Communication energy and bandwidth

# 8. Transmission has a per-bit floor

**Verdict: Floor proved; the optimal-bandwidth prediction is false in the ideal model**

*Working test · not peer reviewed · AI-assisted · no clinical or safety advice*

## The hypothesis, as published

> The energy per bit of a communication link is bounded below by the ratio of
> noise to bandwidth, so widening bandwidth lowers the floor but raises total
> power.

> Prediction: energy per bit should bottom out at an optimal bandwidth.
> Refute: energy per bit falling indefinitely with bandwidth would remove the
> floor.

## What was tested

This is a closed-form calculation on the additive white Gaussian noise channel.
Capacity is C = B log₂(1 + P/(N₀B)). For a fixed rate R, the minimum signal power
is P_min(B) = N₀B(2^(R/B) − 1), and the energy per bit is P_min/R. The limit as
B grows is N₀ ln 2.

The test separates two claims that the published entry runs together: that there
is a floor, and that the floor is reached at an interior optimum in bandwidth.

## Results

| Quantity | Value |
|---|---|
| Shannon floor, N₀ ln 2 | 0.6931 J per bit |
| Floor in dB | -1.59 dB |
| Energy per bit at B = 10⁷ Hz, relative to floor | 1.00000003 |
| Is the curve monotone decreasing? | True |
| Interior minimum without circuit power? | False |
| Spectral-efficiency curve minimum | -1.59 dB |
| Landauer limit at 300 K | 2.87e-21 J per bit |

The floor is real and exact. Energy per bit falls monotonically as bandwidth
grows and approaches N₀ ln 2 = 0.693 J, which is
−1.59 dB in the usual Eb/N₀ units. It never reaches the floor at finite
bandwidth, and it never turns back up. There is therefore no optimal bandwidth
in the information-theoretic problem: the published prediction fails.

An interior optimum appears only when a bandwidth-proportional circuit power is
added. With P_total = P_min + αB:

| Circuit coefficient α | Optimal bandwidth (Hz) | Energy per bit (J) | Above floor (dB) |
|---|---|---|---|
| 0.00 | 1000.00 | 0.693 | +0.00 |
| 0.01 | 5.12 | 0.793 | +0.59 |
| 0.05 | 2.41 | 0.924 | +1.25 |
| 0.20 | 1.30 | 1.176 | +2.29 |

That is an engineering term — amplifier, converter and clock power that scale
with bandwidth — not a consequence of the channel bound. It is the reason real
radios have an optimum, and it is not evidence for the claim as stated.

## Sensitivity

- The floor is independent of rate: changing R rescales P_min but not the
  asymptote.
- The floor is independent of the noise model's bandwidth: it depends only on
  N₀ and the ln 2 factor.
- The circuit-power optimum moves with α but stays finite for any α > 0.

## What this test cannot see

It is an ideal-channel calculation. Real links have intersymbol interference,
finite constellation granularity, coding gap to capacity and non-Gaussian noise,
all of which raise the achievable energy per bit above the floor. The result
therefore bounds the physics, not any particular modem. It also says nothing
about latency or spectral efficiency, which are the constraints that usually
set bandwidth in practice.

## Files

| File | What it is |
|---|---|
| `hypothesis.md` | the published entry, quoted verbatim |
| `run.py` | the calculation, runnable |
| `data.csv` | the required-power curve |
| `results.json` | every number quoted above |
| `figures/fig1.png` | energy per bit vs bandwidth and the Eb/N₀ curve |
| `SOURCE.md` | method note |

**Source.** Calculation from Shannon capacity. No measured dataset.