{"id":"d72c7081-7c93-4cd8-a4e0-e2e727288069","arxiv_id":"2507.14765","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A theory paper claims necessary and sufficient observability conditions for multi-target passive tracking, but the proofs have major mathematical errors and tautological steps.","lead":"This paper derives conditions for a single observer to uniquely track multiple targets from bearing angles, Doppler shifts, or both, including a requirement that bearing angles stay distinct. The derivations contain serious mathematical gaps, so the claimed necessary-and-sufficient conditions are not supported as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The rank-2 condition on P(t) does not establish observability: C(t) is block-diagonal, so distinct bearings between targets cannot prevent a nonzero target state from yielding identically zero pseudo-measurements.","rationale":"The reader's weakest assumption correctly identifies the Section 3 rank argument as the load-bearing flaw: the paper equates the zero-output condition C(t)Phi(t,ti)y=0 for all t with the instantaneous rank of the bearing-angle matrix P(t), but C(t) is block diagonal and the zero-output condition is per-target. A single target on a constant-bearing trajectory produces identically zero pseudo-measurement regardless of range, so pairwise distinct bearings do not restore observability. The proposed static two-target counterexample makes this concrete and shows that Proposition 1 fails even in the simplest first-order case. The reader also correctly flags the Doppler scalar-vector error in equation (24); this is a second independent defect, but the bearing-rank gap alone is sufficient to reject the central claim. I find no reason to change the reader's REJECT verdict. The paper's intuitive idea of bearing distinctness may be a useful heuristic, but the proof provided does not establish a valid necessary and sufficient condition.","tokens_in":9211,"tokens_out":4108,"duration_ms":54539,"concrete_test":"Construct the simplest instance: M=2, pi=0, unforced dynamics, static observer at the origin, and static targets at r1=(1,0) and r2=(0,1). The bearing angles are distinct modulo pi, so the paper's P(t) has rank 2. However, for each target the pseudo-measurement is z_i(t)=x_i cos(theta_i)-y_i sin(theta_i), which is identically zero for r1 and r2. Hence C(t)Phi(t,ti)y=0 for the nonzero initial state y=(r1,r2), violating Definition 1. Equivalently, compute the observability Gramian integral of C(t)^T C(t) over any interval and verify it is singular. This directly falsifies Proposition 1. A correct analysis would instead expand (19) into coefficients of (t-ti)^k for each target and test the rank of the full coefficient matrix, which the paper does not do.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3's central inference is unsound. Definition 1 requires the observability Gramian to be nonsingular, equivalently that C(t)Phi(t,ti)y=0 on [ti,tf] forces y=0. But C(t) in (14) is block diagonal, with each row c_i(t)=[cos(theta_i(t)), -sin(theta_i(t)), 0...]. The zero-output condition therefore decouples per target: c_i(t)Phi_i(t,ti)y_i=0 for every target i. This is exactly the pseudo-linearized measurement x_i(t)cos(theta_i(t))-y_i(t)sin(theta_i(t))=0, which holds for any nonzero state lying along the target's own bearing line (e.g., any range along that line), independent of other targets. The determinant condition (20) on the M x 2 matrix P(t) constrains only pairwise differences of bearing angles and does not enter the zero-output condition. Equation (19) is one scalar polynomial identity per target; its coefficients involve all derivative states and are never separated, so the paper never analyzes the full kernel. Thus a full-rank P(t) is neither necessary nor sufficient for condition (18): a nonzero zero-output state can exist with all bearings pairwise distinct. This is not merely a disagreement with prior consensus; it is an internal gap in the derivation of Proposition 1. The Doppler portion has a similar scalar-vector leap: integrating the scalar Doppler equation (23) yields a scalar range relation, not the vector position relation (24), so the transformation W(t) is not actually derived from the measurement equality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript addresses observability of multiple targets tracked by a single observer. The authors propose two frameworks. First, for bearing-only measurements, they use the observability Gramian of a pseudo-linearized system and claim that the multi-target system is observable iff all bearing angles are distinct modulo π (Proposition 1), independent of the order of target and observer dynamics (Remark 3). Second, they propose a trajectory-ambiguity framework and derive necessary and sufficient conditions for ambiguity under Doppler-only (Proposition 2), bearing-only (Eq. (26)), and combined Doppler-bearing measurements (Proposition 3), based on a transformation matrix W(t). The paper is entirely analytical and contains no simulations.","tokens_in":9597,"tokens_out":5776,"duration_ms":67506,"significance":"The problem is relevant, and a simple geometric criterion would be practically useful. The manuscript is self-contained in defining observability and attempts direct derivations, which is a strength. However, the central derivations contain load-bearing errors: the bearing-only observability conclusion does not follow from the Gramian condition because the measurement matrix is block diagonal and the zero-output condition decouples per target; the Doppler ambiguity condition is obtained by integrating a scalar range-rate equation into a vector position relation, and it introduces an arbitrary time-varying unitary transformation that makes the claimed necessary condition vacuous. These issues invalidate the main claims and cannot be repaired by local edits.","major_comments":[{"comment":"The inference from the zero-output condition to the rank of P(t) is invalid. Condition (18) is equivalent to c_i(t) Φ_i(t,t_i) x_i(t_i) = 0 for every i separately, because C(t) in Eq. (14) is block diagonal. The determinant condition (20) on P(t) involves only pairwise differences of bearing angles and does not enter the per-target zero-output condition. Hence a full-rank P(t) is neither necessary nor sufficient for (18): a nonzero initial state of one target lying on its own bearing line produces identically zero pseudo-measurement regardless of the other targets. Consequently Proposition 1 and Remark 3 are unsupported. In the single-target case (M=1), P(t) has one row and can never be rank 2, so the criterion would declare every single-target bearings-only system unobservable, contradicting the standard maneuver-dependent observability condition in the cited literature [2,3].","section":"Section 3, Eqs. (18)-(20)"},{"comment":"Equation (19) is one scalar polynomial identity per target, whose coefficients involve all derivative states and the time-varying trigonometric functions cos θ_i(t) and sin θ_i(t). The authors never separate coefficients or analyze the kernel of C(t)Φ(t,t_i); the argument jumps from this polynomial identity to the rank of the instantaneous matrix P(t). This missing coefficient-level analysis is the load-bearing step that would be needed to prove condition (18), and it is absent.","section":"Section 3, Eq. (19)"},{"comment":"The integration step leading to Eq. (24) is invalid. Equation (23) equates scalar range-rates, and integrating it yields scalar range as a function of time, not the vector identity s_i(t) = l' s_j(t) + b' + c(1-l')(t-t_i). Therefore Eq. (24) does not follow from the Doppler measurement equality, and the transformation W(t) in Eq. (22) is not actually derived from the measurements. This invalidates Proposition 2 and the subsequent ambiguity conditions that rely on W(t).","section":"Section 4.1, Eqs. (23)-(24)"},{"comment":"The introduction of an arbitrary unitary transformation U(t) makes condition (22) vacuous. Since both unit vectors û_{s_i}(t) and û_{s_j}(t) have unit norm, for any pair of trajectories there exists a unitary transformation mapping one to the other at each time t. Thus Eq. (22) can always be satisfied by choosing U(t) appropriately, so it cannot be a necessary condition for Doppler ambiguity. This is a circularity in the proof of Proposition 2.","section":"Section 4.1, Eq. (22)"},{"comment":"There is an internal contradiction about the status of condition (22). Proposition 2 states that multiple targets have ambiguous Doppler trajectories iff they satisfy (22), but Remark 4 explicitly says that (22) is necessary but not sufficient and lists additional sufficient conditions, and Appendix B derives those additional conditions. A statement cannot be both an iff condition and merely a necessary condition. In addition, Appendix C concludes that the NECNDSUF condition for Doppler-bearing ambiguity is α'(t)=1, which through Eq. (26) reduces to s_i(t)=s_j(t); the claimed eigenvector condition W(t)s_j(t)=α'(t)s_j(t) therefore collapses to a trivial identical-trajectory condition rather than providing a new observability criterion.","section":"Remark 4, Appendix B, Appendix C"}],"minor_comments":[{"comment":"The notation 'cøs' should be 'cos' (for example in Eqs. (14), (15), and (19)), and the symbol s_i(t) is used both for a vector and for a scalar in Section 4.1, which is confusing.","section":"Throughout"},{"comment":"The dimensions are inconsistent: A_i is described as a 2(p_i+1)×(p_i+1) matrix, but each a_i^k is a 2×1 vector, so A_i should be 2×(p_i+1); the dimensions of t_i and of the block-diagonal matrix in the following line should be reconciled.","section":"Section 3, Eq. (13)"},{"comment":"The definition of Φ_i(t,t_i) is not fully specified, and the displayed rows do not clearly match the polynomial coefficient ordering used in Eq. (12), making the transition matrix ambiguous.","section":"Section 3, Eq. (16)"},{"comment":"Definition 2 requires equality of outputs for all t ≥ t_i, whereas Definition 1 and the subsequent analysis use a finite interval [t_i,t_f]; this interval mismatch should be reconciled.","section":"Definition 2"},{"comment":"The 3D claim that either bearing or elevation angles should be distinct modulo π for observability is stated without derivation or a formal statement, so it is not verifiable from the present analysis.","section":"Remark 2"}],"recommendation":"reject","confidential_remarks":"The central claims of the paper are not supported by the derivations, and the flaws are structural rather than cosmetic: the bearing-only observability criterion ignores the block-diagonal structure of the measurement matrix, and the Doppler ambiguity condition is built on an invalid scalar-to-vector integration and an arbitrary unitary transformation. I see no path to a sound version within the current manuscript's scope. The paper also shows signs of rushed presentation, including undefined acronyms, inconsistent notation, and a contradiction between Proposition 2 and Remark 4."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the main result is not a result. Proposition 1's proof confuses per-target zero-output conditions with a rank condition on a matrix that compares bearings between targets. Because the measurement matrix is block diagonal, each target's pseudo-measurement depends only on that target's own state. So pairwise bearing distinctness cannot rule out nonzero initial states lying along each target's own line of sight. The condition fails already for M=1: it is vacuous there, so it would imply a single target is observable from bearings alone, contradicting standard results. That's a load-bearing flaw, not a gap to be patched.\n\nWhat is worth keeping: the paper raises a real and under-addressed question—what makes multiple targets distinguishable from a single observer. The bearing-distinctness heuristic is intuitive and is the kind of thing sensor placement people might think about. The trajectory ambiguity framing, where two trajectories produce identical measurement histories, is also a reasonable way to set up the problem.\n\nThe soft spots are not minor. Equation (19) is a polynomial identity per target; the paper never separates coefficients and never analyzes the full kernel. The step from (19) to the rank condition on P(t) is unsupported. The Doppler analysis has a scalar-vector error: integrating the scalar range-rate equation (23) cannot produce the vector relation (24). The transformation W(t) is built from an arbitrary unitary U(t), making condition (22) nearly tautological. The sufficiency conditions in Appendix B (W=I, same tonal) render the necessary condition vacuous. The notation is messy throughout, but that is secondary to the fact that the central claim is false as stated.\n\nWho is this for? Not for someone wanting to apply observability tests. The geometric heuristic might be worth a passing mention, but the paper does not provide a proof for it. I would not send this to external review; it is not close to a publishable argument.","headline":"The main observability theorem is false—per-target zero-output conditions don't couple through the bearing matrix—so the paper's central result doesn't survive scrutiny.","tokens_in":10072,"tokens_out":4011,"would_cite":false,"duration_ms":42500,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"One line-of-sight rule decides multi-target observability","keywords":["multi-target observability","bearing-only tracking","trajectory ambiguity","Doppler-only measurements","higher-order dynamics","passive target motion analysis","observability condition"],"falsifier":"Compute the observability Gramian for two targets that maintain distinct, constant bearings toward a stationary observer, for example two targets moving radially outward along different lines of sight. If the Gramian is singular, Proposition 1 is false, because the range and along-sight velocity of each target are unobservable even though the bearings are distinct modulo $\\pi$; equivalently, one can directly inspect equation (19) and seek a nonzero initial state whose polynomial coefficients all vanish without $P(t)$ losing rank.","tokens_in":9019,"feed_emoji":"🎯","tokens_out":6036,"duration_ms":67137,"temperature":0.7,"pith_summary":"This paper claims that a single observer can uniquely estimate the states of several moving targets only when no two targets ever lie on the same line of sight with the observer, meaning their bearing angles are distinct modulo $\\pi$ at every instant. The authors argue this condition is necessary and sufficient regardless of the order of target or observer dynamics, and they derive it through a pseudolinearized observability analysis. They then propose a complementary notion called trajectory ambiguity, in which two targets are indistinguishable if their measurement histories coincide, and they derive necessary and sufficient conditions for such ambiguity under Doppler-only, bearing-only, and combined Doppler-bearing measurements. If correct, the result would reduce a nonlinear, high-dimensional observability problem to a checkable geometric distinctness condition.","feed_headline":"One line-of-sight rule decides multi-target observability","feed_subtitle":"Distinct bearings modulo π are claimed necessary and sufficient for a single observer to track several targets.","key_machinery":"The load-bearing object is the pseudolinearized measurement matrix $C(t)$ together with the block-diagonal state transition matrix $\\tilde{\\Phi}(t, t_i)$ built from Taylor-polynomial blocks for each target's relative position. The observability argument passes through the matrix $P(t)$ with rows $[\\cos\\theta_i(t), -\\sin\\theta_i(t)]$; the authors assert that distinctness of the bearings modulo $\\pi$, i.e., rank 2 of $P(t)$ for all $t$, is necessary and sufficient for the zero-output condition to force a zero initial super-state. For Doppler ambiguity, the central object is the time-varying transformation matrix $W(t) = U(t)\\left[l' + \\frac{b' + c(1 - l')(t - t_i)}{s_j(t)}\\right]$, which maps one target's relative position vector into another's; the combined-measurement result states that ambiguity occurs iff $s_j(t)$ is an eigenvector of $W(t)$ with eigenvalue $\\alpha'(t)$.","core_discovery":"The central claim is Proposition 1: a multi-target system observed by a single passive observer is observable if and only if the bearing angles of the targets are distinct modulo $\\pi$ for all $t$ in the observation interval $[t_i, t_f]$, equivalently no target is ever collinear with the observer and another target. The derivation writes each target's relative position as a Taylor polynomial in $(t-t_i)$ and assembles a block-diagonal state transition matrix $\\tilde{\\Phi}(t, t_i)$; the zero-output observability condition then reduces, in the paper's argument, to a rank condition on the instantaneous matrix $P(t)$ whose rows are $[\\cos\\theta_i(t), -\\sin\\theta_i(t)]$. Full rank of $P(t)$ is equivalent to $\\theta_j(t) - \\theta_i(t) \\neq k\\pi$ for all integer $k$, i.e., all bearing angles distinct modulo $\\pi$. For trajectory ambiguity, the paper states that two targets have identical Doppler measurement histories exactly when $\\tilde{s}_i(t) - \\tilde{s}_j(t) = (W(t) - I)s_j(t)$ holds for all $t$, with a time-varying transformation matrix $W(t)$ constructed in the proof; with bearing-only measurements, ambiguity requires the analogous relation with a scalar $\\alpha'(t)$; and with combined Doppler and bearing measurements, ambiguity occurs precisely when $s_j(t)$ is an eigenvector of $W(t)$ corresponding to the eigenvalue $\\alpha'(t)$.","pith_inferences":["Editorial extension: the move from the polynomial zero-output equation (19) to the pointwise rank-2 condition on $P(t)$ skips a coefficient-separation step; because (19) is a polynomial in $(t - t_i)$, its coefficients involve all derivative states, so distinct bearings at each instant may not by themselves rule out every nonzero initial super-state.","Editorial extension: classical single-target bearing-only observability requires an observer maneuver (nonzero bearing rate), yet Proposition 1 as stated does not invoke any per-target maneuver condition; this suggests the multi-target claim silently assumes each target is individually observable, and a constant-bearing receding-target counterexample would test that.","Testable extension: running an observability Gramian numerical test over randomly generated polynomial trajectories with distinct bearings would reveal whether any nonzero initial super-state yields identically zero output, quantifying how often the geometric condition overstates observability."],"forward_implications":["A tracking algorithm could pre-check a scenario for observability simply by testing whether any two bearing-angle traces differ by an integer multiple of $\\pi$ at any time.","The maneuver requirement for a single observer would reduce to breaking collinearity among targets, independent of the order of target or observer dynamics.","Doppler-only and combined Doppler-bearing trackers could use the eigenvector condition on $W(t)$ to detect when two targets' measurement histories are exactly compatible and cannot be separated.","The NECNDSUF ambiguity conditions give a constructive test: if measured histories satisfy the derived relation, a tracker should declare the two targets indistinguishable and fuse them."],"supporting_citations":[{"why":"Supplies the linear observability definitions and the state-transition/measurement setup used in Section 2.","marker":"[2]"},{"why":"Provides the NECNDSUF single-target bearing-only observability condition that the multi-target result extends.","marker":"[3]"},{"why":"Establishes the Taylor-polynomial treatment of higher-order dynamics and the insufficiency of first-order conditions.","marker":"[5]"},{"why":"Frames observability with combined angle and frequency measurements, the setting for the trajectory-ambiguity analysis.","marker":"[9]"},{"why":"Gives the 3D bearing/elevation observability requirements referenced in Remark 2 for the multi-condition extension.","marker":"[19]"},{"why":"Provides the linear-theory approach used to avoid observability matrices and nonlinear equations in the derivations.","marker":"[20]"}],"fun_headline_variants":["Distinct bearings mod pi decide observability","No two targets collinear: single observer observability","Single passive observer, distinct bearings: tracking possible","Observability rule: bearings distinct modulo pi"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that if the bearing angles of the targets are distinct modulo $\\pi$ at every instant, then the only initial super-state producing identically zero pseudolinearized measurements is the zero vector; this equates a pointwise geometric condition on the measurement matrix with a condition on the whole trajectory over the interval.","fun_headline_variants_meta":{"raw":{"variants":["Distinct bearings mod pi decide observability","No two targets collinear: single observer observability","Single passive observer, distinct bearings: tracking possible","Observability rule: bearings distinct modulo pi"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001231,"raw_usage":{"total_tokens":5054,"prompt_tokens":941,"completion_tokens":4113,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":4055}},"tokens_in":557,"tokens_out":4113,"duration_ms":35961,"temperature":1.0,"reasoning_tokens":4055,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:49:13.895745+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the observability Gramian for two targets that maintain distinct, constant bearings toward a stationary observer, for example two targets moving radially outward along different lines of sight. If the Gramian is singular, Proposition 1 is false, because the range and along-sight velocity of each target are unobservable even though the bearings are distinct modulo $\\pi$; equivalently, one can directly inspect equation (19) and seek a nonzero initial state whose polynomial coefficients all vanish without $P(t)$ losing rank.","supporting_citations":[{"cited_title":"Observability in passive target mo- tion analysis","cited_arxiv_id":null,"evidence_quote":"Supplies the linear observability definitions and the state-transition/measurement setup used in Section 2."},{"cited_title":"Observability criteria for bearings- only target motion analysis","cited_arxiv_id":null,"evidence_quote":"Provides the NECNDSUF single-target bearing-only observability condition that the multi-target result extends."},{"cited_title":"Nth-order dynamics target observability from angle measurements","cited_arxiv_id":null,"evidence_quote":"Establishes the Taylor-polynomial treatment of higher-order dynamics and the insufficiency of first-order conditions."},{"cited_title":"A general approach to tma observability from angle and frequency measurements","cited_arxiv_id":null,"evidence_quote":"Frames observability with combined angle and frequency measurements, the setting for the trajectory-ambiguity analysis."},{"cited_title":"Observability requirements for three-dimensional tracking via angle measurements","cited_arxiv_id":null,"evidence_quote":"Gives the 3D bearing/elevation observability requirements referenced in Remark 2 for the multi-condition extension."},{"cited_title":"Simple linear theory approach to tma observability","cited_arxiv_id":null,"evidence_quote":"Provides the linear-theory approach used to avoid observability matrices and nonlinear equations in the derivations."}],"review_version":1}