{"id":"9c1b0f0d-7d92-4986-b29e-0bfcc1c0bc3b","arxiv_id":"2607.19799","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"With a non-degenerate constitutive relation f=F(x,c), simultaneous recovery of source and sound speed from one boundary measurement is unique and Lipschitz stable under convex-foliation and visibility geometry.","lead":"This paper shows that if the initial source and the sound speed of a wave equation are tied by a known material law, both can be recovered stably from a single boundary measurement. That turns a generally unstable inverse problem into a well-posed source-recovery problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniformity of source-stability constant over the c2-dependent coefficient A is asserted but not shown in Thm 2.6.","rationale":"The paper's algebraic reduction in Lemmas 2.1-2.3 is sound, and the constitutive non-degeneracy condition is an explicitly stated, strong modeling assumption. The reader correctly identified the exact knowledge of F as a practical limitation, but that is not an internal inconsistency. My stress-test focuses on a more technical, load-bearing point: the stability theorems apply a linear source-recovery estimate to an equation whose coefficient A depends on the unknown c2. A uniform Lipschitz estimate requires the source-stability constant to be uniform over the family A_{c2}. The full-boundary proof cites [26] for this uniformity without demonstrating that the cited theorem's hypotheses and constants are indeed uniform in the required sense. Proposition 3.3 gives the corresponding uniformity for the partial-data setting, which indicates the issue is likely repairable, but the full-boundary argument as written is incomplete. The reader's moderate confidence is partly due to the sketched partial-data proof; my concern is distinct and affects both the full-boundary and partial-data stability statements. A conditional acceptance is appropriate until the uniformity of the source-recovery constant is verified or a corrected proof is supplied.","tokens_in":13603,"tokens_out":28329,"duration_ms":284915,"concrete_test":"Read [26, Theorem 3.4] and its proof and verify explicitly whether the stability constant is bounded in terms of ∥a∥_{C^2}, the lower bound |a(0)|≥b0, and the geometry only, with no finer dependence on a. Equivalently, extend the finite-covering argument of Prop. 3.3 to the full-boundary setting: show the family A_{c2} is precompact in C^2 and that the constant is locally uniform in C^2. If the constant instead depends on c2 through Δu_{c2}, recompute (2.16) with the corrected constant; if that constant cannot be made uniform, the claimed Lipschitz stability is not valid as stated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Equations (2.8)-(2.9) reduce the comparison problem to P_{c1}V = α(x) A_{c2}(t,x), where A_{c2} contains B(x) plus ∫(t-s)Δu_{c2} ds. Theorems 2.5-2.6 are then obtained by invoking the source-recovery theorem of [26]. That theorem is linear in the unknown α, but it is applied with a coefficient A that itself depends on the unknown c2. For the claimed uniform Lipschitz estimate (2.16), the constant in [26, Thm 3.4] must be uniform over the family A_{c2} as c2 ranges over the admissible class. The proof asserts this uniformity in one sentence, citing [26], but does not verify that [26]'s constant is determined only by C^2 bounds and |A(0)| ≥ b0. Proposition 3.3 supplies the analogous uniformity argument in the partial-data case, but the full-boundary Theorem 2.6 does not. If the constant in [26] depends on finer data of A — for instance on Δu_{c2} or on second derivatives of A — then Theorem 2.6 does not follow from the cited result as written, and the central stability claim is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers simultaneous recovery of the sound speed c and the initial source f for the scalar wave equation from one boundary measurement, under the constitutive constraint f(x)=F(x,c(x)). The main idea is to reduce the coupled problem to a single inverse source problem: subtracting the two wave equations and taking one time primitive gives a source equation P_{c1}V = α(x)A(t,x) with V initially zero, where α=c_1^2-c_2^2 and A(0,x) satisfies the lower bound |A(0,x)|≥m_0/(2c_+) under the nondegeneracy condition |∂_r F|≥m_0. The authors then apply Stefanov--Uhlmann source-recovery theorems to obtain global uniqueness (Theorem 2.5) and Lipschitz stability (Theorem 2.6) under convex-foliation and visibility conditions, as well as partial-data analogues (Theorems 3.1 and 3.4). The paper is clearly written and the