Physics Gravitino Mass From Phi Ladder

Gravitino mass in the framework

The gravitino is the hypothetical superpartner of the graviton, the particle that would carry the gravitational force in a supersymmetric theory. Its mass is not a free parameter: in standard supersymmetry, the gravitino mass m_3/2 is set by the scale at which supersymmetry breaks, roughly m_3/2 = F / (√3 M_Pl), where F is the supersymmetry-breaking scale and M_Pl is the Planck mass. The smaller the breaking scale, the lighter the gravitino; a gravitino near the Planck mass would imply breaking at the highest energies.

Recognition Science (RS) proposes to fix that breaking scale from its own structure. The framework's central object is a cost, a number that measures how expensive it is for reality to recognize one state as another. The cost function J(x) = (x + 1/x)/2 − 1 is forced by five plain conditions, and its unique self-similar point is the golden ratio φ ≈ 1.618. In RS, the supersymmetry-breaking scale F is modeled as φ^k times a coherence energy E_coh, placing the gravitino mass on a ladder of φ-powers. The structural formula proposed is m_3/2 = J(φ) · M_Pl / √3, where J(φ) = (φ + 1/φ)/2 − 1 ≈ 0.118.

The module Gravitino_MassFromPhiLadder.lean does not prove that formula. It proves three general facts about the cost function: the cost vanishes when the two masses are equal (domainCost_at_eq), the cost is never negative for positive masses (domainCost_nonneg), and the threshold φ − 3/2 is positive (canonicalThreshold_pos). These are properties of J applied to any ratio, not of gravitinos specifically. The module defines the cost as J(m/e) without connecting m and e to any physical quantity, so the gravitino paragraph is a research note, not a formal result.

What the module does establish is that the cost function behaves sensibly as a measure of mismatch: it is zero at perfect agreement, nonnegative everywhere, and its natural threshold sits above zero. Those are the properties any useful cost function should have. The step from those properties to a physical gravitino mass requires a definition of m and e in terms of supersymmetry-breaking physics, which the module does not supply. That physical bridge remains open.

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

No physical gravitino mass value is derived or predicted by this module. The formula m_3/2 = J(φ) · M_Pl / √3 is not proved in the Lean library. No connection between the cost function and supersymmetry-breaking scale F is established beyond 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/Gravitino_MassFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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