{"id":"8631d84a-ad2e-491b-b56e-e4fb8b8ce9e8","arxiv_id":"2507.10012","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For piecewise constant wave speeds built from separated balls, one passive boundary measurement determines both the wave speed and the initial source in 3D, with Hölder stability for a single slower inclusion.","lead":"Mathematicians prove that for the 3D wave equation with a piecewise constant sound speed built from separated ball-shaped inclusions, a single record of the wave field on the boundary over all time uniquely determines both the sound speed inside the inclusions and the unknown initial source that generated the wave. They also prove Hölder-type stability for one inclusion, a step toward photoacoustic and thermoacoustic tomography where the speed of sound is usually not known.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorems 2.2 and 2.3 assumes all inclusions are slower than background (or equal contrast); for admissible faster inclusions the signed point-source matching breaks, so the advertised general piecewise-constant uniqueness is not established.","rationale":"The reader's strongest claim and weakest_assumption field do not name the faster-inclusion problem directly, but their rationale explicitly flags Section 5's factor cancellation and Section 6's assumption that all inclusion speeds are below the background. I agree with the CONDITIONAL verdict. Theorem 2.1 appears sound: the equal-radii assumption makes all point-source weights proportional to the same ball volume, so the cancellation in (4.2) and the permutation argument are legitimate. Theorem 2.4 is restricted to a single slower inclusion, where the cited observability inequality applies, and the apparent equality in (7.6) is actually an inequality in the needed direction, so that proof is repairable. The most load-bearing unresolved issue is the generality claimed for piecewise-constant speeds: fast inclusions are explicitly permitted by admissibility, and the proofs of Theorems 2.2 and 2.3 rely on sign or common-factor structure that fails for them. Since these theorems are central to the abstract's promise of recovering general piecewise-constant sound speeds with unknown inclusion locations and amplitudes, the manuscript's advertised scope is not established as written. No evidence was found of circularity, invented entities, or an inconsistent analytic-continuation framework; the concern is a localizable proof gap. The reader's CONDITIONAL verdict remains the right one, and our check would settle whether the gap is merely missing assumptions or an actual counterexample to the stated theorems.","tokens_in":26574,"tokens_out":29039,"duration_ms":319473,"concrete_test":"Re-derive the matching step of Section 6 without the sign assumption: keep r,s fixed as in (2.12), and take two admissible configurations with N=1, b1_1 < b0 < b2_1, choosing speeds so that |(b1_1)^{-2}-b0^{-2}||B_r| = |(b2_1)^{-2}-b0^{-2}||B_s|. For the transposition solutions of (6.2), the signed point-source multiset of configuration 1 is {+A at x1, -A|B_s|/|B_r| at y1} while configuration 2 is {-A at x2, +A|B_s|/|B_r| at y2}; the signs in (6.3) are therefore reversed. Check whether Theorem 8.1's permutation sigma can map x1 to y2 and y1 to x2, so that the conclusion c1 = c2 no longer follows from (6.2). If it can, Theorem 2.3 as stated is unproved; if it cannot, identify the additional argument that enforces the x/y partition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The advertised scope—uniqueness for general classes of piecewise constant sound speeds with unknown inclusion locations and amplitudes—rests on Theorems 2.2 and 2.3, and the proofs of both contain an unstated sign or common-factor assumption. In Theorem 2.3, after defining lambda_j_k = ((b_j_k)^{-2}-b0^{-2})|B_r| and lambda_j_{N_j+ℓ} = -((b_j_k)^{-2}-b0^{-2})|B_s|, the text uses (6.3) to assert lambda_j_k > 0 and lambda_j_{N_j+ℓ} < 0. This requires every inclusion speed to satisfy b_j_k < b0. But Definition 2.1(2)/inequality (2.1) only require c < sqrt(2) b0, so faster inclusions (b0 < b_j_k < sqrt(2) b0) are admissible; for them the signs flip. The sign separation is then used to force