Cosmology Dark Matter Xenonprediction Dm Cross Section Ratio

The predicted cross-section band

The declaration dmCrossSectionRatio is a definition in the Recognition Science framework's machine-checked library of formal statements. It sets a number, the ratio of a dark matter particle's interaction cross-section to some reference scale, equal to the value of a specific cost function at the golden ratio. The cost function, written J(x) = (x + 1/x)/2 - 1, is the framework's central object: a measure of the price of a recognition event, forced by five plain conditions rather than chosen freely. At the golden ratio, this function evaluates to a number between 0.11 and 0.13.

What the formal statement proves is narrow but exact. The framework's library contains a proof that this cross-section ratio is positive, and a second proof that it lies strictly between zero and 0.13. These are statements about the defined number itself, not about any measured particle. The definition also carries a companion mass ratio: the framework's dark matter module sets the dark matter mass to 1/45 of the W boson mass, which predicts a mass around 1.78 GeV. The cross-section ratio is defined at that same point, tying the two predictions together in the formal structure.

The framework's own documentation reports that the XENONnT experiment's current exclusion limits at a dark matter mass of 1.78 GeV sit above the predicted band, meaning the prediction is not yet ruled out. That statement is a report from the framework's notes, not a formal statement. The formal library proves only the internal consistency of the definitions: the ratio is positive and less than 0.13. Whether any real dark matter particle exists with this mass and cross-section is an empirical question the framework's formal proofs do not touch.

In Recognition Science, this definition is one piece of a larger chain that derives physical constants from the cost function. The framework models the dark matter cross-section ratio as a consequence of its forced cost structure, but the link from that formal number to an actual experimental measurement remains open. The declaration establishes a precise mathematical object and its provable bounds; it does not establish that dark matter has been detected, that the predicted mass is correct, or that the cross-section band matches any real observation.

/-- DM cross-section ratio at J(φ). -/
noncomputable def dmCrossSectionRatio : ℝ := Jcost phi

/-- Cross-section ratio is positive. -/
theorem dmCrossSection_pos : 0 < dmCrossSectionRatio :=
  Jcost_pos_of_ne_one phi phi_pos phi_ne_one

/-- The prediction is not yet excluded: cross-section is in J(phi) band. -/
theorem dmCrossSection_in_band : 0 < dmCrossSectionRatio ∧ dmCrossSectionRatio < 0.13 := by
  constructor
  · exact dmCrossSection_pos
  · unfold dmCrossSectionRatio
    rw [Constants.Jcost_phi_val]
    linarith [phi_lt_onePointSixTwo]

/-- Predicted DM mass / W mass = 1/45. -/
noncomputable def dmMassRatio : ℝ := 1 / 45

structure DarkMatterXENONCert where
  dm_mass_ratio : dmMassRatio = 1 / 45
  cross_section_pos : 0 < dmCrossSectionRatio
  cross_section_band : 0 < dmCrossSectionRatio ∧ dmCrossSectionRatio < 0.13

What this page does not claim

The declaration does not claim that dark matter has been detected or that any experiment has confirmed the predicted mass or cross-section. The formal proofs establish bounds on a defined number, not on any measured physical quantity. The framework's documentation report on XENONnT is not a formal statement and carries no proof of the experimental status.

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/DarkMatterXENONPrediction.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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