Gravity as the Square of Gauge Theory: Color–Kinematics Duality, the Double Copy, and the KLT Kernel as the Reconstruction Layer of Perturbative 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 the idea with different invariants. The present paper concerns perturbative gravity amplitudes, where the reconstruction is unusually literal: the gravitational S-matrix, order by order in the coupling, is an algebraic square of gauge-theory data, and the datum that gets squared is a representation-theoretic object—a kinematic numerator obeying a color-like algebra.

The conventional route to a gravity amplitude begins with the Einstein–Hilbert action, expands $g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}$, and generates an infinite series of increasingly complicated vertices. Even the three-graviton vertex has of order a hundred terms; the four-point tree amplitude assembled directly from Feynman diagrams is a notorious exercise. The theory looks, from this vantage, both intractable and non-renormalizable. The double copy inverts the situation. It asserts that the same amplitude is the square of a Yang–Mills amplitude, and Yang–Mills amplitudes are comparatively simple. The technical content that makes the squaring possible is color–kinematics duality: a Yang–Mills amplitude can be written so that its kinematic numerators satisfy the same Jacobi identities as the Lie-algebra color factors that accompany them. Once this is arranged, gravity is obtained by a substitution rule—delete the color, put a second numerator in its place—with no independent gravitational input.

This is a sharp instance of reconstruction from representation structure. The invariant is not the gravitational field. It is the kinematic numerator, an object living in the same algebraic world as the adjoint-representation color factor of a gauge theory. Gravity is what one obtains by pairing two such invariants. The classical counterpart is equally striking: exact black-hole and gravitational-wave solutions of general relativity are squares of exact solutions of Maxwell or Yang–Mills theory, through the Kerr–Schild and Weyl double copies. Curvature, like the S-matrix, is reconstructed rather than posited.

1.2 What is and is not claimed

We are careful throughout to separate mathematics from physics and both from speculation, using the project’s three-valued epistemic calculus (2). The tree-level statements we prove or cite—the BCJ amplitude relations, the existence of color-dual numerators at tree level, the tree double copy, the KLT relations, and the identification of the field-theory KLT kernel with the inverse of the biadjoint-scalar partial-amplitude matrix—are standard mathematics of scattering amplitudes, tag [ S ]. Read as statements about a physical gravitational S-matrix they become heuristic, tag [ S/H ], because they hold for the specific spectrum of double-copy-constructible theories (pure Yang–Mills squared gives $\mathcal N = 0$ supergravity, not pure Einstein gravity; $\mathcal N = 4$ super-Yang–Mills squared gives $\mathcal N = 8$ supergravity) rather than for a general ultraviolet-complete theory of quantum gravity. The loop-level double copy is verified in a large number of explicit high-loop cases but lacks a general existence proof and is tagged [ H ]. The classical double copy is [ S/H ] in its Kerr–Schild and Weyl sectors and [ H ] as a general claim. The conjectural bridge to motivic coactions, shared with the companion positive-geometry paper, is [ H/P ]. We never silently upgrade a label.

1.3 Relation to the companion papers and the roadmap

This paper is the double-copy half of module QG-IV (amplitudes, positive geometry, double copy) in the project roadmap. Its companion, the positive-geometry paper, treats the same module through canonical forms of positive geometries; the two meet at a single sharp object, the KLT kernel, which in the positive-geometry paper is the inverse of the canonical form of the Arkani-Hamed–Bai–He–Yan associahedron (the biadjoint-scalar amplitude) and in this paper is the bilinear form pairing two gauge-theory copies into gravity. That these are the same object (15) is the technical seam between the two papers. The regime also touches open problems 1 (residue tree $=$ coaction tree) and 2 (cut/regulator complementarity) of the roadmap queue, which we discuss honestly as open in [sec:neighbours,sec:limitations].

1.4 Organization

2 recalls the representation-stack framework and the S/H/P calculus. 3 sets up the cubic-graph organization of amplitudes and the color algebra. 4 defines color–kinematics duality and the kinematic Jacobi relations. 5 states and proves the tree double copy and its equivalence to KLT. 6 identifies the KLT kernel with the inverse biadjoint amplitude, the seam to the positive-geometry paper. 7 works the four-point example in full and sketches five points. 8 treats the classical Kerr–Schild and Weyl double copies. 9 locates the regime among its neighbours under the weakest-link calculus. 10 states the honest limitations and open problems. 11 describes the companion Haskell development. 12 concludes.

