Maximal green sequences for the Chekhov–Shapiro Aₙ-quiver and for its double, and a deck involution in the doubled Donaldson–Thomas transformation

The Chekhov-Shapiro Aₙ-quiver of arXiv:2601.18636v3 Section 3 as an integer exchange matrix on n(n-1)/2 vertices, built from the Proposition 3.3.4 adjacency of the main cycles: at n = 4 it has exactly the twelve arrows the source draws (1->2, 1->5, 2->3, 2->6, 3->4, 3->5, 4->1, 4->6, 5->2, 5->4, 6->1, 6->3), no arrow between the two vertices of the innermost length-2 cycle (the two arrows of that cycle cancel), and no multiple arrows; and the cycle mutation of equation (3.4) produces exactly the source's own words tau₁ = mu₁mu₂mu₄mu₃mu₂mu₁ and tau₂ = mu₆mu₅

omega₀ = (tau₁ tau₂... tauₘ)ᵐ has exactly n²(n-2)/2 mutations (plus m² label switches) on the n(n-1)/2 vertices of the Aₙ-quiver: 16, 72, 192, 400, 720, 1176, 1792 mutations on 6, 15, 28, 45, 66, 91, 120 vertices at n = 4, 6, 8, 10, 12, 14, 16

omega₀ is a maximal green sequence for the Aₙ-quiver at n = 4, 6, 8, 10: every one of its 16 / 72 / 192 / 400 mutations is at a green vertex (the corresponding column of the C-matrix is nonnegative), the exchange matrix returns unchanged, and the final C-matrix is exactly -I, so by not_greenₒf_final no green vertex remains and the sequence is maximal

omega₀ is a maximal green sequence for the A₁2-quiver: all 720 mutations of (tau₁... tau₆)⁶ on the 66-vertex Chekhov-Shapiro quiver are at green vertices, the exchange matrix returns, and C^(omega₀) = -I. The first even n past the data arXiv:2601.18636v3 reports, and a new confirming case of its conjecture that omega₀ is a maximal green sequence for every even n

omega₀ is a maximal green sequence for the A₁4-quiver: all 1176 mutations of (tau₁... tau₇)⁷ on the 91-vertex Chekhov-Shapiro quiver are at green vertices, the exchange matrix returns, and C^(omega₀) = -I

omega₀ is a maximal green sequence for the A₁6-quiver: all 1792 mutations of (tau₁... tau₈)⁸ on the 120-vertex Chekhov-Shapiro quiver are at green vertices, the exchange matrix returns, and C^(omega₀) = -I

Negative controls for the green-sequence criterion at n = 6: (a) transposing the first two mutations of omega₀ – same multiset of steps, different word – puts a mutation on a red vertex AND loses both C = -I and the quiver; (b) deleting the last step keeps every mutation green but loses C = -I, so no proper prefix of omega₀ is already reddening; (c) running omega₀ twice returns the quiver but is not green and does not end at -I, which is maximality witnessed as a failure; (d) after the first mutation of omega₀ vertex 0 is red, so the greenness test is not vacuously true; (e) the exchange matrix is not zero, and the two vertices of the innermost cycle of A₄ really are non-adjacent

MEASUREMENT: outside Lean, the same check runs to n = 30 – for every even n from 4 to 30 all n²(n-2)/2 mutations of omega₀ are green, C^(omega₀) = -I and the quiver returns; the largest case (n = 30, 435 vertices, 12600 mutations) takes 13 s of one core. No c-vector is ever sign-mixed during any of these runs, the empirical form of the sign-coherence theorem the source quotes as Theorem 7.1.6

omega₀ is a maximal green sequence for the A₁8-quiver: all 2592 mutations of (tau₁... tau₉)⁹ on the 153-vertex Chekhov-Shapiro quiver are at green vertices, the exchange matrix returns, and C^(omega₀) = -I

omega₀ is a maximal green sequence for the A₂0-quiver: all 3600 mutations of (tau₁... tau₁0)¹0 on the 190-vertex Chekhov-Shapiro quiver are at green vertices, the exchange matrix returns, and C^(omega₀) = -I

The round invariant of omega₀ = wᵐ, w = tau₁ tau₂... tauₘ: after r rounds the nonpositive (red) columns of the C-matrix are exactly the vertices of the innermost r main cycles of the Aₙ-quiver, the nonnegative (green) columns are exactly the rest, every entry of Cᵣ is -1, 0 or 1, and Cᵣ has exactly n(n-1)/2 negative entries – at every round r and every n = 4, 6, 8, 10, 12, 14, 16. So one round of w reddens the innermost still-green cycle and leaves the others green; at n = 10 the red-column counts over the six rounds are 0, 5, 15, 25, 35, 45. Stated for the C-matrix itself at n = 6, n = 12 and n = 16

The doubled Aₙ-quiver of arXiv:2601.18636v3 (definition at main.tex:1631, main cycles at Proposition 3.4.1,:1664) as an integer exchange matrix on n(n-1) vertices, for BOTH parities, together with the word s₀ = (s₁... sₘ)ᵐ of Remark 7.3.5 (:4667) built from the cycle mutation (3.4) of:1208 and the definition of sᵢ at:1676: the model reproduces all three doubled quivers the source draws – 12 arrows for the doubled A₃ (:1639), 50 and 78 for the doubled A₅ and A₆ (Figure 15,:1477-:1627) – arrow for arrow, in the source's own vertex names U_(i,j) / tilde-U_(i,j); and dnv n = n(n-1), mutCount (s0 n) = n(n-1)² for 3 <= n <= 18

