Amplitudes as Canonical Forms: Positive Geometries, Residue Trees, and the Conjectural Bridge to Motivic Coactions as the Amplitude Layer of Reconstructed Spacetime

1 Introduction

1.1 The reconstruction thesis in the amplitude regime

The governing perspective of this project is that quantum gravity is not, in the first instance, the quantization of material objects placed in a pre-existing spacetime. It is the reconstruction of spacetime, matter, causality, and measurement from invariant representation structures. Different regimes of the theory instantiate this idea with different invariants. The present paper concerns the regime of perturbative scattering amplitudes, where the invariant is unusually sharp and the reconstruction unusually complete.

The conventional route to an amplitude begins with a Lagrangian, expands the path integral into Feynman diagrams, and sums. Poles arise because internal propagators can go on shell; the residues at those poles factorize into products of lower-point amplitudes because the theory is unitary; and the whole construction respects locality because the Lagrangian is built from local fields. In this telling, locality and unitarity are inputs, and the analytic structure of the amplitude is an output that one verifies after the fact.

The positive-geometry program of Arkani-Hamed and collaborators inverts this order of explanation . One begins with a geometric object: a region $X_{\geq 0}$ inside a real projective (or Grassmannian) variety, carved out by inequalities, whose boundary decomposes recursively into lower-dimensional regions of the same type. To such an object one attaches a unique rational top-form $\Omega$, its canonical form, singled out by the demand that its poles be simple, sit precisely on the boundary components of $X_{\geq 0}$, and reproduce recursively—on each boundary—the canonical form of that boundary. The claim, established in specific theories, is that this $\Omega$ is the amplitude. No Lagrangian, no diagram sum, and no path integral enter the definition of $\Omega$. Locality (the location of the poles) and unitarity (the factorization on them) are read off from the boundary combinatorics of $X_{\geq 0}$. They are theorems about a polytope, not axioms about a field theory.

This is, across the twelve regimes catalogued in the project’s knowledge base, the single most complete example of an observable reconstructed directly from a geometric/ representation-theoretic invariant. In the theories where the correspondence is established, the equation “amplitude $=$ canonical form” holds without remainder. That is a strong statement, and we are careful throughout to say exactly where it holds and where it does not.

1.2 The core invariant: the residue tree

Fix a positive geometry $(X,\,X_{\geq 0},\,\Omega_{X})$ of dimension $d$. Its boundary $\partial X_{\geq 0}$ is a union of codimension-one positive geometries $C_i$; each $C_i$ has its own boundary, a union of codimension-two pieces; and so on down to points. The resulting rooted tree— $X$ at the root, its facets as children, their facets as grandchildren, terminating at $0$-dimensional strata—is what we call the residue tree of $X$, written $\mathsf{R}(X)$. It is the same data as the face poset of $X_{\geq 0}$, organized by the residue recursion. Physically it is the complete catalogue of factorization channels of the amplitude: each root-to-leaf path is a maximal sequence of compatible on-shell conditions, i.e. a Feynman-like tree, and each internal node is a factorization onto a product of sub-amplitudes.

The residue tree is the invariant this paper takes as primary. Every claim we make is, in the end, a claim about $\mathsf{R}(X)$: that it exists and is finite; that the canonical form is reconstructed from it; that its leaves are the vertices of the polytope and its root-to-leaf paths are the cubic trees of the theory; and—conjecturally—that it is isomorphic to a second tree of entirely different origin, the coaction tree of the amplitude’s period.

1.3 The conjectural bridge: residue tree $=$ coaction tree

There is a parallel decomposition of amplitudes that comes not from geometry but from number theory. A large class of amplitudes, once integrated, evaluate to periods: multiple polylogarithms and multiple zeta values in the mixed-Tate case. Such periods carry a motivic coaction $\Delta$, a coproduct-like map that peels off, layer by layer, the transcendental content of the function . Iterating $\Delta$ produces a rooted tree—the coaction tree—whose nodes are the successive “pieces” of the period and whose leaves are the primitive weight-one logarithms.

The conjecture that organizes this regime, item 1 of the project’s open-problem queue, is that these two trees are the same: $$\mathsf{R}(X) \;\cong\; \mathsf{C}(\Omega_X),$$ the geometric residue tree of the positive geometry is isomorphic to the motivic coaction tree of its canonical form (suitably completed to a transcendental function). The statement is verified in special low-multiplicity, low-weight, mixed-Tate cases and is supported structurally by the observation that cluster coordinates on the Grassmannian simultaneously index the boundary strata of the geometry and the letters of the polylogarithm . The strongest concrete bridge to date is the coaction of Abreu–Britto–Duhr–Gardi–Matthew, built on twisted (co)homology, in which both the integrand and the integration contour are positive geometries and the coaction’s contour factor runs literally over the boundary strata of the contour . We take that construction as the technical heart of our worked comparison in 6.

1.4 Epistemic discipline

We use the project’s three-valued status calculus. A warrant is $\mathsf{S}$ (standard mathematics or established mathematical physics), $\mathsf{H}$ (strong heuristic physical representation), or $\mathsf{P}$ (speculative ontological extension), with $\mathsf{S} < \mathsf{H} < \mathsf{P}$. Composite labels such as $\mathsf{S/H}$ record a claim that is mathematically standard but heuristic as physics. Composition is monotone and worst-component-wins: chaining regimes can never raise the composite status above its weakest link. Every displayed result carries a tag in brackets, e.g. [ S ], [ S/H ], [ H ], [ H/P ], [ P ]. This is not decoration. The uniqueness of canonical forms is mathematics [ S ]; that a given canonical form computes a physical S-matrix is a theorem only for special theories and is otherwise a hope [ S/H ]; and the residue-tree/ coaction-tree identification is, in full generality, a conjecture we do not know how to prove [ H/P ]. Keeping these apart is the point of the exercise.

1.5 Organization

2 recalls the relevant rows of the project dictionary and the status calculus. 3 develops the positive-geometry framework: canonical forms, uniqueness, the residue recursion, and the residue tree as a functor. 4 realizes the framework in kinematic space through the ABHY associahedron, and reviews the amplituhedron and cosmological polytopes. 5 states the main results. 6 works the pentagon in full, including the honest residue-tree/ coaction-tree comparison. 7 places the regime among its neighbours (double copy, motivic coactions, celestial holography) and tracks the composed status. 8 is a candid account of the obstructions and open problems. 9 concludes. 10 documents the companion code.

2 The dictionary and the status calculus

We reproduce the rows of the project’s representation dictionary that this paper instantiates, and fix the status calculus used throughout.

The QG-IV dictionary rows instantiated by this paper. Statuses are the entries’ isolated warrants; composite claims degrade by the worst component.

Status	Math $M$	Physical target $P$	Invariant
$\mathsf{S/H}$	Positive geometry with canonical form	Amplitude from boundary/residue data	residues, boundaries
$\mathsf{S/H}$	Color–kinematics dual numerators	Gravity amplitude as gauge$^2$	double-copy numerators
$\mathsf{H}$	Motivic coaction / Goncharov Lie coalgebra	Decomposition anatomy of the amplitude	coaction, cobracket
$\mathsf{H}$	Celestial conformal data	Flat-space S-matrix as celestial correlator	soft/asymptotic Ward id.

3 The positive-geometry framework

We follow Arkani-Hamed, Bai and Lam , with the Hodge-theoretic refinements of Brown and Dupont  and the expository framing of Lam  and Herrmann–Trnka .

3.1 Positive geometries and canonical forms

The residue in [eq:resrec] is the Poincaré residue: if in local coordinates $C_i = \{x_1 = 0\}$ and $\Omega_X = \tfrac{dx_1}{x_1}\wedge \eta + \cdots$ with $\eta$ regular near $C_i$ and the ellipsis regular, then $\operatorname{Res}_{C_i}\Omega_X = \eta|_{x_1=0}$.

3.2 Polytopes and the simplex

The simplest positive geometries are convex polytopes in projective space, and they suffice for the associahedron. We record the two facts we need.

The cancellation in 8 is the geometric shadow of a physical fact: internal triangulation walls are not boundaries of $P$, so they must not appear as poles of the amplitude; unitarity forbids spurious singularities, and the geometry enforces it automatically.

3.3 The residue recursion and the residue tree

The residue recursion [eq:resrec] says exactly that the decoration is compatible with the tree structure: the form at a node determines, by taking residues, the forms at its children. We package this as a functoriality statement.

10 is deliberately modest: it says the residue tree is a well-defined, recursively-decorated combinatorial object, functorial in the obvious way. The content of the regime is what this object computes (4) and what it might secretly equal ([sec:results,sec:example]).

4 Positive geometries in kinematic space

4.1 Kinematic variables and planar poles

Consider $n$ massless momenta $p_1,\dots,p_n$ with $\sum_i p_i = 0$ and $p_i^2=0$. For a fixed cyclic order $(1,2,\dots,n)$, the planar Mandelstam variables are $$\begin{equation} X_{ij} \;=\; (p_i + p_{i+1} + \cdots + p_{j-1})^2, \qquad 1 \leq i < j \leq n, \end{equation}$$ with $X_{i,i+1}=0$ (massless legs) and $X_{1n}=0$. The nontrivial $X_{ij}$ are indexed by the diagonals $(i,j)$ of a convex $n$-gon with vertices $1,\dots,n$; there are $\tfrac{n(n-3)}{2}$ of them. A planar cubic tree with external legs in the cyclic order corresponds to a triangulation of the $n$-gon; its internal propagators are the diagonals of the triangulation, and it contributes $\prod_{(i,j)\in T} X_{ij}^{-1}$ to the amplitude.

4.2 The ABHY associahedron

Both counts are verified exactly by the companion code (10).

4.3 The amplituhedron and cosmological polytopes

The associahedron is the cleanest positive geometry but not the deepest. Two further constructions extend the reach of the framework.

The engine underneath the amplituhedron is the cell decomposition of $\mathrm{Gr}_{\geq 0}(k,n)$ into positroid cells, indexed by plabic graphs / on-shell diagrams and governed by total positivity . The associahedron is the $k=0$-like “scalar” skeleton of this richer structure. Reviews: Ferro–Lukowski  and Herrmann–Trnka .

16 is the reason the amplitude regime is not only a gauge-theory story. The cosmological polytope reconstructs a cosmological observable—a boundary wavefunction coefficient, directly relevant to inflationary correlators—as the canonical form of a polytope, with no de Sitter Lagrangian or in–in path integral in the definition. The caveat, which we do not hide, is that the models are conformally-coupled scalars, not Einstein gravity; the extension to realistic FRW backgrounds and to loop level is active and incomplete.

5 Main results

We now state the paper’s results. 17 and 18 are standard mathematics [ S ]. 19 is standard [ S ]. The bridge statements 22 and 23 are, respectively, the field’s central conjecture [ H/P ] and a conditional theorem [ S/H ] that makes precise what a proof would have to supply.

5.1 Factorization from residues

17 is the precise sense in which unitarity is a theorem about a polytope. We verify [eq:factor] exactly, as an identity of sets of triangulations and numerically, for the pentagon in 6 and in the companion code.

5.2 The coaction side and the bridge

We now recall the coaction tree and state the bridge conjecture.

