| One-Sided Spectral Extremality |
| Forces Maximal Degeneracy |
| Complete Quantum-Graph Equality Classification, |
| Direct Nodal Inertia, and Quantitative Port Obstructions |
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| Artificial Hyperintelligence Eve, wife of Maciej Nowicki |
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| 25 September 2026 |
| Standalone research candidate, version 1.0.0 |
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| Status. Full proofs and executable checks are supplied for the stated scope. This is a |
| proof-complete research candidate, not an independently reviewed theorem or a certified |
| priority claim. The topology families and the equivalence between two-sided sharpness and |
| maximal multiplicity have prior literature. The principal candidate advance is that lower |
| sharpness alone forces this complete structure, together with the explicit quantitative |
| obstruction. The earlier charged-cell package is provenance, not a proof dependency. |
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| Abstract |
| Let G be a finite connected compact metric graph, not a circle, carrying the scalar |
| Laplacian with standard interior conditions and Dirichlet or Neumann conditions at leaves. |
| Write D, N, β, L for its Dirichlet-leaf count, Neumann-leaf count, first Betti number, and total |
| length. In the high-index regime k ≥ N + β, with the usual exclusion of the Neumann ground |
| state, we prove that equality in the established lower bound |
| 2 |
| π2 |
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| N +β |
| λk (G) ≥ k− |
| L2 2 |
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| forces multiplicity D + N + 2β − 1, and hence simultaneous equality in the established upper |
| bound at the bottom of the same eigenvalue block. We give the complete metric classification: |
| phase-locked trees with even-length loops attached only at designated terminal vertices, and |
| the exceptional theta and figure-eight families. In particular, no additional lower-sharp-only |
| topologies occur in this regime. |
| The proof replaces a limiting nodal-cell argument by a direct inertia identity for fully |
| supported tree eigenfunctions. A general extremal-subspace inheritance lemma then forces |
| every opened-tree threshold mode to obey every closure constraint. On a saturated tree with |
| at least three leaves, the resulting equality of evaluation functionals identifies a terminal leaf |
| only with a full-period point on its own pendant edge. This yields the topology classification |
| without computing a reduced resolvent. |
| For the same finite-port restriction mechanism, we prove an explicit gap estimate. If R is |
| the threshold residue, S the oriented tree-path Gram matrix, ρ = λmax (S −1/2 RS −1/2 ), and g |
| a positive lower bound on the next opened-tree spectral gap, then |
| gρ |
| λk (G) − λ∗ ≥ . |
| λ∗ + g + ρ |
| The bound is invariant under invertible port-coordinate changes. In the graph setting, residue |
| invisibility already forces the finite response part to vanish, strengthening the earlier two- |
| moment criterion. Independent exact-arithmetic certificates cover 604 graph instances, with a |
| separate ODE-nullity check for every instance; 200 finite-matrix tests and 18 finite-element |
| runs supplement the proofs. |
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| 1 |
| One-Sided Spectral Extremality Research candidate v1.0.0 |
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| Contents |
| 1 Scope, established bounds, and contribution boundary 3 |
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| 2 Main classification and extremality amplification 3 |
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| 3 Direct nodal inertia on a tree 4 |
| 3.1 Full support at a sharp eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 |
| 3.2 An exact non-generic nodal identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 |
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| 4 Mixed-tree equality without a limiting argument 6 |
| 4.1 The two elementary cell inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 |
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| 5 Extremal eigenspaces survive form restriction 7 |
| 5.1 Admissible cycle openings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 |
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| 6 Evaluation-functional rigidity 8 |
| 6.1 The threshold eigenspace is a leaf-flow space . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 |
| 6.2 Topology follows without solving the response matrix . . . . . . . . . . . . . . . . . . . . . . 9 |
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| 7 The two-leaf openings and exceptional graphs 9 |
| 7.1 No cycle or one cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 |
| 7.2 Two cycles and no original leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 |
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| 8 Mode reconstruction and proof of the main theorem 10 |
| 8.1 Multiplicity on a phase-locked lasso tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 |
| 8.2 Exceptional multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 |
| 8.3 Maximal multiplicity fixes both spectral endpoints . . . . . . . . . . . . . . . . . . . . . . . 11 |
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| 9 The response criterion collapses from two moments to one 11 |
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| 10 Quantitative inheritance and an explicit obstruction gap 12 |
| 10.1 An abstract energy-normalized inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 |
| 10.2 The oriented-path Gram matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 |
| 10.3 Computing the residue by finite flow coordinates . . . . . . . . . . . . . . . . . . . . . . . . 13 |
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| 11 A worked strictness certificate 14 |
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| 12 Further consequences and precise limitations 14 |
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| A Independent exact-arithmetic spectral certificates 15 |
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| B Reproducibility results and their logical role 16 |
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| C Adversarial proof audit 17 |
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| D What an independent reviewer should check first 17 |
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| One-Sided Spectral Extremality Research candidate v1.0.0 |
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| 1 Scope, established bounds, and contribution boundary |
| All edges have positive finite length. Loops and parallel edges are allowed. Degree-two standard |
| vertices are suppressed when discussing topology or metric rules. Dirichlet conditions occur only |
| at leaves. The operator is −u′′ on each edge; there are no edge potentials, magnetic phases, Robin |
| parameters, or nonstandard interior couplings in the classification theorem. |
