Time as Bruhat-Tits Tree Depth: Discrete $p$-Adic Temporal Ordering and Tree-Geodesic Causality

In the paper's words

We develop a conjectural but internally precise model in which physical time is not a continuous real parameter but a discrete depth coordinate on a Bruhat-Tits tree: an event is a vertex of a rooted (p+1)-regular tree, and its time is the integer depth dinN, identified conjecturally with the p-adic valuation v_p(x). The gap addressed is that no existing framework derives quantization of time, partial temporal order, and acyclic causality from a single non-Archimedean geometric structure with exact, checkable combinatorics. Our approach is purely analytical: every consequence is an exact count on the rooted tree, derived with fully shown arithmetic and no simulation.

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

Numbers it reports

1. The asymptotic formula gives $f_{10} \approx 2\cdot1\cdot10/(3\cdot1024) = 20/3072 \approx 6.5104\times10^{-3}$, against the exact $f_{10} = 27651/4710915 \approx 5.8696\times10^{-3}$ computed above; the asymptotic form overestimates by a factor $6.5104/5.8696 \approx 1.109$ at $D=10$, as expected for a large-$D$ approximation used slightly below its asymptotic regime.

   from 4.5 Fraction of comparable pairs and its asymptotics

2. Free and unfixed; all counts scale with $p$, and the cone counts $62$ versus $363$ at $\Delta=5$ differ by a factor $\approx 5.85$, so the choice of $p$ is empirically consequential if any of these counts becomes observable.

   from 6. Discussion

How it is done

1. Our approach is purely analytical: every consequence is an exact count on the rooted tree, derived with fully shown arithmetic and no simulation.

   from Abstract

2. We adopt, as an explicitly labeled modeling postulate, that no operational procedure resolves time below the Planck scale, and that the real line is therefore not forced as the temporal substrate; this postulate is not established here.

   from 1. Introduction

3. For a spinor field theory on the tree $\mathrm{T}_{p}$, this work calculates the action and partition function of the boundary theory by integrating out the interior of the tree, following the scalar-field method of taking a cut-off boundary $E_{R}$ and the limit $R\to+\infty$, in which the boundary becomes $\mathbb{Q}_{p}$ [6].

   from 2. Background and Related Work

4. All results are obtained by exact combinatorial enumeration on the rooted $(p+1)$-regular tree.

   from 3.2 Method of analysis

What would falsify it

1. In practice, the work gives quantum-gravity researchers a falsifiable, exactly solvable skeleton in which discreteness, partial order, and acyclicity are theorems of the proposed structure rather than assumptions, with continuum recovery stated only as a labeled projection.

   from Abstract

2. (iii) A mapping of the Bruhat-Tits-tree literature onto the questions the conjecture must answer, including the boundary-theoretic route by which a $p$-adic continuum emerges from tree bulk degrees of freedom (Section 2), and an explicit statement of who could use the result, at what readiness level, and what the smallest falsifying test is (Section 7).

   from 1. Introduction

3. This reinforces [6] from a second direction: different tree subspaces yield the same boundary conformal field theory, suggesting a robustness of the tree-to-$\mathbb{Q}_{p}$ limit that a tree-time model would want to exploit — the observable content of time may live at the boundary, where the exponential bulk is projected out. [8] $p$-adic equiangular lines and the $p$-adic van Lint-Seidel bound.

   from 2. Background and Related Work

Where the claim stops

1. The identification $t \leftrightarrow v_p(x)$ is a conjecture, not a derivation; nothing in the combinatorics above forces the root to be a cosmological origin or one depth step to be a Planck time (Projection P2 is explicitly assumption-laden).

   from 6. Discussion

2. If empirical causal structure requires a continuum of causal directions at every event, the single-trunk tree model fails, and one would need either a product structure (tree $\times$ internal degrees of freedom) or a building rather than a tree — the direction suggested by the tree-to-building transfer maps of [5], which construct maps from Bruhat-Tits trees to Bruhat-Tits buildings but were not developed there for causal structure.

   from 6. Discussion

3. The $p$-adic Gibbs-measure machinery of [3] shows that ensemble statistics on regular trees can be solved exactly, but importing it would require a Hamiltonian or constraint, which this paper does not supply.

   from 6. Discussion

Where to read

1. Abstract207 words · 1 min
2. 1. Introduction640 words · 3 min
3. 2. Background and Related Work1,118 words · 5 min
4. 3. Methods274 words · 1 min
5. 4. Analysis1,014 words · 4 min
6. 5. Results322 words · 1 min
7. 6. Discussion720 words · 3 min
8. 7. Practical Implications and Engineering Pathways651 words · 3 min
9. 8. Conclusion237 words · 1 min
10. Provenance and AI Involvement225 words · 1 min