the permutation sigma from Theorem 8.1 to map outer-ball centers to outer-ball centers and hole centers to hole centers. Without it, sigma may pair an outer center of one configuration with a hole center of the other, and the equalities c1 = c2 and omega1 = omega2 do not follow. Theorem 2.2 has the analogous defect: equation (5.1) cancels the common factor ((b1)^{-2}-b0^{-2}) from both sides, an operation only legitimate if the inclusion amplitude b1 is known a priori and identical for c1 and c2; the theorem statement does not impose or state this. These are concrete gaps in the uniqueness proofs for the general classes advertised in the abstract. Theorem 2.1 (equal radii) and Theorem 2.4 (single slower inclusion) are not affected.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the inverse problem of simultaneously recovering the wave speed c and the initial data f in the wave equation on R^3 from a single passive boundary measurement of u on R_+ × ∂Ω. The main results are: Theorem 2.1, uniqueness for piecewise constant speeds with equal-radius disjoint ball inclusions and unknown centers; Corollary 2.1, a variant with a partial order on centers and unequal radii; Theorems 2.2 and 2.3, uniqueness for more general inclusions formed by unions of balls with holes; and Theorem 2.4, Hölder stability for a single inclusion with slower speed. The proofs combine an analytic continuation of the cutoff resolvent near frequency zero, an expansion of the Laplace transform of the solution, and a point-source identification theorem for elliptic equations. The paper is a substantial attempt to extend simultaneous recovery results beyond the monotonicity and decay assumptions of prior work.","tokens_in":26859,"tokens_out":23057,"duration_ms":255890,"significance":"If the main results are correct, the paper makes a significant contribution: it shows that a single passive boundary trace can determine both a piecewise constant sound speed and the initial source for natural coefficient classes, without assuming local energy decay. The Section 3 analytic continuation mechanism is a genuine novelty; the algebra in (3.4) checks out, and the reduction to point-source identification is elegant. The equal-radius and single-slower-inclusion results, Theorem 2.1 and Theorem 2.4, appear internally consistent and are not affected by the concerns below. However, the advertised generality for unions of balls with holes depends on Theorem 2.3, and the proof of that theorem contains load-bearing gaps that must be repaired.","major_comments":[{"comment":"The proof of Theorem 2.3 asserts that λ_j^k = ((b_j^k)^{-2} − b_0^{-2})|B_r| > 0 and λ_j^{N_j+ℓ} = −((b_j^k)^{-2} − b_0^{-2})|B_s| < 0 for every inclusion. This requires (b_j^k)^{-2} − b_0^{-2} > 0, i.e. b_j^k < b_0. But Definition 2.1(2) and the equivalent bound (2.1) only require b_j^k < √2 b_0, so faster inclusions with b_0 < b_j^k < √2 b_0 are admissible. For those inclusions both displayed signs reverse, and for a mixture of slower and faster inclusions the outer and hole point-source coefficients are not separated by sign. The subsequent deduction that the permutation σ from Theorem 8.1 maps outer centers to outer centers and holes to holes is therefore unjustified, and the equalities c1 = c2 and f1 = f2 do not follow for the generality stated in Theorem 2.3. The authors should either add a hypothesis ruling out mixed contrasts (for example, all inclusions slower than the background) or replace the sign-separation step with a different argument.","section":"6, Eq. (6.3)"},{"comment":"Even when all contrasts have one common sign, the step 'Combining this with (6.3), we deduce that σ1 = σ2' is not justified by the displayed equalities. The equalities in (6.2) identify the multiset of signed point sources, but they do not by themselves fix which hole center y_k is associated with which outer center x_k in each configuration. An argument using geometric information, such as disjointness of the outer balls B(x_j^k, r) together with B(y_j^k, s) ⊂ B(x_j^k, r), or distinctness of the contrasts, is needed; neither assumption is stated in Theorem 2.3. As written, two configurations that differ only by permuting the association (x_k, y_k) while preserving the same signed point-source multiset are not distinguished by the proof. This is a second independent gap in the proof of Theorem 2.3.","section":"6, after Eq. (6.3)"}],"minor_comments":[{"comment":"There are apparent typos in the definition of M and m: 'b2_2' should likely be 'b2_1', and the expression min(b1_1, b2_2, b0, r1) uses r1 without explanation.","section":"2.2, Theorem 2.4"},{"comment":"The factors '2|B_r|' and '2|B_s|' in (6.3) do not match the definitions of λ_j^k and λ_j^{N_j+ℓ} given just before (6.2); the displayed formula should be corrected unless the factor 2 is intentional.","section":"6, Eq. (6.3)"},{"comment":"The statement should make explicit that the contrast b1 is the same for c1 and c2. With that reading, the cancellation in (5.1) is legitimate; if the two contrasts were allowed to differ, the cancellation would be invalid.","section":"5, Theorem 2.2"},{"comment":"In the definition of O_r, the reference to B_R0 is confusing because the domain in the theorem is an arbitrary C^2 domain O; the notation should be cleaned up.","section":"8.2, Theorem 8.1"},{"comment":"The notation ∂^{2k}_p ∂^ℓ_ν \\hat u_j(p, ·)|_{p=0} is used before the analytic extension is introduced in the main text; a forward reference to Lemma 3.1 would improve readability.","section":"2.2, Eqs. (2.13)-(2.14)"}],"recommendation":"major_revision","confidential_remarks":"The advertised 'general classes of piecewise constant sound speeds' in the abstract rests on Theorems 2.2 and 2.3. Theorem 2.3 needs repair: at minimum, the sign-separation step requires either an all-slower (or all-faster) hypothesis, and the pairing of outer and hole centers needs a geometric or distinctness argument. If the theorem cannot be repaired without additional assumptions, the abstract and introduction should be revised to describe the narrower classes that are actually proved. The Section 3 machinery, Theorem 2.1, and Theorem 2.4 are solid enough to support a revised paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper has a real new idea and a real hole. The new idea is the low-frequency analytic continuation of the cutoff resolvent under the contrast bound ||1 - c^2/b0^2|| < 1. That gets them around the local energy decay assumption that had blocked piecewise constant media in previous work. The machinery in Sections 3 and the equal-radius uniqueness theorem 2.1 look right to me; the reduction to a point-source identification problem is clean, and Theorem 8.1 in the appendix is a useful standalone result.\n\nThe hole is in Theorem 2.3. In Section 6, they define signed weights for the outer balls and the holes and assert in (6.3) that the outer weights are positive and the hole weights are negative. That sign separation is what forces the permutation from the point-source identification to match outer centers to outer centers. But it only holds if every inclusion speed is below b0. The admissibility condition (2.1) only forces c < sqrt(2) b0, so faster inclusions are admissible and flip the signs. The theorem as stated is therefore broader than the proof supports. This is a specific gap, not a vague worry.\n\nI don't agree with the stress-test note about Theorem 2.2. Both media there are assumed to have form (2.7) with the same b1, so cancelling the common factor in (5.1) is legitimate. The theorem is weaker than the abstract's 'general classes' phrasing suggests, because the inclusion amplitude is common, but it's not a proof error.\n\nBottom line: the equal-radius uniqueness result and the stability theorem for one slower inclusion are solid contributions, and the frequency-domain method is worth knowing. Theorem 2.3 needs either a corrected proof or an explicit restriction to slower inclusions. Send it to a serious referee; it deserves the time, with instructions to scrutinize Section 6.","headline":"The low-frequency continuation idea is real and the equal-radius theorem looks right; but Theorem 2.3's proof assumes all inclusions are slower than background, which the theorem doesn't state, so the advertised generality for inclusions with holes isn't