>[!warning] >This content has not been peer reviewed. **Pillar:** Relational Substrate Theory (RST) as the **informational origin** of MOND. The MONDian acceleration threshold **a₀** is derived from first principles (no independent universal constant); the interpolation function **μ** is the geometric projection of the relational signal onto a total workload (L_n-norm); cores arise as a mandatory high-SNR state; and **a₀** is **epoch-dependent** (a₀ ∝ H(z)), yielding a falsifiable JWST-era signature distinct from ΛCDM and static MOND. --- ## 1. The MOND programme and the a₀ coincidence (standard view) - **MOND:** Modified Newtonian Dynamics reproduces galaxy rotation curves and scaling relations (Tully–Fisher, etc.) with an acceleration threshold **a₀** below which dynamics deviate from Newton/GR. - **Cosmic coincidence:** The observed value a₀ ≈ c H₀/(2π) has long been noted but not derived; in standard MOND, a₀ is an independent constant. - **RST role:** RST derives a₀ from the **Relational Landauer Principle**: physical persistence is an active, resource-constrained process on a finite informational substrate. The macroscopic noise floor is identified with the cosmic expansion scalar **θ**; the threshold follows from the geometry of the substrate. --- ## 2. RST derivation of a₀: noise floor from cosmology By treating the substrate as a 3D volumetric process and the relational “signal” as projected onto a 1D channel, the acceleration threshold is: $a_0 = \frac{c\,\theta}{6\pi} \approx \frac{c\,H(z)}{2\pi}.$ - **θ** is the cosmic expansion scalar (Hubble rate in appropriate units). - **a₀** is thus **not** an independent universal constant; it is fixed by the global expansion state. This provides a rigorous physical derivation for the cosmic coincidence and removes a₀ as a free parameter. The same scale appears in the RST field equation ([[expanded theory/Resource Triangle]], [[expanded theory/Relational Substrate Theory (RST)]]) as the **noise floor** in the fidelity relation μ(q'/a₀, n) q' = g_N. --- ## 3. Interpolation function μ and the relational backbone The **μ** function in RST is the geometric projection of the relational signal onto a total workload magnitude defined by an **L_n-norm**: $\mu(\eta,\, n) = \frac{\eta}{(1 + \eta^n)^{1/n}},\qquad \eta = \frac{q'}{a_0}.$ - **Calibration:** Fits to **171 SPARC galaxies** measure **n₀ = 1.25 ± 0.05**, confirming the value derived from the Pure Axiom Substrate (thermodynamic limit). - **Interpretation:** μ is not an ad hoc interpolation; it is the projection of the signal onto the workload budget set by the finite substrate. The exponent n encodes the effective dimensionality of the relational structure. Together with a₀ = c H₀/(2π), the gravity sector is fixed by theory; SPARC confirms the derived n; no free fit parameters for the interpolation. --- ## 4. Zero-parameter core recovery RST **natively forbids central cusps**: - In **high-acceleration** regions (η ≫ 1), substrate **fidelity** μ → 1, so the rendered acceleration is strictly the baryonic source: q' = g_N. - Cores emerge as the **mandatory high-SNR state**: where the signal dominates the noise floor, the substrate adds no extra workload. No baryonic feedback or dark-matter profile is required to produce a core. This aligns with the Tremaine core–cusp application ([[Tremaine Core-Cusp/Tremaine Core-Cusp (RST)]]): the same fidelity limits that give MOND-like behaviour in the outskirts give Newtonian behaviour in the centre. --- ## 5. Redshift evolution: the JWST prediction A critical departure from **static MOND** is that in RST **a₀ is epoch-dependent**: $a_0 \propto H(z).$ - **At z = 2:** The MONDian transition threshold was approximately **three times higher** than the current local value. - **Implication:** For a fixed baryonic mass, the flat rotation velocity **v_f** is predicted to be **∼32% higher** at z = 2. This can contribute to explaining the accelerated formation of massive disks observed by JWST. This scaling is a **clean, falsifiable signature** to distinguish relational gravity from both ΛCDM and from MOND with a constant a₀. --- ## 6. Relativistic consistency The relativistic implementation (**RAWT 1.6**) maps to the simplest surviving **Horndeski** class (pure **K-essence**), ensuring: - **GW170817:** Gravitational-wave speed c_gw = c (no extra scalar-mediated propagation). - **Solar System:** PPN bounds are satisfied through a one-parameter **Einstein–Aether** family. RST thus provides the ontological “why” for MONDian dynamics: the “dark sector” is reinterpreted as the substrate’s **resource-management ledger**, while remaining consistent with local and cosmological tests. --- ## 7. Result (figure and numbers) The script **[[Milgrom MOND - Code]]** was run with flat ΛCDM (Ωm = 0.3, ΩΛ = 0.7), Hernquist M = 10¹¹ M☉ and a = 3 kpc, and RST n ≈ 1.24 (derived). Output:  **Figure:** 2×2 panels — (1) a₀(z) = c H(z)/(2π) vs z; (2) μ(η) with n ≈ 1.24 (derived); (3) V_bar and V_obs(r) (Hernquist + RST); (4) v_flat(z)/v_flat(0) with z = 2 marker (~31% higher). Full description: **[[Milgrom MOND Results]]**. --- ## 8. Code and reproducibility The script **[[Milgrom MOND - Code]]** (`rst_milgrom_mond.py`) plugs into **`rst_engine`** (`solve_omega`, `mu_rst`, `DEFAULT_N`) and produces a single 2×2 figure: a₀(z), μ(η), rotation curve (Hernquist + RST), and v_flat(z)/v_flat(0). From the workspace root: ``` python "expanded theory applied/further applications/Milgrom MOND/rst_milgrom_mond.py" ``` Output: `milgrom_mond.png` in the application folder. M, a, cosmology (Ωm, ΩΛ), and n can be changed in the script to test sensitivity. --- ## 9. Summary and links | Item | RST content | |:---|:---| | **a₀** | Derived: a₀ = c θ/(6π) ≈ c H(z)/(2π); not an independent constant. | | **μ** | Geometric projection (L_n-norm); n₀ ≈ 1.24 (derived); SPARC confirms. | | **Cores** | Mandatory where η ≫ 1 (μ → 1); no feedback required. | | **Redshift** | a₀ ∝ H(z) → higher v_f at high z; JWST falsifiable. | | **Relativistic** | Horndeski (K-essence), GW170817, PPN via Einstein–Aether. | - **Results (figure + caption):** [[Milgrom MOND Results]] - **Code (script + figure):** [[Milgrom MOND - Code]] - **RST field equation, Resource Triangle:** [[expanded theory/Resource Triangle]], [[expanded theory/Relational Substrate Theory (RST)]] - **Core–cusp (same fidelity limits):** [[Tremaine Core-Cusp/Tremaine Core-Cusp (RST)]] - **SPARC (same equation, calibration):** [[expanded theory/sparc evaluation/SPARC evaluation - Code]] - **Fidelity inversion (gravity ↔ materials duality):** [[expanded theory/Fidelity Inversion — Gravity and Materials]] - **Roadmap:** [[../../Applications Roadmap]]