What question did this study set out to answer?

To formulate the mathematical foundation of Harmonic Relational Theory and demonstrate how gravity can be derived from coherence in relational fields.

March 12, 2026Open Access

Harmonic Relational Theory: Mathematical Foundation and Derivation of Emergent Gravity

Key Points

To formulate the mathematical foundation of Harmonic Relational Theory and demonstrate how gravity can be derived from coherence in relational fields.
Developed mathematical formalism for Harmonic Relational Theory (HRT)
Derived field equations from Hamiltonian governing fields ψ and C
Proved relativistic covariance and extracted an effective metric
Analyzed impacts on galaxy formation using JADES-GS-z14-0 data
Tested predictions against NANOGrav gravitational wave data
Modified Einstein equations established with coherence-scaled coupling
Coherence feedback predicted to produce massive early galaxies faster than standard models
Explained rotation curves without dark matter through morphological coherence
Predicted organized galaxies exhibit stronger 'dark matter' signatures than irregular ones
Recent gravitational wave data suggests consistency with HRT's predictions

Abstract

I present the complete mathematical formalism for Harmonic Relational Theory (HRT), deriving how gravity emerges from coherence in a relational field rather than spacetime curvature. Starting from fields ψ and C governed by Hamiltonian Hψ, C, I derive field equations, extract an effective metric, prove relativistic covariance, and recover Einstein's equations in the appropriate limit. Key results: (1) Modified Einstein equations G_μν = 8πG₀⟨C⟩T_μν with coherence-scaled coupling, (2) Application to JADES-GS-z14-0 showing coherence feedback produces massive early galaxies faster than ΛCDM, (3) Rotation curve explanation without dark matter via morphological coherence, (4) Testable prediction: organized galaxies show stronger "dark matter" signatures than irregular galaxies of equal mass. Recent NANOGrav gravitational wave data (Chen et al. , 2026) disfavors dark matter spikes, consistent with HRT's smooth coherence gradients. Highest-priority test: morphology-dark matter correlation in SPARC database. Theoretical insights from Scott L. Costello, mathematical formalization assisted by Claude (Anthropic). Full disclosures in manuscript.

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