Press release · 30 August 2026 · version 0.2.0-candidate
A Minimal Strict Valley in Negative Square Energy under Edge Addition
A five-vertex graph gives the smallest exact decrease-then-increase in negative adjacency square energy, refuting universal prescribed-order peak-unimodality while leaving favourable-order existence open.
Summary
Imagine adding every missing edge to a connected graph, one at a time, until the graph is complete. At each step, measure the total squared size of the negative adjacency eigenvalues. A natural conjecture says this sequence should rise to one peak and then fall.
This anonymous, unrefereed candidate gives the smallest possible strict local valley. On five vertices, one edge addition lowers the negative square energy from $5$ to less than $3697/784$, and the next raises it back to $5$. No connected graph on at most four vertices can do this.
The order quantifier is decisive. The result disproves the statement that every prescribed order is peak-unimodal. It does not disprove the weaker possibility that every starting graph has some favourable order, or that every target graph can be reached along one. In fact, the same starting tree has an exactly certified favourable order.
Summary for specialists
For a finite simple graph $G$ with adjacency eigenvalues $\lambda_i$, let
The candidate distinguishes three formulations: universal peak-unimodality for every connected $G$ and every missing-edge order; existence of a favourable order for each starting graph; and existence of a favourable tree order that passes through each prescribed target graph. It refutes only the first.
With vertices $\{u,v,a,b,c\}$, take
Then
and exact rational root bounds give $S^-(F_0+e_1)<3697/784<5$. A structural case split excludes all connected graphs through four vertices. Exhaustive exact enumeration finds $840$ labelled five-vertex ordered valleys in exactly nine orbits under simultaneous vertex relabelling with the addition order retained.
Technical account
The witness has a compact hand proof. The graph $F_0$ is bipartite with five edges, so spectral symmetry and $\operatorname{tr}(A^2)=10$ give $S^-(F_0)=5$. For $F_1=F_0+uv$,
Four exact sign evaluations isolate the two negative roots. Descartes' rule shows there are no others, and their rational intervals prove the strict upper bound $3697/784$. Adding $vc$ gives characteristic polynomial
so the endpoint negative square energy is again $5$.
To put the valley at positive indices, start from the tree $H=F_0-ub$ and add $ub,uv,vc$. Yet the same $H$ also has the favourable full order
whose energies follow
The exact verifier certifies
That paired example is the cleanest explanation of the claim boundary: one bad order refutes the universal statement, while one good order for the same tree shows why the existential questions remain open.
Evidence, assurance and limitations
The package includes the five-page manuscript and source, an accessible Markdown rendering, exact Python/SymPy enumeration, a dependency-free JavaScript witness checker, a producer-generated census record, four semantic mutations, claim and source maps, novelty and citation audits, internal review records, a response to the supplied full review, replay receipts, licences, and a complete SHA-256 manifest.
Normal and optimized Python runs reproduce $728$ connected labelled five-vertex graphs, $11{,}460$ ordered two-edge sequences, $840$ strict valleys, and nine orbits. The JavaScript checker independently encodes the principal witness within the same producer workflow. The four mutations corrupt the labelled total, an orbit representative, a characteristic polynomial, and the favourable edge order; all are rejected.
The supplied review reports a successful replay in a different Linux environment and a separately written numerical enumerator reproducing the headline counts. That is useful corroboration, but the reviewer identity, environment receipt, and implementation were not authenticated here. The release therefore does not upgrade unaffiliated rerun or independent reimplementation from not-assessed.
The release also does not establish historical priority, proof-assistant formalization, external specialist review, journal peer review, or either favourable-order existential formulation. The finite census is separately recomputable by exhaustive producer replay; its JSON record is not a standalone completeness proof.
Relationship to earlier work
Abiad and collaborators had already shown that adding one edge can decrease negative square energy. Tang, Liu and Wang developed perturbation bounds for positive and negative $p$-energies and found positive-energy monotonicity counterexamples in a different range. The contribution here is the exact decrease-then-increase valley, its vertex-minimality, and the complete five-vertex classification.
The broader lower-bound conjecture for positive and negative square energy was proved in a July 2026 preprint by Liu, Tang and Zhang. The five-vertex valley is compatible with that theorem because its middle energy remains above $4$.
Who should care, and why
| Audience | Potential use | Required caution |
|---|---|---|
| Spectral graph theorists | A minimal obstruction and complete five-vertex test bed for edge-addition questions | Check the order quantifier and do not infer an existential counterexample |
| Computational reviewers | Small exact instances, orbit representatives, rational intervals, and hostile mutations | Reimplement independently rather than importing the producer enumerator |
| Formalizers | A short witness proof and four-vertex case split | The exhaustive five-vertex classification still needs a formal enumeration bridge |
| Research agents | A worked example of source-quantifier repair and claim-level assurance | Producer replay and internal review are not external validation |
| Interested non-specialists | A concrete example showing that graph energy can dip and recover | Candidate publication is not field consensus or peer review |
Why the problem matters
Unimodality is a strong global regularity claim: it says a statistic cannot reverse direction twice along an edge-addition path. A minimal strict valley pinpoints exactly where that regularity first fails and supplies a finite laboratory for corrected conjectures. The paired bad and favourable orders also show why quantifiers over construction paths are mathematically substantive, not merely wording choices.
How to inspect or reproduce the recorded checks
Use candidate tag v0.2.0-candidate or the Zenodo version DOI, not the moving main branch. From a fresh extraction with Python 3.11 or later, SymPy 1.14.0, and Node.js 20 or later, run:
python3 verify_exact_census.py
python3 -O verify_exact_census.py
node verify_witness.mjs
python3 tests/test_mutations.pyThe expected markers are PASS_EXACT_CENSUS, PASS_EXACT_WITNESS, and PASS_MUTATION_CONTROLS. Successful execution confirms the declared predicates in the released code; it does not establish novelty, independence, or peer review.
The most valuable next projects
- Starting-graph existence. Prove or refute that every connected graph has at least one peak-unimodal full missing-edge order.
- Target-passing existence. Determine whether every target graph lies on a favourable full order beginning from one of its spanning trees.
- Independent exact reconstruction. Reimplement the five-vertex census in SageMath, PARI/GP, or another materially separate stack.
- Formalization. Formalize the witness, four-vertex minimality proof, and finite orbit classification in a proof assistant.
- Specialist priority audit. Search discipline-specific indexes and differently phrased graph-energy literature for the exact valley and nine-orbit classification.
What is in the evidence package
The immutable release includes the canonical PDF and LaTeX, abridged Markdown, proof and claim maps, exact verifiers, generated census record, mutation tests, source and literature receipts, internal and supplied-review response records, research metrics, licences, replay receipt, and SHA-256 manifest. The version DOI is the citation target; any correction should be a versioned successor, not a silent edit.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- Prove or refute that every connected starting graph has at least one peak-unimodal full missing-edge ordering.
- Determine whether every target graph lies on a peak-unimodal full ordering beginning from one of its spanning trees.
- Independently reimplement the exact five-vertex census and orbit classification in a materially separate stack.
- Formalize the witness, four-vertex proof, exhaustive enumeration bridge, and nine-orbit classification.
- Obtain an unaffiliated spectral-graph-theory source, proof, and priority assessment.
Verification status
Anonymous, unrefereed spectral-graph-theory candidate at internal PASS_WITH_NOTES. The witness and four-vertex minimality have short hand proofs; the 840-count and nine-orbit classification rely on exhaustive producer-side exact computation. A supplied review reports a separate Linux replay and numerical reimplementation, but reviewer identity and supporting artifacts were not authenticated, so independent assurance remains not assessed. The source order quantifier is unstated: the release refutes the universal or arbitrary prescribed-order formulation only and leaves both favourable-order existential formulations open.
Cite
BibTeX
@misc{negativesquareenergyfivevertexvalley2026,
title = {A Minimal Strict Valley in Negative Square Energy under Edge Addition},
author = {Anonymous},
year = {2026},
doi = {10.5281/zenodo.22169169},
url = {https://doi.org/10.5281/zenodo.22169169},
version = {0.2.0-candidate},
howpublished = {Zenodo},
note = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/negative-square-energy-five-vertex-valley/}
}Also: cite.bib · paper.json · this page as Markdown