{"id":"3a404a71-1104-43fe-a835-3cab1abbbdca","arxiv_id":"2510.14089","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The mean evolved projected density field is a 2D Zel'dovich-evolved field times a Gaussian damping factor that quantifies information loss from unconstrained line-of-sight modes.","lead":"This paper derives a model for how the two-dimensional projected matter density field evolves from a fixed initial projected field, treating the unknown three-dimensional line-of-sight modes statistically. It predicts an exponential damping of small-scale information and tests the prediction against N-body simulations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Key result Eq. 2.24 rests on an unjustified claim that the q∥ integral in Eq. 2.23 is a constant; the integral is actually k⊥-dependent.","rationale":"The reader identified the same weakest assumption: the step from Eq. 2.23 to Eq. 2.24 is asserted rather than demonstrated. My analysis confirms this is the most load-bearing concern because it underlies the central claim that the mean evolved projected field is a Zeldovich-evolved field times a Gaussian damping factor. In fact, the q∥ integral in Eq. 2.23 is not a constant but a k⊥-dependent prefactor (or, with the sign error, divergent), which would alter the predicted suppression and the interpretation of Σ². The simulation comparison in Fig. 3 empirically supports a pure exponential, but that does not repair the derivation; it suggests either a hidden cancellation or an erratum. The paper should either supply a correct derivation or demonstrate numerically that the exact q∥ integral yields the claimed form. This does not change the reader's CONDITIONAL verdict: the paper is promising but requires a fix or clarification before the central formula can be trusted.","tokens_in":10357,"tokens_out":11712,"duration_ms":91462,"concrete_test":"Evaluate the q∥ integral in Eq. 2.23 exactly (without the quadratic expansion Eq. 2.20) using the top-hat window W(k∥)=sinc(k∥L/2) and the simulation power spectrum, then apply the projection P̂. Numerically compute the resulting mean field and its power spectrum, and form the ratio P_me/P_ee over the same k⊥ range as Fig. 3. If the ratio is consistent with exp(−½ k⊥² Σ²) (no power-law trend), the step may be salvageable; if it shows a k⊥−2 or other deviation, Eq. 2.24 must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result Eq. 2.24 is obtained from Eq. 2.23 by asserting that the q∥ integral 'just changes the overall normalization' and that the projection operator behaves likewise. This assertion is internally inconsistent. In Eq. 2.23 the q∥ integral is ∫ dq∥ exp[−½ D² k⊥² q∥² Σ_W^{(2)}], which—if evaluated as a Gaussian—equals √(2π/(D² k⊥² Σ_W^{(2)})), a factor ∝ 1/|k⊥|, not a constant. Moreover, the sign is suspect: combining Eq. 2.20 (fullΣ_W = Σ_W² − ½ q∥² Σ_W^{(2)}) with the exponent in Eq. 2.14 yields +¼ D² k⊥² q∥² Σ_W^{(2)} in the exponent, i.e. a growing exponential, not the decaying one shown. Even if the sign is a typo, the k⊥−1 prefactor survives the projection operator, since the k∥ integral in P̂ only acts on the k∥-dependent Gaussian and leaves the k⊥-dependent prefactor untouched. Thus the predicted mean projected field is A(k⊥) exp(−½ D² k⊥² Σ²) Z(Δ0,t) with A(k⊥) ∝ 1/|k⊥|, not the pure exponential of Eq. 2.24. This would change the predicted ratio P_me/P_ee by a k⊥−2 factor, which is not observed in Fig. 3 (the ratio approaches unity at low k). The paper does not resolve this discrepancy; the derivation of the flagpole formula is missing a demonstrated step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses what can be predicted for the two-dimensional projected matter density field from the initial projected matter density field alone. At linear order the relation is deterministic; at nonlinear order the unconstrained 'bulk' modes contribute through mode coupling. Using Lagrangian perturbation theory in the Zeldovich approximation, the authors derive an expression for the ensemble mean of the evolved projected field conditioned on fixed initial projected modes: Eq. (2.24), a 2D Zeldovich-evolved field multiplied by a Gaussian damping exp(-1/2 D^2 k⊥^2 Σ^2). This prediction is tested with 100 small-box N-body simulations that share the same k_z=0 initial modes but differ in all other modes, by comparing the power spectrum of the mean field to that of a simulation evolved only from the projected modes. The reported agreement is good in shape, with the predicted damping amplitude matching the measured one to 7% at z=3, 20% at z=1, and 30% at z=0. The paper concludes by sketching a hybrid field-level likelihood in which the mean field is modeled deterministically and residual bulk-mode effects are described statistically.","tokens_in":10782,"tokens_out":23679,"duration_ms":194593,"significance":"If the central formula is correct, it gives a parameter-free, falsifiable prediction for how much information about initial projected modes survives into the nonlinear projected field, and it provides a natural functional form for a projected-field likelihood. The simulation protocol is a clean and honest test: the suppression exponent is not fit in the theory curves, and the 'best fit' in Fig. 3 is used only as a comparison diagnostic. The degrading agreement with redshift is reported transparently. The main issues are an asserted rather than demonstrated step in the LPT derivation, an apparent inversion of the growth-factor dependence in the quoted Σ² values, and a mismatch between the abstract and the actual content regarding the HMC implementation.","major_comments":[{"comment":"The step from Eq. (2.23) to Eq. (2.24) is not demonstrated and, as written, is not correct. Evaluating the q∥ integral in Eq. (2.23) as a Gaussian gives a prefactor proportional to 1/|k⊥|, not a constant. Even after correcting the coefficient from Eqs. (2.14)–(2.20) — the characteristic-function exponent should contain −¼ D² k⊥² q∥² Σ_W^(2), not −½ — the |k⊥|⁻¹ factor survives because the projection operator P̂ acts only on k∥ and leaves the k⊥-dependent prefactor untouched. The exact q∥ integral before the quadratic expansion is not the Gaussian shown; over a finite window it approaches a constant at low k⊥ and crosses to a 1/k⊥ scaling only at large k⊥, with corrections that are not computed. Thus Eq. (2.24) is not a proven consequence of Eq. (2.23). This is the central derivation of the paper and needs to be fixed or replaced with a controlled computation of the q∥ integral.","section":"§2.2, Eqs. (2.23)–(2.24)"},{"comment":"The quoted suppression values are internally inconsistent with Eq. (2.24). Equation (2.24) contains the combination D² Σ², so the effective Gaussian width at redshift z should scale as D(z)² Σ²(z=0), decreasing toward high redshift. Yet the text reports predicted/measured Σ² = 264/246, 103/84, 25.1/17.13 (Mpc/h)² at z=3, 1, 0 — the opposite trend. These numbers are close to Σ²(z=0)/D(z)², suggesting an inversion of the growth factor. Because this comparison is the main quantitative test of the model, the definition of Σ² in Eq. (3.1) and the role of D(t) must be clarified or corrected.","section":"§3.2, Fig. 3 and text"},{"comment":"The abstract states that the approach is implemented in a likelihood code and that HMC sampling reconstructs initial fields in the presence of non-trivial masks. The main text contains no such implementation, results, or code description; §4 only sketches the likelihood in Eq. (4.1) and lists it as future work. Either the implementation should be added or the abstract claim removed. This is a substantial mismatch between the advertised scope and the actual content.","section":"Abstract vs. §4"}],"minor_comments":[{"comment":"The exponent of the q∥ Gaussian appears to be off by a factor of 2 relative to the derivation in Eqs. (2.20)–(2.22); please check the sign and coefficient.","section":"Eq. (2.23)"},{"comment":"Table 1 is labeled as z=0, but the text then quotes values at z=3 and z=1 without an explicit formula relating them. State the growth-factor convention explicitly so the reader can reproduce the quoted numbers.","section":"Table 1 and §3.2"},{"comment":"The replacement of the Zeldovich operator by a fully nonlinear field is acknowledged as a 'common swindle.' It would be helpful to state explicitly that Eq. (2.24) is therefore a resummed/phenomenological model rather than a strict LPT result.","section":"§2.2, 'common swindle'"},{"comment":"Typographical issues: 'preform' should be 'perform' (§3.1); 'Efstathion' should be 'Efstathiou'; the notation 'Mpc/ℎ2' throughout is unformatted.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The q∥–integral gap and the inverted redshift dependence of Σ² are real technical problems that must be addressed; the abstract/body mismatch on the HMC demonstration is also a scope issue. The paper has a useful idea and a clean simulation protocol, so major revision seems feasible rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Kevin, here's my take on 2510.14089. The new thing is the calculation of the mean evolved projected density field conditioned on fixed initial projected modes. Within Zeldovich/LPT, they get a 2D Zeldovich-evolved field times a Gaussian damping exp(-½ D² k⊥² Σ²), with Σ² = Σ_Z² - Σ_W² + Σ_W2². That's a concrete, testable statement about information loss in projected fields, and it's not in the earlier field-level inference papers I know.\n\nThe simulation setup is the right way to test it: 100 small boxes with identical projected initial conditions and different bulk modes. At z=3 the predicted damping matches the measured P_me/P_ee to 7%; at z=0 it drifts to 30%, which they note and attribute to higher-order corrections. That's honest.\n\nThe weak spots are real but manageable. The step between Eq. 2.23 and 2.24 is the one place where the derivation is asserted rather than demonstrated. The stress-test note worried that the q∥ integral leaves a 1/k⊥ prefactor; I don't think that survives, because the subsequent k∥ integral in the projection operator convolves with the Gaussian in k∥ and gives a constant at leading order. But the authors should show that calculation instead of saying 'just changes the overall normalization.' Also, the abstract promises an HMC likelihood implementation and mask tests; the body has neither. That's an overclaim and needs to be fixed, either by adding the material or rewording the abstract. Finally, the quoted Σ² values at z=3,1,0 — 264,103,25.1 Mpc/h² — look backwards if they're supposed to scale as D²(z) Σ²(z=0). Given the table's z=0 value of 25.1, either the list or the normalization is off. That needs a clarifying edit.\n\nWho's this for? People building field-level likelihoods for weak lensing and photometric clustering, and anyone interested in how much of the initial projected field survives nonlinear evolution. It's a subfield contribution, not a breakthrough, but it's a solid one. With the abstract fixed and the derivation gap filled, I'd send it to a referee.","headline":"A useful and honest paper on the conditional mean of evolved projected fields; the abstract oversells an HMC implementation that isn't in the body, but the core derivation and simulation test are worth referee time.","tokens_in":11177,"tokens_out":16209,"would_cite":true,"duration_ms":122145,"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":"The paper derives a closed-form prediction for the mean evolved projected matter field at fixed initial projected modes: it is a 2D Zel'dovich-evolved field multiplied by a Gaussian damping factor whose width is set by the variance of the u","keywords":["field-level likelihood","projected density field","Lagrangian perturbation theory","Zel'dovich approximation","initial conditions","information loss","N-body simulations"],"falsifier":"Evaluate Eq. (2.23) numerically without dropping the q_parallel integral, and compare it with Eq. (2.24) for a given window function; if the projection produces any k_perp-dependent correction beyond a constant normalization, or if the measured P_me/P_ee in a fixed-projected-modes N-body suite deviates from exp(-k_perp^2 Sigma^2) on scales where the expansion was assumed convergent, the central relation is wrong.","tokens_in":10265,"feed_emoji":"🌌","tokens_out":6256,"duration_ms":53917,"temperature":0.7,"pith_summary":"For a 2D projected cosmological density field, the evolved map is not a deterministic function of the initial projected map beyond linear order: non-projected 'bulk' modes also drive the evolution. This paper asks what an observer who knows only the initial projected modes can still predict. Using Lagrangian perturbation theory, it derives the ensemble mean of the evolved projected field conditioned on fixed initial projected modes: a 2D Zel'dovich-evolved field multiplied by a Gaussian damping exp(-1/2 D^2 k_perp^2 Sigma^2). The damping width is set by the variance of the bulk modes' displacements, with window-function corrections. A suite of 100 N-body simulations with identical projected initial conditions confirms the Gaussian suppression, implying that a field-level likelihood for 2D surveys can recover initial projected information only on large scales.","feed_headline":"Projected cosmic fields forget initial modes exponentially","feed_subtitle":"Theory and 100 fixed-projection simulations agree: a 2D density map retains its initial projected state only on large scales.","key_machinery":"The load-bearing object is Eq. (2.24), which packages the whole argument as a 2D Zel'dovich-evolved field times a Gaussian damping factor. It is obtained by writing initial conditions conditioned on fixed projected modes, expanding the evolved field in Zeldovich displacements, and Gaussian-averaging over the unconstrained line-of-sight modes. The three variance terms in Sigma^2 arise, respectively, from the full 3D displacement variance, the variance of the projected component of the residual modes, and the cross-term that depends on the line-of-sight window; an expansion of that window term in q_parallel turns the result into a pure Gaussian damping.","core_discovery":"The central claim is Equation (2.24): the average over all initial conditions sharing a fixed projected density field d(k_perp) gives the mean evolved projected field Delta(Delta0,t) = exp(-1/2 D^2 k_perp^2 Sigma^2) Z(Delta0,t), where Z(Delta0,t) is the 2D Zel'dovich evolution of the initial projected modes and Sigma^2 = Sigma^2_Z - Sigma^2_W + Sigma^2_W2. In words, the deterministic part of the evolution is a 2D Zel'dovich map, while the unknown bulk modes act as a Gaussian random smearing that exponentially suppresses memory of the initial projected state on small scales. The paper tests this against 100 small N-body simulations with fixed projected initial conditions and finds the predict","pith_inferences":["Inference: Because the derivation expands cos(k_parallel q_parallel) only to leading order, the exact Gaussian shape is an approximation; for narrow window functions one should see the next term as a k^4 correction to the log-ratio, a measurable prediction.","Inference: For broad weak-lensing kernels that integrate over cosmic time, Sigma^2 cannot be a single number; a tomographic generalisation of Eq. (2.24) would need a redshift-dependent damping applied bin-by-bin.","Inference: The formula suggests a natural upgrade path: replace the Zel'dovich 2D evolution with a fast dedicated 2D non-linear solver and test whether the residual damping stays Gaussian with the same Sigma^2, which would make the hybrid likelihood fully non-linear.","Inference: The information-loss curve makes a concrete survey-design prediction: the recoverable information at scale k_perp is controlled by the combination Sigma^2(L_window), so photometric bin widths and window shapes can be optimised to maximise the retained initial-projected information."],"forward_implications":["On scales where k_perp^2 Sigma^2 becomes large, the mean evolved projected field loses almost all memory of the initial projected modes; information is exponentially suppressed, so a field-level likelihood for 2D surveys mainly constrains large scales.","The paper proposes a hybrid likelihood: evolve the initial projected field deterministically via Eq. (2.24), then model the residual from bulk modes statistically (e.g. Gaussian with covariance P - P_mm), and use summary statistics of residuals to capture the statistical information in bulk modes.","The suppression has the same physical origin as BAO damping in 3D but applies to the one-point field, with a factor of 1/2; the window-correction terms reduce the damping because some modes are evolved explicitly.","At z=0 the measured suppression departs from growth-factor-squared scaling by about 20%, indicating higher-order Lagrangian corrections matter at low redshift."],"fun_headline_variants":["2D cosmic fields exponentially forget initial modes","Projected density maps lose initial info exponentially on small scales","Initial 2D modes suppressed exponentially in evolved maps","Evolved 2D density forgets initial projected state exponentially"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The step from Eq. (2.23) to Eq. (2.24) is asserted rather than demonstrated: it assumes the line-of-sight integral in Eq. (2.23) only changes the normalization and that, after projection, the damping is exactly the Gaussian exp(-1/2 D^2 k_perp^2 Sigma^2) with amplitude fixed by linear theory.","fun_headline_variants_meta":{"raw":{"variants":["2D cosmic fields exponentially forget initial modes","Projected density maps lose initial info exponentially on small scales","Initial 2D modes suppressed exponentially in evolved maps","Evolved 2D density forgets initial projected state exponentially"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000697,"raw_usage":{"total_tokens":2984,"prompt_tokens":737,"completion_tokens":2247,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":2192}},"tokens_in":481,"tokens_out":2247,"duration_ms":14300,"temperature":1.0,"reasoning_tokens":2192,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T09:37:50.356290+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate Eq. (2.23) numerically without dropping the q_parallel integral, and compare it with Eq. (2.24) for a given window function; if the projection produces any k_perp-dependent correction beyond a constant normalization, or if the measured P_me/P_ee in a fixed-projected-modes N-body suite deviates from exp(-k_perp^2 Sigma^2) on scales where the expansion was assumed convergent, the central relation is wrong.","supporting_citations":[],"review_version":1}