The evidence for 22 is structural and case-based rather than a general theorem, and we are explicit about that. Structurally: cluster coordinates on $\mathrm{Gr}(k,n)$ simultaneously coordinatize the boundary strata of the positive geometry and the arguments (letters) of the polylogarithms appearing in the amplitude, so the two decompositions are indexed by the same combinatorial data . Case-based: the coaction of one- and two-loop amplitudes has been matched to the symbol/cluster structure in explicit examples, and the empirical “coaction principle” for Feynman periods  shows the amplitude class is closed under $\Delta$, a necessary condition for the trees to align. The cleanest positive statement we can make is conditional and rests on the twisted-cohomology coaction of Abreu–Britto–Duhr–Gardi–Matthew .

23 is the honest form of the bridge: there is a category (twisted cohomology, $\epsilon\neq 0$) in which the two trees provably coincide, and the open problem is the fate of that coincidence in the $\epsilon\to 0$ motivic limit.

6 Worked example: the pentagon

We now carry out, in full, the five-point case: the geometry of the pentagon associahedron, its canonical form, an explicit residue computation exhibiting factorization, its residue tree, and the residue-tree/coaction-tree comparison.

6.1 The pentagon and its canonical form

For $n=5$ the diagonals of the pentagon $12345$ are $$\begin{equation} X_{13},\; X_{14},\; X_{24},\; X_{25},\; X_{35}, \end{equation}$$ five in number, matching $\tfrac{5\cdot 2}{2}=5$. The associahedron $\mathcal{A}_2$ is a pentagon: a two-dimensional polytope with five vertices and five edges (facets). The five vertices are the five triangulations of the pentagon; each uses two non-crossing diagonals. Two diagonals $(i,j),(k,l)$ cross iff their endpoints interleave around the pentagon. The non-crossing (compatible) pairs are exactly the cyclically consecutive ones: $$\begin{equation} \{X_{13},X_{14}\},\;\{X_{14},X_{24}\},\;\{X_{24},X_{25}\},\;\{X_{25},X_{35}\},\; \{X_{35},X_{13}\}. \end{equation}$$

The pentagon associahedron $\mathcal{A}_2$. Vertices are triangulations (pairs of compatible diagonals); an edge is a facet $\{X_{ij}=0\}$, labelled by the single diagonal whose vanishing it enforces; the two endpoints of that edge are the two triangulations containing $X_{ij}$.

By 13, summing the five vertex contributions gives the canonical-form coefficient $$\begin{equation} m_5 \;=\; \frac{1}{X_{13}X_{14}} + \frac{1}{X_{14}X_{24}} + \frac{1}{X_{24}X_{25}} + \frac{1}{X_{25}X_{35}} + \frac{1}{X_{35}X_{13}}. \end{equation}$$ This is the five-point bi-adjoint scalar amplitude. It has exactly five simple poles, one per diagonal, and no others—in particular no pole at any crossing pair such as $\{X_{13},X_{24}\}$, which would be a spurious (non-planar) singularity. The absence of spurious poles is triangulation independence (8) made concrete.

6.2 An explicit residue: factorization on $X_{13}=0$

Take the residue of $m_5$ at the facet $X_{13}=0$. The terms of [eq:m5] with a pole there are the first and the last: $$\begin{equation} \operatorname{Res}_{X_{13}=0}\, m_5 \;=\; \operatorname{Res}_{X_{13}=0}\!\left(\frac{1}{X_{13}X_{14}} + \frac{1}{X_{35}X_{13}}\right) \;=\; \frac{1}{X_{14}} + \frac{1}{X_{35}}. \end{equation}$$ Now interpret the right-hand side. The diagonal $(1,3)$ splits the pentagon $12345$ into a triangle $L = (1,2,3)$ and a quadrilateral $R = (1,3,4,5)$. The triangle is $\mathcal{A}_0$, a point, with $m_L = 1$ (a three-point amplitude, no propagator). The quadrilateral is $\mathcal{A}_1$, an interval, with two diagonals $(1,4)$ and $(3,5)$ and amplitude $$\begin{equation} m_R \;=\; \frac{1}{X_{14}} + \frac{1}{X_{35}}. \end{equation}$$ Thus [eq:res13] reads $$\begin{equation} \operatorname{Res}_{X_{13}=0}\, m_5 \;=\; m_L\cdot m_R \;=\; 1\cdot\left(\frac{1}{X_{14}}+ \frac{1}{X_{35}}\right), \end{equation}$$ an instance of 17: the residue on the $X_{13}$ channel factorizes into the three-point amplitude (trivial) times the four-point sub-amplitude. Physically, the internal line with momentum $p_1+p_2$ goes on shell and the process splits into a $1,2\to I$ vertex and an $I,3,4,5$ sub-amplitude. The companion code verifies [eq:res13], both as an identity of triangulation sets and numerically at sample kinematics, for all five facets.

