Physics Graviton Mass3 From Jcost

A module named for a bound

The graviton is the hypothetical quantum of the gravitational field, the particle that a quantized theory of gravity would exchange. Experiments have never detected it directly, so physicists quote only an upper bound on its mass: less than 1.76 x 10^-23 eV/c^2, a limit derived from observations of gravitational waves. In the Recognition Science (RS) framework, the file physics-graviton-mass3-from-jcost was planned as a step toward deriving that bound from the framework's cost function, but the machine-checked content does not do that.

The file defines a cost function, a rule that assigns a nonnegative number to a ratio of two quantities, here written as Jcost(m/e). The three theorems it proves are general facts about that function: it equals zero when the two quantities are equal, it is never negative for positive inputs, and a certain threshold value phi - 3/2 is positive. These are true for any positive numbers m and e, with no reference to gravity, mass, or physics. The file's own documentation says so plainly: the proofs establish nothing specific to this subject, because the cost function is defined without connecting m and e to any physical quantities.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 is the unique function forced by five plain conditions, a result proved in the framework's machine-checked library. The framework's research note records an intended application: a graviton mass formula m_g = J(phi) * E_coh / phi^k, where k depends on a coherence energy E_coh, and a claim that the graviton is massless in RS because the gauge sector has zero Noether charge. That note is a research plan, not a theorem. The file's own status line says the paragraph is a research note recording where the idea was meant to go, not a result.

What the module does establish, in plain language, is a small set of properties of the cost function that would hold for any ratio of positive real numbers. The graviton mass bound appears only in the documentation as a stated target, with no formal connection to the proofs. A reader who wants a derived graviton mass in RS will not find it here; the file is a template that could be specialized to a physical subject if someone defined m and e in that subject's own terms.

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]

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)

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

What this page does not claim

This module proves no bound on the graviton mass. The graviton mass upper bound of 1.76 x 10^-23 eV/c^2 is an experimental limit, not a framework result. The framework's claim that the graviton is massless is a research note, not a theorem in this file.

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/Graviton_Mass3_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

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