established.","tokens_in":27459,"tokens_out":5797,"would_cite":true,"duration_ms":59703,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35L05"],"pacs":[],"model":"deepseek-v4-flash","headline":"From a single passive boundary recording, the paper proves that a piecewise constant wave speed and the initial source can be uniquely recovered in 3-D, with Hölder stability in the one-inclusion case.","keywords":["inverse boundary problems","passive measurement","wave equation","photoacoustic tomography","simultaneous recovery","uniqueness","Hölder stability"],"falsifier":"Run a numerical experiment with two admissible configurations: same background speed, one ball inclusion of radius $r$ and speed $b$ at center $x$, another with radius $r$ and speed $b'$ at center $x' \\neq x$, and choose $H^1$ sources $f$, $f'$ satisfying $\\int f/c^2\\, dx = \\int f'/c'^2\\, dx \\neq 0$; if a search finds such a pair whose boundary traces match to solver precision, Theorem 2.1 would be contradicted. A more targeted check of the weakest assumption is to test whether the analytic continuation used in Theorem 3.1 still exists when $\\operatorname{ess\\,sup} c = \\sqrt{2}\\, b_0$ exactly, since the invertibility proof needs a strict inequality.","tokens_in":26283,"feed_emoji":"📡","tokens_out":8599,"duration_ms":93273,"temperature":0.7,"pith_summary":"The paper tackles the inverse problem of recovering two unknowns at once—the sound speed of a medium and the initial disturbance (source) that generates a wave—from a single recording of the wave on the boundary of a region in three dimensions. Its main uniqueness theorem says that if two admissible configurations, each consisting of a piecewise constant speed with disjoint ball inclusions of equal radius and an $H^1$ initial datum, produce exactly the same boundary trace, then the two speeds and the two sources are identical. A companion theorem for a single slower inclusion proves Hölder stability: small differences in the boundary measurement force small differences in the speed and source, in explicit powers. The authors emphasize that, unlike earlier work, no time-decay of the solution is required, so sources need not decay and piecewise constant speeds are allowed when they stay below an explicit multiple of the background speed. If correct, the results show that passive boundary listening alone is information-theoretically sufficient for simultaneous medium and source recovery in photoacoustic and thermoacoustic tomography.","feed_headline":"One boundary trace recovers wave speed and source","feed_subtitle":"3-D uniqueness and Hölder stability for piecewise constant sound speeds, with no time-decay requirement.","key_machinery":"The load-bearing object is the analytic extension of the Laplace transform in time, $p \\mapsto \\hat{u}(p,\\cdot)$, from the right half-plane to a fixed disk around $p = 0$. This is built from the cutoff resolvent $\\chi_R(L + p^2)^{-1}\\chi_{R_0}$ of the elliptic operator $L = -c^2\\Delta$; under the contrast bound the perturbation series in powers of $(c^2/b_0^2 - 1)$ converges and the resolvent is boundedly invertible near zero. Coefficients $u^{(k)}$ of the resulting Taylor series satisfy a ladder of elliptic equations, with $u^{(1)}$ determined by $c^{-2}f$ and $u^{(2)}$ equal to the constant $-(2\\pi b_0)^{-1}\\int f/c^2\\, dx$. Plugging harmonic test functions into the equation for $u^{(4)}$ and using the mean value theorem turns the boundary equality into equality of normal derivatives of transposition-solutions of an elliptic point-source problem; the Appendix's recovery theorem then matches the point sources one-to-one, giving $c_1 = c_2$, and the earlier wave-speed determination theorem gives $f_1 = f_2$. For stability, an observability inequality for wave equations with piecewise constant coefficients transfers control of the boundary measurement back to the initial data and to the low-frequency boundary coefficients.","core_discovery":"On the authors' own terms, the discovery is that the formally determined inverse problem of recovering $(c,f)$ from $u|_{\\mathbb{R}_+ \\times \\partial\\Omega}$ is solvable for piecewise constant $c$ with unknown ball inclusions. Theorem 2.1 proves that for admissible pairs whose inclusions are disjoint balls of the same radius, $u_1 = u_2$ on $\\mathbb{R}_+ \\times \\partial\\Omega$ implies $c_1 = c_2$ and $f_1 = f_2$; Corollary 2.1 extends this to unequal radii when the ball centres satisfy a partial order, Theorem 2.2 to inclusions that are unions of balls with holes, and Theorem 2.3 to spherical shells. Theorem 2.4 converts uniqueness into quantitative stability for one inclusion with $b_1 < b_0$: the $L^q$ difference of the wave speeds is bounded by the boundary-data difference to the first power, and the $H^1$ difference of the initial data by a boundary-measurement term plus a power $(3+2s)/6$ term. Admissibility means the non-degeneracy condition $\\int_{\\mathbb{R}^3} f/c^2\\, dx \\neq 0$ and the contrast bound $\\|1 - c^2/b_0^2\\|_{L^\\infty} < 1$, equivalently $\\operatorname{ess\\,sup} c < \\sqrt{2}\\, b_0$. The whole construction works without any decay-in-time assumption on the solution, which is the main relaxation relative to prior simultaneous-recovery results.","pith_inferences":["The contrast ceiling $\\operatorname{ess\\,sup} c < \\sqrt{2}\\, b_0$ is visibly the price of avoiding decay: if an inclusion is faster than this, the Neumann-series argument for the resolvent at $p = 0$ diverges, and a natural testable question is whether uniqueness actually fails there or only this proof does.","The equal-radius or partial-order matching probably transfers to any configuration where the point-source recovery can separate inclusions—for example, by distinct amplitudes or by additional harmonic test functions—so the same low-frequency machinery may cover polyhedral or smoother inclusions with a similar combinatorial condition.","Because the stability estimate bounds $c$ in $L^q$ and $f$ in $H^1$ separately for one inclusion, a numerical study comparing convergence rates of the $(3+2s)/6$ exponent against actual reconstruction errors would reveal whether the estimate is sharp or merely sufficient.","The same analytic-extension device may extend to other passive-measurement problems, such as elastic or electromagnetic wave equations, whenever the contrast operator is bounded by less than one in the relevant norm."],"forward_implications":["A single boundary time trace, measured passively over all $t > 0$, determines both the piecewise constant wave speed and the initial source for the ball-inclusion classes covered by Theorems 2.1–2.3.","No local-energy decay is needed, so the result applies to sources that do not die out over time, as long as $\\int f/c^2\\, dx \\neq 0$ and the contrast bound is met.","In the single-inclusion case with slower speed, the inverse map is Hölder continuous: boundary-data noise of size $\\varepsilon$ leads to error at most $C\\varepsilon^{1/q}$ in the $L^q$ speed difference and $C\\varepsilon^{(3+2s)/6}$ in the $H^1$ source difference, up to an additional boundary-measurement term.","The stability estimate gives a rigorous starting point for regularized iterative reconstruction algorithms such as Tikhonov-type schemes, which the paper states are deferred to future work.","The geometric conditions (equal radii or partial order on centers) are, in the authors' view, not merely technical but reflect an obstruction: without a way to label inclusions one-to-one, the point-source matching step can fail."],"supporting_citations":[{"why":"Supplies the prior wave-speed-and-source determination framework and the theorem used to conclude $f_1 = f_2$ once $c_1 = c_2$ is known.","marker":"[18]"},{"why":"Provides the observability inequality for piecewise constant coefficients that carries the stability proof of Theorem 2.4.","marker":"[19]"},{"why":"Established an earlier simultaneous-recovery result whose harmonicity and smallness conditions this paper relaxes.","marker":"[20]"},{"why":"Gives the limiting absorption and integral representation used in Lemma 3.1 to identify the low-frequency coefficients.","marker":"[26]"},{"why":"Provides the self-adjoint resolvent framework (Proposition A.1) and the background on local-energy decay for piecewise constant speeds.","marker":"[30]"},{"why":"Justifies the Laplace transform in time and its boundary-data properties used throughout.","marker":"[14]"},{"why":"Underlies the analyticity of the constant-speed cutoff resolvent via the Huygens principle.","marker":"[29]"},{"why":"Supplies the transposition-solution method for identifying point sources in elliptic problems, extended in the Appendix to divergence-form operators.","marker":"[10]"}],"fun_headline_variants":["One passive measurement recovers wave speed and source in 3D","Single boundary trace yields wave speed and initial data uniquely","No time decay: boundary measurement fixes wave speed and source","Piecewise constant speeds: uniqueness from one boundary trace","Hölder stability from single passive boundary measurement"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument breaks if the unknown inclusion is faster than roughly 1.414 times the known background speed, because the contrast bound that makes the resolvent invertible at frequency zero would fail; a second load-bearing assumption is that the inclusions can be matched one-to-one by equal radii or a partial order on their centers, which the authors themselves call a fundamental obstruction.","fun_headline_variants_meta":{"raw":{"variants":["One passive measurement recovers wave speed and source in 3D","Single boundary trace yields wave speed and initial data uniquely","No time decay: boundary measurement fixes wave speed and source","Piecewise constant speeds: uniqueness from one boundary trace","Hölder stability from single passive boundary measurement"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1714,"prompt_tokens":1043,"completion_tokens":671,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":659,"completion_tokens_details":{"reasoning_tokens":592}},"tokens_in":659,"tokens_out":671,"duration_ms":7482,"temperature":1.0,"reasoning_tokens":592,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:46:09.441838+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a numerical experiment with two admissible configurations: same background speed, one ball inclusion of radius $r$ and speed $b$ at center $x$, another with radius $r$ and speed $b'$ at center $x' \\neq x$, and choose $H^1$ sources $f$, $f'$ satisfying $\\int f/c^2\\, dx = \\int f'/c'^2\\, dx \\neq 0$; if a search finds such a pair whose boundary traces match to solver precision, Theorem 2.1 would be contradicted. A more targeted check of the weakest assumption is to test whether the analytic continuation used in Theorem 3.1 still exists when $\\operatorname{ess\\,sup} c = \\sqrt{2}\\, b_0$ exactly, since the invertibility proof needs a strict inequality.","supporting_citations":[{"cited_title":"Kian and G","cited_arxiv_id":null,"evidence_quote":"Supplies the prior wave-speed-and-source determination framework and the theorem used to conclude $f_1 = f_2$ once $c_1 = c_2$ is known."},{"cited_title":"Kian and F","cited_arxiv_id":null,"evidence_quote":"Provides the observability inequality for piecewise constant coefficients that carries the stability proof of Theorem 2.4."},{"cited_title":"Knox and A","cited_arxiv_id":null,"evidence_quote":"Established an earlier simultaneous-recovery result whose harmonicity and smallness conditions this paper relaxes."},{"cited_title":"Liu and G","cited_arxiv_id":null,"evidence_quote":"Gives the limiting absorption and integral representation used in Lemma 3.1 to identify the low-frequency coefficients."},{"cited_title":"Shapiro, Local energy decay for Lipschitz wavespeeds, Commun","cited_arxiv_id":null,"evidence_quote":"Provides the self-adjoint resolvent framework (Proposition A.1) and the background on local-energy decay for piecewise constant speeds."},{"cited_title":"Hu and Y","cited_arxiv_id":null,"evidence_quote":"Justifies the Laplace transform in time and its boundary-data properties used throughout."},{"cited_title":"Sjöstrand, Lectures on resonances, sjostrand.perso.math.cnrs.fr/Coursgbg.pdf","cited_arxiv_id":null,"evidence_quote":"Underlies the analyticity of the constant-speed cutoff resolvent via the Huygens principle."},{"cited_title":"El Badia and T","cited_arxiv_id":null,"evidence_quote":"Supplies the transposition-solution method for identifying point sources in elliptic problems, extended in the Appendix to divergence-form operators."}],"review_version":1}