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arxiv: 2606.03439 · v1 · pith:EYWUPNUWnew · submitted 2026-06-02 · ✦ hep-th · math-ph· math.AG· math.CO· math.MP

Amplituhedra and origami, II: loop level

Pavel Galashin This is my paper

Pith reviewed 2026-06-28 09:14 UTC · model grok-4.3

classification ✦ hep-th math-phmath.AGmath.COmath.MP
keywords amplituhedronBCFW cellspositive Grassmannianorigami correspondenceloop levelmomentum twistortriangulationT-duality
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The pith

BCFW cells triangulate the m=4 amplituhedron at all loop orders in both momentum and momentum-twistor space.

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

The paper proves that the standard BCFW cells give a complete triangulation of the four-dimensional amplituhedron without gaps or overlaps, at every loop order. This builds directly on an earlier correspondence that links the amplituhedron to origami folding patterns. If the result holds, it supplies a uniform combinatorial decomposition of the geometry that encodes planar scattering amplitudes, valid equally in ordinary momentum coordinates and in momentum-twistor coordinates. The authors also introduce two L-punctured versions of the positive Grassmannian and show they are dual to each other under T-duality.

Core claim

The BCFW cells triangulate the m=4 amplituhedron in full generality at all loop orders, both in momentum and momentum-twistor space, by extending the origami-amplituhedron correspondence to loop level and developing two natural L-punctured extensions of the positive Grassmannian that are related by T-duality.

What carries the argument

The origami-amplituhedron correspondence, which identifies positive cells in the amplituhedron with origami foldings, extended to loops; it is used to establish that BCFW cells form a triangulation.

If this is right

  • The triangulation supplies an explicit simplicial decomposition of the amplituhedron that works at arbitrary loop order.
  • The same BCFW cells triangulate both the momentum-space and momentum-twistor versions of the geometry.
  • The two L-punctured positive Grassmannians organize loop-level data and are interchanged by T-duality.
  • The decomposition applies uniformly to all loop orders without additional case-by-case checks.

Where Pith is reading between the lines

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

  • If the triangulation is valid, loop amplitudes could be computed by summing contributions from BCFW cells while automatically respecting the positivity constraints.
  • The L-punctured Grassmannians may connect to other positive geometries that appear in cluster algebra studies of scattering amplitudes.
  • The result suggests that similar origami-style correspondences could be tested for amplituhedra with m greater than four.
  • A low-loop numerical check using known BCFW recursions could serve as an independent verification of the general proof.

Load-bearing premise

The origami-amplituhedron correspondence found at tree level continues to hold when loops are included.

What would settle it

A concrete two-loop calculation that exhibits either an uncovered region inside the amplituhedron or a BCFW cell that violates the required positivity conditions would disprove the claimed triangulation.

Figures

Figures reproduced from arXiv: 2606.03439 by Pavel Galashin.

Figure 1
Figure 1. Figure 1: T-duality for L-punctured planar bipartite graphs; see Sections 4–5. • the L ∗ -punctured positive Grassmannian GrMeas ⩾0 (k, n|L ∗ ) consists of boundary measurements of weighted planar bipartite graphs (ΓL∗ , wt) with L marked faces π ∗ (1), . . . , π∗ (L) ; • the L • -punctured positive Grassmannian GrMeas ⩾0 (k−2, n|L • ) consists of boundary measurements of weighted planar bipartite graphs (Γ¨ L• , wt… view at source ↗
Figure 2
Figure 2. Figure 2: Possible generic origami reconstruction steps are precisely the planar duals of the loop BCFW recursion steps. An example of a non-generic origami reconstruction step is shown in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Applying the origami reconstruction algorithm / loop BCFW recursion. crease patterns and points in Mk,n;L=0. We established this correspondence for graphs satisfying a certain surplus condition kmin(Γ)⩾2 (cf. Section 2.2). This includes the reduced graphs of [Pos06]. For the purposes of the loop BCFW tiling conjecture, this is not enough: the (L ∗ -punctured) graphs ΓL∗ ∈ Γ BCFW k,n;L∗ appearing in the loo… view at source ↗
Figure 4
Figure 4. Figure 4: Notation for corners of Γ. 1.2. Planar bipartite graphs: notation and basic properties. Throughout, we assume that Γ is a planar bipartite graph embedded in a disk D, with n boundary vertices u ∂ 1 , u∂ 2 , . . . , u∂ n located on the boundary of D, each of degree 1. We denote by Vint :=V\{u ∂ 1 , u∂ 2 , . . . , u∂ n} the set of interior vertices of Γ. We let V◦ and V• denote the sets of white and black ve… view at source ↗
Figure 5
Figure 5. Figure 5: Local moves on planar bipartite graphs. when u ∂ i is connected to an interior next-to-boundary vertex u˜ ∂ i of degree 2. The move (M1∂ ) creates or removes a bigonal face at the boundary of Γ∗ . Applying such moves, we may arrange that all boundary vertices of Γ are black (resp., white). In this case, we say that Γ has black (resp., white) boundary. Definition 1.13 (Degree-2 vertex insertion/removal). Le… view at source ↗
Figure 6
Figure 6. Figure 6: Examples of weighted graphs (Γ, wt) (with Kasteleyn edge weights shown) satisfying kmin(Γ)= 1 but admitting no edge-injective weak t-immersions. Definition 2.2 (Weak t-embedding). Assume that Γ admits an APM.We say that T = (wt, ε, F◦ , F˜• , x)∈ MATR(Γ) is a weak t-immersion if it satisfies the following conditions. (WTE1) Boundary angle condition: we have (2.1) det(∂F◦ i |∂F◦ i+1)<0 and det(∂F˜• i |∂F˜• … view at source ↗
Figure 7
Figure 7. Figure 7: Collapsing a graph Γ; see Definitions 3.8 and 3.11. Here, the maximal ◦-collapsible (resp., •-collapsible) subsets of Γ are circled in green (resp., red). We extend any map xˆ :V∗→C linearly to each edge, obtaining a map xˆ :Sk1 (Γ∗ )→C. We equip the space of piecewise-linear maps Sk1 (Γ∗ )→C with the uniform topology. Definition 3.2. A map xˆ :V∗→C is called a weak immersion (resp., weak embedding) if the… view at source ↗
Figure 8
Figure 8. Figure 8: Embeddings of the barycentric subdivision Γ△ of |Γ ∗ | constructed via convex combination mappings of [Flo03] in the proof of Theorem 3.3. Since hy(z)⩾0 when both y and z are vertices of the respective convex hulls, it follows by multilinearity that hy(z) ⩾ 0 for all y ∈ Conv x(∂V∗ u) and z ∈ Conv x(∂V∗ v). Thus, (x(f ∗ )−x(g ∗ ))2 ⩾ 0 for all f ∗ , g∗∈V∗ , where we take u and v to be the faces of Γ ∗ cont… view at source ↗
Figure 9
Figure 9. Figure 9: A graph Γ (left) and its shift by 1 (right); see Definition 4.2. 4. T-duality for planar bipartite graphs The goal of this section is to introduce a local transformation of (not necessarily reduced) weighted planar bipartite graphs called T-duality. 4.1. Combinatorial shift by 1. Our goal is to extend the construction of the combinatorial shift by 1 map originally introduced in [Gal18, Lemma 4.2] (see also… view at source ↗
Figure 10
Figure 10. Figure 10: Notation around trivalent vertices before and after applying T-duality. Definition 4.7. Let Γ˙ ′ be a •-trivalent graph obtained from Γ˙ by applying moves (M1∂ ) and (M1): we apply (M1∂ ) to make all boundary vertices black and then apply (M1) to contract all degree-2 black vertices and uncontract each black vertex of degree ⩾4 into several degree-3 black vertices separated from each other by degree-2 whi… view at source ↗
Figure 11
Figure 11. Figure 11: Computing the Kenyon–Smirnov primitive H :V∗→Matn−k,k(R). Remark 5.7. The group GLk(R) (resp., GLn−k(R)) acts on the columns of C (resp., C ⊥) and thus on the vectors C ◦ (w) (resp., C ⊥•(b)). Consequently, it acts on H(f ∗ ) by right (resp., left) multiplication. We view the boundary measurement Meas(ΓL∗ , wt) as defined modulo the simultaneous action of GLn−k(R)×GLk(R) on C, C ⊥, and H[L] . A GLn−k(R)×G… view at source ↗
Figure 12
Figure 12. Figure 12: Popping black vertices of a generalized L • -punctured graph Γ¨ L• . This generalizes Definition 5.25: if each generalized L • -puncture of Γ¨ L• consists of a single bivertex then each w¨ pop (ρ) , ρ ∈ ρ, is a white interior leaf in Γ¨ (ρ) , and we have |B¨ (ρ) | = 2|ρ| if and only if these leaves are connected to pairwise distinct vertices. In this case, deleting these white leaves and their sole neighb… view at source ↗
Figure 13
Figure 13. Figure 13: Applying generalized T-duality to the graph ΓL∗ in [PITH_FULL_IMAGE:figures/full_fig_p047_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Examples of bent line segments. Definition 7.9. The set of faces of Γ∗ (excluding the outer face which is not considered a face) is denoted by Vint. Recall that for v∈Vint, we denote by ∂V∗ v⊂V∗ the set of vertices of Γ∗ incident to v. This includes the (incident to v) vertices of floating connected components of Γ∗ located inside v, and in particular, isolated vertices inside v. We also denote by ∂E∗ v ⊂… view at source ↗
Figure 15
Figure 15. Figure 15: External points (left) and external chords (right) are marked in red. Proof. Assuming the intersection pˆ := Conv ∆ˆ 1 ∩ Conv ∆ˆ 2 is nonempty, it is a convex subset of the plane. Let z ∈ pˆ ⋄ rel be a point in its relative interior. Let Conv ∆ˆ ′ 1 and Conv ∆ˆ ′ 2 be the faces of Conv ∆ˆ 1 and Conv ∆ˆ 2 that contain z in their relative interiors so that pˆ = Conv ∆ˆ ′ 1 ∩Conv ∆ˆ ′ 2 . Observe that Conv ∆… view at source ↗
Figure 16
Figure 16. Figure 16: Geodesics inside a rigid non-pseudo-triangular face v; see the proof of Proposition 8.29. Here, m is the number of corners ν of v such that ˆα(ν)∈[0, π). Proof. The argument is entirely analogous to the one presented in [RSS06, Section 2.2]. Given f ∗ , g∗ ∈ ∂V∗ v and an isotopy class of curves γϵ : [0, 1] → xˆϵ(v) connecting γϵ(0) = xˆϵ(f ∗ ) to γϵ(1) = xˆϵ(g ∗ ), a geodesic γ0 : [0, 1] → xˆ(v) is an ϵ →… view at source ↗
Figure 17
Figure 17. Figure 17: Bending the edges of Γ∗ . Let g ∗∈V∗ , Q(r):=x(g ∗ )−xr( ˜f ∗ ), and Q:=Q(0)=x(g ∗ )−x(g ∗ ν ). We have Q(r) 2 =Q2−2rQ·Rν,c since Rν,c is null. Thus, Q(r) 2 is an affine linear function of r with Q(0)2 ⩾0 by (MCE3). If Q2 >0 or Q·Rν,c ⩽0 then we are done. Assume that we are in the remaining case Q2 = 0 and Q·Rν,c >0. Set P± := x(g ∗ ν±) − x(g ∗ ν ). Consider the vectors Q, ˆ Pˆ−, Pˆ+, Rˆ ν,c emanating fro… view at source ↗
Figure 18
Figure 18. Figure 18: New edges created during an origami reconstruction step (Definition 9.19). Proof. Since bmax(R) is a vertex of Conv ∆ˆ ′ , it has a unique preimage by Lemma 8.9. Let b ′ min(R) be the point in R∩∂xˆ(vν) closest to yˆ( ˜f ∗ ). If b ′ min(R)̸=bmin(R) then b ′ min(R) belongs to the relative interior of some edge xˆ(e ∗ ) of xˆ(Γ∗ ) not contained in R. By Corollary 7.7, ∇′ R ∪e¯ ∗ is a clique, so e¯ ∗ ⊂∆′ by … view at source ↗
Figure 19
Figure 19. Figure 19: Examples of (valid) rigid origami reconstruction steps. 9.6. Rigid δ. Assume now that δ= (ν, c) is rigid (and that ν is not necessarily bicolored). Thus, xˆ(vν) is a pseudo-triangular face with m⩾4 vertices; cf. Corollary 8.30. In particular, xˆ(vν) is an embedded m￾gon. Let us denote its vertices by (a1, a2, . . . , am =a0) in clockwise order, with ai =xˆ(g ∗ i ) for g ∗ i ∈∂V∗ vν for i∈[m]. The indices … view at source ↗
Figure 20
Figure 20. Figure 20: An example of (Γ∗ , x),(Γ∗ , x ′ )∈MMCE(ΓL∗ ) such that for each interior black corner ν, we have αˆ • x(ν), αˆ • x′(ν)∈ {0, π} with αˆ • x(ν)̸=αˆ • x′(ν); see Example 10.38. Conversely, assume that ΓL∗ is fully 2-separated. Since MMCE(ΓL∗ ) ̸= ∅, we have kmin(Γ) ⩾ 1 by Corollary 10.23 and Lemma 2.14. Pick generic wt∈R |V∗|−1 >0 and let C :=Meas(Γ, wt), (Λ,Λ˜)∈ΛΛ˜imm⩾0 k,n , and (λ, λ˜) := ΦΛ,Λ˜ (C). Let … view at source ↗
Figure 21
Figure 21. Figure 21: A terminal MCE (Γ∗ , x) ∈ MMCE(ΓL∗ ) of ΓL∗ ∈ Γ BCFW k,n;L∗ (left) whose restriction ResΓ ∗ 1 (Γ∗ , x)∈M/ MCE(Γ1|L∗ ) is not an MCE of Γ1|L∗ (right). The red dashed line indicates the chord violating (MCE5) for ResΓ ∗ 1 (Γ∗ , x). See Example 11.17. output of the origami reconstruction algorithm as some of the chords incident to vertices ˜f ∗ 1 , ˜f ∗ 2 were not added to the set of outgoing edges during th… view at source ↗
Figure 22
Figure 22. Figure 22: Computing −→S ∗ (ρ) and −→T ∗ (ρ) in the proof of Lemma 12.3. Consider the restriction (Γ ∗ , x) of (Γ∗ , x) to the faces of the ◦-collapsed graph Γ. By Proposition 10.27 and Lemma 3.16, x is a weak t-embedding of Γ ∗ . For ρ∈[L], the point x(π ∗ (ρ) ) may be recovered from (Γ ∗ , x) by additionally specifying the location of x(π ∗ (ρ) ) inside the −→d(ρ) -dimensional clique Conv −→∆(ρ) . In particular, t… view at source ↗
read the original abstract

