Cosmology Cosmic Strings4 From Jcost Cosmic Strings4 Cert

Cosmic string certification

Cosmic strings are hypothetical one-dimensional cracks in the fabric of spacetime, left over from phase transitions in the early universe. If they exist, they would be extremely thin but enormously dense, and their gravitational pull could bend light from distant galaxies. A key observable quantity is the string tension, often written as Gμ, which measures how much energy a string carries per unit length. Current observational limits require Gμ to be less than about 10⁻⁷.

In the Recognition Science framework, a ledger (a discrete record of recognition events) assigns a cost to comparing two quantities. The framework's central cost function is J(x) = (x + 1/x)/2 − 1. The declaration CosmicStrings4Cert is a machine-checked certificate, a formal proof object in the framework's library of verified theorems. It establishes three general facts about this cost function applied to a ratio: the cost is zero when the two quantities are equal, the cost is never negative for positive inputs, and a certain threshold value involving the golden ratio is positive.

These three facts are completely general properties of the cost function. They hold for any positive real numbers m and e. The certificate does not define what m and e mean in the context of cosmic strings. It does not specify a string tension, a mass scale, or any physical parameter. The framework's own documentation states this plainly: the proof proves nothing specific to cosmic strings, because the cost function is defined without reference to one.

What the certificate does not claim is as important as what it proves. It does not claim that cosmic strings exist, that the framework's model of them is correct, or that any predicted tension matches observation. The research note attached to the framework sketches an intended application: a predicted tension Gμ = 0.118 × (v/M_Pl)², which at a symmetry-breaking scale of 10¹⁶ GeV gives about 8 × 10⁻⁶, a value marginally excluded by current limits. But this is a note about where the idea was meant to go, not a result of the certificate.

In plain terms, CosmicStrings4Cert is a small, correct piece of mathematics about a cost function. It is a building block that could be used in a future, more complete model of cosmic strings, but it is not itself such a model. The certificate's value lies in what it rigorously proves about the cost function's basic behavior, not in any claim about the physical universe.

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 certificate does not prove that cosmic strings exist. The certificate does not claim that the framework's model of cosmic string tension matches observation. The certificate does not establish any value for the string tension Gμ.

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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