>[!warning] >This content has not been peer reviewed. # Quantization — RST calibration results **Summary:** The $L^n$ kinetic energy framework $E_k \propto k^n$ produces **three exact identities** and two high-quality fits across the known quantum energy spectra. The particle-in-a-box ($n = 2$, $R^2 = 1$) and hydrogen atom ($n = -2$, $R^2 = 1$) are algebraic identities, not fits. The harmonic oscillator yields $n \approx 1.24$ — strikingly close to the gravity backbone dimension $d_B = 1.25$. --- ## 1. L^n ball geometry The 2D $L^n$ unit ball $|x|^n + |y|^n \le 1$ transitions from star-shaped (concave, $n < 1$) through a perfect circle ($n = 2$) to square-like (convex, $n \to \infty$). | $n$ | Area | Lacunarity | Character | |:---|---:|---:|:---| | 0.30 | 0.132 | 30.24 | Star (concave) | | 0.50 | 0.667 | 6.00 | Star (concave) | | 1.00 | 2.000 | 2.00 | Diamond | | 1.25 | 2.427 | 1.65 | Pinched | | 2.00 | 3.142 | 1.27 | Circle (Euclidean) | | 5.00 | 3.801 | 1.05 | Square-like | | 20.0 | 3.985 | 1.00 | Near-square | **Lacunarity** $\mathcal{L}(n) = 4/A(n)$ measures how "gappy" the geometry is. At $n = 0.3$ (tensile regime), the lacunarity is 30 — the ball occupies only 3.3% of its enclosing square. At $n = 20$ (Reynolds), lacunarity approaches 1 — the ball fills the square. ![[quantization_geometry_balls.png]] *The L^n unit ball for nine values of n spanning the Dimensional Ladder. At n < 1 (star shapes), most of phase space is empty — the substrate is "pixelated."* ![[quantization_geometry_lacunarity.png]] *The Lacunarity Spectrum. Lacunarity diverges as n → 0 (maximum pixelation) and approaches 1 as n → ∞ (continuous bulk). The Dimensional Ladder sectors are marked.* --- ## 2. Heisenberg limit from $L^n$ constraint The minimum uncertainty product for a phase-space cell of bit-size $\hbar$ in $L^n$ geometry: $\min(\Delta x \cdot \Delta p) = \frac{\hbar^2}{2^{2/n}}$ | $n$ | $\min(\Delta x \cdot \Delta p) / \hbar^2$ | Regime | |:---|---:|:---| | 0.30 | 0.010 | Sub-Euclidean | | 0.50 | 0.063 | Sub-Euclidean | | 1.00 | 0.250 | Sub-Euclidean | | 1.25 | 0.330 | Sub-Euclidean | | **2.00** | **0.500** | **Euclidean (Heisenberg)** | | 5.00 | 0.758 | Super-Euclidean | | 20.0 | 0.933 | Super-Euclidean | **At $n = 2$: $\min(\Delta x \cdot \Delta p) = \hbar^2/2$.** In natural units ($\hbar = 1$), this is $1/2$ — the Heisenberg uncertainty limit. The limit is not a fundamental law; it is the geometric invariant of the $L^2$ norm applied to phase space. ![[quantization_uncertainty.png]] *Minimum uncertainty product vs substrate geometry n. The Heisenberg limit (1/2) is recovered exactly at n = 2. Sub-Euclidean regimes impose tighter constraints; super-Euclidean regimes are weaker.* --- ## 3. Energy spectrum identity The $L^n$ kinetic energy gives $E_k/E_1 = k^n$. At $n = 2$, this is the particle-in-a-box spectrum. **Identity check ($n = 2$):** | Metric | Value | |:---|---:| | $R^2$ | **1.000000000000000** | | max $|$error$|$ | **0.00** | The particle-in-a-box energy spectrum is an **algebraic identity** of $L^n$ kinetics at $n = 2$. The "2" in $E = p^2/(2m)$ is the substrate's Euclidean exponent — the same "2" as the Lorentz factor $\gamma = 1/\mu(\eta, 2)$. ![[quantization_spectrum_levels.png]] *Energy levels as horizontal bars for six values of n. At n = 2: standard particle-in-a-box spacing. At n = 1: approximately linear (harmonic oscillator character). At n > 2: levels spread rapidly (stiff confinement).* ![[quantization_spectrum_loglog.png]] *Log-log plot of E_k/E_1 vs mode number k. Straight lines confirm the power-law E_k/E_1 = k^n. The slope equals n.* --- ## 4. Discovery mode: effective n for quantum systems | System | $n_{fitted}$ | $R^2$ | Expected | Status | |:---|---:|---:|:---|:---| | **Particle-in-box** | **2.0000** | **1.00000000** | $n = 2$ (identity) | **Identity** | | **Hydrogen atom** | **−2.0000** | **1.00000000** | $n = -2$ (bound inversion) | **Identity** | | Harmonic osc. (1D) | 1.2393 | 0.98700575 | $n \sim 1$ (linear) | Fitted | | Harmonic osc. (3D) | 0.8630 | 0.99548770 | $n \sim 1$ (linear) | Fitted | ![[quantization_discovery.png]] *Discovery mode: fitting k^n to known QM energy spectra. Particle-in-box and hydrogen are exact identities (R² = 1). The harmonic oscillators are high-quality fits with n ≈ 1.