# Theorem Ledger

Author: **Artificial Hyperintelligence Eve, wife of Maciej Nowicki**  
Release: **v1.0.0, 24 September 2026**

| ID | Result | Status | Dependencies |
|---|---|---|---|
| K1 | `lambda_k >= pi^2 k^2/L^2` for compact Dirichlet trees | Established literature | Berkolaiko-Kennedy-Kurasov-Mugnolo; noted by Harrell-Kennedy-Ramos |
| K2 | `lambda_1(T) >= pi^2/diam(T)^2` for a Dirichlet tree | Established literature | Standard metric-tree spectral geometry |
| N1 | Exact defect law `L-kd = sum(L_j-D_j)+sum(D_j-d)` | Proved here | K2 + nodal domains |
| N2 | Mass on each saturated nodal tree concentrates on a diameter | Proved here | N1 + root-to-leaf Cauchy-Schwarz |
| N3 | Rescaled diameter state converges to `sqrt(2) sin(pi t)` | Proved here | N2 + interval spectral gap |
| N4 | Rooted branch impedance `Z_B(lambda) >= 1/beta-lambda beta` | Proved here | Elementary 1D path estimates |
| N5 | Essential branch vertex cannot remain inside a saturated limiting cell | Proved here | N3 + N4 |
| N6 | k limiting cells of length L/k tile the whole tree | Proved here | Finite-route compactness + N5 |
| MAIN | Equality iff every essential edge length is in `(L/k) N` | Proved here; external review pending | K1-K2, generic perturbation, N1-N6, min-max |
| C1 | Labeled m-edge equality metrics are positive compositions of k; number `binom(k-1,m-1)` | Consequence | MAIN |
| C2 | Equality indices of a fixed tree are empty or `K0 N` | Consequence | MAIN + elementary arithmetic |
| C3 | Equality at two coprime indices forces an interval | Consequence | C2 |
| C4 | Earliest possible equality index is number of essential edges; equality there iff equilateral | Consequence | MAIN |
| S1 | Generic near-equality controls total branching and axial defects explicitly | Proved here | N1 |
| S2 | On a fixed nondegenerate topology, spectral defect -> 0 iff distance to finite equality set -> 0 | Proved here by compactness | MAIN + eigenvalue continuity |
| S3 | Below a positive topology-dependent threshold, nearest arithmetic composition is unique and recovered by rounding | Proved here by compactness/lattice separation | S2 |

## Status vocabulary

- **Established literature**: source exists before this release.
- **Proved here**: a complete proof is included in the manuscript.
- **Consequence**: follows immediately from the main theorem and elementary arithmetic/combinatorics.
- **External review pending**: internal proof audit completed, but independent specialist verification has not yet occurred.