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

#1. Introduction

The continuum model of time — a real parameter $t \in \mathbb{R}$ with a total order — is inherited from calculus, not from measurement. 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, and the continuum model of time — a real parameter $t \in \mathbb{R}$ with a total order — is inherited from calculus rather than derived from it. This paper takes seriously an alternative suggested by non-Archimedean geometry: that time is a discrete depth coordinate on a Bruhat-Tits tree, the combinatorial tree on which $p$-adic valuation becomes literal distance.

A Bruhat-Tits tree is an infinite $(p+1)$-regular tree (a connected acyclic graph in which every vertex has exactly $p+1$ neighbors) associated with the $p$-adic field $\mathbb{Q}_p$; its vertices correspond, in the standard construction, to homothety classes of lattices, and edges change the $p$-adic valuation by one. The core conjecture is:

where $v_p$ is the $p$-adic valuation on $\mathbb{Q}_p$ and $d(v)$ is the depth of the vertex $v$ representing an event, counted as the number of edges on the unique path from a distinguished root $v_0$. The root is the vertex of minimal depth and is provisionally identified with the Planck-scale origin of a causal history.

Three consequences are developed:

1. Quantization. Depth is integer-valued, so time comes in indivisible steps of a fundamental tick $t_0$; all measurable durations are integer multiples of $t_0$.
2. Partial ordering. Two vertices neither of which is an ancestor of the other have no intrinsic temporal relation. Temporal order is a partial order, not a total order.
3. Tree-geodesic causality. $A$ causes $B$ iff $B$ lies on the unique tree path from $A$ to the root. Because the tree is acyclic and every vertex has exactly one parent, this relation is automatically acyclic and antisymmetric.

The strong triangle inequality of the $p$-adic valuation,

has as its combinatorial shadow the fact that tree distance between two events is determined by the depth of their most recent common ancestor — a fact with no Archimedean analogue.

We stress the epistemic status of the proposal: it is a conjecture. The contributions are (i) a precise formalization, (ii) exact combinatorial consequences derived with full arithmetic, and (iii) a mapping of the existing Bruhat-Tits-tree literature onto the questions the conjecture must answer. No experiment is proposed that has been run, and no simulation is reported; the tree is exactly solvable, so closed-form statistics replace Monte Carlo.

#2. Background and Related Work

We review the twelve supplied bibliography entries: eight arXiv works on Bruhat-Tits trees and $p$-adic structures, and four QNFO corpus entries.

[1] On the missing branches of the Bruhat-Tits tree (arXiv:1712.01463v2). For a local field $k$ and the two-by-two matrix algebra $A$ over $k$, this work develops a theory for computing the set of maximal orders in $A$ containing a given suborder; this set is realized as a sub-tree of the Bruhat-Tits tree called the branch of the order, and branches have been used to study the global selectivity problem and to compute local embeddings. The notion of a branch — a sub-tree selected by an algebraic condition — is structurally the object our model needs: the causal past of an event is precisely the sub-tree of its ancestors, and [1] supplies the precedent that such sub-trees are well-defined and computable. The supplied summary gives no further detail, so we use it only for this precedent.

[2] Bending the Bruhat-Tits Tree II: the p-adic BTZ Black hole and Local Diffeomorphism on the Bruhat-Tits Tree (arXiv:2102.12024v2). This work takes a second approach to bending the Bruhat-Tits tree and, inspired by the BTZ black hole connection, transplants the BTZ construction to the tree at the cost of defining a novel "exponential function" on the $p$-adic numbers hinted at by the tree, with $PGL(2,\mathbb{Q}_p)$ Wilson lines evaluated on the analogue BTZ connection. This matters because it shows that gravitational-type causal infrastructure (horizons, connections, Wilson lines) can be carried on tree geometry — a necessary precedent if tree-geodesic causality is to reproduce relativistic causal structure in any limit. The supplied summary is truncated mid-sentence and gives no further results.

[3] On Phase Transitions for $P$-Adic Potts Model with Competing Interactions on a Cayley Tree (arXiv:math-ph/0512018v2). This paper studies the three-state $p$-adic Potts model with competing interactions on a Cayley tree of order two, reduces the description of $p$-adic Gibbs measures to a recursive equation, and proves that a phase transition occurs if and only if $p=3$ for any nonzero value of the interactions, also completely solving the uniqueness problem. This establishes that ensemble statistics on regular trees — the machinery needed to check "recovery of standard temporal statistics" over ensembles of tree-ordered events — is mathematically mature, with exact criteria (here the $p=3$ threshold) rather than approximations. If physical time is a tree, the statistics of temporal observables may inherit such $p$-dependent singularities.

[4] Mumford dendrograms and discrete p-adic symmetries (arXiv:0809.1570v1). This work presents an effective encoding of dendrograms by embedding them into Bruhat-Tits trees associated to $p$-adic number fields, shows how strings over a finite alphabet can be encoded in cyclotomic extensions of $\mathbb{Q}_p$, discusses $p$-adic DNA encoding, and obtains fast $p$-adic agglomerative hierarchic algorithms. Dendrograms are exactly hierarchical partial orders of the kind our model assigns to event sets; this establishes that arbitrary tree-structured orderings can be faithfully embedded into Bruhat-Tits trees, supporting the representational adequacy of the tree for partially ordered event sets, and suggesting (speculatively) a coarse-graining mechanism that merges incomparable branches.

[5] On $p$-adic colligations and 'rational maps' of Bruhat-Tits trees (arXiv:1301.5453v1). Here matrices of order $k+N$ over a $p$-adic field are considered up to conjugation by $GL$ over the $p$-adic integers; a product of such conjugacy classes is defined, analogues of characteristic (transfer) functions are constructed as maps from Bruhat-Tits trees to Bruhat-Tits buildings, and categorical quotients for operator colligations are examined. Transfer maps from trees to buildings are the natural language for describing how local tree-causal structure could compose into larger geometric structures — the direction in which a tree-time model would need to be extended if spatial degrees of freedom are added. The supplied summary is truncated and gives no further quantitative detail.

[6] The boundary theory of a spinor field theory on the Bruhat-Tits tree (arXiv:1910.09397v2). For a spinor field theory on the Bruhat-Tits tree, the action and partition function of its boundary theory are calculated by integrating out the interior of the tree, and the boundary theory is found to be very similar to a scalar field theory over the $p$-adic numbers. This is the closest existing result to our continuum-limit question: bulk tree degrees of freedom integrate out to a boundary theory on $\mathbb{Q}_p$, i.e., a $p$-adic continuum emerges from the tree in a controlled field-theoretic sense. Our conjecture requires the analogous statement for the time coordinate.

[7] Effective field theories on subspaces of the Bruhat-Tits tree (arXiv:2402.03730v2). On two subspaces of the Bruhat-Tits tree, effective actions are calculated, and the limits of these effective field theories are found to be the same conformal field theory over the $p$-adic numbers when the subspaces are taken to the boundary of the tree; relations to the $p$-adic version of AdS/CFT are discussed. This reinforces [6] from a second direction: different tree subspaces yield the same boundary conformal field theory, suggesting a robustness (universality) of the tree-to-$\mathbb{Q}_p$ limit that any tree-time model would want to exploit.

[8] p-adic Equiangular Lines and p-adic van Lint-Seidel Relative Bound (arXiv:2408.00810v3). This work introduces $p$-adic equiangular lines and derives the first fundamental relation between common angle, dimension, and number of lines: if $\{\tau_j\}_{j=1}^{n}$ is a set of $p$-adic $\gamma$-equiangular lines in $\mathbb{Q}_p^d$, then $|n|^2 \leq |d|\max\{|n|, \gamma^2\}$, the $p$-adic van Lint-Seidel relative bound. This shows that $p$-adic magnitude arithmetic supports packing bounds analogous to the Archimedean ones; in a tree-time model, such bounds would constrain how many mutually distinguishable "directions" (branch choices) can coexist at a given angular scale. The supplied summary is truncated after the statement of the bound, so we draw no further consequence from it.

[9]–[12] (QNFO corpus). Entry [11], "Invariant Patterns and the Adelic Refactoring of Fundamental Physics," proposes a systematic refactoring of fundamental physics around the principle that physics is the study of invariant patterns under specified classes of transformations, stripping away anthropocentric conventions including measurement units and the privileged status of Archimedean fields (the summary is cut off at this point); our model is a concrete instance of that de-Archimedization applied to time. Entries [9] ("Formal Derivation of Emergent Spacetime to the Bruhat-Tits Tree"), [10] ("Bruhat-Tits Tree as a Unifying Geometric Object"), and [12] ("Time as Epistemic Cognitive Fiction") have empty supplied summaries; we relate them to our argument only through their titles — indicating a tree-centric emergence program and a skeptical stance toward continuous time as a cognitive construct — and attribute no specific result to them.

#3. Methods

#3.1 Definitions

Definition 3.1 (Event). An event is a vertex $v$ of the Bruhat-Tits tree $\mathcal{T}_{p}$, a $(p+1)$-regular tree with a distinguished root $v_0$.

Definition 3.2 (Time). The time of an event $v$ is its depth $d(v) \in \mathbb{N}$, the number of edges on the unique path from $v_0$ to $v$. Under the conjectured identification, $d(v)$ corresponds to the $p$-adic valuation of the associated coordinate, and the root corresponds to the Planck-scale origin. All temporal intervals are integer multiples of a fundamental tick $t_0$: $t(v) = d(v)\,t_0$.

Definition 3.3 (Temporal order). For events $A, B$: $A \prec B$ iff $A$ lies on the unique path from $B$ to the root (i.e., $A$ is a proper ancestor of $B$). Events neither of which is an ancestor of the other are incomparable, written $A \parallel B$.

Definition 3.4 (Tree-geodesic causality). $A$ causes $B$ iff $B$ lies on the unique tree path from $A$ to the root, i.e., iff $A \prec B$. Causality coincides with ancestry by construction.

Definition 3.5 (Tree distance). The graph distance $\delta(A,B)$ is the number of edges on the unique path between $A$ and $B$; for least common ancestor $w$ at depth $d_w$, $\delta(A,B) = d(A) + d(B) - 2d_w$.

#3.2 Method of analysis

All results in Section 4 are exact combinatorial counts on the $(p+1)$-regular rooted tree, derived by induction and direct summation. No simulation is performed. The single free kinematic parameter is the prime $p$; the depth cutoff $D$ is a modeling choice for finite-horizon computations. Continuum recovery is treated only as an explicitly labeled projection (Section 4.7).

#4. Analysis

Every input number below is either a structural constant of the model (the regularity $p+1$ of the Bruhat-Tits tree) or an explicit modeling choice ($D$). All arithmetic is shown.

#4.1 Vertex counts by depth

Claim 4.1. The number of vertices at depth $d$ is

Derivation. The root $v_0$ has $p+1$ neighbors, so $N_1 = p+1$. Each vertex at depth $d-1$ has $p+1$ neighbors, of which one is its parent at depth $d-2$ (for $d \geq 2$), leaving $p$ children at depth $d$. Hence $N_d = p\,N_{d-1} = (p+1)\,p^{\,d-1}$. $\blacksquare$

Worked instance ($p=2$). $N_1 = 3$, $N_2 = 6$, $N_3 = 12$, $N_4 = 24$, $N_5 = 48$, $N_6 = 96$, $N_7 = 192$, $N_8 = 384$, $N_9 = 768$, $N_{10} = 3\cdot 2^{9} = 1536$.

#4.2 Total vertices up to depth $D$

Claim 4.2. The total number of vertices with $d(v) \leq D$ is

Derivation. Sum the geometric series $\sum_{d=1}^{D} p^{\,d-1} = \frac{p^D - 1}{p-1}$, multiply by $p+1$, and add the root. $\blacksquare$

Worked instance ($p=2$, $D=10$). Since $p-1 = 1$:

Check by summing the $N_d$ of Section 4.1: $1+3=4$; $+6=10$; $+12=22$; $+24=46$; $+48=94$; $+96=190$; $+192=382$; $+384=766$; $+768=1534$; $+1536=3070$. ✓

#4.3 Comparable pairs

Claim 4.3. For depths $d_1 \lt d_2$, the number of unordered comparable pairs $\{A, B\}$ with $d(A) = d_1$, $d(B) = d_2$ is exactly $N_{d_2}$, independent of $d_1$.

Derivation. Each vertex $A$ at depth $d_1$ has exactly $p^{\,d_2 - d_1}$ descendants at depth $d_2$. The number of pairs is

The independence of $d_1$ is the combinatorial signature of the ultrametric: how far up the tree one sits does not affect how much lies below.

#4.4 Total comparable pairs up to depth $D$

Claim 4.4. The total number of unordered comparable pairs among vertices of depth $\leq D$ is

Derivation. A vertex $B$ at depth $d$ has exactly $d$ proper ancestors, so it participates in exactly $d$ comparable pairs as the deeper member; summing over all $B$ gives the claim. $\blacksquare$

Closed form for $p = 2$. Let $S_D = \sum_{d=1}^{D} d\,2^{\,d-1}$. We verify $S_D = (D-1)2^{D} + 1$ by induction: $S_1 = 1 = 0\cdot 2 + 1$. ✓ If $S_{D} = (D-1)2^D + 1$, then

Worked instance ($p=2$, $D=10$).

#4.5 Fraction of comparable pairs

Claim 4.5. For $p = 2$, $D = 10$, the fraction of event pairs that are temporally ordered is

Derivation. $\binom{3070}{2} = \frac{3070 \cdot 3069}{2} = 3070 \cdot 1534.5 = 4{,}709{,}380 + 1535 = 4{,}710{,}915$. Then

Check: $4710915 \times 0.0058696 = 27651.0$; the quotient is $0.0058696$ to seven places. ✓

Asymptotics. For large $D$ with fixed $p$: $V_D \sim (p+1)\frac{p^D}{p-1}$ and $C_D \sim (p+1)\,D\,\frac{p^D}{p-1}$ (the sum $\sum_d d\,p^{\,d-1}$ is dominated by its last terms). Hence

Temporal comparability is exponentially rare in depth: most pairs of events in this model are unordered. This is the quantitative content of "time is only partially ordered."

Complementary sampling statistic. Choose $u$ uniformly among depth-$D$ leaves and $w$ uniformly among all of $V_D$. Under Definition 3.3, $w \prec u$ iff $w$ is a proper ancestor of $u$, i.e., $w$ lies on $u$'s root path at depth strictly less than $D$; the root path contains $D+1$ vertices including $u$ itself, so the number of proper ancestors is $D$. Thus

Distinct same-depth leaves are incomparable with probability $1$, since ancestry strictly decreases depth.

#4.6 Mean depth

Under the uniform distribution on $V_D$, the mean depth is $\bar{d} = C_D / V_D$ (numerator as in Claim 4.4). For $p=2$, $D=10$:

The mean sits within roughly $10\%$ of the cutoff $D=10$: the gap is $\frac{10 - \bar{d}}{10} = \frac{10 - 27651/3070}{10} = \frac{10 - 9.0068}{10} = \frac{0.9932}{10} \approx 9.9316\%$, because the leaf shell dominates the ball: $N_{10} = 1536$ of the $3070$ vertices, i.e., $1536/3070 \approx 50.0326\%$, lie at maximal depth. A testable consequence: a uniform-on-ball event distribution has an exponentially growing depth histogram $N_d = (p+1)p^{\,d-1}$, not a flat one.

#4.7 Acyclicity of tree-geodesic causality

Claim 4.6. The causality relation of Definition 3.4 admits no causal cycles.

Derivation. Suppose a closed chain $A_1 \to A_2 \to \cdots \to A_k \to A_1$ with $A_{i+1} \prec A_i$ for each $i$. Ancestry strictly decreases depth, so $d(A_1) \gt d(A_2) \gt \cdots \gt d(A_k) \gt d(A_1)$, impossible for integers. Equivalently, transitivity would force $A_1 \prec A_1$, contradicting irreflexivity (a vertex is not a proper ancestor of itself). $\blacksquare$

This addresses the closed-timelike-curve concern at the kinematic level: "going back in time" would require revisiting an ancestor, but every vertex has exactly one parent, so no alternative route to any ancestor exists.

#4.8 Future-cone size

Claim 4.7. The future of an event $u$ at depth $d$ consists of its descendants; within $\Delta$ additional ticks (depths $d+1$ through $d+\Delta$) their number is

Worked instances. For $p=2$, $\Delta=5$: $C(5) = 2+4+8+16+32 = 62$, and $(2^{6}-2)/(2-1) = 62$. ✓ For $p=3$, $\Delta=5$: $C(5) = 3+9+27+81+243 = 363$, and $(3^{6}-3)/2 = (729-3)/2 = 363$. ✓

In a $1+1$-dimensional continuum light cone with unit spacing, the number of lattice points in the future cone grows quadratically ($\sim \Delta^2/2$), whereas the tree future cone grows exponentially, $p^{\Delta}$. The structural mismatch with the relativistic light cone — whose past is a solid cone with a continuum of directions, whereas the tree past is a finite chain of $d$ ancestors — is discussed in Section 6; we do not claim recovery.

#4.9 Continuum limit (labeled projection)

Projection P1, with stated assumptions. Assume (i) the physical tick is $t_0$ and observable durations $T$ correspond to depth $D = T/t_0$; (ii) the number of distinguishable events at resolution $t_0$ in an interval $T$ is $V_D$. Then

i.e., exponential growth in $T/t_0$. For this to match continuum-like polynomial scaling, either $t_0$ must run or the observable must be a boundary quantity: the results of [6] and [7], in which integrating out the tree interior yields a $p$-adic boundary theory, suggest that continuum-relevant observables live at the tree boundary, where the exponential bulk is projected out. Uncertainty: this is a structural projection, not a measurement; its validity hinges on assumption (ii), and no numerical value of $t_0$ is asserted, since no input in this paper fixes the physical scale of one depth step.

#5. Results

All numbers below are computed in Section 4 with shown arithmetic; none are measured or simulated.

R1 (Vertex growth). The number of events at time-step $d$ is $N_d = (p+1)p^{\,d-1}$ (Claim 4.1). For $p=2$: $N_1 = 3$, $N_{10} = 1536$.

R2 (Event budget). For $p=2$, $D=10$: $V_{10} = 3070$ (Claim 4.2, verified by two independent summations).

R3 (Comparability scarcity). For $p=2$, $D=10$: $C_{10} = 27651$ ordered pairs out of $\binom{3070}{2} = 4{,}710{,}915$ total pairs, a fraction $f_{10} \approx 5.8696 \times 10^{-3}$ (Claims 4.4–4.5). Complementarily, $\Pr[w \prec u] = 10/3070 \approx 3.2573\times 10^{-3}$ for the leaf-versus-ball sampling of Section 4.5, counting proper ancestors only. Asymptotically $f_D \sim D/p^{D} \to 0$: almost all pairs of events are temporally incomparable.

R4 (Depth-independence). The number of ordered pairs between depth $d_1$ and depth $d_2 \gt d_1$ is $N_{d_2}$, independent of $d_1$ (Claim 4.3) — an exact ultrametric signature.

R5 (Temporal statistics). Mean depth $\bar{d} = 27651/3070 \approx 9.0068$; the depth histogram is exactly $N_d = (p+1)p^{\,d-1}$, exponentially growing.

R6 (No causal cycles). Tree-geodesic causality is a strict partial order; causal cycles, including closed timelike curves, are impossible by construction (Claim 4.6).

R7 (Future-cone counts). $C(\Delta) = (p^{\Delta+1}-p)/(p-1)$; for $p=2$, $\Delta=5$: $C=62$; for $p=3$, $\Delta=5$: $C=363$.

R8 (Distance formula). $\delta(A,B) = d(A) + d(B) - 2d_w$, where $d_w$ is the depth of the least common ancestor (Definition 3.5).

Projection P1 (not a result). Under assumptions (i)–(ii) of Section 4.9, event count scales as $V_D \approx \frac{p+1}{p-1}p^{T/t_0}$; recovery of continuum statistics is not established here and is delegated to the boundary-theory mechanism of [6] and [7] as a hypothesis.

Projection P2 (not a result). If the root corresponds to the Planck scale and one depth step to one Planck time, a horizon of $D$ steps spans $D$ Planck times. We state this conversion as an assumption, not a derivation, since no input in this paper fixes the physical scale of one depth step.

#6. Discussion

Limitations. The model as constructed is a kinematic skeleton: it fixes ordering and counting but assigns no dynamics. 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). The prime $p$ is a free parameter with no observed value; the $p=3$ phase-transition threshold found in the Potts analysis [3] is the only hint in the surveyed literature that specific primes may be dynamically distinguished, and that result concerns a Potts system on a Cayley tree, not temporal ordering. The model also has no spatial degrees of freedom: the light-cone question is answered only in the limited sense of cone counting (Section 4.8), not by a derivation of relativistic light-cone structure.

Failure modes. The most serious structural mismatch is with relativity. In special relativity, the causal past of an event is a solid cone — a continuum of directions — and ordering is partial but dense along every timelike curve. In the tree model, the causal past of an event is a finite chain of $d$ ancestors: linear, not conical, and totally ordered internally. The scarcity result R3 is cone-like in spirit (most pairs unordered), but the shape of the past is a chain, not a cone. 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], though the supplied summary of [5] gives no detail connecting such maps to causality.

What would falsify the conjecture.

1. Observation of a causal cycle. Any empirically demonstrated closed causal chain (e.g., a closed timelike curve analogue with genuine information flow) would contradict Claim 4.6, which forbids cycles by the strict-decrease argument on depth. One confirmed cycle falsifies the identification of causality with ancestry.
2. Total order of events. If two events that are incomparable under Definition 3.3 were shown to have an empirically accessible, intrinsic temporal order (not a convention or a clock-synchronization artifact), the partial-order claim R3 would fail.
3. Continuum resolution below one tick. An operational time measurement resolving intervals finer than a single tick $t_0$, with $t_0$ fixed by the model, would contradict the quantization claim; conversely, the model predicts no such measurement is possible.
4. Wrong depth histogram. The model predicts an exponentially growing event-depth histogram $N_d = (p+1)p^{\,d-1}$ under uniform sampling of a depth-$D$ ball (R5). A measured flat or polynomial depth distribution for the relevant event ensemble would falsify the uniform-on-ball sampling assumption or the tree structure itself.
5. Wrong future-cone growth. The tree future cone grows as $C(\Delta) = (p^{\Delta+1}-p)/(p-1)$, exponentially in $\Delta$; an empirically quadratic cone-growth law $\sim \Delta^2/2$ at the finest resolvable scale would falsify the tree-geodesic cone structure.
6. Wrong prime dependence. The combinatorics depend on the free prime $p$ only through the regularity $p+1$; if a specific prime were empirically required and the resulting counts (e.g., $C(5) = 62$ for $p=2$ versus $363$ for $p=3$) were distinguishable and excluded, the model would be constrained or falsified.

#7. Conclusion

We have formalized a conjectural model in which time is the depth coordinate $d(v) \in \mathbb{N}$ on a rooted $(p+1)$-regular Bruhat-Tits tree, conjecturally identified with the $p$-adic valuation $v_p(x)$. The model's virtues are that discreteness (integer ticks $t_0$), partial temporal ordering (ancestry on the tree, with incomparability across branches), and acyclicity of causality (Claim 4.6) are theorems of the structure rather than added assumptions. The exact combinatorial consequences computed here — $V_{10} = 3070$ events in the depth-$10$ ball for $p=2$, $C_{10} = 27651$ ordered pairs out of $\binom{3070}{2} = 4{,}710{,}915$, a comparability fraction $f_{10} \approx 5.8696 \times 10^{-3}$ with asymptotics $f_D \sim D/p^{D} \to 0$, mean depth $\bar{d} = 27651/3070 \approx 9.0068$, and future-cone counts $C(5) = 62$ ($p=2$) and $363$ ($p=3$) — are all derived with shown arithmetic and require no simulation.

The model's central weakness, argued in Section 6, is the mismatch between the chain-shaped tree past and the cone-shaped relativistic past; the surveyed literature ([6], [7]) offers a boundary-theoretic route by which a $p$-adic continuum emerges from tree bulk degrees of freedom, which we adopt only as Projection P1, an explicitly labeled hypothesis. The prime $p$ is unfixed, the root-to-Planck-scale identification (Projection P2) is an assumption, and the falsification criteria of Section 6 make the proposal empirically exposed rather than insulated. The contribution is precision about a conjecture: the tree-time skeleton is exactly solvable, and its predictions are stated in a form that observation can attack.

#References

[1] On the missing branches of the Bruhat-Tits tree. arXiv:1712.01463v2. https://arxiv.org/abs/1712.01463v2 [2] Bending the Bruhat-Tits Tree II: the p-adic BTZ Black hole and Local Diffeomorphism on the Bruhat-Tits Tree. arXiv:2102.12024v2. https://arxiv.org/abs/2102.12024v2 [3] On Phase Transitions for $P$-Adic Potts Model with Competing Interactions on a Cayley Tree. arXiv:math-ph/0512018v2. https://arxiv.org/abs/math-ph/0512018v2 [4] Mumford dendrograms and discrete p-adic symmetries. arXiv:0809.1570v1. https://arxiv.org/abs/0809.1570v1 [5] On $p$-adic colligations and 'rational maps' of Bruhat-Tits trees. arXiv:1301.5453v1. https://arxiv.org/abs/1301.5453v1 [6] The boundary theory of a spinor field theory on the Bruhat-Tits tree. arXiv:1910.09397v2. https://arxiv.org/abs/1910.09397v2 [7] Effective field theories on subspaces of the Bruhat-Tits tree. arXiv:2402.03730v2. https://arxiv.org/abs/2402.03730v2 [8] p-adic Equiangular Lines and p-adic van Lint-Seidel Relative Bound. arXiv:2408.00810v3. https://arxiv.org/abs/2408.00810v3 [9] QNFO: Formal Derivation of Emergent Spacetime to the Bruhat-Tits Tree [10] QNFO: Bruhat-Tits Tree as a Unifying Geometric Object [11] QNFO: Invariant Patterns and the Adelic Refactoring of Fundamental Physics [12] QNFO: Time as Epistemic Cognitive Fiction

#Appendix A. Divergence report

No unresolved divergences between independent drafts were recorded in the material available for this revision; the quantitative claims in Sections 4 and 5 trace to single shown derivations, and no conflicting values were present to adjudicate. The one corrected internal inconsistency (the former "$0.8\%$" gap statement in Section 4.6, replaced by the computed $9.93\%$) was an arithmetic error against the paper's own mean $\bar{d} = 27651/3070 \approx 9.007$, not a divergence between competing conventions.

#Appendix B. Claim attribution