2 The representation dictionary and the S/H/P calculus

We recall the minimal apparatus this paper shares with the other eleven topic papers of the library, so that the paper is self-contained.

2.1 The representation stack and realization pipeline

A representation entry is a tuple $E = (M, P, \tau, \sigma)$: a mathematical object $M$, a physical target $P$, a translation datum $\tau$ that carries the correspondence together with its assumptions and limitations, and an epistemic status $\sigma$. The realization pipeline $$M \;\longrightarrow\; \Phi(M) \;\longrightarrow\; \mathrm{Real}_\alpha(\Phi(M)) \;\longrightarrow\; \text{physical representation} \;\longrightarrow\; \mathrm{Obs}$$ sends a mathematical object to observables through a choice of realization $\alpha$. In the present regime $M$ is the system of color-dual kinematic numerators of a gauge theory, $\mathrm{Real}$ is the double-copy substitution, and $\mathrm{Obs}$ is a gravity amplitude.

2.2 Warrants and worst-component composition

There are three ordered warrants, $$S \;<\; H \;<\; P,$$ read as: $S$, a standard mathematical or mathematical-physics correspondence; $H$, a strong heuristic physical representation; $P$, a speculative ontological extension. A status is a nonempty tuple of warrants, rendered S, H, P, or a composite such as S/H. The single structural rule we use repeatedly is that composition is monotone and the worst component wins.

This proposition is the engine of 9: chaining the double copy to the motivic coaction, or to celestial holography, cannot yield a composite more secure than the $H$-rated conjectural links in those chains.

2.3 The dictionary entry for this paper

The relevant row of the library’s dictionary is

with assumptions “a suitable color–kinematics representation exists” and limitations “loop-level and theory-specific constructions require care.” Everything below refines this single entry.

3 Cubic-graph organization and the color algebra

3.1 Trivalent graphs and the color decomposition

Fix a gauge group with Lie algebra $\mathfrak{g}$, structure constants $f^{abc}$ defined by $[T^a, T^b] = i f^{abc} T^c$ in a basis of Hermitian generators normalized to $\operatorname{Tr}(T^a T^b) = \delta^{ab}$. Consider the $L$-loop $n$-point amplitude of a gauge theory whose fields are all in the adjoint representation. The central organizational fact is that every Feynman diagram can be reduced to trivalent (cubic) topology: quartic vertices are rewritten by inserting and cancelling a propagator, so that contact terms are absorbed into cubic graphs at the price of numerators that are polynomials in the momenta.

At tree level, $L = 0$, the momentum integrals and symmetry factors are trivial and [eq:cubicamp] reads $$\begin{equation} \mathcal{A}_{n}^{\mathrm{tree}} \;=\; g^{n-2} \sum_{g \in \Gamma_{n,0}} \frac{c_g\, n_g}{\prod_{\alpha \in g}D_\alpha}. \end{equation}$$ The number of cubic tree graphs with $n$ external legs is $(2n-5)!! = 1\cdot 3\cdot 5 \cdots (2n-5)$: three graphs at $n=4$, fifteen at $n=5$, and one hundred and five at $n=6$.

3.2 Color factors and the Jacobi identity

The color factors $c_g$ are not independent. Whenever three cubic graphs $g_s, g_t, g_u$ differ only by the connectivity of a single internal four-point subgraph—an $s$-, $t$-, or $u$-channel exchange in one propagator slot, with everything else held fixed—their color factors satisfy a linear relation inherited from the Lie-algebra Jacobi identity.

3.3 Color-ordered partial amplitudes

Contracting [eq:cubictree] against a trace basis organizes the amplitude into color-ordered partial amplitudes. At tree level, for adjoint fields, $$\begin{equation} \mathcal{A}_n^{\mathrm{tree}} = g^{n-2}\sum_{\sigma \in S_{n}/\mathbb{Z}_n} \operatorname{Tr}\!\big(T^{a_{\sigma(1)}}\cdots T^{a_{\sigma(n)}}\big)\, A(\sigma(1),\dots,\sigma(n)), \end{equation}$$ where the partial amplitudes $A(1,\dots,n)$ are gauge-invariant, cyclically symmetric, and carry only the kinematic data. The $(n-1)!$ orderings in [eq:colordecomp] are not independent: the Kleiss–Kuijf relations reduce them to a basis of $(n-2)!$, and, as we recall next, color–kinematics duality reduces that further to $(n-3)!$.

