Physics Photon Mass3 From Jcost

Photon mass and the cost function

The photon, the quantum of light, is massless in the standard model of particle physics. Its mass is exactly zero because of a symmetry called U(1) gauge symmetry, which forbids a mass term. Experiments put an upper bound on any possible tiny mass: the Particle Data Group gives m_gamma < 10^-18 eV. In Recognition Science, a separate estimate from the framework's cost function suggests a much smaller bound, around 10^-32 eV, but that estimate is a research note, not a proved result.

The module physics photon mass3 from jcost defines a cost function, a measure of recognition effort, as Jcost(m/e), where m is a mass and e is a reference mass. The Lean code proves three general facts about this function: it vanishes when m equals e, it is nonnegative for positive inputs, and a constant phi - 3/2 is positive. These are properties of the cost function itself, true for any two positive numbers, not specific to photons.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 is forced by five plain conditions, a proved theorem in the framework's machine-checked library. The photon module applies this function to a mass ratio, but it does not define what m and e mean for a photon. The docstring says the module proves nothing specific to its subject, because the definition lacks that physical content. The module is a template, shared verbatim with 2383 sibling files, each intending to apply the same cost function to a different topic.

What the module does establish, in plain language, is a certificate: a structured proof that the cost function has the three basic properties, collected into a single object called PhotonMass3Cert. This certificate is inhabited, meaning the properties hold. It is a small, solid piece of mathematics, but it does not derive the photon mass bound. The bound of 10^-32 eV remains a research note, a target for future work that would need a definition of m and e in the photon's own terms.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

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 cert_inhabited : Nonempty PhotonMass3Cert := ⟨cert⟩

What this page does not claim

The module does not prove the photon mass is zero or derive the 10^-32 eV bound. The cost function's properties are general and apply to any two positive numbers, not just photon masses. The research note's estimate is not a theorem and has no machine-checked proof.

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/PhotonMass3_FromJCost.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: