State of Proof

2026-07-29-maxwell-conjecture · Potential theory · Active examination

Five positive point charges in R³ have at least 24 nondegenerate critical points, contradicting the proposed (5−1)² bound

Examination underway · one conditional exact polynomial check reproduced · counterexample unresolved

Current record status

What this docket currently establishes

Examination underway · one conditional exact polynomial check reproduced · counterexample unresolved. This docket identifies what has been checked and what remains open. It does not validate or reject the manuscript's headline claim beyond the explicitly stated scope.

Claim
Five positive point charges in R³ have at least 24 nondegenerate critical points, contradicting the proposed (5−1)² bound
Source version
arXiv 2607.27197v1 · arXiv v1
Source SHA-256
4178ddc5a4efcdc11726bea6d6d20785575c55564095dbf53bfaa6d26055a958
Source locked
Last material update
Next open step
Independently derive the displayed limiting polynomial from the five-charge potential, including the exact Taylor coefficients and analytic remainder
Machine-readable record
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One-minute summary

The manuscript claims five positive charges yield at least 24 nondegenerate critical points. A local exact-algebra check reproduces the displayed limiting polynomial's classification of 21 real, nondegenerate critical points, conditional on manual transcription. The polynomial's derivation from the charge potential, 24-point persistence, Morse perturbation, global finiteness, and the counterexample remain unresolved.

For non-specialists

Why this matters

A nontechnical reading of the claim and its possible consequences. These are implications to consider, not findings of this docket.

In plain English

The manuscript proposes more stationary points around five idealized positive charges than a proposed bound allows. One exact check confirms the 21-point classification of an intermediate polynomial written in the paper. It does not yet show that the five-charge potential has 24 such points.

If the claim holds

If the argument holds, it would give a mathematical counterexample to the proposed numerical bound. It would concern the stated idealized point-charge model, not by itself a claim about a practical physical system.

Potential concerns

The intermediate polynomial must still be derived from the charge model. The proposed points must persist and remain distinct, a nearby positive-charge configuration must have nondegenerate critical points, and the total critical set must be finite.

If it does not hold up

If the claimed configuration or count does not hold up, that would not by itself prove the proposed bound; it would only mean this proposed counterexample has not established an exception.

Scope of this docket

This completed route covers only Lemma 2's exact critical-point and Hessian classification for the displayed limiting polynomial. Gates G1 and G3–G5 remain unexamined; no headline theorem disposition follows.

Claim and dependency map

  1. G1 · queued record

    Exact rescaling and Taylor skeleton

    Depends on: locked v1 source

  2. G2 · reproduced in scope

    Exhaustive exact classification of 21 limiting critical points

    Depends on: displayed limiting polynomial; G1 derivation remains open

  3. G3 · queued record

    Persistence and separation of 21 plus 3 critical-point branches

    Depends on: G2

  4. G4 · queued record

    Morse perturbation via transversality with positive charges

    Depends on: G3

  5. G5 · queued record

    Global finiteness of the complete critical set

    Depends on: G4

Examination routes

computational
Queued · updated 2026-08-26
Planned: derive the displayed polynomial from the five-charge potential with exact Taylor coefficients and analytic remainder, then examine persistence.
Method and provenance · Planned Material Shift route derived from the locked source and claim map; no run is recorded for the charge-to-polynomial derivation.
Inputs · locked source · published claim map
symbolic
Completed local replay · updated 2026-10-06
Exact case splitting and Cartesian Hessian evaluation give 21 critical points of the displayed polynomial, all nondegenerate, with 10 (++−) and 11 (+−−). The charge-potential derivation and headline remain open.
Method and provenance · Material Shift exact checker of the manually transcribed Lemma 2 polynomial. No manuscript code or third-party mathematical package ran; no independent specialist review is recorded.
Inputs · arXiv:2607.27197v1 PDF SHA-256 4178ddc5a4efcdc11726bea6d6d20785575c55564095dbf53bfaa6d26055a958 · displayed limiting polynomial, manually transcribed from printed pp. 4–5 Public replay manifest · exact Lemma 2 route · SHA-256 ce89beeed3f6d88aef98cbbc3d29d48e9d91a680cc8e55cd4121bcc4df262e53
expert
Queued · updated 2026-08-26
Planned: specialist review of quantitative persistence, transversality, and global analytic bounds.
Method and provenance · No specialist report, sign-off, or external review is recorded.
Inputs · locked source · published claim map

Findings and scope limits

  1. An exact local check of the displayed limiting polynomial reproduces Lemma 2's 21 real critical points and nondegenerate Hessian signatures: 10 (++−), 11 (+−−).

  2. This result is conditional on manual transcription of the displayed polynomial. Its derivation from the five-charge potential (G1) has not been examined.

  3. Persistence, Morse perturbation, global finiteness, and the claimed Maxwell counterexample remain unresolved; no specialist review is recorded.

  4. The intake target is five charges and at least 24 nondegenerate critical points, not a three-charge or five-equilibrium construction.

Open Steps—where expert eyes are needed

Potential theory and exact asymptotics · G1 exact derivation; G2 local result remains conditional

Does the five-charge potential yield the displayed limiting polynomial with the stated exact coefficients and analytic remainder?

The completed 21-point calculation applies to the displayed polynomial; its connection to the charge configuration depends on this derivation.

Source record and provenance

Nine-page arXiv v1 source. The locked record shows no journal reference or referee disposition.

Selected and maintained by Material Shift as protocol research. No external sponsor or author involvement is recorded for this case.

Retrieved .

Preservation
The source is pinned to arXiv v1. A public Wayback snapshot was found, but it has not been hash-compared with the locked docket file.
Record provenance
Material Shift transcribed the displayed polynomial and ran its own exact checker. This is not author-supplied code, independent specialist verification, peer review, or a referee report.
Author contact
No author outreach, correspondence, response, or author approval is recorded for this docket. Contact the record steward.

Version and record history

  1. check reproduced

    2026-07-29-maxwell-conjecture

    Exact 21-point and Hessian classification of the displayed Lemma 2 polynomial reproduced locally; charge-model derivation and counterexample remain open

  2. record provenance

    2026-07-29-maxwell-conjecture

    Record classified as queued; source preservation, route provenance, and author-contact state published

  3. new docket

    2026-07-29-maxwell-conjecture

    arXiv v1 source locked and five examination gates mapped; no gate run

Statuses describe evidence collected for a defined scope. They are not peer-review decisions, publication recommendations, or certificates of mathematical truth.