Six is minimal

Summary

In a simple mathematical model of a neural network, a group of active neurons can settle into a stable pattern. A natural conjecture said that such a group must be a clique: every neuron in the group points to every other, with a further condition on the inactive neurons outside it. Geneson’s six-neuron counterexample showed that the conjecture was false. The remaining small question was sharp: could five neurons already break the rule?

This release answers no, under the model’s stated legal-parameter and nondegeneracy assumptions. Six is minimal. The proof covers every directed graph on five or fewer vertices and the whole allowed parameter region—not just a grid of simulated networks.

The candidate also gives exact rational parameters for the known six-neuron graph, describes its complete full-support stability region, and constructs larger examples on 6, 11, 16, … neurons. These larger examples use different parameters at different sizes. Their stability margin shrinks; the result is not a claim of uniform robustness.

Summary for specialists

For a nondegenerate combinatorial threshold-linear network (CTLN) with uniform positive drive and legal weights

the minimum size of a stable nonclique fixed-point support is six. Stable supports of size at most five are exactly target-free cliques in this setting. The lower-bound certificate covers 9,846 isomorphism classes, accounting for 1,052,741 labelled loopless digraphs of orders one through five.

The upper witness is Geneson’s existing graph, reconstructed exactly at $\varepsilon=1/12$, $\delta=11/120$. The graph itself is not a new contribution. For every integer $r\ge0$, the candidate gives a globally nondegenerate stable nonclique full support on $5r+6$ vertices at

No construction at every integer size, common parameter pair for all sizes, or uniform positive decay margin is asserted.

Technical account

Convert stability into exact polynomial obstructions

Let $N_{ij}=1$ when the directed edge from neuron $j$ to neuron $i$ is missing, with zero diagonal. Define

The active Jacobian is $-\varepsilon K$. Positive fixed-point coordinates and eigenvalues of $K$ with positive real parts are separate requirements. Writing $u=\beta-2$ and $v=ut-1$ turns the entire legal parameter region into $u>0,v>0$.

For each small graph, the checker reconstructs determinant, Cramer and characteristic polynomials and verifies an obstruction on this whole quadrant. Some cases fail positivity; others fail necessary stability conditions. The final exceptional quintics are eliminated by an explicit square identity and coefficient bounds. The manuscript explains the reductions and the certificate inventory; finite numerical scans do not supply the lower bound.

Grow the seed without hiding its weak direction

The larger family is a nonuniform clique expansion of the six-neuron seed, with component sizes $(1,3r+1,2r+1,1,1,1)$. It uses established composite-graph and simply-added-split ideas from Curto, Geneson and Morrison. The new certificate concerns this particular expansion, its five-class weighted quotient, and explicit parameters. The quotient is not an ordinary five-neuron CTLN, so it does not contradict the lower bound.

Positive-coefficient polynomials prove quotient stability for all $r\ge0$. A parity argument establishes global nondegeneracy. Meanwhile, $5r+1$ transverse eigenvalues of the physical Jacobian are

They approach zero. The parameter schedule is a sufficient construction, not an optimal or necessary one.

Locate the seed’s stable region

For the six-neuron seed, the candidate gives the exact full-support phase region

where $u_\star$ is a specified sextic root in $(0.166355027849,0.166355027850)$ and $T(u)$ is the unique positive root of the displayed quartic Hurwitz determinant. The package gives exact root isolators. This classifies full-support stability; it does not assert global nondegeneracy at every point or establish a nonlinear Hopf bifurcation.

Evidence, assurance and limitations

The package combines written reductions with exact, standard-library Python calculations. The finite checker reconstructs polynomials by signed-permutation determinant expansions rather than trusting the generator’s Newton-identity output. The supplied regression suite compares three arithmetic routes and requires nine corrupted certificates to fail. Shared low-level utilities remain part of the trust boundary.

The supplied AI-generated review contains separately written audit programs covering the finite enumeration, family, phase and appendix identities. Its recorded input hash matches the submitted archive. This is useful implementation diversity, not authenticated human specialist review or unaffiliated reproduction. The revision response records every review point, including the stronger prior-art comparison and shrinking-margin qualification.

This remains an unrefereed candidate. No end-to-end proof-assistant verification, exhaustive novelty guarantee, biological validation, global convergence theorem or classification of all larger CTLNs is claimed. Positive drive, legal weights and the stated nondegeneracy boundary matter.

Relationship to earlier work

Geneson’s six-neuron counterexample supplies the sharp upper witness. Curto, Geneson and Morrison’s stable-fixed-point paper supplies earlier small-support results and composite-graph machinery; Section 2.2 and Lemma 2.6 address simply-added splits and inherited eigenvalues, while Section 5 treats composite graphs. Their 2019 fixed-point paper supplies the broader competitive-network framework.

The contribution assessed here is the exact exclusion through five over the full legal domain, together with the stated family and seed-phase certificates. Clique expansion and inherited transverse eigenvalues are not presented as new general principles. The source audit is bounded and cannot settle exhaustive priority.

Who should care, and why

Why the problem matters

A small counterexample overturns a rule; a sharp minimum explains where that rule stops working. Knowing that every smaller support obeys the clique classification restricts the search for exceptional stable patterns and gives a precise boundary for graph-based reasoning about these networks.

The connection to neuroscience is through an idealised mathematical model. Nothing here measures neural tissue or shows that biological networks realise these parameter choices.

How to inspect or reproduce the recorded checks

Start with the manuscript, then the AI index and versioned source and Zenodo archive.

Python 3.9 or later, standard library only:

python3 code/verify_all.py --regenerate --read-only

The command runs in a temporary copy, regenerates the finite certificates, checks the family and phase results, runs mutation regressions, and requires byte-identical mathematical certificate output. It leaves the downloaded release unchanged. Assertions must be enabled: python3 -O is intentionally rejected. Compare the output with REPLAY_RECEIPT.md; a replay is not a formal proof of the prose reductions.

The most valuable next projects

1. Independent reconstruction. Rebuild the all-parameter exclusions and check the dynamical interpretation without importing the supplied utilities.
2. Common-parameter families. Determine whether unbounded stable nonclique supports can be obtained at one fixed legal parameter pair. The present schedule does not answer this.
3. Robustness and expansion conditions. Characterise which multiplicity choices preserve stability and whether a uniformly positive physical stability margin is possible.

What is in the evidence package

The banner and synthetic audio explain the result. They do not add scientific evidence.

@misc{ctlnsixisminimal2026,
  title        = {Six is minimal},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.23089682},
  url          = {https://doi.org/10.5281/zenodo.23089682},
  version      = {1.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/ctln-six-is-minimal/}
}