Holographic QEC as AdS/CFT RG Flow on Bruhat–Tits Trees

In the paper's words

Tasks X3.1 and X3.2 established bosonic QEC as RG fixed-point subspaces on Bruhat–Tits trees. Task X3.3 completes the trilogy by showing that this structure is holographic: the Bruhat–Tits tree T_p is the p-adic analog of anti-de Sitter space, tensor networks on T_p are holographic QEC codes, and the AdS/CFT correspondence is the statement that the bulk RG flow (tree depth) is dual to the boundary conformal field theory (tree boundary). We establish: - The **Bruhat–Tits

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The Ryu–Takayanagi formula for entanglement entropy on $\mathcal{T}_p$ is discrete: the minimal surface is a vertex cut at depth $\ell$, and $S_{\text{EE}}(A) \propto \ell \cdot \log p$ — the $p$-adic valuation of the boundary interval length. p-Adic AdS/CFT is the natural setting for the Cross-Domain program: the Bruhat–Tits tree unifies the RG (tree depth), QEC (tree subtrees as codespaces), holography (tree as bulk geometry), and the SM mass spectrum (Pythagorean lattice as diagonal embedding of $\mathcal{T}_{2,3,5}$).

   from Abstract

2. Property Value Code type Holographic (HaPPY / random tensor network) Geometry $\mathcal{T}_2 \times \mathcal{T}_3 \times \mathcal{T}_5$ Logical dimension $d_L = 1$ (single logical qubit at the root) Physical dimension $d_P = \prod_{p} p^{L_p}$ (for trees truncated at depth $L_p$) Code rate $1 / \prod_p p^{L_p}$ (vanishes as $L_p \to \infty$) Code distance $\min_p p^{L_p}$ (determined by the shallowest tree) Erasure threshold $50\%$ of boundary sites (holographic code property) The code rate vanishes as the trees are taken deeper — this is the holographic encoding of finitely many bulk degrees of freedom in infinitely many boundary degrees of freedom.

   from 6.2 Code Properties

3. [CAL-X3-3-01, 2029] — The entanglement entropy of a subset of $L$ Fock states in any bosonic QEC code must exhibit $p$-adic stepwise scaling: $S_{\text{EE}}(L)$ should be approximately constant on intervals $[p^k, p^{k+1}-1]$ and jump by $\approx \log p$ at each power of $p$.

   from 9. Calibration

4. If entanglement entropy varies monotonically and continuously with $L$ with no $p$-adic steps, the $\mathcal{T}_p$ holographic interpretation is disconfirmed. [CAL-X3-3-02, 2030] — The boundary CFT effective central charge extracted from bosonic QEC entanglement scaling should satisfy $c \propto \log p$ for $\mathcal{T}_p$-based codes.

   from 9. Calibration

How it is done

1. Holographic renormalization is a finite procedure — the tree has finite depth, so UV divergences are automatically regularized.

   from 5.1 The $p$-Adic Holographic Correspondence

2. The discretuum of the tree provides a natural regulator without any need for a cutoff or renormalization procedure.

   from 5.2 Finite $p$ vs. the $p \to \infty$ Limit

3. Heavier particles (larger $a, b, c$ in $2^a \cdot 3^b \cdot 5^c$) require deeper circuits to prepare — a holographic complexity bound.

   from 8.3 Tensor Network State Complexity

4. A bosonic QEC experiment that couples resonators at different prime moduli and measures cross-talk between $\operatorname{ord}_2$ and $\operatorname{ord}_3$ syndromes should find zero correlation.

   from 9. Calibration

What would falsify it

1. For $\mathcal{T}_p$: $$c \propto \log p$$ For the joint tree $\mathcal{T}_{2,3,5}$, the total central charge is: $$c_{\text{total}} \propto \log 2 + \log 3 + \log 5 = \log 30$$ This is a prediction for the effective central charge of the "boundary theory" describing the SM particle spectrum. [speculative]

   from 8.2 CFT Central Charge from Tree Branching

2. If entanglement entropy varies monotonically and continuously with $L$ with no $p$-adic steps, the $\mathcal{T}_p$ holographic interpretation is disconfirmed. [CAL-X3-3-02, 2030] — The boundary CFT effective central charge extracted from bosonic QEC entanglement scaling should satisfy $c \propto \log p$ for $\mathcal{T}_p$-based codes.

   from 9. Calibration

3. If circuit depth and mass are uncorrelated, the holographic encoding hypothesis is disconfirmed. [CAL-X3-3-04, 2028] — [not yet falsifiable] The product structure $\mathcal{T}_2 \times \mathcal{T}_3 \times \mathcal{T}_5$ predicts that errors in different $p$-adic sectors are independent.

   from 9. Calibration

Where the claim stops

1. For a general $L$, the $p$-adic valuation $\operatorname{ord}_p(L)$ determines the tree depth: $$\ell(A) = \operatorname{ord}_p(L) + 1 \quad \text{(approximately)}$$ with corrections for boundary effects.

   from 4.2 $p$-Adic Valuation as Entanglement Measure

2. The continuum limit $p \to \infty$ is a classical approximation — the true quantum geometry is at finite $p$.

   from 5.2 Finite $p$ vs. the $p \to \infty$ Limit

3. [CAL-X3-3-01, 2029] — The entanglement entropy of a subset of $L$ Fock states in any bosonic QEC code must exhibit $p$-adic stepwise scaling: $S_{\text{EE}}(L)$ should be approximately constant on intervals $[p^k, p^{k+1}-1]$ and jump by $\approx \log p$ at each power of $p$.

   from 9. Calibration

Where to read

1. Abstract270 words · 1 min
2. 1. Introduction — Three Pillars, One Geometry252 words · 1 min
3. 2. The Bruhat–Tits Tree as $p$-Adic AdS387 words · 2 min
4. 3. Tensor Networks as RG Transformations on $\mathcal{T}_p$568 words · 2 min
5. 4. The Ryu–Takayanagi Formula on $\mathcal{T}_p$288 words · 1 min
6. 5. $p$-Adic AdS/CFT376 words · 2 min
7. 6. The Universal Holographic QEC Code290 words · 1 min
8. 7. The Full Adelic Dictionary190 words · 1 min
9. 8. Physical Predictions234 words · 1 min
10. 9. Calibration219 words · 1 min
11. 10. Conclusion279 words · 1 min