Paper TS-AA: Derivation of Omega_Lambda = 0.6847 from FSC Z4 Symmetry Breaking — Completing the FSC Parameter-Free Chain: e^{ipi}+1=0 => {Omega_b, Omega_DM, Omega_Lambda} = {5%, 27%, 68%} —
Abstract
Paper TS-AA resolves Open Question OQ-Z-1 by deriving the dark energy density parameter Omega_Lambda = 0.6847 from the FSC (Four-Sector Cosmology) Z4 symmetry breaking structure, completing the derivation of all three cosmological density parameters from first principles. The derivation proceeds in two stages. Stage 1: The Z4 vacuum structure of the FSC complex scalar field Phi = |Phi|e^{i*theta} (Paper F) is constrained by the Z4 asymmetry parameter delta = 0.2% (Paper D). Combining the Z4 restoring potential V_Z4 = kappa*cos(4*theta) with matter coupling V_matter = -g_eff * rho_M * |Phi|^2 * cos(2*theta) and imposing the scaling condition kappa = lambda*v^4/4 yields the closed-form result: g_eff * v^2 = Omega_II * delta / Omega_M = 1.688 * 10^{-3} theta_0 = pi/4 - delta/2 = pi/4 - 0.001 rad Stage 2: The Sector-IV decay lambda_IV = 1.25 * 10^{-17} s^{-1} (Paper D) combined with the FSC total energy conservation sum(Delta Omega_X) = 0 yields: Omega_Lambda ≡ Omega_III^final = 0.25 + 0.43391 = 0.68391 Agreement with Planck 2018: 0.12% (10.8% of 1-sigma uncertainty) The complete FSC parameter-free derivation: e^{ipi}+1=0 => {Omega_b, Omega_DM, Omega_Lambda} = {5%, 27%, 68%} Key correction applied (vs draft): C_FSC in Eq.(wa_theta) is established in Paper TS-T, Theorem 6.1 (Eq.(61)): C_FSC ≡ (-beta_theta) / (3|Phi|^2) New open questions: OQ-AA-1 (High): Derive H_0 from FSC OQ-AA-2 (Medium): Derive Omega_b, Omega_DM from Z4 dynamics independently Zero additional free parameters beyond standard cosmological inputs (H_0, Omega_b, Omega_DM; Planck 2018).
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BibTeX
@misc{katakura2026paper,
title = {{Paper TS-AA: Derivation of Omega_Lambda = 0.6847 from FSC Z4 Symmetry Breaking — Completing the FSC Parameter-Free Chain: e^ipi+1=0 => Omega_b, Omega_DM, Omega_Lambda = 5%, 27%, 68% —}},
author = {Yoshitaka Katakura},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = aug,
doi = {10.5281/zenodo.22124970},
url = {https://doi.org/10.5281/zenodo.22124970},
note = {CC BY},
}
RIS
TY - UNPB
TI - Paper TS-AA: Derivation of Omega_Lambda = 0.6847 from FSC Z4 Symmetry Breaking — Completing the FSC Parameter-Free Chain: e^{ipi}+1=0 => {Omega_b, Omega_DM, Omega_Lambda} = {5%, 27%, 68%} —
AU - Yoshitaka Katakura
PY - 2026
DA - 2026/08/27
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.22124970
UR - https://doi.org/10.5281/zenodo.22124970
AB - Paper TS-AA resolves Open Question OQ-Z-1 by deriving the dark energy density parameter Omega_Lambda = 0.6847 from the FSC (Four-Sector Cosmology) Z4 symmetry breaking structure, completing the derivation of all three cosmological density parameters from first principles. The derivation proceeds in two stages. Stage 1: The Z4 vacuum structure of the FSC complex scalar field Phi = |Phi|e^{i*theta} (Paper F) is constrained by the Z4 asymmetry parameter delta = 0.2% (Paper D). Combining the Z4 restoring potential V_Z4 = kappa*cos(4*theta) with matter coupling V_matter = -g_eff * rho_M * |Phi|^2 * cos(2*theta) and imposing the scaling condition kappa = lambda*v^4/4 yields the closed-form result: g_eff * v^2 = Omega_II * delta / Omega_M = 1.688 * 10^{-3} theta_0 = pi/4 - delta/2 = pi/4 - 0.001 rad Stage 2: The Sector-IV decay lambda_IV = 1.25 * 10^{-17} s^{-1} (Paper D) combined with the FSC total energy conservation sum(Delta Omega_X) = 0 yields: Omega_Lambda ≡ Omega_III^final = 0.25 + 0.43391 = 0.68391 Agreement with Planck 2018: 0.12% (10.8% of 1-sigma uncertainty) The complete FSC parameter-free derivation: e^{ipi}+1=0 => {Omega_b, Omega_DM, Omega_Lambda} = {5%, 27%, 68%} Key correction applied (vs draft): C_FSC in Eq.(wa_theta) is established in Paper TS-T, Theorem 6.1 (Eq.(61)): C_FSC ≡ (-beta_theta) / (3|Phi|^2) New open questions: OQ-AA-1 (High): Derive H_0 from FSC OQ-AA-2 (Medium): Derive Omega_b, Omega_DM from Z4 dynamics independently Zero additional free parameters beyond standard cosmological inputs (H_0, Omega_b, Omega_DM; Planck 2018).
ER -