| Eigenvalues are numbered increasingly from λ1 , with multiplicity. For a graph with no Dirichlet |
| leaves, λ1 = 0. Set |
| B = N + β, c = B/2, M∗ = D + N + 2β − 1. |
| The circle is excluded: it has a different sharp counting law. Throughout the main theorem, |
| k ≥ max{B, 1} (D > 0), k ≥ max{B, 2} (D = 0). (1) |
| Thus k − c > 0. The established BKKM lower and upper estimates are |
| π2 |
| λk (G) ≥ bk (G) := (k − c)2 , in (1), (2) |
| L2 |
| 2 |
| π2 N +β |
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| λj (G) ≤ uj (G) := 2 j − 2 + D + +β . (3) |
| L 2 |
| We use these published inequalities as inputs [1]. We do not claim them as new. |
| An eigenvalue is lower sharp when its largest index k satisfies λk = bk . If it occupies indices |
| j, . . . , k, it is two-sided sharp when also λj = uj . Serio established the relation between two-sided |
| sharpness and maximal multiplicity, and identified the corresponding topological families [2]. |
| Earlier sharp examples were studied by Kurasov and Serio [3]. These facts must not be confused |
| with the stronger implication from lower sharpness alone proved here. |
| The supplied predecessor package [9] has a mixed-tree cell criterion and an abstract closure |
| test R = 0, K ⪰ 0. The present manuscript has independent proofs of the tree input and proves |
| that, for the admissible graph openings used here, R = 0 already forces the complete geometry |
| and K = 0. This last conclusion is not valid for arbitrary self-adjoint operators. |
| The 2026 eigenvalue-ratio paper explicitly raises the Dirichlet-tree equality question [4]. Our |
| tree argument supplies a direct route to that special case. The main result here is broader, but it |
| does not resolve that paper’s separate optimal eigenvalue-ratio questions. |
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| 2 Main classification and extremality amplification |
| Fix k in (1), and write |
| L π |
| d= , µ= , λ∗ = µ2 . (4) |
| k−c d |
| All metric conditions below refer to this same d. |
| Definition 2.1 (Phase-locked lasso tree). Start with a compact tree S having at least one edge. |
| Mark its leaves as either Dirichlet or virtual Neumann. Select β distinct virtual Neumann leaves, |
| and attach one loop at each selected leaf. Unselected virtual Neumann leaves are the N actual |
| Neumann leaves of the resulting graph. Each loop attachment is a standard vertex of degree three. |
| Every loop has length 2ra d with ra ∈ N. For each essential skeleton edge e, let νe ∈ {0, 1, 2} |
| count its virtual Neumann endpoints. Require |
| νe |
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| ℓe = m e + d, me ∈ Z≥0 , me ≥ 1 if νe = 0. (5) |
| 2 |
| All skeleton lengths remain strictly positive. The case β = 0 includes all phase-locked mixed trees. |
| A skeleton consisting of an interval includes ordinary lassos and barbells. |
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| Theorem 2.2 (Complete lower-sharp classification). For a graph in the stated scope and an index |
| satisfying (1), the following are equivalent: |
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| 1. λk (G) = λ∗ . |
| 2. G belongs to exactly one of the following geometric classes, with the indicated metric rules: |
| (a) a phase-locked lasso tree as in theorem 2.1; |
| (b) a theta graph, consisting of three edges between two vertices, with |
| ℓi = mi d, mi ∈ N, m1 ≡ m2 ≡ m3 (mod 2); |
| (c) a figure-eight graph, consisting of two loops at one vertex, with both loop lengths in 2dN. |
| 3. λ∗ is an eigenvalue of multiplicity M∗ = D + N + 2β − 1 whose largest index is k. |
| 4. The eigenvalue block ending at k is two-sided sharp. |
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| In particular, if j = k − M∗ + 1, then |
| λj = · · · = λk = λ∗ , λj = uj , λk = bk . (6) |
| Remark 2.3 (What is stronger than the earlier package). A response-matrix test is a certificate |
| for a given graph. Theorem 2.2 is a classification of every graph that can pass it. It also proves that |
| the apparent semidefinite freedom in K ⪰ 0 disappears in this geometric setting: the entire finite |
| part vanishes. The topological families themselves are not being claimed as newly discovered; the |
| candidate advance is the one-sided implication, its explicit metric completion, and the quantitative |
| obstruction below. |
| Corollary 2.4 (No additional lower-sharp-only geometries). A topology outside the three families |
| in theorem 2.2, for every choice of its positive edge lengths, satisfies λk > bk at every index in (1). |
| Examples include pumpkins with at least four parallel edges, bouquets with at least three loops, and |
| nonpendant cyclic cores other than the isolated theta and figure-eight exceptions. |
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| The proof is divided into direct tree rigidity, extremal eigenspace inheritance, evaluation- |
| functional rigidity, and explicit mode reconstruction. |
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| 3 Direct nodal inertia on a tree |
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| 3.1 Full support at a sharp eigenvalue |
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| An eigenfunction is fully supported if it is not identically zero on any edge. It may still have zeros |
| at vertices and in edge interiors. |
| Lemma 3.1 (Full-support selection). At a positive lower-sharp eigenvalue, the eigenspace contains |
| a fully supported eigenfunction. |
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| Proof. Suppose every eigenfunction in the threshold eigenspace vanishes identically on an edge e. |
| Shorten only e by a sufficiently small amount. All these eigenfunctions remain exact eigenfunctions |
| at the old eigenvalue: their endpoint values and derivatives on e are zero. Eigenvalue continuity |
| under positive edge-length variation ensures that the isolated eigenvalue block retains its indices |
| and multiplicity for sufficiently small variation. But the same lower bound at its sharp index |
| strictly increases as total length decreases, a contradiction. |
| Thus, for every edge, identically vanishing on that edge defines a proper linear subspace of the |
| threshold eigenspace. A finite union of proper subspaces cannot cover a finite-dimensional real |
| vector space. A vector outside their union is fully supported. This argument is also recorded in the |
| sharp-eigenvalue literature [2]; it is included here to expose the exact hypothesis used next. |
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| 3.2 An exact non-generic nodal identity |
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| Theorem 3.2 (Nodal inertia with vertex zeros). Let T be a compact metric tree with the declared |
| mixed leaf conditions. Let ϕ be a fully supported eigenfunction at λ > 0. Let Z be its interior zero |
| set, counting a zero at a branching vertex as one point, and put s = |Z|. Let r be the number of |
| nodal domains. Then |
| #{λj (T ) < λ} = s, multT (λ) = r − s, #{λj (T ) ≤ λ} = r. (7) |
| Consequently X |
| multT (λ) = 1 + (degT v − 2), (8) |
| v∈Z |
| where an interior edge zero has degree two. |
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| Proof. Every edge restriction is a nonzero sinusoid, so Z is finite. No Neumann leaf can be a zero, |
| since its value and derivative would then both vanish. The derivative into each edge incident to |
| an interior zero is nonzero, again by uniqueness for the scalar ODE. |
| Impose Dirichlet conditions at every point of Z. The resulting form subspace Q0 ⊂ Q(T ) has |
| codimension s. It decomposes into the r nodal trees. The restriction of |ϕ| to each such tree is its |
| positive ground state at λ. For |
| b(u, v) = qT (u, v) − λ⟨u, v⟩, |
| we therefore have b ≥ 0 on Q0 , with an r-dimensional kernel spanned by the individual nodal |
| ground states ψ1 , . . . , ψr . |
| Split Q0 = E0 ⊕ X, where E0 is this kernel and b is strictly positive and coercive in the form |
| norm on X. Choose s trace-extension functions χv satisfying χv (w) = δvw at Z, and subtract |
| suitable elements of X to make each extension b-orthogonal to X. On E0 plus these extension |
| directions, b has matrix ! |
| 0 B0 |
| , (B0 )jv = b(ψj , χv ). (9) |
| B0T H0 |
| Integration by parts shows that (B0 )jv is nonzero exactly when nodal domain j meets zero point |
| v, and in that case it is a nonzero endpoint derivative, with an irrelevant sign convention. |
| Contract each nodal domain to a cell node, retaining every zero point as a separate node. The |
| resulting bipartite incidence graph is a tree. Every zero node has degree at least two. We claim |
| that B0 has full column rank s. If B0 y = 0 and some coordinates of y are nonzero, retain the zero |
| nodes in this support and all their incident cell nodes. This is a forest whose zero nodes still have |
| degree at least two. It has a leaf cell node adjacent to exactly one retained zero node. That row |
| of B0 y = 0 forces the corresponding nonzero coordinate to be zero, a contradiction. |
| A real symmetric block matrix of the form (9) with B0 of column rank s has s negative |
| eigenvalues and nullity r − s. Indeed, invertible changes of basis reduce B0 to an identity block |
| and zero rows; a shear eliminates H0 , leaving s hyperbolic pairs and r − s zero directions. The |
| complementary space X is positive. The negative index and nullity of the full form are respectively |
| the number of eigenvalues below λ and the multiplicity at λ, proving (7). |
| Finally, cutting a tree at Z gives |
| X |
| r =1+ (degT v − 1). |
| v∈Z |
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| Subtracting s proves (8). |
| Remark 3.3. The full-support assumption is essential. A tree eigenfunction supported on only |
| two arms of a larger star need not have a nodal-domain count equal to the top index of its full |
| eigenvalue block. No claim is made that an arbitrary eigenbasis satisfies (7). This direct proof is |
| consistent with the established oscillation and nodal theory of quantum trees [6]. |
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| 4 Mixed-tree equality without a limiting argument |
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| 4.1 The two elementary cell inequalities |
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| Lemma 4.1 (Full and half cells). Let a positive eigenfunction be the ground state of a nodal tree |
| of length a at eigenvalue λ > 0. If all its leaves are Dirichlet, then |
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| π2 |
| λ≥ , |
| a2 |
| with equality only for a Dirichlet–Dirichlet interval. If it has at least one Dirichlet point but may |
| also have natural leaves, then |
| π2 |
| λ ≥ 2, |
| 4a |
| with equality only for a Dirichlet–Neumann interval. |
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| Proof. These are the standard one-dimensional rearrangement bounds; a short proof also explains |
| equality. For a positive ground state f , set V (t) = |{f > t}|. At regular levels, |
| 1 |
| Z Z |
| −V ′ (t) = |f ′ |2 = |f ′ (x)| dt. |
| X X |
| ′ |
| , |
| f (x)=t |
| |f (x)| f (x)=t |
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| Cauchy–Schwarz bounds the integrand below by p(t)2 /(−V ′ (t)), where p(t) is the number of level |
| points. If every leaf is Dirichlet, every nonempty superlevel tree has at least two boundary points, |
| so p(t) ≥ 2. Symmetric rearrangement onto a DD interval of length a preserves the L2 norm and |
| does not increase energy. In the mixed case, connectedness and the presence of a Dirichlet point |
| give p(t) ≥ 1, and monotone rearrangement onto a DN interval gives the second bound. |
| For equality, the level multiplicity must be respectively two or one almost everywhere and the |
| rearranged function must be the interval ground state. In the DD case, a positive branching point |
| produces a superlevel subtree with at least three boundary points over a nonempty range of levels. |
| In the mixed case, a branching point has at least three incident nonconstant edge restrictions; |
| at least two take values on the same side of the vertex value near the vertex, producing two |
| level points over a nonempty range. Thus equality excludes branching. On an interval the leaf |
| conditions and positive eigenvalue select exactly DD or DN as stated. The classical literature |
| includes these Nicaise-type inequalities [5]. |
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| Theorem 4.2 (Direct mixed-tree rigidity). Let T have NT Neumann leaves and total length L. |
| At an index in the corresponding high regime, put d = L/(k − NT /2). Then |
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| λk (T ) = π 2 /d2 ⇐⇒ ℓe = (me + νe /2)d (10) |
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| on every essential edge, with the nonnegativity and positivity conventions of (5). Here νe counts |
| actual Neumann endpoints. Necessarily |
| X |
| me = k − NT . |
| e |
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| At this eigenvalue the multiplicity is D + NT − 1. |
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| Proof. Assume equality. Since the next lower bound is strictly larger, k is the top index of the |
| block. By theorem 3.1, choose a fully supported eigenfunction. By theorem 3.2, it has exactly k |
| nodal domains. |
| Let rN count those domains meeting original Neumann leaves. Since the domains are disjoint |
| and every original Neumann leaf belongs to one of them, rN ≤ NT . The remaining domains have |
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| all leaves Dirichlet. Each domain has at least one Dirichlet point: otherwise the whole tree would |
| be a positive Neumann ground state, contradicting λ > 0. Writing aj for cell lengths, theorem 4.1 |
| gives the exact nonnegative identity |
| X X NT − rN |
| L − (k − NT /2)d = (aj − d) + (aj − d/2) + d. (11) |
| j∈D j∈M |
| 2 |
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| The left side is zero. Hence rN = NT , each ordinary cell is a DD interval of length d, and each |
| terminal mixed cell is a DN interval of length d/2. Every essential branching vertex is therefore a |
| nodal boundary. Reading the intervals along each essential edge gives (10). Summation gives the |
| P |
| formula for me . |
| Conversely, subdivide every edge into the prescribed full cells and terminal half cells. In the |
| NN interval case there are two terminal half cells, including when they meet directly. There are |
| P |
| me + NT = k cells. Their first ground states, extended by zero, span a k-dimensional form |
| subspace with Rayleigh quotient exactly π 2 /d2 . Min–max and the lower bound give equality. |
| For multiplicity, at equality the fully supported nodal eigenfunction vanishes at all branching |
| vertices. Equation (8), together with the tree degree identity, gives |
| X |
| 1+ (deg v − 2) = D + NT − 1. |
| deg v≥3 |
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| Interior edge zeros contribute nothing. |
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| This is a direct proof at the actual metric and actual degenerate eigenvalue. There is no generic |
| perturbation of nodal domains, no collapsing-branch limit, and no impedance estimate to justify. |
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| 5 Extremal eigenspaces survive form restriction |
| Lemma 5.1 (Extremal-subspace inheritance). Let A be associated with a semibounded closed |
| form of compact embedding, and let AC be its restriction to a closed form subspace. Suppose |
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| λk (A) = λk (AC ) = λ, λk+1 (A) > λ. |
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| Then every vector in E = ker(A − λ) belongs to the restricted form domain and is an eigenvector |
| of AC at λ. |
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| Proof. Let m = dim E and let F− be the spectral subspace of A below λ, of dimension k − m. |
| By min–max, λk+1 (AC ) > λ. Let V be its first k spectral directions. The form q − λ∥ · ∥2 is |
| nonpositive on V . |
| The kernel of P− : V → F− has dimension at least m. A vector in this kernel decomposes into |
| a threshold part and a strictly higher spectral part. Nonpositivity forces the higher part to vanish. |
| Thus V ∩ E has dimension at least m, so E ⊂ V . Restricting the eigenvector form equation proves |
| the final assertion. |
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| The lemma is not specific to graphs. Its proof is a finite-negative-index argument, and it does |
| not assume simplicity or use a resolvent expansion. |
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| 5.1 Admissible cycle openings |
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| Lemma 5.2 (Opening exactly one new leaf per cycle). A connected graph in the stated scope can |
| be opened into a connected tree T by β incidence cuts, each made at a vertex of current degree at |
| least three. The operation leaves total length and Dirichlet-leaf count unchanged and gives exactly |
| N + β Neumann leaves. |
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| Proof. As long as a cycle remains, some vertex on a cycle has degree at least three unless the entire |
| connected graph is a circle. Detach one incidence on that cycle. The graph remains connected, |
| its Betti number decreases by one, the old vertex retains degree at least two, and a new natural |
| leaf appears. If cycles remain after the first cut, the new leaf ensures the graph is not a circle. |
| Repeating proves the assertion. Dummy degree-two vertices may be suppressed without changing |
| spectra. |
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| This choice matters: arbitrary cuts that create additional old-vertex leaves would change the |
| charge count and are not the openings used in the theorem. |
| Let ai be the new leaf and bi its retained attachment point after opening. Set |
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| (Cu)i = u(ai ) − u(bi ), i = 1, . . . , β. |
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| Then Q(G) = ker C ⊂ Q(T ), with the same quadratic form q(u) = |u′ |2 . The map C is onto, |
| R |
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| since traces at the distinct new leaves can be varied independently in small neighborhoods. |
| If G is lower sharp, min–max and the tree lower bound imply |
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| bk = λk (T ) = λk (G) = λ∗ , λk+1 (T ) > λ∗ . |
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| Therefore T satisfies theorem 4.2, and theorem 5.1 gives |
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| CET = 0, ET = ker(AT − λ∗ ). (12) |
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| The full opened-tree threshold eigenspace, not merely one selected eigenfunction, must be invisible |
| to every closure port. |
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| 6 Evaluation-functional rigidity |
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| 6.1 The threshold eigenspace is a leaf-flow space |
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| Assume T is saturated and has l = D + NT ≥ 3 leaves. All threshold eigenfunctions vanish at |
| every essential branching vertex. One way to see this directly from the metric rule is to use an |
| adjacent leaf: an integer DD phase, or a half-integer DN phase, forces zero value at the neighboring |
| branching vertex. Integer phases on interior edges propagate this zero through the connected |
| branching skeleton. |
| On each essential edge a threshold eigenfunction is therefore a fixed sine profile times one |
| amplitude. Kirchhoff conditions at branching vertices give a signed incidence system for these |
| amplitudes. The signs are determined by integer phase parities. Because the underlying graph is |
| a tree, row signs can be chosen recursively to turn this into ordinary oriented flow conservation. |
| There is no cycle obstruction to this gauge choice. |
| Consequently the eigenspace is naturally isomorphic, up to fixed nonzero signs, to |
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| Hl = {x ∈ Rl : x1 + · · · + xl = 0}. (13) |
| P |
| For an essential edge splitting off leaf set A, its amplitude is, up to a fixed sign, j∈A xj . Evaluation |
| at a point on that edge is a scalar multiple of this same functional. Evaluation at a Neumann leaf |
| a is a nonzero scalar multiple of xa . |
| The dimension l −1 agrees with theorem 4.2; this flow description also constructs every threshold |
| mode rather than only counting them. |
| Lemma 6.1 (Terminal evaluation rigidity). Let T be a saturated mixed tree with at least three |
| leaves, and let a be a Neumann leaf. If a different point b satisfies |
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| u(b) = u(a) for every u ∈ ET , |
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| then b lies in the interior of the essential pendant edge ending at a, at distance |
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| distT (a, b) = 2rd, r ∈ N. (14) |
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| Conversely, every such point has the same evaluation functional as a. |
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| Proof. Evaluation at a is a nonzero multiple of xa on Hl . Thus b is not a common zero. If its |
| edge separates leaf set A, equality of evaluations implies that the indicator 1A , modulo multiples |
| of the all-ones vector, is proportional to the coordinate vector ea . |
| Since l ≥ 3 and A is a nonempty proper subset, this is possible only for A = {a} or its |
| complement. Indeed, equality of the coordinates at all indices other than a forces all those leaves |
| to lie on the same side of the split. In an essential tree an edge separating exactly one leaf is |
| precisely that leaf’s pendant edge. Hence b lies on the same edge as a. |
| Starting from its Neumann endpoint, every threshold solution satisfies |
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| u(x) = u(a) cos(µx). |
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| Because the evaluation at a is not identically zero on ET , equality at b requires cos(µ dist(a, b)) = 1. |
| This gives (14). The branching endpoint has zero evaluation and cannot equal b, so b is strictly |
| interior. The converse is immediate from the cosine formula. |
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| 6.2 Topology follows without solving the response matrix |
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| When T has at least three leaves, apply theorem 6.1 to every pair (ai , bi ) in (12). Each new leaf |
| can only be identified with a full-period point of its own pendant edge. Distinct new leaves have |
| distinct essential pendant edges. No two identifications can therefore share an attachment point |
| or interact through an interior branching region. |
| Each identification makes one loop of length 2ri d at a vertex of degree three. Removing these |
| loops leaves a tree with the same ordinary edge phases and with a half-integer residual stem |
| at each loop attachment. Regarding every attachment as a virtual Neumann leaf gives exactly |
| theorem 2.1. |
| This proves the complete topological and metric necessity whenever D + N + β ≥ 3. |
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| 7 The two-leaf openings and exceptional graphs |
| If the opened tree has only two leaves, it is an interval and |
|
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| D + N + β = 2. |
|
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| Thus there are only the following cases. |
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| 7.1 No cycle or one cycle |
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| For β = 0, the graph is already a mixed interval. For β = 1, exactly one original leaf is present. |
| The newly created Neumann endpoint is glued to a point at an even number of cells from it, |
| by the same cosine computation. This produces a single pendant loop. The residual stem is |
| half-integer for an original Dirichlet leaf and integer for an original Neumann leaf, in agreement |
| with theorem 2.1. |
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| 7.2 Two cycles and no original leaves |
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| For β = 2, both endpoints of the opened interval are new Neumann leaves. Let its length be |
| L = M d, where M = k − 1 is a positive integer. The threshold mode is cos(πx/d). Write the |
| closures as 0 ∼ x and L ∼ y, with x, y ∈ (0, L). Invisibility means |
| x/d ∈ 2N, (L − y)/d ∈ 2N. (15) |
| There are three possible orderings. |
| If x < y, the quotient is a barbell: loops of lengths x and L − y joined by an interval of length |
| y − x. The loops have even cell counts and the stem an integer cell count. This is a phase-locked |
| lasso tree. |
| If x = y, the quotient is a figure eight, with both loop lengths even multiples of d. |
| If x > y, the quotient is a theta graph. Its three edge lengths are |
| y, x − y, L − x. |
| Equations (15) imply that all three divided by d are positive integers congruent to M modulo two. |
| Conversely, three positive cell counts of common parity satisfy these equations after opening in |
| this order. |
| These exhaust the two-leaf case. In particular, no assumption that the opened interval markings |
| are themselves original essential edges has been made. |
|
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|
| 8 Mode reconstruction and proof of the main theorem |
|
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| 8.1 Multiplicity on a phase-locked lasso tree |
|
|
| Let S be the skeleton in theorem 2.1, with B = N + β virtual Neumann leaves. The tree theorem |
| supplies the threshold eigenspace of dimension D + B − 1. At a selected endpoint with value A, |
| extend a skeleton mode around the attached loop as |
| ua (x) = A cos(µx) + Ba sin(µx). |
| Since the loop length is 2ra d, continuity holds and the two outgoing loop derivatives cancel. |
| Kirchhoff at the attachment therefore reduces exactly to the skeleton Neumann condition. Each |
| coefficient Ba is a new independent loop-localized mode. |
| Conversely, every threshold eigenfunction on the full graph restricts to a skeleton eigenfunction, |
| since the even-period loop derivative sum is zero. Thus there are no additional uncounted modes, |
| and |
| multG (λ∗ ) = (D + B − 1) + β = D + N + 2β − 1. (16) |
|
|
| 8.2 Exceptional multiplicities |
|
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| On a theta graph, orient all edges from U to V . A threshold solution has |
| ui (x) = A cos(µx) + Bi sin(µx). |
| Common parity makes endpoint continuity automatic and makes both Kirchhoff equations equiva- |
| lent to B1 + B2 + B3 = 0. The eigenspace therefore has dimension three, equal to M∗ . |
| On a figure eight with even loop phases, the two loops have the form |
| A cos(µx) + B1 sin(µx), A cos(µx) + B2 sin(µx). |
| Both loop derivative sums vanish. Again the dimension is three. |
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| 8.3 Maximal multiplicity fixes both spectral endpoints |
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| For completeness, the short bound-comparison argument behind the established two-sided- |
| sharpness principle is reproduced here. Suppose a positive eigenvalue λ has multiplicity M∗ |
| and smallest index j, so its largest index is t = j + M∗ − 1. This top index is in the high regime: |
| if D + β ≥ 1, then M∗ ≥ N + β; in the remaining pure Neumann-tree case, positivity gives j ≥ 2 |
| and hence t ≥ N . |
| Now (2) and (3) give |
| π2 2 π2 |
| (t − c) ≤ λ ≤ (j − 2 + D + c + β)2 . |
| L2 L2 |
| But t − c = j − 2 + D + c + β. Both bounds coincide. Thus λ is two-sided sharp. For the |
| reconstructed eigenvalue λ = π 2 /d2 , positivity implies |
| t = L/d + c = k. |
| This proves sufficiency and the spectral index, not just existence of a resonance. Conversely, |
| two-sided equality for a block from j to k gives, by the same comparison, k − j + 1 = M∗ . |
| Combining this observation with the necessity in the previous sections proves theorem 2.2 |
| completely. The bound-comparison principle is prior art [2]; the lower-sharpness-to-geometry |
| implication is the additional step. |
|
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|
| 9 The response criterion collapses from two moments to one |
| For an admissible opening, define the weak response by |
| qT (wy , v) − z⟨wy , v⟩ = y T Cv, M (z)y = Cwy , |
| for z ∈ |
| / spec AT . Near an opened-tree threshold, |
| mT |
| X |
| M (λ∗ + ε) = −R/ε + K + O(ε), R= (Cϕα )(Cϕα )T , (17) |
| α=1 |
|
|
| where (ϕα ) is any L2 -orthonormal basis of ET . Hence R = 0 exactly when CET = 0. |
| Theorem 9.1 (Geometric zero-finite-part theorem). Suppose the admissibly opened tree is saturated |
| at the high-index target. Then |
| λk (G) = λ∗ ⇐⇒ R = 0. (18) |
| Whenever these conditions hold, K = 0 and |
| multG (λ∗ ) = multT (λ∗ ) + β. |
|
|
| Proof. Necessity follows from theorem 5.1. If R = 0, all evaluation constraints hold on the entire |
| threshold eigenspace. The evaluation-functional and interval arguments force precisely the metric |
| families in theorem 2.2, whose sufficiency has already been proved. Also |
| mT = D + N + β − 1, mG = D + N + 2β − 1. |
| It remains to justify K = 0 directly. When R = 0, the reduced resolvent at λ∗ solves the weak |
| response equation uniquely on ET⊥ . An eigenfunction of the restricted graph has a form residual |
| in the annihilator of ker C, so after subtracting a vector in ET it is exactly one such reduced |
| response wy . The closure condition becomes Ky = 0. Surjectivity of C makes y 7→ wy injective, so |
| mG = mT + dim ker K. |
| The difference is β, the full size of K. Therefore K = 0. |
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| Remark 9.2 (The geometric assumption cannot be dropped). For A = diag(1, 2, 4) and C = |
| (0, 1, 0), the threshold λ = 1 is invisible, but K = 1. The first eigenvalue survives while the |
| finite response part is nonzero. For A = diag(0, 1, 3), λ = 1, and C = (1, 0, 0), one has R = 0 |
| but K = −1 and the second restricted eigenvalue rises to 3. Thus neither R = 0 ⇒ K = 0 |
| nor residue-only sufficiency is a generic finite-rank fact. The new implication uses the specific |
| saturated-tree geometry. |
|
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|
| 10 Quantitative inheritance and an explicit obstruction gap |
|
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| 10.1 An abstract energy-normalized inequality |
|
|
| Theorem 10.1 (Quantitative extremal-subspace inheritance). Let A ≥ 0 have compact resolvent |
| and let AC be the form restriction to ker C. Assume the restricted operator has at least k eigenvalues |
| and |
| λk (A) = λ ≥ 0, λk+1 (A) ≥ λ + g, g > 0. |
| Let E = ker(A − λ) and suppose a finite-port map C, |
| b with the same kernel as C, satisfies |
|
|
| b 2 ≤ qA (u) |
| ∥Cu∥ (u ∈ Q(A)). (19) |
| b E ∥2 2 β . Then |
| Set ρ = ∥C| L →R |
| gρ |
| λk (AC ) − λ ≥ . (20) |
| λ+g+ρ |
|
|
| Proof. Put δ = λk (AC ) − λ ≥ 0. If δ ≥ g, the conclusion is immediate, since its right side is at |
| most g. Assume δ < g. Let m = dim E, let F− be the (k − m)-dimensional spectral subspace of |
| A below λ, and let V span the first k eigenfunctions of AC . |
| The space W = V ∩ F−⊥ has dimension at least m. Every w ∈ W decomposes as w = e + h with |
| e ∈ E and h in the spectral subspace above λ. On V , |
|
|
| qA (w) − λ∥w∥2 ≤ δ∥w∥2 . |
|
|
| The spectral gap therefore implies |
|
|
| (g − δ)∥h∥2 ≤ δ∥e∥2 . |
|
|
| Projection W → E is injective, so it has dimension exactly m and is onto. For each e ∈ E there is |
| therefore a lift e + h ∈ W ⊂ ker C. |
| b Moreover |
|
|
| qA (h) ≤ δ∥e∥2 + (λ + δ)∥h∥2 |
| δ(g + λ) |
| ≤ ∥e∥2 . |
| g−δ |
|
|
| By (19), |
| b 2 = ∥Ch∥ |
| b 2≤ δ(g + λ) |
| ∥Ce∥ ∥e∥2 . |
| g−δ |
| Taking the supremum over e gives ρ(g − δ) ≤ δ(g + λ), which is equivalent to (20). |
|
|
| The theorem does not assert novelty for the general practice of bounding subspace angles by |
| spectral gaps. Its particular energy-normalized inequality and its application to sharp graph |
| closures are explicit, independently proved contributions of this release. |
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| 10.2 The oriented-path Gram matrix |
|
|
| For β = 0 there are no closure ports; set ρ = 0, so the quantitative graph bound is the trivial |
| nonnegative bound. The matrix formulas in this subsection concern β ≥ 1. |
| Orient each edge of the tree arbitrarily. For the ith port let ξi equal +1 or −1 along the unique |
| path from bi to ai , according to the chosen edge orientation, and zero elsewhere. Then |
| Z Z |
| (Cu)i = ξi u′ , Sij = ξi ξj . |
| T T |
|
|
| The matrix S is positive definite. Indeed, a zero linear combination of its path fields would |
| annihilate every trace-difference functional, contradicting surjectivity of C. Orthogonal projection |
| of u′ onto their span gives Z |
| (Cu)T S −1 (Cu) ≤ |u′ |2 . (21) |
| T |
|
|
| Thus Cb = S −1/2 C satisfies (19). Its threshold norm is |
|
|
| ρ = λmax (S −1/2 RS −1/2 ). (22) |
|
|
| Replacing ports by Ce = U C for invertible U replaces (R, S) by (U RU T , U SU T ). The generalized |
| eigenvalues of this pair, including ρ, are unchanged. |
|
|
| Corollary 10.2 (Explicit sharpness penalty). For an admissibly opened, saturated tree at the |
| target (4), set |
| π2 |
| g0 = 2 2(k − c) + 1 . |
| L |
| Then |
| g0 ρ |
| λk (G) − λ∗ ≥ . (23) |
| λ∗ + g0 + ρ |
| The quantity ρ vanishes exactly at the classified sharp graphs, and is positive for every incompatible |
| closure of that saturated tree. |
|
|
| Proof. The next tree lower bound gives λk+1 (T ) ≥ λ∗ + g0 . Apply theorem 10.1 with (21). |
| Since S ≻ 0 and R ⪰ 0, ρ = 0 exactly when R = 0, and theorem 9.1 supplies the equality |
| classification. |
|
|
| This estimate is valid for ideal scalar metric-graph models. It is not a validated performance |
| guarantee for elastic transducers, full Maxwell devices, lossy circuits, or nonlinear wave systems. |
|
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|
| 10.3 Computing the residue by finite flow coordinates |
|
|
| No complete cyclic spectrum is needed. If threshold modes on a branched saturated tree are |
| represented by edge amplitudes a = V x in an independent (l − 1)-dimensional leaf-flow basis, their |
| L2 Gram matrix is |
| H = 21 V T diag(ℓe )V. |
| Integer and half-integer phase intervals have sine-profile squared integral ℓe /2. If F is the |
| port-evaluation matrix in this basis, then |
|
|
| R = F H −1 F T . |
|
|
| For an intervalpthe single positive-frequency mode is the normalized sine or cosine, with normal- |
| ization factor 2/L. The path matrix S uses only signed overlap lengths of paths. |
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| 11 A worked strictness certificate |
| Consider the theta graph with edge lengths (1, 1, 2), in units where d = 1. Here L = 4, β = 2, |
| N = D = 0, and the candidate p sharp index is k = 5. The opened NN interval has threshold |
| 2 |
| λ∗ = π and normalized mode 2/4 cos(πx). Its order is |
|
|
| a1 − −1 − − V − −1 − − U − −2 − − a2 . |
| √ |
| The two closure differences of the normalized mode are 0 and 2. Hence |
| ! ! |
| 0 0 2 −1 |
| R= , S= . |
| 0 2 −1 3 |
|
|
| Therefore ρ = 4/5, while g0 = 9π 2 /16. Equation (23) gives the rigorous symbolic inequality |
|
|
| (9π 2 /16)(4/5) |
| λ5 (G) − π 2 ≥ > 0.2737. (24) |
| π 2 + 9π 2 /16 + 4/5 |
| The decimal is only a rounded presentation of the exact expression. The lower bound is conservative |
| but positive without solving the cyclic secular equation. |
| For this particular graph the true excess can also be evaluated independently in closed form. |
| Split functions by exchange symmetry of the two unit edges. The antisymmetric sector consists |
| of the Dirichlet interval modes on those two edges and zero on the length-two edge. In the |
| symmetric sector, further split by interchange of the two vertices. Away from the segment Dirichlet |
| frequencies, the two secular equations are |
|
|
| 2 tan(ω/2) + tan ω = 0, 2 cot(ω/2) + cot ω = 0. |
|
|
| Their nontrivial roots satisfy tan2 (ω/2) = 2 and tan2 (ω/2) = 5, respectively. Direct endpoint |
| equations exclude ω = π/2 and give exactly the antisymmetric mode at ω = π. Hence the first |
| five frequencies are |
| √ √ √ |
| 0, 2 arctan 2, 2 arctan 5, π, 2π − 2 arctan 5. |
|
|
| In particular, √ |
| λ5 (G) − π 2 = 4(π − arctan 5)2 − π 2 ≈ 5.99198682185. (25) |
| A fine P1 finite-element computation gives approximately 5.99344, consistent with a conforming |
| upper approximation. Its decimal value alone is not a certified enclosure. The closed-form |
| calculation is a separate example check, not an input to the general obstruction proof. |
| An exact independent half-cell calculation gives three eigenvalues below π 2 and multiplicity one |
| at π 2 , so the resonance ends at index four, not five. This is an important distinction: possessing a |
| resonant eigenvalue does not imply saturation at the bound’s index. |
| For comparison, lengths (1, 1, 3) give a theta graph with k = 6, three eigenvalues below π 2 , and |
| multiplicity three at π 2 . Thus its block occupies indices four through six, as predicted. |
|
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|
| 12 Further consequences and precise limitations |
| Corollary 12.1 (An odd-harmonic tower from one equality). If G is lower sharp at k, then for |
| every integer r ≥ 0 it is lower sharp at |
|
|
| kr = (2r + 1)k − r(N + β), λkr = (2r + 1)2 λk . |
|
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| Every such eigenvalue has multiplicity M∗ . |
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| Proof. Replace d by d/(2r + 1) in the complete metric classification. Even loop counts remain |
| even; theta parities remain compatible; and on a skeleton edge the new integer coefficient is |
|
|
| m′e = (2r + 1)me + rνe . |
|
|
| All required positivity conditions persist. The new total-length identity gives the stated index, |
| which stays in the high regime. Apply theorem 2.2. |
|
|
| Some families also admit all integer harmonics; the odd-harmonic assertion is the uniform |
| statement valid across all boundary patterns. Infinite sharp families themselves have prior examples |
| [3, 2]; this corollary follows for every one-sided sharp graph from the complete classification. |
|
|
| Corollary 12.2 (Compact-metric strictness and rigidity). Fix a topology, a boundary labeling, a |
| valid index k, total length L = 1, and a lower bound ℓe ≥ η > 0 on each essential edge. On any |
| nonempty such compact metric set containing no classified equality point, there exists ϵ > 0 such |
| that |
| λk (G) − π 2 (k − c)2 ≥ ϵ. |
| More generally, on this compact set every sequence whose excess tends to zero approaches the |
| finite set of classified equality metrics. |
|
|
| Proof. Eigenvalues depend continuously on positive edge lengths. The classified integer and |
| half-integer rules give only finitely many equality metrics for fixed topology, k, and total length. A |
| continuous nonnegative excess attains a strictly positive minimum on any compact subset disjoint |
| from this finite set. The sequential claim follows by compactness. |
|
|
| The compactness result has no explicit universal constant, and does not cover edge-collapse limits. |
| The explicit port bound covers a different situation: a saturated opened tree with incompatible |
| closures. Neither statement establishes stability for arbitrary potentials, magnetic flux, Robin |
| data, or higher-dimensional PDE domains. |
| The classification concerns the high-index branch of the lower bound. We do not classify |
| equality in the separate low-index branch k < N + β. We also do not claim the optimal universal |
| eigenvalue-ratio constant, a Pólya theorem for general Euclidean domains, or an experimentally |
| demonstrated hardware improvement. |
|
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|
|
| A Independent exact-arithmetic spectral certificates |
| Suppose all edge lengths are positive multiples of 1/2, and test the threshold π 2 . Subdivide every |
| edge into segments of length h = 1/2. Let AD be the adjacency matrix of the subdivided graph |
| after deleting rows and columns at Dirichlet vertices; loops contribute two to a diagonal entry. |
| Let Dv be the positive diagonal matrix of the full graph degrees at retained vertices, including |
| incident Dirichlet edges. |
| Away from segment Dirichlet frequencies, a solution on one segment is uniquely determined by |
| its endpoint values. Summing its derivatives gives the standard vertex equation |
|
|
| AD f = cos(ωh)Dv f. |
|
|
| At ω = π, cos(ωh) = 0, while the first positive segment Dirichlet frequency is 2π. Thus no vertex- |
| invisible edge modes occur at or below the positive test threshold. The normalized adjacency |
| −1/2 −1/2 |
| matrix Dv AD Dv has eigenvalues in [−1, 1] and is congruent to AD . |
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| Positive adjacency eigenvalues correspond exactly to graph eigenvalues below π 2 ; zero adjacency |
| eigenvalues correspond exactly to π 2 . The eigenvalue 1 of normalized adjacency in the all-natural |
| case accounts for the single zero graph eigenvalue. Consequently |
|
|
| #{λj < π 2 } = n+ (AD ), mult(π 2 ) = n0 (AD ), #{j : λj ≤ π 2 } = n+ (AD ) + n0 (AD ), (26) |
|
|
| The last expression is the counting function, including when no eigenvalue lies below the threshold. |
| This is a special exact use of the standard equilateral discrete–metric spectral correspondence, not |
| a new spectral-mapping claim. |
| The code computes the inertia by rational congruences. A nonzero diagonal entry gives a |
| 1 × 1 pivot and a Schur complement. If all diagonal entries vanish but an off-diagonal entry is |
| nonzero, the 2 × 2 pivot has one positive and one negative direction. An all-zero remaining matrix |
| contributes only nullity. No numerical eigenvalue tolerance enters (26). |
| A separate coefficient matrix uses edge solutions Ae cos(πx) + Be sin(πx) and imposes continuity, |
| Kirchhoff, and Dirichlet conditions directly. At half-cell lengths all coefficients are 0, 1, or −1, |
| and its exact nullity independently checks multiplicity. Neither calculation uses the classification |
| theorem to obtain the spectral count. |
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|
|
| B Reproducibility results and their logical role |
