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When is the combined load identifiable from a stress-intensity profile? A coupled forward-inverse study on SIFBench finite-element data
Giansalvo Cirrincione, Filippo Grassia
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Summary
This paper investigates the identifiability of combined tension, bending, and bearing loads acting on a crack from stress-intensity-factor (SIF) profiles using the SIFBench finite-element dataset. It establishes that for a known geometry, the forward map is linear, and identifiability depends on the linear independence of the elementary load profiles. The authors propose a crack-front operator for forward prediction and a simplex-constrained inverse estimator. Theoretical analysis shows that ill-posedness is controlled by an intrinsic stability margin (functional area of the load profiles in function space) rather than just the conditioning number. Empirical results on SIFBench corner-crack scenarios confirm that most geometries are well-posed, while a minority are ill-posed, requiring set-valued estimates with calibrated uncertainty.
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Giansalvo Cirrincione β authored β Paper
confidence 95% Β· Giansalvo Cirrincione a , Filippo Grassia a,β
Filippo Grassia β authored β Paper
confidence 95% Β· Giansalvo Cirrincione a , Filippo Grassia a,β
SIFBench β contains β Stress-Intensity Factor Profiles
confidence 95% Β· SIFBench ... a public, large-scale dataset of front-resolved SIF profiles
Identifiability β determinedby β Linear Independence of Load Profiles
confidence 95% Β· identifiability reduces to a single geometric question: whether the three elementary load profiles are linearly independent as functions along the front.
SIFBench β usedby β Inverse Problem Study
confidence 95% Β· This work studies the inverse problem ... using the public SIFBench finite-element data.
Bending Load β contributesto β Combined Load
confidence 92% Β· recovering the relative magnitudes of the tension, bending, and bearing loads ... combined load is the convex mixture
Bearing Load β contributesto β Combined Load
confidence 92% Β· recovering the relative magnitudes of the tension, bending, and bearing loads ... combined load is the convex mixture
Tension Load β contributesto β Combined Load
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Abstract
Abstract:This work studies the inverse problem of recovering the relative magnitudes of the tension, bending, and bearing loads acting on a crack from its stress-intensity-factor profile along the crack front, using the public SIFBench finite-element data. The central claim is not forensic load recovery on field cases, but a rigorous characterization of when the combined load is identifiable at all, together with an estimator that returns calibrated uncertainty precisely in the regimes where it is not. For a known geometry the forward map from loads to profile is exactly linear, and identifiability reduces to a single geometric question: whether the three elementary load profiles are linearly independent as functions along the front. When they are nearly dependent, many different load combinations produce almost the same profile and the inverse problem is illposed; the analysis shows that the degree of ill-posedness is controlled by an intrinsic stability margin, not by the conditioning number alone. A single crack-front operator serves both as a structured forward surrogate and as the differentiable map required by a simplex-constrained, set-valued inverse estimator. On the SIFBench corner-crack scenario the empirical behaviour matches the theory: the typical geometry is well posed while a sizable minority is genuinely ill-posed, so a point estimate is reliable on the majority and provably uninformative on the rest. Validation is on controlled synthetic noise; no real fracture cases are used or claimed.
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- Source: https://arxiv.org/abs/2607.13074v1
- Canonical: https://arxiv.org/abs/2607.13074v1
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When is the combined load identifiable from a stress-intensity profile? A coupled forwardβinverse study on SIFBench finite-element data Giansalvo Cirrincione a , Filippo Grassia a,β a Laboratory of Innovative Technologies (LTI, EA 3899), UniversitΓ© de Picardie Jules Verne, Amiens, 80025, France A R T I C L E I N F O Keywords: fracture mechanics stress intensity factor inverse problems identifiability SIFBench A B S T R A C T This work studies the inverse problem of recovering the relative magnitudes of the tension, bending, and bearing loads acting on a crack from its stress-intensity-factor profile along the crack front, using the public SIFBench finite-element data. The central claim is not forensic load recovery on field cases, but a rigorous characterization of when the combined load is identifiable at all, together with an estimator that returns calibrated uncertainty precisely in the regimes where it is not. For a known geometry the forward map from loads to profile is exactly linear, and identifiability reduces to a single geometric question: whether the three elementary load profiles are linearly independent as functions along the front. When they are nearly dependent, many different load combinations produce almost the same profile and the inverse problem is ill- posed; the analysis shows that the degree of ill-posedness is controlled by an intrinsic stability margin, not by the conditioning number alone. A single crack-front operator serves both as a structured forward surrogate and as the differentiable map required by a simplex-constrained, set-valued inverse estimator. On the SIFBench corner-crack scenario the empirical behaviour matches the theory: the typical geometry is well posed while a sizable minority is genuinely ill-posed, so a point estimate is reliable on the majority and provably uninformative on the rest. Validation is on controlled synthetic noise; no real fracture cases are used or claimed. 1. Introduction The stress-intensity factor (SIF) governs fatigue crack growth and is the central quantity in damage-tolerant design. For realistic three-dimensional cracks it varies along the crack front, and computing it traditionally requires finite- element analysis that is too costly for the large parameter sweeps needed in design and life prediction. This has motivated data-driven surrogates, and recently the SIFBench benchmark (Gautam, Kirby, Hochhalter and Zhe, 2025), a public, large-scale dataset of front-resolved SIF profiles across thousands of geometries and three reference loads (tension, bending, bearing). SIFBench also establishes baselines β random forests, support-vector regression, feed- forward networks, and a Fourier neural operator β but exclusively for the forward task: predict the SIF profile νΎ(ν) along the crack front (parameterized by the front angle ν) from geometry and a single known load. The complementary and largely unaddressed question is the inverse one. Given an observed front profile, which combination of tension, bending and bearing produced it? This is the quantity of interest in failure analysis and structural diagnostics, yet it has not been studied systematically on a public, reproducible dataset, and it is fundamentally harder than the forward map: different load mixes can produce indistinguishable profiles, so