Reflexive Gravity and the Diagonal Universe: Technical Derivations
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2025
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| author | Broomhead, Geoffrey Sovereign Trust Node: Broomhead Private Sovereign Trust geoffreybroomhead.eth |
| author_facet | Broomhead, Geoffrey Sovereign Trust Node: Broomhead Private Sovereign Trust geoffreybroomhead.eth |
| contents | <p><em>Reflexive Gravity — Technical Derivations</em> is the mathematical companion to <em>Reflexive Gravity and the Diagonal Universe</em>, providing the complete proofs and derivations that underpin the Reflexive Gravity framework.<br>It rigorously establishes the <strong>reflexive diagonal operator</strong></p> <p> g⁽ⁿ⁺¹⁾ = g⁽ⁿ⁾ − α (∇∇Φ⁽ⁿ⁾ − λ g⁽ⁿ⁾□Φ⁽ⁿ⁾),</p> <p>and demonstrates that this update performs a gradient descent on the curvature-redundancy functional Q.<br>Through the Lyapunov theorem (Vₙ₊₁ − Vₙ < 0), the paper proves monotonic increase of survivability S, ensuring curvature stability and information conservation.<br>A DeTurck-gauged continuous limit confirms the flow’s parabolic well-posedness, while analytic tests verify compliance with the Null and Weak Energy Conditions.<br>Further sections derive the FLRW bounce criterion, the quantised horizon-entropy increment (ΔS_BH = ½ log 2 ≈ 0.347 nats), and convergence of the discrete recursion scheme.<br>The closing <em>Recursive Lineage</em> appendix traces the mathematics back through Euclid, Eratosthenes, Ramanujan, and Cantor, revealing a single overflow inequality (Δτ > 0, ΔQ < 0, ΔS > 0) running through the entire F-Series lineage.</p> <p>This paper forms the technical foundation for the <strong>Reflexive Gravity Series</strong> and ensures the reproducibility of its results across analytical, numerical, and phenomenological domains.</p> <p> </p> <h3><strong>Related Works and DOI Network</strong></h3> <p><strong>Reflexive Gravity Series</strong><br>Part I — <em>From Survivability to Spacetime</em> — DOI 10.5281/zenodo.17393561<br>Part II — <em>Reflexive Gravity and the Diagonal Universe</em> — DOI 10.5281/zenodo.17393586<br>Part IIa — <em>Reflexive Gravity and the Diagonal Universe: Technical Derivations</em> — DOI 10.5281/zenodo.17393606<br>Part III — <em>Experiments and Observables of Reflexive Gravity</em> — DOI 10.5281/zenodo.17393683</p> <p><strong>Foundational Infinity (F∞) Series</strong><br>F∞.1 — <em>Euclid’s Infinite Field – Proof, Survival, and the Birth of Recursion</em> — DOI 10.5281/zenodo.17336162<br>F∞.2 — <em>Eratosthenes’ Recursive Sieve</em> — DOI 10.5281/zenodo.17345449<br>F∞.3 — <em>Ramanujan’s Hidden Field</em> — DOI 10.5281/zenodo.17336202<br>F∞.4 — <em>Cantor’s Diagonal Overflow</em> — DOI 10.5281/zenodo.17354915</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17429633 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | Reflexive Gravity and the Diagonal Universe: Technical Derivations Broomhead, Geoffrey Sovereign Trust Node: Broomhead Private Sovereign Trust geoffreybroomhead.eth Reflexive Gravity Geometric Flows Lyapunov Stability Deturck Parabolic Structure Energy Conditions Flrw Bounce Quantised Entropy Recursive Lineage <p><em>Reflexive Gravity — Technical Derivations</em> is the mathematical companion to <em>Reflexive Gravity and the Diagonal Universe</em>, providing the complete proofs and derivations that underpin the Reflexive Gravity framework.<br>It rigorously establishes the <strong>reflexive diagonal operator</strong></p> <p> g⁽ⁿ⁺¹⁾ = g⁽ⁿ⁾ − α (∇∇Φ⁽ⁿ⁾ − λ g⁽ⁿ⁾□Φ⁽ⁿ⁾),</p> <p>and demonstrates that this update performs a gradient descent on the curvature-redundancy functional Q.<br>Through the Lyapunov theorem (Vₙ₊₁ − Vₙ < 0), the paper proves monotonic increase of survivability S, ensuring curvature stability and information conservation.<br>A DeTurck-gauged continuous limit confirms the flow’s parabolic well-posedness, while analytic tests verify compliance with the Null and Weak Energy Conditions.<br>Further sections derive the FLRW bounce criterion, the quantised horizon-entropy increment (ΔS_BH = ½ log 2 ≈ 0.347 nats), and convergence of the discrete recursion scheme.<br>The closing <em>Recursive Lineage</em> appendix traces the mathematics back through Euclid, Eratosthenes, Ramanujan, and Cantor, revealing a single overflow inequality (Δτ > 0, ΔQ < 0, ΔS > 0) running through the entire F-Series lineage.</p> <p>This paper forms the technical foundation for the <strong>Reflexive Gravity Series</strong> and ensures the reproducibility of its results across analytical, numerical, and phenomenological domains.</p> <p> </p> <h3><strong>Related Works and DOI Network</strong></h3> <p><strong>Reflexive Gravity Series</strong><br>Part I — <em>From Survivability to Spacetime</em> — DOI 10.5281/zenodo.17393561<br>Part II — <em>Reflexive Gravity and the Diagonal Universe</em> — DOI 10.5281/zenodo.17393586<br>Part IIa — <em>Reflexive Gravity and the Diagonal Universe: Technical Derivations</em> — DOI 10.5281/zenodo.17393606<br>Part III — <em>Experiments and Observables of Reflexive Gravity</em> — DOI 10.5281/zenodo.17393683</p> <p><strong>Foundational Infinity (F∞) Series</strong><br>F∞.1 — <em>Euclid’s Infinite Field – Proof, Survival, and the Birth of Recursion</em> — DOI 10.5281/zenodo.17336162<br>F∞.2 — <em>Eratosthenes’ Recursive Sieve</em> — DOI 10.5281/zenodo.17345449<br>F∞.3 — <em>Ramanujan’s Hidden Field</em> — DOI 10.5281/zenodo.17336202<br>F∞.4 — <em>Cantor’s Diagonal Overflow</em> — DOI 10.5281/zenodo.17354915</p> |
| title | Reflexive Gravity and the Diagonal Universe: Technical Derivations |
| topic | Reflexive Gravity Geometric Flows Lyapunov Stability Deturck Parabolic Structure Energy Conditions Flrw Bounce Quantised Entropy Recursive Lineage |
| url | https://doi.org/10.5281/zenodo.17429633 |