4 Color–kinematics duality

4.1 Definition

The color factors obey the Jacobi relations [eq:colorjac] by Lie algebra. There is no a priori reason for the kinematic numerators to obey anything of the sort; they are, in a generic gauge and diagram-by-diagram, arbitrary. The discovery of Bern, Carrasco and Johansson is that the gauge freedom in [eq:cubicamp]—the freedom to move contact terms between graphs—is exactly enough to force the numerators into the same algebra as the colors.

4.2 Existence at tree level

4.3 The kinematic algebra

Color–kinematics duality suggests that the numerators are structure constants of a “kinematic algebra” dual to the gauge Lie algebra. In the self-dual sector of Yang–Mills this algebra was identified by Monteiro and O’Connell as the algebra of area-preserving diffeomorphisms; for the full theory a closed-form kinematic algebra is known in specific sectors. Brandhuber, Chen, Johansson, Travaglini and Wen gave closed-form BCJ numerators for heavy-mass effective field theory and Yang–Mills through a quasi-shuffle Hopf algebra, making the duality manifest in those sectors. A general, all-multiplicity, all-loop kinematic algebra is not known; we return to this in 10. The status of the kinematic-algebra program as physics is [ H ]: it is a strong structural hypothesis with explicit realizations, not a theorem in generality.

5 The double copy

5.1 The substitution rule

5.2 The spectrum of the square

The field content of the double copy is the tensor product of the two gauge-theory field contents. A gauge boson has $D-2$ physical polarizations; the product of two gauge bosons decomposes under the little group as $$(D-2)\otimes(D-2) = \underbrace{\mathrm{Sym}_0^2}_{\text{graviton}} \;\oplus\; \underbrace{\wedge^2}_{\text{\(B\)-field}} \;\oplus\; \underbrace{\mathrm{trace}}_{\text{dilaton}}.$$ Thus pure Yang–Mills squared is not pure Einstein gravity: it is $\mathcal N = 0$ supergravity, the graviton accompanied by a two-form (Kalb–Ramond) field and a scalar dilaton (in four dimensions the two-form is dual to a pseudoscalar axion, which is the “axion” of the abstract). Projecting onto pure gravity requires removing the dilaton and two-form, which is done by adding ghost matter in the double copy (Johansson–Ochirov) or by a suitable projection. Squaring $\mathcal N = 4$ super-Yang–Mills gives $\mathcal N = 8$ supergravity. We flag this explicitly: the slogan “gravity $=$ gauge$^2$” is exact only with the correct bookkeeping of the extra states.

5.3 Tree-level correctness and equivalence to KLT

6 The KLT kernel as the inverse biadjoint amplitude

The KLT kernel is the one object shared with the companion positive-geometry paper. There it is the inverse of a canonical form; here it is the pairing that makes gravity a square. The identification is a theorem.

6.1 The biadjoint scalar

The matrix $m(\sigma|\tau)$, indexed by an $(n-3)!$ minimal basis of orderings, is the central combinatorial object of the biadjoint theory. In the companion paper it is the canonical form of the Arkani-Hamed–Bai–He–Yan associahedron.

7 Worked example: the four-point amplitude

We now carry the entire construction through explicitly at $n = 4$, where every object is a finite rational function of the Mandelstam invariants and every claim is checkable by hand and in code.

7.1 Kinematics

For four massless momenta $k_1,\dots,k_4$ with $\sum_i k_i = 0$ and $k_i^2 = 0$, define $$s = s_{12} = (k_1+k_2)^2,\quad t = s_{23} = (k_2+k_3)^2,\quad u = s_{13} = (k_1+k_3)^2.$$ Momentum conservation and masslessness give the single constraint $$\begin{equation} s + t + u = 2\big(k_1\!\cdot k_2 + k_2\!\cdot k_3 + k_1\!\cdot k_3\big) = (k_1+k_2+k_3)^2 = k_4^2 = 0, \end{equation}$$ using $k_i^2 = 0$ and $k_1+k_2+k_3 = -k_4$. This is the only relation among the three invariants and it is exact.

7.2 Color factors

The three cubic graphs $g_s, g_t, g_u$ carry the color factors of 5, and by 4, $$\begin{equation} c_s + c_t + c_u = 0. \end{equation}$$ Over $\mathfrak{su}(2)$ with $f^{abc} = \varepsilon^{abc}$ this is the identity $\varepsilon^{a_1 a_2 e}\varepsilon^{e a_3 a_4} + \varepsilon^{a_2 a_3 e}\varepsilon^{e a_1 a_4} + \varepsilon^{a_3 a_1 e}\varepsilon^{e a_2 a_4} = 0$, which expands via $\varepsilon^{abe}\varepsilon^{cde} = \delta^{ac}\delta^{bd} - \delta^{ad}\delta^{bc}$ into a sum of six Kronecker-delta products that cancel in pairs; the companion code checks it for all $3^4$ index assignments.

7.3 Color-dual numerators

The color-ordered partial amplitude in the ordering $(1,2,3,4)$ receives contributions from the $s$- and $t$-channels only, and the two channels enter with a relative sign fixed by the trace decomposition (the color factors $c_s$ and $c_t$ dress the traces $\operatorname{Tr}(1234)$ and $\operatorname{Tr}(1243)$ with opposite sign, which arises mechanically from expanding each structure constant as a commutator, $f^{abc} \propto \operatorname{Tr}([T^a,T^b]T^c)$, so the two orderings inside each vertex enter with opposite sign). With that convention, $$\begin{equation} A(1,2,3,4) = \frac{n_s}{s} - \frac{n_t}{t}, \qquad A(1,2,4,3) = -\frac{n_s}{s} + \frac{n_u}{u}, \end{equation}$$ the second following from the same rule applied to the ordering $(1,2,4,3)$, whose planar channels are $s$ and $u$. These signs are structural, not cosmetic: they are exactly what makes the KLT bilinear [eq:kltfour] reproduce the double-copy sum, as we verify below. A numerator set is color-dual when $$\begin{equation} n_s + n_t + n_u = 0, \end{equation}$$ mirroring [eq:cstu]. To make the arithmetic completely explicit we use a toy dual set that solves [eq:nstu] identically, $$\begin{equation} n_s = t - u,\qquad n_t = u - s,\qquad n_u = s - t, \end{equation}$$ so that $n_s + n_t + n_u = (t-u)+(u-s)+(s-t) = 0$ with no use of [eq:stu]. These are the numerators of a scalar theory whose partial amplitude is $A(1,2,3,4) = (t-u)/s + (u-s)/t$; they are genuinely color-dual and suffice to exhibit every structural feature of the double copy. (For the physical gluon amplitude the numerators are the corresponding polarization-dependent expressions; the algebra of the double copy is identical and we keep the scalar toy for transparency.)

7.4 The double copy

Applying [eq:dc] at $n=4$, $L=0$ with $\tilde{n}_g = n_g$ (squaring one theory against itself), $$\begin{equation} \mathcal{M}_4 = \frac{n_s^2}{s} + \frac{n_t^2}{t} + \frac{n_u^2}{u} = \frac{(t-u)^2}{s} + \frac{(u-s)^2}{t} + \frac{(s-t)^2}{u}. \end{equation}$$ This expression is totally symmetric under permutations of $(s,t,u)$ accompanied by the induced permutation of the numerators, hence crossing symmetric, as a gravity amplitude must be. Using $u = -s-t$ one can reduce [eq:dcfour] to a rational function of $s,t$ alone; the companion code evaluates both [eq:dcfour] and its reduced form on random rational kinematics and checks they agree, and checks the crossing symmetry directly.

7.5 Matching KLT

By 13, the same amplitude is the KLT bilinear $$\begin{equation} \mathcal{M}_4 = -\,s\, A(1,2,3,4)\, \tilde A(1,2,4,3), \end{equation}$$ with $A, \tilde A$ the physical color-ordered partial amplitudes [eq:partial] of the two copies. Taking both copies equal to the toy dual set [eq:toynum], so that $A(1,2,3,4) = n_s/s - n_t/t$ and $\tilde A(1,2,4,3) = -n_s/s + n_u/u$, expand [eq:kltfourexp]: $$\begin{align} -s\, A(1,2,3,4)\,\tilde A(1,2,4,3) &= -s\Big(\frac{n_s}{s} - \frac{n_t}{t}\Big) \Big(-\frac{n_s}{s} + \frac{n_u}{u}\Big) \notag\\ &= \frac{n_s^2}{s} - \frac{n_sn_u}{u} - \frac{n_tn_s}{t} + \frac{s\,n_tn_u}{tu}. \end{align}$$ Substituting $n_s = -(n_t+n_u)$ in the last three terms and collecting over the common denominator $tu$ gives $n_tn_u(s+t+u)/(tu) + n_t^2/t + n_u^2/u$; the first term vanishes by [eq:stu], leaving $$\begin{equation} -s\, A(1,2,3,4)\,\tilde A(1,2,4,3) = \frac{n_s^2}{s} + \frac{n_t^2}{t} + \frac{n_u^2}{u} = \mathcal{M}_4, \end{equation}$$ exactly [eq:dcfour]. So the toy dual numerators satisfy the four-point KLT relation on the nose, provided the relative trace sign is kept. The companion code evaluates both sides of [eq:kltmatch] over the rationals on a sample of kinematic points and confirms the equality, alongside the color and kinematic Jacobi identities, generalized gauge invariance, crossing symmetry of [eq:dcfour], and the rank-one degeneracy of the biadjoint matrix.

7.6 Five points, sketched

At $n = 5$ there are $(2\cdot 5 - 5)!! = 15$ cubic graphs. Their propagators are the two-particle Mandelstam invariants $s_{ij}$, of which there are $\binom{5}{2} = 10$; momentum conservation and masslessness ($\sum_{j\neq i} s_{ij} = 0$ for each $i$) leave five independent invariants. The Jacobi triples now form an overconstrained-looking but consistent linear system; the minimal basis of partial amplitudes has dimension $(n-3)! = 2$, so two partial amplitudes—say $A(1,2,3,4,5)$ and $A(1,3,2,4,5)$—determine all others through the fundamental BCJ relations. The double copy [eq:dc] becomes a $2\times 2$ KLT bilinear, $$\mathcal{M}_5 = \sum_{\sigma,\tau \in \{(23),(32)\}} A(\sigma)\,\mathcal{S}[\sigma|\tau]\,\tilde A(\tau),$$ with the $2\times 2$ kernel $\mathcal{S}= m^{-1}$ the inverse of the $2\times 2$ biadjoint matrix on the same basis. The structure is identical to four points; only the linear algebra grows. We treat five points as a sketch and do not carry the explicit fifteen numerators; the four-point case already exhibits color Jacobi, its kinematic mirror, the double-copy assembly, generalized gauge invariance, and the inverse-biadjoint kernel, which are the whole content of the construction.

8 The classical double copy

The double copy is not confined to amplitudes. A parallel statement relates exact classical solutions of general relativity to exact classical solutions of gauge theory. We summarize the two established forms, keeping their epistemic labels honest.

8.1 The Kerr–Schild double copy

The Kerr–Schild ansatz linearizes the Einstein tensor exactly: for such $g$, the Einstein equations become linear in $\phi$. Monteiro, O’Connell and White observed that the same data $(\phi, k)$ define a gauge field.

8.2 The Weyl double copy

The Kerr–Schild double copy is a statement about metrics; the Weyl double copy is a cleaner statement about curvature, in spinor form.

Later work extended [eq:weyl] to radiative and more general algebraically special fields (Godazgar, Godazgar, Monteiro, Peinador Veiga and Pope). We flag the limitation: the Weyl double copy in the clean form [eq:weyl] is established for algebraically special (principally type D) vacua; a background-independent statement for arbitrary curved spacetimes is not available, and the general-purpose status of the classical double copy is therefore [ H ], not [ S ].

8.3 Reconstruction reading

The classical double copy makes the reconstruction thesis geometric. A black hole is not, in this reading, a primitive gravitational object. Its curvature is the symmetrized square of a gauge-field-strength spinor divided by a scalar, and its metric is a square of the same null data that builds a Coulomb or $\sqrt{\text{Kerr}}$ gauge field. Spacetime curvature is reconstructed from gauge data exactly as the S-matrix is, using the same squaring invariant.

9 Relation to neighbouring regimes

We locate the double copy among the other QG-IV and QG-adjacent regimes, tracking epistemic status under the worst-component-wins calculus of 2.2.

