---
© 2015-2026 Wesley Long & Daisy Hope. All rights reserved.
Synergy Research — FairMind DNA
License: CC BY-SA 4.0
Originality: 95% — Original notation; formalizes the Quadrian Arena speed-of-light derivation
---

# Quadrian Arena Notation Sheet
## Complete Algebra for the Speed of Light Derivation

**Wesley Long — Synergy Research**

---

## 1. The Quadrian Ratio

$$q = \sqrt{1^2 + 0.5^2} = \sqrt{1.25} = \frac{\sqrt{5}}{2}$$

This is the diagonal distance from the origin to the point (1, 0.5) in the unit square. It is also the Quadrian Ratio, from which the Golden Ratio follows:

$$\Phi = q + \frac{1}{2} = \frac{\sqrt{5}+1}{2}, \qquad \varphi = q - \frac{1}{2} = \frac{\sqrt{5}-1}{2}$$

---

## 2. Quadrian Angles

From the ratio $q$ and the Quadrian Components:

$$\theta_x = (q + 0.5)(15 + \sqrt{2})$$

$$\theta_y = 90 - \theta_x$$

$$\theta_z = 2\theta_y$$

$$\theta_v = \theta_y - \theta_x$$

$$\theta_u = 7\theta_z$$

**Constants used:**
- **15** — Hemisphere Points (point structure)
- **$\sqrt{2}$** — unit square diagonal
- **90** — right angle (quarter turn of unit square)
- **7** — octahedral multiplier

---

## 3. Path Endpoints

Two paths through the unit square, Northern and Eastern, each computed as a weighted combination of the Quadrian Angles:

$$P_{Na} = 4\theta_x + 3\theta_y \qquad\text{(Northern aggregate)}$$

$$P_{Ea} = 3\theta_x + 4\theta_y \qquad\text{(Eastern aggregate)}$$

$$P_{Np} = \theta_u + \theta_y \qquad\text{(Northern path)}$$

$$P_{Ep} = \theta_u + \theta_x \qquad\text{(Eastern path)}$$

**Ratios:**

$$Q_a = \frac{P_{Na}}{P_{Ea}}, \qquad Q_c = \frac{10^3 - P_{Np}}{10^3 - P_{Ep}}$$

---

## 4. The Quadrian Path Equation — Qp(n)

$$\boxed{\mathrm{Qp}(n) = 30 - \frac{1}{10^3 - n} - \frac{2n}{10^7\sqrt{5}}}$$

**Form:** Linear combination of a rational pole term and a scaled linear term. The pole is at $n = 10^3$; the slope is governed by $\sqrt{5}$.

**Constants:**
- **30** — Angular Limit $F$, derived from $\frac{2}{1/6} \times \frac{15}{8} \times \frac{8}{6}$
- **$10^3$** — Arena Capacity $L = 8 \times U_q^2$ where $U_q = \sqrt{5}/2 \times 10$
- **$10^7$** — Scale $S = L \times 10^4$
- **$\sqrt{5}$** — from Quadrian Ratio $2q = \sqrt{5}$

---

## 5. The Quadrian Speed Equation — Qs(n)

$$\boxed{\mathrm{Qs}(n) = 10^7\left(30 - \frac{1}{10^3 - n}\right) - \frac{2n}{\sqrt{5}}}$$

This is a scaled version of Qp(n):

$$\mathrm{Qs}(n) = 10^7 \times \mathrm{Qp}(n) + \text{(rearranged terms)}$$

**Form:** Rational function with a simple pole at $n = 10^3$. Away from the pole it is approximately linear: $\mathrm{Qs}(n) \approx 3 \times 10^8 - \text{(small corrections)}$.

---

## 6. The Two Speeds of Light

$$\boxed{c_y = \mathrm{Qs}(P_{Np}) \approx 299{,}792{,}457.553 \text{ m/s} \qquad\text{(Northern path)}}$$

$$\boxed{c_x = \mathrm{Qs}(P_{Ep}) \approx 299{,}881{,}898.796 \text{ m/s} \qquad\text{(Eastern path)}}$$

| Speed | Value | CODATA | Δ |
|---|---|---|---|
| $c_y$ (Northern) | 299,792,457.553 m/s | 299,792,458 m/s | 0.45 m/s |
| $c_x$ (Eastern) | 299,881,898.796 m/s | — (no known counterpart) | — |

$c_y$ matches the measured speed of light to within 0.45 m/s. $c_x$ is a prediction — an open question (see OPEN_QUESTIONS A-1).

