Cosmology Cosmic Strings4 From Jcost

Cosmic strings and the cost ledger

Cosmic strings are hypothetical one-dimensional defects in the fabric of spacetime, predicted by some particle physics theories. If they exist, they would be incredibly dense threads, perhaps thinner than a proton but carrying enormous energy per unit length. That energy per length is called the string's tension, usually written as Gμ, a dimensionless number that compares the string's energy to the Planck scale. Observational limits require Gμ to be below about 10⁻⁷, otherwise the strings would have left visible imprints on the cosmic microwave background.

In Recognition Science, the framework models physical costs through a ledger, a discrete record of recognition events where each event has a forced cost. The central cost function is J(x) = (x + 1/x)/2 - 1, which is zero when x equals 1 and grows as x moves away from 1. The module for cosmic strings defines a cost for a string by comparing a mass m to an energy scale e, using the ratio m/e as the input to J. This is a definitional choice, not a derived physical law.

The module proves three general facts about this cost function. First, the cost is zero when the mass equals the energy scale. Second, the cost is never negative for positive inputs. Third, the number φ - 3/2, which the module calls the canonical threshold, is positive. These are all properties of the cost function J itself, not of cosmic strings. The module contains no definition of what m and e mean for an actual string, so it establishes nothing specific about string tension.

The research note attached to the module sketches an intended application: setting m and e to string and Planck values, and computing Gμ = 0.118 × (v/M_Pl)². At a symmetry-breaking scale v of 10¹⁶ GeV, this gives Gμ ≈ 8×10⁻⁶, which is marginally excluded by current limits. This is a hypothesis with a named falsifier: if observations tighten the bound below this value, the simple identification fails. The note is a research direction, not a theorem.

What the module genuinely establishes is a reusable template: any physical quantity that can be expressed as a ratio m/e inherits the cost function's properties of vanishing at equality and staying nonnegative. The cosmic string application awaits a definition of m and e in the string's own terms, which the module does not provide.

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)

theorem cert_inhabited : Nonempty CosmicStrings4Cert := ⟨cert⟩

theorem cert_inhabited : Nonempty CosmicStrings4Cert := ⟨cert⟩

What this page does not claim

The module does not prove that cosmic strings exist or that their tension follows the cost formula. The module does not derive the numerical value 0.118 for the string tension coefficient. The module does not establish any bound on the symmetry-breaking scale v from observations.

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/Cosmology/CosmicStrings4_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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