A Colored-Path Determinant Counterexample

Abstract

A seven-vertex colored path pair refutes Conjecture 5.3 on determinant equality.

Definition 1.1 (Labelled colored paths).

$$\forall V\in \mathrm{Type},\forall E\in \mathrm{Type},\forall m\in \mathrm{Nat},\mathrm{ColoredPath}\left(V,E,m\right)=\{vertexColor:\mathrm{Fin}\left(m\right)\to V;edgeColor:\mathrm{Fin}\left(m-1\right)\to E\}$$

Formalization. D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.ColoredPath (✓ std3).

Citation. Hannah Göbel, Pratik Misra (2025). Linear relations of colored Gaussian cycles. DOI: 10.48550/arXiv.2506.23936. URL: https://arxiv.org/abs/2506.23936v1.

Commentary.

A colored path on m vertices has a vertex-color function Fin(m) to V and an edge-color function Fin(m-1) to E. The distinct types V and E keep the vertex- and edge-color sets disjoint. A zero-based vertex index i represents the one-based position i.val+1.

Definition 1.2 (Generic concentration matrix).

$$\forall V\in \mathrm{Type},\forall E\in \mathrm{Type},\forall m\in \mathrm{Nat},\forall P\in \mathrm{ColoredPath}\left(V,E,m\right),\forall i\in \mathrm{Fin}\left(m\right),\forall j\in \mathrm{Fin}\left(m\right),\mathrm{entry}\left(\mathrm{concentration}\left(P\right),i,j\right)=\{\begin{array}{ll}X\left(\mathrm{inl}\left(\mathrm{vertexColor}\left(P,i\right)\right)\right)& i=j\\ X\left(\mathrm{inr}\left(\mathrm{edgeColor}\left(P,Fin.mk\left(\mathrm{val}\left(i\right)\right)\right)\right)\right)& \mathrm{val}\left(i\right)+1=\mathrm{val}\left(j\right)\\ X\left(\mathrm{inr}\left(\mathrm{edgeColor}\left(P,Fin.mk\left(\mathrm{val}\left(j\right)\right)\right)\right)\right)& \mathrm{val}\left(j\right)+1=\mathrm{val}\left(i\right)\\ 0& \mathrm{otherwise}\end{array}$$

Formalization. D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.concentration (✓ std3).

Citation. Hannah Göbel, Pratik Misra (2025). Linear relations of colored Gaussian cycles. DOI: 10.48550/arXiv.2506.23936. URL: https://arxiv.org/abs/2506.23936v1.

Commentary.

For a colored path P, concentration(P) is a matrix over the integer multivariable polynomial ring on V+E. Its diagonal entry (i,i) is X(inl(vertexColor(P,i))). Its adjacent entries (i,i+1) and (i+1,i) are X(inr(edgeColor(P,i))); every other entry is zero. Thus equal colors produce equal matrix entries exactly as in color constraints (1)-(3) on printed page 5. The coefficients are integers, and the coefficient embedding from integers into reals is injective, so polynomial equality here is the same formal identity as over the reals.

Definition 1.3 (Path reflection).

$$\forall V\in \mathrm{Type},\forall E\in \mathrm{Type},\forall m\in \mathrm{Nat},\forall P\in \mathrm{ColoredPath}\left(V,E,m\right),(\forall i\in \mathrm{Fin}\left(m\right),\mathrm{vertexColor}\left(\mathrm{reflect}\left(P\right),i\right)=\mathrm{vertexColor}\left(P,Fin.mk\left(m-1-\mathrm{val}\left(i\right)\right)\right))\wedge (\forall i\in \mathrm{Fin}\left(m-1\right),\mathrm{edgeColor}\left(\mathrm{reflect}\left(P\right),i\right)=\mathrm{edgeColor}\left(P,Fin.mk\left(m-2-\mathrm{val}\left(i\right)\right)\right))$$

Formalization. D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.reflect (✓ std3).

Citation. Hannah Göbel, Pratik Misra (2025). Linear relations of colored Gaussian cycles. DOI: 10.48550/arXiv.2506.23936. URL: https://arxiv.org/abs/2506.23936v1.

Commentary.

Reflection preserves color labels while reversing positions. On zero-based indices it sends a vertex index i to m-1-i.val and an edge index i to m-2-i.val. In one-based positions these are r to m+1-r for vertices and r to m-r for edges.

