Foundation Proton Electron Mass Ratio3 From Jcost

The proton electron mass ratio

The proton to electron mass ratio is one of the plainest facts in physics: the proton, a composite particle made of quarks and gluons, weighs 1836.15 times the electron, a fundamental lepton. The measured value, 1836.15, sits between the fifteenth and sixteenth powers of the golden ratio, phi, where phi^15 is 1364 and phi^16 is 2207. The logarithm of the ratio in base phi is 15.4, a number that lands between integer rungs of a phi-power ladder.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, and a cost, the forced price of recognizing one thing as another. The cost function J(x) = (x + 1/x)/2 - 1 is proved unique under five plain conditions in a machine-checked library of formal theorems. The library also proves that the golden ratio phi is the unique self-similar scaling, which gives the phi-ladder a structural role.

The module named ProtonElectronMassRatio3FromJcost defines a domain cost as J(m/e), the cost of recognizing a mass m as a mass e. It proves three general facts: the cost vanishes when m equals e, it is nonnegative for positive inputs, and phi minus 3/2 is positive. These are true for any positive masses, not specifically for protons and electrons. The module proves nothing about the actual proton electron ratio because the definition of m and e is left abstract.

The research note attached to the module records the intended target: the ratio 1836.15 as phi^15.4, consistent with a phi-rung between 15 and 16. That note is not a theorem. The gap between the note and a proof is the missing definition of m and e in terms of the framework's own structure. The machine-checked facts establish the cost function's behavior, not the mass ratio.

What the module does establish is a template: any positive mass ratio has a well-defined cost, the cost is zero only when the masses are equal, and the cost is never negative. This is a foundation for future work, not a derivation of the measured ratio. The measured ratio remains an empirical input, and the phi-ladder remains a hypothesis about where it sits.

noncomputable def cert : ProtonElecMassRatio3Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

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

The proton electron mass ratio is not derived from the cost function in this module. The phi-ladder placement of the ratio is not proved; it is a research note. The module does not establish any property specific to protons or electrons.

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/ProtonElectronMassRatio3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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