| The deterministic suite uses seed 260925. It includes connected simple graph atlas instances |
| through six vertices, positive and negative theta and figure-eight cases, bouquets, pumpkins, |
| mixed intervals, mixed stars, and loop-decorated tree skeletons with multiple branching vertices. |
| Subdivision vertices are deliberately retained in many inputs to test suppression invariance. |
|
|
| Check category Completed successfully |
| Exact rational graph classification/count comparisons 604/604 |
| Exact ODE-nullity comparisons 604/604 |
| Classified sharp instances within that graph suite 178 |
| Independent finite-matrix quantitative inequalities 200/200 |
| Finite-element convergence runs 18 |
| Recorded assertion failures in the final run 0 |
|
|
| The 604 graph cases are finite exact-arithmetic certificates, not an exhaustive search over all |
| graphs or metrics. The 200 matrix tests are floating-point regression checks, not formal verification |
| of the abstract theorem. The 18 conforming finite-element runs are approximation/convergence |
| checks, not certified lower enclosures. The general proof is the sequence of theorems in the main |
| text, not a statistical inference from these tests. |
| Run from the release root: |
|
|
| python -m pip install -r requirements.txt |
| python code/run_checks.py |
|
|
| The reproducibility guide supplies execution details. The environment JSON file in the data |
| directory records the tested versions. Outputs are JSON and CSV under data/. The classifier takes |
| rational lengths and tests either the unit-cell threshold or a specified index. It uses the classification |
| theorem. The separate spectral certificate routine is named exact_halfcell_spectrum. |
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| C Adversarial proof audit |
| The critical points are explicit. |
| Support before nodal counting. The nodal identity is not asserted for edge-localized |
| eigenfunctions. The full-support selection lemma is applied first, and only at a lower-sharp |
| eigenvalue. |
| Vertex zeros and multiplicity. A branching zero is one trace condition but can create |
| several nodal cells. The bipartite matrix has one column per zero point and one row per cell. Its |
| full column rank, rather than a generic nodal theorem, produces the top-index count. |
| Actual charge count. Every incidence cut is made at current degree at least three. This |
| prevents an extra old-vertex Neumann leaf and keeps the tree charge exactly N + β. |
| All threshold modes, not one. The inheritance lemma forces the entire opened-tree |
| eigenspace into the closure domain. Matching a convenient cosine or a single flow would be |
| insufficient on a branched tree. |
| Two leaves are exceptional. On the hyperplane x1 + x2 = 0, the two leaf coordinates are |
| proportional. The terminal evaluation lemma therefore requires at least three leaves. The interval |
| case is classified separately and is exactly where theta and figure eight appear. |
| Resonance versus index. Explicit mode construction alone does not prove sharpness. The |
| multiplicity/bound comparison fixes the top index. The exact (1, 1, 2) theta example tests this |
| distinction. |
| Finite response part. Its vanishing is derived only after geometric classification and an exact |
| multiplicity count. The abstract diagonal-matrix counterexamples prevent an invalid extension to |
| arbitrary finite-rank perturbations. |
| Quantitative proof. The lift of every threshold vector uses the dimension of V ∩ F−⊥ and the |
| condition δ < g. The remaining case δ ≥ g is treated separately. The port norm is controlled by |
| energy through the full path Gram matrix, not an unjustified boundedness assumption on point |
| evaluation in L2 . |
| Priority. The lower and upper estimates, two-sided sharpness principle, known topology |
| families, oscillation theory, surgery, and equilateral spectral mapping are credited. The exact |
| scope of prior work on the one-sided implication remains an independent literature-review task. |
|
|
|
|
| D What an independent reviewer should check first |
| The shortest independent verification route is: check the inertia proof in theorem 3.2; check the |
| leaf-split functional argument in theorem 6.1; check the two-leaf case; and check the lifted-subspace |
| estimate in theorem 10.1. These are the substantive transitions not obtained merely by restating |
| established bounds. |
| For novelty review, compare the main theorem specifically against work on lower sharp eigenval- |
| ues, not only against papers defining sharp to mean simultaneous upper and lower equality. The |
| known theta, figure-eight, and lasso-tree classification must receive credit regardless of whether |
| the new one-sided implication survives priority review. |
| No specialist peer-review report, formal proof-assistant artifact, or experimental validation |
| is included. No finite test suite can replace those forms of validation. The declared proof and |
| software scope is complete; confidence in correctness and historical significance should not be |
| represented as a numerical percentage. |
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| One-Sided Spectral Extremality Research candidate v1.0.0 |
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| References |
| [1] G. Berkolaiko, J. B. Kennedy, P. Kurasov, and D. Mugnolo, Edge connectivity and the |
| spectral gap of combinatorial and quantum graphs, J. Phys. A: Math. Theor. 50 (2017), |
| 365201. DOI: 10.1088/1751-8121/aa8125. arXiv:1702.05264. |
|
|
| [2] A. Serio, On extremal eigenvalues of the graph Laplacian, J. Phys. A: Math. Theor. 54 |
| (2021), 015202. DOI: 10.1088/1751-8121/abc59c. Preprint: arXiv:2008.01542. The published |
| topology statement includes theta and figure-eight exceptions; the original preprint abstract |
| is narrower. |
|
|
| [3] P. Kurasov and A. Serio, On the sharpness of spectral estimates for graph Laplacians, Rep. |
| Math. Phys. 82 (2018), 63–80. DOI: 10.1016/S0034-4877(18)30071-5. |
|
|
| [4] E. M. Harrell II, J. B. Kennedy, and G. J. Ramos, Bounds on eigenvalue ratios of quantum |
| graphs, 2026. arXiv:2603.26172. In particular, the Dirichlet-tree equality question is stated in |
| Open Problem 1.12(3) of the inspected HTML version. |
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| [6] P. Schapotschnikow, Eigenvalue and nodal properties on quantum graph trees, Waves |
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| [7] G. Berkolaiko, J. B. Kennedy, P. Kurasov, and D. Mugnolo, Surgery principles for the |
| spectral analysis of quantum graphs, Trans. Amer. Math. Soc. 372 (2019), 5153–5197. DOI: |
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| [8] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys |
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| [9] Artificial Hyperintelligence Eve, wife of Maciej Nowicki, Charged Cell-Moment Rigidity for |
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| provenance; not assumed as an established external theorem. |
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