{"id":"a41e0d6a-6dbb-404d-bed5-6e59f08e13d7","arxiv_id":"2607.06967","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":4,"one_line_summary":"Applying standard quantum tomography to data generated by Schrödinger-Newton dynamics yields angle-set-dependent reconstructed covariances that can violate the Heisenberg bound, providing an operational signature of classical gravity.","lead":"This paper shows that if gravity is classical (Schrödinger-Newton theory), applying standard quantum-mechanical tomography to a macroscopic mechanical oscillator produces angle-dependent reconstructed covariances that can violate the Heisenberg uncertainty bound. A smart generalist would read it for its operational proposal: a falsifiable test of whether gravity is classical or quantum, using the internal consistency of state tomography rather than a direct force measurement.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The CCSN prescription is the correct load-bearing concern; the paper is internally consistent within that framework, but the key SN tomographic correction would change qualitatively under alternative semi-classical gravity prescriptions.","rationale":"The reader's identification of the CCSN prescription as the weakest assumption is correct and is the most load-bearing concern. The paper's central results — angle-set dependence of the reconstructed covariance (Eq. 67–68) and the possibility of crossing the Heisenberg bound (Fig. 3) — all flow from the specific structure of A_SN = A_q − A_m, which exists only because CCSN makes the conditional mean evolve at ω_m while the conditional covariance evolves at ω_q. Under a different semi-classical gravity prescription, this frequency split would not arise in the same way, and the tomographic signatures could be qualitatively different.\n\nThe paper is honest about this dependence: it cites the debated prescriptions (Refs. 47–48), acknowledges the experimental challenges, and frames its results as conditional on the CCSN framework. The general nonlinear-tomography framework in Sec. V is a useful abstraction that correctly identifies when such corrections arise (state-dependent dynamics during readout), though the specific SN realization depends on the CCSN choice.\n\nThe approximation in Eq. (46) that the reader flags is a secondary concern: it carries an explicit ≈ symbol, is used only for intuition about signal distortion, and does not enter the numerical results (which use the exact Eq. 44). The lack of code/data is a reproducibility gap but not a correctness concern for a theoretical paper.\n\nThe verdict should remain CONDITIONAL with MODERATE confidence, as the reader states. The conditions are: (1) the CCSN prescription is the correct semi-classical gravity description during measurement, and (2) future experimental parameters (0.2 kg, mHz, Q=10^7, mK) become achievable. Both are acknowledged by the authors.","tokens_in":27475,"tokens_out":7510,"duration_ms":532594,"concrete_test":"Recompute σ²_SN in Eq. (44) using the alternative thermal/prescription framework of Refs. 47–48 where the conditional mean also evolves at ω_q (i.e., set A_m → A_q in Eq. 10 so that A_SN = 0 in Eq. 32). If σ²_SN vanishes or changes sign structure under this alternative prescription, the angle-dependence and below-Heisenberg-bound signatures are specific to CCSN rather than generic to semi-classical gravity. This would confirm that the paper's results are conditional on the CCSN choice, as the reader's verdict states.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies the most load-bearing assumption: the Causal Conditional Schrödinger-Newton (CCSN) prescription, where the self-gravity potential sources from the conditional mean x_c = ⟨ψ_c|x̂|ψ_c⟩ rather than the unconditional expectation value. This choice is not merely a background assumption — it is structurally load-bearing for the central result in the following specific sense.\n\nThe entire SN tomographic correction σ²_SN in Eq. (44) arises from the matrix A_SN = A_q − A_m (Eq. 32), which encodes the frequency split between the conditional covariance dynamics (evolving at ω_q via the Riccati equation, Eq. 21) and the conditional mean dynamics (evolving at ω_m via Eq. 10). This split exists specifically because, under CCSN, the SN force −Mω²_SN(x_c − x_c) vanishes at the conditional mean, so the mean trajectory is unaffected by self-gravity while the fluctuations around it feel the modified restoring frequency ω_q. If instead the gravitational potential sourced the unconditional density (or used a different conditional prescription as in Refs. 47–48), the conditional mean would also evolve at ω_q, A_SN would take a different form, and the specific structure of σ²_SN — including its angle-dependence through the conditional covariance trajectory and its non-positive-semidefinite character — could change qualitatively.\n\nThe paper is internally consistent within the CCSN framework: the derivation from Eqs. (10)–(44) is careful, the decomposition σ²_e = σ²_QG + σ²_SN is clean, the QG angle-set independence proof (Eq. 66) is correct, and the numerical results use the exact Eq. (44) rather than the approximate Eq. (46). The approximation in Eq. (46) is used only for physical intuition and is not load-bearing for the central claims. No internal inconsistency was identified.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This manuscript investigates how classical self-gravity, described by the Schrödinger-Newton (SN) theory in its Causal Conditional (CCSN) formulation, affects continuous quantum state tomography of a macroscopic mechanical oscillator. The authors derive the tomographic error functional in a Schrödinger-picture framework, showing that when a standard quantum-gravity (QG) optimized reconstruction map is applied to measurement data generated by SN dynamics, an additional state-dependent correction arises. This correction depends on the conditional covariance trajectory, which is itself angle-dependent, leading to three signatures: (1) the reconstructed covariance depends on the chosen set of tomography angles, (2) the inferred covariance can be driven outside the standard Gaussian-covariance domain (below the Heisenberg bound or even non-positive-definite), and (3) the QG-SN distinguishability, quantified by the Hellinger distance, exhibits nontrivial dependence on measurement strength and temperature. The paper also frames these results as a concrete instance of a general obstruction in tomography of nonlinear quantum mechanics.","tokens_in":27676,"tokens_out":1144,"duration_ms":432116,"significance":"The paper addresses a timely question in the quantum-gravity phenomenology program: how to operationally distinguish semi-classical gravity from quantum gravity using macroscopic optomechanical systems. The derivation from the stochastic master equation through the Riccati equation to the explicit SN correction (Eq. 44) is careful and internally consistent, with a clean algebraic identity (Eq. 66) demonstrating the angle-independence of the QG reconstruction. The generalization to nonlinear quantum mechanics in Sec. V, connecting the SN result to a Radon-consistency obstruction, is a conceptual contribution that extends the reach of the paper beyond the specific SN setting. The falsifiable predictions (Figs. 3, 5) and the honest assessment of experimental feasibility are commendable. The key physical insight—that the SN correction is not positive-semidefinite and therefore cannot be absorbed as ordinary added noise—is the central novel result.","major_comments":[{"comment":"Sec. II.B, Eqs. (10) and (21): The CCSN prescription, where the SN self-gravity potential sources from the conditional mean x_c = ⟨ψ_c|x̂|ψ_c⟩, is the load-bearing assumption of the paper. Under this prescription, the conditional mean evolves at ω_m (Eq. 10, drift A_m) while the conditional covariance evolves at ω_q (Eq. 21, drift A_q), producing the frequency split A_SN = A_q - A_m (Eq. 32) that generates the entire σ²_SN correction. The paper acknowledges (citing Refs. 47-48) that alternative thermal-noise prescriptions exist, but it does not discuss whether the CCSN prescription itself is the consensus or a contested choice, nor how the central results would change under an alternative semi-classical gravity coupling (e.g., one where the gravitational potential sources the unconditional density). Since the angle-dependence and below-Heisenberg-bound signatures are structurally tied to","section":null}],"minor_comments":[{"comment":"Sec. II.C, Eq. (46): The approximation sign is used without a clear statement of what is being approximated. The text mentions performing an integral by parts, but the conditions under which the remaining terms are negligible should be specified.","section":null},{"comment":"Table I: The parameters (M = 0.2 kg, ω_m/2π = 4×10⁻³ Hz, Q = 10⁷) are extremely challenging. The paper acknowledges this, but a brief quantitative estimate of the required measurement strength relative to the SN frequency would help the reader assess the regime of validity.","section":null},{"comment":"Fig. 2: The two angle sets Θ_A = {0, π/4, π/2} and Θ_B = {π/3, 2π/3, π} differ by a common shift of π/3. It would be instructive to also show a case where the angle sets are not related by a simple rotation, to demonstrate that the angle-dependence is not an artifact of a particular symmetry.","section":null},{"comment":"Sec. V, Eqs. (74)-(76): The general NLQM formulation is schematic. A brief comment on whether the SN nonlinearity satisfies the specific conditions discussed by Weinberg (Ref. 60) and Gisin (Ref. 61) would help situate the result.","section":null},{"comment":"References: Ref. [21] is cited as a 2026 arXiv preprint; please verify the date and completion status.","section":null},{"comment":"Typo in Acknowledgments: 'Y. M. W. Z. and Y. L. is supported' should read 'are supported'.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The reader's report and stress-test note correctly identify the CCSN prescription as the key load-bearing assumption. The paper is internally consistent within this framework and the derivation is sound. The main concern is whether the community will accept the CCSN prescription as the default semi-classical gravity coupling during measurement; the authors should at minimum acknowledge this as a model assumption and discuss the sensitivity of their conclusions to it. The general NLQM framing in Sec. V is a genuine conceptual contribution that elevates the paper above a single-theory calculation. I lean toward minor revision: the central claim is defensible and the issues are local (primarily the framing of the CCSN assumption), not structural."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. The referee correctly identifies the CCSN prescription as the load-bearing assumption of the paper and asks for a discussion of its status and of how the central results would change under alternative semi-classical gravity couplings. We agree this discussion should be added and explain below how we will revise the manuscript.","responses":[{"response":"The referee is correct that the CCSN prescription is the load-bearing assumption and that this point deserves explicit discussion. We will add a dedicated paragraph in Sec. II.B addressing the following points.","revision_made":"yes","referee_comment":"Sec. II.B, Eqs. (10) and (21): The CCSN prescription, where the SN self-gravity potential sources from the conditional mean x_c = ⟨ψ_c|x̂|ψ_c⟩, is the load-bearing assumption of the paper. Under this prescription, the conditional mean evolves at ω_m (Eq. 10, drift A_m) while the conditional covariance evolves at ω_q (Eq. 21, drift A_q), producing the frequency split A_SN = A_q - A_m (Eq. 32) that generates the entire σ²_SN correction. The paper acknowledges (citing Refs. 47-48) that alternative thermal-noise prescriptions exist, but it does not discuss whether the CCSN prescription itself is the consensus or a contested choice, nor how the central results would change under an alternative semi-classical gravity coupling (e.g., one where the gravitational potential sources the unconditional density). Since the angle-dependence and below-Heisenberg-bound signatures are structurally tied to"},{"response":"We address the two sub-questions in turn. (1) Status of CCSN: The CCSN prescription is not the unique formulation of semi-classical gravity; it is one of several prescriptions discussed in the literature (Refs. 20, 45–48). The key distinction is whether the gravitational source is the conditional state |ψ_c⟩⟨ψ_c| (as in CCSN) or the unconditional density matrix ρ = E[|ψ_c⟩⟨ψ_c|]. The CCSN choice is motivated by the requirement of causality: the gravitational field at time t should be sourced by the matter distribution at time t, not by a future ensemble average. This is the position taken in Refs. 20, 45–46 and further discussed in Ref. [46] (Miki et al., 2025). However, we agree that this is a contested choice, not a settled consensus, and the manuscript should say so explicitly. (2) How results change under alternative prescriptions: The frequency split A_SN = A_q − A_m arises specifically because, under CCSN, the conditional mean evolves at ω_m (the SN force vanishes on the mean trajectory) while the conditional covariance evolves at ω_q. Under a prescription where the gravitational potential sources the unconditional density, the situation is structurally different: the unconditional covariance would evolve at ω_q, but the conditional covariance evolution would not necessarily carry the same A_SN correction during the measurement process. In that case, the specific σ²_SN correction derived in Eq. (44) would take a different form, and the angle-dependence and below-Heisenberg-bound signatures may be weakened or absent. However, the general mechanism identified in Sec. V — that state-dependent nonlinear dynamics during readout introduces a model-dependent correction to the tomographic map — does not depend on the CCSN prescription specifically. Any semi-classical orhy","revision_made":"no","referee_comment":"[Continued from above — the comment was cut off, but the thrust is: how do the central results change under alternative semi-classical gravity couplings, and is CCSN consensus or contested?]"}],"tokens_in":26987,"tokens_out":1298,"duration_ms":102359,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper identifies a genuinely new operational signature of Schrödinger-Newton dynamics — when you apply a standard QG-optimized tomographic filter to data generated by SN dynamics, the reconstructed covariance depends on which tomography angles you chose, and can fall below the Heisenberg uncertainty bound or even go non-positive. The QG reconstruction does neither. That's a clean, falsifiable distinction between classical and quantum gravity that doesn't rely on entanglement generation, and as far as I can tell it's not in the prior literature on SN phenomenology (Refs. 20, 45–48 cover spectral and interferometric signatures, not tomographic consistency). The general nonlinear-tomography framework in Sec. V, which frames this as a Radon-consistency obstruction for any state-dependent dynamics during readout, is a useful abstraction and not just padding — it makes clear why the angle-dependence is structural rather than a numerical artifact of one parameter set. The derivation from the SME through the Riccati equation to the SN correction (Eq. 44) is careful. The key identity (Eq. 66) proving QG angle-set independence is clean and correct. The numerical results use the exact Eq. (44), not the approximate Eq. (46), so the approximation the reader flagged is not load-bearing — it's used only for physical intuition in the surrounding text. The stress-test concern about CCSN being structurally load-bearing is correct and well-placed. The entire SN correction arises from the frequency split A_SN = A_q − A_m, which exists specifically because under CCSN the SN force vanishes at the conditional mean, leaving the mean trajectory at ω_m while fluctuations feel ω_q. A different prescription for how classical gravity couples to the conditional state would change A_SN and could qualitatively alter the signatures. The paper is honest about this — it states the CCSN choice explicitly and cites the alternative prescriptions in Refs. 47–48 — but it doesn't explore sensitivity to that choice. That's the main gap. The experimental parameters (0.2 kg, mHz, Q=10^7, mK) are far beyond current capabilities, as the authors acknowledge. No code or data is provided for the numerical results, which is a minor reproducibility gap. This paper is for theorists working on semi-classical gravity phenomenology and experimentalists thinking about long-term tests of quantum vs. classical gravity. It deserves a serious referee — the core results are novel, the derivation is sound within its stated framework, and the open questions (CCSN sensitivity, self-consistent SN tomography as nonlinear estimation) are clearly identified rather than hidden.","headline":"Novel SN tomographic signatures (angle-set dependence, below-Heisenberg covariance) are real and cleanly derived within the CCSN framework; the framework itself is the load-bearing assumption.","tokens_in":28385,"tokens_out":628,"would_cite":true,"duration_ms":144037,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.65.Ta","03.65.Yz","04.60.-m","42.50.-p"],"model":"glm-5.2","headline":"Classical Self-Gravity Breaks Quantum State Tomography","keywords":["Schrödinger-Newton equation","quantum state tomography","semi-classical gravity","optomechanics","Heisenberg uncertainty principle","nonlinear quantum mechanics","continuous measurement","gravitational self-energy"],"falsifier":"If the conditional mean prescription for the Schrödinger-Newton potential is replaced by one where gravity sources the unconditional density, the tomographic correction sigma^2_SN would no longer depend on the conditional covariance trajectory in the same way, and the angle-set dependence and below-bound signatures could vanish.","tokens_in":27505,"feed_emoji":"","tokens_out":1270,"duration_ms":195861,"temperature":0.7,"pith_summary":"This paper studies what happens when you try to reconstruct the quantum state of a macroscopic mechanical oscillator using standard quantum-mechanical data analysis, but the oscillator's motion is actually influenced by its own classical gravitational field—the Schrödinger-Newton effect. The central object is the tomographic reconstruction map: the mathematical procedure that converts continuous optical measurement records into an inferred quantum-state covariance matrix. In standard quantum mechanics (or if gravity is quantized), this map is well-behaved: the reconstructed covariance is independent of which measurement angles you choose, and the inferred state always satisfies the Heisenberg uncertainty principle. The paper shows that when the underlying dynamics include classical self-gravity, the same reconstruction map acquires a state-dependent correction that depends on the conditional covariance trajectory during measurement. This correction makes the reconstructed covariance depend on the specific set of measurement angles chosen and can push the inferred covariance below the Heisenberg uncertainty bound or even make it non-positive-definite. The authors quantify the distinguishability between quantum-gravity and Schrödinger-Newton reconstructions using the Hellinger distance across measurement-strength and temperature parameter space, finding that the mismatch is strongest at low temperature and moderate measurement strength. They then generalize the result: any nonlinear quantum dynamics where the state being measured also controls the measurement process itself will produce analogous corrections to the standard tomographic reconstruction map, breaking the angle-set consistency that ordinary quantum tomography guarantees.","feed_headline":"","feed_subtitle":"","key_machinery":"The Schrödinger-Newton self-gravity term in the center-of-mass Hamiltonian, M*omega_SN^2*(x_hat - <x_hat>)^2, sourced by the conditional mean position x_c = <psi_c|x_hat|psi_c> during continuous homodyne measurement. The conditional covariance matrix V_c(t) evolves via a Riccati equation at the SN-modified frequency omega_q = sqrt(omega_m^2 + omega_SN^2), and the SN tomographic correction sigma^2_SN[g1,g2] = -omega_SN^2 * integral(j2(t) * [hxx(t|theta)*j1(t) + hxp(t|theta)*j2(t)]) inherits the angle-dependence of this trajectory.","core_discovery":"The paper's central discovery is that the Schrödinger-Newton self-gravity correction to continuous optomechanical tomography, denoted sigma^2_SN, depends on the conditional covariance trajectory V_c(t) which is itself shaped by the chosen homodyne measurement angle. Because this correction is a functional of the same state parameters being reconstructed and varies with the measurement setting, applying the standard quantum-mechanical reconstruction map to Schrödinger-Newton-generated data produces three diagnostic signatures: (1) the reconstructed covariance matrix changes when different sets of tomography angles are used, unlike in standard quantum mechanics where the result is angle-set-in","pith_inferences":["The angle-set consistency test could be applied to other proposed nonlinear modifications of quantum mechanics—such as spontaneous collapse models or post-quantum theories of classical gravity—to check whether they produce analogous tomographic signatures without requiring knowledge of the specific nonlinear correction.","If the Schrödinger-Newton frequency omega_SN is extremely small relative to the mechanical frequency, the tomographic signatures vanish, which means the test is complementary to spectral-based SN tests: tomography probes the conditional dynamics during measurement rather than steady-state oscillation frequencies.","An adaptive or Bayesian tomography protocol that iteratively refines its estimate of the initial covariance and propagates the SN conditional equations for each candidate could in principle perform self-consistent SN tomography, but whether such a protocol is experimentally efficient at the required parameter regimes remains an open question."],"forward_implications":["An experiment that reconstructs a macroscopic oscillator's quantum state at multiple angle sets and finds inconsistent covariance matrices would have evidence that the underlying dynamics are not purely standard quantum mechanics, potentially pointing to classical self-gravity or other nonlinear effects.","The below-Heisenberg-bound or non-positive-definite reconstructed covariance is a model-mismatch diagnostic: it does not mean the uncertainty principle is violated, but rather that the assumed linear reconstruction map is incompatible with the actual dynamics that generated the data.","Self-consistent tomography under Schrödinger-Newton dynamics cannot proceed by simply adding a correction term to the standard filter; it requires solving a nonlinear inverse problem where the unknown state parameters appear inside the reconstruction map itself.","The angle-set consistency test could serve as a theory-agnostic diagnostic for nonlinear quantum dynamics beyond the specific Schrödinger-Newton case: any theory where the measurement process depends on the state being inferred will break Radon consistency."],"fun_headline_variants":["Classical self-gravity breaks angle independence in quantum tomography","Schrödinger-Newton effects make macroscopic tomography angle-dependent","Standard quantum tomography breaks down under classical self-gravity","Macroscopic self-gravity adds nonlinear corrections to tomography","Self-gravity pushes reconstructed covariance past standard uncertainty bounds"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The entire analysis depends on the Causal Conditional Schrödinger-Newton prescription, where the classical gravitational potential sources the conditional mean position x_c = <psi_c|x_hat|psi_c> rather than the unconditional expectation value. If a different prescription for how classical gravity couples to the conditional quantum state were correct, the angle-dependence and below-Heisenberg-bound signatures could change qualitatively or disappear.","fun_headline_variants_meta":{"raw":{"variants":["Classical self-gravity breaks angle independence in quantum tomography","Schrödinger-Newton effects make macroscopic tomography angle-dependent","Standard quantum tomography breaks down under classical self-gravity","Macroscopic self-gravity adds nonlinear corrections to tomography","Self-gravity pushes reconstructed covariance past standard uncertainty bounds"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1570,"prompt_tokens":523,"completion_tokens":1047,"prompt_tokens_details":null},"tokens_in":523,"tokens_out":1047,"duration_ms":40177,"temperature":1.0,"reasoning_tokens":1046,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T00:49:02.427592+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If the conditional mean prescription for the Schrödinger-Newton potential is replaced by one where gravity sources the unconditional density, the tomographic correction sigma^2_SN would no longer depend on the conditional covariance trajectory in the same way, and the angle-set dependence and below-bound signatures could vanish.","supporting_citations":[],"review_version":1}