reduction in Lemmas 2.1--2.3 is internally consistent.","tokens_in":13905,"tokens_out":4363,"duration_ms":40830,"significance":"If the results hold, they give a positive structural answer to a simultaneous recovery problem that is known to be unstable in general (Stefanov--Uhlmann [27]). The constitutive-reduction idea is elegant and potentially transferable to other two-parameter inverse problems. The paper explicitly identifies the nondegeneracy condition as the mechanism restoring well-posedness, and carefully separates the geometric assumptions inherited from the source-recovery framework from the new constitutive assumption. The partial-data localization in Section 3 is a useful contribution, as it allows the constitutive law to degenerate outside the recoverable region. However, the central stability claim in Theorem 2.6 relies on an unverified uniformity assertion, which currently prevents the main theorem from being fully established as written.","major_comments":[{"comment":"The Lipschitz stability estimate requires the constant in the source-recovery theorem [26, Thm 3.4] to be uniform over the family A_{c2}(t,x)=B(x)+∫_0^t(t-s)Δu_{c2}(s,x)ds as c2 varies in the admissible class. The proof asserts this uniformity in one sentence ('where C is uniform as c2 varies in the admissible class') without verifying that the constant in [26] depends only on the C^2 bounds of A and the lower bound |A(0)|≥b0. Since A_{c2} contains Δu_{c2}, the dependence is nontrivial and not covered by the class C(K,T,b,M) defined in Section 2.1, which controls only time regularity and |a(0)|. If the constant in [26] depends on finer data of A, Theorem 2.6 does not follow from the cited result as written. The authors should quote the exact uniformity statement in [26] or prove it by adapting the finite-covering/C^2-precompactness argument used in Proposition 3.3. This issue is load-bea","section":"Theorem 2.6, Eq. (2.16)"}],"minor_comments":[{"comment":"The definition of the class C(E,T,b,M) uses C^2([0,T]; C(Ω)); it may be clearer to state explicitly the spatial regularity of the coefficient a, since A(t,x) contains Δu_{c2} and the source stability estimates in [26] require spatial smoothness assumptions. The current formulation only specifies time regularity.","section":"Section 2.1, Lemma 2.2"},{"comment":"There are several typographical artifacts, e.g., 'w(s, x), ds' and 'Z t 0' in Lemmas 2.2 and 2.3; these should be cleaned up.","section":"Throughout"},{"comment":"The notation DΓ versus D_Γ is inconsistent; also the definition of τ(y) and the cone condition could be restated more explicitly to avoid ambiguity about the initial time slice.","section":"Theorem 3.1 and Definition (3.1)"},{"comment":"Reference [24] is cited for both the source-stability argument and the exterior cone condition; the precise statements used in Proposition 3.3 would be easier to verify if the citation included theorem numbers.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The reduction and uniqueness results are sound and the paper is a good candidate for publication once the uniformity gap in Theorem 2.6 is addressed. Since Proposition 3.3 already contains a template for a uniform-constant argument, the fix is likely straightforward; I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper does something genuinely new: it shows that if the initial source and the sound speed are tied by a prescribed constitutive law f(x) = F(x, c(x)) with |∂_r F| ≥ m0, then the simultaneous recovery problem collapses to a single inverse source problem. The reduction is elegant. The time-primitive lemma (Lemma 2.2) is the key trick, and it works: the difference of two solutions is transformed into a source problem with a coefficient A(t,x) that is bounded below at t=0. Lemmas 2.1–2.3 check out algebraically. The uniqueness theorems (2.4, 2.5) then follow directly from Stefanov–Uhlmann, and the geometric hypotheses are the standard foliation and visibility conditions. The partial-data version (Theorem 3.1) is a nice local formulation: the constitutive law only needs to be non-degenerate along the swept recoverable region, not globally. The motivating examples (cure monitoring, thermoelastic generation) are plausible and make the strong structural assumption less artificial.