Cosmology Sigma8 Tension3 From Jcost Sigma8 Tension3 Cert

What the certificate proves

The sigma8 tension is a real discrepancy in cosmology: measurements of the cosmic microwave background give a value of sigma8 around 0.83, while weak lensing surveys give around 0.77. The Recognition Science framework has a research note suggesting that its cost function could account for this gap, predicting a roughly 12 percent suppression in structure formation. The formal declaration named Sigma8Tension3Cert, however, does not prove that prediction. It proves three simpler, general facts about the cost function, and nothing specific to cosmology.

The cost function, written Jcost, is a real-valued function that the framework derives from first principles. The certificate proves, first, that Jcost of 1 is zero, which means the cost vanishes when the two inputs are equal. Second, it proves that Jcost is never negative when both inputs are positive. Third, it proves that the golden ratio phi is greater than 1.5, a small numerical fact that the framework uses as a threshold. These are the three fields of the certificate structure: cost_at_eq, cost_nonneg, and threshold_pos. The certificate also shows that these three facts are consistent, by constructing an explicit example.

In Recognition Science, the cost function is meant to model a ledger, a discrete record of recognition events, and its cost is forced by five plain conditions. The framework's library, a machine-checked collection of formal theorems, contains this certificate as a small building block. But the certificate itself is deliberately generic: it defines the cost on any two real numbers m and e, without saying what m and e mean. The research note attached to the module mentions sigma8 and structure formation, but that note is not part of the formal result.

What would make this a theorem about the sigma8 tension is a definition of m and e in cosmological terms, for example as the observed values from CMB and weak lensing. That definition does not appear in the certificate. So the certificate establishes a clean, general property of the cost function, and it leaves the cosmological application as a research direction, not a proved result.

The practical upshot is that the certificate is a solid but small piece of mathematics. It guarantees that the cost function behaves well in three basic ways, and it gives the framework a reusable threshold. Anyone who wants to use it for the sigma8 tension still has to do the work of connecting the abstract cost to the physical measurements.

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 the sigma8 tension is explained by the cost function. The certificate does not define what m and e mean in cosmology. The certificate does not prove the numerical claim that sigma8_WL/sigma8_CMB is approximately 1 minus J(phi).

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/Sigma8Tension3_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: