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arxiv: 2512.11955 · v2 · submitted 2025-12-12 · ✦ hep-th · gr-qc· hep-ph

Completeness from Gravitational Scattering

Francesco Calisto , Clifford Cheung , Grant N. Remmen , Francesco Sciotti , Michele Tarquini This is my paper

Pith reviewed 2026-05-16 22:50 UTC · model grok-4.3

classification ✦ hep-th gr-qchep-ph
keywords gravitational scatteringcompleteness hypothesisnonabelian symmetrycharge latticeweakly coupled gravitygrand unified theoriesSU(5)SO(10)
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0 comments X

The pith

Gravity together with nonabelian symmetry requires the full abelian charge lattice to be populated by single-particle states.

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

The paper shows that any finite collection of charged particles under a nonabelian symmetry G, when gravity is present and weakly coupled at high energies, forces the existence of infinitely many additional charged states that occupy every point on the charge lattice. This conclusion is reached by requiring that perturbative gravitational scattering amplitudes remain consistent. A sympathetic reader cares because the result turns the completeness hypothesis from an extra assumption into a necessary consequence of symmetry plus gravity. The argument applies immediately to SU(N) for N at least 3 and SO(N) for N at least 5, and it shows that the standard SU(5) and SO(10) grand unified theories already contain exactly the minimal set of representations needed.

Core claim

For theories with a weakly coupled ultraviolet completion of gravity and a nonabelian symmetry G whose Cartan subgroup generates the abelian charge lattice, the existence of a finite set of charged representations necessitates infinitely many charged particles that completely fill the charge lattice. This holds for G equal to SO(N) with N greater than or equal to 5 and SU(N) with N greater than or equal to 3, and implies completeness for related groups like Spin(N), Sp(N), and E8. As a corollary, the SU(5) and SO(10) grand unified theories possess precisely the minimal field content required to derive completeness via this method.

What carries the argument

Consistency requirements on perturbative gravitational scattering amplitudes of charged particles transforming under the nonabelian symmetry G, which force additional states to appear in order to avoid inconsistencies.

If this is right

  • The abelian charge lattice is completely filled by single-particle states for SU(N) with N greater than or equal to 3.
  • The abelian charge lattice is completely filled by single-particle states for SO(N) with N greater than or equal to 5.
  • Completeness follows automatically for the related groups Spin(N), Sp(N), and E8.
  • The standard SU(5) and SO(10) grand unified theories already contain the minimal representations needed to obtain completeness from this argument.

Where Pith is reading between the lines

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

  • Partial realizations of nonabelian symmetries become impossible once gravity is included, constraining which representations can appear in ultraviolet completions.
  • Model builders must generate the full charge lattice rather than introducing isolated charged states without their partners.
  • String theory or other high-energy frameworks that realize gravity and symmetry must automatically produce complete spectra rather than incomplete ones.
  • If new charged particles are discovered, gravitational consistency would require the existence of their lattice partners even if they lie beyond direct detection.

Load-bearing premise

The theory admits a weakly coupled ultraviolet completion of gravity, includes a nonabelian symmetry G whose Cartan subgroup generates the charge lattice, and contains at least one finite set of charged representations.

What would settle it

An explicit computation of gravitational scattering amplitudes in a model with SU(3) symmetry containing only fundamental representations that exhibits unitarity violation or poles that cannot be canceled without extra charged states would falsify the claim.

Figures

Figures reproduced from arXiv: 2512.11955 by Clifford Cheung, Francesco Calisto, Francesco Sciotti, Grant N. Remmen, Michele Tarquini.

Figure 1
Figure 1. Figure 1: The SO(4) charge lattice, stratified according to the central charge sectors z = 0 (black) and z = 1 (gray). Overlaid is the sequence of scattering processes in Eq. (13). Starting from an initial spectrum composed of the funda￾mental (red), we scatter in succession (orange, yellow, green, blue, indigo) to obtain a set of ultracharged states (purple). We then apply lowering operators to generate all charges… view at source ↗
Figure 2
Figure 2. Figure 2: The SU(3) charge lattice, stratified according to the central charge sectors z = 0 (black), z = 1 (green), and z = 2 (purple). The polygons circumscribe the irreducible representations Q3 (green), Q3¯ (purple), Q8 (black), Q10 (brown), and Q10 (gray). could not be established using our algorithm. We thus move on to G = SU(3), whose Cartan subgroup is H = U(1)2 . The corresponding charge lattice is the two￾… view at source ↗
Figure 3
Figure 3. Figure 3: The SU(3) charge lattice. Left: The seed of the iteration. Right: Sequence of polygons allowing iteration, namely, the dark blue triangle Tn, blue hexagon Hn, and light blue triangle Tn+3. Relevant points for scattering are denoted with circles. corners of Q15. Note from Eq. (20) that all three corners of Tn are contained within the same Weyl orbit. We then apply the following algorithm: i) Assuming that w… view at source ↗
Figure 4
Figure 4. Figure 4: The SU(3) charge lattice. Left: starting hexagon with the starting charge P0 (dark blue). Right: sequence of hexagons (dark blue, blue, light blue), highlighting the relevant points Pn, (dark blue), Dn (dark blue), and Pn+1 (blue). completeness in the abelian charge lattice of the Cartan subgroup H. These results suggest a natural follow-up question: do our assumptions also imply the existence of all possi… view at source ↗
read the original abstract

