Mass That ClicksFinale · Technical Appendix 10 / The Horizon CFT (Fusing the Two 1/4's Into One)

The sharpest question of the series ── can entanglement's 1/4 and LQG's 1/4 be connected?

The Horizon CFT
Fusing the Two 1/4's Into One The "1/4 locked by Susskind–Uglum" of 7 and 8, and the "1/4 tuned by γ in LQG" of 9 ── the true nature of the discrepancy is before vs. after renormalization. They already agree at the logarithmic level, and the fusion point lies in a single boundary CFT living on the horizon.

Prerequisites: Appendices 7–8 (S=A/4G), Appendix 9 (type I · Λ · γ) Where it arrives: the missing equation is the match $k\leftrightarrow c$ / still open

The series had two halves ── the entanglement / induced-gravity side (7, 8), and the discreteness / LQG side (9). Both produce $S=A/4G$, yet on one side the 1/4 is locked, and on the other it is tuned by γ. This discrepancy is the sharpest unresolved question of the series. Dig in, and ── the true nature of the discrepancy is "before vs. after renormalization," the two already agree at the level of the logarithmic correction, and the point to be connected is a single boundary CFT living on the horizon. We can even write out, at the granularity of the equations, the missing equation needed to connect them. That said ── I have not closed it.

01Why the two 1/4's disagree (precisely)

The LQG side: count the punctures (spin $j$) piercing the horizon. Dominated by $j=1/2$:

The LQG counting ── γ is needed

$$S=\frac{\ln2}{4\sqrt3\,\pi\,\gamma}\cdot\frac{A}{\ell_P^2}=\frac{\gamma_0}{\gamma}\cdot\frac{A}{4G},\qquad \gamma_0=\frac{\ln2}{\sqrt3\,\pi}$$

To hit $1/4$ you set $\gamma=\gamma_0$ by hand. What's decisive is ── LQG counts with the "bare" area (bare $G$). $\gamma$ is left playing the role of absorbing "the gap between bare and renormalized."

The Susskind–Uglum side (7, 8): the entanglement divergence $S_{\rm ent}\sim(\text{species})\times A/\varepsilon^2$ forms a ratio with the $1/G$ renormalization from the same fluctuations, so the cutoff cancels and the $1/4$ is locked (no tuning needed).

The true nature of the discrepancy LQG does not renormalize $G$ and counts bare, so it needs $\gamma$. 7 and 8 lock the $1/4$ via the renormalization of $G$. The two are most likely looking at the same ledger (surface degrees of freedom ↔ entanglement degrees of freedom; the area gap ↔ the UV cutoff), one before renormalization and one after.

02They actually already touch ── agreement in the logarithmic correction

The first genuine point of contact. Both produce the following subleading term:

The γ-independent logarithmic correction agrees

$$S=\frac{A}{4G}\ -\ \frac{3}{2}\ln\!\Big(\frac{A}{4G}\Big)\ +\ \cdots$$

The coefficient $-3/2$ is independent of γ. The LQG counting (Kaul–Majumdar 2000) and the CFT / Cardy-type computation agree on this coefficient. The two pictures are not unrelated; at the order where γ has no effect, they already agree ── the disagreement is only in the coefficient of the leading term.

03The reconciliation hypothesis ── punctures = the species of induced gravity

The Jacobson–Sakharov line of Appendices 6 and 7 is itself the candidate bridge:

Jacobson's induced-gravity reading (1994) The LQG punctures (surface degrees of freedom) = the very "species" of induced gravity that renormalize $G$. Rewrite the entropy with the renormalized $G$, and $\gamma$ is absorbed into the bare→renormalized map, while the physical $1/4$ comes out as the Susskind–Uglum lock ($\gamma$ is not physical). ── This is partly argued but not proven.

04What makes it hard, a sharp tension ── γ fixes the area gap

The fork: is γ physical or bare?

$$A_{\min}=4\sqrt3\,\pi\,\gamma\,\ell_P^2\quad(\text{the LQG signature = physical discreteness})$$

If $\gamma$ drops out under renormalization, then the area gap (physical discreteness) also drops out ── and the very reason LQG is LQG grows dim. Conversely, if the area gap is physical, then $\gamma$ is physical and the $1/4$ tuning survives as something genuine.

