Paper TS-O: Scaling Exponent Duality and Self-Consistent Matter Density in the FSC Model — Time–Parameter Duality of α, α_c, α_J, and the Relation α + α_c + α_J = 1 + Ω_M
Abstract
We present two results within the Four-Sector Cosmology (FSC) framework, continuing the analytic programme initiated in Papers O through TS-N. Result 1 (Theorem TS-O-1): We establish a time–parameter duality among three scaling exponents of the FSC model: the time-axis exponent α = 0.527 (rate of Ω_IV convergence; Theorem O-4), the parameter-axis exponent α_c = 5/12 = 0.417 (logarithmic sensitivity of the closed-form constant c to Ω_M; Theorem T-5/12), and a newly derived bridge exponent α_J = 0.373 (logarithmic sensitivity of J_first to Ω_M; Lemma TS-O-1). All three exponents are projections of the single function J_first(Ω_M) = (2/3) arcsinh(√R)/√Ω_Λ onto the time axis and parameter space. The asymmetry between axes, δ_dual ≈ 0.21, originates in the acoustic coupling factor (√3−1)/(2e) present in α but absent in α_c and α_J. Corollary TS-O-1a: Under the flatness condition Ω_M + Ω_Λ = 1, the three exponents satisfy: α + α_c + α_J = 1 + Ω_M + O(ε_O, δ²) Numerically: 0.527 + 0.417 + 0.373 = 1.317 = 1 + 0.317. This provides a fourth independent determination of Ω_M within FSC, complementary to CMB, BBN+BAO, and Conjecture T-6. Result 2 (Theorem TS-O-2): The self-consistent matter density Ω_M^**, obtained by combining the ε_O-corrected Conjecture T-6 with the β_V feedback fixed-point equation, converges to: Ω_M^** = 0.28209 (Conj. T-6) + 0.00329 (ε_O) + 0.03156 (β_V feedback) = 0.31694 in agreement with the Planck 2018 value Ω_M^Planck = 0.3170 ± 0.0057 to within 0.019%. A key structural finding is that the β_V feedback alone has a strict upper bound Ω_M^** ≤ 0.3137, and the ε_O correction from Paper O is essential to bridge to the observed value, revealing a previously unrecognised structural connection between Paper O and Conjecture T-6 of Paper T. Open Problem TS-O-3 (flatness from FSC symmetry) is formulated with three candidate routes; the cyclic universe periodicity condition (Route III) is identified as most promising. Eight open problems are carried forward to Paper TS-P.
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BibTeX
@misc{katakura2026paper,
title = {{Paper TS-O: Scaling Exponent Duality and Self-Consistent Matter Density in the FSC Model — Time–Parameter Duality of α, α_c, α_J, and the Relation α + α_c + α_J = 1 + Ω_M}},
author = {Yoshitaka Katakura},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = aug,
doi = {10.5281/zenodo.21911880},
url = {https://doi.org/10.5281/zenodo.21911880},
note = {CC BY},
}
RIS
TY - UNPB
TI - Paper TS-O: Scaling Exponent Duality and Self-Consistent Matter Density in the FSC Model — Time–Parameter Duality of α, α_c, α_J, and the Relation α + α_c + α_J = 1 + Ω_M
AU - Yoshitaka Katakura
PY - 2026
DA - 2026/08/13
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.21911880
UR - https://doi.org/10.5281/zenodo.21911880
AB - We present two results within the Four-Sector Cosmology (FSC) framework, continuing the analytic programme initiated in Papers O through TS-N. Result 1 (Theorem TS-O-1): We establish a time–parameter duality among three scaling exponents of the FSC model: the time-axis exponent α = 0.527 (rate of Ω_IV convergence; Theorem O-4), the parameter-axis exponent α_c = 5/12 = 0.417 (logarithmic sensitivity of the closed-form constant c to Ω_M; Theorem T-5/12), and a newly derived bridge exponent α_J = 0.373 (logarithmic sensitivity of J_first to Ω_M; Lemma TS-O-1). All three exponents are projections of the single function J_first(Ω_M) = (2/3) arcsinh(√R)/√Ω_Λ onto the time axis and parameter space. The asymmetry between axes, δ_dual ≈ 0.21, originates in the acoustic coupling factor (√3−1)/(2e) present in α but absent in α_c and α_J. Corollary TS-O-1a: Under the flatness condition Ω_M + Ω_Λ = 1, the three exponents satisfy: α + α_c + α_J = 1 + Ω_M + O(ε_O, δ²) Numerically: 0.527 + 0.417 + 0.373 = 1.317 = 1 + 0.317. This provides a fourth independent determination of Ω_M within FSC, complementary to CMB, BBN+BAO, and Conjecture T-6. Result 2 (Theorem TS-O-2): The self-consistent matter density Ω_M^**, obtained by combining the ε_O-corrected Conjecture T-6 with the β_V feedback fixed-point equation, converges to: Ω_M^** = 0.28209 (Conj. T-6) + 0.00329 (ε_O) + 0.03156 (β_V feedback) = 0.31694 in agreement with the Planck 2018 value Ω_M^Planck = 0.3170 ± 0.0057 to within 0.019%. A key structural finding is that the β_V feedback alone has a strict upper bound Ω_M^** ≤ 0.3137, and the ε_O correction from Paper O is essential to bridge to the observed value, revealing a previously unrecognised structural connection between Paper O and Conjecture T-6 of Paper T. Open Problem TS-O-3 (flatness from FSC symmetry) is formulated with three candidate routes; the cyclic universe periodicity condition (Route III) is identified as most promising. Eight open problems are carried forward to Paper TS-P.
ER -