{"id":"ea9a4562-e6b7-4d53-8739-28f2c206a32c","arxiv_id":"1908.02015","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two boundary flux measurements, at angular separation not a rational multiple of pi, uniquely recover (up to scale) a heat source f(x,t)=p(x)q(t) where q is a step function and p has fractional Sobolev regularity.","lead":"This paper proves that two boundary sensors can uniquely determine both the spatial and time-dependent parts of a heat source, up to a scale factor, under restrictions on the source shape and switching behavior. The result matters because it shows minimal sensing can identify a product-form source in a severely ill-posed inverse problem, with numerical reconstructions demonstrated at low noise.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.6's proof rests on an unjustified boundedness assertion: the limit of S1(s) as Re s→∞ does not imply S1 is bounded on C+, so the Liouville step fails; since Theorem 1 invokes Lemma 3.6 twice, the uniqueness proof has a genuine gap.","rationale":"The reader's formal weakest_assumption is the regularity p∈D((−Δ)^γ), which is a legitimate scope restriction and not an internal flaw. However, the reader's rationale already flags Lemma 3.6 as containing a 'genuine derivation gap', and that is exactly the most load-bearing concern in this paper. The central claim is plausible and the overall strategy is coherent; the gap is localized to the boundedness assertion for S1 on C+, which is used to apply Liouville's theorem. Because the lemma is invoked twice in the proof of Theorem 1, the uniqueness theorem is not fully rigorous as written. I do not see a reason to move the verdict from CONDITIONAL: the gap is serious but appears repairable by a Laplace-support or Dirichlet-series argument, and no counterexample to the theorem itself is apparent. The agreement is partial rather than full because the reader's headline 'weakest assumption' emphasizes the regularity condition rather than the Lemma 3.6 derivation gap, though the gap is acknowledged in the reader's rationale.","tokens_in":18707,"tokens_out":25328,"duration_ms":274622,"concrete_test":"Independently re-derive Lemma 3.6 without the disputed boundedness step: from the hypothesis lim_{Re s→∞}e^{εs}P_l(s)=0, prove or disprove that the Dirichlet series F_l(t)=Σ_n a_n(z_l)p_n e^{-λ_n t} vanishes on (0,ε). If F_l vanishes there, analyticity on (0,∞) gives F_l≡0 and the grouped coefficients vanish, completing the lemma. The check is whether this replacement proof requires only Assumption 2.1 or needs an extra condition such as absolute convergence of Σ a_n p_n λ_n; if the latter, the gap is substantial and the theorem's proof must be amended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Lemma 3.6 the authors define S1,l(s) = ∫_0^ε e^{(ε-t)s} [Σ_n a_n(z_l)p_n(t−λ_n^{-1}+λ_n^{-1}e^{-λ_n t})] dt and, after showing lim_{Re s→∞}(L1−S2)=0, assert 'This implies that S1,l(s) is bounded on C+.' This is not a logical consequence: S1,l is an entire function of exponential type ε, and on C+ it can grow like e^{ε Re s}; a limit along the positive real axis controls only one ray, not the half-plane. The separate bound given for Re s<0 does not cover Re s≥0. The subsequent conclusion that S1,l is a bounded entire function, hence constant and then zero by the limit, is therefore unjustified. This is load-bearing because Lemma 3.6 is used twice in the proof of Theorem 1: first to deduce p=0 from lim_{Re s→∞}e^{εs}q1P_l(s)=0 when c1≠c~1, and again to obtain q1p−q~1p~=0 after c1=c~1. If the lemma is not established, the uniqueness theorem is not proven as written. The lemma's statement may still be true—a direct proof might use the fact that P_l(σ)=O(e^{-εσ}) forces the Dirichlet series F_l(t)=Σ a_n(z_l)p_n e^{-λ_n t} to vanish on (0,ε), and analyticity then gives F_l≡0—but that is not the argument in the paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the inverse problem of recovering a separated source term f(x,t)=p(x)q(t) in the heat equation on the unit disc, with zero Dirichlet data and zero initial data, from normal-derivative traces recorded at two boundary points for all positive times. Under Assumption 2.1, which requires p to belong to D((-Δ)^γ) for some γ>0 and q to be a piecewise-constant L^1 function with a positive minimum gap between consecutive jump times, the main theorem (Theorem 1) claims that the pair (p,q) is determined uniquely up to the multiplicative ambiguity (p,q)↦(C p, C^{-1}q), provided the angular separation of the two measurement points is not a rational multiple of π. The proof combines a harmonic-function representation of the flux, a Laplace transform identity, and several auxiliary lemmas on Dirichlet series; a numerical section presents an alternating Tikhonov iteration with total-variation regularization for step-function sources and shows reconstructions for several test examples.","tokens_in":19060,"tokens_out":18860,"duration_ms":182376,"significance":"If Theorem 1 is established, the paper makes a meaningful contribution: it demonstrates that two time-resolved boundary flux measurements can determine a two-component product source up to scaling, extending the two-point result of Hettlich and Rundell to a genuinely time-dependent factor. The proof is largely self-contained, the regularity assumptions and the resulting limitations are stated candidly, and the numerical experiments illustrate the practical behavior of a reconstruction scheme. However, the central proof contains a specific gap in Lemma 3.6, which is invoked twice in the proof of Theorem 1. Because the gap is load-bearing for the uniqueness claim, the paper cannot be accepted in its present form; nevertheless, the statement of Lemma 3.6 is plausible and a repair appears feasible within the scope of the manuscript, so a major revision is the appropriate outcome.","major_comments":[{"comment":"The inference 'This implies that S_l^1(s) is bounded on C+' is not justified. The preceding limit, lim_{Re s→∞} S_l^1(s)=0, controls the behavior only along a ray (or at best uniformly in vertical strips if the limit is interpreted that way); it does not preclude growth such as e^{ε Re s} in other parts of the half-plane. The separately derived bound for Re s<0 does not cover Re s≥0, so the function has not been shown to be bounded on C. Consequently, the Liouville-theorem step that S_l^1 is constant, and then zero, is not established. This is a genuine gap because Lemma 3.6 is used twice in the proof of Theorem 1: once to conclude p=0 from lim_{Re s→∞} e^{εs} q_1 P_l(s)=0 when c_1≠c~_1, and again to obtain q_1 p - q~_1 p~ = 0 after c_1=c~_1. The lemma's statement may still be true, but a different argument is needed, for example showing directly from the assumption that the Dirichlet series F_l(t)=∑ a_n(z_l)p_n e^{-λ_n t} vanishes on (0,ε), or obtaining a genuine uniform bound for S_l^1 on C+ by a more refined estimate.","section":"Lemma 3.6, proof, after the decomposition of L1 into S_l^1 and S_l^2"},{"comment":"The step from 'the union of the sets of zeros of the two factors covers C+' to 'we can find an open connected nonempty subset C1⊂C+ such that P~_l(s)≡0 on C1' is terse. The intended argument is a Baire category argument: the zero set of the finite exponential sum has no accumulation points and is therefore nowhere dense, so if the union of the zero sets covers C+, the zero set of P~_l must have nonempty interior. This requires the additional observation that P~_l is not identically zero, which follows from Lemma 3.6 and Assumption 2.1. Please spell out this argument; as written, the conclusion does not follow from the preceding sentence alone.","section":"Proof of Theorem 1, final case K < K~"}],"minor_comments":[{"comment":"The claim that the eigenfunctions form a complete basis for L^2(Ω) and that their restrictions to ∂Ω also form a complete set is inaccurate as written: Dirichlet eigenfunctions vanish on the boundary. If the intended object is instead the normal derivatives, or the traces of the harmonic functions ξ_j, the statement should be corrected and a proof or reference supplied.","section":"Remark 3.1"},{"comment":"The notation lim_{Re s→∞} is ambiguous. If the limit is taken only along the real axis, the boundedness inference in Lemma 3.6 is even more clearly invalid; if it is intended uniformly for all s with Re s→∞, that should be stated explicitly, since the subsequent arguments rely on the meaning.","section":"Lemma 3.6 and Theorem 1"},{"comment":"When applying Lemma 3.5 to the equation ∑ a_n p_n (1−e^{-λ_n t})=0 on (0,ε), the text says 'the conditions of Lemma 3.5 are satisfied' but does not explicitly note that the constant term ∑ a_n p_n must be included as a zero-exponent term in the Dirichlet series. This is a minor omission; please clarify.","section":"Proof of Lemma 3.6, application of Lemma 3.5"},{"comment":"The condition 'k(θ1−θ2) ≠ jπ for any integers j,k' should specify k≠0; when k=0 the inequality is false for j=0. The intended condition is that m(θ1−θ2) is not an integer multiple of π for every nonzero integer m.","section":"Section 4.2, paragraph on measurement points"},{"comment":"There are several typographical and grammatical issues: in the Introduction, 'we have unable to allow' should be 'we have been unable to allow'; in Section 4.1, 'saves the edge-preserving property' should be 'has the edge-preserving property'; in Lemma 3.2, 'Cauchy-Schwartz' should be 'Cauchy-Schwarz'.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The sole serious obstruction is the gap in Lemma 3.6. If the authors can replace the invalid boundedness inference with a correct argument, the central theorem would be established and the paper would likely be acceptable. The rest of the proof is careful, and the numerical section, while limited, is not a barrier. I would not reject the paper on the basis of this gap, because the lemma's statement appears believable and a repair is plausibly within scope, but the current proof is incomplete."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe main result here is a real extension: two boundary flux traces on the unit disk uniquely determine a product source p(x)q(t) up to reciprocal scaling, provided p is in D((-Δ)^γ) and q is piecewise constant with a minimum gap. That goes beyond Hettlich-Rundell, which only handled q=1. The machinery is mostly clean: the harmonic-function representation, the absolute convergence lemma, and the analytic continuation of the Laplace transform are all carefully done. The numerical section is honest about being heuristic, and the authors flag that their assumptions exclude characteristic functions, though they approximate them.\n\nThe problem is Lemma 3.6, and it is load-bearing. The proof asserts that because lim_{Re s→∞} S1(s)=0, S1 is bounded on C+. That inference is not valid. S1 is entire of exponential type ε; on the right half-plane it can grow like e^{ε Re s}. A limit along one ray (the positive real axis) does not bound the half-plane. The separate bound for Re s<0 covers only the left half-plane. So the Liouville step fails. Since Theorem 1 uses Lemma 3.6 twice—once to conclude p_n=0 and again after matching c1 to get q1 p_n - \\tilde q1 \\tilde p_n =0—the uniqueness theorem is not rigorously proven as written.\n\nI don't think the paper is sunk. The lemma statement is probably true, and the stress-test note sketches a direct proof using the Dirichlet series F_l(t)=Σ a_n(z_l)p_n e^{-λ_n t}: the decay of P_l(σ) forces F_l to vanish on (0,ε), and analyticity gives F_l≡0. That would avoid the flawed boundedness claim. So the gap is real but repairable.\n\nWho is this for? Inverse problems people who care about identifiability from minimal boundary data. It deserves peer review, not a desk reject. Send it out, and ask for a corrected proof of Lemma 3.6. With that in place it's a solid, if niche, contribution.\n\nBest,","headline":"The two-point sparse-data uniqueness result is plausible and the paper mostly clean, but Lemma 3.6 contains a genuine, load-bearing boundedness error that needs fixing before the proof is complete.","tokens_in":19564,"tokens_out":4365,"would_cite":false,"duration_ms":43305,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","65M32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two boundary flux measurements can determine both factors of a heat source term.","keywords":["inverse source problem","heat equation","sparse boundary measurements","uniqueness","Laplace transform","source identification","regularization","time-dependent source"],"falsifier":"Look for a nonzero $p\\in D((-\\Delta)^\\gamma)$ whose boundary flux at two sensors separated by an irrational multiple of $\\pi$ is identically zero for all $t>0$; if such a $p$ exists, then $(p,q_1)$ and $(p,q_2)$ give identical data for any two distinct admissible step functions $q_1,q_2$, contradicting Theorem 1. The calculation reduces to checking whether the infinite matrix $\\{a_n(z_\\ell)\\}_{\\ell=1,2,\\,n\\ge1}$ has a nontrivial null vector, and the paper's Lemma 3.6 asserts it does not under the angular condition.","tokens_in":18518,"feed_emoji":"🔥","tokens_out":16081,"duration_ms":148069,"temperature":0.7,"pith_summary":"The paper asks whether a heat source that separates as $f(x,t)=p(x)q(t)$ can be recovered from flux data measured over time at just two points on the boundary. It proves a uniqueness theorem: if the two sensors are placed so that their angular separation is not a rational multiple of $\\pi$, and if $p$ has mild extra regularity while $q$ is a step function with separated jump times, then equal two-point flux records force $p=C_0\\tilde p$ in $L^2(\\Omega)$ and $q=C_0^{-1}\\tilde q$ on $[0,\\infty)$ for a nonzero constant $C_0$. This matters because real monitoring problems often have very few sensors at a distance from a source whose spatial shape and time-varying strength are both unknown, and the result says the two factors can still be disentangled in principle despite the severe ill-posedness. The paper also supplies an alternating iterative algorithm and numerical tests with noisy data that recover both factors.","feed_headline":"Two flux readings can pin down a heat source's shape and timing","feed_subtitle":"Both the spatial shape and the time schedule of a heat source become unique, up to one scale factor.","key_machinery":"The carrying identity is the Laplace-domain representation (12),\n$$$s^{2}$\\mathcal{L}\\left(-\\int_0^t \\frac{\\partial u}{\\partial n}(z,\\tau)\\,d\\tau\\right)(s)=\\left(\\sum_{k=1}^K q_k $e^{{-c_k s}}$\\right)\\left(\\sum_{n=1}^\\infty a_n(z)p_n\\frac{\\lambda_n}{s+\\lambda_n}\\right),$$\nwhere $a_n(z)$ is the boundary value of the $n$-th eigenfunction at $z$ (in the disc, an angular trigonometric factor times a Bessel normalization), $p_n$ the Fourier coefficient of $p$, and $\\{c_k,q_k\\}$ the jumps and amplitudes of $q$. The first factor is a Dirichlet-type series encoding $q$; the second encodes $p$. The proof compares two such products at two sensors, using the angular separation condition to keep the $2\\times2$ coefficient matrix nonsingular (Lemma 3.4), using analyticity and a zero-accumulation lemma for absolutely convergent series of exponentials to peel off the factors of $q$ one by one, and using an entire-function argument (Lemma 3.6) to show the only way a certain limit at infinity vanishes is that $p$ itself vanishes. The numerical scheme mirrors this split: alternating Tikhonov updates for $p$ and total-variation-regularized updates for the step function $q$.","core_discovery":"Theorem 1 is the core discovery: under Assumption 2.1, two boundary flux observations uniquely determine $(p,q)$ up to multiplication, provided $\\theta_1-\\theta_2\\notin\\pi\\mathbb{Q}$. Precisely, if two admissible pairs produce the same flux traces $\\partial u/\\partial n(z_\\ell,\\cdot)$ for $\\ell=1,2$ on $t>0$, then there is a nonzero constant $C_0$ with $p=C_0\\tilde p$ in $L^2(\\Omega)$ and $q=C_0^{-1}\\tilde q$ on $[0,\\infty)$. The admissible class requires $p\\in D((-\\Delta)^\\gamma)$ for some $\\gamma>0$ and $q$ a finite or infinite linear combination of Heaviside steps whose jump times are separated by a fixed positive gap. The scaling ambiguity is intrinsic to the product structure $f=pq$ and cannot be removed from any data. The paper notes that the regularity on $p$ excludes characteristic functions of subdomains, the case treated in earlier work on discontinuous sources, but such sources can be approximated arbitrarily closely.","pith_inferences":["An implication the paper leaves implicit is that the irrational-separation condition is not a practical obstacle: for the finitely many eigenmodes that matter numerically, one can avoid rational separations with small denominators by choosing sensors from the degree-scale angular gaps the paper maps out.","The Laplace-transform structure suggests a testable extension to fractional diffusion, where $e^{-\\lambda_n t}$ is replaced by a Mittag-Leffler function; the same analyticity and zero-accumulation arguments should yield an analogous uniqueness theorem, with different short-time decay altering numerical conditioning.","The intrinsic scale ambiguity means any practical inversion must fix a normalization such as $\\|p\\|_{L^2}=1$; without that choice the data-to-solution map is locally flat along the one-parameter scaling family even though the quotient problem is unique.","If the angular condition fails, the coefficient matrix in the proof becomes singular on infinitely many eigenmodes, so the condition is likely necessary as well as sufficient; constructing a nonzero $p$ whose two-point flux vanishes at a rational separation would demonstrate this sharply."],"forward_implications":["Two pointwise flux sensors, placed so that their angular separation is not in $\\pi\\mathbb{Q}$, are enough in principle to fix both $p$ and $q$ up to the intrinsic scaling; dense boundary data is not needed for uniqueness.","The uniqueness is global within the admissible class, not merely local near a known source, extending the earlier two-sensor result for a known uniform source to unknown time dependence.","The regularity assumption $p\\in D((-\\Delta)^\\gamma)$ excludes discontinuous characteristic-function sources, but such sources can be approximated arbitrarily well, and the paper's numerical experiments treat them successfully with a different reconstruction scheme.","The proof carries over to smooth bounded domains in $\\mathbb{R}^2$ and self-adjoint elliptic operators $L=-\\nabla\\cdot(a\\nabla u)+q_0u$, provided the measurement points avoid zeros of boundary traces of eigenfunctions, as stated in Remark 3.1.","The alternating algorithm with Tikhonov and total-variation regularization reconstructs both factors from noisy flux data at 1% to 5% noise in the reported experiments."],"supporting_citations":[{"why":"Establishes the two-measurement setting for a discontinuous source $p=\\chi(D)$ and provides the harmonic-basis technique and local-injectivity baseline this paper generalizes.","marker":"[6]"},{"why":"Proof of the no-common-positive-zeros result for integer-order Bessel functions, used to justify the eigenvalue multiplicity structure of the disc's Dirichlet Laplacian.","marker":"[12]"},{"why":"Trace theorem used in Lemma 2.2 to pass from interior Sobolev regularity of $u$ to boundary regularity of the flux.","marker":"[8]"},{"why":"Fractional Sobolev embedding used in Lemma 2.2 to conclude the boundary flux is Hölder continuous on $\\partial\\Omega$.","marker":"[3]"},{"why":"Earlier inverse-source result for the heat equation with a spatially compact source, supplying a predecessor for the separated-source model.","marker":"[2]"},{"why":"Fractional-diffusion inverse-source paper whose computational approach is used for the discontinuous-support experiments.","marker":"[11]"},{"why":"Reference for total-variation regularization used in the $q$-update of the iterative reconstruction scheme.","marker":"[9]"}],"fun_headline_variants":["Two probes, one scaling ambiguity: heat source identified","Sparse-flux identifiability: heat source from two boundary points","Two-point flux data: heat source p(x)q(t) up to a constant","Unique heat source from two lateral sensors, up to scaling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that $p$ lies in $D((-\\Delta)^\\gamma)$ for some $\\gamma>0$, which makes the spectral series $\\sum_n a_n(z)p_n$ converge absolutely and underpins the Laplace-transform representation; the paper notes this excludes characteristic functions of subdomains, and without it the coefficient-peeling argument is not justified.","fun_headline_variants_meta":{"raw":{"variants":["Two probes, one scaling ambiguity: heat source identified","Sparse-flux identifiability: heat source from two boundary points","Two-point flux data: heat source p(x)q(t) up to a constant","Unique heat source from two lateral sensors, up to scaling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000703,"raw_usage":{"total_tokens":3161,"prompt_tokens":922,"completion_tokens":2239,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":2164}},"tokens_in":538,"tokens_out":2239,"duration_ms":16703,"temperature":1.0,"reasoning_tokens":2164,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:58:25.571984+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a nonzero $p\\in D((-\\Delta)^\\gamma)$ whose boundary flux at two sensors separated by an irrational multiple of $\\pi$ is identically zero for all $t>0$; if such a $p$ exists, then $(p,q_1)$ and $(p,q_2)$ give identical data for any two distinct admissible step functions $q_1,q_2$, contradicting Theorem 1. The calculation reduces to checking whether the infinite matrix $\\{a_n(z_\\ell)\\}_{\\ell=1,2,\\,n\\ge1}$ has a nontrivial null vector, and the paper's Lemma 3.6 asserts it does not under the angular condition.","supporting_citations":[{"cited_title":"Hettlich and W","cited_arxiv_id":null,"evidence_quote":"Establishes the two-measurement setting for a discontinuous source $p=\\chi(D)$ and provides the harmonic-basis technique and local-injectivity baseline this paper generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proof of the no-common-positive-zeros result for integer-order Bessel functions, used to justify the eigenvalue multiplicity structure of the disc's Dirichlet Laplacian."},{"cited_title":"Lions and E","cited_arxiv_id":null,"evidence_quote":"Trace theorem used in Lemma 2.2 to pass from interior Sobolev regularity of $u$ to boundary regularity of the flux."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier inverse-source result for the heat equation with a spatially compact source, supplying a predecessor for the separated-source model."},{"cited_title":"Rundell and Z","cited_arxiv_id":null,"evidence_quote":"Fractional-diffusion inverse-source paper whose computational approach is used for the discontinuous-support experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reference for total-variation regularization used in the $q$-update of the iterative reconstruction scheme."}],"review_version":1}