# Volume C — Three verification investigations

*Incoprea Research — Expedition V: The Hidden Discovery Laboratory*

| | Hypothesis | Outcome | Status |
|---|---|---|---|
| **C1** | H152 — 1/f noise: number or mobility fluctuation? | Both scale as $1/N$; a geometry series at fixed carrier number separates them, with slopes $-1$ versus $0$ | **Computational model result** |
| **C2** | H231 — Laplace pressure displaces a droplet's transition | $\Delta T = 2\gamma T_m/(\rho\Delta H\,r)$, and $\Delta T\cdot r$ constant is the discriminating signature | **Derivation + computation** |
| **C3** | H007 — Butler–Volmer vs Marcus at high overpotential | BV is exactly linear in Tafel; Marcus peaks at $\eta^*=\lambda$ and reverses | **Derivation + computation** |

All three are theory/computation. **No empirical claim is made anywhere in this volume** — no data
were available in the execution environment, and none is simulated in place of data.

---

## C1 — separating two noise mechanisms

The point of the investigation is that the usual discriminator does not work. **Both** the McWhorter
(number) and Hooge (mobility) models give $S_I/I^2 \propto 1/N$, so a doping series or a bias series
cannot separate them. What does separate them is geometry at **fixed carrier number**: vary the film
thickness and adjust the area so $N$ is constant, which changes the trap count (area) while leaving
$\alpha_H$ untouched.

| thickness | area | number-noise | mobility-noise |
|---|---|---|---|
| 5 nm | 200 µm² | 1.000 | 1.000 |
| 20 nm | 50 µm² | 0.250 | 1.000 |
| 100 nm | 10 µm² | 0.050 | 1.000 |
| 500 nm | 2 µm² | 0.010 | 1.000 |

A log–log slope of $-1$ indicates number fluctuation, $0$ indicates mobility fluctuation.
The superposition of log-uniform Lorentzians was verified to give a $1/f$ slope of **−0.992**.

**Limitations.** This is a model, not a measurement. Real contacts, surface depletion and series
resistance all break the fixed-$N$ assumption, and the report says so.

---

## C2 — the Laplace shift, and how to separate it from chemistry

$$\Delta T = \frac{2\gamma T_m}{\rho \Delta H\, r}$$

For a lipid-like droplet with $\gamma = 2$ mN/m, $\rho = 900$ kg/m³, $\Delta H = 2\times10^4$ J/kg:
13.3 K at 5 nm, 3.3 K at 20 nm, 0.67 K at 100 nm, 0.07 K at 1 µm.

The discriminating signature is **not** the magnitude but the form: a Laplace shift gives
$\Delta T\cdot r$ constant, while a surfactant-chemistry effect gives a size-independent offset.
At 20 versus 40 nm the two differ by 1.67 K, so a resolution better than 0.33 K gives a 5σ
separation. A shift that persists above 1 µm is not Laplace pressure at all.

---

## C3 — where Butler–Volmer and Marcus part

Butler–Volmer is exactly linear in a Tafel plot (constant slope $2.303/\alpha f$). Marcus is a
downward parabola in $\log i$ versus $\eta$, peaking at $\eta^* = \lambda$ (in volts, for a
one-electron transfer) and then *falling*.

| $\lambda$ | BV/Marcus ratio at 0.6 V | Marcus slope at 0.6 V |
|---|---|---|
| 0.5 eV | $1.4\times10^{5}$ | −4.09 dec/V |
| 0.7 eV | $1.4\times10^{5}$ | +2.64 dec/V |
| 1.0 eV | $5.6\times10^{5}$ | +7.69 dec/V |

The sign of the Marcus slope at a given overpotential is the decisive observable, and it depends on
$\lambda$ — which is what makes the measurement informative rather than merely confirmatory.

**A correction to the hypothesis.** H007 quoted a 1100× separation at 0.6 V. The computed separation
is four to five orders of magnitude larger and depends strongly on $\lambda$ and $\alpha$; the quoted
figure is not generic. The *sign* test survives; the magnitude claim does not.

---

## Reproducibility

```
cd 03_VOLUME_C/C1_noise_mechanism && python3 c1_noise.py
cd 03_VOLUME_C/C2_laplace_shift    && python3 c2_laplace.py
cd 03_VOLUME_C/C3_bv_marcus        && python3 c3_bv_marcus.py
```

Deterministic; numpy only.