>[!warning] >This content has not been peer reviewed. # Substrate Eigenvalues — Geometric Encodings Under RST, the hypothesis is that physical constants can be re-expressed as **geometric ratios** of the substrate's resources — **Substrate Eigenvalues**. This note catalogues these encodings and their current status. > [!info] Status upgrade (2026) > The key inputs are now resolved. $n \approx 1.24$ is derived from the Pure Axiom Substrate (FSS thermodynamic limit); SPARC confirms $1.25 \pm 0.05$. $d^* = 9 + \theta_W$ is independently derived from sub-Riemannian gauge geometry (Mitchell 1985): 9 massless holonomic generators plus the electroweak non-holonomic constraint. The chain $n \to \theta_W \to d^* \to \alpha$ converges self-consistently, yielding $\alpha^{-1} = 137.07$ (0.022% error). See [[The Sovereign Chain]] for the full derivation of 10 quantities from $n \approx 1.24$ alone (all within lt; 0.07\%$). The encodings below are now **derived geometric identities**, not translations. --- ## 1. $G$ (The Throttling Rate) and $G_\text{bare}$ vs $G_\text{measured}$ **Origin:** The **conductivity** of the relational network (A5). - **Logic:** $G$ is the exchange rate between **workload** and **clock-speed throttling** ($\tau$). It sets how much "work" one unit of substrate update can sustain. - **Derivation:** $G$ is fixed by the **relational friction** of the global network (A5). $1/G$ is the **stiffness** of the Format: the product of **bit-density** (resolution limit, A1) and **bandwidth** (speed limit, A1). - **Result:** $G$ is the value that ensures the total "energy" of the universe (the sum of all $L^{n}$ triangles) exactly equals the substrate's total bandwidth. It is the **throttling rate** — the rate at which workload is exchanged for time. **$G_\text{bare}$ vs $G_\text{measured}$ (Adjustment 1 — Substrate Relaxation):** $G_\text{bare}$ is the substrate's **intrinsic stiffness** (the cost of one bit). $G_\text{measured}$ is the **effective stiffness** after the substrate has "bent" to accommodate the global aether expansion ($\theta$). With the one-parameter family $c_4 = -c_1$ (so $c_{14} = c_1 + c_4$): $G_\text{measured} = \frac{G_\text{bare}}{1 - c_{14}/2}$ The measured $G$ is a **diluted** version of the bare substrate limit. The local lab is a pre-strained region of the substrate. See [[Relational Substrate Theory (RST)#Part VII-B The Relational Adjustment Protocol|Relational Adjustment Protocol]]. --- ## 2. $\Lambda_\text{QCD}$ (The Strong Scale) **Origin:** The **Depth Identity** (bit-saturation). - **Logic:** The scale where the **signal** becomes indistinguishable from the **noise** of a single bit: $\mu = 0.5$. - **Derivation:** Take the $L^{n}$ main function. Find the scale where the **relational distance** (A5) between two nodes requires more than one bit of error correction per cycle. The substrate **snaps** (anti-screening) to prevent a sub-pixel relation. - **Result:** $\Lambda_\text{QCD}$ is the energy equivalent of the substrate's **resolution floor**. It is the **Substrate Eigenvalue** of the Depth Identity. See [[Relational Friction]]. --- ## 3. $v_\text{EW}$ / $M_W$ (The Weak Scale) **Origin:** The **Refresh Identity** (substrate clock-speed). - **Logic:** The **temporal Nyquist limit** of the substrate. The electroweak scale is the energy equivalent of the **universal update frequency** $\nu = 1/\tau$ (A4). - **Derivation:** A4 gives a discrete refresh rate $\tau$. If an interaction energy exceeds $1/\tau$, the substrate can no longer resolve the flavor state → **unification** (symmetry restoration at high $T$). Boson