An honest status report on the two developments introduced in Episode 4 ── do the work by hand and you see what is missing

Not a CFT yet At the end of Episode 4 we introduced holography for flat spacetime and soft hair.
Both were labelled "in progress" and "not accepted," but that is no diagnosis.
Where exactly does it stop? Celestial CFT breaks the moment you try to build a four-point function.
And that breakage has an anticlimactically simple physical reason.

Discipline #5 of this series was "do not publish a conjecture with no falsification condition." Episode 4 wrote that "celestial holography is unfinished" and "soft hair has objections," but that is just sticking labels on.
This episode does the work by hand. Run the Mellin transform, try to build a four-point function, and watch it break. Then track down why.
The conclusion first ── the breakage is very clean, and its physical meaning is clear. Not "it cannot be done yet" but "we understand why it has this shape." That is progress.

01First, what does work ── a plane wave becomes a conformal primary

Let us confirm celestial holography's starting point exactly, in one line.

Split a massless momentum into "energy × a point on the celestial sphere."

A Lorentz transformation acts on \(z\) as a Möbius transformation and on \(\omega\) as a scaling. So just diagonalize the scaling of \(\omega\) ── that is the Mellin transform.

A plane wave has turned into a conformal field on the sphere. So far, perfect. Because the four-dimensional Lorentz group \(SO(3,1)\cong SL(2,\mathbb{C})\) acts as the two-dimensional conformal group, this happens automatically.

02Try to build a four-point function and it breaks

The problem shows up when you transform a scattering amplitude rather than one particle.

In an ordinary 2d CFT, a four-point function is a smooth function of the cross ratio \(z\). Conformal symmetry fixes three points, leaving one degree of freedom: the cross ratio. The correlator can be written as a function of it.

Celestial CFT should take the same form ── so, expecting that, let us actually drop a 2→2 massless scattering event into celestial coordinates and compute the cross ratio.

assemble 4 momenta in the centre-of-mass frame, drop the directions onto celestial coordinates z_i = tan(θ/2)e^{iφ}
and compute the cross ratio z = (z12·z34)/(z13·z24)

 scattering angle   cross ratio z   |Im z|          -s/t
   30°      14.928203      1.7e-15      14.928203
   60°       4.000000      3.7e-16       4.000000
   90°       2.000000      1.2e-16       2.000000
  120°       1.333333      4.1e-17       1.333333

taking 4 random points on the celestial sphere (= ignoring scattering)
   Re(cross ratio)  |Im|
       0.5861     0.4765
       2.0586     2.1025
      -0.1783     0.4818

Two things become clear at once.

② is a happy find. The cross ratio of an abstract place like the celestial sphere corresponds to the most basic quantity of scattering.

But ① is the problem. Momentum conservation pins the cross ratio to the real axis. The cross ratio ought to have two dimensions' worth of freedom as a complex number, yet only one is actually reachable. Hence ──

The celestial four-point function is not a smooth function but a distribution. A different kind of object from an ordinary CFT correlator.

03Why it breaks ── because the scattering is planar

Track down the reason and it is anticlimactically simple.

Picture 2→2 scattering in the centre-of-mass frame. The two incoming momenta point exactly opposite each other, and so do the two outgoing ones. The four momenta necessarily lie in a single plane.

And directions within a plane through the origin trace out a great circle on the celestial sphere. Furthermore ──

So the chain is this.

momentum conservation → scattering is planar → a great circle on the sphere → the cross ratio is real → the correlator is a distribution

Celestial CFT is a "weird CFT" because 2→2 scattering is planar. Not a deep pathology but elementary kinematics.

Return the non-planarity to zero and the four points necessarily lie on one circle, with the imaginary part dropping to zero. Lift it even slightly and they leave the circle and an imaginary part appears ── and that configuration does not satisfy momentum conservation. It is unreachable.

04So what is missing?

Now we can write the diagnosis. When celestial holography is said to be "not a CFT yet," what concretely is lacking?

The third row is the essential one. What makes AdS/CFT strong is that the boundary side, \(\mathcal{N}=4\) supersymmetric Yang–Mills, is a theory defined independently of gravity, so both sides can be computed separately and cross-checked.

Celestial CFT has no such "independent definition" yet. It is at the stage of calling a rewriting of four-dimensional amplitudes by two-dimensional names. The rewriting itself is correct and useful, but it is not an instance of a duality.

05The other proposal, which did not survive ── soft hair

Let us properly diagnose the soft hair touched on in Episode 4.

The claim (Hawking–Perry–Strominger 2016): there are infinitely many supertranslation charges. Then a black hole should have infinitely many pieces of "soft hair," and the information that fell in may be recorded there. The no-hair theorem's "three hairs" was only about hard hair.

An attractive line. Indeed the existence of soft hair is broadly granted. The dispute is about what comes next.

The current view is this. Soft hair is real. But it does not solve the information paradox.

06So what did actually work?

For fairness, let us note that on the information-paradox side there was major progress somewhere else.

The island formula and replica wormholes (2019–20). From the gravitational path integral, the Page curve of the entanglement entropy of black-hole radiation was computed. Rising, turning over, and falling ── the behaviour of information coming back out actually came out.

And what matters for this series: that computation does not use string theory. Only the semiclassical gravitational path integral and the replica method. The main battleground was two-dimensional JT gravity.

07Diagnosis ── grading by the series' discipline

Let us line up the three things seen here against this series' discipline (falsification conditions, prior work, negative results).

Episode 1 wrote that "beauty is the entrance to a hypothesis, not evidence for it." Soft hair is exactly that case ── the line is beautiful and the structure real, and yet it did not do the job required of it.