# Theorem ledger

**Status vocabulary.** “Proof supplied” means a full mathematical argument is present in this release; it does not mean independently peer reviewed or formally verified. “Prior input” means an established external result used with attribution. “Finite check” means only the recorded instances were tested.

| ID | Claim | Status | Principal assumptions | Location |
|---|---|---|---|---|
| P1 | BKKM total-length lower and upper eigenvalue estimates | Prior input, cited | Scalar compact metric graph, declared leaf conditions, not a circle | Section 1 |
| P2 | Two-sided sharpness is equivalent to maximal multiplicity | Prior art, proof reproduced in the needed regime | Same graph class | Sections 1 and 8 |
| P3 | Full-support selection at a lower-sharp eigenvalue | Prior argument, proof reproduced | Positive lower-sharp eigenvalue | Section 3 |
| T1 | Exact tree nodal inertia with vertex zeros | Proof supplied; no broad claim to new oscillation theory | Fully supported eigenfunction, tree, positive eigenvalue | Section 3 |
| T2 | Mixed-tree integer/half-integer equality rigidity | Independent direct proof supplied | High-index regime; D/N only at leaves | Section 4 |
| T3 | Entire extremal eigenspace survives a form restriction | Proof supplied | Same kth eigenvalue; positive next spectral gap | Section 5 |
| T4 | Admissible cycle opening adds exactly one Neumann leaf per cycle | Proof supplied | Connected, noncircle metric graph | Section 5 |
| T5 | Equality of terminal evaluation functionals forces the same pendant edge and an even period | Proof supplied | Saturated tree with at least three leaves | Section 6 |
| T6 | Complete two-leaf opening classification | Proof supplied | Opened interval; zero, one, or two cycles | Section 7 |
| T7 | Lower sharpness alone forces full geometry, maximal multiplicity, and two-sided sharpness | Main candidate theorem; full proof supplied | Complete scope of Theorem 2.2 | Sections 2–8 |
| T8 | On a saturated admissible opening, R=0 is sufficient and forces K=0 | Proof supplied; graph-specific | Same geometry, positive threshold | Section 9 |
| T9 | Energy-normalized quantitative inheritance: delta >= g*rho/(lambda+g+rho) | Proof supplied | Nonnegative compact-resolvent form, controlled constraints, next gap g>0 | Section 10 |
| T10 | Explicit graph gap from path Gram matrix and residue | Proof supplied | Opened tree is saturated | Section 10 |
| T11 | Every one-sided equality generates an odd-harmonic tower | Proof supplied | Main classification assumptions | Section 12 |
| T12 | Uniform strictness on compact nondegenerate metric sets away from equality | Proof supplied | Fixed topology, index, total length; positive minimum edge length | Section 12 |
| C1 | 604 exact rational graph comparisons | Finite checks passed | Half-cell rational instances in saved data | Appendix A, code and data |
| C2 | Independent ODE-nullity crosscheck of each of 604 instances | Finite checks passed | Exact coefficient matrices at pi frequency | Appendix A, code and data |
| C3 | 200 abstract quantitative matrix tests | Floating-point checks passed | Positive diagonal test operators and random constraints | Appendix B |
| C4 | 18 conforming finite-element runs | Approximation/convergence evidence | Six example graphs, three mesh densities | Appendix B |

## Not established by this release

No proof is supplied for the separate low-index equality classification, arbitrary Robin or magnetic conditions, general potentials, degenerate edge-collapse stability, the universal optimal eigenvalue-ratio constant, or any higher-dimensional Pólya conjecture. No publication acceptance, external referee approval, formal proof-assistant artifact, hardware validation, or certified priority claim exists in this release.