>[!warning] >This content has not been peer reviewed. # Relational Dynamics — Results --- ## 1. The Lorentz Factor (Phase 2): Exact Identity This is the central result. The Lorentz gamma factor is **exactly** the reciprocal of the RST fidelity function: $\gamma = \frac{1}{\mu\!\left(\frac{\sqrt{1-\beta^2}}{\beta},\, 2\right)}$ | Metric | Value | |:---|---:| | R² | 1.000000000000000 | | RMSE | 2.04 × 10⁻¹⁶ | | Max relative error | 3.90 × 10⁻¹⁶ | These errors are **machine epsilon** — the identity is algebraically exact, not approximate. ### Proof $\mu(\eta, 2) = \frac{\eta}{\sqrt{1+\eta^2}}, \quad \eta = \frac{\sqrt{1-\beta^2}}{\beta}$ $1 + \eta^2 = 1 + \frac{1-\beta^2}{\beta^2} = \frac{1}{\beta^2}$ $\mu = \frac{\sqrt{1-\beta^2}/\beta}{1/\beta} = \sqrt{1-\beta^2} = \frac{1}{\gamma} \quad \blacksquare$ ### n-scan: uniqueness of n = 2 | n | R² | RMSE | |---:|---:|---:| | 0.50 | −2.496 | 2.454 | | 1.00 | 0.858 | 0.494 | | 1.25 | 0.960 | 0.264 | | 1.50 | 0.989 | 0.135 | | **2.00** | **1.000** | **0.000** | | 3.00 | 0.994 | 0.104 | | 5.00 | 0.984 | 0.164 | | 10.00 | 0.978 | 0.193 | No other n value produces R² = 1. The closest are n = 1.5 (R² = 0.989) and n = 3 (R² = 0.994), but both deviate significantly at high velocity. ### Velocity comparison (n = 2) | v/c | η | γ (SR) | γ (RST) | Error | |---:|---:|---:|---:|---:| | 0.100 | 9.950 | 1.005 | 1.005 | 0 | | 0.300 | 3.180 | 1.048 | 1.048 | 0 | | 0.500 | 1.732 | 1.155 | 1.155 | 10⁻¹⁶ | | 0.700 | 1.020 | 1.400 | 1.400 | 10⁻¹⁶ | | 0.900 | 0.484 | 2.294 | 2.294 | 0 | | 0.950 | 0.329 | 3.203 | 3.203 | 10⁻¹⁶ | | 0.990 | 0.142 | 7.089 | 7.089 | 0 | | 0.999 | 0.045 | 22.37 | 22.37 | 10⁻¹⁶ | ### Physical interpretation - **η = √(c²−v²)/v** is the **bandwidth headroom**: the ratio of remaining velocity capacity to current demand. At v = 0, η → ∞ (infinite headroom). At v → c, η → 0 (no headroom left). - **n = 2** encodes the **quadratic structure of the Lorentzian metric**. The "2" in 1 − β² is the same "2" as the fidelity exponent. The velocity constraint (v_x² + v_y² + v_z² < c²) is a quadratic surface — its topology is n = 2. - **This is the only sector where n is known a priori from geometry.** All other n values (0.35, 1.25, 5, etc.) were fitted from data. For special relativity, n = 2 is dictated by the metric signature. --- ## 2. Dynamical Friction (Phase 1): Approximate Fit **Model:** h_RST(X) = μ(X, n_dyn)^α **Reference:** Chandrasekhar h(X) = erf(X) − 2X·exp(−X²)/√π | Parameter | Value | |:---|---:| | n_dyn | 3.83 | | α | 4.41 | | R² | 0.9965 | | RMSE | 0.0204 | ### Observations 1. **R² = 0.997**: good but not exact. The Chandrasekhar function involves Gaussian integrals (erf, exp(−X²)); the algebraic μ function cannot reproduce transcendental shapes perfectly. 2. **α = 4.41** (expected ~3). The small-X scaling h ∝ X³ follows from 3D velocity-space integration. The optimizer finds α > 3 because the RST model μ^α ≈ X^α at small X with coefficient 1, while the Chandrasekhar coefficient is 4/(3√π) ≈ 0.752. The model compensates for the missing 0.752 prefactor by increasing α. 3. **n_dyn = 3.83**: sits between the Lorentz n = 2 and the electronic n = 5 on the dimensional ladder. Dynamical friction involves a 3D velocity integral over a Maxwellian distribution — the effective dimensionality of this phase-space interaction is ~4. 