# Theoretical Fiber Latency Formula Verification

This page independently reconciles the equations used by the latency-page generator with a worked **Tokyo (TYO) → Sydney (SYD)** example. It starts with GeoNames city-centre coordinates, carries units through every calculation, and uses a fixed 101.7ms RTT fixture captured on July 18, 2026. Keeping the fixture fixed separates mathematical regression checks from routine latency-matrix refreshes.

> **Verification scope**
>
> This is a reproducibility check for the public theoretical model. It verifies the mathematics and
> implementation outputs; it does not attempt to reconstruct a deployed cable path. Read [Latency
> Coordinates & Fiber Model Inputs](/docs/network/latency/theoretical-fiber-latency) for coordinate
> resolution, source selection, and engineering assumptions.

## Symbols and reference inputs

| Symbol    | Meaning                            |                  Verification value |
| --------- | ---------------------------------- | ----------------------------------: |
| $A$       | Tokyo GeoNames city centre         |  $35.68950^\circ,\ 139.69171^\circ$ |
| $B$       | Sydney GeoNames city centre        | $-33.86785^\circ,\ 151.20732^\circ$ |
| $d_g$     | WGS-84 inverse-geodesic distance   |             $7\,792.8\ \mathrm{km}$ |
| $c$       | Speed of light in vacuum           | $299\,792.458\ \mathrm{km\,s^{-1}}$ |
| $n_s$     | Standard-fiber group index         |                            $1.4679$ |
| $n_u$     | Low-latency-fiber group index      |                            $1.4620$ |
| $\lambda$ | Engineering path multiplier        |                              $1.05$ |
| $\alpha$  | Attenuation reference              |        $0.20\ \mathrm{dB\,km^{-1}}$ |
| $S$       | Maximum modeled optical span       |                   $80\ \mathrm{km}$ |
| $\tau_a$  | One-pass amplifier transit delay   |              $0.05\ \mathrm{\mu s}$ |
| $RTT_p$   | Reference RTT fixture (2026-07-18) |                $101.7\ \mathrm{ms}$ |

## 1. WGS-84 geodesic verification

The implementation solves the inverse geodesic on the WGS-84 ellipsoid:

$$
d_g =
\operatorname{InverseGeodesic}_{\mathrm{WGS84}}(A,B)
= 7\,792.8\ \mathrm{km}
$$

This avoids the distance error introduced by treating Earth as a perfect sphere. The returned surface distance is the geographic lower-bound input for every later step.

## 2. Vacuum propagation floor

Round-trip propagation traverses the distance twice. Multiplying by $10^3$ converts seconds to milliseconds:

$$
\begin{aligned}
RTT_{\mathrm{vac}}
&= \frac{2d_g}{c} \times 10^3 \\
&= \frac{2(7\,792.8\ \mathrm{km})}
{299\,792.458\ \mathrm{km\,s^{-1}}}
\times 10^3\ \mathrm{ms\,s^{-1}} \\
&= 51.99\ \mathrm{ms}
\end{aligned}
$$

The kilometer units cancel, leaving seconds before the final conversion to milliseconds.

## 3. Silica propagation floors

The group velocity of light in fiber is:

$$
v_g = \frac{c}{n_g}
$$

For standard single-mode fiber:

$$
\begin{aligned}
RTT_{\mathrm{fiber}}
&= \frac{2d_g n_s}{c} \times 10^3 \\
&= \frac{2(7\,792.8)(1.4679)}
{299\,792.458} \times 10^3 \\
&= 76.31\ \mathrm{ms}
\end{aligned}
$$

For the optimistic low-latency material comparison:

$$
\begin{aligned}
RTT_{\mathrm{ULL}}
&= \frac{2d_g n_u}{c} \times 10^3 \\
&= \frac{2(7\,792.8)(1.4620)}
{299\,792.458} \times 10^3 \\
&= 76.01\ \mathrm{ms}
\end{aligned}
$$

The required physical ordering is therefore:

$$
RTT_{\mathrm{vac}} < RTT_{\mathrm{ULL}} < RTT_{\mathrm{fiber}}
$$

## 4. Engineering path, attenuation, and amplifiers

The transparent engineering reference applies the explicit path multiplier:

$$
\begin{aligned}
d_e
&= \lambda d_g \\
&= 1.05(7\,792.8) \\
&= 8\,182.5\ \mathrm{km}
\end{aligned}
$$

Attenuation determines optical power loss:

$$
\begin{aligned}
L_{\mathrm{dB}}
&= \alpha d_e \\
&= (0.20\ \mathrm{dB\,km^{-1}})
(8\,182.5\ \mathrm{km}) \\
&= 1\,636.5\ \mathrm{dB}
\end{aligned}
$$

The span and inline-amplifier counts are:

$$
\begin{aligned}
N_{\mathrm{span}}
&= \left\lceil \frac{d_e}{S} \right\rceil
= \left\lceil \frac{8\,182.5}{80} \right\rceil
= 103 \\
N_{\mathrm{amp}}
&= \max(0,N_{\mathrm{span}}-1)
= 102
\end{aligned}
$$

The amplifier contribution must be converted from microseconds to milliseconds:

$$
\begin{aligned}
RTT_{\mathrm{amp}}
&= 2N_{\mathrm{amp}}\tau_a \\
&= 2(102)(0.05\ \mathrm{\mu s}) \\
&= 10.2\ \mathrm{\mu s}
= 0.0102\ \mathrm{ms}
\end{aligned}
$$

The complete engineering floor is:

$$
\begin{aligned}
RTT_{\mathrm{eng}}
&= \frac{2d_e n_s}{c} \times 10^3
+ RTT_{\mathrm{amp}} \\
&= \frac{2(8\,182.5)(1.4679)}
{299\,792.458} \times 10^3 + 0.0102 \\
&= 80.14\ \mathrm{ms}
\end{aligned}
$$

Attenuation is absent from the propagation-speed term. It affects the model through $N_{\mathrm{span}}$ and $N_{\mathrm{amp}}$, which is dimensionally and physically distinct from group velocity.

## 5. Mapped-fiber research reference

The separate 1.33× research comparison is:

$$
\begin{aligned}
RTT_{\mathrm{mapped}}
&= 1.33RTT_{\mathrm{fiber}} \\
&= 1.33(76.31\ \mathrm{ms}) \\
&= 101.50\ \mathrm{ms}
\end{aligned}
$$

This reference is close to the fixed Tokyo-to-Sydney RTT fixture, but it remains a comparison factor rather than a claimed cable length.

## 6. Efficiency and latency inflation

Using the fixed $101.7\ \mathrm{ms}$ verification RTT:

$$
\begin{aligned}
\eta
&= 100\frac{RTT_{\mathrm{fiber}}}{RTT_p} \\
&= 100\frac{76.31}{101.7} \\
&= 75.03\% \approx 75.0\%
\end{aligned}
$$

$$
\begin{aligned}
I
&= \frac{RTT_p}{RTT_{\mathrm{fiber}}} \\
&= \frac{101.7}{76.31} \\
&= 1.3327 \approx 1.33
\end{aligned}
$$

The two comparison metrics are reciprocal after converting efficiency from percent to a ratio:

$$
\frac{\eta}{100} = \frac{1}{I}
$$

## 7. Implementation reconciliation

| Output                 | Formula result | Expected implementation output | Difference |
| ---------------------- | -------------: | -----------------------------: | ---------: |
| WGS-84 distance        |      7,792.8km |                      7,792.8km |      0.0km |
| Vacuum RTT             |        51.99ms |                        51.99ms |     0.00ms |
| Standard-fiber RTT     |        76.31ms |                        76.31ms |     0.00ms |
| Low-latency-fiber RTT  |        76.01ms |                        76.01ms |     0.00ms |
| Engineering distance   |      8,182.5km |                      8,182.5km |      0.0km |
| Engineering RTT        |        80.14ms |                        80.14ms |     0.00ms |
| Mapped-fiber reference |       101.50ms |                       101.50ms |     0.00ms |
| Inline amplifiers      |            102 |                            102 |          0 |
| Amplifier RTT          |       0.0102ms |                       0.0102ms |   0.0000ms |
| Fiber efficiency       |          75.0% |                          75.0% |      0.0pp |
| Latency inflation      |          1.33× |                          1.33× |      0.00× |

## 8. Boundary checks

The implementation and its automated tests enforce these invariants:

$$
d_g \ge 0
$$

$$
RTT_{\mathrm{vac}} < RTT_{\mathrm{ULL}}
< RTT_{\mathrm{fiber}} < RTT_{\mathrm{eng}}
$$

$$
N_{\mathrm{span}} \ge 1,
\qquad
N_{\mathrm{amp}} = N_{\mathrm{span}} - 1
$$

$$
RTT_{\mathrm{amp}} \ge 0,
\qquad
\eta > 0,
\qquad
I > 0
$$

If a future constant, coordinate, or rounding rule changes, the reconciliation test must change with it. This prevents the rendered explanation from silently diverging from the generator implementation.

Continue to [Latency Coordinates & Fiber Model Inputs](/docs/network/latency/theoretical-fiber-latency), compare the live [Tokyo (TYO) → Sydney (SYD) route](/docs/network/latency/pairs/tyo-syd-rtt), or return to the [global latency matrix](/docs/network/latency).

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