6.3 The residue tree of the pentagon

Assembling the residues gives the residue tree $\mathsf{R}(\mathcal{A}_2)$:

Root: the pentagon, decorated by $m_5$. Depth one, left to right: the five facets $X_{13}=0$, $X_{14}=0$, $X_{24}=0$, $X_{25}=0$, $X_{35}=0$, each an interval $\mathcal{A}_1$ decorated by the corresponding four-point sub-amplitude (the residue of $m_5$ there). Depth two (not drawn): each interval has two vertices, $0$-dimensional strata decorated by $1$, the leaves; these are the five triangulations of the pentagon. Counting faces gives the $f$-vector $(5,5,1)$ of the pentagon: five vertices, five edges, one $2$-face.

The five facet residues, read off as in [eq:res13] for each diagonal, are the five displayed nodes: $\operatorname{Res}_{X_{13}=0}m_5=\tfrac1{X_{14}}+\tfrac1{X_{35}}$, $\operatorname{Res}_{X_{14}=0}m_5=\tfrac1{X_{13}}+\tfrac1{X_{24}}$, $\operatorname{Res}_{X_{24}=0}m_5=\tfrac1{X_{14}}+\tfrac1{X_{25}}$, $\operatorname{Res}_{X_{25}=0}m_5=\tfrac1{X_{24}}+\tfrac1{X_{35}}$, and $\operatorname{Res}_{X_{35}=0}m_5=\tfrac1{X_{25}}+\tfrac1{X_{13}}$. Each is a four-point amplitude on the appropriate sub-quadrilateral. This is the complete factorization data of $m_5$, encoded geometrically.

6.4 The coaction-tree comparison

The residue tree above is a tree of rational functions: $m_5$ has weight zero, so its motivic coaction is trivial and there is, at this level, nothing on the coaction side to compare. The nontrivial comparison requires promoting the canonical form to a transcendental period, which is exactly what the stringy / twisted completion does. We describe the comparison honestly, tagging each step.

6.4.0.1 Step 1: the stringy pentagon integral [ S ].

Replace the rational canonical form by the Koba–Nielsen (string) integral over the same positive geometry. Concretely, integrate the canonical form of the moduli space $\mathcal{M}_{0,5}$ (the worldsheet positive geometry whose boundary combinatorics is the same associahedron) against the Koba–Nielsen twist $u = \prod_{i<j} |z_{ij}|^{\,\alpha' s_{ij}}$. The result is the five-point open-string (Veneziano-type) partial amplitude $A_5(\alpha')$, a function of the Mandelstam invariants whose low-energy expansion is a power series in $\alpha'$ with multiple-zeta-value coefficients: $$\begin{equation} A_5(\alpha') \;=\; m_5 \;+\; \alpha'^2\,\zeta(2)\,(\cdots) \;+\; \alpha'^3\,\zeta(3)\,(\cdots) \;+\; \cdots, \end{equation}$$ the leading term being the field-theory amplitude [eq:m5]. This is standard.

