Flat and Celestial Dictionaries: Unitarity, Completeness, and Status

Flat-boundary holography now contains several controlled dictionaries but no single construction should be credited with components it has not supplied. This page compares celestial and Carrollian proposals against the same fixed-theory questions: states and inner product, operator algebra, dynamics, infrared completion, massive sectors, loops, reconstruction, unitarity, factorization, and nonperturbative definition. The evidence cutoff is 10 August 2026.

Required background. Infrared-Dressed Scattering States and Boundary Dictionaries fixes the physical asymptotic states; Celestial OPEs, Loop Corrections, and Infrared Factorization supplies perturbative operator data; and Carrollian and Other Flat-Boundary Holography Proposals: Evidence and Obstructions supplies the null-boundary formulation.

Helpful background. AdS Flat Limits as Candidate Flat-Holography Dictionaries gives a controlled extraction from a complete AdS pair, while Claim Status, Freshness, and Research Handoffs explains how a changing frontier conclusion should be maintained.

Celestial dictionaries

Established components. Massless one-particle states admit a principal-series conformal basis with a Mellin inverse. Transformed amplitudes carry exact Lorentz covariance and retain momentum-conservation distributions. Soft and collinear limits generate current and OPE-like singularities; explicit loop calculations display infrared factors, higher poles, and mixing. These results are reviewed with their domains in Pasterski 2021, while a concrete loop analysis appears in Krishna 2024.

Open components. An infrared-complete interacting Hilbert space and conjugation have not been established for general gravitational scattering. Massive conformal-primary bases exist, but incorporating every massive and bound sector into one interacting operator algebra is a further task. Collinear expansions do not yet supply a globally controlled local OPE, and the Mellin inverse reconstructs the input S-matrix rather than independently deriving all bulk observables. No generally accepted nonperturbative fixed-theory celestial dynamics is selected by these facts alone.

Carrollian dictionaries

Established components. Null infinity has canonical Carrollian geometry, BMS acts conformally upon it, and sourced Carrollian Ward identities can reproduce gravitational flux-balance laws. Integral transforms connect appropriate Carrollian correlators to celestial ones, as shown by Donnay et al. 2023. Explicit free and symmetry-controlled models continue to clarify boundary states and currents.

Open components. Matching Ward identities does not uniquely determine interacting dynamics or an inverse reconstruction of all news, Coulombic, corner, and timelike-infinity data. Infrared zero modes are structurally important: Fredenhagen, Prohazka, and Tiefenbacher 2026 find, in explicit free Carrollian systems, sectors with nonregular distinguished states or a nonseparable zero-mode factor. These model results sharpen rather than remove the need for a physical inner product and full gravitational state space.

A matched comparison

Both programs organize exact asymptotic symmetries and scattering constraints. Celestial variables diagonalize boosts and compress retarded-time dependence into conformal weights; Carrollian variables retain null time and make sourced flux laws local on $\mathscr I$. Their integral relation can be useful without making their axioms identical. Neither covariance nor an invertible transform establishes fixed-theory factorization, black-hole sectors, or nonperturbative completeness.

First application. Compare one celestial and one Carrollian proposal against the same fixed-theory dictionary checklist, citing a primary calculation and an independent or contrary assessment for every substantive claim. Mark a component “shown” only when the cited calculation constructs it; record symmetry evidence, basis completeness, perturbative dynamics, and full duality as distinct conclusions.

Stress tests

Adversarial control. Require both proposals to include loop-level gravitational infrared effects, a massive external state, and a fixed-theory optical theorem. Then request two inverse maps: one back to the complete dressed S-matrix and one to asymptotic observables not used as input. A proposal that succeeds only for massless tree amplitudes remains a controlled partial dictionary rather than a complete dual.

A second test compares two scattering processes with identical chosen charges but different news or hard amplitudes. Charge or current data that cannot distinguish them are not complete observables. Conversely, if full celestial amplitudes distinguish them only because the S-matrix was supplied, the transform has preserved rather than derived the information.

Evidence ceiling

The literature through the stated cutoff establishes rich, mutually connected symmetry, basis, factorization, and Ward-identity structures. It does not establish a complete unitary and nonperturbative boundary theory for four-dimensional asymptotically flat quantum gravity. Future progress should update the missing component it actually resolves rather than retroactively promoting every existing map.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Donnay, Laura, Adrien Fiorucci, Yannick Herfray, and Romain Ruzziconi. “Bridging Carrollian and Celestial Holography.” Physical Review D 107 (2023): 126027. DOI; Open PDF.
• Fredenhagen, Stefan, Stefan Prohazka, and Robert Tiefenbacher. “Carrollian Quantum States and Flat Space Holography.” (2026). arXiv:2604.22745.
• Krishna, Hare. “Celestial Gluon and Graviton OPE at Loop Level.” Journal of High Energy Physics 2024, no. 3 (2024): 176. DOI; Open PDF.
• Pasterski, Sabrina. “Lectures on Celestial Amplitudes.” The European Physical Journal C 81 (2021): 1062. DOI; Open PDF.
• Ruzziconi, Romain. “Carrollian Physics and Holography.” (2026). arXiv:2602.02644.