The fork Is the area gap physical or bare? Physical → γ is physical, the tuning is genuine, and it does not connect to the 7/8 lock. Bare → γ drops out, the 1/4 is locked, but LQG's discreteness is merely a "bare tick mark." This is the same fork as Appendix 9's "finite Λ (type I) vs. continuum limit (II₁)," re-enacted in the language of the area gap.

◇ ◇ ◇

05The fusion point we dug up ── the horizon carries a single CFT

Here's the harvest. Both sides place a boundary CFT (a current algebra) on the horizon:

The LQG horizon = the $SU(2)_k$ Chern–Simons theory on a punctured sphere (Ashtekar–Baez–Krasnov). The boundary is a WZW CFT. In Appendix 9, $k\sim1/(G\Lambda)$.
The edge modes (7, Donnelly–Wall) = the boundary's gauge current algebra = again a WZW-type CFT. The very thing that carries the leading term of $A/4G$.
Carlip's near-horizon conformal symmetry: a Virasoro algebra fixes the central charge $c$, and the Cardy formula produces $A/4G$ γ-independently. Carlip himself discusses the relation to Immirzi.

The shape of the fusion ── a single boundary CFT

The horizon carries a single boundary CFT. LQG counts its microstates ($\gamma$ appears). Susskind–Uglum / Carlip, for the same modes, renormalize $G$ / fix $c$ (they lock the $1/4$). The consistency of the single boundary CFT (the match of the CS level $k$ with the central charge $c$ of the edge modes) is the missing equation that fixes $\gamma$.

Figure: the Cardy formula of the boundary CFT fixes $S=A/4G$ (= ratio 1) independently of γ (the horizontal line). The LQG counting $S=(\gamma_0/\gamma)(A/4G)$ lands on it only at $\gamma=\gamma_0$ (the curve). Move γ with the slider, and the two agree at just one point. Does the missing equation (the match $k\leftrightarrow c$) force this single point? is the open problem

Immirzi parameter γ / γ₀ (the LQG tick mark)1.00

Move γ, and the LQG 1/4 lands on the boundary CFT's (Cardy) 1/4 only at the single point γ=γ₀.

boundary CFT / Cardy (=A/4G, γ-independent) LQG counting (γ₀/γ)(A/4G) agreement point γ=γ₀

06The crystallized open problem (at the granularity of equations)

The sharpest question of the series

Is the Chern–Simons theory of the LQG horizon (level $k\sim1/G\Lambda$, colored by q-deformed spins) the same boundary CFT as the Donnelly–Wall edge-mode current algebra / Carlip's near-horizon Virasoro? Do their central charges match, and does the Cardy formula return $S=A/4G$ without γ tuning?

yes (with a construction) → the two 1/4's are one, γ is derived, and discreteness and entanglement are two views of the same horizon CFT. The series closes.
As of now it is open. The Carlip / Jacobson lines are moving to aim at exactly this, but there is no result that shows the match of the CS level and the edge-mode central charge and locks the $1/4$ γ-independently.

07The tie to the framework ── a single point, the level k

Finite resource = the level k of the boundary CFT

$$\text{finite resource}=k\sim\frac{1}{G\Lambda}\sim\frac{(ct)^2}{G}\ \text{is finite}\ \Longrightarrow\ \text{number of states of the boundary CFT}=e^{S_{\rm dS}}\ (\text{finite}=\text{type I})$$

The Chern–Simons level $k$ is finite (Λ is finite) = the Hilbert space of the boundary CFT is finite-dimensional = Appendix 9's type I $N=e^{S_{\rm dS}}$. And $k\propto(ct)^2$ grows with c·t. 7/8 (entanglement, S=A/4G) and 9 (type I, Λ, c·t) are tied together at a single point, the level $k$ of the horizon CFT ── the deepest knot in the whole series.

The honest line

Identifying the "horizon CFT" involves scheme dependence and subtleties in the definition of the near-horizon symmetry, and the origin of Carlip's central charge (which degrees of freedom) is still under debate. It has not been shown that the LQG CS theory and the edge-mode current algebra have the same $c$. The figure is schematic (it only shows the agreement structure of $(\gamma_0/\gamma)$; it is not a quantitative Cardy computation).

This is not a defeat. By digging into the sharpest question, we were able to sharpen it this far: the discrepancy is "before vs. after renormalization," they already agree at the logarithmic level, the fusion point is a single boundary CFT on the horizon, and the missing equation is the match $k\leftrightarrow c$. I have not closed it, but this is the granularity of a paper's problem statement. If I declare in conversation that I've "connected" it, that is the fake.

