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REVIEW 3 major objections 4 minor 14 references

One gravity theory on the two-dimensional p-adic space and conjectures on $p$-adic AdS/CFT

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that a discrete gravitational theory on the p-adic plane Q_p^2, built from a subgraph-dependent distance and curvature R(a)=1-deg(a), reproduces the curvature signs of dS, Euclidean, and AdS spaces, and conjectures that…

desk verdict The distance-function core is a real contribution; the gravity-action construction is not yet well-defined because the measure (48) violates the additivity axiom it depends on. read the letter →

arxiv 2608.07919 v1 pith:4Y7QTINQ submitted 2026-08-08 hep-th

Feng Qu This is my paper · ORCID
classification hep-th
keywords p-adicnumbersBruhat-TitstreeAdS/CFTdiscretegravitycurvatureontreeschordaldistanceQ_p^2
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish a gravitational theory on the two-dimensional p-adic space $\mathbb{Q}_p^2$ without introducing a metric or line element. It embeds $\mathbb{Q}_p^2$ as the boundary of an infinite tree, defines a distance that depends on a chosen subgraph $\Omega$, and shows that three natural choices of $\Omega$ produce p-adic counterparts of de Sitter, Euclidean, and anti-de Sitter spaces. The curvature is defined on vertices by $R_\Omega(a)=1-\deg(a)$, giving positive, zero, and negative values in those three cases, and the proposed action is $S=\sum_a (1-\deg(a))\sum_{e\sim a} l_e$. If this construction holds, it supplies a discrete gravity side for p-adic AdS/CFT and motivates the paper's two conjectures: that p-adic AdS/CFT can be computed in arbitrary spaces with essentially the same results, and that p-adic CFT is essentially unique.

What carries the argument

The carrying object is the subgraph-dependent regularized distance $D(x,y)=p^{d(x,y)-d(x,\Omega)-d(y,\Omega)}$, together with the associated ball measure $\mu(a)=\sum_{e\sim a}l_e$. Vertices of the tree are interpreted as balls in $\mathbb{Q}_p^2$, and the boundary measure $dx$ is fixed by requiring $\int_c dx=\mu(c)$ for every ball $c$. The curvature $R_\Omega(a)=1-\deg(a)$, extended to boundary points by $R_\Omega(x)=R_\Omega(a)$ when $x$ lies in the ball of $a$, turns the action into $S=\sum_{a\in\Omega}R_\Omega(a)\mu(a)$. This is the machinery that lets the paper go from distances to an action without ever writing a line element.

What would settle it

Take $\Omega=T_p$ inside the tree $T'_p$, truncate the tree at depth $N$, compute $S_N=\sum_{a}(1-\deg(a))\sum_{e\sim a}l_e$ with all edge lengths equal, and let $N$ grow; if the partial sums diverge, or if the partition identity $\sum_{a\in\Omega}\int_a dx=\int dx$ fails because the balls overlap, the proposed action is undefined without an extra regularization.

Watch

Extended reading notes

Core claim

The central claim is that the pair $(\mathbb{Q}_p^2,\Omega)$ is a self-contained gravitational system: the boundary carries the space, the subgraph $\Omega$ carries the curvature, and the action $S=\sum_{a\in\Omega}(1-\deg(a))\sum_{e\sim a}l_e$ defines gravity on both at once. The derivation passes through the identity $S=\int dx\,R_\Omega(x)=\sum_{a\in\Omega}R_\Omega(a)\mu(a)$, with $\mu(a)=\sum_{e\sim a}l_e$ as the measure of the ball represented by vertex $a$. For $\Omega$ a single vertex the curvature is positive and the distance is the p-adic analogue of the dS chordal distance; for $\Omega$ a single boundary point the curvature is zero; for $\Omega=T_p$ the curvature is negative. The paper concludes that different choices of $\Omega$ turn the same boundary $\mathbb{Q}_p^2$ into different spaces, and that the action allows the edge lengths inside $\Omega$ to fluctuate while edge lengths outside remain unchanged.

Load-bearing premise

The load-bearing premise is that for every vertex $a$ of the chosen subgraph $\Omega$, the balls $b(a)$ form a disjoint partition of the boundary and that $\mu(a)=\sum_{e\sim a}l_e$ is additive; for $\Omega=T_p$ the balls are nested rather than disjoint, so the action sum is not obviously finite.

Editorial extensions

If this is right

  • If the construction is correct, gravity on the p-adic plane can be written as a discrete sum over vertices of a subgraph, giving a concrete bulk-like model for p-adic AdS/CFT.
  • The curvature sign $1-\deg(a)$ automatically matches the real-space expectations: positive for dS, zero for Euclidean, negative for AdS.
  • The action makes edge lengths inside $\Omega$ the dynamical variables, while edges outside $\Omega$ remain fixed and equal.
  • Under the paper's identification of measures and distances, the three boundary effective actions in Eq. (50) coincide, which is the basis for the conjecture that p-adic AdS/CFT results are the same in arbitrary spaces.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A stress test would be to regularize the action for $\Omega=T_p$ on a finite truncation of the tree and check convergence; the paper does not perform this check.
  • If the uniqueness conjecture is right, all boundary correlation functions computed with different subgraphs should agree after identifying measures, not just the three examples displayed in Eq. (50).
  • The same embedding construction could be applied to $\mathbb{Q}_p^n$ for $n>2$ by assigning curvatures from vertex degrees of a higher-dimensional tree-like complex.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper proposes a subgraph-dependent regularization of the distance between boundary points of the Bruhat-Tits trees T_p and T'_p, leading to p-adic analogues of de Sitter, Euclidean, and anti-de Sitter chordal distances. It then defines a vertex curvature R_Omega(a) = 1 - deg(a), an action S = sum_{a in Omega} (1 - deg(a)) sum_{e ~ a} l_e, and claims that this is a gravitational theory on Q_p^2. The paper closes with two conjectures: that p-adic AdS/CFT computations can be performed in arbitrary spaces with essentially the same results, and that p-adic CFT is essentially unique.

Significance. The distance formulas in Section 3 and their comparison with real chordal distances are a useful and partially convincing observation; the sign pattern of R_Omega in Eq. (43) matches the expected dS/Euclidean/AdS curvatures, and Eq. (49) is an explicit, concrete proposal that could in principle be tested. However, the central construction is not yet a well-defined gravity theory: the measure underlying the action is not additive, the boundary decomposition used in Eq. (47) fails for the tree subgraph Omega = T_p, and the conjectures rest on an equality quoted from the author's own prior work without independent verification in this paper.

major comments (3)
  1. [Sec. 4, Eqs. (47)-(49)] The derivation of the action is invalid as it stands. The first equality in Eq. (47) requires that the balls b(a) for a in Omega form a disjoint cover of the boundary and that mu be additive, but the proposed measure mu(a) = sum_{e ~ a} l_e in Eq. (48) violates the additivity axiom stated in Sec. 4. For example, in a finite subtree r-a-{b1,b2} with unit edge lengths, additivity for the children of a requires mu(a) = mu(b1) + mu(b2) = 2, whereas Eq. (48) gives mu(a) = 3 because it also counts the edge to the parent. For the pAdS2 case Omega = T_p, the balls b(a) are nested rather than disjoint, so boundary points whose geodesic stays in T_p lie in infinitely many balls and boundary points outside T_p lie in none; with l_e = 1 the sum in Eq. (49) evaluates to -p(p+1)|V(T_p)|, which diverges. The boundary integral in Eq. (47) and the vertex sum in Eq. (49) are therefore different objects, and the proposed gravitational action is not well-defined without an additive measure or a regularization.
  2. [Sec. 4, Eq. (42)] The curvature R_Omega(a) = 1 - deg(a) is introduced to match the sign pattern of dS/Euclidean/AdS spaces, but it is not derived from the distance or embedding structure, and no variational principle or equations of motion are given for the action in Eq. (49). In particular, the absence of a line element means that the claim that R_Omega is a scalar curvature cannot be checked under the allowed transformations. The paper would need at least a justification that Eq. (42) is a natural curvature for the discrete structure, and some statement of how the edge lengths l_e are varied, before Eq. (49) can be called a gravity theory.
  3. [Sec. 5, Eq. (51)] The two conjectures at the end of the paper rest on the equality d(1)x d(1)y / |x-y|_1^2 = d(2)x d(2)y / |x-y|_2^2 = d(3)x d(3)y / |x-y|_3^2 = dx dy / |x-y|_p^2, quoted from the author's previous paper [14]. This equality is not derived or independently checked here, and the notation d(i) is not fully specified. Since the conjectures are the main conceptual conclusion of the paper, the equality should at least be verified for representative cases in this manuscript, and the meaning of 'essentially unique' should be made precise.
minor comments (4)
  1. [Sec. 3.2, Eq. (19)] The piecewise distance formulas for the finite subgraph Omega = {c_k} are stated after 'following analogous procedures' for most of the nine types; a derivation or at least an exhaustive case table should be provided so that the formulas can be checked.
  2. [Sec. 3.6 and Table 2] There are typos: 'choral distance' should be 'chordal distance', and 'Bruhat-Tits' should be capitalized consistently. The table also uses D for both the regularized and unregularized distances; please clarify which quantity is tabulated.
  3. [Sec. 4, Eq. (45)] The definition R_Omega(x) = R_Omega(a) for x in a in Omega is ambiguous because a boundary point can belong to several nested balls when Omega contains ancestor and descendant vertices; the domain of the map a -> b(a) needs to be made precise.
  4. [Sec. 5, Eq. (50)] The notation d(i), i = 1,2,3 is used without defining the normalization or the precise domain of integration; this makes the quoted equality (51) hard to verify independently.

Circularity Check

3 steps flagged · score 5.0 of 10

Curvature sign agreement is built into the definition of R_Omega and the naming in Table 1, while the two conjectures lean on the author's own [14]; the action itself is an explicit ansatz rather than a forced result.

  1. self definitional [Section 4, Eqs. (42)-(43), following Table 1]
    "Considering that the curvatures of dS space, Euclidean space and AdS space should be positive, zero, and negative respectively, we introduce a curvature on vertices of Ω as RΩ := 1 − deg(a), a ∈ Ω. ... pdS2 : RΩ(a) = 1 − 0 = 1 > 0, a ∈ Ω = {c(0, 1, ∞)}; pAdS2 : RΩ(a) = 1 − (1 + p) = −p < 0, a ∈ Ω = T p."

    The sign pattern is not discovered but imposed. The names pdS2 and pAdS2 were assigned in Table 1 because the distance formulas resemble the real dS and AdS chordal distances; RΩ is then defined as 1 − deg(a), which by the degree counts of those Ω's automatically gives positive and negative values. The 'result' that p-adic dS curvature is positive and p-adic AdS curvature is negative is therefore a restatement of the definition together with the earlier naming, not an independent check. Since this sign agreement is the paper's stated justification of the theory's reasonableness, the justification is circular.

  2. other [Section 4, Eqs. (47)-(49)]
    "S = ∫ dx RΩ(x) = Σ_{a∈Ω} ∫_a dx RΩ(a) = Σ_{a∈Ω} RΩ(a) μ(a). ... We now consider the case of μ(a) := Σ_{e∼a} le, where le is the length of e. Now the action can be written as S = Σ_{a∈Ω} RΩ(a) Σ_{e∼a} le = Σ_{a∈Ω} (1 − deg(a)) Σ_{e∼a} le."

    The purported reformulation of the boundary action as the vertex sum (49) depends on the balls of Ω partitioning the boundary and on μ being additive, which the paper itself lists as condition (3) for a measure. The newly introduced μ(a) = Σ_{e∼a} l_e is not additive at a branching vertex, so Eq. (47) does not actually hold for this μ. For Ω = T_p, the balls are nested rather than disjoint, and with l_e = 1 the sum diverges. Consequently Eq. (49) is a separately postulated lattice action, not a consequence of the measure construction. It is a definitional ansatz, and any later appeal to (49) as a predicted gravitational action would reduce to that choice rather than to independent input.

1 more flagged steps
  1. self citation load bearing [Section 5, Eq. (51) and the two conjectures]
    "It is further pointed out in [14] that d(1)x d(1)y / |x−y|^2_1 = d(2)x d(2)y / |x−y|^2_2 = d(3)x d(3)y / |x−y|^2_3 = dxdy / |x−y|^2_p ... And it lead us to make the following conjectures: (1) the computation of p-adic AdS/CFT can be performed in arbitrary spaces, and the final results are essentially the same; (2) the p-adic CFT is essentially unique."

    The evidence connecting three effective actions to a single p-adic expression is quoted from the author's own earlier paper [14], and the conjectures—especially 'p-adic CFT is essentially unique'—are advanced on the strength of that self-cited equality. No independent derivation, numerical test, or external theorem is offered here; the load-bearing support for the conjectures is a self-citation. The conjectures are admittedly speculative, which softens the charge, but it is still a case of the paper's advertised conclusions resting on the author's prior work rather than on independent content.

full rationale

The paper is not wholly circular: the pAdS2 chordal distance is taken from the independent reference [4], and the comparison with real-number chordal distances in Table 2 is an external analogy. However, the central validation step for the curvature is circular by construction: R_Omega is chosen as 1 − deg(a) so that the pre-named pdS2, pE2, and pAdS2 cases have positive, zero, and negative sign, respectively. The action (49), while not fitted to data, is also not derived from the stated measure axioms because μ(a)=Σ l_e is non-additive; it is an explicit ansatz, so any claim that it is 'justified' by the curvature comparison reduces to the definition. Finally, the two conjectures are motivated by an equality from the author's own [14], making the speculative conclusions self-citation-dependent. These issues are real but limited: they concern how the theory is justified and named, not a prediction that is secretly a fit. Score 5 reflects partial circularity in the validation and conjecture chain, while leaving room for the genuinely imported external results.

Assumptions & free parameters 3 free parameters · 5 assumptions · 3 invented entities

The central construction depends on several choices that are either pulled from prior literature or posited without independent evidence: the embedding tree T'_p, the distance regularization scheme from [4], the curvature R=1-deg(a), and the non-additive measure mu(a)=sum of edge lengths. The conjectures rest on the author's own prior equality in [14].

free parameters (3)
  • subgraph Omega = various: single vertex, boundary point, whole tree
    The distance D_Omega, curvature R_Omega, and action S all depend on the choice of Omega; this is the input that selects pdS, pE, or pAdS.
  • edge length L = not fixed
    Introduced in Sec. 3.6 to give D the dimension of length squared; treated as a free scale.
  • curvature offset (the 1 in R=1-deg(a)) = 1
    Chosen by hand in Eq. (42) so that pdS2 has curvature +1 and pE2 curvature 0; no derivation is provided.
assumptions (5)
  • domain assumption The boundary of the tree T'_p is Q_p^2
    Sec. 3.1 postulates the (1+p^2)-regular tree T'_p and identifies its boundary with Q_p^2; the entire construction rests on this embedding.
  • domain assumption The regularized distance D(x,y)=p^{d(x,y)-d(x,Omega)-d(y,Omega)} is the correct p-adic analogue of chordal distance
    Eq. (7), adopted from [4]; the paper does not prove uniqueness or independence from the regularization scheme.
  • ad hoc to paper The combinatorial curvature R_Omega(a)=1-deg(a) is a valid scalar curvature for the boundary
    Eq. (42); introduced solely to match the sign pattern of real dS, Euclidean, and AdS curvatures.
  • domain assumption The boundary integral can be rewritten as the sum over a in Omega of R_Omega(a) mu(a)
    Eq. (47) assumes the balls of vertices in Omega partition the boundary; this fails for Omega = T_p where balls are nested and the sum diverges.
  • standard math The equality d^(1)x d^(1)y/|x-y|_1^2 = d^(2)x d^(2)y/|x-y|_2^2 = d^(3)x d^(3)y/|x-y|_3^2
    Eq. (51), cited from [14]; underpins the conjectures and is not re-derived in this paper.
invented entities (3)
  • Tree T'_p with boundary Q_p^2
    purpose: Provides a higher-dimensional embedding on which distance, measure, and curvature are defined
    Defined in Sec. 3.1; a mathematical construction with no falsifiable physical prediction.
  • Curvature formula R_Omega(a)=1-deg(a)
    purpose: Assigns scalar curvature to boundary points through vertices of Omega
    Introduced in Eq. (42); only justification is sign matching with real spaces.
  • Gravity action S=sum over a of (1-deg(a)) times sum over e~a of l_e
    purpose: Proposed gravitational theory on Q_p^2
    Defined in Eq. (49); no dynamics, no matter coupling, and no testable predictions.

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Cite this review

Pith. "Pith review of One gravity theory on the two-dimensional p-adic space and conjectures on $p$-adic AdS/CFT." pith.science (2026). https://pith.science/paper/4Y7QTINQ

@misc{pith2026260807919,
  author       = {Pith},
  title        = {Pith review of: One gravity theory on the two-dimensional p-adic space and conjectures on $p$-adic AdS/CFT},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4Y7QTINQ}},
  note         = {Machine review of arXiv:2608.07919}
}
abstract

By means of a higher-dimensional embedding approach, together with the study of distance, measure, and curvature, we construct a gravitational theory on the two-dimensional space over $p$-adic numbers. Before obtaining the final gravitational theory, we supplement some calculations based on existing literature and justify the reasonableness of the gravitational theory by comparison with the real-number case-for instance, the curvatures of de Sitter(dS), anti-de Sitter(AdS), and Euclidean spaces should be positive, negative, and zero, respectively. Finally, we put forward two conjectures: (1) $p$-adic AdS/CFT can be computed in arbitrary spaces, not only in $p$-adic AdS space($p$AdS), and the results are essentially the same; (2) $p$-adic CFT is essentially unique.

Figures

Figures reproduced from arXiv: 2608.07919 by the authors.

Figure 1
Figure 1. The subgraph-dependent regularized distance on the boun [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Imagining this tree as lying in three-dimensional space, paralle [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The relation between |x − y|p and the lowest vertex on line (x − y) an = 1, it deviates to the right in the direction parallel to the page; if bn = 0, it simultaneously deviates outward in the direction perpendicular to the page; and if bn = 1, it deviates inward in the direction perpendicular to the page. Eventually, the polyline reaches one of the four adjacent vertices at level n + 1. The values of the remaining … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Different positional relations between Ω = [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Tp (drawn in red) as the subgraph Ω of T′ p 3.5 Distances on Qp On the Tp tree, three boundary points 0, 1, ∞ determine a vertex of the tree, which is the common vertex (noted by c) of three lines (0−1),(0−∞) and (1−∞). In the case of Ω = {c}, similar to the previous t…
Figure 6
Figure 6. Figure 6: Different positional relations between Ω = [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Vertices as balls in Q2 p absolute value carries the dimension of length, then we can reinstate L into [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Details near the boundary and the modified subgraph. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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Reviewed August 12, 2026 · model on record in the stance chip above.