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arxiv: 2412.07680 · v2 · submitted 2024-12-10 · ✦ hep-th · math.AG

BPS Dendroscopy on Local mathbb{P}¹times mathbb{P}¹

Bruno Le Floch , Boris Pioline , Rishi Raj This is my paper

Pith reviewed 2026-05-23 06:59 UTC · model grok-4.3

classification ✦ hep-th math.AG
keywords BPS statesscattering diagramattractor flow treesPi-stabilitylocal F0Split Attractor Flow Tree ConjectureCalabi-Yau threefoldmodular group action
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The pith

The scattering diagram on local F0 determines all BPS indices from attractor indices via ray intersection consistency.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs the scattering diagram for BPS states in type II strings on the total space of the canonical bundle over P1 times P1, first near the orbifold point and at large volume, then along the physical Pi-stability slice. Consistency conditions at intersections of rays fix the full set of BPS indices in terms of the attractor indices on the initial rays. This yields a sketch of the Split Attractor Flow Tree Conjecture, valid in a restricted range of central charge phase, extending earlier results on local P2 while accounting for an extra mass parameter and ramification points.

Core claim

In the scattering diagram for local F0, an arrangement of rays in the space of stability conditions marks loci where BPS states of given charge and central charge phase exist; the consistency of the diagram when rays intersect determines all BPS indices in terms of the attractor indices carried by the initial rays, allowing a sketch of the proof of the Split Attractor Flow Tree Conjecture for a restricted range of the central charge phase.

What carries the argument

The scattering diagram, an arrangement of real codimension-one rays in stability condition space whose intersections enforce consistency relations on BPS indices.

If this is right

  • All BPS indices on the Pi-stability slice are fixed once the attractor indices on the initial rays are known.
  • The scattering diagram carries an action of a Z^4 extension of the modular group Gamma0(4).
  • The construction combines the quiver description near the orbifold point with the large-volume limit to cover the physical slice.
  • The extra mass parameter and ramification points on the Pi-stability slice complicate the diagram relative to the local P2 case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same ray-consistency method could be applied to other non-compact toric Calabi-Yau threefolds once their attractor indices are known.
  • The presence of ramification points may indicate that the modular action on the stability slice is richer than in simpler geometries.
  • If the restricted-phase sketch extends without new obstructions, it would give a systematic way to compute the full BPS spectrum from a small set of attractor data.

Load-bearing premise

The consistency arguments and conjecture sketch hold only for a restricted range of the central charge phase.

What would settle it

A mismatch between BPS indices computed directly at a point in the restricted phase range and the values fixed by enforcing consistency on the constructed scattering diagram would falsify the determination claim.

Figures

Figures reproduced from arXiv: 2412.07680 by Boris Pioline, Bruno Le Floch, Rishi Raj.

Figure 1
Figure 1. Figure 1: Initial rays for the large volume scattering diagram in the (s, t) plane at m = 1/2 scattering of these rays. The scattering diagram along the slice Π is much more complicated, since there are initial rays for all Γ1(3) images of OP2 , emanating from every τ = p q ∈ Q with (p, q) = 1, q ̸= 0 mod 3. This includes in particular all homological shifts OP2 (m)[k] at τ = m. However, as long as |ψ| is less than … view at source ↗
Figure 2
Figure 2. Figure 2: Quiver associated to the Ext-exceptional collection (1.3), sometimes known as phase II or orbifold quiver. This is accompanied by the quartic super￾potential W in (3.4). γ1 γ2 γ3 γ4 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scattering diagram of the orbifold quiver for m = 1/2, keeping rays with height P i Ni ≤ 8. branch while m keeps track of the 5D gauge coupling, in units of the circle radius. The study of the mirror curve in Appendix A shows that the Coulomb branch is a double cover of H/Γ0(4), ramified over a point τB determined in terms of m by the condition (1.5) J4(τB) = −8 cos πm, J4(τ ) := 8 +  η(τ ) η(4τ ) 8 wher… view at source ↗
Figure 4
Figure 4. Figure 4: Fundamental domain F (in blue) for the action of Γ0(4) on the upper half-plane, and its images under τ 7→ τ−1 4τ−3 and τ 7→ 3τ−1 4τ−1 . The cusps at τ = (i∞, 0, 1 2 ) correspond to the large volume limit z = 0, conifold point z = 1 4(1+√ λ) 2 and dual conifold point z = 1 4(1− √ λ) 2 , while the branch point τB corresponds to z = ∞. The dotted lines show the cuts joining τB and its images to τ = 1 2 along … view at source ↗
Figure 5
Figure 5. Figure 5: (Putative) boundary of the space of geometric stability conditions, for m = 0.4 + 0.3i. The red solid line is the locus ℑT = 0. Along the blue, yellow, green and orange lines, the central charge associated to O(0, 0), O(1, 0), O(0, 1) (or rather its image O(1, 0) through the cut) and O(1, 1) become real and negative, destabilizing skyscraper sheaves. The green and orange lines intersect at a point τb where… view at source ↗
Figure 6
Figure 6. Figure 6: Initial rays emitted from τ = 0 (left) and τ = 1 2 (right), for m = 1/2 and ψ = 0. Rays with positive and negative rank r are shown in magenta and blue, respectively. with m = 0 and (1.7) C(τ, 0) = η(τ ) 4 η(2τ ) 6 η(4τ ) 4 = 1 − 4 X n≥1 n 2χ4(n)q n 1 − q n where χ4(n) is the Dirichlet character modulo 4, equal to ±1 for n = ±1 mod 4 and 0 otherwise. In the fundamental domain and its translates F + Z, the … view at source ↗
Figure 7
Figure 7. Figure 7: Scattering diagram DΠ m,ψ along the Π-stability slice for m = 0.4, ψ = 0. Only primary scatterings are shown. -1.0 -0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Scattering diagram DΠ m,ψ for m = 0.4, ψ = 0.4 below the first critical value ψcr ≃ 0.545. Only primary scatterings are shown. DΛ ψ,m as τ → i∞, to the orbifold scattering diagram Do m as τ → τB, and by requiring invariance under the monodromy group Γ. In particular, it includes an infinite set of rays emanating from every conifold point τ = p q with (p, q) = 1, q ̸= 0 mod 4. The structure of the initial r… view at source ↗
Figure 9
Figure 9. Figure 9: Scattering diagram DΠ m,ψfor m = 0.4, ψ = 0.68 between the first two critical values ψcr ≃ 0.545 and ψ˜ cr ≃ 0.832. Only primary scatterings are shown. -1.0 -0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Scattering diagram DΠ m,ψ for m = 0.4, ψ = 0.98 above ψ˜ cr ≃ 0.832. Only primary scatterings are shown. rays manage to escape to i∞ (see Figs. 8-10). In particular, for ψ bigger than both11 (1.9) ψcr = arctan  ℑV(m) ℜV(m)  , ψ˜ cr = arctan ℑ[V˜(m) + i(1 − m)] ℜ[V˜(m) + i(1 − m)]! , where V(m) and V˜(m) are the ‘quantum volumes’ defined in (A.84) and (A.107), the rays R(γi) associated to the simple obje… view at source ↗
Figure 11
Figure 11. Figure 11: Embedding of the orbifold scattering diagram into the Π-stability scattering diagram around τB − 1 = i−1 4 , for m = 1/2 and ψ = 0.72. The initial rays R(γi) for the exceptional collection (5.6) are shown in green, blue, red and orange. The scatterings of {γ1, γ2} and {γ2, γ3} take place on one sheet, and those of {γ3, γ4} and {γ4, γ1} on another sheet, glued along the branch cut in red dashed line. 0.0 0… view at source ↗
Figure 12
Figure 12. Figure 12: Contour lines of s = ℑTD ℑT for m = 1/2. our computations, in particular evaluating the central charge on the principal sheet over any point in the Poincaré upper half-plane, is provided in a publicly available repository. 13 Acknowledgements: We are grateful to Pierrick Bousseau, Andrea Brini, Cyril Closset, Pierre Descombes, Horia Magureanu, Jack Huizenga, Jan Manschot and Thorsten Schimannek for useful… view at source ↗
Figure 13
Figure 13. Figure 13: Lower bound δDLP(η, a, b) on the discriminant of non-exceptional H-stable sheaves for η = 1, obtained by retaining exceptional sheaves with rank up to 19. Fig. adapted from [33, [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Approximate shape of δ(s, η, M1) for η = 1, M1 = 1/4. The red dots indicate the corresponding lowest values arising from exceptional stable bundles. C := C( 3 4 , 1 4 ), C˜ := C( 1 4 , 1 4 ), and extend them elsewhere by acting with the autoequivalences STO(0,0) and O(1, 1) ⊗ −, which generate the group Γ0(4). As for the inequality ℑT > 0, the q-series expansion (A.66) with m = 0 shows that it holds at th… view at source ↗
Figure 15
Figure 15. Figure 15: For m = 0 (hence λ = 1), the contours with fixed s = ℑTD ℑT in the strip τ1 ∈ (0, 1), shown in orange, all converge to τ = 1 2 . The contours with fixed (and real) t = √ 2w − s 2 approach τ = 0 and τ = 1 when t → 0 (shown in dashed line), but stay away from τ = 1 2 . In the region above the dashed line, Π stability is related to large volume stability by GL(2, R) +. -2 -1 1 2 s -0.5 0.5 1.0 w [PITH_FULL_… view at source ↗
Figure 16
Figure 16. Figure 16: Image of the fundamental domain F and its translates into the (s, w) plane for m = 0. The dashed line is the parabola w = 1 2 s 2 , the red vertical segments are the images of the half-circles on the boundary of the fundamental domain and its translates and the segments in magenta correspond to the values of (s, w) around the dual conifolds τ ∈ 1 2 +Z. The blue curve denotes the (continuous part of the) l… view at source ↗
Figure 17
Figure 17. Figure 17: For m = 1 2 , the contours with fixed s = ℑTD ℑT in the strip τ1 ∈ (0, 1), shown in orange, converge either to τ = 1+i 4 or τ = 3+i 4 . The contours with fixed (and real) t = q 2w − (s − 1 4 ) 2 stay well above the conifold and branch points In the region above the dashed line t = 0, Π stability is related to large volume stability by GL(2, R) +. conditions (A.70) implies that s = ℜτ along the vertical li… view at source ↗
Figure 18
Figure 18. Figure 18: Image of the fundamental domain F and its translates into the (s, w) plane, shown for m = 1/4. The dashed line is the parabola w = 1 2 (s − m 2 ) 2 , the red vertical segments are the images of the half-circles on the boundary of the fundamental domain and its translates and the segments in magenta correspond to the values of (s, w) on an infinitesimal circle around the branch points. The blue curve denot… view at source ↗
Figure 19
Figure 19. Figure 19: Contours with fixed s (orange) or fixed t = √ 2w1 (black) in the τ half-plane, for m = 0.4 + 0.3i. In the region above the dashed line, corresponding to t = 0, Π stability is related to large volume stability by GL(2, R) +. The blue, yellow, green, dark orange and red lines delimit (putatively) the domain of geometric stability conditions, as in [PITH_FULL_IMAGE:figures/full_fig_p023_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Values of w1 (defined in (2.30)) along the various components of the boundary of the space of geometric stability conditions, including either side of the locus ℑT = 0, compared to the sufficient lower bound −2/(9η) (dashed line) following from δ(s, η, M1) > 4 9 . The value of m and the color coding for the boundary components is the same as in Figs. 5 and 19. dual20 to the Ext-exceptional collection (3.2… view at source ↗
Figure 21
Figure 21. Figure 21: Scattering diagram Do µ for the orbifold quiver at µ = 1/4. The shaded area corresponds to the generalized ‘collimation chamber’ (3.14). 1 ′ 2 ′ 3 ′ 4 ′ [PITH_FULL_IMAGE:figures/full_fig_p027_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Quiver associated to the Ext-exceptional collection (3.21), sometimes known as phase I quiver. This is accompanied by the cubic superpotential W in (3.23). 4 autoequivalence MZ4 , but it is invariant under fiber-base duality. The Chern vectors of the objects E i and dual objects Ei are (3.22) γ ′1 = [1, 0, 0, 0] γ ′2 = [1, 1, 0, 0] γ ′3 = [1, 0, 1, 0] γ ′4 = [1, 1, 1, 1] γ ′ 1 = [1, 0, 0, 0] = γ1 γ ′ 2 = … view at source ↗
Figure 23
