{"id":"44e6b796-5bd9-4125-ba58-e0822a67d8a8","arxiv_id":"2608.06465","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For d=3, Δ=2 Einstein-scalar gravity with sinusoidal sources on two R^3 boundaries, the large wormhole dominates the path integral whenever wormholes exist, and there is no Hawking-Page-like subdominant regime.","lead":"This paper computes Euclidean wormhole and disconnected saddles in Einstein-scalar gravity with sinusoidal boundary sources, and finds that whenever wormhole solutions exist, the large wormhole dominates the two-boundary path integral. A generalist might read it because it challenges the standard picture that wormhole dominance is preceded by a Hawking-Page-like regime, with consequences for ensemble averaging in holography.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The dominance claim rests on unverified stability and contour status of the large wormhole; the paper itself concedes both in Section 6, so the central claim is conditional.","rationale":"The reader's weakest_assumption identifies precisely the point on which the central claim hinges. The paper's Section 6 openly concedes both required conditions: stability of the wormhole saddles is not analyzed, and the integration contour is assumed to pass through them. Without these, the action comparison in Section 5.3 only shows that, among the three saddles considered and if they all contribute, the large wormhole has the lowest on-shell action. It does not show that the large wormhole actually contributes to the path integral or that it is a local minimum of the action. A negative mode would make the saddle a maximum or saddle point rather than a dominant contribution, and a contour that misses the saddle would remove it from the semiclassical sum entirely, leaving the disconnected geometry as the only contribution. These are not mere technicalities; they are the difference between 'the large wormhole dominates' and 'the large wormhole has the lowest action among saddles that may or may not contribute.' The reader's CONDITIONAL verdict is therefore appropriate, and no change is needed.","tokens_in":28151,"tokens_out":4720,"duration_ms":46660,"concrete_test":"Compute the spectrum of quadratic fluctuations around the large R^3 wormhole by adapting the linearized stability analysis of Marolf–Santos [23] (Appendix C.2) to continuous perturbation wavenumbers, and determine whether any negative mode exists. In parallel, perform a numerical Picard–Lefschetz check on the finite-dimensional minisuperspace of Z2-symmetric throat data (a0, φ0) to see whether the large-wormhole saddle lies on the original integration contour. If a negative mode appears, or if the contour does not pass through the large-wormhole saddle, the dominance claim and the exponential variance prediction collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract asserts that once wormhole solutions exist, the large wormhole is always the dominant saddle in the two-boundary path integral and that the normalized variance is exp(∆S). This is a saddle-point statement, and it requires two things that are not established: the large wormhole must be perturbatively stable, and it must lie on the integration contour of the Euclidean gravitational path integral. The paper does not compute the quadratic fluctuation spectrum for the R^3 wormholes; it argues by analogy with the T^3 wormholes of [23] (Section 6: 'While we did not carry out a stability analysis...'). It also states explicitly in Section 6 that 'we have implicitly assumed that the integration contour for the gravitational path integral passes through the wormhole saddles,' and notes that a Picard–Lefschetz analysis could reveal that they are not picked up. If the large wormhole has a negative mode or lies off the original contour, then the on-shell action comparison in Section 5.3 is not enough to establish dominance, and the exponential variance in Eq. (6.1) has no basis. This is not an internal inconsistency, but it is a load-bearing missing verification for the paper's strongest claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies Euclidean Einstein-scalar gravity in d=3 with a conformally coupled scalar (Δ=2), turning on identical sinusoidal scalar sources on two planar AdS boundaries. It numerically constructs one-boundary (disconnected) solutions and Z2-symmetric wormhole solutions, maps their parameter space (small and large branches meeting at a critical wormhole), and interprets both families as holographic RG flows: the disconnected geometry as a boomerang flow returning to the same CFT, and the folded wormhole as a gapped flow. The paper then computes renormalized on-shell actions and compares saddles, concluding that above the critical source strength the large wormhole is always the dominant saddle, with no Hawking–Page-like subdominant window. Under an ensemble interpretation, the normalized variance is claimed to jump from zero to exp(ΔS) at the threshold.","tokens_in":28418,"tokens_out":6464,"duration_ms":60735,"significance":"If the central claim is correct, the paper is a significant counterpoint to the Marolf–Santos bottom-up wormhole results: it would establish a two-boundary semiclassical path integral with a sharp transition directly from disconnected-only to large-wormhole dominance, and an ensemble that is strongly non-self-averaging whenever wormholes exist. The paper is transparent about its numerical workflow, derives the holographic counterterms in an appendix, and gives a falsifiable prediction for the phase diagram and for the normalized variance. These are genuine strengths. However, the headline dominance claim is conditional on two unverified premises—perturbative stability of the large wormhole and placement of the integration contour through the wormhole saddles—and the numerical evidence for the sharp transition lacks error estimates. The result is therefore significant but not yet fully established.","major_comments":[{"comment":"The central claim that for eJ ≥ eJcrit the large wormhole is always the dominant saddle presupposes that this saddle is perturbatively stable and lies on the integration contour of the Euclidean gravitational path integral. The paper explicitly states that no stability analysis was carried out and that the contour was assumed to pass through the wormhole saddles (Section 6: “While we did not carry out a stability analysis...” and “we have implicitly assumed that the integration contour for the gravitational path integral passes through the wormhole saddles”). A negative mode or an off-contour saddle would invalidate the dominance statement and the exponential variance in Eq. (6.1). The analogy with the T^3 wormholes of [23] is suggestive but not a substitute for the R^3 calculation, since compactness of the transverse space changes the perturbation spectrum. I request a quadratic-fluctuation analysis for at least the large wormhole branch and a Picard–Lefschetz discussion of contour placement, or a substantially qualified formulation of the headline claim.","section":"Section 6, Eq. (6.1), Abstract"},{"comment":"The numerical action differences Δs are presented without estimates of numerical uncertainty (shooting tolerances, sensitivity to the choice of cutoff window, or fitting errors in the ϵ→0 extrapolation described around Eq. (5.22)). The sharp conclusions that Δs is positive at eJcrit and that Δs ∼ J^4/k follow from a linear best fit through sampled points, but no residuals or fit ranges are reported. Moreover, the text concedes near the bifurcation that the saddle-point approximation cannot clearly resolve the dominant saddle (Section 6). Without error bars or a dedicated near-critical analysis, the claimed discontinuity (‘zeroth-order phase transition’) and the absence of a Hawking–Page-like regime are not demonstrated to the precision required for the paper's strongest claim.","section":"Section 5.3, Fig. 10"},{"comment":"The dominance comparison is performed on the action density Δs after factoring out the infinite volume Vol(R^d). As a result, the statements “the large wormhole dominates” and “the normalized variance is exp(ΔS)” are statements about intensive free-energy densities, not about the actual finite path-integral weights exp(−S). This is a standard maneuver for planar boundaries, but the manuscript should state it explicitly and justify why the intensive comparison controls the saddle-point approximation in the noncompact case; otherwise the exponential factors exp(−Δs Vol(R^d)) appearing in the text are purely formal.","section":"Section 5.3, Eq. (5.21)"}],"minor_comments":[{"comment":"The caption states J/k = 2, but the surrounding text and Fig. 7 use J/k = 50; this appears to be a typo and should be corrected.","section":"Fig. 8 caption"},{"comment":"The linear fit leading to Δs ∼ J^4/k should specify the fit range, whether the fit is on a log-log or linear plot, and the residuals; otherwise the functional form is hard to assess.","section":"Section 5.3, Eq. (5.23)"},{"comment":"The statement in the abstract and Section 5.3 that “there is no regime where wormholes are subdominant” is slightly stronger than the text in Section 6, which notes that near the critical wormhole the dominant saddle cannot be clearly resolved; the authors should either resolve this region or qualify the global statement.","section":"Section 6, Fig. 10"},{"comment":"The claim that no wormhole solutions exist below (J/k)crit is based on numerical root-finding; this should be stated as a numerical result, with the caveat that complex or otherwise non-geometric saddles are not excluded.","section":"Section 4.1"},{"comment":"The estimate of the truncation error in Eq. (5.7) is helpful, but the paper should also report the analogous convergence check for the wormhole action integrals and for the extraction of J from the near-boundary fit in Eq. (4.8).","section":"Section 5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest and well written, and I found no evidence of circular reasoning or self-citation inflation. My concern is that the headline claim is sharper than the evidence: the missing stability and contour analysis is a load-bearing gap, and the numerical phase diagram would benefit from error bars and a near-critical treatment. These issues are fixable within the scope of the manuscript, so I recommend major revision rather than rejection. If the authors can supply a stability check or clearly qualify the dominance claim, I would be willing to revisit favorably."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nHere is my read of arXiv:2608.06465.\n\nThe genuinely new thing is the phase diagram in Fig. 10 for d=3, Delta=2 planar wormholes sourced by sinusoidal scalars: below Jcrit only the disconnected saddle exists; above, large and small wormholes appear, and the large one is always the dominant saddle. No Hawking–Page-like transition, unlike Marolf–Santos. If true, that removes the wormhole-subdominant regime from this class and implies a strongly non-self-averaging ensemble once wormholes exist. Also new are the boomerang RG flows with multivalued beta functions and the folded-wormhole interpretation as gapped flows. Those are clean, interesting results.\n\nWhat the paper does well: the numerical work is careful. It derives holographic counterterms from scratch, uses a cutoff-independent extraction via the epsilon-expansion, checks that truncation errors are negligible, and compares saddles on the same footing using the extracted source. The paper is also transparent: it explicitly flags where it has not checked something. That honesty is real and welcome.\n\nSoft spots, in proportion:\n\n1. The central dominance claim is a saddle-point statement. It requires the large wormhole to be perturbatively stable and to lie on the integration contour. Neither is established. Section 6 concedes both: no stability analysis was carried out, and the paper assumes the contour passes through the wormhole saddles. The analogy with the T^3 wormholes of [23] is plausible but not a proof. If the large wormhole has negative modes, or if the contour does not pick it up, Eq. (6.1) has no basis.\n\n2. There are no error bars or convergence data on the action differences. The claim that Delta s is positive and monotonically increasing rests on a finite set of numerically sampled points. This matters most near the critical wormhole where the two branches meet and the actions are close.\n\n3. The fit Delta s ~ J^4/k is presented as a linear best fit, not derived. That is fine, but it is post-hoc, not a prediction.\n\nThese are not fatal flaws. The phase diagram may well be correct, and the T^3 stability analogy is reasonable. But the abstract's unconditional wording runs ahead of what is actually verified.\n\nWho should read this: anyone working on AdS wormholes, factorization, or ensemble averaging in dimensions greater than two. The RG-flow sections also deserve attention from the holographic RG community.\n\nRecommendation: send it to peer review. A serious referee should push on stability and contour. The author should either supply the quadratic fluctuation analysis or soften the claims. With those caveats addressed, this would be a solid paper; as it stands, it is a legitimate advance.\n\nBest,\n[You]","headline":"A careful numerical study of planar sinusoidal-scalar wormholes whose central claim—large wormhole always dominant, no Hawking–Page transition—is plausible but awaits stability and contour analysis.","tokens_in":28916,"tokens_out":1937,"would_cite":true,"duration_ms":18182,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Sinusoidal scalar sources in AdS make the large wormhole the dominant saddle of the two-boundary path integral whenever it exists.","keywords":["Euclidean wormholes","AdS/CFT","factorization puzzle","ensemble averaging","holographic renormalization group","boomerang RG flows","multivalued beta-functions","sinusoidal scalar sources"],"falsifier":"Compute the spectrum of quadratic fluctuations around the large $\\mathbb{R}^3$ wormhole in $d=3$, $\\Delta=2$, or carry out a Picard--Lefschetz deformation of the gravitational path-integral contour. A negative mode in the fluctuation operator, or a steepest-descent contour that does not pass through the wormhole saddle, would overturn the claim that the large wormhole dominates whenever it exists.","tokens_in":27973,"feed_emoji":"🕳️","tokens_out":9748,"duration_ms":77539,"temperature":0.7,"pith_summary":"This paper studies a bottom-up holographic model in which identical sinusoidal scalar sources are turned on two asymptotic anti-de Sitter boundaries. It finds that wormhole solutions exist only above a critical source strength, and that whenever they exist the large wormhole is the dominant saddle of the two-boundary Euclidean gravitational path integral. There is no intermediate regime in which wormholes exist but are subdominant, unlike earlier examples with inhomogeneous matter on compact boundaries. Under the ensemble interpretation of the gravitational path integral, the model therefore predicts that the dual ensemble is exactly self-averaging below the threshold and strongly non-self-averaging above it, with normalized fluctuations growing exponentially in the action difference. The same geometries also give holographic renormalization-group flows with multivalued $\\beta$-functions: boomerang flows for disconnected boundaries and gapped flows ending at the throat for wormholes.","feed_headline":"No Hawking–Page transition: large wormhole always wins","feed_subtitle":"Sinusoidal scalar sources in AdS leave no regime where wormholes exist but lose to disconnected geometry.","key_machinery":"The sinusoidal scalar ansatz is the load-bearing construction: $d$ complex scalars $\\Phi_I(r,\\vec x)=\\phi(r)e^{ikx^I}$, each a plane wave in one boundary direction, give a stress tensor whose phases cancel, so the inhomogeneous matter sources a homogeneous and isotropic geometry and the Einstein--Klein--Gordon system reduces to coupled ODEs for the scale factor and radial profile. The saddle comparison is carried by holographically renormalized on-shell actions, with the $d=3$, $\\Delta=2$ counterterms derived in the appendix; the dimensionless source strength $\\tilde J=J/k^{d-\\Delta}$ parametrizes all physical solutions. For the RG interpretation, the folding trick maps a $Z_2$-symmetric wormhole to a one-sided flow on $\\mathbb{R}^d\\times S^0$, whose throat is a finite-depth endpoint.","core_discovery":"Working in $d=3$ with a conformally coupled scalar ($\\Delta=2$) and identical sinusoidal sources $J_I=Je^{ikx^I}$ on two $\\mathbb{R}^3$ boundaries, the paper constructs numerically the fully backreacted disconnected and $Z_2$-symmetric wormhole saddles. Below $\\tilde J\\equiv J/k\\simeq 22.5$ no wormhole exists; above it, a small and a large wormhole coexist, and the renormalized on-shell action difference $\\Delta s=s_{\\mathrm{disc.}}-s_{\\mathrm{conn.}}$ is positive and monotonically increasing for both branches, with the large wormhole always ahead. The central claim is that above threshold the large wormhole dominates the semiclassical path integral and there is no Hawking--Page-like regime of subdominant wormholes; equivalently, the normalized variance of the putative dual ensemble jumps from zero to $\\exp(\\Delta S)$ at the threshold. The paper also claims the disconnected geometry is dual to a boomerang RG flow that returns to the same CFT, while the folded wormholes are dual to gapped flows ending at the throat.","pith_inferences":["Editorial: If the large wormhole turns out to be free of negative modes, the model becomes a higher-dimensional example where non-factorization is not exponentially suppressed, and restoring factorization would require non-geometric cancellations of the same order as the wormhole action itself.","Editorial: A direct check of the paper's suspected mechanism is to build the disconnected counterpart of the $T^3$ wormholes of [23] and compare the action difference; if it is discontinuous there too, the missing compact boundary scale is the culprit, while a smooth Hawking--Page-like transition would point to boundary topology.","Editorial: The multivalued $\\beta$-function is likely generic for any periodic boundary source, not just a single sinusoid; testing other periodic profiles or different operator dimensions would show whether boomerang flows with two $\\beta$ branches are a universal feature of inhomogeneous sources.","Editorial: The wormhole's UV invisibility gives a concrete field-theoretic signature: any candidate dual ensemble must reproduce the finite cross-boundary correlator, and this could be checked numerically in a lattice model before a full gravitational dual is known."],"forward_implications":["For $\\tilde J\\equiv J/k^{d-\\Delta}$ below the critical value, only the disconnected saddle exists, so the two-boundary partition function factorizes to leading order and the would-be ensemble is exactly self-averaging.","For $\\tilde J\\ge \\tilde J_{\\mathrm{crit}}$, the large wormhole dominates immediately; the transition is zeroth-order (the action difference jumps to positive), not a Hawking--Page-like first-order transition.","The normalized variance of the ensemble jumps from 0 to $\\exp(\\Delta S)$, where $\\Delta S>0$ is $O(G_N^{-1})\\mathrm{Vol}(\\mathbb{R}^d)$, so a typical member of the ensemble is not represented by the mean.","The disconnected geometry is dual to a boomerang RG flow that leaves and returns to the same CFT, with a multivalued $\\beta$-function whose branch point is a turning point, not a fixed point.","Folded wormholes describe a single CFT on $\\mathbb{R}^d\\times S^0$ that flows to a gapped theory at the throat, with cross-boundary two-point functions decaying exponentially with $\\sqrt{\\lambda_0}$."],"supporting_citations":[{"why":"Supplies the sinusoidal-scalar wormhole construction and the bottom-up models whose Hawking--Page-like behavior this paper contrasts; also provides the $T^3$ stability analogy.","marker":"[23]"},{"why":"Adapts the ansatz to planar boundaries and gives the ordinary-wormhole equations and coordinate setup used here.","marker":"[36]"},{"why":"Establishes the JT-gravity/random-matrix ensemble interpretation that underlies the paper's variance reading of wormholes.","marker":"[8]"},{"why":"Provides the holographic renormalization method used to obtain finite on-shell actions for the saddles.","marker":"[75]"},{"why":"Gives the lecture-level framework for holographic renormalization used in the counterterm derivation and cutoff expansion.","marker":"[76]"},{"why":"Defines the holographic $\\beta$- and $c$-functions and the $c$-theorem framework that the RG-flow interpretation relies on.","marker":"[38]"},{"why":"Gives the Sturm--Liouville spectrum and exponential correlator form used to identify folded-wormhole flows as gapped.","marker":"[68]"},{"why":"Introduced boomerang RG flows, which the disconnected-saddle flows are compared to.","marker":"[55]"}],"fun_headline_variants":["Wormholes always dominate once they appear","No Hawking-Page: large wormhole always wins","Sinusoidal scalars kill subdominant wormholes","Large wormhole always wins, no Hawking-Page","Multivalued beta-functions from sinusoidal scalars"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the large wormhole is a genuine saddle of the Euclidean gravitational path integral---perturbatively stable and lying on the integration contour. The paper does not compute its quadratic fluctuation spectrum; it argues by analogy with the $T^3$ wormholes of [23] and explicitly assumes the contour passes through the wormhole saddles.","fun_headline_variants_meta":{"raw":{"variants":["Wormholes always dominate once they appear","No Hawking-Page: large wormhole always wins","Sinusoidal scalars kill subdominant wormholes","Large wormhole always wins, no Hawking-Page","Multivalued beta-functions from sinusoidal scalars"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1402,"prompt_tokens":986,"completion_tokens":416,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":339}},"tokens_in":602,"tokens_out":416,"duration_ms":4264,"temperature":1.0,"reasoning_tokens":339,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:33:27.850069+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the spectrum of quadratic fluctuations around the large $\\mathbb{R}^3$ wormhole in $d=3$, $\\Delta=2$, or carry out a Picard--Lefschetz deformation of the gravitational path-integral contour. A negative mode in the fluctuation operator, or a steepest-descent contour that does not pass through the wormhole saddle, would overturn the claim that the large wormhole dominates whenever it exists.","supporting_citations":[],"review_version":2}