The Cykloid-Adelic Recursive Expansive Field Equation (CARE)
<p dir="ltr">The full expression lives alongside the original paper RED34Section </p><p dir="ltr">1: Mathematical Framework Definitions Core Functions – r₁(t)=exp(t·α), r₂(t)=tᵅ, d(t)=2 sin(t·α)+3 Time Derivatives – ṙ₁=α exp(t α), ṙ₂=tᵅ·(α/t), ḋ=2 α cos(t α) <...
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| _version_ | 1849927625912352768 |
|---|---|
| author | Del Bel, Julian (21403013) |
| author_facet | Del Bel, Julian (21403013) |
| author_role | author |
| dc.creator.none.fl_str_mv | Del Bel, Julian (21403013) |
| dc.date.none.fl_str_mv | 2025-11-25T18:50:30Z |
| dc.identifier.none.fl_str_mv | 10.5281/zenodo.15750370 |
| dc.relation.none.fl_str_mv | https://figshare.com/articles/online resource/The_Cykloid-Adelic_Recursive_Expansive_Field_Equation_CARE_/30559742 |
| dc.rights.none.fl_str_mv | CC BY 4.0 info:eu-repo/semantics/openAccess |
| dc.subject.none.fl_str_mv | Mathematical physics not elsewhere classified Numerical and computational mathematics not elsewhere classified recursive expansive dynamics Sobolev norm regularization Adelic adelic cosmology |
| dc.title.none.fl_str_mv | The Cykloid-Adelic Recursive Expansive Field Equation (CARE) |
| dc.type.none.fl_str_mv | Text Online resource info:eu-repo/semantics/publishedVersion text |
| description | <p dir="ltr">The full expression lives alongside the original paper RED34Section </p><p dir="ltr">1: Mathematical Framework Definitions Core Functions – r₁(t)=exp(t·α), r₂(t)=tᵅ, d(t)=2 sin(t·α)+3 Time Derivatives – ṙ₁=α exp(t α), ṙ₂=tᵅ·(α/t), ḋ=2 α cos(t α) </p><p dir="ltr">Section 2: Partial Derivatives of the Area Function Auxiliary terms θ₁…θ₄ defined from r₁, r₂, d and Δ=√(θ₁θ₂θ₃θ₄) Exact formulas for ∂A/∂r₁, ∂A/∂r₂, ∂A/∂d, each featuring: – a Heron-type square‐root term – a cyclic product of three θ’s divided by 2 θ₁θ₂θ₃θ₄ – an arccos term – additional correction factors </p><p dir="ltr">Section 3: Total Time Derivative of Area dA/dt = (∂A/∂r₁)ṙ₁ + (∂A/∂r₂)ṙ₂ + (∂A/∂d)ḋ Substitution of ṙ₁, ṙ₂, ḋ yields a sprawling multi-line expression combining all three partial contributions </p><p dir="ltr">Section 4: Key Mathematical Constructs Golden ratio φ = (1+√5)/2 ≈ 1.618 Fractal Hausdorff dimension D_H ≈ 3.48 Tribonacci constant ≈ 1.839 Adelic prime-based product structures Recursive operator ℛ(x)=limₙ→∞φ⁻ⁿ P_stratumₙ∘Tⁿx Eigenvergence at rate O(φ⁻ⁿ) </p><p dir="ltr">Section 5: Validation Metrics χ²/ν = 1.03 (ν=112) Gelman–Rubin R̂ = 1.002 ± 0.0003 Energy conservation: d(E_epic+E_epitro)/dt = 0 CMB multipoles C_ℓ ∼ ℓ^{–α} cos(2π ℓ φ) Gravitational-wave echo frequencies fₙ = f₀ φ⁻ⁿ </p><p dir="ltr">We develop a unified field framework, the Cykloid-Adelic Recursive Expansive Field Equation (CARE), which scaffolds stratified geometric manifolds with adelic number-theoretic dynamics via recursive cycloidal parameter spaces. This approach rigorously defines a hierarchy of embedded strata, governed by golden-ratio scaling, and constructs a convergence-proof action principle on a fractal manifold. CARE introduces novel mechanisms for field trifurcation into matter, interaction, and geometric sectors; formulates curvature-based nexus point theory with discrete quantization; and derives a p-adically regulated cosmological constant. The framework delivers testable predictions in gravitational wave echoes, CMB multipole anomalies, and dark matter fractal distributions, while grounding the theory in weighted Sobolev spaces and distributional analysis on singular stratified spaces.</p> |
| eu_rights_str_mv | openAccess |
| id | Manara_83f8aad0e2b07965d6def8c7788b8920 |
| identifier_str_mv | 10.5281/zenodo.15750370 |
| network_acronym_str | Manara |
| network_name_str | ManaraRepo |
| oai_identifier_str | oai:figshare.com:article/30559742 |
| publishDate | 2025 |
| repository.mail.fl_str_mv | |
| repository.name.fl_str_mv | |
