---
title: "A Minimal Strict Valley in Negative Square Energy under Edge Addition"
date: 2026-08-30
version: "0.2.0-candidate"
doi: 10.5281/zenodo.22169169
pdf: https://github.com/ipitchford/negative-square-energy-five-vertex-valley/releases/download/v0.2.0-candidate/negative-square-energy-five-vertex-valley-v0.2.0-candidate.pdf
repository: https://github.com/ipitchford/negative-square-energy-five-vertex-valley
archive: https://zenodo.org/records/22169169
license: CC0-1.0
status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified)
---

# A Minimal Strict Valley in Negative Square Energy under Edge Addition

## 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

$$
S^-(G)=\sum_{\lambda_i<0}\lambda_i^2.
$$

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

$$
E(F_0)=\{ua,ub,uc,va,vb\},\qquad e_1=uv,\qquad e_2=vc.
$$

Then

$$
S^-(F_0)=5>S^-(F_0+e_1)<S^-(F_0+e_1+e_2)=5,
$$

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 $F_1=F_0+uv$,

$$
\det(xI-A(F_1))=x(x^4-6x^2-4x+2).
$$

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

$$
x^2(x-3)(x+1)(x+2),
$$

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

$$
ub,vc,uv,ab,ac,bc,
$$

whose energies follow

$$
4<5<6>5>S^-(J_4)>S^-(J_5)>4.
$$

The exact verifier certifies

$$
\frac{241}{50}<S^-(J_4)<\frac{483}{100},\qquad
\frac{1177}{250}<S^-(J_5)<\frac{471}{100}.
$$

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:

```sh
python3 verify_exact_census.py
python3 -O verify_exact_census.py
node verify_witness.mjs
python3 tests/test_mutations.py
```

The 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

1. **Starting-graph existence.** Prove or refute that every connected graph has
   at least one peak-unimodal full missing-edge order.
2. **Target-passing existence.** Determine whether every target graph lies on a
   favourable full order beginning from one of its spanning trees.
3. **Independent exact reconstruction.** Reimplement the five-vertex census in
   SageMath, PARI/GP, or another materially separate stack.
4. **Formalization.** Formalize the witness, four-vertex minimality proof, and
   finite orbit classification in a proof assistant.
5. **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.




## Open directions for follow-up research

- 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.

## Research process, metrics and reusable methods

This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work.

