domainCost as J(m/e) but never defines what m and e mean for an axion, so the Lean code proves nothing specific to axions.

The research note attached to the module records where the idea was meant to go: set m to the axion mass and e to some reference mass, then J(phi) times a scale mass gives a prediction. At the Z boson mass scale, that yields 10.8 GeV; at a grand unified theory scale of 10^16 GeV, it yields 1.18e15 GeV. These numbers are structural targets, not derived results, because the connection between the cost function and the axion's actual mass is not established in the code.

What the module does establish, cleanly and generally, is that the cost function J has three useful properties: it costs nothing when the two inputs are equal, it never assigns a negative cost to positive inputs, and it has a built-in threshold at phi - 3/2. These are theorems about J itself, not about axions. The axion-specific prediction remains a research note, a hope for what the framework might one day derive if someone defines m and e in axion terms.

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 module does not derive the axion mass from the cost function. The module does not define m or e in terms of any axion property. The numbers 10.8 GeV and 1.18e15 GeV are not theorems; they are structural targets from a research note.

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/Axion_Mass3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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