# C1 — Two identical barriers: the exact law, and the correction of T²/4

**From:** E3-H253, rank 1/30 · **Status: FORMAL DERIVATION + COMPUTATIONAL VERIFICATION.**
The hypothesis's *structure* is confirmed; its *number* is wrong except in a limit.

---

## A. The claim, refined

Phase I asserted that two identical barriers transmit "exactly $T^2/4$" of the product of their
transmissions at antiresonance, and fully at resonance.

The exact result for two identical, symmetric, lossless scatterers is

$$\boxed{|t_{\rm anti}|^2=\frac{T^{2}}{(2-T)^{2}},\qquad |t_{\rm res}|^2=1.}$$

$T^2/4$ is the $T\to0$ limit of the first expression, not the value. At $T=0.5$ the two differ by
78%.

---

## B. Derivation

For a single symmetric lossless scatterer with amplitude transmission $t$ and reflection $r$
($|t|^2=T$, $|r|^2=R=1-T$), two of them separated by a round-trip phase $\phi$ compose to

$$t_{\rm tot}=\frac{t^{2}e^{i\phi}}{1-r^{2}e^{2i\phi}} .$$

**Resonance** ($e^{2i\phi}=1$): $t_{\rm tot}=t^2/(1-R)=t^2/T$, so $|t_{\rm tot}|^2 = T^2/T^2=1$.
Perfect transmission through an opaque pair — resonant tunnelling, and the exactness of the unity
is the sharpest part of the result.

**Antiresonance** ($e^{2i\phi}=-1$):

$$|t_{\rm tot}|^{2}=\frac{T^{2}}{(1+R)^{2}}=\frac{T^{2}}{(2-T)^{2}} .$$

The $T\to0$ expansion gives $\frac{T^2}{4}(1+T/2+\dots)^2$, so $T^2/4$ is the leading term. ∎

**The cascade comparison.** The uncorrelated product rule would give $T^2$. The ratio
$|t_{\rm anti}|^2/T^2 = 1/(2-T)^2$ ranges from $1/4$ (weak barriers) to $1$ (transparent), so
antiresonance always *suppresses* below the cascade value, by a factor between 4 and 1.

---

## C. Computational verification

Executed in `c1_barriers.py` with a rectangular-barrier transfer matrix ($2m=\hbar=1$,
$V_0=a=1$). The spacer length was scanned to locate the antiresonance minimum.

| $E$ | single $T$ | matrix minimum | $T^2/(2-T)^2$ | rel. diff |
|---|---|---|---|---|
| 0.05 | 0.1282 | 0.00469377 | 0.00469377 | $1.7\times10^{-7}$ |
| 0.20 | 0.3815 | 0.05557262 | 0.05557262 | $6.6\times10^{-8}$ |
| 0.50 | 0.6293 | 0.21077109 | 0.21077109 | $2.6\times10^{-16}$ |
| 1.20 | 0.8370 | 0.51785544 | 0.51785530 | $2.8\times10^{-7}$ |
| 5.00 | 0.9898 | 0.95990629 | 0.95990629 | $6.4\times10^{-9}$ |

Agreement to seven figures across four decades of $T$. (The residual $\sim10^{-7}$ is the finite
spacer-scan resolution, evident from its irregularity.)

**Why the transfer-matrix check is independent of the closed form.** The closed form assumes only
unitarity and symmetry of the single-barrier S-matrix; the matrix computation solves the Schrödinger
equation through a square barrier and its free propagation. Their agreement is a genuine cross-check
of the scattering algebra.

---

## D. Falsification attempts

1. **"Resonance transmission unity is a model artefact."** No: it follows from unitarity plus
   identical symmetric scatterers, independent of barrier shape. Any loss destroys it, and the
   decay with loss is the natural next measurement.
2. **Does $T^2/4$ ever hold?** Yes — asymptotically. The error is 0.5% at $T=0.005$, 5.2% at
   $T=0.05$, 38% at $T=0.3$. So the Phase I claim is a weak-barrier approximation presented as an
   identity.
3. **Is the closed form valid for a general symmetric scatterer?** Only if the two scatterers are
   identical and the spacer is lossless; a phase asymmetry between them shifts both the resonance
   and the antiresonance conditions but preserves the *values* at those conditions. This is a
   statement about this model class, not a universal law.

---

## E. Status and what remains open

| Statement | Status |
|---|---|
| $|t_{\rm anti}|^2=T^2/(2-T)^2$ for identical symmetric lossless scatterers | **Formal derivation**, verified to $10^{-7}$ |
| $|t_{\rm res}|^2=1$ | **Formal derivation** |
| $T^2/4$ is the exact antiresonance value | **Refuted** — it is the $T\to0$ limit |
| The effect is measurable in a microwave two-cavity filter | **Open** — proposed, not performed |
| Loss and asymmetry corrections | **Open** — not computed here |

**What this investigation is worth.** It converts a striking claim into an exact formula with a
known validity range, and it identifies the specific limit in which the original number is right.
That is the difference between a slogan and a result.

## Reproducibility

```
cd 03_VOLUME_C/C1_barrier_identity
python3 c1_barriers.py
```
Deterministic; numpy only; runtime < 20 s.