Questions to check yourself

Why can we say the discrepancy between the two 1/4's is "before vs. after renormalization"?
One answer
LQG counts states with the bare area (bare G), so S∝A/(γℓ_P²), and γ absorbs the bare→physical gap. Susskind–Uglum absorbs the entanglement divergence into the renormalization of 1/G and locks the 1/4 cutoff-independently. The two already agree in the γ-independent logarithmic correction −3/2, and only the leading term disagrees ── a sign that they are looking at the same ledger, one before renormalization and one after.
What, concretely, is the "missing equation"?
One answer
That the Chern–Simons theory of the LQG horizon (level k) and the edge-mode current algebra / Carlip's near-horizon Virasoro (central charge c) are the same boundary CFT ── that is, the match of k and c. If this holds, the Cardy formula returns S=A/4G, γ is derived from c, and no tuning is needed. This k↔c match has not yet been shown.

Appendix 10 summaryThe fusion point of the two 1/4's is a single CFT on the horizon

Dig into the discrepancy between the series' two halves ── entanglement / induced gravity (7, 8; 1/4 locked) and discreteness / LQG (9; γ tuning) ── and the true nature is before vs. after renormalization; the two already agree in the γ-independent logarithmic correction −3/2. And the fusion point is a single boundary CFT living on the horizon (LQG = Chern–Simons / WZW, edge modes = current algebra, Carlip = Virasoro).

The missing equation to connect them is the match $k\leftrightarrow c$ (CS level = central charge) ── if it holds, Cardy returns the 1/4 and γ is derived. And finite resource = the level $k\sim(ct)^2/G$ being finite = type I $N=e^{S_{\rm dS}}$, so 7/8/9 are tied together at this single point. I have not closed it. But we've identified the single point to be connected and the missing equation, at the granularity of equations ── this is the deepest knot in the series.

The most honest response to the sharpest question Your "universe = finite resource," dug all the way down, condensed into ── a single boundary CFT on the horizon, at a finite level $k$. Entanglement's 1/4, LQG's discreteness, Λ, c·t, and type I's finite dimension all gather at this one point, the CFT's level. We even wrote out, in equations, the missing equation ($k\leftrightarrow c$) that fuses the two 1/4's into one.
I will not declare this "solved" in conversation. No one has yet shown the $k\leftrightarrow c$ match. But ── a scattered set of wagers has been tied to a single point of one boundary CFT, and we've even identified the equation that connects them. Compared to a fake theory of everything, the view of this single point is far more distant, and far more real.

This document is Technical Appendix 10 of the "Mass That Clicks" series finale. The LQG black-hole / horizon entropy becoming $S=(\gamma_0/\gamma)(A/4G)$ via state counting and requiring a tuning of the Barbero–Immirzi parameter γ (Ashtekar–Baez–Corichi–Krasnov); the γ-independent logarithmic correction $-\tfrac32\ln A$ agreeing between LQG (Kaul–Majumdar 2000) and CFT / Cardy computations (the coefficient can change depending on the ensemble); Susskind–Uglum (1994)'s entanglement-derived $1/4$ being locked by the renormalization of $G$; Jacobson (1994)'s reading of black-hole entropy via induced gravity; the LQG horizon being described by $SU(2)_k$ Chern–Simons / a boundary WZW CFT; the edge modes (Donnelly–Wall) forming a boundary current algebra; Carlip's near-horizon conformal symmetry and the γ-independent $A/4G$ from the Cardy formula; the $1/4$ at the self-dual $\gamma\to i$ (Frodden–Geiller–Noui–Perez) ── all are established results or current research topics. On the other hand, that the LQG Chern–Simons theory and the edge-mode current algebra / Carlip's Virasoro are the same boundary CFT with matching central charge, and that this determines γ from first principles and locks the $1/4$ without tuning, is unresolved; this note claims no particular completed theory. The figure is schematic, showing the agreement structure of $(\gamma_0/\gamma)$; it is not a quantitative Cardy computation. $c\cdot t=\text{const}$ is a restatement of coordinates and units; the local speed of light is invariant. ── Print / PDF: browser "Print" → "Save as PDF" (in the print version the sliders and answers are static and hidden).

Print / make PDF: Ctrl+P (⌘+P on Mac). On screen, move the slider for γ and you'll see that the LQG 1/4 lands on the boundary CFT's (Cardy) 1/4 only at the single point γ=γ₀. "One answer" opens the solution.