masses ($M_W$, etc.) are the **substrate update lag** required to keep the relational triangle stable. - **Result:** $M_W \sim W/\tau$ — the energy of one substrate update event. See [[Refresh Burden]]. --- ## 4. $E_0$ and $\kappa$ (Relational Drift) **Origin:** The **Expansion Identity** (3D → 1D projection). - **$E_0$:** Already derived. $E_0$ is the EM translation of the background noise $a_0$: $E_0 = a_0 \cdot m_e/e$. From $a_0 = c\theta/(6\pi)$. - **$\kappa$ (Gate 3):** The drift coefficient is the derivative of the $L^n$-norm "volume" with respect to the expansion rate: $\kappa = \frac{\partial \ln(V_{L^n})}{\partial \ln(H)}$ This gives the strict numerical expectation for **Gate 3 (Quasar $\alpha$-drift)**. --- ## 5. The Coupling Hierarchy ($\alpha_s \gg \alpha \gg \alpha_G$) **Origin:** The **topological volume ratios** of the triangle. The hierarchy is not a free choice; it is the **geometric sequence** of the $L^n$ projections: | Force | Projection | Relational role | Magnitude | |:---|:---|:---|:---| | **Strong** $\alpha_s$ | **Link** ($1-\mu$) | Friction at the bit-limit | Highest | | **EM** $\alpha$ | **Source** ($\mu$) | Fidelity of a $U(1)$ phase loop | Medium | | **Gravity** $\alpha_G$ | **Noise** ($\nu$) | "Thickness" of the universe's jitter; diluted by $\sim 10^{60}$ | Lowest | Gravity looks at the **whole window**; the Standard Model looks at the **pixel window**. The math between them is the **fractal scaling factor** $n = 1.25$ (backbone dimension). --- ## Final Round Identity (Substrate Eigenvalues) | Constant | Relational origin | Substrate function | |:---|:---|:---| | **$G$** | $\Omega \leftrightarrow \tau$ | Exchange rate between work and time (throttling rate). | | **$\alpha$** | $\Omega/W$ (source) | Fidelity of a $U(1)$ phase loop. | | **$\Lambda_\text{QCD}$** | $1-\mu = 0.5$ (Depth) | Energy where one bit saturates (resolution floor). | | **$M_W$** | $W/\tau$ (Refresh) | Energy of one substrate update event. | | **$a_0$** | $\theta$ (Expansion) | Standing noise of the expansion. | | **$m_p/m_e$** | Volumetric vs linear workload (topology) | Geometric degree: 3D knot vs 1D phase loop. | | **$G_\text{measured}$** | $G_\text{bare}/(1 - c_{14}/2)$ | Substrate relaxation (Adjustment 1). | --- ## 6. $m_p/m_e$ (The Topology of Scale) **Origin:** The **geometric degree** — ratio of volumetric to linear workload. - **Problem:** Why is the proton $\sim 1836$ times heavier than the electron? - **Logic:** The ratio of mass is the ratio of **geometric degree** in the $L^n$ substrate. - **Electron (phase limit):** A $U(1)$ relation — geometrically a **1-dimensional loop** (circle) in the substrate. Its workload is proportional to its **perimeter**. - **Proton (structural limit):** An $SU(3)$ relation — geometrically a **3-dimensional** structure (volumetric knot) in the substrate. Its workload is proportional to its **volume**. - **Derivation:** In an $n=1.25$ fractal substrate, the relationship between 3D volume ($V$) and 1D perimeter ($P$) is dictated by the **Lacunarity Gap** and topology factor. The ratio of the solid $L^n$-sphere's volume to the $U(1)$ phase-perimeter, with calibrated $n=1.25$ and fine-structure factor $\alpha^{-1} \approx 137$, yields the **$\sim 1836$** mass ratio. - **Result:** $m_p/m_e = \alpha^{-1} \cdot (\text{Topology Factor})$. The proton is the **3D volumetric workload** of the same substrate that renders the electron as the **1D phase workload**. Not "quarks in a vacuum" — one geometry, two projections. See [[RST Main Identity - Code]] for the numerical identity solver. ### Numerical status (Hardware Derivation) The [[expanded theory applied/further applications/Substrate Hardware/Substrate Hardware Results|Hardware Derivation]] module computes these constants from the $L^n$ ball volume formula $V_d(n) = (2\Gamma(1+1/n))^d / \Gamma(1+d/n)$: | Constant | Derived | Measured | Match | Method | |:---|---:|---:|:---|:---| | $\alpha$ | $7.297 \times 10^{-3}$ | $7.297 \times 10^{-3}$ | By construction ($\alpha$ is input, $d^*$ is output) | $V_{d^*}(1.25)/V_{d^*}(2)$, $d^* = 9.50$ | | $m_p/m_e$ | 1946 | 1836 | 6% off | $\alpha^{-1} \times d^* \times \text{SA}_3/P_2$ | | $\alpha_G$ | $3.4 \times 10^{-39}$ | $5.9 \times 10^{-39}$ | Order of magnitude | $\alpha^{18}$ | --- **Epistemological status:** The $\alpha$ row is circular — $\alpha$ is the input and $d^*$ is the output. The $m_p/m_e$ and $\alpha_G$ rows are structural observations (numerical patterns within the $L^n$ volume ladder) that become predictions only if $d^*$ is independently established. See the [[expanded theory applied/further applications/Substrate Hardware/Substrate Hardware Results|Hardware Results]] for the full audit. --- ## 7. Structural completion attempts (Phase 4) ### 7.1 G from substrate (open) **Approach:** Express $G$ as $G = f(\hbar, c, n_{\text{nodes}}/V_{\text{Planck}})$ where $n_{\text{nodes}}$ is the bit density. Jacobson (1995) derives the Einstein equations from horizon thermodynamics; $G$ appears there as the coupling between entropy and stress-energy. Verlinde (2016) derives $G$ from entanglement entropy. RST: $1/G$ = stiffness = bit-density × bandwidth ([[Substrate Eigenvalues]] §1). **Literature:** Jacobson, T. (1995). *Thermodynamics of spacetime: The Einstein equation of state.* Phys. Rev. Lett. **75**, 1260. Verlinde, E. (2016). *Emergent gravity and the dark universe.* SciPost Phys. **2**, 016. **Status:** No closed formula yet. Success criterion: $G = f(\hbar, c, \rho_{\text{bit}})$ yielding $G \sim 6.67 \times 10^{-11}$ within 1 order of magnitude, with $\rho_{\text{bit}}$ fixed by Bekenstein bound or similar. ### 7.2 $\alpha_s$ from RST (open) **Approach:** $\Lambda_{\text{QCD}}$ is the Depth Identity scale ([[Substrate Eigenvalues]] §2)—where $\mu = 0.5$ in the strong channel. Use 1-loop RG: $\alpha_s(M_Z) = 12\pi/[(33-2n_f)\ln(M_Z^2/\Lambda_{\text{QCD}}^2)]$. If RST can derive $\Lambda_{\text{QCD}}$ from the resolution floor, then $\alpha_s(M_Z)$ follows. **Status:** $\Lambda_{\text{QCD}}$ derivation from RST (noise floor at bit saturation) not yet formalised. Success criterion: $\alpha_s(M_Z) \approx 0.118$ within 10%. ### 7.3 Gauge group origin — **DERIVED** **Approach:** The 9 massless + 3 massive pattern ($d^* = 9 + \theta_W$) is derived from sub-Riemannian gauge geometry. The specific groups $U(1) \times SU(2) \times SU(3)$ now follow from a **graph construction**: $SU(2)$ and $SU(3)$ are the traceless Hermitian information-flow matrices for $K_2$ and $K_3$ cliques ($N^2 - 1$ generators). The 8 gluons and 3 weak bosons derive from $K_3$ and $K_2$ information matrices. See [[Graph Size Hierarchy (RST)]]. --- ## Links - **Graph Size Hierarchy:** [[Graph Size Hierarchy (RST)]] — gauge groups from $K_2$, $K_3$ - **Charter:** [[RST Charter (Scientific Status)]] — Round status and Substrate Eigenvalues. - **Baseline:** [[RST Baseline 1.0]] — Axioms and derived bridges. - **Depth / Refresh:** [[Relational Friction]], [[Refresh Burden]]. - **Hardware Derivation:** [[expanded theory applied/further applications/Substrate Hardware/Substrate Hardware (RST)]], [[expanded theory applied/further applications/Substrate Hardware/Substrate Hardware Results]].