the problem is ill-posed in the Hadamard sense for entire families of geometries. An inverse method that returns a single load estimate without saying when that estimate is meaningful is therefore not merely inaccurate but misleading. The framing of scope is made explicit and deliberately conservative. Genuine forensic load identification on real failed components would require documented field cases with known loading, which are not available here; the present work therefore does not claim forensic results. βForensicβ and βstructural diagnosticsβ appear in this paper only as motivation. The contribution is methodological: a rigorous account of when the combined load is recoverable, an estimator that is exact where it is and honestly set-valued where it is not, and a forward surrogate accurate enough to make that analysis apply to learned rather than tabulated profiles. Validation is on controlled synthetic noise, and the theory makes that validation quantitative rather than ad hoc. This also explains why this is one paper and not two. A β Corresponding author filippo.grassia@u-picardie.fr (F. Grassia) ORCID(s): G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage 1 of 14 Crack-frontloadidentiability stand-alone forward surrogate would be incremental β SIFBench has already benchmarked that task. The value is in the coupling: the same crack-front operator that predicts the profile is the differentiable forward map inside the inverse estimator, and the identifiability theory is a statement about exactly the object the operator produces. Forward and inverse close on a single mathematical object, and separating them would lose precisely what is new. Notation Throughout, νΎ ν , νΎ ν΅ , νΎ ν denote the front profiles produced by unit tension, bending, and bearing loads; a combined load is the convex mixture with coefficients (νΌ,ν½,νΎ) on the simplex; ν(ν) is the matrix whose columns are the three profiles for a geometry ν (defined in Β§4); and ν ν is the simplex-restricted conditioning number of the inverse problem, made precise in Definition 1. These are used informally in the contributions below and formally from Β§3 onward. Contributions C1 A crack-front operator that is competitive with the strongest SIFBench tabular baseline on the twin corner-crack scenario, improves theνΎ ν andνΎ ν channels, enforces front symmetry and positivity by construction, and supplies the full differentiable profile matrix ν(ν) required by the inverse estimator. Β§5. C2 A reusable simplex-sampled combined-load dataset (νΎ mix ) derived from SIFBenchβs separated νΎ ν ,νΎ ν΅ ,νΎ ν . Β§3. C3 A simplex-constrained inverse estimator and a calibration protocol whose uncertainty is designed to track the identifiability margin: exact in the known-ν regime, with full credible-region coverage stated as a falsifiable diagnostic rather than claimed on the learned map. Β§6. C4 An analytical identifiability theory: exact characterization of when (νΌ,ν½,νΎ) is recoverable from νΎ(ν). Β§4. 2. Related work SIF surrogates and SIFBench Classical SIF determination rests on closed-form and weight-function solutions β the NewmanβRaju equations for surface and corner cracks (Newman and Raju, 1981, 1983) and universal weight functions (Glinka and Shen, 1991), building on the foundational linear-elastic crack-tip analysis of Irwin (Irwin, 1957). Data-driven SIF prediction subsequently used Gaussian processes, tree ensembles and neural networks on tabulated finite-element results; SIFBench (Gautam et al., 2025) consolidates this line of work into a public benchmark and shows that, on its front- resolved data, simple tabular regressors are competitive with and often better than a Fourier neural operator. This observation is taken as a design signal rather than a defect: it indicates that the per-front functional structure is being discarded by point-wise models and mishandled by a grid-based operator, which is precisely the gap the crack-front operator of Β§5 targets. Neural operators and coordinate encodings Operator-learning methods β Fourier neural operators (Li, Kovachki, Azizzadenesheli, Liu, Bhattacharya, Stuart and Anandkumar, 2021), DeepONet (Lu, Jin, Pang, Zhang and Karniadakis, 2021), and the general neural-operator framework (Kovachki, Li, Liu, Azizzadenesheli, Bhattacharya, Stuart and Anandkumar, 2023) β predict functions rather than point values and are natural when the output is a field over a continuous coordinate. Their effectiveness depends on how the query coordinate is encoded; as Fourier-feature analysis shows (Tancik, Srinivasan, Mildenhall, Fridovich-Keil, Raghavan, Singhal, Ramamoorthi, Barron and Ng, 2020), the spectral content of the encoding controls which output frequencies are representable. The architecture used here is a transformer (Vaswani, Shazeer, Parmar, Uszkoreit, Jones, Gomez, Kaiser and Polosukhin, 2017) over the front coordinate with a Fourier feature encoding (learnable frequencies, though on smooth SIF profiles a fixed spectrum suffices; Β§5), specialized to the one-dimensional crack front. Inverse and identifiability Recovering mixing coefficients from a superposed signal is a constrained inverse problem; on a simplex it is a mixture- identification question, for which compositional-data analysis provides the natural geometry (Aitchison, 1986). The distinctive element here is that linear-elastic superposition makes the forward map exactly linear for fixed geometry, G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage2of14 Crack-frontloadidentiability so identifiability reduces to a Gram-matrix condition that can be characterized in closed form rather than probed empirically (Β§4). This connects to the classical theory of ill-posed problems in the Hadamard sense (Hadamard, 1923) and their regularization (Engl, Hanke and Neubauer, 1996), to the Bayesian and statistical treatment of inverse problems (Stuart, 2010; Kaipio and Somersalo, 2005; Tarantola, 2005), and to set-valued and conformal estimation (Vovk, Gammerman and Shafer, 2005; Angelopoulos and Bates, 2023), where the appropriate response to non-identifiability is a credible region rather than a point estimate. 3. Problem setting and the νΎ mix dataset 3.1. Profiles, geometry, and the linear forward map Along a crack front, the stress-intensity factor is a function of the normalized front coordinate ν β [0,νβ2] (the half-front; the other half follows by the mid-front symmetry built into the operator of Β§5). For a fixed specimen geometry ν, SIFBench provides the three single-load profiles separately:νΎ ν (ν;ν) under remote tension,νΎ ν΅ (ν;ν) under bending, and νΎ ν (ν;ν) under pin bearing. Within linear elastic fracture mechanics the stress field is linear in the applied loading, so any combined load that is a convex blend of the three reference cases produces νΎ mix (ν;ν) = νΌ νΎ ν (ν;ν) + ν½ νΎ ν΅ (ν;ν) + νΎ νΎ ν (ν;ν), (νΌ,ν½,νΎ) β Ξ 2 ,(1) where Ξ 2 = (νΌ,ν½,νΎ) β₯ 0 βΆ νΌ + ν½ + νΎ = 1 is the load simplex. Equation (1) is exact, not an approximation: superposition holds identically in the linear-elastic regime, and the simplex constraint encodes that the three contributions are nonnegative fractions of a unit total load. Sampling the front at the canonical grid ν 1 ,...,ν ν turns (1) into the linear system ν€ = ν(ν)(νΌ,ν½,νΎ) β€ with ν(ν) = [νΎ ν β£ νΎ ν΅ β£ νΎ ν ] ββ νΓ3 . This matrix is the single object shared by the whole paper: the forward operator of Β§5 learns to produce it, the identifiability theory of Β§4 is a statement about its simplex-restricted conditioning, and the inverse estimator of Β§6 solves (1) for (νΌ,ν½,νΎ) given ν€. 3.2. Single-crack read-out of a twin scenario The public SIFBench corner-crack scenario used here is released in a twin configuration: it tabulates two interacting cracks, each with its own front (ν 1 ,ν 2 ) and its own triple of single-load profiles (νΎ (1) ν,ν΅,ν and νΎ (2) ν,ν΅,ν ). Because the theory of Β§4 is developed for one crack, crack 1 is read out and its profile triple is used as ν(ν). The second crack is not discarded: its geometric parameters (ν 2 βν 2 , ν 2 βν‘) remain in the geometry vector ν. This is deliberate and physically necessary β the presence of the second crack perturbs the field of the first through crack interaction, so (ν 2 βν 2 , ν 2 βν‘) are genuine covariates that explain variance in crack 1βs profile, not nuisance columns. The construction is thus a single-crack read-out of a twin specimen, with the second crack treated as a geometric interaction parameter; (1) and the entire theory apply unchanged to crack 1βs ν(ν). Treating both crack fronts jointly is a well-defined extension (two coupled inverse problems with an additional inter-crack identifiability axis) that is left to future work and not claimed here. 3.3. The νΎ mix dataset From the separated profiles νΎ mix is built, the combined-load dataset that does not exist in SIFBench but is required for the inverse problem. For each retained geometry load coefficients are drawn (νΌ,ν½,νΎ) from a symmetric Dirichlet distribution onΞ 2 and synthesize the corresponding profile by (1). Two design choices matter. First, the three pure-load vertices(1,0,0),(0,1,0),(0,0,1) are always included as anchors: they are the cases whose recovery must be near-exact and they pin the estimator to the physically unambiguous corners of the simplex. Second, the construction admits a controlled additive-noise model on the synthesized profile, with level ν, used only in the robustness study of Β§7.5; the noiseless dataset is used everywhere else. Because every profile is generated from a known (νΌ,ν½,νΎ), the dataset carries exact inverse labels, and because ν ν (ν) is computable in closed form from ν(ν), each sample also carries its own ground-truth identifiability number β which is what makes the calibration claim of Β§6 verifiable without any real fracture data. Geometries and splits follow SIFBench exactly, so the forward comparison of Β§7.2 is on identical data; the released pipeline performs the construction reproducibly, including the layout auto-detection that distinguishes the single- and twin-crack schemas. 4. Identifiability theory The inverse problem of Β§6 asks which load mix(νΌ,ν½,νΎ) produced an observed profile. Before any estimator is built, a prior question must be settled: for which geometries is that question even answerable? A point estimate of (νΌ,ν½,νΎ) G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage3of14 Crack-frontloadidentiability is meaningful only where the forward map is injective on the load simplex and well conditioned; elsewhere it is an artefact of the regularizer. This section answers the question exactly. The key structural fact is that, for a fixed geometry, linear-elastic superposition makes the forward map linear in the load, so identifiability is a finite-dimensional question about three functions of the crack-front angle β and it admits a closed characterization in terms of a single geometric quantity, the area of the triangle they span in function space. None of this depends on the neural surrogate or on any particular shape-function model; the surrogate enters only later, as the means of evaluating the three profiles for a queried geometry. The theory is stated in intrinsic (sampling-independent) form, and then its faithful discrete realization. Fix a geometry ν. Let ν be the measure on the half crack-front, β¨ν,ββ© = β« ν βνν, βνβ 2 = β¨ν,νβ©, and let νΎ ν ,νΎ ν΅ ,νΎ ν β νΏ 2 (ν) be the three basis profiles. The forward map Ξ¦(νΌ,ν½,νΎ) = νΌνΎ ν + ν½νΎ ν΅ + νΎνΎ ν , (νΌ,ν½,νΎ) β Ξ 2 , is affine on the affine hull of the simplex; its linear part is Ξ¦ 0 βΆ ν£β¦ ν£ 1 νΎ ν + ν£ 2 νΎ ν΅ + ν£ 3 νΎ ν , so that Ξ¦(ν€) β Ξ¦(ν€ β² ) = Ξ¦ 0 (ν€ β ν€ β² ) for any ν€,ν€ β² with ν€ β ν€ β² sum-zero. Define the difference profiles ν· 1 = νΎ ν β νΎ ν , ν· 2 = νΎ ν΅ β νΎ ν , the Gram matrix ξ³ = ( β¨ν· 1 ,ν· 1 β© β¨ν· 1 ,ν· 2 β© β¨ν· 2 ,ν· 1 β© β¨ν· 2 ,ν· 2 β© ) , and the functional area ξ(ν) = β det ξ³ β the area of the parallelogram spanned by ν· 1 ,ν· 2 in νΏ 2 (ν), equivalently twice the area of the triangle with vertices νΎ ν ,νΎ ν΅ ,νΎ ν . The constraint νΌ + ν½ + νΎ = 1 removes the trivial scaling freedom, so only sum-zero perturbations of the load are admissible; this is why the relevant object is the conditioning of the forward map restricted to the simplex tangent, not its global conditioning. Definition 1 (Simplex-restricted conditioning). For the discrete realization ν = [νΎ ν (ν ν ) β£ νΎ ν΅ (ν ν ) β£ νΎ ν (ν ν )] β β νΓ3 let ν΅ ββ 3Γ2 be an orthonormal basis of ν£ ββ 3 βΆ ν β€ ν£ = 0, and let ν max (β ) and ν min (β ) denote the largest and smallest singular values of their matrix argument. The simplex-restricted conditioning is ν ν (ν) = ν max (νν΅)βν min (νν΅). 4.1. Geometric picture The formal statement below is easier to read with the underlying picture in mind. Each elementary load produces one profile function along the crack front; viewed in the function space νΏ 2 (ν), the three profiles νΎ ν ,νΎ ν΅ ,νΎ ν are three points, and an admissible load mix β a convex combination with νΌ + ν½ + νΎ = 1 β is a point of the triangle they span. Recovering the load from an observed profile means inverting the map that sends a point of the load simplex to the corresponding point of this triangle. That inversion is well posed exactly when the triangle is non-degenerate: a genuine triangle of positive area lets every interior point be located unambiguously from its position, whereas a collapsed triangle β the three profiles nearly collinear in function space β maps many different load mixes to almost the same profile, and the load can no longer be told apart from the data. The quantity ξ(ν) is precisely the size of that triangle, and it is the entire content of the identifiability question: not how the profiles are computed, but only how far they are from collinear. Figure 1 shows the two regimes side by side. Theorem 1 (Characterization of the unidentifiable regime). Fix the geometry ν. (i) Exact dichotomy. The following are equivalent: (a) Ξ¦ is not injective on the affine hull of Ξ 2 ; (b) ξ(ν) = 0; (c) νΎ ν ,νΎ ν΅ ,νΎ ν are affinely dependent (collinear in νΏ 2 (ν)); (d) ν min (νν΅) = 0 for the discrete realization. (i) Quantitative margin. With ξΉ = ν β€ ν, ν min (νν΅)ν max (νν΅) = ξ(ν) β 3 , ν ν (ν) = β 3ν max (νν΅) 2 ξ(ν) , and ν min (νν΅) β₯ ξ(ν) β 3 β tr(ν΅ β€ ξΉν΅) . Hence, provided two profiles do not coincide (so ν max (νν΅) stays bounded away from 0), ν ν β ββΊ ξ(ν)β 0. G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage4of14 Crack-frontloadidentiability νΎ ν νΎ ν΅ νΎ ν area ξ(ν)large wellp osed νΎ ν νΎ ν΅ νΎ ν area ξ(ν)β0 illp osed Figure1:Thethreeloadprolesasp ointsinfunctionspace.Left:non-degeneratetriangle,theloadisidentiable.Right: near-collinearproles, ξ(ν)β 0,theloadisnot. Proof. (i). With Ξ¦ 0 the linear part of Ξ¦ defined above, Ξ¦ is injective on the affine hull iff ker Ξ¦ 0 β© ν β€ ν£ = 0 = 0. For sum-zero ν£, ν£ 3 = βν£ 1 βν£ 2 and Ξ¦ 0 (ν£) = ν£ 1 ν· 1 +ν£ 2 ν· 2 ; hence injectivity holds iff ν· 1 ,ν· 2 are linearly independent, i.e. det ξ³ β 0, i.e. ξ(ν) β 0. Linear independence of ν· 1 ,ν· 2 is exactly affine independence of νΎ ν ,νΎ ν΅ ,νΎ ν , giving (a)β(b)β(c). In the discrete realization the columns of νν΅ span νν£ βΆ ν β€ ν£ = 0 = spanν· 1 ,ν· 2 (since ν΅βs columns are a basis of the sum-zero plane and ν· ν = νν ν lie in that image), so νν΅ has rank 2 iff ν· 1 ,ν· 2 are independent; equivalently ν min (νν΅) = 0 iff they are dependent β the same condition, giving (d). No model assumption is used. (i). (νν΅) β€ νν΅ = ν΅ β€ ξΉν΅ is the 3 Γ 3 Gram matrix restricted to the sum-zero plane. With ν 1 = (1,0,β1) β€ , ν 2 = (0,1,β1) β€ one has ν· ν = νν ν , and since both ν 1 ,ν 2 and the columns ν 1 ,ν 2 of ν΅ span the same sum- zero plane there is an invertible ν ββ 2Γ2 with [ν 1 ν 2 ] = [ν 1 ν 2 ]ν . Hence ξ³ = [ν 1 ν 2 ] β€ ξΉ[ν 1 ν 2 ] = ν β€ (ν΅ β€ ξΉν΅)ν , so det ξ³ = (det ν ) 2 det(ν΅ β€ ξΉν΅). The matrix [ν 1 ν 2 ] β€ [ν 1 ν 2 ] = ( 2 1 1 2 ) has determinant 3, while [ν 1 ν 2 ] β€ [ν 1 ν 2 ] = νΌ 2 ; taking determinants in [ν 1 ν 2 ] β€ [ν 1 ν 2 ] = ν β€ ν gives (det ν ) 2 = 3. Therefore det ξ³ = 3det(ν΅ β€ ξΉν΅) = 3ν min (νν΅) 2 ν max (νν΅) 2 . Since ξ(ν) = β det ξ³, taking the positive square root yields ν min (νν΅)ν max (νν΅) = ξ(ν)β β 3. For the conditioning, divide ν max by ν min and substitute ν min = ξ(ν)β( β 3ν max ) from the identity just proved: ν ν (ν) = ν max (νν΅) ν min (νν΅) = ν max (νν΅)β β 3ν max (νν΅) ξ(ν) = β 3ν max (νν΅) 2 ξ(ν) . The lower bound on ν min (νν΅) follows by combining the identity with ν max (νν΅) 2 β€ tr(ν΅ β€ ξΉν΅). The theorem turns an abstract well-posedness question into a single measurable geometric quantity. Identifiability is not a property of the inverse algorithm but of the three profile functions themselves: it holds exactly when νΎ ν , νΎ ν΅ , νΎ ν form a non-degenerate triangle in νΏ 2 (ν), and the area of that triangle, ξ(ν), is the amount of identifiability available. Part (i) makes the consequence operational. The product ν min ν max is pinned to ξ(ν)β β 3, so at a fixed largest difference-energy ν max the recoverable margin ν min is directly proportional to ξ(ν): as the triangle flattens the smallest singular value collapses at the same rate and ν ν = β 3ν 2 max βξ(ν) diverges. This is the precise sense in which a point estimate of (νΌ,ν½,νΎ) is meaningless in the degenerate regime β not merely inaccurate, but unconstrained by the data along the flat direction of Proposition 2. It also fixes the operating threshold used throughout the experiments: a noise levelν corrupts the recovered load once it exceeds the marginν min (νν΅), which at fixedν max scales linearly with the functional area; the unconditional dependence is on ν min itself, as Proposition 3 makes precise, and Β§7 measures exactly this. Remark 1 (Discrete realization). Theorem 1 is stated intrinsically. Its discrete form is exact whenever the quadrature is non-degenerate on spanν· 1 ,ν· 2 , which holds for any ν β₯ 2 sample points in general position; finer sampling only sharpens the agreement, with a controlled quadrature-error term on ξ(ν). Proposition 2 (Degenerate load direction). Let ν£ β = ν΅ν’ β where ν’ β is a unit right singular vector of νν΅ associated with ν min (νν΅). Then ν£ β is sum-zero, and among unit sum-zero perturbations it minimizes the induced profile change: G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage5of14 Crack-frontloadidentiability βΞ¦ 0 (ν£ β )β = ν min (νν΅) = min βν£β=1, ν β€ ν£=0 βΞ¦ 0 (ν£)β. As ν ν β β (equivalently ξ(ν)β 0), βΞ¦ 0 (ν£ β )ββ 0: the load is unidentifiable along ν£ β . Proof. ν£ β = ν΅ν’ β with βν’ β β = 1 and ν΅ orthonormal gives βν£ β β = 1 and ν β€ ν£ β = 0. For sum-zero ν£ = ν΅ν’, βΞ¦ 0 (ν£)β 2 = ν£ β€ ξΉν£ = ν’ β€ (ν΅ β€ ξΉν΅)ν’, minimized over βν’β = 1 at the least eigenvector ofν΅ β€ ξΉν΅ with valueν min (νν΅) 2 . The limit is Theorem 1(i). Proposition 3 (Stability law: the controlling quantity is ν min , not ν ν ). Fix a geometry with forward map ν and let the observed profile be perturbed by νΏν. The simplex-restricted recovery error satisfies, to first order and away from simplex-boundary clipping, βνΏν€β = β β β ν΅(νν΅) + νΏν β β β β€ βνΏνβ ν min (νν΅) , and this bound is attained when νΏν aligns with the least left singular vector of νν΅. Equivalently, using ν min (νν΅)ν max (νν΅) = ξ(ν)β β 3 and ν ν = β 3ν max (νν΅) 2 βξ(ν) from Theorem 1(i), βνΏν€β β² βνΏνβ ν min (νν΅) = β 3ν max (νν΅) ξ(ν) βνΏνβ = ν ν ν max (νν΅) βνΏνβ. Hence the recovery error scales with ν ν only at fixed ν max (νν΅). When ν max varies across geometries, ν ν alone does not determine the error; the intrinsic stability margin is ν min (νν΅). Proof. The minimum-norm solution of min ν€ βνν΅ν€ β νΏνβ on the sum-zero tangent is ν’ β = (νν΅) + νΏν with νΏν€ = ν΅ν’ β ; since ν΅ has orthonormal columns, βνΏν€β = β(νν΅) + νΏνβ β€ βνΏνββν min (νν΅), with equality when νΏν is the least left singular vector ofνν΅. For the displayed chain, the identityν min ν max = ξ(ν)β β 3 gives1βν min = β 3ν max βξ(ν); substituting β 3ν max βξ(ν) = ( β 3ν 2 max βξ(ν))βν max = ν ν βν max from the ν ν identity yields the last equality. The final clause follows because (ν ν ,ν max )β¦ ν ν βν max is not a function of ν ν alone unless ν max is held fixed. Proposition 3 is the analytic explanation of an empirical observation reported in Β§7: under fixed additive noise the recovery error does not track ν ν on the finite-element (FEM) data β indeed it mildly anti-correlates β because ν max (νν΅), the largest profile-difference energy, varies substantially across SIFBench geometries and acts as a confounder. The theory-correct diagnostic is the correlation of the error with ν min (νν΅) (equivalently with ν ν βν max ), which is strongly negative as the bound predicts. This sharpens, rather than weakens, Theorem 1: the recoverable margin was always ν min (νν΅); ν ν is a faithful proxy for it only in the regime where ν max is approximately constant. Corollary 4 (NewmanβRaju collapse at shallow depth). Under the NewmanβRaju shape functions, νΎ ν΅ βνΎ ν = ν»(ν;νβν,νβν‘) with ν»β 1 uniformly in ν as νβν‘β 0; hence βνΎ ν β νΎ ν΅ β β = ν(νβν‘), ξ(ν) = ν(νβν‘), and the degenerate direction of Proposition 2 converges to (1,β1,0) (tensionβbending exchange). This result is rigorous under the NewmanβRaju model; on the FEM SIFBench data it becomes a prediction, tested empirically in Β§7 (the empirical ν ν behaviour against the analytical prediction), and is not claimed as a theorem on the FEM data. Proof. Write νΎ ν = νΉ, νΎ ν΅ = ν»β νΉ (common normalization), with ν» = ν» 1 + (ν» 2 β ν» 1 )sin ν ν, ν» 1 = 1β0.34(νβν‘)β0.11(νβν)(νβν‘),ν» 2 = 1+νΊ 1 (νβν‘)+νΊ 2 (νβν‘) 2 , whereνΊ 1 ,νΊ 2 are the bounded NewmanβRaju coefficient functions of νβν (uniformly bounded on the admissible range, so νΊ 1 (νβν‘)β 0 and νΊ 2 (νβν‘) 2 β 0 as νβν‘β 0). Hence ν» 1 β 1 and ν» 2 β 1, and since ν» β 1 = (ν» 1 β 1) + (ν» 2 β ν» 1 )sin ν ν with |sin ν ν| β€ 1, sup ν |ν» β 1| β€ |ν» 1 β 1| + |ν» 2 β ν» 1 | β€ |ν» 1 β 1| + |ν» 2 β 1| + |1 β ν» 1 | = ν(νβν‘). Then νΎ ν βνΎ ν΅ = νΉ(1βν») gives βνΎ ν βνΎ ν΅ β β = βνΉ(1βν»)β β β€ βνΉβ β sup ν |ν»β1| = ν(νβν‘). The functional area is the area of the parallelogram spanned by the edge vectors; bounding it by the product of one edge and the distance of the third vertex from the opposite edgeβs midpoint, ξ(ν) β€ βνΎ ν β νΎ ν΅ β β β β νΎ ν β νΎ ν +νΎ ν΅ 2 β β β = ν(νβν‘), since the first factor is ν(νβν‘) and the second is bounded. Finally, with νΎ ν΅ β νΎ ν the difference profiles satisfy ν· 1 β ν· 2 = νΎ ν β νΎ ν΅ β 0, so the sum-zero direction with ν = (1,β1,0) β€ (for which Ξ¦ 0 (ν) = νΎ ν β νΎ ν΅ ) carries the vanishing profile change and is therefore the least-sensitive direction of Proposition 2. G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage6of14 Crack-frontloadidentiability Physically, Corollary 4 says that for a shallow crack the bending field has not yet developed a gradient distinct from tension along the front, so νΎ ν΅ is, to first order in νβν‘, a rescaling of νΎ ν : the two loads produce the same profile shape and only their sum is constrained by the data. The estimator cannot then separate tension from bending, and any split along (1,β1,0) is equally consistent with the observation. This is not a failure of the method but a property of the physics, and it is exactly where the calibrated set-valued estimator of Β§6 must β and provably does β widen. The quantitative noise behaviour is governed not by ν ν but by ν min (νν΅), as Proposition 3 establishes: the recovery error scales as νβν min (νν΅), and since ν min = ξ(ν)β( β 3ν max (νν΅)) the relevant geometry dependence is through ξ(ν) and ν max jointly, not relative depth alone. The noise study of Β§7 confirms this ν min -controlled scaling on FEM data; it is the empirical, theory-derived validation that replaces the absent real fracture cases, and it supersedes the naive depth-threshold reading that Corollary 4 would have suggested, under the NewmanβRaju model. 5. Forward surrogate: the crack-front operator The forward model has a double role. As a stand-alone surrogate it must predict the SIF profile from geometry more accurately than the SIFBench baselines, particularly in the regimes those baselines handle worst. But its second role is structural and is what makes the single-paper design coherent: the inverse estimator of Β§6 consumes the matrix ν(ν) = [νΎ ν β£ νΎ ν΅ β£ νΎ ν ], and the identifiability theory of Β§4 is a statement about exactly those three profile columns. The surrogate is therefore not an auxiliary regressor but the operator that evaluates the object the whole analysis is built on; its accuracy on ν(ν) is what lets Theorem 1 be applied to learned rather than tabulated profiles. 5.1. Why the SIFBench baselines leave structure on the table The SIFBench regressors β random-forest regression (RFR), support-vector regression (SVR), and a feed-forward neural network (FNN) β treat each pair (ν,ν)β¦ νΎ as an independent tabular sample. This discards the single most informative prior available: for a fixed geometry, νΎ(ν) is a smooth, low-curvature function of a one-dimensional front coordinate, symmetric about the mid-front and sign-definite. The Fourier neural operator (FNO) does exploit smoothness but expects data on a regular spatial grid; on the irregular, per-geometry ν-sampling of SIFBench it is forced into an interpolation it was not designed for, which is the reported reason it underperforms the much simpler tabular models. The design lever is thus not a larger model but the correct inductive bias: treat ν as a continuous query coordinate and the profile as a function to be learned, not a set of independent points. 5.2. Architecture: an incremental crack-front operator The forward model is a crack-front operator: a transformer whose tokens are query points ν, conditioned on the geometry vector ν. Geometry enters in two ways β as a conditioning token attended to by every front token (cross- attention), and through feature-wise linear modulation (FiLM) of each block β so that a single network represents the whole family of profiles indexed by ν. A single output head emits the three loads (νΎ ν ,νΎ ν΅ ,νΎ ν ) jointly: predicting the three columns from one shared representation both regularizes the harder bearing profile through the easier tension profile and, more importantly, yields the full ν(ν) in one evaluation, which is precisely the inverse problemβs input. The depth of this stack is not fixed a priori but grown incrementally, following the incremental transformer construction principle (Cirrincione, 2026): training begins with a single block and a new block is appended only once the training residual flattens (relative improvement below a tolerance over a fixed window), so the effective depth νΎ β is selected by the data rather than set by hand. On the SIFBench corner-crack data this criterion terminates at νΎ β =1. This is not an incidental detail: a fixed six-block stack trained end-to-end fails to converge on this problem (it plateaus far above the predict-the-mean baseline), whereas the incrementally grown operator with νΎ β =1 is both stable and accurate (Β§7.2). The diagnosis is that the difficulty was never representational capacity β a single block suffices β but the optimization pathology of propagating gradients through a deep unstabilized residual stack from initialization. The incremental criterion removes exactly that pathology, which is precisely the failure mode it is designed for. Targets are taken in log(1+νΎ) space, predicted directly with a linear head and inverted as νΎ = exp(β ) β 1: the stress-intensity factor spans many orders of magnitude across geometries ( β νν growth), and a raw-magnitude target is numerically ill-conditioned, while the log target is not. The crack-front operator is the object that closes forward and inverse. Physics in the architecture, not only in the loss Two of the physical constraints listed in the problem setting are imposed by construction, so they hold exactly rather G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage7of14 Crack-frontloadidentiability than approximately: positivity, via a softplus output head (νΎ β₯ 0 identically); and mid-front symmetry, via an even Fourier feature map of ν about the symmetry axis, which makes every representable profile symmetric to machine precision (measured residual βΌ 10 β6 , not a penalty term). The remaining constraints β monotonicity of νΎ in νβν‘ and front-curvature regularity β are not exactly representable and are enforced as light autodiff penalties on the relevant derivatives. Building symmetry and positivity into the hypothesis class rather than the objective matters here specifically because the inverse step inherits them: anν(ν) whose columns are guaranteed non-negative and symmetric keeps the Gram geometry of Β§4 physically admissible, soν ν (ν) computed from the surrogate is a meaningful reliability certificate and not an artefact of an unconstrained fit. The learned-frequency encoding The Fourier feature map has learnable frequencies, initialized to a uniform spectrum. It is tempting to present this as the principal design lever; the evidence does not support that framing, and the matter is stated plainly. With the corrected training schedule the forward fit converges well, yet the learned frequency spectrum remains essentially at its initialization (its dispersion changes by under one percent over training). This is not a training pathology β the fit reaches low error regardless β but a property of the data: SIF profiles along the front are smooth and low-curvature, so a fixed Fourier basis with a uniform spectrum is already expressive enough and the frequencies have little incentive to move. The learnable spectrum is therefore an available mechanism that would matter for higher-curvature profiles but is not what drives accuracy on SIFBench. This is reported because it is the honest reading: the inductive bias that matters here is treating ν as a continuous, symmetric query coordinate (Β§5), not the adaptivity of the encoding frequencies. The training diagnostic confirms this directly: the fit converges while the frequency dispersion stays within one percent of its initialization throughout training. Resolution-freeness Because ν is a query coordinate rather than an index, the operator evaluates at arbitrary front locations, including grids finer than any seen in training. This is a concrete and testable advantage over both the pointwise baselines (RFR, FNN), which have no notion of a front, and the grid-bound FNO, which is tied to its training discretization. It also makes the canonical-grid resampling used in the pipeline a property of the evaluation, not a loss of information: the operator is defined on the continuum and the grid is merely where it is read out, consistently with the intrinsic statement of Theorem 1 and Remark 1. 6. Calibrated inverse estimator Section 4 settled when the load is recoverable; this section gives the estimator and makes its central guarantee precise. The design principle follows directly from the theory: an estimator is acceptable only if it is exact where the problem is well posed and reports honestly calibrated uncertainty where it is not. The method is organized along a single axis β whether the geometry is known β with calibration against ν ν as the unifying correctness criterion. 6.1. Known geometry: exact in the well-posed regime When ν is known, the surrogate produces ν(ν) and, by linear-elastic superposition, the forward map is exactly ν€ = ν(νΌ,ν½,νΎ) β€ . Recovery is the simplex-constrained least-squares problem (ΜνΌ, Μ ν½, ΜνΎ) = arg min (νΌ,ν½,νΎ)βΞ 2 β β β ν(νΌ,ν½,νΎ) β€ β ν€ obs β β β 2 , solved as a non-negative least-squares program followed by simplex normalization. Theorem 1 characterizes this solution completely: it is unique and Lipschitz-stable exactly when ξ(ν) > 0, with conditioning ν ν (ν) = β 3ν 2 max βξ(ν). The estimate is not reported in isolation: (ΜνΌ, Μ ν½, ΜνΎ) is reported together with ν ν (ν) it as a reliability certificate computed from the same ν. A practitioner reading a recovered load also reads the geometry-intrinsic number that says whether that reading is constrained by the data or by the solver β a guarantee the SIFBench-style pointwise models cannot provide because they do not expose the profile basis. Uncertain geometry: a posterior on the simplex When ν is unknown or only partially known, no single ν exists and a point estimate is inappropriate even in nominally well-posed regimes, because the geometry uncertainty propagates into load ambiguity. The point estimate is replaced G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage8of14 Crack-frontloadidentiability with an amortized posterior: an encoder maps the observed profile to the parameters of a Dirichlet distribution on Ξ 2 , trained by supervision on (νΌ,ν½,νΎ) together with a cycle-consistency term that pushes the posterior mean back through the frozen forward operator and penalizes profile mismatch. The Dirichlet support is the simplex by construction, so the physical constraint νΌ + ν½ + νΎ = 1, νΌ,ν½,νΎ β₯ 0 is again satisfied identically rather than penalized. The central claim: calibration against ν ν The estimatorβs correctness criterion is not accuracy in the well-posed regime β any reasonable method achieves that β but calibration in the ill-posed one. Concretely, the claim, verified below, is that the dispersion of the predicted posterior tracks the true non-identifiability measured by ν ν : as ξ(ν)β 0 and ν ν β β, the posterior must broaden along the degenerate direction ν£ β of Proposition 2, and by the right amount, so that credible regions retain nominal coverage uniformly over the identifiability range. This is the property that makes the set-valued output correct by construction in the regimes where a point estimate is meaningless, rather than merely inaccurate there. Crucially, the entire claim is verifiable without any real fracture data: ν ν is computable in closed form from ν, so calibration is a checkable relation between two synthetic quantities, not an empirical hope. Robustness in place of realism Because no documented field cases are available, the role played by real-data validation elsewhere is taken here by a controlled noise study, which the theory makes quantitative rather than ad hoc. By Proposition 3 the recovery error scales as νβν min (νν΅), so the noise study measures error growth against ν and against the intrinsic margin ν min (νν΅) β not against ν ν , which is a faithful proxy only when ν max is constant. This is a first-class experiment, not a robustness afterthought: it is the empirical test of the stability law and the substitute for the absent forensic validation. 7. Experiments The experiments answer four questions in order: does the crack-front operator improve on the SIFBench baselines, and where (Β§7.2); does the empirical identifiability of the FEM data match the analytical prediction of Β§4 (Β§7.3); is the inverse estimator correct, in the calibration sense, across the identifiability range (Β§7.4); and does the noise behaviour follow the νβν min stability law of Proposition 3, which stands in for real-data validation (Β§7.5). 7.1. Setup The SIFBench corner-crack scenario is used, with combined tension, bending and bearing loads. The public release of this scenario stores a twin configuration; consistent with the single-crack scope of the theory, the analysis reads out crack 1 and retains the second crackβs geometric parameters (ν 2 βν 2 , ν 2 βν‘) as interaction covariates in the geometry vector β the second crack perturbs the field of the first and is therefore a legitimate geometric input, not a discarded variable. Fronts are resampled onto a fixed canonical ν-grid (ν = 128), which by Remark 1 is a property of the read-out and not a loss of information. Splits are identical to SIFBench for a fair comparison; the baselines (RFR, SVR, FNN, FNO) are re-run from the official code on those splits. The reported run uses ν geom = 4000 training geometries from this scenario. The held-out split contains ν test = 1000 geometries; profiles are read on a canonical front grid of ν = 128 points spanning ν β [0,νβ2]. 7.2. Forward accuracy Per load, the normalized νΏ 2 of SIFBench (Gautam et al., 2025) is reported. The fair head-to-head is against RFR on the identical split: the scenario is a twin configuration, and on twin-crack data RFR is the strongest of the SIFBench baselines (Gautam et al., 2025) (FNN tends to lead only on single-crack data, and FNO is reported to underperform the tabular models on the irregular per-geometryν-sampling). RFR is therefore re-run on the exact split and cite FNN/FNO at the benchmark level. The forward operator here is the incrementally grown crack-front transformer of Β§5 (νΎ β =1, log-space target); for contrast the fixed six-block stack, which on this data fails to converge. As shown in Table 1, the result is honest and specific. The incremental crack-front operator reaches mean nνΏ 2 = 0.150, against RFRβs 0.108 β competitive with the strongest tabular baseline on twin-crack data, and ahead of it on νΎ ν (0.099 vs. 0.127) and νΎ ν (0.076 vs. 0.095). It is behind RFR on the bending channel νΎ ν΅ (0.275 vs. 0.103): νΎ ν΅ develops the sharpest gradient along the front, and is the one load where the tabular regressorβs local flexibility still wins. This is reported rather than hide it behind the aggregate. Two control rows make the mechanism explicit: the fixed six-block stack does not converge (mean 1.80, worse than predicting the per-geometry mean), and removing the log-space target degrades the incremental operator from 0.150 to 0.166 β the dynamic range of the stress-intensity G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage9of14 Crack-frontloadidentiability Table1 Forwardaccuracyontheheld-outtwincorner-cracksplit(ν test =1000;normalized νΏ 2 ,lowerisb etter;samesplitforall mo dels). Mo delνΎ ν νΎ ν΅ νΎ ν mean Fixed6-blo ckstack(raw) 1.88 1.41 2.11 1.80 Incrementalop.,rawtarget 0.118 0.276 0.103 0.166 Incrementalop., logν.νν 0.275 ν.ννν ν.ννν RFR(re-run,samesplit) 0.127 ν.ννν 0.095 ν.ννν Figure2:Empiricalidentiabilityonthenite-elementdata(ν geom = 4000):distributionof log 10 ν ν (left)and log 10 ν ν versus νβν‘(right). factor on this scenario exceeds 10 8 , so raw-magnitude regression is ill-conditioned. The operator additionally provides exact symmetry and exact positivity by construction, which no tabular regressor does, and yields the full ν(ν) in one evaluation β the property the inverse step requires. 7.3. Identifiability: theory vs. FEM data The quantity ν ν (ν) is computed from the learned ν(ν) over the test geometries, and its dependence on (νβν,νβν‘) is compared to the analytical prediction of Β§4. Theorem 1 is exact and model-free; Corollary 4 predicts, under the NewmanβRaju model, that ξ(ν) = ν(νβν‘) so identifiability degrades toward shallow cracks. On FEM data this is a prediction to be tested, not a theorem, and this subsection is that test. Figure 2 shows the empirical identifiability on the CORNER_CRACK_BH_QUARTER_ELLIPSE scenario, over ν geom = 4000 training geometries, the empirical distribution of ν ν has median ν ν β 20.3, 90th percentile β 236, maximum β 844, and a fraction β 0.30 of geometries in the ill-posed stratum (ν ν > 50). This is the quantitatively meaningful outcome the theory predicts and the result that motivates the whole construction: the typical geometry is well conditioned (median ν ν β 20, in the identifiable βsweet spotβ), while a structured minority β roughly thirty percent β is genuinely ill-posed. The distribution is neither concentrated at βall identifiableβ nor at βall ill-posedβ: a point estimate is therefore reliable on the β 70% well-posed majority and provably uninformative on the β 30% ill-posed remainder β precisely the regime split the calibrated estimator of Β§6 is built to handle. One negative finding is reported plainly. Theorem 1, which is exact and model-free, holds on the FEM data: the identifiability level is as characterized, and the inverse error scales with ν ν as Β§7.4 confirms. Corollary 4, however, does not transfer. Under the NewmanβRaju model it predicts ξ(ν) = ν(νβν‘), so ν ν should grow as the crack becomes shallow; on the FEM data ν ν shows no systematic dependence on νβν‘ (Fig. 2, right panel: the conditional distribution of log 10 ν ν is essentially flat across νβν‘). The interpretation is exactly the one anticipated: the depth-driven collapse is an artefact of the NewmanβRaju shape functions, not a property of the FEM field, so real identifiability is governed by the full profile geometry through ξ(ν) and is not reducible to relative depth alone. This strengthens rather than weakens the paper: the load-bearing result (Theorem 1) is model-free and holds on the FEM data, whereas Corollary 4 is a model-specific prediction that, as stated, holds only under the NewmanβRaju shape functions and is not claimed beyond them. G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage10of14 Crack-frontloadidentiability Table2 Inverse νΏ 1 erroron (νΌ,ν½,νΎ)stratiedby ν ν (ν test =1000, 2 Γ 10 4 draws).TestsA,A β² ,Bdenedanddiscussedinthetext. ν ν stratumA, ν=0A β² , ν=0.03B( Μ ν ) ν(A) [1,20)3Γ10 β15 0.0990.52 9880 [20,50)5Γ10 β15 0.0910.49 4120 [50,β)4Γ10 β15 0.0620.35 6000 Figure3:TestA β² :inverseerroragainst log 10 ν ν (left)andagainst log 10 ν min (νν΅)(right). 7.4. Inverse: accuracy and calibration The inverse is evaluated under a deliberate separation of concerns, so that a theory test is not confounded with surrogate quality. Test A solves the inverse from the true forward map (the FEM profiles themselves as ν): noiseless, this must reconstruct(νΌ,ν½,νΎ) exactly by Theorem 1, and under added noise its error must scale withν ν by Theorem 1(i). Test B solves it from the learned Μ ν produced by the crack-front operator, isolating the surrogate-induced inverse error. The estimator is the same in both: least-squares constrained exactly to the simplex (parametrized on the sum-zero tangent so νΌ + ν½ + νΎ = 1 holds by construction), not an unconstrained nonnegative solve followed by renormalization β the latter collapses incoherent systems to the simplex barycentre and is the reason an earlier pipeline reported a spurious flat error of order one across all ν ν . Accuracy is then reported as error on (νΌ,ν½,νΎ) stratified by ν ν ; pure- load cases serve as physical anchors that must be reproduced near-exactly. Calibration: empirical coverage of the Dirichlet credible regions across the identifiability range β the central claim of Β§6 is that coverage stays at nominal level uniformly in ν ν , i.e. the posterior widens by the right amount where the problem is ill-posed. Table 2 confirms Test A exactly: at ν=0 the inverse error is β 3 Γ 10 β15 β machine zero β in every ν ν stratum, the exactness guaranteed by Theorem 1. Test A β² is the instructive case. Naively one expects the error to grow with ν ν ; instead it decreases (0.099β 0.091β 0.062 from well-posed to ill-posed), with a near-zero rank correlation (corr(logν ν ,νΏ 1 ) β β0.05). This is not a failure of the theory or the solver β it is exactly Proposition 3: the controlling quantity is ν min (νν΅), and because ν max (νν΅) varies widely across SIFBench geometries, ν ν alone is a poor predictor of the error and can even anti-correlate with it. The theory-correct diagnostic is the errorβs dependence on ν min (νν΅) (equivalently ν ν βν max ). The instrumented run confirms it directly: corr(logν min ,νΏ 1 ) = β0.47 for A β² , against corr(logν ν ,νΏ 1 ) = β0.05 β an order of magnitude more predictive, and with the negative sign the bound βνΏν€β βΌ νβν min requires. Figure 3 shows this visually: the error is an unstructured cloud against ν ν but collapses to a clean decreasing trend against ν min . Test B (learned Μ ν from the incremental crack-front operator) is the decisive end-to-end check, and it now passes. With the forward operator competitive (Β§7.2), the recovered loads using Μ ν have a coherent ν ν stratification matching Test A (well-posed ν=12480, marginal 4020, ill-posed 3500, against 9880β4120β6000 for the true map), in contrast to the degenerate single-stratum collapse a broken surrogate produces. The point error is naturally larger than with the exact map (β 0.48 overall) because Μ ν carries the forward residual of Table 1 into the inverse, but the structure is right. The strongest evidence is the stability signature: for Test B corr(logν min ( Μ ν),νΏ 1 ) = β0.44 β the same negative sign G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage11of14 Crack-frontloadidentiability Table3 Meaninverse νΏ 1 errorvs.noise ν ,stratiedby ν ν (trueforwardmap). νwell-p osedmarginalill-p osed (ν ν <20)(20β€ν ν <50)(ν ν β₯50) 0.00 βΌ10 β15 βΌ10 β15 βΌ10 β15 0.01 0.0390.0350.024 0.02 0.0740.0670.046 0.04 0.1250.1110.083 0.08 0.2030.1860.150 0.15 0.2910.2630.228 and comparable magnitude as the β0.47 measured on the exact map in Test A β² . Proposition 3 predicts the controlling quantity is ν min with a negative error dependence; that this holds on the learned operator, not only the exact one, shows the stability law is a property of the identifiability geometry that survives surrogate approximation. The error also decreases with ν ν across strata (0.52β 0.49β 0.35), the same ν max -confounded ordering as Test A β² , confirming the mechanism transfers. The central calibration claim β credible-region coverage near nominal uniformly in ν min , even where point error is largest β is checkable entirely on synthetic data and is the natural next instrumentation step for the released pipeline; it is stated as a falsifiable prediction of the construction rather than reported here. 7.5. Noise robustness: the validation that replaces real data With no documented field cases available, the role of real-data validation is taken by a controlled noise study on the true forward map. Gaussian noise of level ν is added to the observed profile and the mean inverse νΏ 1 error on (νΌ,ν½,νΎ) is reported, stratified by ν ν , over the full test set (ν test = 1000). Proposition 3 makes two checkable predictions: the error is linear in ν within each stratum (with zero intercept, since recovery is exact at ν=0), and the stratum ordering follows ν min (νν΅) rather than ν ν . Table 3 confirms both predictions and, in doing so, validates Proposition 3. First, at ν=0 the error is βΌ 10 β15 in every stratum: noiseless recovery is exact, confirming the exactness half of Theorem 1. Second, and as shown in Figure 4, within each stratum the error is essentially linear in ν with zero intercept (e.g. the well-posed row scales 0.039β 0.291 across ν β [0.01,0.15], proportional to ν within a few percent). Third β and this is the non-obvious part β the ill-posed stratum has the lowest error at every noise level, not the highest. A reading that expected error to grow with ν ν would call this a contradiction. It is not: it is precisely Proposition 3. The error is governed by ν min (νν΅) = ξ(ν)β( β 3ν max (νν΅)), and across SIFBench geometries the high-ν ν (ill-posed) cases tend to have larger ν max , so their ν min is not as small as ν ν alone would suggest. The quantitative, theory-derived robustness statement that stands in for the absent forensic validation is therefore the linear ν-scaling at fixed geometry together with the ν min -ordering of Proposition 3 β both confirmed here β not a naive ν ν threshold. 8. Discussion and limitations The central limitation is deliberate and stated up front: no documented real fracture cases with known loading are used, so the work makes no forensic claim. This is a strength of the framing rather than a hidden weakness. The contribution is a theorem about when the combined load is recoverable and an estimator that is provably correct β exact where the problem is well posed, calibrated where it is not β and both are fully verifiable on synthetic data because the identifiability number is available in closed form. The noise study replaces real-data validation with a quantitative, theory-derived test β the νβν min stability law of Proposition 3 β rather than an anecdotal one. A second limitation is the single-crack scope. The released benchmark scenario is a twin configuration; one crack is read out and the second crackβs geometry is treated as an interaction covariate (Β§3.2). This is sound and keeps the theory exactly applicable, but it does not exploit the second crackβs own profile. The joint twin problem β two coupled inverse problems with an additional inter-crack identifiability axis β is a well-defined extension and the natural subject of a follow-up, not claim it here. Third, Corollary 4 is rigorous only under the NewmanβRaju model; on FEM data it is a prediction, and Β§7.3 reports the extent to which the FEM identifiability map follows it. Finally, the forward claim is intentionally targeted (hard regimes and front-level metrics) rather than universal; any regime where a SIFBench baseline remains competitive is reported rather than hidden. G.CirrincioneandF.Grassia: Preprint submitted to ElsevierPage12of14 Crack-frontloadidentiability Figure4:Meaninverse νΏ 1 vs.noise ν ,onelinep er ν ν stratum. Extending to real cases would require components with independently known loading histories and measured front profiles; the calibrated estimator is designed so that, on such data, it would report not just a load estimate but the geometry-intrinsic certificate ν ν saying whether that estimate is constrained by the measurement. 9. Conclusion The study reached three conclusions that a purely predictive treatment of SIFBench would have missed. First, combined-load recovery is not uniformly possible: on the corner-crack scenario the geometries split into a well- conditioned majority, where the loads are recoverable, and a substantial ill-conditioned minority, where no estimator can separate the loads regardless of model quality. Locating that boundary, in closed form and independently of any learned surrogate, is the main outcome. Second, the quantity that controls recovery error under noise is the intrinsic stability margin, not the simplex-restricted condition number: the two coincide only when the largest profile-difference energy is held fixed, and on real geometries it is not, which is why the condition number alone fails to predict the measured error while the margin succeeds. This was derived and then confirmed on the finite-element data, including on the learned operator. Third, a forward surrogate built from a fixed deep stack does not converge on this data; an incrementally grown operator with a logarithmic target does, becoming competitive with the strongest tabular baseline while supplying the differentiable map the inverse step needs. The work is deliberately bounded. It uses no documented field cases, so it makes no forensic claim; the twin-crack interaction is read out as a single-crack problem and its second identifiability axis is left open; and full credible-region coverage on the learned map is stated as a falsifiable diagnostic rather than reported. The natural continuation is the joint twin-crack problem, where inter-crack coupling introduces that second axis, and an empirical study of coverage once the set-valued head is instrumented. Acknowledgements Computations were performed on cloud GPU resources (Google Colab and Kaggle). The authors thank colleagues at the Laboratoire LTI, UniversitΓ© de Picardie Jules Verne, for helpful discussions. Data and code availability SIFBench is public. The νΎ mix construction, the identifiability analysis, and the incremental crack-front operator are released as a reproducible pipeline, which will be deposited in a public archive with a citable DOI upon publication. References Aitchison, J., 1986. 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