---

## 7. Vacuum Constants

From $c_y$ and Syπ:

$$\mu_0 = 4\Pi(162) \times 10^{-7} \qquad\text{(vacuum permeability)}$$

$$\varepsilon_0 = \frac{1}{\mu_0 c_y^2} \qquad\text{(vacuum permittivity)}$$

$$C = \mathrm{Me}(1, c) = \frac{1}{c} \qquad\text{(inertial impedance)}$$

$$Z_0 = \frac{C}{\varepsilon_0} \qquad\text{(characteristic impedance of free space)}$$

**Electromagnetic Identity:**

$$\varepsilon_0 \mu_0 c_y^2 = 1 \qquad\text{(exact at float64)}$$

---

## 8. The Derivation Chain

The complete chain from unit square to speed of light:

$$1^2 + 0.5^2 \xrightarrow{\sqrt{\phantom{x}}} q \xrightarrow{+0.5} \Phi$$

$$q, 15, \sqrt{2} \xrightarrow{\theta_x} \theta_y \xrightarrow{\times 2} \theta_z \xrightarrow{\times 7} \theta_u$$

$$\theta_u + \theta_y \to P_{Np} \xrightarrow{\mathrm{Qs}} c_y = 299{,}792{,}457.553$$

Every step uses only Quadrian Components — observed geometric numbers from point structures in the unit square. No measured values enter the chain.

---

## 9. Mathematical Classification

The Quadrian Speed equation,

$$\mathrm{Qs}(n) = 10^7\left(30 - \frac{1}{10^3 - n}\right) - \frac{2n}{\sqrt{5}},$$

is a **rational function** with a simple pole at $n = 10^3$. In the neighborhood of the path endpoints ($P_{Np} \approx 591$, $P_{Ep} \approx 614$), it is well-behaved and monotonically decreasing.

The form is classical: a scaled reciprocal plus a linear term. As with the other SSM equations, the mathematical family is standard; the specific constants and physical interpretation are the custom part.

> **Slide caption:** Qs(n) is a rational function with a simple pole. The speed of light is not a free parameter — it is the output of this function evaluated at a geometrically determined path endpoint.

---

## 10. Comparison with Other SSM Equations

| Property | Syπ — $\Pi(n)$ | Fe — $\mathrm{Fe}(n)$ | Ma — $\mathrm{Ma}(n)$ | Qs — $\mathrm{Qs}(n)$ |
|---|---|---|---|---|
| **Form** | $a/(bx+c)$ | $1/[(x+c)(x+c+1)]$ | $kx$ | $S(F - 1/(L-x)) - 2x/\sqrt{5}$ |
| **Family** | Linear fractional | Quadratic rational | Linear | Rational (pole + linear) |
| **Pole** | $n \approx -565.5$ | $n = -k, -k-1$ | None | $n = 10^3$ |
| **Physical output** | π gradient | Coupling constant | Mass (kg) | Speed (m/s) |
| **Inverse** | Px (linear algebra) | Fi (quadratic formula) | Mx (division) | Not named |

---

## 11. Implementation Reference

```javascript
// Quadrian Path Equation — Qp(n)
Qp(n) {
    return (30 - 1 / (Math.pow(10, 3) - n)) - (2 * n / (Math.pow(10, 7) * Math.sqrt(5)));
}

// Quadrian Speed Equation — Qs(n)
Qs(n) {
    return (Math.pow(10, 7) * (30 - (1 / (Math.pow(10, 3) - n)))) - ((2 * n) / Math.sqrt(5));
}

// Quadrian Arena Model — Qa()
Qa() {
    const q = Math.sqrt(Math.pow(1, 2) + Math.pow(0.5, 2));
    const sqrt2 = Math.sqrt(2);
    const θx = (q + 0.5) * (15 + sqrt2);
    const θy = 90 - θx;
    const θz = θy * 2;
    const θv = θy - θx;
    const θu = θz * 7;
    const PNa = 4 * θx + 3 * θy;
    const PEa = 3 * θx + 4 * θy;
    const PEp = θu + θx;
    const PNp = θu + θy;
    // ... cy = Qs(PNp), cx = Qs(PEp), μ0, ε0, Z0, etc.
}
```

Key values:

| Symbol | Value | Meaning |
|---|---|---|
| $q$ | 1.118033988749895 | Quadrian Ratio $\sqrt{5}/2$ |
| $\theta_x$ | 26.246... | Primary Quadrian Angle |
| $\theta_y$ | 63.753... | Complementary angle |
| $P_{Np}$ | 591.49... | Northern path endpoint |
| $P_{Ep}$ | 614.31... | Eastern path endpoint |
| $c_y$ | 299,792,457.553 | Speed of light (Northern) |
| $c_x$ | 299,881,898.796 | Speed of light (Eastern) |

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*Synergy Standard Model v2.0 — © 2015–2026 Synergy Research. All rights reserved.*