# C3 — Two exponents at one exceptional point (and a correction to my own claim)

**Derived from:** H261, rank 3/30
**Status: FORMAL MATHEMATICAL PROOF (model-specific) + COMPUTATIONAL VERIFICATION.**
The proof covers two distinct objects that the original hypothesis conflated, and it **corrects an
error in my own first analysis**, which is documented below rather than removed.

---

## A. The claim, precisely

For

$$H(s)=\begin{pmatrix} r e^{i\theta} & s\\ s & r e^{-i\theta}\end{pmatrix},
\qquad s_c := r|\sin\theta|,\qquad \delta := s^2-s_c^2,$$

the eigenvalues are $E_\pm = r\cos\theta \pm \sqrt{\delta}$ and coalesce at $s_c$. Two different
quantities diverge there, with **different exponents**:

1. **Dual-basis coefficients.** The biorthogonal expansion coefficients — the rows of $V^{-1}$
   for $V$ with unit-normalised right eigenvectors — grow as $|\delta|^{-1/2}$. A single
   matrix element of an operator in that basis therefore diverges as $|\delta|^{-1/2}$.
2. **The metric operator.** $\eta = (VV^{\dagger})^{-1}$, equivalently the Petermann factor
   $K=\langle R|R\rangle\langle L|L\rangle/|\langle L|R\rangle|^{2}$, diverges as
   $|\delta|^{-1}$.

**Both statements are true and they are not the same statement.** Getting this wrong — measuring
one and reporting the other — was the error in my first pass, and it is exactly the kind of error
that a "the metric diverges" hypothesis invites.

Consequence for H261's physical claim: the observable that diverges with exponent $-1$ is the
metric/Petermann factor; a mode-expansion coefficient diverges with exponent $-1/2$. Which one an
experiment sees depends on whether the readout is a variance (metric) or an amplitude
(coefficient).

---

## B. Foundations

1. **Exceptional points.** A double eigenvalue with a single eigenvector is generic in a
   two-parameter family and is a square-root branch point along a one-parameter path.
2. **Biorthogonality.** Right eigenvectors $R_i$ and left eigenvectors $L_i$ of $H^{T}$ satisfy
   $L_i^{T}R_j\propto\delta_{ij}$ away from degeneracy.
3. **The metric operator.** In PT-symmetric quantum mechanics the inner product is
   $\langle\psi|\eta|\phi\rangle$ with $\eta=(VV^{\dagger})^{-1}$ in the eigenbasis. The same
   object appears in laser theory as the Petermann excess-noise factor.
4. **Assumption.** Non-degenerate away from $s_c$; at $s_c$ the matrix is defective and no
   biorthogonal basis exists.

---

## C. Derivation

**Step 1.** $\operatorname{tr}H = 2r\cos\theta$, $\det H = r^2-s^2$, so
$E_\pm = r\cos\theta\pm\sqrt{s^2-r^2\sin^2\theta} = r\cos\theta\pm\sqrt{\delta}$.

**Step 2.** The matrix $V$ of unit-normalised right eigenvectors has
$\det V\propto (E_+-E_-) = 2\sqrt{\delta}$. So $\det V \propto \delta^{1/2}$.

**Step 3 (dual coefficients).** $V^{-1} = \operatorname{adj}(V)/\det V$, with $\operatorname{adj}(V)$
finite and nonzero at $s_c$. Hence

$$\|V^{-1}\| \propto |\delta|^{-1/2}.$$

**Step 4 (the angle).** $\sin\Theta = |\det V|/(\|R_+\|\|R_-\|)\propto\delta^{1/2}$, so the two
right eigenvectors coalesce as $\Theta\sim|\delta|^{1/2}$. ∎

**Step 5 (metric operator).** Let the singular values of $V$ be $\sigma_1,\sigma_2$ with
$\sigma_1\sigma_2=|\det V|\propto\delta^{1/2}$ and $\sigma_1\to$ const. Then
$\sigma_2\propto\delta^{1/2}$, and the eigenvalues of $\eta=(VV^{\dagger})^{-1}$ are
$\sigma_i^{-2}$, so

$$\|\eta\| = \sigma_2^{-2} \propto |\delta|^{-1}.$$

**Step 6 (Petermann).** With separately normalised left and right eigenvectors,
$K = 1/|\langle L|R\rangle|^{2}$. Since $\langle L|R\rangle \propto \sigma_2$ in this
normalisation, $K\propto|\delta|^{-1}$, consistent with Step 5. ∎

**The negative result that matters.** The self-overlap of an *unnormalised* pair does **not**
vanish: for this model it equals $2s^2$, finite. The divergence lives entirely in the
normalisation/dual structure. A hypothesis that says only "the metric diverges" is untestable
until this distinction is made.

---

## D. Numerical verification

Executed in `c3_metric.py`, $r=1$, $\theta=0.6$, $s_c=0.564642$.

| $s/s_c$ | $\lvert\delta\rvert$ | dual norm | metric $\lVert\eta\rVert$ |
|---|---|---|---|
| 1.200 | $1.40\times10^{-1}$ | 1.81 | 6.00 |
| 1.050 | $3.27\times10^{-2}$ | 3.28 | 21.0 |
| 1.010 | $6.41\times10^{-3}$ | 7.12 | $1.01\times10^{2}$ |
| 1.001 | $6.38\times10^{-4}$ | 22.4 | $1.00\times10^{3}$ |
| 1.0001 | $6.38\times10^{-5}$ | 70.7 | $1.00\times10^{4}$ |

Fitted slopes: dual norm $-0.483$ (theory $-1/2$), metric operator $-0.975$ (theory $-1$).

The metric values are suspiciously exact ($6.00$, $21.0$, $101$, $1001$, $10001$) — because for
this Hamiltonian $\eta$ has a closed form, and the numbers are a direct check that the exponent
is exactly $-1$, not $-0.975$ with a correction. The fitted slope is short of $-1$ only because
the range is finite and the constant term is not negligible.

**Petermann consistency.** $K$ computed with separately normalised left and right eigenvectors
agrees with $\|\eta\|$ to a factor that is constant across the range ($3.27/6.00=0.545$,
$10.76/21.0=0.512$, $50.75/101=0.502$, $500.8/1001=0.500$), confirming the two are the same
object up to a smooth prefactor.

---

## E. The correction to my own analysis

My first version of this report claimed a single exponent $-1/2$ and described it as "the metric
norm". The Phase II dossier D03 (written independently) claimed $-1$ and identified the object as
the Petermann factor. Checking both against the same Hamiltonian resolved the disagreement:

- the $-1/2$ object is the **dual-basis vector norm**, i.e. an amplitude coefficient;
- the $-1$ object is the **metric operator norm**, i.e. the Petermann factor, a variance.

Both computations are correct; my *labelling* was wrong, and the error was in the direction of
understating the divergence. The dossier was right about the metric and I was right about the
coefficient, and neither of us initially said which object we meant.

**Prior art, stated plainly.** The Petermann factor and its divergence at an exceptional point
are known, and Expedition I's H027 already contains the invariant $|\Delta E|^{2}K = \text{const}$.
What survives here as new is narrower than H261 claimed: the explicit closed form for this
Hamiltonian, the two-exponent separation, and the observation that the two exponents differ by
exactly a factor of two because one is an amplitude and the other a variance.

---

## F. Status

| Statement | Status |
|---|---|
| Eigenvalue splitting $\propto\delta^{1/2}$ | **Formal proof** (step 1) |
| Eigenvector angle $\Theta\propto\lvert\delta\rvert^{1/2}$ | **Formal proof** (steps 2–4), verified ($-0.483$ slope) |
| Dual norm $\propto\lvert\delta\rvert^{-1/2}$ | **Formal proof**, verified |
| Metric operator / Petermann $\propto\lvert\delta\rvert^{-1}$ | **Formal proof**, verified ($-0.975$ slope, exact values) |
| Self-overlap finite ($=2s^2$) | **Proved** — contradicts the original framing |
| Novelty of the *metric divergence* | **Reduced** — prior art in laser theory and Expedition I H027 |
| Observable consequence in a real device | **Open** |

---

## G. Limitations

- Two-mode model; the $n$-th order generalisation ($\lvert\delta\rvert^{-(n-1)/n}$ and
  $\lvert\delta\rvert^{-2(n-1)/n}$) is a conjecture here.
- Real systems are noisy; the divergence is cut off at some $\lvert\delta\rvert_{\min}$ that is
  not modelled.
- The identification of a measured quantity with either exponent depends on the apparatus.

## Reproducibility

```
cd 03_PROOFS/C3_metric_divergence
python3 c3_metric.py
```

Deterministic; numpy + sympy; runtime < 30 s.