| repository_id_str | |
| rights_invalid_str_mv | CC BY 4.0 |
| spelling | The Cykloid-Adelic Recursive Expansive Field Equation (CARE)Del Bel, Julian (21403013)Mathematical physics not elsewhere classifiedNumerical and computational mathematics not elsewhere classifiedrecursive expansive dynamicsSobolev norm regularizationAdelicadelic cosmology<p dir="ltr">The full expression lives alongside the original paper RED34Section </p><p dir="ltr">1: Mathematical Framework Definitions Core Functions – r₁(t)=exp(t·α), r₂(t)=tᵅ, d(t)=2 sin(t·α)+3 Time Derivatives – ṙ₁=α exp(t α), ṙ₂=tᵅ·(α/t), ḋ=2 α cos(t α) </p><p dir="ltr">Section 2: Partial Derivatives of the Area Function Auxiliary terms θ₁…θ₄ defined from r₁, r₂, d and Δ=√(θ₁θ₂θ₃θ₄) Exact formulas for ∂A/∂r₁, ∂A/∂r₂, ∂A/∂d, each featuring: – a Heron-type square‐root term – a cyclic product of three θ’s divided by 2 θ₁θ₂θ₃θ₄ – an arccos term – additional correction factors </p><p dir="ltr">Section 3: Total Time Derivative of Area dA/dt = (∂A/∂r₁)ṙ₁ + (∂A/∂r₂)ṙ₂ + (∂A/∂d)ḋ Substitution of ṙ₁, ṙ₂, ḋ yields a sprawling multi-line expression combining all three partial contributions </p><p dir="ltr">Section 4: Key Mathematical Constructs Golden ratio φ = (1+√5)/2 ≈ 1.618 Fractal Hausdorff dimension D_H ≈ 3.48 Tribonacci constant ≈ 1.839 Adelic prime-based product structures Recursive operator ℛ(x)=limₙ→∞φ⁻ⁿ P_stratumₙ∘Tⁿx Eigenvergence at rate O(φ⁻ⁿ) </p><p dir="ltr">Section 5: Validation Metrics χ²/ν = 1.03 (ν=112) Gelman–Rubin R̂ = 1.002 ± 0.0003 Energy conservation: d(E_epic+E_epitro)/dt = 0 CMB multipoles C_ℓ ∼ ℓ^{–α} cos(2π ℓ φ) Gravitational-wave echo frequencies fₙ = f₀ φ⁻ⁿ </p><p dir="ltr">We develop a unified field framework, the Cykloid-Adelic Recursive Expansive Field Equation (CARE), which scaffolds stratified geometric manifolds with adelic number-theoretic dynamics via recursive cycloidal parameter spaces. This approach rigorously defines a hierarchy of embedded strata, governed by golden-ratio scaling, and constructs a convergence-proof action principle on a fractal manifold. CARE introduces novel mechanisms for field trifurcation into matter, interaction, and geometric sectors; formulates curvature-based nexus point theory with discrete quantization; and derives a p-adically regulated cosmological constant. The framework delivers testable predictions in gravitational wave echoes, CMB multipole anomalies, and dark matter fractal distributions, while grounding the theory in weighted Sobolev spaces and distributional analysis on singular stratified spaces.</p>2025-11-25T18:50:30ZTextOnline resourceinfo:eu-repo/semantics/publishedVersiontext10.5281/zenodo.15750370https://figshare.com/articles/online resource/The_Cykloid-Adelic_Recursive_Expansive_Field_Equation_CARE_/30559742CC BY 4.0info:eu-repo/semantics/openAccessoai:figshare.com:article/305597422025-11-25T18:50:30Z |
| spellingShingle | The Cykloid-Adelic Recursive Expansive Field Equation (CARE) Del Bel, Julian (21403013) Mathematical physics not elsewhere classified Numerical and computational mathematics not elsewhere classified recursive expansive dynamics Sobolev norm regularization Adelic adelic cosmology |
| status_str | publishedVersion |
| title | The Cykloid-Adelic Recursive Expansive Field Equation (CARE) |
| title_full | The Cykloid-Adelic Recursive Expansive Field Equation (CARE) |
| title_fullStr | The Cykloid-Adelic Recursive Expansive Field Equation (CARE) |
| title_full_unstemmed | The Cykloid-Adelic Recursive Expansive Field Equation (CARE) |
| title_short | The Cykloid-Adelic Recursive Expansive Field Equation (CARE) |
| title_sort | The Cykloid-Adelic Recursive Expansive Field Equation (CARE) |
| topic | Mathematical physics not elsewhere classified Numerical and computational mathematics not elsewhere classified recursive expansive dynamics Sobolev norm regularization Adelic adelic cosmology |