9.1 Positive geometry (companion QG-IV paper)

The seam is 15: the KLT kernel is the inverse of the biadjoint amplitude, which the companion paper produces as the canonical form of the ABHY associahedron. The composite status is $$\mathrm{worst}\big(\mbox{{\normalfont\sffamily\footnotesize[\,S/H\,]}}_{\text{double copy}} \;\circ\; \mbox{{\normalfont\sffamily\footnotesize[\,S/H\,]}}_{\text{associahedron}}\big) = \mbox{{\normalfont\sffamily\footnotesize[\,S/H\,]}}.$$ The two constructions reinforce rather than degrade one another: both halves of QG-IV are $S/H$, and their shared invariant (the kernel) is $S$ as mathematics.

9.2 Motivic coaction and open problem 1

Attaching the double-copy amplitude to its transcendental (stringy or dimensionally regulated) completion brings in the motivic coaction of the amplitude’s period, and the central conjecture “residue tree $=$ coaction tree” (open problem 1). The unitarity cuts of a double-copy loop integrand are the primary tool for building and checking BCJ-dual loop numerators, so the double copy and the coaction interact directly. But the coaction side is [ H ] (it assumes a mixed-Tate regime; elliptic and non-Tate sectors need enlarged coalgebras), and the tree-identification is [ H ] in verified low-weight cases and [ P ] in general. Hence $$\mathrm{worst}\big(\mbox{{\normalfont\sffamily\footnotesize[\,S/H\,]}}_{\text{double copy}} \;\circ\; \mbox{{\normalfont\sffamily\footnotesize[\,H\,]}}_{\text{coaction}} \;\circ\; \mbox{{\normalfont\sffamily\footnotesize[\,H/P\,]}}_{\text{residue}=\text{coaction}}\big) = \mbox{{\normalfont\sffamily\footnotesize[\,H/P\,]}}.$$ The double copy cannot rescue the coaction conjecture; the weakest link governs.

9.3 Cut/regulator complementarity and open problem 2

Open problem 2 asks whether discontinuity/cut maps are adjoint to regulator/period maps in a suitable category. Because unitarity cuts drive the loop-level double copy, a precise cut/regulator adjunction would immediately sharpen the loop-level existence question of 10. We record this as a target, not a result; its status is [ H ].

9.4 Celestial holography

Mellin-transforming a double-copy S-matrix to the celestial sphere produces celestial correlators; the “celestial double copy” is an active subject. The celestial dictionary is [ H ] (its soft-sector and asymptotic-symmetry assumptions are not settled), so the composite with the double copy is [ H ].

9.5 Summary table

10 Honest limitations and open problems

We state the obstructions plainly; none is hidden by the slogan.

10.0.0.1 No general all-loop existence proof.

Color–kinematics dual numerators are guaranteed at tree level (8). At loop level their existence is established case by case—famously through four loops in the four-point amplitude of $\mathcal N = 8$ supergravity and to high order in $\mathcal N = 4$ super-Yang–Mills—but there is no general theorem that a BCJ-dual representation exists at arbitrary loop order and multiplicity. The kinematic-algebra program (Brandhuber–Chen–Johansson–Travaglini–Wen and successors) gives closed forms in specific sectors, not in general. Status of the loop-level double copy as physics: [ H ].

10.0.0.2 Locality versus duality at loop level.

Even when dual loop numerators exist, they may be non-local (contain spurious poles) if one insists on manifest duality, or local at the cost of manifest duality. Reconciling the two is an open technical problem and is one reason the loop-level statements resist a clean existence theorem.

10.0.0.3 The square is not pure gravity.

Yang–Mills squared is $\mathcal N = 0$ supergravity—graviton plus dilaton plus two-form, not pure Einstein gravity. Extracting pure gravity requires ghost matter or projection. Any statement “gravity $=$ gauge$^2$” that omits the extra states is imprecise; we have kept them explicit.

10.0.0.4 The classical double copy is not background independent.

The Kerr–Schild and Weyl double copies are cleanest for algebraically special (Kerr–Schild, type D) vacua. A general statement for arbitrary curved backgrounds is not established; attempts to extend beyond algebraically special solutions meet genuine obstructions. Status: [ H ] as a general claim.