Figure 23. Figure 23: Scattering diagram DI µ for the phase I quiver at µ = 1/2, keeping rays with height P i N′ i ≤ 8, with max(n1, n2) ≤ 4 at each intersection. Up to truncation effects, it coincides with the scattering diagram for the orbifold quiver in [PITH_FULL_IMAGE:figures/full_fig_p029_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Scattering diagram in the (x, y) plane at m = 1 2 , including only primary scattering of initial rays. For ψ = 0, the real part of T coincides with the coordinate x, and m1 with µ, so we set T = x + it, m = µ + im2. The coordinate s = ℑTD ℑT determining the heart is s = x + µ 2 + m2x t , so coincides with x + m 2 when m is real. The rays R0(γ) satisfy (4.5) ℜZ(γ) = r(t 2 − x 2 ) + (d1 + d2 − µr)x + µd2 − … view at source ↗
Figure 25
Figure 25. Figure 25: Initial rays for the large volume scattering diagram in the (x, t) plane at m = 1/4. The shaded blue and red regions correspond to the validity of the Ext-exceptional collection (3.21) tensored with O(k, k) and O(k, k − 1), respectively. starts at (min(d1 − µ, d2), 0) and bends to the left, while the ray R0(O(d1, d2))[1] starts at (max(d1 − µ, d2), 0) and bends to the right. In the (x, y) plane, the rays … view at source ↗
Figure 26
Figure 26. Figure 26: Images of the phase I quiver under mutations µ1 and µ2µ3. whose first node is the same {O(0, −1), GV (0, 1, −1)} for any m > 0. For m < 0, we have the same trees upon setting d = ⌊−m⌋ and applying fiber/base duality. 4.2. m ∈ Z. When m is an integer, the initial rays at x ∈ Z and x ∈ Z−Fr(m) coalesce and the structure of the scattering diagram becomes more complicated. In particular, assuming m = 0 withou… view at source ↗
Figure 27
Figure 27. Figure 27: Initial rays in the (x, t) plane for m = 0. The orange rays are obtained from the phase I collection by iterating the mutations µ1 and µ2µ3. The dense yellow rays arise from the scattering of R(O(1, 0)) and R(O(0, −1)[1]) with |⟨γ1, γ2⟩| = 4. -1.0 -0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 [PITH_FULL_IMAGE:figures/full_fig_p035_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Initial rays for the large volume scattering diagram DΛ m,0 for fixed m = 0.4 + 0.3i, and their primary scatterings. The red and magenta lines denote the walls of marginal stability for the binary trees {O(1, 0), O(0, −1)[1]} and {O(1, 1), O(0, −1)[1]}. 5.1. Uncovering the scattering diagram. By the same arguments as in [1], we know that initial rays can only be emitted from conifold points along the boun… view at source ↗
Figure 29
Figure 29. Figure 29: Scattering diagram DΠ m,ψ for m = 0, ψ = 0.01, keeping only primary scatterings. Note that rays related by fiber-based duality are indistinguishable, with further degeneracies when r = 0. As ψ → 0, the intersection point between O(0, −1)[1], O(−1, 0)[1] and O(0, 0) coalesces with τ = 0, similar to [PITH_FULL_IMAGE:figures/full_fig_p037_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Vψ and V˜ ψ as a function of ψ, for m = 1/4. The values ψ + cr and ψ˜+ cr correspond to the transitions where Vψ = 0 and V˜ ψ = 1 − µ, respectively. For m = 1/2, V˜ ψ = Vψ + 1 2 and the two critical values coincide. §3.1, the rays R(γ1) and R(γ2) no longer come from infinity, but start at τ = 0 and τ = − 1 2 (approached on the principal sheet), which map to (5.8) γ1 : (u, v) = µ 2 + Vψ, 1−µ 2  , γ2 : (u,… view at source ↗
Figure 31
Figure 31. Figure 31: Left: Domain of validity of the quiver descriptions based on the collections (5.6), (5.9), for m = 0.3. The solid green and red lines are the boundaries of ∆ψ and ∆˜ ψ and correspond to straight lines in the (x, y) plane. Right: the same regions in the (x, y) plane, together with some of their translates; the overlaps correspond to different stability conditions that map to the same values of x, y coordin… view at source ↗
Figure 32
Figure 32. Figure 32: Top: Trees contributing to the structure sheaf index Ω([1, 0, 0, 0]) as function of m (real) and ψ. Bottom: Trees in τ -plane for m = 0.35, varying ψ. Note that the tree T4 does not arise for this value. 0  𝓞1, 𝓞1, 0 1  𝓞-1, 01, 𝓞 3 2  𝓞-1, 01, 𝓞0, -21, 2 𝓞0, -1 -1  𝓞1, 01, 𝓞2, 0 -2  𝓞2, 0, 𝓞1, 2, 2 𝓞1, 11 -3 1 4 1 2 3 4 1 m -π 2 -π 3 -π 6 0 π 6 π … view at source ↗
Figure 33
Figure 33. Figure 33: Top: Trees contributing to the D2 index Ω([0, 1, 0, 0]) as function of m (real) and ψ. Bottom: Trees in τ -plane for m = 0.5, varying ψ [PITH_FULL_IMAGE:figures/full_fig_p044_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Convergence regions of the expansions around τ = i∞, 0, 1, 1 2 (half￾plane and disks intersected with the fundamental domain, shaded in blue, orange, green, red, respectively), and domains where each of these expansions are used, as well as expansion around images of the branch point τB (black dot, connected to τ = 1 2 by the cut). Different values of m require different subsets of the expansions. Left: m… view at source ↗
read the original abstract

BPS states in type II string compactified on a Calabi-Yau threefold can typically be decomposed as moduli-dependent bound states of absolutely stable constituents, with a hierarchical structure labelled by attractor flow trees. This decomposition is best understood from the scattering diagram, an arrangement of real codimension-one loci (or rays) in the space of stability conditions where BPS states of given electromagnetic charge and fixed phase of the central charge exist. The consistency of the diagram when rays intersect determines all BPS indices in terms of the `attractor indices' carried by the initial rays. In this work we study the scattering diagram for a non-compact toric CY threefold known as local $\mathbb{F}_0$, namely the total space of the canonical bundle over $\mathbb{P}^1\times \mathbb{P}^1$. We first construct the scattering diagram for the quiver, valid near the orbifold point, and for the large volume slice, valid when both $\mathbb{P}^1$'s have large (and nearly equal) area. We then combine the insights gained from these simple limits to construct the scattering diagram along the physical slice of $\Pi$-stability conditions, which carries an action of a $\mathbb{Z}^4$ extension of the modular group $\Gamma_0(4)$. We sketch a proof of the Split Attractor Flow Tree Conjecture in this example, albeit for a restricted range of the central charge phase. Most arguments are similar to our early study of local $\mathbb{P}^2$ [arXiv:2210.10712], but complicated by the occurence of an extra mass parameter and ramification points on the $\Pi$-stability slice.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper constructs the scattering diagram for BPS states on the non-compact toric Calabi-Yau threefold local F0 (total space of the canonical bundle over P1 x P1). It first builds the diagram near the orbifold point from the quiver and at large volume for nearly equal large P1 areas, then combines these to the Pi-stability slice carrying a Z^4 extension of Gamma0(4). Diagram consistency at ray intersections is claimed to determine all BPS indices from the attractor indices on initial rays. A sketch of the Split Attractor Flow Tree Conjecture is provided, but only for a restricted range of central charge phase, with complications from an extra mass parameter and ramification points relative to the local P2 case studied in arXiv:2210.10712.

