{"id":"dc158ad6-23f7-422f-83cf-3e8961e1386b","arxiv_id":"2608.07919","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A discrete gravity action on the boundary of a tree representing Q_p^2 is proposed, together with two conjectures on the universality and uniqueness of p-adic AdS/CFT.","lead":"This paper embeds two-dimensional p-adic numbers as the boundary of a tree, defines distances and curvature there, and proposes a discrete gravity action. It also conjectures that p-adic AdS/CFT computations are independent of the chosen space and that p-adic CFT is unique.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central action (49) is not well-defined: the measure (48) violates the additivity axiom (3), so (47) does not follow, and for Omega=T_p the action diverges.","rationale":"The reader's weakest assumption identifies the measure/partition problem, and my analysis confirms it is the load-bearing flaw. The distance tables are mostly standard and the paper is honest about the conjectural status, but the central construction fails: even a finite check shows mu is not additive, and the pAdS2 case has divergent action. I would keep the REJECT verdict without moving it. I also considered whether Eq. (49) could be taken as a standalone discrete functional; even then it is linear in l_e and has generically no stationary point on a branched tree, which reinforces rather than replaces the primary concern.","tokens_in":10605,"tokens_out":12712,"duration_ms":153852,"concrete_test":"Use p=2 and the finite subtree of T_p with vertices r, a, b1, b2 and edges r-a, a-b1, a-b2, each of length 1. Compute mu from Eq. (48): mu(a)=3, mu(b1)=mu(b2)=1. Axiom (3) for the ball decomposition a=b1 u b2 requires mu(a)=mu(b1)+mu(b2)=2, so the measure is non-additive. Then evaluate Eq. (49) for Omega=T_p with l_e=1: the sum is -p(p+1) times the number of vertices and diverges. Either computation settles that the step (47)->(49) is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the reformulation S = integral dx R_Omega(x) = sum_{a in Omega} R_Omega(a) mu(a) in Eq. (47). For this equality to hold, the balls b(a) for a in Omega must form a disjoint partition of the boundary and mu must be additive. The new measure mu(a) := sum_{e~a} l_e in Eq. (48) is not additive: in a finite subtree r-a-{b1,b2} of T_p with all l_e=1, additivity for the sub-balls b1,b2 of a requires mu(a)=mu(b1)+mu(b2)=2, but Eq. (48) gives mu(a)=3 (edges to r,b1,b2). Hence the boundary integral in Eq. (47) and the vertex sum in Eq. (49) are different objects; the action is not derived from a measure on Q_p^2. For the pAdS2 case Omega=T_p, the balls b(a) are not a disjoint cover: boundary points whose geodesic stays in T_p lie in infinitely many nested balls and points outside T_p lie in none. With l_e=1, Eq. (49) evaluates to -p(p+1)|V(T_p)|, which diverges. Without a corrected additive measure or a regularization, the proposed gravitational action on Q_p^2 is undefined, independent of whether the sign of R_Omega matches dS/AdS/Euclidean expectations.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10919,"tokens_out":5390,"duration_ms":62821,"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":[{"comment":"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.","section":"Sec. 4, Eqs. (47)-(49)"},{"comment":"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.","section":"Sec. 4, Eq. (42)"},{"comment":"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.","section":"Sec. 5, Eq. (51)"}],"minor_comments":[{"comment":"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.","section":"Sec. 3.2, Eq. (19)"},{"comment":"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.","section":"Sec. 3.6 and Table 2"},{"comment":"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.","section":"Sec. 4, Eq. (45)"},{"comment":"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.","section":"Sec. 5, Eq. (50)"}],"recommendation":"reject","confidential_remarks":"The conjectures in Section 5 rely heavily on Eq. (51) from the author's own prior paper [14], with no independent confirmation presented here; the editor may wish to weigh whether self-citation alone provides adequate support for the main conceptual claims. The measure-defect in the central action is a technical obstruction that would require substantial revision, not a mere presentation issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the distance-function part of this paper is a real contribution; the gravity action is not. The central derivation of (49) breaks at Eq. (47): the measure (48) does not satisfy the additivity axiom (3) introduced two pages earlier, so the boundary integral and the vertex sum are different objects. For Omega = T_p the balls are nested, not disjoint, and the sum diverges. That is the load-bearing flaw, and it sits exactly where the paper claims its main new result.\n\nWhat is new and good: the two-dimensional embedding Q_p^2 = boundary of T'_p is a natural step, and the distance formulas (19), (20), (24) are worked out carefully with the 9-type case analysis and then compared to real chordal distances. Those tables are honest and useful. The observation that different subgraphs Omega produce pdS, pE, pAdS distances, with curvature signs matching the real case, is suggestive and worth stating.\n\nSoft spots, in order: first, the curvature R=1-deg(a) is posited, not derived; it reproduces the desired signs, but there is no variational principle and no EOM, so calling it \"gravity\" is premature. Second, even granting R, the action (49) is not obtained from a measure on Q_p^2. The author acknowledges that the action is intentionally not written as an integral, but then the earlier boundary-integral motivation is misleading. Either define a proper additive measure or present (49) as a graph action on Omega without claiming it descends from the boundary. Third, the two conjectures rest on the equality (51) from the author's own earlier paper; they are honestly labeled as conjectures, so this is a circularity of motivation more than a defect of the calculation. The paper also leaves the variable-edge-length generalization of the distance undeveloped, which is fine for a first pass.\n\nIs it serious? Yes, I think the author is thinking carefully and the distance calculations are reproducible. But the central gravity construction currently does not stand up. I would not reject without referee input: there is enough new material here that a referee can mark up the measure issue and the author can either fix it or reframe the claim. Send it to peer review, but expect major revision. If the measure cannot be repaired, the publishable residue is the distance-formula work, which is a solid short paper on its own.","headline":"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.","tokens_in":11502,"tokens_out":2154,"would_cite":false,"duration_ms":25810,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["p-adic numbers","Bruhat-Tits tree","AdS/CFT","p-adic AdS/CFT","discrete gravity","curvature on trees","chordal distance","Q_p^2"],"falsifier":"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.","tokens_in":10334,"feed_emoji":"🧮","tokens_out":8885,"duration_ms":79822,"temperature":0.7,"pith_summary":"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.","feed_headline":"One curvature formula turns p-adic space into dS, AdS, or Euclidean","feed_subtitle":"Embedded as a tree boundary, p-adic space admits one action matching the curvature signs of dS, AdS, and Euclidean.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Bruhat-Tits tree embedding, the subgraph chordal distance, and the vertex ball measure that the paper reuses in Eqs. (24) and (40).","marker":"[4]"},{"why":"Sets the AdS/CFT duality framework that the paper's p-adic gravitational theory is meant to fit into.","marker":"[1]"},{"why":"Provides the non-Archimedean boundary effective action that appears as one of the three equivalent cases in Eq. (50).","marker":"[12]"},{"why":"Computes the effective field theory on a finite boundary of the Bruhat-Tits tree, used in Eq. (50).","marker":"[13]"},{"why":"Establishes the equality of the three measures and distances in Eq. (51), which directly motivates the paper's two conjectures.","marker":"[14]"}],"fun_headline_variants":["One p-adic action yields dS, AdS, and Euclidean curvature","Subgraph choice sets p-adic curvature to dS, AdS, or flat","p-adic gravity: same boundary tree, three curvature signs","Curvature sign from subgraph: p-adic gravity unifies dS, AdS, flat","One action, three spaces: p-adic boundary curvature signs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["One p-adic action yields dS, AdS, and Euclidean curvature","Subgraph choice sets p-adic curvature to dS, AdS, or flat","p-adic gravity: same boundary tree, three curvature signs","Curvature sign from subgraph: p-adic gravity unifies dS, AdS, flat","One action, three spaces: p-adic boundary curvature signs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000826,"raw_usage":{"total_tokens":3604,"prompt_tokens":931,"completion_tokens":2673,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":2570}},"tokens_in":547,"tokens_out":2673,"duration_ms":20932,"temperature":1.0,"reasoning_tokens":2570,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:41:37.297191+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Nonarchimedean Strings and Bruhat-tits Tre es,","cited_arxiv_id":null,"evidence_quote":"Provides the non-Archimedean boundary effective action that appears as one of the three equivalent cases in Eq. (50)."},{"cited_title":"Effective field theory on a finite boundary of the Bruhat-Tits tree","cited_arxiv_id":"2103.02882","evidence_quote":"Computes the effective field theory on a finite boundary of the Bruhat-Tits tree, used in Eq. (50)."},{"cited_title":"Effective field theories on subspaces of the Bruhat-Tits tree","cited_arxiv_id":"2402.03730","evidence_quote":"Establishes the equality of the three measures and distances in Eq. (51), which directly motivates the paper's two conjectures."}],"review_version":1}