Cosmology Track4 Acert Track4 Acert Inhabited

The Track 4.A certificate

The declaration track4ACert_inhabited is a formal statement in the Recognition Science framework's machine-checked library of formal theorems. It establishes that a particular certificate, called Track 4.A, exists: that is, a bundle of five claims about the dark-energy fraction of the universe and the baryon-to-photon ratio can be simultaneously proved. The certificate is not a single new physical law; it is a packaging of three pre-existing theorem-grade results into one verified object.

The first claim concerns the baryon-to-photon ratio, the number of protons and neutrons for every photon in the early universe. The certificate states that the integer -44, which appears as an exponent in the ratio's golden-ratio rung, is forced by the fact that space has three dimensions. The proof offers three independent routes to this conclusion: a gap-from-dimension calculation, a chirality-times-torsion argument, and a fermionic degrees-of-freedom count. All three converge on the same integer.

The second and third claims concern the dark-energy fraction Ω_Λ, the portion of the universe's energy density attributed to dark energy. The certificate proves the formula Ω_Λ = 11/16 − α/π, where 11/16 is a structural seed from the three-dimensional ledger structure and α/π is an electromagnetic correction using the measured fine-structure constant α as a single input. It then proves the interval 0.683 < Ω_Λ < 0.686. The fourth claim checks this prediction against the Planck 2018 satellite measurement of 0.6889 ± 0.0056 and finds agreement at the 2σ level. The fifth claim provides an explicit witness for the gap-from-dimension route, showing the calculation 1 − D²(D+2) at D = 3 yields -44.

In Recognition Science, this certificate upgrades several items in the master plan's audit rows from OPEN to THEOREM. The dark-energy fraction is no longer merely a target; it is a structural prediction with a proved interval and a stated consistency with external data. The baryon-to-photon rung integer is likewise no longer a free parameter but a forced consequence of three-dimensionality.

The certificate does not claim to address the vacuum-fluctuation discrepancy known as Track 4.B: proving that the cost-minimum vacuum mode-sum is precisely the phi^(-44) rung value, not 10^120 times it, is a separate result not included here. It also does not address Track 4.C, which concerns the dark-energy tension and equation-of-state predictions. The full Λ_RS · ℓ_P² band in RS units, anchored at a separate declaration, is independent of this certificate and not bundled with it.

/-- The Track 4.A certificate is verified. -/
noncomputable def track4ACert : Track4ACert where
  etaB_rung_forced := etaBExactRungCert
  omegaLambda_formula := by
    -- omega_lambda = omega_raw - em_correction; omega_raw = 11/16; em_correction = α/π.
    unfold omega_lambda omega_raw em_correction
    rfl
  omegaLambda_band := omega_lambda_interval
  planck_2sigma := rs_consistent_with_planck
  etaB_dimension_route := eta_B_rung_from_dimension_at_D3

theorem track4ACert_inhabited : Nonempty Track4ACert :=
  ⟨track4ACert⟩

/-- The Track 4.A certificate is verified. -/
noncomputable def track4ACert : Track4ACert where
  etaB_rung_forced := etaBExactRungCert
  omegaLambda_formula := by
    -- omega_lambda = omega_raw - em_correction; omega_raw = 11/16; em_correction = α/π.
    unfold omega_lambda omega_raw em_correction
    rfl
  omegaLambda_band := omega_lambda_interval
  planck_2sigma := rs_consistent_with_planck
  etaB_dimension_route := eta_B_rung_from_dimension_at_D3

/-- The Track 4.A certificate is verified. -/
noncomputable def track4ACert : Track4ACert where
  etaB_rung_forced := etaBExactRungCert
  omegaLambda_formula := by
    -- omega_lambda = omega_raw - em_correction; omega_raw = 11/16; em_correction = α/π.
    unfold omega_lambda omega_raw em_correction
    rfl
  omegaLambda_band := omega_lambda_interval
  planck_2sigma := rs_consistent_with_planck
  etaB_dimension_route := eta_B_rung_from_dimension_at_D3

What this page does not claim

The certificate does not prove that the vacuum mode-sum is precisely the phi^(-44) rung value. The certificate does not address the dark-energy tension or equation-of-state predictions. The certificate does not bundle the full Λ_RS · ℓ_P² band in RS units.

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/Track4ACert.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: