Time as Epistemic Emergence on a $p$-Adic Bruhat–Tits Tree: Static Constraint, Counting Laws, and the Observer-Generated Arrow

In the paper's words

The problem of time in quantum gravity blocks any straightforward union of general relativity, in which the time coordinate is dynamical, with quantum mechanics, in which evolution requires an external background parameter; until it is resolved, a non-perturbative quantum theory of gravity cannot specify what its observables are. Existing timeless proposals keep a continuum configuration space and leave the emergence of an experienced arrow unquantified. We formulate a conjectural but internally precise alternative: the fundamental ontology is a static, rooted (p+1)-regular Bruhat–Tits tree carrying a quantum state Ψ satisfying a Wheeler–DeWitt-type constraint HΨ=0 with H=A-(p+1)I built from the adjacency operator A; experienced time is a subsystem-relative ordering of node traversals, not an ontology-level parameter.

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

Numbers it reports

1. Let $\mathcal{T}_p=(X^0,X^1)$ be the rooted $(p+1)$-regular Bruhat–Tits tree: a non-empty connected graph without circuits in which every vertex belongs to $p+1$ edges [3], with a distinguished root $r$ of depth $d(r)=0$ and distance $d(x,y)$ the edge-count of the unique path [3].

   from 3.1 Ontology and notation

2. With all arithmetic shown, we derive for $p=2$: the depth-$10$ ball contains $V_{10}=3070$ events, of which only $C_{10}=27651$ ordered pairs out of $\binom{3070}{2}=4710915$ are temporally comparable, a fraction $f_{10}\approx 5.8696\times 10^{-3}$ with asymptotics $f_D\sim 2(p-1)D/((p+1)p^{D})\to 0$; the future cone of a depth-$5$ vertex holds $62$ events against $5$ ancestors at horizon $10$, growing to $65534$ against $5$ at horizon $20$; and a single observer path costs $10$ bits against $3070$ bits for the full depth-$10$ state, a compression factor of $307$.

   from Abstract

3. The boundary of such a tree is a $p$-adic projective line [11], and tree geodesics compute $p$-adic conformal correlators with a cross-ratio $u=p^{-d(c_1,c_2)}$ built from graph distance [8]; the tree is therefore a geometry on which Archimedean time is already absent, making it a natural substrate for a timeless ontology.

   from 1. Introduction

4. Second, node-to-node amplitude structure admits a semiclassical limit: the adjacency spectrum is bounded by the spectral radius $2\sqrt{p}$, which plays the role of a maximal propagation speed in the manner of the Lieb–Robinson bound invoked in [12], and the boundary $\partial\mathcal{T}_p\cong\mathbb{P}^1(\mathbb{Q}_p)$ supplies the real-line surrogate in a continuum limit, in line with the boundary constructions of [2].

   from 1. Introduction

5. A compression argument with numbers: a depth-$10$ observer path costs $10$ bits against $3070$ bits for the full ball, a factor $307$, making linear narrative the optimal code (Section 4.7).

   from 1. Introduction

6. A semiclassical speed bound $\rho(\hat{A})=2\sqrt{p}=2.8284271\ldots$ for $p=2$, adopted as the tree analogue of the Lieb–Robinson velocity of [12] (Section 4.8).

   from 1. Introduction

How it is done

1. With all arithmetic shown, we derive for $p=2$: the depth-$10$ ball contains $V_{10}=3070$ events, of which only $C_{10}=27651$ ordered pairs out of $\binom{3070}{2}=4710915$ are temporally comparable, a fraction $f_{10}\approx 5.8696\times 10^{-3}$ with asymptotics $f_D\sim 2(p-1)D/((p+1)p^{D})\to 0$; the future cone of a depth-$5$ vertex holds $62$ events against $5$ ancestors at horizon $10$, growing to $65534$ against $5$ at horizon $20$; and a single observer path costs $10$ bits against $3070$ bits for the full depth-$10$ state, a compression factor of $307$.

   from Abstract

2. Second, node-to-node amplitude structure admits a semiclassical limit: the adjacency spectrum is bounded by the spectral radius $2\sqrt{p}$, which plays the role of a maximal propagation speed in the manner of the Lieb–Robinson bound invoked in [12], and the boundary $\partial\mathcal{T}_p\cong\mathbb{P}^1(\mathbb{Q}_p)$ supplies the real-line surrogate in a continuum limit, in line with the boundary constructions of [2].

   from 1. Introduction

3. That work formalises the tree in the Lean 4 theorem prover on top of mathlib, using the Cartan decomposition $\mathrm{GL}_n(K)=\bigsqcup_{t\in T^-}\mathrm{GL}_n(R)\,t\,\mathrm{GL}_n(R)$ as the entry point, and verifies a result on harmonic cochains [1].

   from 2. Background and Related Work

4. We adopt as a standard spectral input (Assumption A1, used only in Section 4.8) that the spectrum of $\hat{A}$ is $[-2\sqrt{p},\,2\sqrt{p}]$, so the spectral radius is $\rho(\hat{A})=2\sqrt{p}$; this is the quantity identified with the dimensionless speed bound in [12].

   from 3.3 The constraint operator

What would falsify it

1. Limitations, falsifiability criteria, and a concrete next-step test (Sections 6–7).

   from 1. Introduction

2. Failure modes and falsification.

   from 6. Discussion

3. If a Lorentz-invariant observable ordering cannot be recovered from boundary correlators in the continuum limit, the model fails as physics regardless of the correctness of the counting.

   from 6. Discussion

Where the claim stops

1. The result rests on the following assumptions, each with a distinct sensitivity.

   from 6. Discussion

2. If records could be erased, the monotonicity of Section 4.6 fails and the arrow disappears.

   from 6. Discussion

3. The compression factor $307$ assumes one bit per branch choice and one bit per vertex for the ball.

   from 6. Discussion

Where to read

1. Abstract227 words · 1 min
2. 1. Introduction660 words · 3 min
3. 2. Background and Related Work1,236 words · 5 min
4. 3. Methods860 words · 4 min
5. 4. Analysis964 words · 4 min
6. 5. Results220 words · 1 min
7. 6. Discussion693 words · 3 min
8. 7. Practical Implications and Engineering Pathways571 words · 2 min
9. 8. Conclusion252 words · 1 min
10. Provenance and AI Involvement225 words · 1 min