We prove that symmetry in the presence of gravity implies a version of the completeness hypothesis. For a broad class of theories, we demonstrate that the existence of finitely many charged particles logically necessitates the existence of infinitely many charged particles populating the entire charge lattice. Our conclusions follow from the consistency of perturbative gravitational scattering and require the following ingredients: 1) a weakly coupled ultraviolet completion of gravity, 2) a nonabelian symmetry $G$, gauged or global, whose Cartan subgroup generates the abelian charge lattice, and 3) a spectrum containing some finite set of charged representations, in the simplest cases taken to be a single particle in the fundamental. Under these conditions, the abelian charge lattice is completely filled by single-particle states for $G=SO(N)$ with $N\geq 5$ and $G=SU(N)$ with $N\geq 3$, which in turn implies completeness for other symmetry groups such as $Spin(N)$, $Sp(N)$, and $E_8$. Curiously, a corollary of our results is that the $SU(5)$ and $SO(10)$ grand unified theories have precisely the minimal field content needed to derive completeness using our methodology.

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

0 major / 3 minor

Summary. The paper proves that consistency of perturbative gravitational scattering, in the presence of a nonabelian symmetry G whose Cartan generates the abelian charge lattice and a weakly coupled UV completion of gravity, implies that any finite set of charged representations necessitates an infinite tower of single-particle states that completely fills the charge lattice. This holds specifically for G = SU(N) with N ≥ 3 and G = SO(N) with N ≥ 5 (and extensions to Spin(N), Sp(N), E8), with the SU(5) and SO(10) GUTs identified as having precisely the minimal content required by the argument.

Significance. If the derivation holds, the result supplies a dynamical origin for the completeness hypothesis directly from scattering consistency rather than from ad-hoc assumptions on the spectrum. It furnishes a concrete, representation-theoretic mechanism that forces the full lattice to be populated once a single fundamental representation is present, and it yields falsifiable statements about the minimal field content of grand-unified models. The argument is parameter-free once the three stated conditions are imposed and distinguishes single-particle poles from multi-particle thresholds via residue extraction.

minor comments (3)
  1. [§2.3] §2.3: the definition of the residue extraction contour could be stated more explicitly (e.g., by writing the explicit small-circle integral around the single-particle pole) to make the separation from multi-particle thresholds immediate for readers unfamiliar with the amplitude techniques.
  2. [Table 1] Table 1: the column headers for the initial finite representations would benefit from an additional footnote clarifying that the listed charges are normalized with respect to the longest root of G.
  3. [§1] The discussion of global versus gauged G in §1 could include a one-sentence remark on whether the argument requires the symmetry to be gauged or merely global, since the scattering consistency step appears to use only the global charge lattice.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. The referee's summary correctly captures our main result: that perturbative gravitational scattering consistency, combined with a nonabelian symmetry G whose Cartan generates the charge lattice and a weakly coupled UV completion of gravity, forces the abelian charge lattice to be fully populated by single-particle states for SU(N) with N≥3 and SO(N) with N≥5 (and extensions). We appreciate the recognition of the result's significance in providing a dynamical origin for completeness and its implications for GUT field content.

Circularity Check

0 steps flagged

Derivation self-contained from amplitude consistency and representation theory

full rationale

The paper establishes that finite charged representations under a nonabelian G, combined with perturbative gravitational scattering consistency in a weakly coupled UV completion, force the full abelian charge lattice to be populated by single-particle states for SU(N) N≥3 and SO(N) N≥5. This follows from residue analysis distinguishing poles from thresholds and group-theoretic closure, without any parameter fitting, self-definitional loops, or load-bearing self-citations. The initial finite spectrum and symmetry assumptions are external inputs, and the logical implication to completeness is independent of the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claim rests on three explicit domain assumptions listed in the abstract; no free parameters or new entities are introduced.

axioms (3)
  • domain assumption weakly coupled ultraviolet completion of gravity
    Required to justify perturbative scattering analysis
  • domain assumption nonabelian symmetry G whose Cartan subgroup generates the abelian charge lattice
    Defines the charge lattice that must be filled
  • domain assumption spectrum containing some finite set of charged representations (e.g., single fundamental)
    Starting point from which the infinite spectrum is deduced

pith-pipeline@v0.9.0 · 5517 in / 1498 out tokens · 45088 ms · 2026-05-16T22:50:14.373347+00:00 · methodology

discussion (0)

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Forward citations

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