\n\nThe main soft spot is the stability estimate, Theorem 2.6. The reduced equation is P_{c1} V = α A_{c2}, where A depends on c2 through u_{c2}. The proof asserts that the constant in the cited source-recovery theorem [26, Thm 3.4] is uniform over the family A_{c2}, but it does not demonstrate this. The class C(E,T,b,M) is defined precisely to capture the needed uniform bounds, so the claim is plausible, but the verification is missing. Notice that the partial-data analogue (Proposition 3.3) contains exactly the compactness and continuity argument needed to make the constant uniform, but that argument is not carried out for the full-boundary case. This is a gap in the written proof, not a fatal flaw. A referee should ask the author to fill it in or point to a version of [26] that already proves uniformity over such a class. A second, minor issue is that Proposition 3.3 is fairly sketched, leaning heavily on [24] and [26]; that is acceptable for a specialist journal but not fully self-contained.\n\nOverall, the central mechanism is correct and the paper is a real advance for a class of material-constrained inverse problems. The constitutive assumption is strong but explicitly stated, and the paper does not oversell its practical reach. It deserves a serious referee; with the stability uniformity argument patched, it would be a solid publication.\n\nRecommendation: engage with it. Send it to a referee who knows the Stefanov–Uhlmann machinery and ask specifically about the uniformity step.","headline":"A clean reduction of a coupled source-speed problem to an inverse source problem under a constitutive constraint, with the main unresolved step being a uniformity claim in the full-boundary stability estimate.","tokens_in":14352,"tokens_out":2993,"would_cite":true,"duration_ms":30805,"reading_group":"maybe","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":"A known material link between source and sound speed makes simultaneous recovery of both from one boundary measurement stable.","keywords":["inverse wave problem","simultaneous recovery","initial source","sound speed","constitutive constraint","Lipschitz stability","single boundary measurement","photoacoustic tomography"],"falsifier":"Run a numerical experiment in a domain satisfying the stated convex-foliation and visibility conditions: generate two admissible pairs (c1,f1=F(·,c1)) and (c2,f2=F(·,c2)) with |∂_rF| ≥ m0 > 0 over the admissible speed range, and compute the ratio (‖c1−c2‖_{L2(K)}+‖f1−f2‖_{L2(K)}) / ‖Λ_{f1,c1}−Λ_{f2,c2}‖_{L2((0,T)×∂Ω)} over a bounded smooth admissible class. If the ratio is unbounded, the Lipschitz claim is false. Alternatively, test necessity: choose F with ∂_rF = 0 on an open set, search for two distinct positive speeds in the admissible range with equal initial sources and equal boundary tra","tokens_in":1614,"feed_emoji":"🌊","tokens_out":1707,"duration_ms":70641,"temperature":0.7,"pith_summary":"The paper claims that if the initial source and the sound speed are coupled by a known constitutive relation f(x)=F(x,c(x)) whose derivative with respect to speed is bounded away from zero, then one boundary measurement of the wave field determines both unknowns simultaneously, with Lipschitz stability. This matters because recovering either quantity alone is classical, while simultaneous recovery was previously open in general and known to be unstable at the linearized level. The mechanism is a reduction: the two-unknown problem is rewritten as a single inverse source problem for the squared-speed difference, with a time-dependent coefficient that is uniformly nonzero at t=0. Under strictly convex foliation and geodesic visibility conditions, global uniqueness and stability follow; a local version recovers both quantities in the swept visible region from partial boundary data.","feed_headline":"One material link restores stable recovery of source and speed","feed_subtitle":"With a nondegenerate constitutive relation, equal boundary traces force both unknowns to match.","key_machinery":"The core device is the second time primitive of the wave difference, V(t,x)=∫_0^t (t−s)(u_{c1}−u_{c2})(s,x) ds. Lemma 2.2 shows that V satisfies P_{c1}V = α(x)A(t,x) with α = c1^2 − c2^2, where A(t,x)=B(x)+∫_0^t (t−s)Δu_{c2}(s,x) ds and B is the divided difference (F(x,c1)−F(x,c2))/(c1^2−c2^2). Nondegeneracy forces |B(x)| ≥ m0/(2c+), so A(0,x) is uniformly nonzero. Together with the trace identity ∂_t^2V|_{∂Ω} = Λ_{f1,c1} − Λ_{f2,c2}, this converts the two-unknown comparison into the one-unknown inverse source problem that microlocal and Carleman source-recovery methods already solve.","core_discovery":"The paper establishes that the simultaneous inverse problem becomes well-posed when f and c are linked by a prescribed smooth constitutive law F with |∂_rF(x,r)| ≥ m0 > 0 on the recovery set. In that case, the difference of two candidate wave fields, after one time integration, satisfies