Encyclopedia Cosmology Cosmology Track4 Acert Track4 Acert
ARTICLE 4 claims 4 theorems
Cosmology Track4 Acert Track4 Acert
A machine-checked certificate bundles two cosmological predictions, the dark-energy fraction and the baryon-to-photon ratio, into one theorem.
The Track 4.A certificate
The dark-energy fraction, written ΩΛ, is the share of the universe's energy density that drives its accelerated expansion. The baryon-to-photon ratio, written ηB, compares the number of ordinary matter particles to the number of photons in the cosmic microwave background. The Track 4.A certificate is a formal statement in the Recognition Science framework's machine-checked library of formal theorems that bundles three pre-existing theorem-grade closures into one result.
The certificate states that the integer −44 governing the baryon-to-photon ratio's φ-rung is forced by the framework's derivation of three spatial dimensions, via three structurally independent derivations that converge. It further states that the dark-energy fraction equals 11/16 − α/π, where 11/16 is a structural seed from the three-dimensional ledger and α/π is an electromagnetic correction using the measured CODATA α as one input. The proved interval is ΩΛ ∈ (0.683, 0.686). This prediction overlaps Planck 2018's measured value 0.6889 ± 0.0056 within 2σ.
The certificate does not claim to derive the fine-structure constant α; within the framework, exact α is a free boundary datum. It also does not address Track 4.B, the vacuum-fluctuation discrepancy, nor Track 4.C, the dark-energy equation-of-state predictions. The full ΛRS · ℓP² band in RS units is independent of this certificate and not bundled here.
THEOREM Track4ACert · IndisputableMonolith/Cosmology/Track4ACert.lean
/-- **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
THEOREM track4ACert · IndisputableMonolith/Cosmology/Track4ACert.lean
/-- 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 · IndisputableMonolith/Cosmology/Track4ACert.lean
theorem track4ACert_inhabited : Nonempty Track4ACert :=
⟨track4ACert⟩
THEOREM track4A_headline · IndisputableMonolith/Cosmology/Track4ACert.lean
/-- **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
The certificate does not derive the fine-structure constant α. The certificate does not address Track 4.B or Track 4.C. The certificate does not bundle the full Λ_RS · ℓ_P² band.
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:
- What is the vacuum-fluctuation discrepancy that Track 4.B addresses?
- What dark-energy equation-of-state predictions does Track 4.C make?
- What is the full Λ_RS · ℓ_P² band in RS units?
MACHINE LAYER · GROUNDED CLAIM TABLE · CLICK TO EXPAND
THEOREM Track4ACert · IndisputableMonolith/Cosmology/Track4ACert.lean
/-- **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 = -44The certificate states that the integer −44 governing the baryon-to-photon ratio's φ-rung is forced by the framework's derivation of three spatial dimensions, via three structurally independent derivations that converge. Track4ACert · IndisputableMonolith/Cosmology/Track4ACert.leanTHEOREM track4ACert · IndisputableMonolith/Cosmology/Track4ACert.lean
/-- 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_D3It further states that the dark-energy fraction equals 11/16 − α/π, where 11/16 is a structural seed from the three-dimensional ledger and α/π is an electromagnetic correction using the measured CODATA α as one input. track4ACert · IndisputableMonolith/Cosmology/Track4ACert.leanTHEOREM track4ACert_inhabited · IndisputableMonolith/Cosmology/Track4ACert.lean
theorem track4ACert_inhabited : Nonempty Track4ACert := ⟨track4ACert⟩The proved interval is Ω<sub>Λ</sub> ∈ (0.683, 0.686). track4ACert_inhabited · IndisputableMonolith/Cosmology/Track4ACert.leanTHEOREM track4A_headline · IndisputableMonolith/Cosmology/Track4ACert.lean
/-- **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_eqThis prediction overlaps Planck 2018's measured value 0.6889 ± 0.0056 within 2σ. track4A_headline · IndisputableMonolith/Cosmology/Track4ACert.lean