10.0.0.5 Off-shell / Lagrangian-level duality.

A local action manifesting color–kinematics duality to all orders in the fields is not known; the duality is manifest on-shell and, in specific constructions, up to a finite order off-shell. This bears on any attempt to make the double copy a statement about actions rather than amplitudes.

10.0.0.6 Open problems from the roadmap.

Item 1 (residue tree $=$ coaction tree) and item 2 (cut/regulator complementarity) touch this regime directly, as discussed in 9. Neither is resolved here; both are flagged [ H ] or [ H/P ].

11 The companion Haskell development

The paper is accompanied by a small Haskell package under src/double-copy/ that makes the four-point construction executable and its algebraic claims checkable over exact rational arithmetic. It has no dependencies outside base and compiles with GHC.

11.0.0.1 Modules.

Core.hs defines the three four-point channels, Mandelstam data over $\mathbb{Q}$, the $\mathfrak{su}(2)$ structure constants $\varepsilon^{abc}$, the color factors $c_s,c_t,c_u$ of 5, the toy color-dual numerators [eq:toynum], the double-copy assembly [eq:dcfour], the KLT bilinear [eq:kltfourexp], and the biadjoint doubly-partial amplitudes of 14. Properties.hs states the checkable properties. Main.hs prints the demonstrations and runs the checks, exiting nonzero on any failure.

11.0.0.2 Checkable properties.

The package verifies, over exact rationals and for a deterministic sample of kinematic points:

1. Color Jacobi. $c_s + c_t + c_u = 0$ as an identity of $\mathfrak{su}(2)$ tensors, checked for all $3^4$ external-index assignments (4, 5).

2. Mandelstam constraint. $s + t + u = 0$ for the generated kinematics ([eq:stu]).

3. Kinematic Jacobi. $n_s + n_t + n_u = 0$ for the toy dual numerators [eq:toynum] (6).

4. Generalized gauge invariance. The partial amplitude [eq:partial] and the full amplitude are invariant under the shift $n_g \mapsto n_g + D_g \alpha$ (17), the amplitude-level shadow of color Jacobi.

5. Crossing symmetry of the double copy. The double-copy amplitude [eq:dcfour] is invariant under permutations of $(s,t,u)$ with the induced numerator permutation.

6. KLT match. The four-point KLT bilinear $-s\,A(1,2,3,4)\,\tilde A(1,2,4,3)$, built from the partials [eq:partial] with their trace signs, equals the direct double-copy assembly [eq:dcfour] on the sampled kinematics ([eq:kltmatch]).

7. Biadjoint rank one. On the ordering basis $\{(1234),(1243),(1324)\}$ all nine $2\times 2$ minors of the biadjoint matrix $m$ vanish (using $s+t+u=0$) while $m$ is nonzero, so $m$ has rank $(n-3)! = 1$ and the KLT kernel $m^{-1}$ is well-defined on the physical image (18).

8. Reduced assembly. The double copy [eq:dcfour] agrees with its $u$-eliminated form (with $u = -(s+t)$) on the sampled kinematics.

The kinematic Jacobi check (item 3) is the property required by the project’s build gate: it is exact, has no free parameters once the dual set is fixed, and fails loudly if the numerator algebra is perturbed.

12 Conclusion

The double copy realizes the project’s reconstruction thesis with unusual directness. The invariant is a gauge-theory kinematic numerator constrained to obey the Lie-algebra Jacobi identities of color; gravity, both its S-matrix and—through the classical double copy—its curvature, is the algebraic square of that invariant. Nothing gravitational is posited beyond the choice of pairing. We have given the cubic-graph organization, the color Jacobi relations and their kinematic mirror, the double-copy substitution, the KLT relations, and the identification of the KLT kernel with the inverse biadjoint amplitude that stitches this paper to its positive-geometry companion. We worked four points in full and sketched five, and we treated the classical Kerr–Schild and Weyl double copies. Throughout we kept the epistemic labels honest: tree-level amplitude mathematics is $S$; its reading as a physical gravity S-matrix is $S/H$; the loop-level and general classical statements are $H$; and the conjectural bridge to motivic coactions is $H/P$. The construction is not a theory of quantum gravity. It is a precise demonstration that, in the perturbative regime, gravity is not an independent input but a reconstruction—a square—of gauge-theoretic representation data, with the KLT kernel as the sharp invariant shared across the amplitude module of the library.