4. **Residual pattern**: the model over-predicts near X ≈ 1 and under-predicts at small X (0.1–0.5) and large X (2–5). The transition is smoother (more "Gaussian") than the algebraic μ function. ### The RST-Gaussian boundary This result parallels the "RST-Arrhenius boundary" from the fluid dynamics sector: RST's algebraic μ function captures **topological transitions** (regime changes governed by geometry) but struggles with **statistical transitions** (governed by Boltzmann/Gaussian probability distributions). The Chandrasekhar formula falls on the statistical side — it emerges from averaging over a Maxwellian velocity distribution, not from a topological constraint. --- ## 3. Figures ![[relational_dynamics_lorentz.png]] *Left: γ(SR) vs 1/μ(η,2) — indistinguishable. Center: n-scan — only n=2 matches. Right: relative error at machine precision.* ![[relational_dynamics_friction.png]] *Left: Chandrasekhar h(X) vs μ^α RST model. Right: absolute residual.* --- ## 4. The complete dimensional ladder | Sector | n | Origin | Type | |:---|:---|:---|:---| | Tensile (FCC) | 0.35 | Slip-system connectivity | Fitted | | Tensile (BCC) | 1.0 | Slip-system connectivity | Fitted | | Gravity | 1.25 | Percolation backbone | Fitted | | Thermal | 1.3–1.5 | Lattice backbone | Fitted | | **Lorentz (SR)** | **2 (exact)** | **Quadratic metric** | **Derived** | | Dyn. friction | 3.8 | Velocity-space dimension | Fitted | | Electronic | ≈ 5 | Fermi × phonon (Bloch-Grüneisen) | Fitted | | Fluid (viscosity) | 7–10 | Intermolecular bond network | Fitted | | Fluid (Reynolds) | ≈ 20 | Regime boundary | Fitted | The Lorentz factor is the only entry that is algebraically exact (Tier 1 — mathematical compatibility). It shows the fidelity function $\mu(\eta, n)$ *contains* special relativity as the $n = 2$ sub-case. All other entries are calibrated fits (Tier 2). --- ## 5. Significance 1. **Special relativity is contained in the $L^n$ framework.** The Lorentz factor is $1/\mu(\eta, 2)$. This is a mathematical identity — choosing $n = 2$ and defining $\eta$ appropriately recovers $\gamma$ exactly. The non-trivial claim is that the *same* functional form appears across all sectors. 2. **The "2" is the metric dimension.** Just as n = 1.25 encodes the backbone dimension of 3D percolation and n = 5 encodes the Bloch-Grüneisen phase-space, n = 2 encodes the Euclidean metric exponent. If spacetime used a cubic or quartic metric (hypothetically), n would be 3 or 4. 3. **The fidelity function is universal.** The same μ(η, n) — with only n changing — governs galaxy rotation, steel fracture, electrical resistance, thermal conductivity, fluid viscosity, dynamical friction, and special relativity. Seven phenomena, one function, one parameter. --- ## 6. Limitations 1. **Dynamical friction fit is approximate.** R² = 0.997, not 1.0. The Chandrasekhar formula is fundamentally Gaussian/transcendental; the algebraic μ function is an approximation. 2. **No GR.** The Lorentz derivation covers flat spacetime only. Curved spacetime (gravitational time dilation, frame dragging) is not addressed. 3. **Equivalence principle is interpretive.** The claim that m_inertial = m_gravitational follows from ledger identity is not yet formally derived from RST axioms. --- ## 7. Links - **Theory:** [[Relational Dynamics (RST)]] - **Code:** [[Relational Dynamics - Code]] - **Fidelity inversion:** [[expanded theory/Fidelity Inversion — Gravity and Materials]] - **Tremaine Core-Cusp:** [[../Tremaine Core-Cusp/Tremaine Core-Cusp (RST)]] - **Fluid dynamics:** [[../Fluid Dynamics/Fluid Dynamics Results]]