a single-source equation whose spatial factor is the squared-speed difference α = c1^2 − c2^2 and whose coefficient A(t,x) satisfies |A(0,x)| ≥ m0/(2c+) > 0. Equality of full boundary measurements then implies c1 = c2 and f1 = f2 under a convex-foliation condition, and the estimate ‖c1−c2‖_{L2(K)} + ‖f1−f2‖_{L2(K)} ≤ C‖Λ_{f1,c1} − Λ_{f2,c2}‖_{L2((0,T)×∂Ω)} holds under an added visibility condition. The same","pith_inferences":["The reduction suggests a design principle for imaging modalities: calibrate source amplitude and wave speed to a common scalar material parameter, such as cure degree or thermoelastic stress; the nondegeneracy condition is then an experimentally testable slope condition on the calibrated response curves.","The method converts a known instability into a stable problem by restricting to a submanifold of admissible pairs; the same strategy may apply to other coupled parameter pairs, such as density and source or attenuation and speed, whenever a similar monotone calibration is available.","The divided-difference lower bound |B| ≥ m0/(2c+) implies that stability degrades as m0 approaches zero; numerical experiments sweeping a family of constitutive laws toward degeneracy could test this predicted blow-up of the constant.","The local theorem implies that a mis-calibrated constitutive law in one region does not spoil recovery elsewhere, as long as nondegeneracy holds along the propagation path from the observed boundary to the target—relevant for multi-material samples."],"forward_implications":["When a nondegenerate constitutive relation holds, simultaneous recovery of the initial source and sound speed from a single boundary measurement is unique and Lipschitz stable in the full-data geometry.","The stability estimate uses the measured boundary trace difference itself, with no differentiation of the data lost.","The partial-data version shows that the constitutive law needs to be nondegenerate only in a neighborhood of the swept visible region, not in the whole medium.","In the thermoelastic calibration example f(x)=ρ(x)ε_T(x)c(x)^2, nondegeneracy holds wherever ρ ε_T ≥ q0 > 0 and c ≥ c− > 0, so the theorem applies to that concrete material class.","Local partial-data uniqueness and stability hold in compact visible regions satisfying the stated convex foliation, cone, and microlocal visibility conditions."],"fun_headline_variants":["Coupled unknowns stabilize source-speed recovery from one boundary trace","One boundary measurement recovers linked source and speed","Constitutive link turns ill-posed wave recovery into stable problem","Nondegenerate material law yields unique recovery of source and speed","Material law reduces coupled wave inversion to single source problem"],"cache_read_input_tokens":15744,"weakest_assumption_plain":"The whole argument rests on the material law f=F(x,c) being exactly known, smooth, and having a speed-derivative bounded away from zero in the region where recovery is sought; if the law is misspecified or its derivative vanishes locally, the reduction to a single inverse source problem fails and the known instability of simultaneous recovery returns.","fun_headline_variants_meta":{"raw":{"variants":["Coupled unknowns stabilize source-speed recovery from one boundary trace","One boundary measurement recovers linked source and speed","Constitutive link turns ill-posed wave recovery into stable problem","Nondegenerate material law yields unique recovery of source and speed","Material law reduces coupled wave inversion to single source problem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001272,"raw_usage":{"total_tokens":5045,"prompt_tokens":752,"completion_tokens":4293,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":4226}},"tokens_in":496,"tokens_out":4293,"duration_ms":26033,"temperature":1.0,"reasoning_tokens":4226,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T11:41:22.204933+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a numerical experiment in a domain satisfying the stated convex-foliation and visibility conditions: generate two admissible pairs (c1,f1=F(·,c1)) and (c2,f2=F(·,c2)) with |∂_rF| ≥ m0 > 0 over the admissible speed range, and compute the ratio (‖c1−c2‖_{L2(K)}+‖f1−f2‖_{L2(K)}) / ‖Λ_{f1,c1}−Λ_{f2,c2}‖_{L2((0,T)×∂Ω)} over a bounded smooth admissible class. If the ratio is unbounded, the Lipschitz claim is false. Alternatively, test necessity: choose F with ∂_rF = 0 on an open set, search for two distinct positive speeds in the admissible range with equal initial sources and equal boundary tra","supporting_citations":[],"review_version":1}