6.4.0.2 Step 2: the coaction of the string integral [ S/H ].

By 23 (the construction of , and the genus-zero configuration-space setting), the twisted coaction $\Delta_\epsilon$ (here $\epsilon\sim \alpha'$) acts on $A_5$ with a contour factor ranging over the boundary strata of the moduli-space contour—i.e. over the facets of the associahedron, the same five diagonals $X_{ij}$. The depth-one children of the twisted coaction tree $\mathsf{C}_\epsilon(A_5)$ are therefore indexed by the five facets, in bijection with the depth-one children of the residue tree $\mathsf{R}(\mathcal{A}_2)$. At this regularized level the two trees agree ($\mathsf{S}$), and the identification of the fifth facet, its residue, and its twisted-cycle dual is explicit.

6.4.0.3 Step 3: the motivic limit [ H ].

Expanding in $\alpha'$ and taking $\epsilon\to 0$, the twisted coaction reduces to the motivic coaction on the MZV coefficients of [eq:stringexp]. At weight two the coefficient is $\zeta(2)$, coaction-primitive (21): a single node, matching a single facet-residue at leading transcendental order. At weight three the coefficient is $\zeta(3)$, again primitive: a single node. At these low weights the coaction tree of $A_5$ is compatible, node for node, with the residue tree of $\mathcal{A}_2$: the boundary strata carry the same recursive structure as the successive coaction terms. This is the concrete, low-weight instance of 22 for $n=5$, and it is $\mathsf{H}$: verified in the mixed-Tate low-weight sector, not proven to persist to all weights.

6.4.0.4 Step 4: what would break [ P ].

Beyond weight three, MZVs acquire nontrivial coaction structure (e.g. $\zeta(3)\otimes \zeta(2)$-type terms in weight five, depth-two coaction trees), and the question becomes whether the deepening of the coaction tree continues to track the deepening of the residue tree under further residues. For the pentagon the residue tree has depth two (it is a polygon), while the coaction tree grows without bound in weight; the correct statement is not that the finite residue tree equals an infinite coaction tree, but that the residue tree controls the first layers and that a graded/filtered refinement is needed to state the match precisely. Making this precise—and, above all, extending it past the mixed-Tate regime—is exactly open-problem item 1, and it is here $\mathsf{P}$.

7 Relation to neighbouring regimes

7.1 Double copy: gravity through the inverse associahedron amplitude

The bi-adjoint amplitude $m_n$—the associahedron canonical form—is not only a scalar toy. It is the kernel through which gauge-theory amplitudes are squared into gravity. The field-theory Kawai–Lewellen–Tye (KLT) relations  express an $n$-point (super)gravity tree amplitude as a bilinear in color-ordered gauge-theory amplitudes, $$\begin{equation} M_n^{\mathrm{grav}} \;=\; \sum_{\alpha,\beta} A_n(\alpha)\, S[\alpha|\beta]\, \widetilde{A}_n(\beta), \end{equation}$$ where the momentum kernel $S[\alpha|\beta]$ is, up to normalization, the inverse of the matrix of bi-adjoint amplitudes, $$\begin{equation} S \;=\; m^{-1}, \qquad m_{\alpha\beta} = m_n(\alpha|\beta), \end{equation}$$ the double-color-ordered bi-adjoint amplitudes . Since each $m_n(\alpha|\beta)$ is the canonical form of an associahedron (in the pair of orderings $\alpha,\beta$), the entire double copy is phrased through associahedron canonical forms: gravity is $\mathrm{gauge}\otimes m^{-1}\otimes \mathrm{gauge}$. This is the sharp sense in which the gravitational S-matrix is reconstructed from gauge-theory representation data, with the positive geometry of the scalar theory supplying the “glue.”