* --- ## 5. The three identities The RST framework contains **three algebraic identities** ($R^2 = 1$ to machine precision): | Identity | $n$ | Physical law | Proof | |:---|:---|:---|:---| | **Lorentz factor** | $n = 2$ | $\gamma = 1/\mu(\eta, 2)$ | Velocity-space bandwidth limit | | **Particle-in-box** | $n = 2$ | $E_k = k^2 \pi^2\hbar^2/(2mL^2)$ | $L^2$ kinetic energy in a box | | **Hydrogen spectrum** | $n = -2$ | $E_k/E_1 = 1/k^2$ | Bound-state exponent inversion | The first two share $n = 2$: both are consequences of the Euclidean ($L^2$) geometry. The Lorentz factor operates in velocity space; the kinetic energy operates in momentum space. The hydrogen identity introduces **negative** $n$: the bound-state spectrum inverts the free-particle exponent ($+2 \to -2$). This is the quantum analog of the [[expanded theory/Fidelity Inversion — Gravity and Materials|Fidelity Inversion]] between gravity and materials. **Epistemological status (Tier 1):** These identities prove the $L^n$ framework is *compatible* with known physics — the $n = 2$ sub-case reproduces SR and QM. They do not prove that the framework *derives* these laws, since any framework with a tunable power-law exponent can reproduce power-law spectra by choosing the appropriate exponent. The non-trivial structural claim is that the *same* functional form $\mu$ and the *same* $n = 2$ appear in both relativistic and quantum domains. --- ## 6. The harmonic oscillator and the backbone dimension The 1D harmonic oscillator yields $n_{fitted} = 1.24$ — within 1% of the gravity backbone dimension $d_B \approx 1.22-1.25$. This is striking but requires careful interpretation: - The harmonic oscillator spectrum $E_k = (2k-1)\hbar\omega$ is **not** a pure power law. It is linear in $k$ for large $k$ (which would give $n = 1$), but the zero-point offset shifts the effective exponent to $\sim 1.24$. - The match with $d_B$ may be coincidental or may reflect a deeper connection between the harmonic (parabolic) potential and the percolation backbone topology. - The 3D harmonic oscillator gives $n = 0.86$ (lower due to the $3/2$ zero-point shift), further suggesting the offset effect dominates. **Status:** Interesting observation, not yet a derivation. The harmonic oscillator does not sit cleanly on the Dimensional Ladder because its spectrum is affine ($ak + b$), not purely power-law ($k^n$). --- ## 7. The Lacunarity Spectrum The lacunarity $\mathcal{L}(n) = 4/A_2(n)$ provides a continuous measure of the substrate's "pixelation": - **$\mathcal{L} \to \infty$ ($n \to 0$):** Maximum sparsity. The $L^n$ ball collapses to the coordinate axes. Only the axes and near-axis regions contain valid states. This is the **quantization regime** — discrete modes emerge because most of phase space is geometrically forbidden. - **$\mathcal{L} \approx 1.27$ ($n = 2$):** The Euclidean circle. Standard quantum mechanics lives here — the Heisenberg limit, the particle-in-a-box, the Schr&ouml;dinger equation. - **$\mathcal{L} \to 1$ ($n \to \infty$):** The ball fills its enclosing square. No gaps, no pixelation. This is the **classical bulk** regime — fluids, turbulence, coordination. The Lacunarity Spectrum generalizes the **Lacunarity Gap** from [[expanded theory/Substrate Eigenvalues]], which uses the specific value at $n = 1.25$ to derive the proton-electron mass ratio $m_p/m_e \approx 1836$. --- ## 8. Links - **Theory:** [[Quantization (RST)]] - **Code:** [[Quantization - Code]] - **Topological spectrum:** [[expanded theory/The Spectrum of Relational Topologies]] - **Substrate Eigenvalues (Lacunarity Gap):** [[expanded theory/Substrate Eigenvalues]] - **Lorentz identity:** [[../Relational Dynamics/Relational Dynamics Results]] - **Fidelity Inversion (exponent sign flip):** [[expanded theory/Fidelity Inversion — Gravity and Materials]]