{"id":"15bcb672-998c-4de1-91be-34f19213eb72","arxiv_id":"2607.21388","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For two-phase Stefan problems, the paper derives spectral reconstruction formulas and claims existence and uniqueness of recovered time-dependent source and reaction coefficients.","lead":"A preprint works out formulas for recovering time-dependent heat-source and reaction coefficients in two-phase melting/solidification problems from boundary data. If correct, it would let engineers identify thermal parameters in phase-change materials without extra interior sensors, but key proofs are missing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reconstruction formulas treat F_i,n as known data although F_i,n contains the unknown R_i through 1/R_i terms in h_i; the claimed Volterra structure and uniqueness are not established.","rationale":"The reader's weakest assumption is the same as the one I find decisive, so I agree. The manuscript's whole reconstruction machinery depends on the claim that the Fourier coefficients F_i,n can be treated as known inputs in Eqs. (24), (26), (45), (51), (84), and (85). But by construction (5)-(6) and (74)-(75), these coefficients contain the unknown R_i (and in the reaction case R_i' as well). This is not a technical gap that can be patched by an assumption like positivity: the functional dependence is structural, because the boundary-data lifting that homogenizes the boundary conditions inevitably produces terms proportional to 1/R_i. Thus the formulas the paper calls Volterra integral equations are actually implicit, nonlinear, denominator-dependent equations. The Gronwall argument in Lemma 3 assumes the kernel is known and bounded independent of R; that assumption fails. The uniqueness proof in Theorem 4 similarly assumes the Volterra form (65) with a fixed kernel K_i(t,tau), which is not available. The numerical examples do not test the generic inverse problem: they choose f_i so that the effective h_i is independent of R_i, corresponding to a special calibration. For these reasons the central claim of explicit recoverability and existence/uniqueness for the stated inverse problems is not established; a REJECT verdict is appropriate. This is not a disagreement with the physical plausibility of the setting, but a specific correctness risk in the derivation.","tokens_in":36007,"tokens_out":5731,"duration_ms":57656,"concrete_test":"Set f1=0, g1(t)=const, s(t)=const in the first problem (or any data satisfying the paper's smoothness assumptions). Compute F1,n(t) from (5) explicitly; it is proportional to 1/R1(t). Substitute into (24) and check whether R1 cancels or the equation reduces to an identity for arbitrary R1. If so, the integral overdetermination condition does not determine R1, contradicting Theorem 1. If it does not cancel, check whether the resulting equation can be rearranged as R1=G+integral(K R1) with G,K independent of R1; if not, the Volterra theory invoked in Lemma 3 is inapplicable. A complementary check: repeat the same substitution for the reaction problem (74)-(75) in (84)-(85).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central reconstruction step fails because the spectral coefficients fed into the inversion formulas are not known data. In (5), h1 = f1 - [g1 - (g1' s + s'(u*-g1))/s^2]/R1, so F1,n(t) = (h1(·s(t),t), phi_n) = f1,n(t) + A1(t) C_n / R1(t) with C_n = integral of phi_n. Hence every occurrence of F1,n in (24) — including the denominator sum F1,2n-1/sqrt(lambda) — depends on the unknown R1(t). Formula (24) is therefore implicit/nonlinear in R1, not a Volterra equation of the second kind with known kernel, and the Gronwall-based boundedness argument in Lemma 3 and Theorem 1 does not apply. The same defect appears in (26) for R2 and in Section 5: h1,h2 in (74)-(75) contain R1'/R1 and R2'/R2, so F1,n,F2,n in (84)-(85) depend on the very functions being recovered. The numerical examples avoid the issue by choosing f_i so that h_i is an explicit known function of (xi,t), but that is only a special calibration, not the general reconstruction claim. Because the existence and uniqueness proofs are delegated to a self-cited preprint, the manuscript does not establish the claimed explicit Volterra solvability, uniqueness, or existence for generic admissible data.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two inverse Stefan-type problems for heat equations. In the first problem, unknown time-dependent source coefficients R1(t), R2(t) are to be recovered together with the temperature fields in a two-phase one-moving-boundary configuration, using integral, pointwise, or nonlocal overdetermination data. The moving domains are mapped to the fixed interval (0,1), the temperature is expanded in Fourier series of the Dirichlet eigenfunctions, and reconstruction formulas (24), (26), (45), (51) are derived. These formulas are claimed to be Volterra integral equations of the second kind, leading to existence, uniqueness, boundedness, and regularity theorems (Theorems 1–5). In the second problem, a two-moving-boundary configuration with Neumann conditions contains unknown reaction coefficients P1(t),P2(t); via the exponential substitution (67)–(68) it is reduced to an inverse source problem, and the two Stefan conditions are claimed to determine both coefficients without additional overdetermination (Theorem 6, formulas (84)–(85)). Numerical experiments for both examples are discussed.","tokens_in":36441,"tokens_out":3923,"duration_ms":41531,"significance":"If the reconstruction formulas and the accompanying existence, uniqueness, and regularity theorems were correct, the paper would constitute a meaningful extension of the one-phase inverse Stefan analysis of NauKhum (2025) to a genuinely two-phase setting and would support the interesting claim that, in the two-moving-boundary configuration, the two Stefan conditions suffice to determine two time-dependent coefficients. However, the central inversion step is not justified: the spectral coefficients F_i,n fed into the reconstruction formulas are defined through the very unknown coefficients R_i, so the purported Volterra equations are self-referential rather than explicit. The analytical claims are also undermined by the systematic deferral of the key proofs to a self-cited preprint. The numerical examples are carefully constructed calibrations in which the source functions are chosen so that F_i,n becomes an explicit known function; they do not test the general reconstruction procedure. On the positive side, the paper demonstrates familiarity with the relevant fixed-domain transformation and spectral-expansion machinery, and the numerical experiments for the calibrated examples are","major_comments":[{"comment":"The reconstruction formula (24) is not an explicit Volterra equation. By (5), h1(x,t) = f1(x,t) - g1(t)/R1(t) + [g1'(t)s(t)+s'(t)(u*-g1(t))]/[s^2(t)R1(t)], so the spectral coefficient F1,n(t) = (h1(·s(t),t), φn)L2 contains a term of the form C_n(t)/R1(t). Consequently the denominator in (24), namely the sum over F1,2n-1/√λ2n-1, depends on R1(t) itself. Equation (24) is therefore an implicit, nonlinear equation for R1(t), not a Volterra equation of the second kind with known kernel. The Gronwall argument in Lemma 3, which assumes |K(τ,t)| ≤ MK with K independent of R_i, does not apply. The same defect appears in (26) for R2, since F2,n inherits R2(t) from (6). This is a load-bearing error: the central reconstruction claim is not established.","section":"§2, Eqs. (5)–(6) and (24)"},{"comment":"The same circularity occurs in the two-moving-boundary problem. In (74) and (75), h1 and h2 contain R1'/R1, R2'/R2, R1, and R2; consequently F1,n and F2,n in (84)–(85) depend on the unknown functions being recovered. Formula (84) expresses R1(t) in terms of R1(τ), R2(t), F1,n(t) (which depends on R1), and F2,n(t) (which depends on R2); formula (85) similarly involves R2 on both sides. The claim that 'each boundary provides its own Stefan condition' sufficient to determine two unknown coefficients is therefore conditional on the solvability of a nonlinear, coupled, implicit system. No existence, uniqueness, or fixed-point argument for this system is supplied. The illustrative Example 2 sidesteps the issue by choosing source functions f1,f2 so that the transformed F_i,n are explicit known functions, which is not the general case covered by Theorem 6.","section":"§5, Eqs. (74)–(75), (84)–(85)"},{"comment":"The main existence and uniqueness theorems are not proved in the manuscript. Theorem 1 states 'The proof can be shown by similar approach in Lemma 2 from NauKhum (2025)', and the same deferral is used for Theorems 2, 3, and 6. The cited item is a preprint by the same author, not part of this submission. Moreover, Lemma 3's boundedness proof for Ri(t) assumes that the kernel Ki(τ,t) in the rewritten equation Ri(t)=Gi(t)+∫Ki(τ,t)Ri(τ)dτ is bounded independently of Ri; but as noted above, the actual kernel depends on Ri through F_i,n. Thus the circularity affects not only the reconstruction formulas but also the regularity and boundedness arguments on which Theorems 4 and 5 rely. The manuscript therefore does not deliver the promised rigorous base for the inverse problems under consideration.","section":"§3, Theorems 1–3 and Lemma 3; §5, Theorem 6"},{"comment":"Assumptions (ii) and (v) in Theorem 1 require F_i,1(t)>0 and F_i,n(t)≥0, and compatibility conditions on F_i. However, F_i is defined via h_i, which contains 1/R_i(t) and R_i-dependent terms; hence these assumptions cannot be verified from the given data (f_i, g_i, s, ω, p, h) before the unknown coefficient is found. The same issue affects Lemma 1, where boundary conditions on F1 are assumed although F1 depends on the a priori unknown R1. This makes the stated sufficient conditions partly circular and not checkable a posteriori, further weakening the claimed inversion theorem.","section":"§3.1, Theorem 1 assumptions"}],"minor_comments":[{"comment":"The abstract contains grammatical errors: 'which sufficient to determine' should be 'which are sufficient to determine'; 'singla data observations' should be 'single data observations'; 'Caushy-Schwarz' is misspelled in several places. The introduction also refers to 'NauKhum (2025)' repeatedly without summarizing the specific Lemma 2 used in proofs.","section":"Abstract and Introduction"},{"comment":"Notation is inconsistent and undefined: the symbols λ2,n and λ1,n appear without definition in (26), and the expression λn/√λn is written where √λn is evidently intended.","section":"Eq. (26)"},{"comment":"The moving boundaries in the problem statement (86) are π√(t+1) and 2π√(t+1), but the change of variables (88)–(89) uses π(t+1) and 2π(t+1), and the transformed equation (90) uses π²(t+1)². This dimensional discrepancy makes the example internally inconsistent as a verification of the general formulas.","section":"Example 2, Eqs. (86)–(89)"},{"comment":"The paper refers to Figures 1–6, but the submitted text contains no actual figures; only captions and descriptions are present. The numerical stability assessment (absolute errors of 4.5×10⁻² and 1.5×10⁻²) is quoted from the text but cannot be independently verified from the manuscript.","section":"Numerical section"}],"recommendation":"reject","confidential_remarks":"The manuscript's central algorithmic claim is vitiated by the dependence of the spectral coefficients F_i,n on the unknown coefficients R_i. This is not a matter of presentation or of a missing minor step; it invalidates the Volterra-equation structure and the subsequent existence and uniqueness proofs. The repeated deferral of the main proofs to a self-cited preprint, the internal inconsistency in Example 2, and the absence of the referenced figures further reduce confidence. In my view this is a load-bearing error that cannot be repaired by local revision within the manuscript's current scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the two-phase, two-boundary extension is genuinely new, but the central reconstruction formula is circular and the existence/uniqueness proofs are not in the manuscript. As written, I wouldn't rely on any of the theorems.\n\nThe genuinely new piece is Section 5. The idea that the two Stefan conditions alone can pin down two time-dependent reaction coefficients is worth taking seriously, and the fixed-domain transformation plus Fourier expansion is a natural toolkit. The numerical section is also honest: it shows the instability of direct differentiation and that Tikhonov regularization with GCV stabilizes the recovery, at least for the examples chosen.\n\nThe soft spots are load-bearing, not cosmetic. In Eqs. (5)–(6), h1 and h2 contain 1/R1 and 1/R2, so the spectral coefficients F1,n and F2,n are functions of the unknown coefficients, not known data. Yet formulas (24), (26), (84), and (85) treat them as known, even in denominators. That makes the inversion implicit and nonlinear, and the Gronwall-based argument in Lemma 3 does not apply. The numerical examples avoid the difficulty by choosing f_i so that h_i is an explicit known function—a special calibration, not the general claim.\n\nThe other major issue is that the main theorems—1, 2, 3, and 6—are not proved. They are delegated to 'a similar approach in Lemma 2 from NauKhum (2025),' the author's own preprint. That is not a substitute for a proof, and it leaves the central claims unsupported. The global existence theorem (Theorem 4) does contain a Galerkin argument, but it assumes the coefficients R_i are already 'uniquely reconstructed,' so it doesn't repair the gap.\n\nWho is this for? An inverse-problems reader might take the two-moving-boundary formulation as a starting point, but should not cite the results until the circularity is fixed and the proofs are written out. The paper deserves a serious referee—the problem is well-motivated and the error is instructive—but it should be rejected in its current form.","headline":"The two-phase extension is real, but the reconstruction is circular and the main theorems are deferred; the central claims don't hold as written.","tokens_in":36805,"tokens_out":4150,"would_cite":false,"duration_ms":40762,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35K20","80A22","35R35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two inverse two-phase Stefan problems are solved by fixed-domain Fourier expansions, recovering time-dependent source and reaction coefficients from Volterra equations and, in the two-boundary case, from Stefan conditions alone.","keywords":["Inverse Stefan problem","two-phase Stefan problem","time-dependent coefficient","spectral expansion","Volterra integral equation","Tikhonov regularization","generalized cross-validation","free boundary"],"falsifier":"Take a problem with a known, nonconstant R1(t) and compute the spectral coefficient F_1,1(t) of h1 from (5) using the true R1. If F_1,1 contains R1 in its denominator, then substituting the true R1 into the right-hand side of (24) will not reproduce R1 unless the equation is merely an identity; a numerical test with Example 1's setting (where F_1,1 is artificially independent of R1) would not detect this, so a problem with a different f1 that makes the R1-dependence explicit would settle whether (24) is truly a closed-form inversion.","tokens_in":35918,"feed_emoji":"🌡️","tokens_out":11144,"duration_ms":97343,"temperature":0.7,"pith_summary":"This paper tries to establish that two inverse two-phase Stefan-type problems, one with a single moving interface and unknown time-dependent source coefficients R1(t), R2(t), and one with two moving interfaces and unknown reaction coefficients P1(t), P2(t), can be solved by freezing each phase onto a fixed interval and expanding in the eigenfunctions of the Dirichlet spectral problem. The reconstruction relations for the source coefficients reduce to Volterra integral equations of the second kind, obtained from integral, pointwise, or nonlocal overdetermination data. In the two-moving-boundary reaction problem, the exponential substitution R_i = exp(−∫P_i) converts the problem back to a source-coefficient form, and the two Stefan conditions themselves supply enough equations to determine both coefficients, so no extra measurement is needed. Under smoothness and compatibility assumptions, the paper claims existence, uniqueness, boundedness, and regularity of weak and strong solutions, and illustrates stable recovery of the reaction coefficients from noisy boundary data after Tikhonov regularization. The principal load-bearing assumption is that the transformed source terms are treated as known data in the reconstruction formulas even though those terms are defined through the coefficients being sought.","feed_headline":"Volterra equations unlock two-phase Stefan coefficients","feed_subtitle":"Fixed-domain Fourier expansions recover both coefficient types; the two Stefan conditions alone provide enough data.","key_machinery":"The machinery is a pair of coordinate transformations ξ_1 = x/s(t), ξ_2 = (x−s(t))/(d−s(t)) that freeze each moving phase onto the fixed interval (0,1), together with the Dirichlet spectral basis φ_n(ξ)=√2 sin(nπξ) (and the mixed Neumann–Dirichlet basis w_n(ξ)=√2 cos((2n−1)πξ/2) for the second problem). Expanding the transformed temperatures in these bases turns the PDEs into ODEs for the spectral coefficients; their explicit solutions, containing exponentials of the time-dependent diffusivities and advection terms, are inserted into the data conditions and differentiated to yield Volterra integral equations of the second kind for the unknown coefficients R_i(t). An exponential substitution","core_discovery":"The central claim is that a two-phase Stefan inverse problem with one moving interface and unknown source coefficients R1(t), R2(t) can be fully solved by (i) subtracting the boundary data, (ii) mapping each moving phase to the unit interval, (iii) expanding in sine eigenfunctions, and (iv) differentiating the overdetermination condition to obtain Volterra equations for R1 and R2 from integral, pointwise, or nonlocal data. For a second problem with two moving interfaces and unknown reaction coefficients P1(t), P2(t), the exponential substitution R_i(t)=exp(−∫_0^t P_i(τ)dτ) converts it into a source-coefficient problem; the two Stefan conditions then determine both auxiliary coefficients simu","pith_inferences":["The paper leaves implicit that the reconstruction formulas (24) and (26) are only explicit linear Volterra equations if the transformed source terms h1 and h2 are independent of the sought R1 and R2; when h1 and h2 contain 1/R1 and 1/R2, the equations become nonlinear and the convergence arguments in Lemma 3 and the theorems (which assume bounded, known kernels) would need a separate justification","The two-boundary result suggests a general counting rule for inverse Stefan problems — the number of recoverable time-dependent coefficients matches the number of independent Stefan conditions — which could guide data requirements in multi-front or multidimensional phase-change problems.","The numerical instability in the reaction problem is concentrated in the differentiation step P_i = −R_i'/R_i, not in recovering R_i; a testable extension would regularize the logarithmic-derivative estimation directly rather than post-differentiating the reconstructed R_i.","The regularity assumptions on the data (fourth-order differentiability and strong boundary-compatibility) mean the reconstruction will exhibit smoothing bias when the true temperature lacks such smoothness; quantifying that bias against noise in realistic data would be a natural next step."],"forward_implications":["Time-dependent source intensities in a two-phase melting or solidification body can be recovered from a single integral measurement, a single interior point temperature, or a nonlocal weighted average, requiring no interior probes beyond those observations.","In a two-phase body bounded by two moving fronts, the Stefan energy-balance conditions alone determine both unknown reaction coefficients, removing the need for separate overdetermination measurements.","The explicit spectral-Volterra reconstruction formulas are stable under noisy boundary data once the numerical differentiation step is regularized, as demonstrated at 1%, 2%, and 5% noise levels.","The existence, uniqueness, and regularity theorems establish that the recovered pairs are genuine weak/strong solutions of the original moving-boundary problem, not merely formal series solutions.","When only finitely many spectral modes are active, the reconstruction is given in closed form, providing concrete test cases for numerical implementations."],"fun_headline_variants":["Two Stefan conditions determine both coefficients without extra data","Volterra equations from Stefan conditions recover two coefficients","Stefan inverse problem solved by its own boundary conditions","Two-phase Stefan coefficients identified from two moving boundaries","Fourier expansions plus Stefan conditions fix both coefficients"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The transformed source terms h1 and h2 in Eqs. (5)–(6) are defined through the unknown coefficients R1(t) and R2(t) (they contain 1/R1 and 1/R2 terms), yet the spectral coefficients F_i,n of these terms are treated as known data in the reconstruction formulas (24) and (26); if F_i,n depends on R_i, the claimed explicit linear Volterra structure is not justified.","fun_headline_variants_meta":{"raw":{"variants":["Two Stefan conditions determine both coefficients without extra data","Volterra equations from Stefan conditions recover two coefficients","Stefan inverse problem solved by its own boundary conditions","Two-phase Stefan coefficients identified from two moving boundaries","Fourier expansions plus Stefan conditions fix both coefficients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000461,"raw_usage":{"total_tokens":2122,"prompt_tokens":698,"completion_tokens":1424,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":1351}},"tokens_in":442,"tokens_out":1424,"duration_ms":11080,"temperature":1.0,"reasoning_tokens":1351,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:34:15.570425+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a problem with a known, nonconstant R1(t) and compute the spectral coefficient F_1,1(t) of h1 from (5) using the true R1. If F_1,1 contains R1 in its denominator, then substituting the true R1 into the right-hand side of (24) will not reproduce R1 unless the equation is merely an identity; a numerical test with Example 1's setting (where F_1,1 is artificially independent of R1) would not detect this, so a problem with a different f1 that makes the R1-dependence explicit would settle whether (24) is truly a closed-form inversion.","supporting_citations":[],"review_version":1}