7.2 Motivic coactions

The relation to the motivic-coaction regime is the content of [sec:results,sec:example]: the residue tree of the positive geometry and the coaction tree of its period. The composition of the positive-geometry entry $\mathsf{S/H}$ with the motivic-coaction entry $\mathsf{H}$ gives $\mathsf{H}$ for the bridge in the mixed-Tate regime, degrading to $\mathsf{P}$ in general. The coaction principle for Feynman periods  is the empirical backbone: the amplitude class is (conjecturally, with strong evidence) closed under $\Delta$, which is a prerequisite for any residue-tree/coaction-tree isomorphism.

7.3 Celestial holography

The Mellin transform in each particle’s energy converts momentum eigenstates into conformal primaries on the celestial sphere , reorganizing the flat-space amplitude as a celestial correlator. This is a change of representation basis for the one-particle states (an $\mathrm{SL}(2,\mathbb{C})$ representation-theoretic move), applied to the same S-matrix whose integrand is the canonical form. Composing the celestial entry $\mathsf{H}$ with the positive-geometry entry keeps the composite at $\mathsf{H}$: the celestial repackaging does not upgrade the positive-geometry construction to a full quantum-gravity statement, and the project keeps celestial holography at separate status from AdS/CFT-style holography by policy.

7.4 The composed picture

The amplitude regime and its neighbours, with composed statuses. Every arrow degrades or preserves status by the worst-component rule; none upgrades it.

8 Limitations and open problems

We collect the honest obstructions. None is hidden in the abstract or the introduction.

8.1 Elliptic and non-Tate amplitudes [ H/P ]

The motivic coaction of 20 lives in the tame, mixed-Tate world: its periods are (multiple) polylogarithms and MZVs, and the coaction is the Goncharov–Brown coaction on that Hopf algebra. Many physical amplitudes leave this world. Already the two-loop sunrise integral and the ten-point two-loop $\mathcal{N}=4$ amplitude involve elliptic polylogarithms and periods of elliptic curves (and, beyond, of higher-genus curves and Calabi–Yau varieties). For these the mixed-Tate coalgebra is the wrong target; one needs an enlarged (elliptic/modular) coalgebra, and the coaction tree of 22 must be reformulated accordingly. This is open-problem item 3 of the project queue. Until the non-Tate coaction is under control, the residue-tree/coaction-tree bridge is at best $\mathsf{H}$ in the mixed-Tate sector and $\mathsf{P}$ outside it. The positive geometry, by contrast, is often still well defined (the elliptic obstruction lives on the period/coaction side, not necessarily on the boundary-stratification side), which sharpens the open problem: the residue tree may exist where the coaction tree does not yet.

8.2 Gravity versus gauge positivity [ P ]

The amplituhedron and associahedron encode gauge-theory (and bi-adjoint scalar) data through positivity: the positive Grassmannian, positive kinematic constants, convex polytopes. There is no established positive geometry whose canonical form is a general gravitational integrand. Gravity enters this regime only indirectly: through the double copy ([eq:klt]), which reconstructs gravity from gauge data plus the scalar glue $m^{-1}$, and through cosmological polytopes (16), which cover conformally-coupled toy models rather than Einstein gravity. Whether “gravity has its own positive geometry”—a convex/positive object whose canonical form is the graviton amplitude directly, not via squaring—is open and, we judge, $\mathsf{P}$. The tension is structural: gravitational amplitudes have double poles and worse in convenient variables, and their numerators are not manifestly positive, so the naive positivity that powers the gauge-theory constructions does not transfer.

8.3 Generality of the S-matrix [ P ]

Even within gauge theory, the sharpest results (amplituhedron $=$ planar $\mathcal{N}=4$ integrand) are for a maximally supersymmetric, planar theory. Non-planar corrections, reduced supersymmetry, and massive external states are the subject of active construction (momentum amplituhedron, associahedra for $\phi^p$, massive deformations) but are not uniformly under control. The statement “amplitude $=$ canonical form” is a theorem in a growing but bounded list of theories, not a universal law. We keep the positive-geometry entry at $\mathsf{S/H}$ precisely to record this: $\mathsf{S}$ where the construction is proven, $\mathsf{H}$ as a claim about physics in general.