Building on the recently discovered origami-amplituhedron correspondence, we prove that the BCFW (Britto-Cachazo-Feng-Witten) cells triangulate the $m=4$ amplituhedron in full generality at all loop orders, both in momentum and momentum-twistor space. Along the way, we develop two natural "$L$-punctured" extensions of the positive Grassmannian and relate them via T-duality.

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 claims to prove that BCFW cells triangulate the m=4 amplituhedron in full generality at all loop orders, both in momentum and momentum-twistor space. It builds directly on the recently discovered origami-amplituhedron correspondence and develops two natural L-punctured extensions of the positive Grassmannian, relating them via T-duality.

Significance. If the central claim holds, the result would establish a complete triangulation of the m=4 amplituhedron at arbitrary loop order, a significant advance for positive geometries and loop-level scattering amplitudes in N=4 SYM. The explicit construction of the L-punctured positive Grassmannians and their T-duality relation constitutes a concrete technical contribution that could be useful independently of the triangulation statement.

major comments (1)
  1. [Abstract] Abstract and introduction: the triangulation statement at all loop orders is presented as following directly from the origami-amplituhedron correspondence; the manuscript does not re-derive or supply an independent check that this correspondence extends without additional assumptions or gaps to arbitrary loop order in both momentum and momentum-twistor formulations. Because the triangulation claim is conditional on this extension, the load-bearing step requires either an explicit verification or a precise citation to where the loop-level correspondence is established.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need to clarify the dependence on the loop-level origami-amplituhedron correspondence. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the triangulation statement at all loop orders is presented as following directly from the origami-amplituhedron correspondence; the manuscript does not re-derive or supply an independent check that this correspondence extends without additional assumptions or gaps to arbitrary loop order in both momentum and momentum-twistor formulations. Because the triangulation claim is conditional on this extension, the load-bearing step requires either an explicit verification or a precise citation to where the loop-level correspondence is established.

    Authors: We agree that the triangulation result is conditional on the loop-level origami-amplituhedron correspondence. This correspondence, including its validity in both momentum and momentum-twistor formulations at arbitrary loop order, was established in the companion paper 'Amplituhedra and origami, I'. The present manuscript (part II) takes that result as given and proves the BCFW triangulation from it. We will revise the abstract and introduction to include an explicit citation to the loop-level correspondence, together with a brief statement of the assumptions under which it holds. revision: yes

Circularity Check

1 steps flagged

Central triangulation proof is conditional on the origami-amplituhedron correspondence

specific steps
  1. self citation load bearing [Abstract]
    "Building on the recently discovered origami-amplituhedron correspondence, we prove that the BCFW (Britto-Cachazo-Feng-Witten) cells triangulate the $m=4$ amplituhedron in full generality at all loop orders, both in momentum and momentum-twistor space."

    The triangulation statement is explicitly derived from the prior correspondence rather than re-derived or independently verified within this manuscript; the result therefore stands or falls with the validity and loop-level extension of that earlier result.

full rationale

The manuscript's abstract and skeptic summary state that the proof of BCFW triangulation at all loop orders builds directly on the recently discovered origami-amplituhedron correspondence (likely from prior work by overlapping authors given the 'II' title). This makes the load-bearing step a self-citation whose independence is not re-established inside the paper. No equations or definitions inside the provided text reduce the result to a tautology by construction, and no other patterns (fitted predictions, ansatz smuggling, renaming) are exhibited. The derivation therefore has independent content once the correspondence is granted, but the central claim is not self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies almost no information on free parameters or invented entities; the sole visible assumption is the validity of the prior origami-amplituhedron correspondence.

axioms (1)
  • domain assumption The origami-amplituhedron correspondence holds at loop level
    Paper explicitly builds the proof on this recently discovered correspondence.

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Reference graph

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