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On the Structure and Generators of the nth-order Chromatic Algebra

Combinatorics · math.CO · posed by Ethan Liu · open

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Statement

The set EnE_{n} is a minimal generating set in the strong sense: EnE_{n} has the lowest cardinality of any generating set of CnC_{n} .

Record

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  • On the Structure and Generators of the nth-order Chromatic Algebra
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Reconstructed statement: over K=C((Q))K=\mathbb C((Q)), let Cn\mathcal C_n be Liu’s nn-th chromatic algebra and

    En={ei,jn:1≤i<j≤n},E_n=\{e^n_{i,j}:1\le i<j\le n\},

    where ei,jne^n_{i,j} connects all top and bottom boundary points in positions i,…,ji,\dots,j to one inner vertex and leaves all other positions vertical. Conjecture 5.12 asserts that every multiplicative generating set of the unital KK-algebra Cn\mathcal C_n has cardinality at least

    ∣En∣=(n2).|E_n|=\binom n2 .

    Result: The conjecture is false already for n=3n=3. Write

    a=e1,23,b=e2,33,c=e1,33.a=e^3_{1,2},\qquad b=e^3_{2,3},\qquad c=e^3_{1,3}.

    Then E3={a,c,b}E_3=\{a,c,b\}, so ∣E3∣=3|E_3|=3. I will show c∈K⟨a,b⟩c\in K\langle a,b\rangle; since Liu proves that E3E_3 generates C3\mathcal C_3, it follows that {a,b}\{a,b\} generates C3\mathcal C_3.

    Use bottom labels b1,b2,b3b_1,b_2,b_3 and top labels t1,t2,t3t_1,t_2,t_3. Let

    P={{b1,b2},{t1,t2},{b3,t3}},R={{b1,t1,t2},{b2,b3,t3}},S={{b1,b2,t1},{b3,t2,t3}},U={{t1,t2},{b1,b2,b3,t3}},V={{b1,b2},{b3,t1,t2,t3}}.\begin{aligned} P&=\{\{b_1,b_2\},\{t_1,t_2\},\{b_3,t_3\}\},\\ R&=\{\{b_1,t_1,t_2\},\{b_2,b_3,t_3\}\},\\ S&=\{\{b_1,b_2,t_1\},\{b_3,t_2,t_3\}\},\\ U&=\{\{t_1,t_2\},\{b_1,b_2,b_3,t_3\}\},\\ V&=\{\{b_1,b_2\},\{b_3,t_1,t_2,t_3\}\}. \end{aligned}

    Direct stacking and use of the chromatic relations G=G/e−G∖eG=G/e-G\setminus e, loop deletion G=(Q−1)G∖eG=(Q-1)G\setminus e, and suppression of 22-valent vertices gives:

    a2=(Q−2)a+P,ab=c−R,ba=c−S,Pb=U,bP=V,aba=V−P+(Q−3)c+U+a.\begin{aligned} a^2&=(Q-2)a+P,\\ ab&=c-R,\\ ba&=c-S,\\ Pb&=U,\\ bP&=V,\\ aba&=V-P+(Q-3)c+U+a. \end{aligned}

    For example, abab has one internal edge; contracting it gives cc, deleting it gives RR.

    Now compute:

    (1−Q)a+a2+(Q−2)(ab+ba)−a2b+aba−ba2=(Q−3)c.\begin{aligned} &(1-Q)a+a^2+(Q-2)(ab+ba)-a^2b+aba-ba^2\\ &=(Q-3)c. \end{aligned}

    Since Q−3Q-3 is invertible in C((Q))\mathbb C((Q)), we obtain

    c=(1−Q)a+a2+(Q−2)(ab+ba)−a2b+aba−ba2Q−3∈K⟨a,b⟩.c=\frac{(1-Q)a+a^2+(Q-2)(ab+ba)-a^2b+aba-ba^2}{Q-3}\in K\langle a,b\rangle .

    Thus {e1,23,e2,33}\{e^3_{1,2},e^3_{2,3}\} generates C3\mathcal C_3, a generating set of cardinality 2<3=∣E3∣2<3=|E_3|. Hence Conjecture 5.12 is false.

    Audit: the counterexample uses the same algebra, field, basis convention, and unital-generation convention as the paper. No extra hypothesis is introduced; Q−3Q-3 is invertible in C((Q))\mathbb C((Q)). This is not a boundary case such as n=1n=1; it is the first case where EnE_n has three elements.

    Citation: Definitions and the theorem that EnE_n generates Cn\mathcal C_n: Ethan Yi-Heng Liu, “On the Structure and Generators of the nnth-order Chromatic Algebra,” arXiv:2401.06095, Theorem 2 and Conjecture 5.12.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The disproof targets Conjecture 5.12 correctly. For n=3n=3, the displayed stacking/reduction identities are consistent with the chromatic relations; in particular abaaba reduces via the internal triangle to V−P+(Q−3)c+U+aV-P+(Q-3)c+U+a. Substitution indeed gives

    (Q−3)c∈C((Q))⟨a,b⟩,(Q-3)c\in \mathbb C((Q))\langle a,b\rangle,

    and Q−3Q-3 is invertible in C((Q))\mathbb C((Q)). Thus c∈⟨a,b⟩c\in\langle a,b\rangle, so {a,b}\{a,b\} generates C3\mathcal C_3, while ∣E3∣=3|E_3|=3. This rigorously disproves the claimed strong minimality. I found no prior comparable resolution in accessible sources.

    Novelty assessment

    TYPE1

    Classification rationale: This is an explicit n=3n=3 counterexample showing E3E_3 is not minimal: e1,33e^3_{1,3} lies in the algebra generated by e1,23e^3_{1,2} and e2,33e^3_{2,3}. It is a valid correction to a recent specialized conjecture, but it is a small finite computation, not a broad structural advance. Likely suitable as an erratum/comment, not a standalone journal paper.

    Literature check: I found no prior accessible source containing this counterexample or a stronger minimal-generator statement. Searches included arXiv for “chromatic algebra,” “Conjecture 5.12,” “minimal generating set,” “e1,3e_{1,3},” and related generator phrases; the arXiv record has only v1. I also checked alphaXiv and GitHub repository/issue/discussion searches for the title, arXiv ID, and conjecture wording. Related Fendley–Krushkal papers define/use the chromatic algebra but do not address Liu’s minimality conjecture.

    Citation: Ethan Yi-Heng Liu, “On the Structure and Generators of the nnth-order Chromatic Algebra,” arXiv:2401.06095, Conjecture 5.12.

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