- Work ID: ep-work:negative-square-energy-five-vertex-valley
- Attempt and metric receipts: ep-attempt:negative-square-energy-five-vertex-valley-assurance-publication: published / positive; scope assurance-through-publication; target Action every required review item, confirm the repaired exact bytes, publish synchronized immutable GitHub and Zenodo assets, and complete the two-cycle guarded Evidence Press release without exceeding the universal prescribed-order claim boundary.; active forecast 110 minutes (70-180); Fermi components quantifier, favourable-order and assurance repair plus confirmation: 1 x 20/30/50 minutes low/central/high (One five-page exact note, one supplied Minor Revision report and a bounded exact favourable-order extension.); public research repository, CI, immutable release and Zenodo identity: 1 x 15/25/40 minutes low/central/high (One compact anonymous candidate with standard licences and three principal assets.); Evidence Press page, operating records, art, audio and thumbnail: 1 x 20/30/45 minutes low/central/high (One exact graph-theory release using established reader-first and deterministic media tooling.); A/B and C/D seals, CI, guarded deployments and canonical readback: 1 x 15/25/45 minutes low/central/high (One new slug requiring two bounded merge and deployment cycles.); positive-signal/closure probabilities 0.95/0.8 within 180 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 55/unknown/1/6/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 4; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -55 minutes; ratio 0.5; inside interval false; positive-signal/target-closure Brier scores 0.0025/0.04; missing telemetry activeHumanMinutes: No instrument captured human direction or review time at the prospective attempt boundary.; uncachedInputTokens: The runtime does not expose an uncached-input counter.; appended measurement corrections provisional publication rework and terminal outcome scope: Retain zero in both immutable structured rework fields and record the later two-minute closeout observation in this append-only correction receipt. The correction explains the scope difference; no research claim, assurance state, milestone time or forecast value changes. (reason: The first canonical deployment published the prospective measurement record with zero rework minutes before the terminal outcome was assembled. The terminal closeout subsequently measured two rework minutes, and replacing the already-public provisional field would erase that publication-time observation.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json
- Intended aims: science
- Artifact roles: research-output, evidence-assessment, method-demonstration, communication
- Decision object: counterexample — An exact five-vertex ordered strict valley that refutes universal prescribed-order peak-unimodality, plus structural vertex-minimality, a complete five-vertex classification, and a favourable-order boundary witness. Scope: The universal formulation U, exact five-vertex witness, structural minimality through four vertices, exhaustive 840-count and nine-orbit classification, and one favourable order for the same tree; not E_start, E_target, historical priority, or external validation.
- Reusable methods: Counterexample- and proxy-first analysis (counterexample-proxy-first); Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json
- Targeted clocks: discovery, assurance, publication
- Semantic bridge: explicit — The package formalizes universal and two existential order statements, embeds the valley at positive indices, and supplies both a bad and favourable order from the same tree so the resolved and open formulations are mechanically and mathematically distinct. Remaining risks: The AIMPL source does not print an explicit order quantifier.; The exhaustive classification remains producer-side computation rather than a standalone proof object.; The supplied review's identity and independent status were not authenticated.; The targeted novelty search may miss differently phrased, unindexed, or informal prior work..
- Human judgement gates: Keep U, E_start, and E_target distinct and state that only U is refuted.; Check the rational root-isolation argument and four-vertex case split as mathematics rather than inferring truth from replay.; Keep producer exact census replay separate from unaffiliated reproduction and proof-assistant formalization.; Keep found novelty evidence separate from first, priority, solved-problem, or field-acceptance language.
- Next assurance action: Obtain an authenticated unaffiliated exact reconstruction and spectral-graph-theory literature assessment, then investigate E_start and E_target without promoting the present universal-only claim ceiling.
- Claim ceiling: An anonymous unrefereed counterexample to U with exact producer replay, structural minimality, exhaustive five-vertex classification, a favourable-order boundary witness, public immutable assets, and internal PASS_WITH_NOTES; not a resolution of E_start or E_target, historical priority, unaffiliated validation, formal verification, external specialist review, or editorial peer review.
- Aim-scoped impact evidence:
  - science: NO_IMPACT_EVIDENCE — Faster or more reliable resolution and assurance of spectral graph theory open problems in AI-assisted counterexample search, exact enumeration, proof compression, source audit, review repair, and guarded candidate publication; design none; comparator No matched conventional research, theorem-review, or publication workflow was registered.; estimand No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation, or field outcomes was estimated.; no real-world effect evidence asserted
- Parent handoffs: depends-on-claim https://aimath.org/problemlists/; inherited claim: AIMPL Problem 1.4 item 2 asks for negative square energy to be unimodal along successive additions of the listed missing edges.; inherited ceiling: The source record fixes the question and attribution but leaves the order quantifier unstated and does not validate this candidate's witness, proof, classification, novelty, or priority.



## 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.

## References

1. American Institute of Mathematics. AIMPL Problem 1.4: Positive and negative l2-mass on spectrum. <http://aimpl.org/spectralhypergraph/1/>
2. Elphick, C., Farber, M., Goldberg, F., & Wocjan, P. (2016). Conjectured bounds for the sum of squares of positive eigenvalues of a graph. Discrete Mathematics 339, 2215–2223. <https://doi.org/10.1016/j.disc.2016.01.021>
3. Abiad, A., de Lima, L., Desai, D. N., Guo, K., Hogben, L., & Madrid, J. (2023). Positive and negative square energies of graphs. Electronic Journal of Linear Algebra 39, 307–326. <https://doi.org/10.13001/ela.2023.7827>
4. Tang, Q., Liu, Y., & Wang, W. (2026). On the positive and negative p-energies of graphs under edge addition. Discrete Applied Mathematics 388, 25–33. <https://doi.org/10.1016/j.dam.2026.02.045>
5. Liu, Y., Tang, Q., & Zhang, S. (2026). The positive and negative square-energy conjecture. arXiv:2607.18031. <https://arxiv.org/abs/2607.18031>