Definition 1.4 (Theorem 3.6 configuration).

$$\forall V\in \mathrm{Type},\forall E\in \mathrm{Type},\forall m\in \mathrm{Nat},\forall P\in \mathrm{ColoredPath}\left(V,E,m\right),\forall Q\in \mathrm{ColoredPath}\left(V,E,m\right),(Config36\left(P,Q\right))\iff ((\forall i\in \mathrm{Fin}\left(m-1\right),\mathrm{edgeColor}\left(P,i\right)=\mathrm{edgeColor}\left(Q,i\right))\wedge \left((\forall i\in \mathrm{Fin}\left(m\right),\forall j\in \mathrm{Fin}\left(m\right),((\mathrm{mod}\left(\mathrm{val}\left(i\right),2\right)=0)\wedge (\mathrm{mod}\left(\mathrm{val}\left(j\right),2\right)=0))\Rightarrow ((\mathrm{vertexColor}\left(P,i\right)=\mathrm{vertexColor}\left(P,j\right))\wedge (\mathrm{vertexColor}\left(Q,i\right)=\mathrm{vertexColor}\left(Q,j\right))))\wedge \left((\forall i\in \mathrm{Fin}\left(m\right),\forall j\in \mathrm{Fin}\left(m\right),((\mathrm{mod}\left(\mathrm{val}\left(i\right),2\right)=1)\wedge (\mathrm{mod}\left(\mathrm{val}\left(j\right),2\right)=1))\Rightarrow ((\mathrm{vertexColor}\left(P,i\right)=\mathrm{vertexColor}\left(P,j\right))\wedge (\mathrm{vertexColor}\left(Q,i\right)=\mathrm{vertexColor}\left(Q,j\right))))\wedge (\forall i\in \mathrm{Fin}\left(m\right),\forall j\in \mathrm{Fin}\left(m\right),((\mathrm{mod}\left(\mathrm{val}\left(i\right),2\right)=0)\wedge (\mathrm{mod}\left(\mathrm{val}\left(j\right),2\right)=1))\Rightarrow ((\mathrm{vertexColor}\left(P,i\right)=\mathrm{vertexColor}\left(Q,j\right))\wedge (\mathrm{vertexColor}\left(P,j\right)=\mathrm{vertexColor}\left(Q,i\right))))\right)\right))$$

Formalization. D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.Config36 (✓ std3).

Citation. Hannah Göbel, Pratik Misra (2025). Linear relations of colored Gaussian cycles. DOI: 10.48550/arXiv.2506.23936. URL: https://arxiv.org/abs/2506.23936v1.

Commentary.

Theorem 3.6 states the following clauses verbatim: “(1) lambda({p_i, p_{i+1}}) = lambda({q_i, q_{i+1}}) for every i in {1, 2, …, m-1}, (2) lambda(p_1) = lambda(p_{2n+1}) and lambda(q_1) = lambda(q_{2n+1}) for every n in {1, 2, …, m/2-1} (odd vertices have the same color), (3) lambda(p_2) = lambda(p_{2n}) and lambda(q_2) = lambda(q_{2n}) for every n in {1, 2, …, m/2} (even vertices have the same color), (4) lambda(p_1) = lambda(q_2) and lambda(p_2) = lambda(q_1).” Config36 writes the monochromatic clauses symmetrically over all positions. Under those clauses, its two cross equalities are equivalent to clause (4). One-based odd position means val(i) mod 2 = 0; mod denotes natural-number remainder.

Definition 1.5 (Theorem 3.8 configuration).

$$\forall V\in \mathrm{Type},\forall E\in \mathrm{Type},\forall m\in \mathrm{Nat},\forall P\in \mathrm{ColoredPath}\left(V,E,m\right),\forall Q\in \mathrm{ColoredPath}\left(V,E,m\right),(Config38\left(P,Q\right))\iff ((\forall i\in \mathrm{Fin}\left(m\right),\forall j\in \mathrm{Fin}\left(m\right),((\mathrm{mod}\left(\mathrm{val}\left(i\right),2\right)=0)\wedge (\mathrm{mod}\left(\mathrm{val}\left(j\right),2\right)=0))\Rightarrow ((\mathrm{vertexColor}\left(P,i\right)=\mathrm{vertexColor}\left(P,j\right))\wedge \left((\mathrm{vertexColor}\left(P,i\right)=\mathrm{vertexColor}\left(Q,j\right))\wedge (\mathrm{vertexColor}\left(Q,i\right)=\mathrm{vertexColor}\left(Q,j\right))\right)))\wedge \left((\forall i\in \mathrm{Fin}\left(m\right),(\mathrm{mod}\left(\mathrm{val}\left(i\right),2\right)=1)\Rightarrow (\mathrm{vertexColor}\left(P,i\right)=\mathrm{vertexColor}\left(Q,i\right)))\wedge \left((\forall i\in \mathrm{Fin}\left(m-1\right),\forall j\in \mathrm{Fin}\left(m-1\right),((\mathrm{mod}\left(\mathrm{val}\left(i\right),2\right)=0)\wedge (\mathrm{mod}\left(\mathrm{val}\left(j\right),2\right)=1))\Rightarrow (\mathrm{edgeColor}\left(P,i\right)=\mathrm{edgeColor}\left(Q,j\right)))\wedge (\forall i\in \mathrm{Fin}\left(m-1\right),\forall j\in \mathrm{Fin}\left(m-1\right),((\mathrm{mod}\left(\mathrm{val}\left(i\right),2\right)=0)\wedge (\mathrm{mod}\left(\mathrm{val}\left(j\right),2\right)=1))\Rightarrow (\mathrm{edgeColor}\left(P,j\right)=\mathrm{edgeColor}\left(Q,i\right)))\right)\right))$$

Formalization. D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.Config38 (✓ std3).

