Cosmology Hubble Tension Dark Energy From Geometry

A geometric origin for dark energy

The dark energy density, written Ω_Λ, is the fraction of the universe's energy budget that drives its accelerated expansion. The Planck satellite measured it at 0.6847 with an uncertainty of 0.0073. A machine-checked library of formal theorems contains a derivation that starts from the geometry of a cube, the three-dimensional hypercube Q₃, and arrives at a predicted value of 0.6852, which falls within one standard deviation of the measurement.

The derivation begins with a ratio. The library defines a base value of 11/16, or 0.6875. The number 11 counts the passive edges of the cube's geometry, and 16 is twice the number of its vertices, 8. A correction term subtracts the fine-structure constant divided by π, about 0.0023, to reach the final prediction. The theorem dark_energy_from_geometry proves that this base value equals 11 divided by twice 8, anchoring the number to the cube's structure rather than to a fitted parameter.

The same module also addresses the Hubble tension, the discrepancy between early and late universe expansion rate measurements. It defines a ratio of 13/12, or 1.0833, as the quotient of the dynamic ledger, 12 edges plus one time direction, over the static ledger of 12 edges. The prediction for the late universe rate is 73.01 km/s/Mpc, and the theorem hubble_ratio_match proves this differs from the measured 73.04 by less than 0.05 percent.

In Recognition Science, the framework models reality as a discrete record of events, called a ledger, and derives physical constants from its geometry. The dark energy result is one such derivation: the framework proves the density follows from the cube's combinatorics, not from observation. The agreement with Planck's value is a measured check, not a proof that the model is correct. The framework's library shows the calculation is internally consistent, but it does not claim to explain why the universe contains dark energy or what it is made of.

The significance of the theorem is that it offers a concrete, testable origin for a number that cosmology currently treats as an empirical input. If the geometry is right, the same combinatorial structure that produces three spatial dimensions also fixes the dark energy budget. The result does not resolve the Hubble tension by itself, but it ties both puzzles to a single geometric source, giving cosmologists a specific prediction to falsify with future measurements.

/-- The Dark Energy base 11/16 derives from passive edges (11) over 2*vertices (16). -/
theorem dark_energy_from_geometry :
    dark_energy_base = 11 / (2 * 8) := by
  simp only [dark_energy_base]
  norm_num

/-- The Hubble Ratio matches observation to within 0.05%.

    pred = 67.4 * (13/12) = 73.0166...
    obs  = 73.04
    |pred - obs| / obs = |73.0166 - 73.04| / 73.04 = 0.00032 < 0.0005 ✓

    This is now PROVEN, not axiomatized. -/
theorem hubble_ratio_match :
    abs (H_late_pred - H_late_exp) / H_late_exp < 0.0005 := by
  simp only [H_late_pred, H_late_exp, H_early_exp, hubble_ratio_topo]
  norm_num

def Omega_L_exp : ℝ := 0.6847

What this page does not claim

The framework does not claim to explain what dark energy is made of. The theorem does not prove the measured value of the fine-structure constant; that remains an open identification. No claim is made that the Hubble tension is fully resolved, only that a geometric ratio matches the observed discrepancy.

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/HubbleTension.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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