Cosmology Sigma8 Tension3 From Jcost

The sigma8 tension

The sigma8 tension is a real disagreement in cosmology. Sigma8 measures how much matter clusters on a scale of about 8 megaparsecs (roughly 26 million light-years). Measurements from the cosmic microwave background, the oldest light in the universe, give sigma8 around 0.83. Measurements from weak gravitational lensing, the bending of light by foreground matter, give around 0.77. The two values differ by about 7 percent, and the gap is larger than the reported errors allow. Cosmologists call this the sigma8 tension because it suggests either an unknown systematic error or new physics beyond the standard model.

In Recognition Science, the proposed explanation is that the ratio of the two measured values should follow a particular cost function. The framework defines a cost function J(x) = (x + 1/x)/2 - 1, which is proved in a machine-checked library of formal theorems to be the unique function satisfying five plain conditions. The idea here is that the ratio sigma8_WL / sigma8_CMB, roughly 0.77 / 0.83 = 0.928, should equal 1 - J(phi), where phi is the golden ratio. That expression evaluates to about 0.882, which predicts a roughly 12 percent suppression in structure formation. The direction matches the observed discrepancy, but the numbers do not align closely: 0.928 versus 0.882.

What does the module itself establish? The module named Sigma8Tension3_FromJCost proves three general facts about the cost function applied to a ratio of two real numbers. It proves that the cost vanishes when the two numbers are equal, that the cost is nonnegative for positive inputs, and that phi - 3/2 is positive. These are all true for any positive inputs, because the module defines the cost as J(m / e) without tying m and e to any specific physical quantity. The module does not prove that sigma8 follows this cost, nor does it derive the measured values. The docstring itself says the paragraph about sigma8 is a research note recording where the idea was meant to go, not a result.

In plain language, the module is a template. It shows that the cost function has the basic properties you would want for a measure of discrepancy: zero when things match, positive when they differ, and a threshold tied to the golden ratio. But it does not connect those properties to the actual sigma8 measurements. The connection remains a hypothesis, not a theorem. The machine-checked part is the general cost behavior, not the cosmology.

What this means for a reader is that the sigma8 tension is real, the proposed ratio is a specific number, and the formal proof covers only the generic cost function. The step from cost function to cosmology is where the idea lives, and it is not yet proved.

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 module does not prove that sigma8 follows the cost function. The module does not derive the measured values 0.83 and 0.77. The 12 percent suppression is a research note, not a proved result.

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)

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