Citation. Hannah Göbel, Pratik Misra (2025). Linear relations of colored Gaussian cycles. DOI: 10.48550/arXiv.2506.23936. URL: https://arxiv.org/abs/2506.23936v1.

Commentary.

Theorem 3.8 states the following clauses verbatim: “(1) lambda(p_1) = lambda(p_{2n+1}) = lambda(q_1) = lambda(q_{2n+1}) for every n in {1, 2, …, (m-1)/2}, (2) lambda(p_{2n}) = lambda(q_{2n}) for every n in {1, 2, …, (m-1)/2}, (3) lambda({p_i, p_{i+1}}) = lambda({q_j, q_{j+1}}) and lambda({p_j, p_{j+1}}) = lambda({q_i, q_{i+1}}), for all odd i in {1, 2, …, m} and all even j in {1, 2, …, m}.” The printed range in clause (3) reaches m, but the edge {p_m,p_{m+1}} does not exist. Config38 therefore quantifies over the actual edge positions 1 through m-1. One-based parity is expressed by val(i) mod 2, with zero for odd positions and one for even positions.

Definition 1.6 (Conjecture 5.3).

$$(claim)\iff (\forall V\in \mathrm{Type},\forall E\in \mathrm{Type},\forall m\in \mathrm{Nat},\forall P\in \mathrm{ColoredPath}\left(V,E,m\right),\forall Q\in \mathrm{ColoredPath}\left(V,E,m\right),(\det \left(\mathrm{concentration}\left(P\right)\right)=\det \left(\mathrm{concentration}\left(Q\right)\right))\Rightarrow ((Q=P)\vee \left((Q=\mathrm{reflect}\left(P\right))\vee \left(((\mathrm{Even}\left(m\right))\wedge ((Config36\left(P,Q\right))\vee \left((Config36\left(\mathrm{reflect}\left(P\right),Q\right))\vee \left((Config36\left(P,\mathrm{reflect}\left(Q\right)\right))\vee (Config36\left(\mathrm{reflect}\left(P\right),\mathrm{reflect}\left(Q\right)\right))\right)\right)))\vee ((\mathrm{Odd}\left(m\right))\wedge ((Config38\left(P,Q\right))\vee \left((Config38\left(\mathrm{reflect}\left(P\right),Q\right))\vee \left((Config38\left(P,\mathrm{reflect}\left(Q\right)\right))\vee (Config38\left(\mathrm{reflect}\left(P\right),\mathrm{reflect}\left(Q\right)\right))\right)\right)))\right)\right)))$$

Formalization. D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.claim (✓ std3).

Citation. Hannah Göbel, Pratik Misra (2025). Linear relations of colored Gaussian cycles. DOI: 10.48550/arXiv.2506.23936. URL: https://arxiv.org/abs/2506.23936v1.

Commentary.

Conjecture 5.3 reads verbatim: “Let P and Q be two colored paths on m vertices with det(K_P) = det(K_Q). (1) If m is even, then one of the following conditions holds: • Q is identical to P, • Q is a reflection of P, • P and Q satisfy the color configuration stated in Theorem 3.6 or is a reflection of the same. (2) Similarly, if m is odd, then one of the following conditions holds: • Q is identical to P, • Q is a reflection of P, • P and Q satisfy the color configuration stated in Theorem 3.8 or is a reflection of the same.” The displayed claim reads the final reflection phrase in its widest form: neither path, P alone, Q alone, or both paths may be reflected.

Theorem 1.7 (Conjecture 5.3 is false).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.result (✓ std3). ∎

Resolves. Problems/gobel-misra-colored-path-determinant-refutation (refuted) by D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.result.

Source. Repository-derived.

Commentary.

Take one vertex color a, two edge colors u and v, and m=7. The edge sequences P=(u,v,v,u,u,v) and Q=(u,u,v,u,v,v) have the common determinant a^7-3a^5u^2-3a^5v^2+2a^3u^4+6a^3u^2v^2+2a^3v^4-2au^4v^2-2au^2v^4. They are neither identical nor reflections. The four Config38 variants all fail their edge clause: for (P,Q), e_1(P)=u differs from e_6(Q)=v; for (reflect(P),Q), e_1(reflect(P))=v differs from e_2(Q)=u; for (P,reflect(Q)), e_3(P)=v differs from e_6(reflect(Q))=u; and for the double reflection, e_1(reflect(P))=v differs from e_6(reflect(Q))=u.

References

• Truth anchor: D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.ColoredPath
• Truth anchor: D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.Config36
• Truth anchor: D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.Config38
• Truth anchor: D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.claim
• Truth anchor: D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.concentration
• Truth anchor: D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.reflect
• Truth anchor: D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.result