Significance. If the restricted-range sketch holds, the work extends BPS dendroscopy methods to a geometry with an extra mass parameter and modular action, giving concrete support for the conjecture via explicit ray-intersection consistency in a toric example beyond local P2. Strengths include the explicit construction combining quiver and large-volume limits on the physical slice and the handling of the Z^4 extension of Gamma0(4). This advances the understanding of attractor flow trees and BPS bound-state hierarchies in Calabi-Yau threefolds.

major comments (1)
  1. [Abstract] Abstract: The central claim that diagram consistency at ray intersections determines all BPS indices from attractor indices is load-bearing for the Split Attractor Flow Tree Conjecture sketch, yet the manuscript explicitly restricts the sketch to a limited range of central charge phase owing to ramification points on the Pi-stability slice and the extra mass parameter; this restriction means the determination is not shown to hold uniformly across the full physical slice.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying this point about the scope of the claims in the abstract. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that diagram consistency at ray intersections determines all BPS indices from attractor indices is load-bearing for the Split Attractor Flow Tree Conjecture sketch, yet the manuscript explicitly restricts the sketch to a limited range of central charge phase owing to ramification points on the Pi-stability slice and the extra mass parameter; this restriction means the determination is not shown to hold uniformly across the full physical slice.

    Authors: We agree that the abstract presents the general principle of scattering diagram consistency while the explicit sketch and determination of indices via ray intersections is restricted to a limited range of central charge phase. The restriction arises precisely from the extra mass parameter and ramification points on the Π-stability slice, as stated in the manuscript. To remove any potential ambiguity, we will revise the abstract to tie the determination claim explicitly to the restricted range in which the Split Attractor Flow Tree Conjecture is sketched. revision: yes

Circularity Check

1 steps flagged

Minor self-citation to prior local P2 study for sketch of conjecture

specific steps
  1. self citation load bearing [Abstract]
    "Most arguments are similar to our early study of local P2 [arXiv:2210.10712], but complicated by the occurence of an extra mass parameter and ramification points on the Pi-stability slice."

    The sketch of the Split Attractor Flow Tree Conjecture is presented as relying on similarity to the authors' own prior paper for the core arguments, with the present work only adapting for added complications; while not fully load-bearing (new constructions are supplied), this constitutes a minor self-citation dependency for the central claim.

full rationale

The paper constructs scattering diagrams explicitly from the quiver near the orbifold point and the large-volume slice, then combines them along the Pi-stability slice using the consistency condition at ray intersections to determine BPS indices from attractor indices. This is a direct mathematical construction from stability conditions and the Split Attractor Flow Tree framework. The sole self-citation notes that most arguments are similar to the authors' prior local P2 work but does not serve as load-bearing justification; the current paper supplies the new elements (extra mass parameter, ramification points, Z^4 extension of Gamma_0(4)) and restricts the sketch to a limited central-charge phase range. No reductions by construction, fitted inputs renamed as predictions, or self-definitional steps appear.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based on abstract only; limited visibility into full set of assumptions.

free parameters (1)
  • extra mass parameter
    Mentioned as an additional complication in the construction compared to local P2.
axioms (1)
  • domain assumption Consistency of the scattering diagram when rays intersect determines all BPS indices from attractor indices
    Stated as the foundational principle for the decomposition.

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