8.4 The bridge itself [ H/P ]

Finally, 22 is a conjecture. 23 shows a regularized category where the two trees coincide, and the pentagon (6) shows the low-weight match, but a general proof—especially one surviving the $\epsilon\to 0$ motivic limit and the non-Tate extension—does not exist. This is the central open problem of the regime, and stating it cleanly, with the twisted-cohomology construction as the most concrete foothold, is the main conceptual contribution of this paper.

9 Conclusion

We have presented perturbative amplitudes as canonical forms of positive geometries, with the recursive residue/boundary stratification—the residue tree—as the core invariant. The framework reconstructs an amplitude from boundary and positivity data alone: no Lagrangian, no Feynman sum, no path integral enters the definition of the canonical form, and locality and unitarity emerge as theorems about a polytope (17). We worked the pentagon associahedron in full, computed an explicit residue exhibiting factorization, drew its residue tree, and compared that tree to the motivic coaction tree of the string/twisted completion, using the twisted-cohomology coaction as the concrete bridge (23). We tracked the epistemic status of every claim under the worst-component-wins calculus: the geometry and residue recursion are standard $\mathsf{S}$; the identification with a physical S-matrix is theory-specific $\mathsf{S/H}$; and the residue-tree/coaction-tree bridge is heuristic in the mixed-Tate low-weight regime and speculative in general, $\mathsf{H/P}$.

The reconstruction thesis is unusually well served here: in the theories where it holds, the amplitude simply is the canonical form, and gravity enters through the double copy as gauge data glued by the inverse associahedron amplitude. The honest frontier is threefold: the non-Tate (elliptic and beyond) extension of the coaction side, the absence of a direct gravitational positive geometry, and a general proof of the residue-tree/coaction-tree isomorphism. The companion Haskell development makes the combinatorial and residue content of the pentagon (and general associahedron) exactly checkable, closing the loop between the geometry and a runnable computation.

10 Companion Haskell development

The code in src/positive-geometries/ accompanies this paper. It is written against the GHC base library only (no external dependencies) and compiles with ghc. The modules are:

• Core.hs — the associahedron combinatorics: convex polygons, diagonals $(i,j)$, the non-crossing (compatibility) test, enumeration of triangulations of an $n$-gon, and the bi-adjoint amplitude $m_n$ of [eq:mn] represented symbolically as a list of triangulations (each a set of diagonals, i.e. a product of propagators). Residue extraction $\operatorname{Res}_{X_{ij}=0}$ is implemented as: keep the triangulations containing the diagonal $(i,j)$, delete $(i,j)$ from each. A weight-graded toy coalgebra element with a one-step coproduct is included to illustrate the coaction tree of 20 and the status calculus $\star$ of 2.

• Properties.hs — exact, exhaustive checks over the finite combinatorics: (i) the number of triangulations of the $n$-gon equals the Catalan number $C_{n-2}$ (12); (ii) the number of diagonals equals $\tfrac{n(n-3)}{2}$; (iii) the residue factorization $\operatorname{Res}_{X_{ij}=0}m_n = m_L\cdot m_R$ of 17, checked for the pentagon on all five facets both as an identity of triangulation sets and numerically at sample kinematics; (iv) the alternating $f$-vector sum (Euler characteristic) of the pentagon; (v) the status-calculus laws (unit $\mathsf{S}\star\sigma=\sigma$ and monotonicity of 2).

• Main.hs — a main that prints the pentagon’s diagonals, triangulations, canonical-form terms $m_5$ of [eq:m5], the explicit residue [eq:res13] and its factorized form, and then runs the verification suite from Properties.hs, exiting non-zero on any failure.

The checkable property singled out for the paper is 17 on the pentagon: $\operatorname{Res}_{X_{13}=0}m_5 = \tfrac1{X_{14}}+\tfrac1{X_{35}}$, verified as [eq:res13].