For every n from 3 to 16, both parities: the C-matrix of s₀ = (s₁... sₘ)ᵐ on the doubled Aₙ-quiver is -P_g, where g is the fixed-point-free deck involution of:1696 that exchanges U_(i,j) and tilde-U_(i,j) (and is an automorphism of the doubled quiver), and it is NOT -I. So under the source's own Definition 7.1.3 (:4432, 'if the C-matrix is exactly -I and the underlying quiver is preserved') the word s₀ of Remark 7.3.5 is a reddening sequence but NOT a cluster DT-transformation, while s₀ followed by g is one. Equivalently s₀^*(Cᵢ) = (tilde-Cᵢ)⁻¹ and s₀^*(tilde-Cᵢ) = Cᵢ⁻¹, not the two identities the remark displays

s₀ = (s₁... sₘ)ᵐ is a MAXIMAL GREEN SEQUENCE for the doubled Aₙ-quiver at the ODD values n = 3, 5, 7, 9, 11, 13, 15: every one of its n(n-1)² mutations is at a green vertex, the quiver comes back, and the seed it ends at has no green vertex, so it cannot be extended. The standard Aₙ-quiver admits no reddening sequence at all for odd n (the source's Theorem 7.3.7,:4728); its double admits a maximal green sequence

s₀ = (s₁... sₘ)ᵐ is a MAXIMAL GREEN SEQUENCE for the doubled Aₙ-quiver at the EVEN values n = 4, 6, 8, 10, 12, 14, 16, where sₘ = fₘ is the cycle mutation over the merged innermost cycle of length n that carries both U_(m,.) and tilde-U_(m,.) (Proposition 3.4.1,:1664): every one of its n(n-1)² mutations is green, the quiver returns, and no vertex of the final seed is green

The round invariant of s₀ = wᵐ, w = s₁... sₘ, on the doubled Aₙ-quiver: after r rounds the nonpositive (red) columns of Cᵣ are exactly the vertices of the innermost r LEVELS – a level being BOTH copies of one doubled main cycle, or (for even n, as the innermost level) the merged cycle – the nonnegative columns are exactly the rest, every entry of Cᵣ is -1, 0 or 1, and Cᵣ has exactly n(n-1) negative entries at every r. The innermost r levels are j: 2n(m-r) <= j, so every level has 2n vertices, at every r and for BOTH parities, and the red-column counts run 0, 2n, 4n,..., n(n-1). Checked at n = 5, 6, 7, 8, 9, 10 and stated for the C-matrix itself at n = 7, 8, 10

Negative controls for the doubled word, at n = 6 AND n = 7 (one of each parity), reported as the triple (every mutation green, final C = -I, quiver returned): (a) replacing sₘ by the OTHER parity's recipe of:1676 gives (false, false, true); (b) deleting the last mutation of s₀ g gives (true, false, false), so no proper prefix already works; (c) running s₀ g twice gives (false, false, true), maximality as a failure; (d) after the first mutation of s₀, vertex 0 is red, so the greenness test is not vacuous; (e) the quiver is nonempty and simple and is a genuine 2:1 cover – eps_(0,1) = 1, eps_(1,0) = -1, eps_(0,6) = 0 for the two copies of X_(1,1); (f) the raw s₀ gives (true, false, true)

omega₀ is a maximal green sequence for the A₂2-quiver: all 4840 mutations of (tau₁... tau₁1)¹1 on the 231-vertex Chekhov-Shapiro quiver are at green vertices, the quiver comes back, the C-matrix is exactly -I, and therefore no vertex of the final seed is green

omega₀ is a maximal green sequence for the A₂4-quiver: all 6336 mutations of (tau₁... tau₁2)¹2 on the 276-vertex Chekhov-Shapiro quiver are green, the quiver returns, and the C-matrix is exactly -I

PROSE: for EVERY n >= 3 and every ordering of the cycle index sets allowed by the source's Proposition 3.4.1 and equation (3.4), the C-matrix of s₀ on the doubled Aₙ-quiver is not -I. Argument: by Theorem 3.3.1 (:1214) tau_J does not depend on the order of its mutations as a cluster transformation, and by Theorem 7.1.2 (:4419) the C-matrix is the tropicalisation of that transformation read in the final labelling; so two orderings of s₀ give C-matrices differing by a column permutation pi which is a product of transpositions (j_(N-1), j_N), each supported inside a single cycle-mutation index set J. For n >= 3 the doubled quiver has at least one doubled main cycle, and for such a cycle J is one copy, which g maps to the other copy, disjoint from J. So pi preserves J setwise while g does not, g o pi is never the identity, and C = -P_(g o pi) is never -I

MEASUREMENT: outside Lean the doubled check runs to n = 20 – for every n from 3 to 20, both parities, all n(n-1)² mutations of s₀ are green, the quiver returns, C = -P_g, g is an automorphism of the doubled quiver, the C-matrix after relabelling by g is -I, and no c-vector is ever sign-mixed during any run; the whole sweep is 12 s of one core in C. Independently, a from-scratch reconstruction of the standard Aₙ-quiver from the Fock-Goncharov triangle quiver (vertices i+j+k = n minus the three corners; arrows w - v in (0,1,-1),(1,-1,0),(-1,0,1) with half-arrows along the boundary; forget Z_(0,i,n-i); amalgamate Z_(i,0,n-i) with Z_(n-i,i,0); relabel by the cycle order at:980) reproduces Defs.arrows n exactly at n = 4, 6, 8, 10, 12, and gives for ODD n = 2m+1 the innermost-cycle chords X_(m,j) -> X_(m,j+m) on top of the main cycle