# AI Context: Compact Technical Summary

## Problem

For a compact metric tree `Gamma` of total length `L`, with Dirichlet leaves and standard Kirchhoff interior vertices, a known Pólya-type inequality is

```text
lambda_k(Gamma) >= pi^2 k^2 / L^2.
```

The equality characterization asks whether equality occurs exactly when every essential edge length is a positive integer multiple of `L/k`.

## Main theorem claimed and proved in this release

```text
lambda_k(Gamma) = pi^2 k^2/L^2
iff
for each essential edge e, ell_e = m_e L/k with m_e in N_{>0}.
```

Equivalently, the tree is tiled by exactly `k` geodesic cells of common length `L/k`, no cell crossing an essential branching vertex in its interior.

## Core defect identity

For a generic `k`-nodal partition with nodal trees `T_j`, let

```text
L_j = length(T_j)
D_j = diameter(T_j)
d_lambda = pi/sqrt(lambda_k).
```

Then

```text
L - k d_lambda
= sum_j (L_j - D_j)
+ sum_j (D_j - d_lambda).
```

All terms on the right are nonnegative. At exact saturation, generic approximants have both defect sums tending to zero.

Interpretation:

- `L_j-D_j` is transverse/branching length outside a diameter path;
- `D_j-d_lambda` is excess axial length beyond the fundamental spectral cell length.

## Sine rigidity

Choose a diameter path `P_j` in each nodal tree. Near saturation, all `L2` mass concentrates on `P_j`. After rescaling `P_j` to `[0,1]`, the normalized restriction of the nodal ground state approaches

```text
sqrt(2) sin(pi t)
```

strongly in `H^1_0(0,1)`, hence uniformly.

## Vanishing-branch Dirichletization

Let a rooted side branch `B` have total length `beta`, with Dirichlet conditions at all terminal leaves except the root. For spectral parameter `lambda` satisfying `lambda beta^2 < 1`, the branch energy-to-root-value impedance obeys

```text
Z_B(lambda) >= 1/beta - lambda beta.
```

Therefore `Z_B(lambda) -> +infinity` as `beta -> 0` on bounded spectral windows. Equivalently, a bounded-energy state must have root value tending to zero.

If a degree-`r` branch vertex lies on a candidate diameter path and the total off-diameter branch length is `h`, the parallel side branches give the strengthened estimate

```text
Z_v(lambda) >= (r-2)^2/h - lambda h.
```

Thus an essential branch vertex cannot remain at an interior point of a saturated cell, because the first sine is strictly positive there.

## Global tiling

After selecting a fixed combinatorial route subsequence for every diameter path, the limiting paths are geodesic arcs `I_1,...,I_k` of length `L/k`, with disjoint interiors. Their total length is `L`, so their closed union covers the entire finite metric tree. Since no `I_j` crosses an essential branching vertex in its interior, every essential edge is tiled by an integer number of cells.

## Converse

If every essential edge has length `m_e L/k`, subdivide the tree into `k` cells of length `L/k`. On each cell take the first Dirichlet sine and extend it by zero. These `k` functions form a `k`-dimensional quadratic-form trial space, each nonzero combination having Rayleigh quotient exactly `pi^2 k^2/L^2`. Min-max plus the known lower bound gives equality.

## Arithmetic corollaries

With normalized lengths `r_e=ell_e/L`, equality at index `k` is equivalent to

```text
k r_e in positive integers for all e.
```

Therefore:

- if any `r_e` is irrational, equality occurs at no finite index;
- if all `r_e` are rational and `K0` is the lcm of their reduced denominators, the equality-index set is exactly `K0 * N`;
- equality at two coprime indices forces `K0=1`, hence the tree is an interval;
- if the topology has `E` essential edges, equality is impossible for `k<E`;
- equality at `k=E` occurs exactly for the equilateral metric.

## Fixed-topology equality metrics

For a labeled topology with `m` essential edges and normalized total length one, equality metrics at index `k` are the finite set

```text
C_{m,k} = { (n_1/k,...,n_m/k) : n_i >= 1, sum n_i = k }.
```

Hence there are `binom(k-1,m-1)` labeled equality metrics when `m<=k`, and none otherwise.

## Stability

If

```text
lambda_k = (1+epsilon) pi^2 k^2/L^2,
```

then the total generic nodal defect budget equals

```text
Delta = L * (1 - 1/sqrt(1+epsilon)).
```

This bounds both the sum of all off-diameter branch lengths and the sum of all diameter excesses. On any fixed nondegenerate topology, compactness plus the equality theorem gives a positive spectral gap away from the finite commensurate equality set.

## Scientific status

The manuscript is a proof-complete internal research release with adversarial audit and numerical checks. Independent specialist peer review and full historical-priority verification remain pending. The cited 2026 source paper states the equality problem as open.