Appendix D: Variables and Constants Reference
E.1 Core Framework Variables
| Symbol | Name | Definition | Units | Introduced |
|---|---|---|---|---|
| \(Z_0\) | Characteristic impedance | \(\sqrt{L/C}\) — determines perception range and density access | \(\Omega\) (model) | Ch 2, 5 |
| \(R\) | Resistance (dissipation) | Energy loss rate — maps to distraction, trauma, environmental noise | \(\Omega\) (model) | Ch 7 |
| \(L\) | Inductance (inertia) | This-lifetime accumulated wisdom/integration (\(L_{body} = L_{wisdom} + L_{integration}\)) | H (model) | Ch 8 |
| \(C\) | Capacitance (reactivity) | Unprocessed shadow material; emotional charge storage | F (model) | Ch 7 |
| \(Q\) | Quality factor | \(Z_0/R\) — sovereignty, discernment, resistance to injection locking | dimensionless | Ch 7 |
| \(f_0\) | Resonant frequency | \(1/(2\pi\sqrt{L_{body}C_{body}})\) — body’s characteristic reception scale (pattern scale, not density level) | Hz (model) | Ch 8 |
| \(f_d\) | Density carrier frequency | \(f_1 \cdot \beta_{cascade}^{(d-1)}\) — carrier frequency of density tier \(d\); scales with \(Z_d\) and \(E_d\) | Hz (model) | Ch 2 |
| \(f_{soul}\) | Spectral centroid | \(\int f \cdot |S(f)|^2 df / \int |S(f)|^2 df\) — characterizes soul developmental level; field property | Hz (model) | Ch 6 |
| \(\sigma\) | Spin coherence | Individual-level phase alignment; ranges 0 (random) to 1 (perfect) | dimensionless | Ch 13 |
| \(r\) | Kuramoto order parameter | Population-level phase synchronization; ranges 0 to 1 | dimensionless | Ch 11 |
E.2 Collective Dynamics Variables
| Symbol | Name | Definition | Units | Introduced |
|---|---|---|---|---|
| \(N\) | Population size | Total number of elements in the array | dimensionless | Ch 11 |
| \(f\) | Coherent fraction | Fraction of population that is phase-coherent | dimensionless | Ch 11 |
| \(f_c\) | Critical coherent fraction | \(\sqrt{T/N}\) — minimum coherent fraction for collective effect | dimensionless | Ch 11 |
| \(T\) | Detection threshold | SNR threshold for coherent signal detection; \(T = 10\) | dimensionless | Ch 11 |
| \(AF\) | Array factor | Complex sum of individual element contributions | dimensionless | Ch 11 |
| \(K\) | Coupling strength | Kuramoto inter-oscillator coupling coefficient | rad/s | Ch 11, 9 |
| \(K_c\) | Critical coupling | \(2/(\pi g(0))\) — minimum coupling for synchronization | rad/s | Ch 11 |
E.3 PLL (Phase-Locked Loop) Variables
| Symbol | Name | Definition | Units | Introduced |
|---|---|---|---|---|
| \(K_d\) | Phase detector gain | Perceptual sensitivity / felt-sense acuity | V/rad | Ch 9 |
| \(K_{vco}\) | VCO gain | Emotional responsiveness (freq shift per unit input) | rad/s/V | Ch 9 |
| \(F(s)\) | Loop filter transfer function | Belief system temporal dynamics | dimensionless | Ch 9 |
| \(V_{control}\) | Control voltage | Integrated emotional/cognitive correction signal | V (model) | Ch 9 |
| \(\theta_e\) | Phase error | Gap between soul purpose and current state | rad | Ch 9 |
| \(\omega_n\) | Loop natural frequency | Characteristic correction speed | rad/s | Ch 9 |
| \(\zeta_{loop}\) | Loop damping factor | Feedback stability (underdamped/overdamped) | dimensionless | Ch 9 |
| \(BW_{loop}\) | Loop bandwidth | Tracking agility vs. noise rejection | Hz | Ch 9 |
| \(\Delta\omega_L\) | Lock range | Range of circumstances where alignment holds | rad/s | Ch 9, 11 |
| \(\Delta\omega_C\) | Capture range | Range where system can acquire new lock | rad/s | Ch 9 |
| \(\alpha_{varactor}\) | Varactor sensitivity | Emotional sensitivity coefficient for capacitance modulation | dimensionless | Ch 9 |
| \(C_{eff}(t)\) | Effective capacitance | \(C_{baseline} + C_0 \cdot \alpha_{varactor} \cdot V_{emotion}(t)\) | F (model) | Ch 9 |
| \(V_{ref}(t)\) | Soul reference signal | Spectral signature carrier (→ Ch 6 §6.5) | V (model) | Ch 9 |
E.3a Matching Network / Convergence Variables
| Symbol | Name | Definition | Units | Introduced |
|---|---|---|---|---|
| \(N(i)\) | Degrees of freedom | Number of degrees of freedom at matching point \(i\) (crown fewest, root most) | dimensionless | Ch 9 |
| \(\mu\) | Adaptation rate | Biological plasticity coefficient; higher in young bodies | dimensionless | Ch 9 |
| \(\text{SNR}(i)\) | Band-specific SNR | \(\text{SNR}_{soul} \times \text{SNR}_{env}\) — joint soul signal clarity × environmental clarity | dimensionless | Ch 9 |
| \(\Gamma_{initial}(i)\) | Initial reflection coefficient | Initial impedance mismatch at band \(i\); set by \(Z_{soul}/Z_{body}\) gap | dimensionless | Ch 9 |
| \(\Gamma_{target}\) | Target reflection coefficient | Acceptable coupling threshold; \(\approx 0.1\) | dimensionless | Ch 9 |
| \(\tau(i)\) | Convergence time | \(N(i)/(2\mu \cdot \text{SNR}(i)) \cdot \ln(\Gamma_{initial}(i)/\Gamma_{target})\) | years (model) | Ch 9 |
E.4 Injection Locking Variables
| Symbol | Name | Definition | Units | Introduced |
|---|---|---|---|---|
| \(\omega_L\) | Locking bandwidth | \(\omega_0/(2Q) \cdot V_{inj}/V_0\) — range of frequencies over which lock holds | rad/s | Ch 12 |
| \(V_{inj}\) | Injection signal amplitude | Strength of the external locking signal | V (model) | Ch 12 |
| \(V_0\) | Free-running amplitude | Natural oscillation amplitude | V (model) | Ch 12 |
| \(\Delta\omega\) | Frequency detuning | \(\omega_{inj} - \omega_0\) — distance between injection and natural frequency | rad/s | Ch 12 |
| \(f_{esc}\) | Escape fraction | Critical fraction for narrative escape; \(\omega_{L,C}/(\omega_{L,C} + K_{mean})\) | dimensionless | Ch 12 |
E.5 Link Budget Variables
| Symbol | Name | Definition | Units | Introduced |
|---|---|---|---|---|
| \(P_S\) | Source power | Transmitted power from Source | dB (model) | Ch 17 |
| \(G_S\) | Source gain | Directivity of Source transmission | dB | Ch 17 |
| \(L_{path}\) | Path loss | Signal attenuation through density cascade | dB | Ch 17 |
| \(L_{paradigm}\) | Paradigm loss | Attenuation from paradigm shielding (Faraday cage) | dB | Ch 16, 14 |
| \(L_{parasitic}\) | Parasitic loss | Energy extracted by parasitic coupling | dB | Ch 15, 14 |
| \(G_{practices}\) | Practice gain | Signal enhancement from spiritual/contemplative practices | dB | Ch 17, 17 |
| \(G_{collective}\) | Collective gain | \(10\log_{10}(N \cdot r^2)\) — phased array gain from group coherence | dB | Ch 11, 14 |
| \(M\) | Link margin | \(P_{received} - P_{threshold}\) — excess signal above minimum | dB | Ch 17 |
| \(SE\) | Shielding effectiveness | Attenuation provided by paradigm cage layers | dB | Ch 16 |
| \(NF\) | Noise figure | Degradation of SNR through processing chain | dB | Ch 16 |
| \(J/S\) | Jamming-to-signal ratio | \(\text{ERP}_J - \text{ERP}_S + 20\log(d_S/d_J) + G_{RJ} - G_R\) | dB | Ch 16 |
| \(R_{BT}\) | Burn-through range | Distance at which truth signal overcomes jammer: \(d_J\) where \(J/S = 0\) dB | distance (model) | Ch 16 |
| \(G_p\) | Processing gain | \(10\log_{10}(BW_{spread}/BW_{info})\) — spread spectrum advantage | dB | Ch 17 |
| \(AJM\) | Anti-jam margin | \(G_p - L_{sys} - \text{SNR}_{out,req}\) — surplus protection above minimum | dB | Ch 17 |
| \(\tau_{pull-off}\) | Pull-off time constant | Time from peak experience to maximum displacement from control narrative | time (model) | Ch 17 |
| \(\tau_{pull-back}\) | Pull-back time constant | Time from peak displacement to gravitational return toward control narrative | time (model) | Ch 17 |
| \(M_{fade}\) | Fade margin | \(P_{received} - P_{threshold} - L_{implementation}\) — surplus coherence above minimum | dB | Ch 17 |
| \(K_{Ricean}\) | Ricean K-factor | \(P_{dominant}/P_{scattered}\) — teacher dependence ratio | dimensionless | Ch 17 |
| \(CR\) | Compression ratio | \(BW \cdot T_{sweep}\) — pulse compression / debrief integration gain | dimensionless | Ch 18 |
E.6 Density and Field Variables
| Symbol | Name | Definition | Units | Introduced |
|---|---|---|---|---|
| \(d\) | Density level | Integer 1–7 indicating density tier | dimensionless | Ch 2 |
| \(Z_d\) | Density impedance | \(Z_1 \cdot \beta_{cascade}^{(d-1)}\) — characteristic impedance of density \(d\) | \(\Omega\) (model) | Ch 2 |
| \(\beta_{cascade}\) | Cascade ratio | Impedance ratio between adjacent densities; ~5 (illustrative) | dimensionless | Ch 2 |
| \(f_d\) | Density carrier frequency | \(f_1 \cdot \beta_{cascade}^{(d-1)}\) — characteristic frequency of density \(d\); scales with \(Z_d\) | Hz (model) | Ch 2 |
| \(E_d\) | Density energy | \(E_1 \cdot \beta_{cascade}^{(d-1)}\) — characteristic energy of density \(d\); scales with \(Z_d\) and \(f_d\) | energy (model) | Ch 2 |
| \(\Gamma_d\) | Reflection coefficient | \((Z_d - Z_0)/(Z_d + Z_0)\) — power coupling efficiency at density boundary | dimensionless | Ch 2 |
| \(\kappa\) | Torsion coupling constant | Spin-torsion coupling strength | varies | Ch 0, 11 |
| \(D_s\) | Spectral dimension | Effective dimensionality at a given energy scale | dimensionless | Ch 2 |
E.7 Key Constants
| Constant | Value | Source |
|---|---|---|
| \(f_c\) (Earth) | \(\approx 0.0035\%\) (~283,000 people) | \(\sqrt{T/N}\) with \(T=10\), \(N=8\times10^9\) |
| Critical influencer count | ~283 | \(f_c \times N / A\) with \(A = 1000\) amplification |
| \(g^*\) (UV fixed point coupling) | \(0.71 \pm 0.02\) | Asymptotic safety calculations (Ch 0, 2) |
| \(g_*\) (running equation coefficient) | \(\approx 0.27\) | Gravitational running (Ch 4) — distinct from \(g^*\) |
| \(a_0\) (MOND acceleration) | \(\approx cH_0/6 \approx 1.2 \times 10^{-10}\) m/s\(^2\) | Ch 4 |
E.8 Notation Conventions
- Sigmoid function: \(S()\) — used throughout instead of \(\sigma()\) to avoid collision with coherence \(\sigma\)
- Imaginary unit: \(j\) (engineering convention)
- Torsion field strength: \(\mathcal{T}\) in equations (Chapter 13+); distinguished from temperature \(T\) and torsion tensor \(T^{\lambda}_{\ \mu\nu}\)
- Coherence: \(\sigma\) for individual spin coherence (Ch 13+); \(r\) for population phase alignment (Ch 11+)
- Critical fraction: \(f_c\) for coherent population threshold (Ch 11); \(f_{esc}\) for narrative escape threshold (Ch 12)
- PLL variables: \(K_d\), \(K_{vco}\), \(\theta_e\), \(\omega_n\), \(\zeta_{loop}\) (Ch 9); distinguished from injection locking variables (\(\omega_L\), \(V_{inj}\), \(\Delta\omega\)) in Ch 12
End of Appendix D: Variables and Constants Reference