Cosmology Track4 Acert

The Track 4.A certificate

Cosmology track 4.A is a named milestone in the Recognition Science (RS) framework's master plan, a roadmap for deriving cosmological parameters from the framework's core axioms. The module itself, Track4ACert, is a machine-checked certificate, a formal object that bundles three previously proven results into a single package. It is a structural theorem, meaning it was verified by a computer without any unproven assumptions specific to RS.

The first result concerns the baryon-to-photon ratio, a cosmological parameter describing the number of ordinary matter particles (baryons) relative to photons in the early universe. The certificate proves that the integer -44, which appears as an exponent in the RS formula for this ratio, is forced by the framework's derivation of three spatial dimensions. The proof proceeds through three independent mathematical routes, all converging on the same number. This is not a fitted value; it is a consequence of the framework's structure.

The second result is a formula for the dark-energy fraction, the proportion of the universe's energy density attributed to the cosmological constant. The certificate states that this fraction, denoted Ω_Λ, equals 11/16 minus α/π. Here, 11/16 is a structural seed derived from the three-dimensional ledger, while α/π is an electromagnetic correction using the measured fine-structure constant, a single input taken from experiment. The certificate proves that this formula yields a value within the interval (0.683, 0.686). This prediction overlaps the value 0.6889 ± 0.0056 reported by the Planck 2018 collaboration within 2σ, meaning the two are statistically consistent.

In Recognition Science, this certificate establishes that two key cosmological parameters, the baryon-to-photon ratio's rung and the dark-energy fraction, are not free parameters but are pinned by the framework's structure plus one measured input. It upgrades these items on the master plan's audit from open or conditional to theorem status. The certificate does not address other tracks, such as the vacuum-fluctuation discrepancy or dark-energy equation-of-state predictions, which remain separate targets.

/-- 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⟩

/-- **TRACK 4.A MASTER CERTIFICATE.**

Five clauses establishing Track 4.A's three closure points:

1. `etaB_rung_forced`: the integer `−44` is forced by `D = 3` via three
   independent routes (gap-from-dimension, chirality × torsion,
   fermionic DOF) that converge.
2. `omegaLambda_formula`: `Ω_Λ = 11/16 − α/π`, with `11/16` the
   structural seed from D=3 ledger structure and `α/π` the EM
   correction using the measured CODATA α (one measured input).
3. `omegaLambda_band`: `Ω_Λ ∈ (0.683, 0.686)`.
4. `planck_2sigma`: the RS prediction is consistent with Planck 2018's
   `0.6889 ± 0.0056` at the 2σ level.
5. `etaB_dimension_route`: explicit witness of the gap-from-dimension
   route giving `−44` from `1 − D²(D+2)` at `D = 3`. -/
structure Track4ACert where
  /-- (1) The η_B rung integer is forced by D = 3. -/
  etaB_rung_forced : EtaBExactRungCert
  /-- (2) Ω_Λ formula: 11/16 - α/π. -/
  omegaLambda_formula :
    omega_lambda = (11 / 16 : ℝ)
      - Constants.ExternalAnchors.alpha_CODATA / Real.pi
  /-- (3) Ω_Λ ∈ (0.683, 0.686). -/
  omegaLambda_band : 0.683 < omega_lambda ∧ omega_lambda < 0.686
  /-- (4) RS consistent with Planck 2018 at 2σ. -/
  planck_2sigma :
    |omega_lambda - omega_lambda_planck2018| < 2 * omega_lambda_planck_err
  /-- (5) Explicit witness: gap-from-dimension at D = 3 yields −44. -/
  etaB_dimension_route :
    eta_B_rung_from_dimension Foundation.GapDerivation.D = -44

/-- **TRACK 4.A HEADLINE THEOREM.**

The cosmological-constant fraction `Ω_Λ` and the baryon-to-photon
ratio rung exponent `−44` are simultaneously pinned by RS structure
(D = 3) plus one measured input (the CODATA α), yielding a structural
prediction `Ω_Λ ∈ (0.683, 0.686)` that overlaps Planck 2018 within 2σ
and three convergent derivations of the rung integer. -/
theorem track4A_headline :
    omega_lambda = (11 / 16 : ℝ)
      - Constants.ExternalAnchors.alpha_CODATA / Real.pi ∧
    0.683 < omega_lambda ∧ omega_lambda < 0.686 ∧
    |omega_lambda - omega_lambda_planck2018| < 2 * omega_lambda_planck_err ∧
    eta_B_rung_from_dimension Foundation.GapDerivation.D = -44 ∧
    eta_B_rung_from_chirality = -44 ∧
    eta_B_rung_from_fermionic = -44 := by
  refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
  · unfold omega_lambda omega_raw em_correction; rfl
  · exact omega_lambda_interval.1
  · exact omega_lambda_interval.2
  · exact rs_consistent_with_planck
  · exact eta_B_rung_from_dimension_at_D3
  · exact eta_B_rung_from_chirality_eq
  · exact eta_B_rung_from_fermionic_eq

What this page does not claim

This certificate does not prove the exact value of the fine-structure constant α, which is used as a measured input. This certificate does not address the dark-energy equation-of-state predictions of Track 4.C. This certificate does not resolve the vacuum-fluctuation discrepancy, which is Track 4.B.

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: