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Generate in the Chart, Not on the Boundary: Function-Symbol Grounding for Hard Constraints in LTN-GANs
Nijesh Upreti, Vaishak Belle
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 8/25/2026, 6:56:46 AM
Summary
This paper introduces Function-Symbol Grounding (FSG) for Logic Tensor Network-Enhanced Generative Adversarial Networks (LTN-GANs) to address the limitations of predicate-based grounding and clamping layers in hard-constrained generation. The authors identify that high-resolution ratios (RR) cause predicate gradients to vanish and clamping layers to collapse the margin distribution, producing valid but unrealistic samples. FSG treats constraints as function symbols that construct valid samples within a coordinate chart of the feasible region, preserving the true distribution of constraint margins. A hybrid approach combines FSG for high-RR continuous constraints with clamping for low-RR or discrete constraints.
Entities (10)
Relation Signals (7)
FSG-LTN-GAN → isbasedon → LTN-GAN
confidence 96% · We call the result the FSG-LTN-GAN, for function-symbol grounding (FSG) in LTN-GANs.
FSG-LTN-GAN → uses → Function Symbol
confidence 95% · In this work, we investigate grounding each axiom as a function symbol inside the LTN framework.
FSG-LTN-GAN → improves → Margin Distribution
confidence 92% · This recovers the margin distribution on four high-resolution datasets, with Kolmogorov–Smirnov distance up to 25× smaller than the constraint layer’s
Chart → enables → Valid by Construction
confidence 91% · Together the function symbols form a chart... where every sample is valid by construction
Constraint Layer → causes → Margin Distribution Collapse
confidence 90% · the clamp pushes every sample onto the boundary, and the margin distribution is lost
Resolution Ratio → predicts → Learning Failure
confidence 88% · the resolution ratio R... is a diagnostic... of which constraints a chosen grounding can learn.
Predicate → suffersfrom → Vanishing Gradient
confidence 87% · When R is large, the predicate receives no learning signal
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Abstract
Abstract:Logic Tensor Network-Enhanced Generative Adversarial Networks (LTN-GANs) inject background knowledge by grounding each logical axiom as a predicate and training the generator to raise its satisfaction, a fuzzy truth value in $[0,1]$. Previous LTN-GAN work grounded every constraint this way, at the predicate level, and improved constraint satisfaction. A predicate, however, only scores a sample, so it cannot embed hard structural constraints, rules such as orderings, positivity, and definitional identities that must hold in every generated sample. In this work, we investigate grounding each axiom as a function symbol inside the LTN framework. We compare against the state-of-the-art alternative, a constraint layer that clamps each violating sample onto the feasible boundary and so produces outputs that are always valid. Our investigation shows that a valid sample is not always a realistic one. An inequality is not merely satisfied or violated. It holds by a margin, and a faithful generator should also reproduce the margin's real distribution. We find that the resolution ratio $R$, the data's scale over the margin's spread, is a diagnostic, computable before training, of which constraints a chosen grounding can learn. When $R$ is large, the predicate receives no learning signal, the clamp pushes every sample onto the boundary, and the margin distribution is lost while every standard metric still looks fine. A function symbol avoids both failures, computing the constrained variable rather than scoring it. Together the function symbols form a chart, a coordinate system inside the feasible region, where every sample is valid by construction and the margin is learned like any other quantity.
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- Source: https://arxiv.org/abs/2608.21605v1
- Canonical: https://arxiv.org/abs/2608.21605v1
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Generate in the Chart, Not on the Boundary: Function-Symbol Grounding for Hard Constraints in LTN-GANs Nijesh Upreti Email: n.upreti@ed.ac.uk Vaishak Belle Email: vbelle@ed.ac.uk Affiliation: The University of Edinburgh, 10 Crichton Street, Edinburgh, EH8 9AB, UK Abstract Logic Tensor Network-Enhanced Generative Adversarial Networks (LTN-GANs) inject background knowledge by grounding each logical axiom as a predicate and training the generator to raise its satisfaction, a fuzzy truth value in [0,1][0,1]. Previous LTN-GAN work grounded every constraint this way, at the predicate level, and improved constraint satisfaction. A predicate, however, only scores a sample, so it cannot embed hard structural constraints, rules such as orderings, positivity, and definitional identities that must hold in every generated sample. In this work, we investigate grounding each axiom as a function symbol inside the LTN framework. We compare against the state-of-the-art alternative, a constraint layer that clamps each violating sample onto the feasible boundary and so produces outputs that are always valid. Our investigation shows that a valid sample is not always a realistic one. An inequality is not merely satisfied or violated. It holds by a margin, and a faithful generator should also reproduce the margin’s real distribution. We find that the resolution ratio R, the data’s scale over the margin’s spread, is a diagnostic, computable before training, of which constraints a chosen grounding can learn. When R is large, the predicate receives no learning signal, the clamp pushes every sample onto the boundary, and the margin distribution is lost while every standard metric still looks fine. A function symbol avoids both failures, computing the constrained variable rather than scoring it. Together the function symbols form a chart, a coordinate system inside the feasible region, where every sample is valid by construction and the margin is learned like any other quantity. This recovers the margin distribution on four high-resolution datasets, with Kolmogorov–Smirnov distance up to 25×25× smaller than the constraint layer’s, and a hybrid of charting and clamping matches or exceeds the constraint layer on its own benchmark. 1 Introduction Deep generative models are a standard tool for producing synthetic tabular and scientific data, used to augment scarce records, share sensitive data, and propose candidate designs. Adversarial and variational generators such as CTGAN and TVAE (39), score-based and diffusion models (17; 18), and relational-structure models (22) learn increasingly accurate approximations of the distributions of data such as trip records, flight records, and molecular property profiles. However, approximating the data distribution well is not enough, because such data obey known rules that an approximation can still violate. For example, a synthetic trip record must have its drop-off after its pick-up, a flight must report its departure as schedule plus delay, and a molecule’s internal energy must rise from 00 K (U0U_0) to room temperature (U) and remain below its enthalpy (H), so that U0<U<HU_0<U<H. Constrained generation therefore asks for samples that are both realistic and provably valid under such rules, which in tabular and scientific data are typically linear orderings between properties, positivity requirements, and definitional identities. Two main families of methods inject this knowledge into a generator. The first adds a penalty to the training loss whenever a sample violates a constraint. Such a constraint is called soft, since the generator can trade it against the rest of the loss, and satisfaction is encouraged but never guaranteed. The Logic Tensor Network-Enhanced Generative Adversarial Network (LTN-GAN) (37) grounds each constraint as a predicate, a differentiable function that scores how well a sample satisfies it, and trains the generator to raise this score alongside its adversarial objective (3). The second family builds the constraints into the model, so that a violating sample cannot be produced. Such a constraint is called hard, since no output can break it. The differentiable constraint layer of 35, which turns a generator into a Constrained Deep Generative Model (C-DGM) and is the state of the art for tabular data, moves each violating sample into the feasible region by clamping and guarantees 100%100\% validity for any conjunction of linear inequalities. Our starting observation is that validity alone does not make constrained data realistic. A valid sample satisfies each inequality by some amount, and validity ignores how these amounts are distributed. In a trip record, the time from pick-up to drop-off is the trip’s duration. A generator can place every drop-off after its pick-up yet produce durations unlike those of real trips. The energy step U−U0U-U_0 can fail the same way. We call each such amount (drop-off minus pick-up, or U−U0U-U_0) the constraint’s margin, and we call its distribution over the real data the distribution of the constraint margin, which a faithful generator must reproduce. Whether a generator can reproduce this distribution depends on the resolution ratio R, the data’s scale over the margin’s spread, computed before training. When R is small, the margin is visible at the data’s scale, so an unconstrained generator places it, the satisfaction score has usable gradient, and a clamp rarely fires. When R is large, the margin is invisible at the data’s scale. For thermochemical energies the step U−U0U-U_0 is five orders of magnitude smaller than the energies themselves. In this regime the discriminator can no longer resolve the margin, the satisfaction score has vanishing gradient, and the generator violates the constraint on a constant fraction of samples. Trained in the loop, the constraint layer moves nearly every sample onto the boundary, collapsing the margin distribution to a point mass while validity reads 100%100\% and every per-feature statistic still matches the real data (margin Kolmogorov–Smirnov (KS) distance near 11; Figure 2). High-R constraints are common in scientific data, yet no standard metric detects the collapse. We show that the LTN framework already contains the mechanism to avoid this collapse. A constraint can also be grounded through a function symbol, a term that computes the constrained variable directly. For an ordering b>ab>a, the generator emits a free value for a and assembles b from a by adding a positive, smoothly parameterised increment. Every sample satisfies the constraint by construction, and the margin becomes a coordinate that the discriminator can resolve and shape at unit scale. The function symbols form a chart of the feasible region, a coordinate system in which the generator produces samples inside the region rather than on its boundary. The chart’s per-variable admissible intervals are exactly those the constraint layer computes by Fourier-Motzkin reduction, and the constraint layer becomes the special case that always clamps. We call the result the FSG-LTN-GAN, for function-symbol grounding (FSG) in LTN-GANs. Since charting helps on high-R continuous constraints while low-R and discrete margins are better clamped, we use a short pre-run to decide per constrained variable, yielding a hybrid model. We show that R acts as a condition number for the constrained density-estimation problem (Section 4). Contributions. (i) We show that hard-constrained generation can distort the distribution of the constraint margin while passing every standard metric, and that the resolution ratio R acts as a condition number that predicts this failure before training. (i) We develop function-symbol grounding, a change of coordinates that generates inside the feasible region with exact validity and subsumes the constraint layer as its clamp-everything special case, and extend it to a per-constraint hybrid. (i) We demonstrate that function-symbol grounding cuts the margin KS by up to 25×25× at equal validity on four real high-resolution datasets, that the hybrid matches or exceeds the constraint layer on the benchmark of 35, that the R-based prediction holds out of sample (nycflights13), and that the method transfers unchanged to CTGAN and TVAE backbones. 2 Background: Logic Tensor Networks and LTN-GANs Real Logic and grounding. Logic Tensor Networks (LTN) interpret a first-order language, Real Logic, in real-valued tensors (3). A grounding G assigns meaning to symbols, and every term becomes a tensor. Each predicate P becomes a map (P)G(P) into the truth interval [0,1][0,1] that scores its arguments. Each function symbol f of arity k becomes a real map (f):ℝDk→ℝDG(f):R^Dk\!→\!R^D, fixed or learnable, that computes a term. A predicate acts on learning only through the truth values it contributes, while a function symbol shapes the object being scored. Logical connectives become fuzzy operators (a t-norm for ∧ , its dual for ∨ , a fuzzy implication), and quantifiers become aggregations (∀ as a generalized mean over errors). The truth value of a closed formula under G is its satisfaction Sat∈[0,1]Sat∈[0,1], with 11 fully true. An axiom is a closed formula asserted to hold of the domain, a knowledge base KBKB is a finite set of axioms, and learning maximizes their aggregated satisfaction Sat(KB)Sat(KB). The LTN-GAN objective. A GAN couples a generator Gθ:→ℝDG_θ:Z\!→\!R^D, mapping latent noise ζ∼pζ p_ζ (a standard Gaussian) to a sample, and a discriminator Dψ:ℝD→[0,1]D_ψ:R^D\!→\![0,1], trained to score real samples near 11 and generated ones near 00, while GθG_θ is trained to fool it. An LTN-GAN treats the generated sample as the grounding of the constrained variables and trains the generator to also satisfy a knowledge base, using the objective ℒG=ℒadv(Gθ,Dψ)+λ(1−Sat(KB))L_G=L_adv(G_θ,D_ψ)+λ\,(1-Sat(KB)), whose axioms encode the constraints. Predicate grounding of a constraint. The standard generator-side LTN-GAN (G-LTN-GAN) (37) grounds an ordering axiom b>ab>a as a predicate, (P>)(a,b)=σ((b−a)/s)G(P_>)(a,b)=σ\! ((b-a)/s ) with σ the logistic, at a band s (a width hyperparameter), and the term λ(1−Sat)λ(1-Sat) in the objective pushes samples toward b>ab>a. This grounding is soft. It encourages, but does not guarantee, satisfaction, and its gradient is informative only where the predicate is unsaturated. Section 4 shows this band covers a fraction Θ(1/R) (1/R) of samples, so the usable gradient vanishes as R grows (RQ4). Our method keeps the LTN-GAN objective but regrounds each structural axiom through a function symbol. The constrained variable becomes a term that (f)G(f) computes rather than a free output that a predicate scores, so Sat=1Sat=1 holds by construction. Soft thus refers to the satisfaction signal, since function-symbol grounding also uses smooth links yet satisfies the axiom for every output. 3 Problem Statement Generator GθG_θζ Π -symbol grounding φ -Motzkin bounds [ℓi,ui][ _i,u_i]e.g. xi=ℓi+softplus(zi)x_i= _i+softplus(z_i)valid sample x, Sat(KB)=1Sat(KB)=1Discriminator DψD_ψreal / fakereal data xxzzzzφ−1(x) ^-1(x) Figure 1: Overview of the FSG-LTN-GAN. The grounding map φ assembles valid samples from the generator’s free coordinates z within the Fourier-Motzkin bounds of Π , and the discriminator operates in the chart coordinates. The hybrid instead clamps low-R and discrete variables in the loop (Appendix B). Let pXp_X be an unknown distribution over X∈ℝDX ^D and D a dataset of N i.i.d. samples. A sample is a vector x=(x1,…,xD)x=(x_1,…,x_D) whose scalar components xk∈ℝx_k are its features. A generative model with parameters θ (for us the generator GθG_θ) induces the distribution pθp_θ of its output Gθ(ζ)G_θ(ζ), ζ∼pζ p_ζ, and learning chooses θ so that pθ≈pXp_θ≈ p_X. The background knowledge is a finite set Π of linear-inequality axioms over the features x1,…,xD\x_1,…,x_D\, each of the form ∑kwkxk+b⊵0 _kw_kx_k+b 0 with ⊵∈≥,> ∈\≥,>\, real coefficients wkw_k, and offset b, following the formulation of 35. A sample x~ x satisfies ϕ∈Πφ∈ if ∑kwkx~k+b⊵0 _kw_k x_k+b 0. A generator is compliant (valid) if all its samples satisfy all of Π . Orderings (xi>xjx_i>x_j), positivity (xi>0x_i>0), and definitional identities (xi=∑jwjxjx_i= _jw_jx_j, encoded as two inequalities) are the structural fragment we study. The margin and its distribution. For ϕ:∑kwkxk+b≥0φ: _kw_kx_k+b≥ 0, its margin on a sample is mϕ(x)=∑kwkxk+bm_φ(x)= _kw_kx_k+b, with mϕ≥0m_φ≥ 0 exactly when ϕφ holds. The right target is the distribution of the constraint margin, the conditional distribution of the margin under the data, pX(mϕ∣mϕ≥0)p_X(m_φ m_φ≥ 0). We quantify this by the Kolmogorov–Smirnov (KS) distance between the generated and real constraint margins on a fixed reference sample of the real data. The resolution ratio. The spread σm=std(mϕ) _m=std_D(m_φ) is the standard deviation of ϕφ’s margin over the dataset. The scale σs=maxk:wk≠0std(xk) _s= _k:\,w_k≠ 0std_D(x_k) is the largest standard deviation among the features ϕφ relates: for ϕ:U−U0≥0φ:U-U_0≥ 0 it is std(U)std(U). Both are computed on the raw, unstandardized data, before training. The resolution ratio of ϕφ is Rϕ=σs/σmR_φ= _s/ _m. When RϕR_φ is large the margin is a tiny difference of large near-equal quantities. In standardized coordinates it occupies a band of relative width about 1/Rϕ1/R_φ, sub-resolution to a Lipschitz discriminator, so a free generator (one trained with no constraint mechanism) that has matched the marginals still violates ϕφ on a constant fraction of samples (RQ4). Section 4 casts RϕR_φ as a scaling condition number for recovering the distribution of the constraint margin. The constraint layer (clamping). C-DGM (35) appends a differentiable constraint layer (CL): given a variable ordering, it computes for each variable an admissible interval [ℓi,ui][ _i,u_i] (piecewise-linear in the already-set variables, by Fourier-Motzkin reduction) and clamps the generated value into it, CL(x~)i=min(max(x~i,ℓi),ui)CL( x)_i= ( ( x_i, _i),u_i), guaranteeing validity. A clamp moves a violating sample to the nearest boundary, where the margin is near zero. Section 4.1 shows this is exactly where the distribution of the constraint margin is lost, at a rate governed by R. 4 Function-Symbol Grounding as a Change of Coordinates Change of coordinates and chart carry their differential-geometric meaning throughout. Inside the affine subspace its identities define, the feasible set ℳ=x:⋀ϕ(x)M=\x: _φ(x)\ of a satisfiable linear system is a relatively open convex polytope, diffeomorphic to ℝdR^d and covered by a single global chart. The map φ below is such a chart, with one coordinate per free variable. We ground each structural axiom through a function symbol (Figure 1). The generator emits free terms z∈ℝdz ^d (one per variable not fixed by an identity), and a grounded map φ assembles the constrained sample by processing the variables in the Fourier-Motzkin order. With Πi+ _i^+ (Πi− _i^-) the reduced constraints in which xix_i has positive (negative) coefficient (35), the admissible interval of xix_i is piecewise-linear in the already-assembled x<ix_<i, ℓi(x<i)=maxϕ∈Πi+εiϕ(x<i),ui(x<i)=minϕ∈Πi−εiϕ(x<i),εiϕ=−(∑k<iwkxk+b)/wi, _i(x_<i)= _φ∈ _i^+ _i^φ(x_<i), u_i(x_<i)= _φ∈ _i^- _i^φ(x_<i), _i^φ=- ( _k<iw_kx_k+b )/w_i, (1) the bounds the constraint layer clamps into (Section 3). Per variable, φ applies the grounding that fits the interval: one-sided: xi -sided: x_i =ℓi+softplus(zi) or ui−softplus(zi), = _i+softplus(z_i)\ or \ u_i-softplus(z_i), box: xi : x_i =ℓi+(ui−ℓi)σ(zi), = _i+(u_i- _i)\,σ(z_i), free: xi : x_i =zi, =z_i, identity: xi : x_i =∑j<iwijxj+wi0. = _j<iw_ij\,x_j+w_i0. (For Alchemy’s U0<U<HU_0<U<H: U0=z1U_0=z_1, U=U0+softplus(z2)U=U_0+softplus(z_2), H=U+softplus(z3)H=U+softplus(z_3).) Each bounded variable is a smooth, monotone function of a unit-scale coordinate ziz_i that stays within its admissible interval, and each identity derives its dependent variable, so the axiom holds by construction (Sat(KB)=1Sat(KB)=1, hence the logical term in the objective vanishes for these axioms). The decoder φ:z↦x :z x is the chart of ℳM defined above, turning unconstrained coordinates z into feasible samples x. It inverts in closed form (for a one-sided variable, zi=softplus−1(xi−ℓi)z_i=softplus^-1(x_i- _i)), which gives the encoder φ−1 ^-1 applied to real data. The discriminator operates on the chart coordinates z, where every margin is unit scale, receiving z for generated samples and φ−1(x) ^-1(x) for real ones, so a standard discriminator can learn the distribution of the constraint margin directly. For heavy-tailed margins we ground through exp rather than softplus (a multiplicative increment), set by a fixed dynamic-range rule (Appendix G). Monotone softplus links enforcing order and positivity go back to 9. Function-symbol grounding deploys them as the groundings of structural axioms inside adversarial training. Proposition 1 (Validity by construction). For any satisfiable finite set Π of linear inequalities and any generator output z, the assembled sample φ(z) (z) satisfies Π . The proof (Appendix C) is the soundness of Fourier-Motzkin elimination. In the elimination order each variable’s admissible interval is non-empty given the finalized earlier variables, and softplus and σ map ℝR into its interior, so every axiom is met. Validity is exact, as for the constraint layer. The two differ only in where in the interval the sample lands, and that is what the distribution of the constraint margin captures. 4.1 Why the chart can preserve the distribution of the constraint margin and the clamp cannot For ϕ:b>aφ:b>a, the clamp sends every violator to b=ab=a, while function-symbol grounding emits the margin as a unit-scale coordinate, mϕ=softplus(z)m_φ=softplus(z). Remark 2 casts R as a condition number, and Proposition 5 quantifies both mechanisms. Remark 2 (R as a condition number). Recovering the distribution of the constraint margin of ϕφ requires resolving a margin of scale σm=std(mϕ) _m=std(m_φ) inside data of scale σs=Rσm _s=R\, _m. A Lipschitz discriminator on the ambient coordinates must resolve a relative magnitude 1/R1/R. We describe this as a condition number Θ(R) (R) in the scaling sense, versus Θ(1) (1) in the chart, where the margin is standardized. Corollary 3 (Definitional identities are measure-zero). Let ℳ==x∈ℝD:c(x)=0M_==\x ^D:c(x)=0\ be the zero set of the identities, with c:ℝD→ℝpc:R^D ^p a C1C^1 map whose Jacobian has full rank p on ℳ=M_=. Then ℳ=M_= is Lebesgue-null, so any generated distribution ν that is absolutely continuous has Prx∼ν[c(x)=0]=0 _x ν[c(x)=0]=0, and no satisfaction loss can raise exact satisfaction above probability 00. Function-symbol grounding derives the dependent variables from their parents, so c≡0c≡ 0 holds with probability 11. Corollary 4 (Predicate-grounded ordering: Θ(1/R) (1/R) gradient). Ground an ordering ϕ:b>aφ:b>a as the predicate Ps=σ((b−a)/s)P_s=σ\! ((b-a)/s ) with band s=Θ(σm)s= ( _m). If the generator has matched the marginals of a and b but not their dependence (margin correlation bounded away from 11), its margin has spread Θ(Rσm) (R\, _m) and density Θ(1/σs) (1/ _s) near b=ab=a (no anomalous concentration). The predicate gradient Ps(1−Ps)/sP_s(1-P_s)/s is Θ(1/s) (1/s) on the band |b−a|≲s|b-a| s and exponentially small outside it, so the fraction of generated samples with usable gradient is Θ(1/R) (1/R), vanishing as R→∞R→∞. Proposition 5 (Margin laws). Let the real margin of ϕ:b>aφ:b>a have CDF F supported on (0,∞)(0,∞), and clamp a free generator with violation rate v post hoc, with offset ε below the real support. (i) The clamped margin distribution is vδε+(1−v)ν+v\, _ +(1-v)\,ν^+, with ν+ν^+ the generator’s satisfying-margin distribution. Under the hypotheses of Corollary 4, its KS distance to F is at least max(v,1−v)−Θ(1/R)≥12−Θ(1/R) (v,1-v)- (1/R)≥ 12- (1/R), tending to 11 as v→1v→ 1 (observed in-loop; Section 5). (i) The chart φ is a bijection from ℝdR^d onto the relative interior of ℳM with closed-form inverse, both smooth off the measure-zero set where the active Fourier-Motzkin bound switches, places no probability on the boundary, and realizes every margin distribution on (0,∞)(0,∞) exactly (proofs in Appendix D). 4.2 Which axioms to ground through function symbols: a hybrid Function-symbol grounding is most effective where clamping is most costly, on high-R constraints the generator cannot meet on its own. But a continuous chart cannot represent a margin with a discrete point mass (an integer count, a two-valued category). The softplus or σ increment smears it, whereas the in-loop clamp, which the generator learns to anticipate, matches it empirically (RQ3). We therefore decide per bounded variable, over the conjunction of the reduced constraints that bound it. A short pre-run of a free generator measures the fraction sis_i of its samples that already lie in variable xix_i’s admissible interval, and the largest single-value frequency of xix_i’s binding margin in D measures the discreteness did_i. We chart xix_i if si<0.9s_i<0.9 and di≤0.2d_i≤ 0.2 (violated and continuous). Otherwise we clamp it in the loop, with the discriminator seeing the clamped value so the generator adapts to it. The resulting generator charts the high-R continuous variables and clamps the rest. Validity remains exact (Proposition 1) since every bounded variable is either charted or clamped. Algorithm B (Appendix B) summarizes the procedure. 5 Experimental Analysis We answer four questions. RQ1: does guaranteeing validity guarantee a realistic constrained quantity? RQ2: does the effect persist across generator architectures? RQ3: does the hybrid generalize to the constraint-layer benchmark of 35? RQ4: can predicate grounding (the standard LTN-GAN satisfaction loss) reach high-R constraints instead? Datasets. We use four real high-resolution datasets, each carrying an ordering whose margin is a small difference of large quantities: Alchemy (7) (U0<U<HU_0<U<H), tmQM (4), Transition1x (32) (reaction barriers), and Taxi (28) (trip duration). Constraints and per-margin R (3.63.6 to 7×1067×10^6) are in Table 4, Appendix F. For RQ3 we use the six-dataset benchmark of 35. Baselines. The constraint layer (C-DGM) on the same MLP-GAN as FSG-LTN-GAN; CTGAN and TVAE (39) with and without it; and post-hoc projection. Metrics. Validity, the margin KS against a fixed reference sample of the real data (Appendix E), and per-property moment error, as means ± std over n=10n=10 seeds with paired Wilcoxon p-values. Full tables and the constraint-layer corrections are in Appendices E, H, I, and J. 5.1 RQ1: validity does not imply a realistic constrained quantity Table 1: Validity does not capture the distribution of the constraint margin (RQ1). Four real high-resolution datasets, n=10n=10, mean ± std. Both methods are 100%100\% valid. Only FSG-LTN-GAN recovers the distribution of the constraint margin, while per-property moment error (lower better) is comparable, so the gap is invisible to it. †paired Wilcoxon p<0.01p<0.01. margin KS ↓ per-prop. moment ↓ Dataset R C-DGM FSG-LTN-GAN C-DGM FSG-LTN-GAN Alchemy 3×1043×10^4 to 7×1067×10^6 1.000±0.0001.000± 0.000 0.040±0.007†0.040± 0.007 0.472±0.0120.472± 0.012 0.327±0.0580.327± 0.058 tmQM 2.3×1042.3× 10^4 1.000±0.0001.000± 0.000 0.045±0.010†0.045± 0.010 0.183±0.0140.183± 0.014 0.192±0.0250.192± 0.025 Transition1x 10310^3 0.999±0.0000.999± 0.000 0.065±0.010†0.065± 0.010 0.115±0.0360.115± 0.036 0.196±0.0380.196± 0.038 Taxi 3.63.6 to 1.1×1031.1×10^3 0.574±0.0050.574± 0.005 0.099±0.010†0.099± 0.010 0.492±0.0240.492± 0.024 0.282±0.0320.282± 0.032 Figure 2: The distribution of the constraint margin on all four datasets. Real margin (grey) against C-DGM (orange, the constraint layer) and FSG-LTN-GAN (blue). All are 100%100\% valid. The clamp collapses every margin to a boundary point mass. FSG-LTN-GAN reproduces the real distribution. The annotated KS is for the displayed high-R margin. Table 1 averages each dataset’s constraints, so its Taxi value also counts the low-R total>faretotal>fare margin, which the clamp handles well. Throughout, the constraint layer runs from the code released by 35, with corrections that only help it (including one for a latent bug in its Fourier-Motzkin reduction; Appendix E). Table 1 shows that on all four datasets the constraint layer and FSG-LTN-GAN are both 100%100\% valid, yet the constraint layer’s margin KS is 0.570.57 to 1.001.00 while FSG-LTN-GAN’s is 0.040.04 to 0.100.10 (p<0.01p<0.01, all ten seeds). Inspecting the raw margins (Figure 2) explains the numbers. The clamp places every sample’s margin at the boundary (about 10−610^-6 at every quantile), a point mass disjoint from the real distribution (the collapse of Proposition 5(i)). FSG-LTN-GAN’s margin quantiles track the real ones. The failure is invisible to the standard metrics we report. Per-property moment error is comparable across methods (the constraint layer is even better on tmQM and Transition1x), so one can match every feature’s marginal while losing the distribution of the constrained quantity. The clamp’s distortion tracks R per margin (KS near 11 on the high-R margins, e.g. Alchemy U−U0U-U_0 at R≈105R≈ 10^5). Taxi’s lower table value averages in a low-R margin the clamp handles well, so the effect tracks R rather than the domain. A final, pre-registered test on a fifth dataset outside both papers’ suites (nycflights13) confirmed every prediction (Appendix J, Table 11). Overall sample quality is preserved. On the constraint layer’s own density and coverage metrics (27), function-symbol grounding is close on the chemistry sets (density modestly favours the constraint layer) and substantially exceeds it on Transition1x and Taxi, where clamping collapses the joint distribution (coverage 0.200.20 and 0.700.70 against 0.050.05; Appendix J). 5.2 RQ2: the effect persists across the architectures we test Table 2: Architecture generality (RQ2). Margin KS ↓ (n=10n=10, mean ± std) for CTGAN++CL, TVAE++CL, the constraint layer on our MLP generator (C-DGM), and FSG-LTN-GAN. All are 100%100\% valid. The constraint layer misses the distribution of the constraint margin on all three generator families. Dataset CTGAN++CL TVAE++CL C-DGM (MLP) FSG-LTN-GAN Alchemy 0.674±0.0360.674± 0.036 0.651±0.0150.651± 0.015 1.000±0.0001.000± 0.000 0.040±0.0070.040± 0.007 tmQM 0.579±0.0520.579± 0.052 0.517±0.0110.517± 0.011 1.000±0.0001.000± 0.000 0.045±0.0100.045± 0.010 Transition1x 0.592±0.0400.592± 0.040 0.510±0.0530.510± 0.053 0.999±0.0000.999± 0.000 0.065±0.0100.065± 0.010 Taxi 0.419±0.0240.419± 0.024 0.348±0.0290.348± 0.029 0.574±0.0050.574± 0.005 0.099±0.0100.099± 0.010 Table 2 applies the constraint layer to the two most-cited tabular generators, CTGAN and TVAE. Both reach 100%100\% validity but leave margin KS at 0.350.35 to 0.670.67, against 0.040.04 to 0.100.10 for FSG-LTN-GAN. The in-loop constraint layer on our MLP generator is worse still (0.570.57 to 1.001.00), consistent with the in-loop clamp removing the generator’s incentive to place the sub-resolution margin. The method transfers across architectures the same way the failure does. Because the chart is a change of coordinates on the data, an unmodified CTGAN or TVAE trained in chart coordinates and decoded through φ is exactly valid and recovers the margin (Appendix J, Table 10). 5.3 RQ3: the hybrid matches or exceeds the constraint layer on the Stoian et al. (2024) benchmark Table 3: The constraint-layer benchmark of 35 (RQ3). Margin KS ↓ , n=10n=10, mean ± std, all methods 100%100\% valid. RmaxR_ is the dataset’s largest per-constraint R. The hybrid outperforms the constraint layer (paired Wilcoxon) on faults (p=0.002p=0.002, the high-R case), url (p=0.03p=0.03), and wids (p=0.004p=0.004), ties on heloc, lcld, news, and never underperforms it. FSG-all is worse than the constraint layer everywhere but faults, showing the selective in-loop clamp is necessary. Dataset RmaxR_ C-DGM FSG-all hybrid faults 1.8×1031.8× 10^3 0.745±0.0140.745± 0.014 0.117±0.0080.117± 0.008 0.117±0.0080.117± 0.008 heloc 22 0.156±0.0120.156± 0.012 0.208±0.0050.208± 0.005 0.158±0.0140.158± 0.014 lcld 11 0.131±0.0080.131± 0.008 0.657±0.0040.657± 0.004 0.133±0.0070.133± 0.007 url 11 0.159±0.0100.159± 0.010 0.196±0.0180.196± 0.018 0.144±0.0210.144± 0.021 news 22 0.294±0.0080.294± 0.008 0.318±0.0070.318± 0.007 0.297±0.0070.297± 0.007 wids 44 0.140±0.0070.140± 0.007 0.319±0.0040.319± 0.004 0.133±0.0080.133± 0.008 Does the method help on the constraint layer’s own benchmark? Yes, wherever R is high. Table 3 reports all six datasets of 35, predominantly low-R tabular-ML tasks where clamping is already adequate. On the one high-R dataset, faults (bounding-box orderings on large sensor coordinates, Rmax≈1.8×103R_ ≈ 1.8× 10^3), the hybrid charts every bounded variable, coinciding with FSG-all, and improves (p=0.002p=0.002, all ten seeds). On the low-R remainder the hybrid clamps and matches the constraint layer, for three wins, three ties, and zero losses across the benchmark, all at 100%100\% validity. The FSG-all ablation, by contrast, is worse than the constraint layer everywhere but faults, since charting a point-mass or sub-resolution margin smears it. The hybrid’s per-constraint selection (Section 4.2) is therefore necessary, and R predicts which datasets it helps before training. The selection is robust to its two thresholds. Only extreme grid corners that chart most of wids’ constraints erode that dataset’s win (Appendix J, Table 15). 5.4 RQ4: predicate grounding does not reach high-R constraints Corollary 4 bounds a predicate grounding’s usable-gradient fraction at Θ(1/R) (1/R). We test three placements of the same well-scaled predicate (the generator loss, i.e. G-LTN-GAN; discriminator re-weighting; discriminator-feature augmentation). We score the per-ordering satisfaction fraction, averaged over the dataset’s orderings (n=10n=10, mean ± std): 0.58±0.020.58± 0.02, 0.51±0.030.51± 0.03, 0.14±0.080.14± 0.08 on Alchemy, where a free generator with no mechanism scores 0.49±0.030.49± 0.03, and 0.59±0.030.59± 0.03, 0.55±0.080.55± 0.08, 0.29±0.070.29± 0.07 on tmQM (free generator 0.53±0.080.53± 0.08), against 1.000±0.0001.000± 0.000 for function-symbol grounding. No placement helps by more than 0.090.09. The failure tracks the coordinates, not the placement (Table 16). 6 Related Work Neuro-symbolic generation and LTNs. Logic Tensor Networks train by maximizing grounded satisfaction (3), and related work realises logic as a differentiable loss (38; 11). All are predicate-style soft constraints (RQ4). Hard-constraint output layers guarantee specific fragments (2; 15; 12). Constrained tabular generation. The constraint layer of 35, our primary baseline, guarantees linear inequalities by clamping. GOGGLE (22) injects only simple correlations, and CTGAN and TVAE (39) give no guarantees. Closest are constrained adversarial networks (8) and a non-convex constraint layer (36); Appendix A surveys the rest and our function symbols’ antecedents (9; 5). 7 Discussion and Conclusions In an LTN-GAN the choice of grounding is decisive. Predicate grounding cannot reach high-resolution constraints, and clamping is valid but collapses the margin distribution, invisibly to standard metrics and predictably from R. Function-symbol grounding recovers that distribution with exact validity, outperforming the constraint layer on four high-resolution datasets, and the hybrid at least matches it on the benchmark of 35, with the same advantage in conditional inverse design (Appendix J). Scope and limitations. Our charts cover the linear fragment, where Fourier-Motzkin gives the admissible intervals. 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Appendix A Extended Related Work This appendix expands Section 6. The body cites only the most directly related work. Logic as a soft training signal. Beyond Logic Tensor Networks (3; 34), many methods inject logic as a differentiable loss. The semantic loss (38) and its pseudo-semantic extension for autoregressive models (1) penalise probability mass on violating assignments. DL2 (11), straight-through estimators (41), distribution-aware objectives (25), and bilevel formulations (21) optimise logical losses directly, and VAEL (26) couples a variational autoencoder with probabilistic logic. These all inject logic as a soft satisfaction signal rather than a hard guarantee. We do not evaluate each of them on high-resolution structural constraints, because the obstruction we identify is a property of soft satisfaction itself. In the ambient coordinates, a structural margin predicate has usable gradient on a Θ(1/R) (1/R) fraction of samples (Corollary 4), and RQ4 confirms this for the three predicate placements we test. Hard constraints by construction. A second family of methods guarantees satisfaction, each for a specific class of constraints. Semantic Probabilistic Layers (2), MultiplexNet (15), and coherent hierarchical classifiers (12) cover classes of logical constraints, and physics-informed networks enforce differential constraints (19; 24). For linear constraints on tabular data, the constraint layer of 35 clamps onto the feasible polytope, recently extended to quantifier-free non-convex constraints (36). We share the exact-validity goal but, rather than clamping, ground the constraint as a change of coordinates that can also preserve the distribution of the constraint margin. Constrained generative models. Many domains have built constraints into generators. Constrained adversarial networks enforce logical requirements during training (8; 16; 6), and related mechanisms appear in interactive image editing (13), physical fields (14; 43), decision diagrams (40), parametric CAD sketches (33; 29), combinatorial design (10), sequence optimisation (23; 42), and visual scene generation (30). Almost all of them enforce the constraints by penalty, rejection, or post-hoc projection. Function-symbol grounding instead reparameterises the output space, so the constraints hold by construction. Monotone link functions that enforce order and positivity in regression networks go back to 9, and probabilistic-programming samplers routinely transform positive, interval, and ordered parameters to unconstrained space before sampling (5). We use the same links, but in a new role. They ground logical axioms, they are assembled along the Fourier-Motzkin order so that any satisfiable linear system can be charted, and the discriminator is trained in the transformed coordinates. Modelling a difference rather than the levels that form it also echoes Δ -machine learning for chemical energies (31), a supervised antecedent of the margin-form baseline of Appendix J. Rejection sampling and chance-constrained formulations. Two further strategies deserve direct comparison. Rejection sampling draws from an unconstrained generator and keeps only the valid samples. It recovers the correct conditional distribution in principle, but its acceptance rate is the unconstrained generator’s own validity, which is exactly what collapses in the high-R regime (that validity is 0.050.05 on Alchemy, Table 5, and decreases as R grows), and for definitional identities it is zero outright. By Corollary 3 an absolutely continuous generator satisfies an identity with probability 00, so no rejection budget suffices. Chance-constrained formulations (23) require constraints to hold with a prescribed probability rather than on every sample. They operate in the same soft-satisfaction regime as predicate grounding and inherit its high-R gradient obstruction (Corollary 4). Function-symbol grounding avoids both: validity is guaranteed, no rejection loop is needed, and the margin is learned at unit scale. Tabular data generation. For tabular data, adversarial and variational generators (CTGAN and TVAE (39), CTAB-GAN (44)), score-based and diffusion models (STaSy (17), TabDDPM (18)), and relational-structure models (GOGGLE (22)) improve marginal and dependency fidelity but provide no validity guarantees. Database-constraint enforcement has also been added to GAN-based synthesis (20). The constraint layer (35) and our function-symbol grounding both add hard guarantees, but only function-symbol grounding recovers the distribution of the constraint margin for high-resolution constraints. Appendix B The Hybrid Procedure tcc⊳ ⊳ algorithm2e[H] constraints Π ; data D; thresholds τs=0.9 _s=0.9 (satisfaction), τd=0.2 _d=0.2 (discreteness) LTN-GAN generator with exact validity on Π 1: reduce Π to per-variable interval bounds Fix a variable order; by Fourier-Motzkin elimination, write each xix_i’s bounds ℓi,ui _i,u_i as maxima and minima of linear functions of the strictly earlier variables 2: choose a grounding per bounded variable (one short pre-run) Train a free generator briefly; measure each bounded variable xix_i’s interval satisfaction sis_i on the generator’s samples and its binding-margin discreteness did_i on D bounded variable xix_i <τs_i< _s and di≤τd_i≤ _d modei←chartmode_i← chart *high-R, continuous modei←clampmode_i← clamp *low-R or discrete 3: train; charted variables’ axioms hold by construction (no satisfaction loss) training step draw noise ζ and set the free coordinates z←Gθ(ζ)z← G_θ(ζ) variable i in Fourier-Motzkin order _i is fixed by an identityxi←∑j<iwijxj+wi0x_i← _j<iw_ijx_j+w_i0 _i is free (highest-order in no axiom)xi←zix_i← z_i =chartmode_i= chart xi←x_i← the link fitting [ℓi,ui][ _i,u_i] (softplus or σ, Section 4; exp , Appendix G) *Sat=1Sat=1 ←min(max(zi,ℓi),ui)x_i← ( (z_i, _i),\,u_i) *in-loop clamp update Gθ,DψG_θ,D_ψ adversarially on the chart features: each charted variable enters as its coordinate ziz_i and each clamped variable as its standardized clamped value, with real samples encoded the same way through φ−1 ^-1 Hybrid function-symbol grounding in an LTN-GAN. Appendix C Proof of Proposition 1 Fourier-Motzkin elimination over Π in the variable order gives, for each xix_i, its lower and upper bounds ℓi,ui _i,u_i as maxima and minima of linear functions of the strictly-earlier variables. Assembling in this order, xix_i’s bounds depend only on finalized variables; if Π is satisfiable the interval [ℓi,ui][ _i,u_i] is non-empty (Fourier-Motzkin completeness). The interior placement also requires the open interval to be non-empty, ℓi<ui _i<u_i: if Π pinches ℓi=ui _i=u_i (an equality implied by the inequalities though not declared as an identity), the variable is routed to the dependent branch below and set to its single admissible value. For a bounded or one-sided variable with a non-degenerate interval, softplus and σ map ℝR into the interior of its admissible interval, so each strict inequality in which xix_i is highest-order holds with positive margin; a dependent variable is set by its identity to the single admissible value ℓi=ui _i=u_i, meeting its two non-strict encoding inequalities with margin 00; and a free variable is highest-order in no constraint, so it imposes nothing. Induction over the order gives φ(z)⊧Π (z) . □ Appendix D The Resolution Ratio as Condition Number Figure 3 gives the intuition behind this appendix. The two regimes differ not in the constraint but in the size of its margin relative to the data scale, and that ratio is what makes the clamp and the predicate succeed or fail. The statements below make this precise. A worked example. Two of our orderings sit at the two ends of the range. Alchemy’s U−U0U-U_0 has R≈9×104R≈ 9× 10^4. The margin’s spread σm _m is about 10510^5 times smaller than the scale σs _s, so after standardization the margin occupies a band of relative width ∼1/R≈10−5 1/R≈ 10^-5, far below what a Lipschitz discriminator can resolve. A free generator that matches every feature’s marginal therefore still violates the orderings on a constant fraction of samples (about half on average; RQ4). Trained in the loop, the constraint layer moves every sample onto U=U0U=U_0, and the distribution of the constraint margin collapses to a point mass at 00. Taxi’s total−faretotal-fare has R≈3.6R≈ 3.6. The margin is about a third of the scale, the discriminator resolves it, the free generator already satisfies the constraint on most samples, the clamp rarely acts, and clamping leaves the distribution largely intact. The hybrid charts the former (high R) and clamps the latter (low R). Figure 3: High versus low resolution ratio R (schematic). The same ordering b>ab>a in two regimes. Top: the data (a,b)(a,b) and the boundary b=ab=a. At low R (left) the feasible margin b−ab-a is a visible fraction of the scale, so a free generator resolves it and the constraint layer’s clamp rarely acts. At high R (right) the margin is tiny relative to the scale and invisible at the data’s magnitude, so every sample lies close to the boundary. Bottom: the margin distribution. At low R the clamp (orange) preserves the distribution of the real constraint margin (grey); at high R the free generator cannot place the sub-resolution margin, so the clamp collapses every sample onto the boundary, a point mass at 00 disjoint from the real distribution. Function-symbol grounding repairs the high-R case by making the margin a unit-scale coordinate (Figure 4). We now prove Corollaries 3 and 4 and Proposition 5. Proof of Corollary 3. Full rank of ∇c∇ c on ℳ=M_= makes 00 a regular value, so by the regular-value theorem ℳ==c−1(0)M_==c^-1(0) is an embedded C1C^1 submanifold of dimension D−pD-p; a submanifold of dimension below the ambient D has Lebesgue measure zero. If ν≪Lebν with density f, then ν(ℳ=)=∫ℳ=fLeb=0ν(M_=)= _M_=f\,d\,Leb=0. Grounding sets the dependent coordinate to the exact value its identity prescribes, so every generated x lies in ℳ=M_= by construction; the generated distribution is then carried by the null set ℳ=M_=, so ν≪̸Lebν , escaping the hypothesis, and c(x)=0c(x)=0 for every sample. □ Proof of Corollary 4. Differentiating, ∂(b−a)σ((b−a)/s)=σ′((b−a)/s)/s=Ps(1−Ps)/s _(b-a)σ\! ((b-a)/s )=σ \! ((b-a)/s )/s=P_s(1-P_s)/s. The logistic derivative σ′(t)=σ(t)(1−σ(t))σ (t)=σ(t)\,(1-σ(t)) is Θ(1) (1) on any fixed band |t|≤c|t|≤ c (peak σ′(0)=14σ (0)= 14) and decays as e−|t|e^-|t| for |t|≫1|t| 1, so the gradient is Θ(1/s) (1/s) exactly for |b−a|≲s|b-a| s and exponentially suppressed otherwise. With b−ab-a at scale σs=Rσm _s=R\, _m and density Θ(1/σs) (1/ _s) near 00, the probability mass on the width-Θ(s)=Θ(σm) (s)= ( _m) active band is Θ(σm/σs)=Θ(1/R) ( _m/ _s)= (1/R). □ Proof of Proposition 5. (i) On the ordering ϕφ, the clamp leaves a satisfying sample’s margin unchanged and moves each violator to the boundary value ε , giving the mixture CDF FCL(t)=v 1[t≥ε]+(1−v)F+(t)F_CL(t)=v\,1[t≥ ]+(1-v)\,F^+(t), with F+F^+ the CDF of ν+ν^+. Two evaluations bound KS=supt|FCL(t)−F(t)|KS= _t|F_CL(t)-F(t)|. At t=εt= , below the real support, FCL(ε)−F(ε)≥vF_CL( )-F( )≥ v. At t=qαt=q_α, the real margin’s α-quantile (so qα=Θ(σm)q_α= ( _m)), F(qα)−FCL(qα)≥α−v−(1−v)F+(qα)F(q_α)-F_CL(q_α)≥α-v-(1-v)\,F^+(q_α), and under Corollary 4’s density hypothesis the generator places mass Θ(qα/(Rσm))=Θ(1/R) (q_α/(R\, _m) )= (1/R) in (0,qα](0,q_α], so (1−v)F+(qα)=Θ(1/R)(1-v)\,F^+(q_α)= (1/R). Taking α→1α→ 1, KS≥max(v, 1−v−Θ(1/R))≥max(v,1−v)−Θ(1/R)≥12−Θ(1/R)KS≥ (v,\,1-v- (1/R) )≥ (v,1-v)- (1/R)≥ 12- (1/R), and the first evaluation alone gives KS→1KS→ 1 as v→1v→ 1. With several constraints the statement reads per ordering through its own clamp step; empirically the atom sits at the boundary on every dataset (Figure 2). (i) Each link is a smooth bijection onto its admissible interval with smooth inverse: softplus maps ℝR onto (0,∞)(0,∞) with inverse softplus−1(y)=log(ey−1)softplus^-1(y)= (e^y-1); the affine logistic map ℓ+(u−ℓ)σ(⋅) +(u- )\,σ(·) maps ℝR onto (ℓ,u)( ,u) with inverse the scaled logit z=logt1−tz= t1-t at t=(x−ℓ)/(u−ℓ)t=(x- )/(u- ); the heavy-tailed exp link maps ℝR onto (ℓ,∞)( ,∞) with inverse log(x−ℓ) (x- ); a free coordinate is the identity. Assembling in the Fourier-Motzkin order, the bounds of xix_i read only x<ix_<i (Appendix C), so the inverse is computed coordinate-wise. Each ziz_i is the link inverse of xix_i at the bounds set by x<ix_<i, defined and smooth exactly when every constrained xix_i is strictly interior, i.e. on the relative interior of ℳM (dependent coordinates are recovered by their identities and contribute no coordinate). Induction along the order gives φ∘φ−1=id ^-1=id on relintℳrelint\,M and φ−1∘φ=id ^-1 =id on ℝdR^d, and interior placement (Appendix C) means φ places no probability on the boundary. Realizability: given any margin distribution μ on (0,∞)(0,∞), the coordinate distribution softplus#−1μsoftplus^-1_\#μ pushes forward under softplus to exactly μ. When several reduced constraints bound the same variable, ℓi _i and uiu_i are maxima and minima of linear forms, hence piecewise-linear. φ and φ−1 ^-1 are then smooth off the measure-zero set where the active bound switches and remain continuous bijections, so the validity, boundary, and realizability claims are unaffected. □ Figure 4 shows the mechanism of Remark 2 in pictures. The change of coordinates that gives Θ(1) (1) conditioning turns the sub-resolution margin of Figure 3 into a unit-scale coordinate the GAN can learn, then decodes it back to valid samples with the distribution of the constraint margin recovered. Figure 4: How function-symbol grounding repairs the high-R collapse (schematic). The same high-R ordering margin as in Figure 3; top row shows the margin distribution at each stage, bottom row the corresponding (a,b)(a,b) geometry. (1) In ambient space the margin b−ab-a is sub-resolution, so the constraint layer clamps it to a point mass at 00. (2) The chart coordinate z=φ−1(x)z= ^-1(x) standardizes the margin to unit scale, where the discriminator resolves it and a standard GAN learns the distribution of the real constraint margin (generator in blue matches real in grey). (3) Decoding x=φ(z)=ℓ+softplus(z)x= (z)= +softplus(z) carries that distribution back to the ambient margin. (4) Geometrically the clamp moves every violating sample onto the boundary b=ab=a (margin 00, the point mass of panel 1). (5) The function symbol b=a+softplus(z)b=a+softplus(z) is positive for every generator output z, so b>ab>a holds by construction (Sat(KB)=1Sat(KB)=1); it is smooth and monotone, so matching the distribution of z matches the margin’s distribution, and asymptotically linear, so generator tails do not diverge. (6) Our decoded samples therefore land inside the feasible region b>ab>a, matching the real distribution with exact validity. Appendix E Constraint-Layer Code Corrections and Evaluation Protocol We run the constraint layer from the code released by 35. For genuine 100%100\% validity (a fair margin comparison) we correct a latent bug in its Fourier-Motzkin reduction, which keeps a stale term for a variable whose coefficients cancel (triggered by equalities encoded as two inequalities), and run the layer in double precision with the strict-inequality ε set above one unit in the last place (its default ε=10−12 =10^-12 underflows here, falsely reporting 0%0\% valid). Our decoder runs in double precision on Taxi, nycflights13, and the RQ3 benchmark, and in single precision on the chemistry sets, whose reference data are single precision. These changes only help the constraint layer, and the distortion it induces is unchanged. All generators are sampled in eval mode. The metrics (KS and satisfaction fractions) cannot be improved by shrinking magnitudes. Margins are scored against a fixed reference sample of the real data. On the four high-resolution datasets and nycflights13, which ship as single files with no canonical split, this is a 20,00020,000-row sample drawn once with a fixed seed from the same data the generators train on. On the RQ3 benchmark it is the held-out validation and test splits. Every method in a comparison is scored against the same reference. The grounding link (softplus or exp ) is set by the fixed dynamic-range rule of Appendix G. A definitional identity is scored satisfied when its residual is within a per-dataset tolerance in raw units (0.050.05 on Alchemy, 10−310^-3 on tmQM, 0.50.5 on nycflights13). Corollary 3 concerns exact satisfaction. The tolerance is why unconstrained validity is small but nonzero on the chemistry sets, while the wider-scaled nycflights13 identity is almost never met and its validity is 0.0000.000. Per-dataset R, sizes, constraints, and full per-method results with standard deviations are given in Appendices F to J. Appendix F Datasets, Constraints, and Resolution Ratios Table 4 summarizes the four real high-resolution datasets used in RQ1, RQ2, and RQ4. Each property is standardized to zero mean and unit variance before training. The resolution ratio R=σs/σmR= _s/ _m (Section 3) is computed on the raw data before training. The constraint-layer benchmark of 35 used in RQ3 is described in Appendix I. Table 4: The four real high-resolution datasets. N rows, D properties; R is the resolution ratio of each evaluated ordering margin. Dataset N D Evaluated margin(s) R Alchemy 202,579 12 U−U0U-U_0; H−UH-U; H−GH-G 9×1049×10^4; 7×1067×10^6; 3×1043×10^4 tmQM 108,541 8 Eelec−EtotE_elec-E_tot 2.3×1042.3×10^4 Transition1x 10,073 3 ETS−EreactE_TS-E_react; ETS−EprodE_TS-E_prod 1.0×1031.0×10^3; 8.1×1028.1×10^2 Taxi 50,000 6 duration; total−faretotal-fare 1.1×1031.1×10^3; 3.63.6 The full structural constraint set per dataset is as follows. Alchemy (7), 12 molecular properties (mu, alpha, homo, lumo, gap, r2, zpve, U0, U, H, G, Cv, following QM9 naming): positivity of mu (dipole moment), gap, and zpve (zero-point vibrational energy); the algebraic identity lumo=homo+gaplumo=homo+gap over the frontier-orbital energies; the thermochemical orderings U0<U<HU_0<U<H and G<HG<H, where U0U_0 and U are the internal energies at 00 K and 298298 K, H the enthalpy, and G the free energy. tmQM (4), 8 properties of transition-metal complexes: the high-R dispersion ordering Etot<EelecE_tot<E_elec; positivity of dipole, gap, and polarizability; the low-R identity LUMO=HOMO+gapLUMO=HOMO+gap. Transition1x (32), reaction energy profiles: the transition state is the maximum, ETS>EreactE_TS>E_react and ETS>EprodE_TS>E_prod. (0.08%0.08\% of raw rows violate ETS>EprodE_TS>E_prod, near-barrierless reverse reactions, and are retained, so the real reference margin carries that small negative mass.) Taxi (28), NYC yellow-cab records (2024-01): the duration ordering dropoff>pickupdropoff>pickup on absolute second-scale timestamps (high R); total>faretotal>fare (low R); positivity of distance and fare. Appendix G Architecture and Training Protocol All methods share one generator architecture so that only the constraint mechanism differs. The generator is a multilayer perceptron with latent dimension 64 and three hidden layers of width 256 with LeakyReLU(0.20.2) and dropout 0.10.1, and batch normalization on the two inner layers, followed by a linear map to the property dimension. Weights use Kaiming-uniform initialization. The discriminator is a multilayer perceptron D→256→128→1D→ 256→ 128→ 1 with LeakyReLU(0.20.2), dropout 0.10.1, and a sigmoid output. Both train with Adam (learning rate 2×10−42×10^-4, β=(0.5,0.999)β=(0.5,0.999)), batch size 256, binary cross-entropy adversarial loss with label smoothing (targets 0.90.9 real, 0.10.1 generated), for 1000 steps. Two numerical safeguards apply throughout. The decoder floors each charted margin at 10−610^-6 in raw units, and the encoder clips the real chart coordinates at their 0.50.5 and 99.599.5 percentiles before standardization. Both act only in a thin boundary layer, and neither affects validity. The analysis of Section 4 concerns the exact map. All experiments use ten seeds (0 to 9, except the Alchemy and Transition1x inverse-design runs, which use 10 to 19). For function-symbol grounding the same generator emits the free coordinates z and the grounding map φ assembles the sample (Section 4). No satisfaction loss is added because charted axioms hold by construction. The constraint layer runs on the identical architecture (Appendix E). The grounding link is softplus by default and exp (a multiplicative increment) for heavy-tailed margins. This is the method’s only added hyperparameter beyond the hybrid’s two thresholds, and it was set once per dataset by a fixed criterion. The criterion measures each charted margin’s dynamic range as the ratio q99/q50q_99/q_50 of its positive part on the training data, and selects the exp link when any charted margin reaches q99/q50≥10q_99/q_50≥ 10. Across the eleven datasets in this paper it selects exp exactly once, on faults, whose bounding-box margins have q99/q50=18.0q_99/q_50=18.0 and 13.013.0. It selects softplus everywhere else (every charted margin outside faults has q99/q50≤5.1q_99/q_50≤ 5.1). The released code records the resulting per-dataset choice. Appendix H Full Per-Method Results (RQ1 and RQ2) Table 5 extends Table 1 with all baselines and their validities; Table 6 extends Table 2 with the raw unconstrained architectures, whose validities are given in its caption. The paired Wilcoxon test for FSG-LTN-GAN versus C-DGM returns p=0.00195p=0.00195 on every dataset, the minimum attainable at n=10n=10. In Table 5, the projection row coincides with the unconstrained row on tmQM for the reason given in Table 6’s caption. Moving violators to the boundary leaves the KS supremum unchanged when the reference margin lies above it. Table 5: Full RQ1 results, n=10n=10 seeds, mean ± std. Validity is the fraction satisfying all constraints. Margin KS is the two-sample KS between the generated margins and a fixed real reference sample (lower is better). C-DGM and FSG-LTN-GAN are both exactly valid. Only FSG-LTN-GAN recovers the distribution of the constraint margin. Dataset Method Validity Margin KS ↓ Alchemy unconstrained GAN 0.053±0.0210.053± 0.021 0.539±0.0140.539± 0.014 projection 1.000±0.0001.000± 0.000 0.664±0.0260.664± 0.026 C-DGM (clamp) 1.000±0.0001.000± 0.000 1.000±0.0001.000± 0.000 FSG-LTN-GAN 1.000±0.0001.000± 0.000 0.040±0.0070.040± 0.007 tmQM unconstrained GAN 0.229±0.0460.229± 0.046 0.575±0.0370.575± 0.037 projection 1.000±0.0001.000± 0.000 0.575±0.0370.575± 0.037 C-DGM (clamp) 1.000±0.0001.000± 0.000 1.000±0.0001.000± 0.000 FSG-LTN-GAN 1.000±0.0001.000± 0.000 0.045±0.0100.045± 0.010 Transition1x unconstrained GAN 0.293±0.3840.293± 0.384 0.833±0.1400.833± 0.140 projection 1.000±0.0001.000± 0.000 0.881±0.0930.881± 0.093 C-DGM (clamp) 1.000±0.0001.000± 0.000 0.999±0.0000.999± 0.000 FSG-LTN-GAN 1.000±0.0001.000± 0.000 0.065±0.0100.065± 0.010 Taxi unconstrained GAN 0.253±0.0870.253± 0.087 0.346±0.0420.346± 0.042 projection 0.483±0.1120.483± 0.112 0.345±0.0420.345± 0.042 C-DGM (clamp) 1.000±0.0001.000± 0.000 0.574±0.0050.574± 0.005 FSG-LTN-GAN 1.000±0.0001.000± 0.000 0.099±0.0100.099± 0.010 Table 6: Full RQ2 (margin KS by architecture, n=10n=10, mean ± std). Raw CTGAN/TVAE validity is below 3%3\% on the chemistry sets and 3030 to 46%46\% on Transition1x and Taxi. The ++CL rows, C-DGM, and FSG-LTN-GAN all reach 100%100\% validity. Only FSG-LTN-GAN recovers the distribution of the constraint margin across the architectures we test. On tmQM and Taxi the raw and ++CL margin-KS entries coincide exactly. This is forced rather than copied, because the clamp maps the violating mass to the boundary point mass at ε while the reference margin lies above ε , so the KS supremum, attained at ε , has the same value before and after. Validity and moment error do move. On Alchemy and Transition1x the chained orderings share endpoints, so clamping shifts the evaluated margins and the KS. Method Alchemy tmQM Transition1x Taxi CTGAN (raw) 0.596±0.0290.596± 0.029 0.579±0.0520.579± 0.052 0.569±0.0570.569± 0.057 0.419±0.0240.419± 0.024 CTGAN + CL 0.674±0.0360.674± 0.036 0.579±0.0520.579± 0.052 0.592±0.0400.592± 0.040 0.419±0.0240.419± 0.024 TVAE (raw) 0.528±0.0120.528± 0.012 0.517±0.0110.517± 0.011 0.478±0.0470.478± 0.047 0.348±0.0290.348± 0.029 TVAE + CL 0.651±0.0150.651± 0.015 0.517±0.0110.517± 0.011 0.510±0.0530.510± 0.053 0.348±0.0290.348± 0.029 C-DGM (CL, MLP) 1.000±0.0001.000± 0.000 1.000±0.0001.000± 0.000 0.999±0.0000.999± 0.000 0.574±0.0050.574± 0.005 FSG-LTN-GAN 0.040±0.0070.040± 0.007 0.045±0.0100.045± 0.010 0.065±0.0100.065± 0.010 0.099±0.0100.099± 0.010 Appendix I The Constraint-Layer Benchmark, Full (RQ3) Table 7 extends Table 3 with each dataset’s constraint count and charted count. KS is averaged over each dataset’s constraints. C-DGM, FSG-all, and the hybrid all reach 100%100\% validity; unconstrained validity ranges from 0.0050.005 (wids) to 0.7360.736 (url). The hybrid charts the constraints its pre-run selects (free-generator satisfaction below 0.90.9, margin not discrete) and clamps the rest. On heloc and lcld it charts nothing and reduces to the constraint layer. On url, news, and wids it charts one or two variables, with smaller but significant gains on url and wids (p=0.03p=0.03 and 0.0040.004) and a statistical tie on news (heloc and lcld are ties as well). On faults (whose high-R margin is the bounding-box ordering Ymax>YminY_ >Y_ ) it charts every variable and improves substantially (paired Wilcoxon p=0.002p=0.002, all ten seeds). The FSG-all ablation, which charts every variable, is worse than the constraint layer on every dataset but faults. Table 7: Full RQ3 results on the benchmark of 35, n=10n=10 seeds, mean ± std, margin KS. “charted” is the number of variables the hybrid charts as function symbols; bold marks where the hybrid significantly outperforms the constraint layer (paired Wilcoxon p<0.05p<0.05). Dataset |Π|| | charted RmaxR_ C-DGM FSG-all hybrid faults 4 4 1.8×1031.8×10^3 0.745±0.0140.745± 0.014 0.117±0.0080.117± 0.008 0.117±0.0080.117± 0.008 heloc 7 0 22 0.156±0.0120.156± 0.012 0.208±0.0050.208± 0.005 0.158±0.0140.158± 0.014 lcld 4 0 11 0.131±0.0080.131± 0.008 0.657±0.0040.657± 0.004 0.133±0.0070.133± 0.007 url 8 1 11 0.159±0.0100.159± 0.010 0.196±0.0180.196± 0.018 0.144±0.0210.144± 0.021 news 5 2 22 0.294±0.0080.294± 0.008 0.318±0.0070.318± 0.007 0.297±0.0070.297± 0.007 wids 31 1 44 0.140±0.0070.140± 0.007 0.319±0.0040.319± 0.004 0.133±0.0080.133± 0.008 Appendix J Additional Analyses Density and coverage. Table 8 reports density and coverage (27), the constraint layer’s own realism metrics, computed on standardized features and shown alongside margin KS. Table 8: Density and coverage (27) (n=10n=10, mean ± std), with margin KS for reference. Higher is better for density and coverage, lower for KS. The better method per column is in bold. Dataset Method Density Coverage Margin KS ↓ Alchemy C-DGM 0.923±0.0310.923± 0.031 0.619±0.0140.619± 0.014 1.000±0.0001.000± 0.000 FSG-LTN-GAN 0.780±0.0340.780± 0.034 0.636±0.0170.636± 0.017 0.040±0.0070.040± 0.007 tmQM C-DGM 0.839±0.0150.839± 0.015 0.752±0.0090.752± 0.009 1.000±0.0001.000± 0.000 FSG-LTN-GAN 0.801±0.0220.801± 0.022 0.735±0.0110.735± 0.011 0.045±0.0100.045± 0.010 Transition1x C-DGM 0.200±0.0390.200± 0.039 0.050±0.0050.050± 0.005 0.999±0.0000.999± 0.000 FSG-LTN-GAN 0.248±0.0300.248± 0.030 0.203±0.0090.203± 0.009 0.065±0.0100.065± 0.010 Taxi C-DGM 0.066±0.0080.066± 0.008 0.052±0.0040.052± 0.004 0.574±0.0050.574± 0.005 FSG-LTN-GAN 0.870±0.0130.870± 0.013 0.704±0.0150.704± 0.015 0.099±0.0100.099± 0.010 The failure is not a variable-ordering artifact. A constraint layer must fix a Fourier-Motzkin elimination order. Table 9 runs the constraint layer on Transition1x under every dependency-valid ordering: all stay near 1.01.0 margin KS, so this failure is intrinsic to clamping. Function-symbol grounding, which fixes one ordering, is at 0.0650.065. All configurations are 100%100\% valid. (The two apex-first orders reduce to the same triangular program and give identical runs.) Table 9: Constraint-layer robustness to variable ordering (Transition1x, n=10n=10, mean ± std, margin KS). Bracketed lists are the Fourier-Motzkin variable elimination orders. Configuration Margin KS ↓ C-DGM, natural order [0,1,2][0,1,2] 0.999±0.0000.999± 0.000 C-DGM, apex-first [1,0,2][1,0,2] 0.999±0.0010.999± 0.001 C-DGM, apex-first [1,2,0][1,2,0] 0.999±0.0010.999± 0.001 C-DGM, apex-last [0,2,1][0,2,1] 0.903±0.0490.903± 0.049 FSG-LTN-GAN 0.065±0.0100.065± 0.010 Other backbones: CTGAN and TVAE in the chart. Table 2 showed the failure on CTGAN and TVAE. The chart supplies the fix for them as well. Because function-symbol grounding is a change of coordinates on the data, any tabular generator can be trained in the chart: encode the training data by φ−1 ^-1, fit the unmodified backbone with its package-default hyperparameters at 100100 epochs (the RQ2 protocol), and decode its samples through φ . Table 10 applies this recipe to CTGAN and TVAE with no architectural change: both become exactly valid and recover the margin, with margin KS 3.63.6 to 8.6×8.6× below their constraint-layer counterparts (paired Wilcoxon p=0.002p=0.002 on every dataset–backbone pair). One backbone-specific effect persists. TVAE’s latent-variance shrinkage, applied in chart coordinates, propagates through the free base variable and inflates ambient per-property moment error on tmQM and Transition1x (CTGAN in the chart does not show this, and margin KS is unaffected either way). Table 10: The method transfers to other generator families (margin KS ↓ , n=10n=10, mean ± std). “in chart” trains the unmodified backbone on φ−1 ^-1-encoded data and decodes through φ . Both chart columns are exactly 100%100\% valid. “++CL” columns from Table 2. Dataset CTGAN++CL CTGAN in chart TVAE++CL TVAE in chart Alchemy 0.674±0.0360.674± 0.036 0.105±0.0410.105± 0.041 0.651±0.0150.651± 0.015 0.127±0.0050.127± 0.005 tmQM 0.579±0.0520.579± 0.052 0.109±0.0610.109± 0.061 0.517±0.0110.517± 0.011 0.060±0.0140.060± 0.014 Transition1x 0.592±0.0400.592± 0.040 0.089±0.0190.089± 0.019 0.510±0.0530.510± 0.053 0.141±0.0370.141± 0.037 Taxi 0.419±0.0240.419± 0.024 0.086±0.0160.086± 0.016 0.348±0.0290.348± 0.029 0.064±0.0140.064± 0.014 Flight records (nycflights13). The four main datasets and the RQ3 benchmark were each chosen by the authors of one of the two papers being compared. We therefore ran a fifth dataset that neither paper had used, the nycflights13 flight records (327,346327,346 rows after cleaning, subsampled to 50,00050,000; six properties), with the ordering arrival>departurearrival>departure on absolute timestamps spanning a year, the exact identity dep=sched+delaydep=sched+delay, and positivity of air time and distance. Before training we computed R=3.0×103R=3.0× 10^3 for the ordering from the raw data and recorded the predicted outcome (constraint-layer margin KS near 11, FSG-LTN-GAN below 0.150.15, both exactly valid; unconstrained validity near 00), together with explicit falsification criteria, in a file included in the code release. The sweep then ran once, with ten seeds. Every prediction held (Table 11). The clamp collapses the flight-duration margin to the boundary exactly as on Taxi and the chemistry sets, and the chart recovers it, at equal exact validity. Table 11: Flight records (nycflights13) (n=10n=10, mean ± std). R and the predicted outcome were recorded before training. Method Validity Margin KS ↓ unconstrained GAN 0.000±0.0000.000± 0.000 0.581±0.0440.581± 0.044 projection 0.530±0.0680.530± 0.068 0.541±0.0410.541± 0.041 C-DGM (clamp) 1.000±0.0001.000± 0.000 1.000±0.0001.000± 0.000 FSG-LTN-GAN 1.000±0.0001.000± 0.000 0.086±0.0110.086± 0.011 Where the margin-KS gain comes from. FSG-LTN-GAN differs from the constraint layer in two coupled ways: samples are valid by construction, and the discriminator operates on the chart coordinates z, where every margin is unit scale, while the constraint layer’s discriminator sees the original, ill-scaled coordinates. To separate the two contributions we train FSG-ambient-D: the generator and the chart φ are unchanged, so validity remains exact, but the discriminator receives the decoded, standardized ambient sample instead of z. Table 12 shows margin KS degrades on every dataset (paired Wilcoxon p=0.002p=0.002 each), and the degradation tracks R (largest on the chemistry sets), yet remains well below the clamp’s. Both mechanisms therefore contribute. Generating inside the feasible region rather than clamping onto its boundary already improves the margin, and on three of the four datasets the larger share of the recovery comes from the discriminator operating at unit scale. On Transition1x the shares reverse, with generating inside contributing most of the difference. The two are parts of the same grounding. The chart supplies the coordinates, and the discriminator is best run in them. (Validity is 1.0001.000 in every run except a single tmQM seed at 0.9990.999, a float32 identity-reconstruction round-off in the ambient arm’s training-time decode.) Table 12: Validity by construction versus discriminating in the chart. Margin KS (n=10n=10, mean ± std). FSG-ambient-D keeps the chart generator (validity exact) but shows the discriminator the decoded ambient sample; C-DGM from Table 5 for reference. Dataset FSG (D in chart) FSG-ambient-D C-DGM (clamp) Alchemy 0.040±0.0070.040± 0.007 0.838±0.1540.838± 0.154 1.000±0.0001.000± 0.000 tmQM 0.045±0.0100.045± 0.010 0.737±0.3390.737± 0.339 1.000±0.0001.000± 0.000 Transition1x 0.065±0.0100.065± 0.010 0.267±0.1060.267± 0.106 0.999±0.0000.999± 0.000 Taxi 0.099±0.0100.099± 0.010 0.340±0.1260.340± 0.126 0.574±0.0050.574± 0.005 Margin-form preprocessing is a manual chart. For a single ordering one can instead transform the data: on Taxi, replace dropoff with duration=dropoff−pickupduration=dropoff-pickup, standardize, train the constraint layer with duration>0duration>0, and decode afterwards. Because the margin becomes its own column, its resolution ratio drops to R=1R=1 (from 1.1×1031.1× 10^3), and the clamp then largely preserves the margin distribution. Margin KS falls 0.574→0.134±0.0200.574→ 0.134± 0.020 (duration KS 1.000→0.118±0.0331.000→ 0.118± 0.033) at exact validity (Table 13). This is further evidence for the coordinate explanation. The preprocessing is function-symbol grounding applied to the data by hand, for one constraint. Three observations make the chart its general form. First, the transform alone confers no validity. The same preprocessing with an unconstrained GAN is 52.1%±5.552.1\%± 5.5 valid (duration can still be generated negative, and the untouched constraints are violated), so a clamp or a chart is still required. Second, rewriting a conjunction into consistent margin form (Alchemy’s U0<U<HU_0<U<H chain together with its lumo identity) must proceed variable by variable in a triangular order, which is exactly the Fourier-Motzkin substitution the chart automates. Third, the chart also standardizes what the discriminator sees for all constraints at once (previous paragraph), which manual rewriting achieves only for the rewritten margins. FSG-LTN-GAN accordingly remains best (0.099±0.0100.099± 0.010; paired Wilcoxon p=0.002p=0.002 against the preprocessed constraint layer, all ten seeds). Table 13: Margin-form preprocessing on Taxi (n=10n=10, mean ± std). Preprocessing the data into margin form is a manual, single-constraint chart: it recovers the clamp’s margin, though not validity (without a constraint mechanism) and not the untransformed margins. Duration KS is the KS of the high-R duration margin alone. Method Validity Margin KS ↓ Duration KS ↓ C-DGM, ambient (Table 5) 1.000±0.0001.000± 0.000 0.574±0.0050.574± 0.005 1.000±0.0001.000± 0.000 margin-form ++ C-DGM 1.000±0.0001.000± 0.000 0.134±0.0200.134± 0.020 0.118±0.0330.118± 0.033 margin-form ++ unconstrained GAN 0.521±0.0550.521± 0.055 0.121±0.0150.121± 0.015 0.111±0.0240.111± 0.024 FSG-LTN-GAN 1.000±0.0001.000± 0.000 0.099±0.0100.099± 0.010 0.081±0.0240.081± 0.024 Conditional inverse design. The margin KS matters when a consumer of the samples needs the constrained quantity to be realistic. In conditional inverse design, we condition the generator on a target value t for a designable property (the HOMO–LUMO gap on Alchemy, the reactant energy on Transition1x, the electronic energy on tmQM) and ask for complete property profiles that are (i) valid under the dataset’s constraints, (i) on target, and (i) realistic in their margins. All arms share the conditional architecture (the discriminator sees the sample–target pair). They differ only in the constraint mechanism: none, a predicate-style satisfaction penalty, post-hoc projection, or the chart. In Table 14, function-symbol grounding is the only configuration that delivers all three at once: exact validity (the tmQM chart value is 0.99990.9999 before rounding, a float32 identity round-off), target error at the unconstrained level, and margin KS an order of magnitude below every alternative. The penalty arm shows the cost of soft satisfaction at high R. Pushed toward satisfaction, it misses the target (target error 22 to 5×5× worse) yet still fails validity. Projection attains validity but inherits the unconstrained margins. Table 14: Conditional inverse design (n=10n=10 seeds, mean ± std, 1500 steps). Target error is the mean absolute deviation of the designable property from its conditioning target (standardized units). Margin KS is as in the main text. Only the chart is simultaneously valid, on target, and realistic in its margins. Dataset Mechanism Validity Target err. ↓ Margin KS ↓ Alchemy unconstrained 0.089±0.0500.089± 0.050 0.067±0.0030.067± 0.003 0.629±0.0720.629± 0.072 (target: gap) penalty 0.019±0.0090.019± 0.009 0.349±0.0910.349± 0.091 0.624±0.0350.624± 0.035 projection 0.999±0.0010.999± 0.001 0.067±0.0030.067± 0.003 0.652±0.0500.652± 0.050 chart (FSG) 1.000±0.0001.000± 0.000 0.073±0.0040.073± 0.004 0.049±0.0080.049± 0.008 Transition1x unconstrained 0.662±0.2350.662± 0.235 0.053±0.0040.053± 0.004 0.477±0.1700.477± 0.170 (target: EreactE_react) penalty 0.797±0.0430.797± 0.043 0.124±0.0090.124± 0.009 0.851±0.0310.851± 0.031 projection 1.000±0.0001.000± 0.000 0.053±0.0040.053± 0.004 0.470±0.1740.470± 0.174 chart (FSG) 1.000±0.0001.000± 0.000 0.060±0.0060.060± 0.006 0.049±0.0100.049± 0.010 tmQM unconstrained 0.269±0.0610.269± 0.061 0.065±0.0040.065± 0.004 0.548±0.0420.548± 0.042 (target: EelecE_elec) penalty 0.036±0.0160.036± 0.016 0.253±0.0200.253± 0.020 0.608±0.0510.608± 0.051 projection 1.000±0.0001.000± 0.000 0.065±0.0040.065± 0.004 0.548±0.0420.548± 0.042 chart (FSG) 1.000±0.0001.000± 0.000 0.073±0.0080.073± 0.008 0.038±0.0060.038± 0.006 Sensitivity of the hybrid’s thresholds. The hybrid’s two thresholds are fixed once (τs=0.9 _s=0.9, τd=0.2 _d=0.2) and shared by every dataset in the paper. Sweeping the grid τs∈0.70,0.80,0.85,0.90,0.95×τd∈0.05,0.10,0.20,0.30,0.50 _s∈\0.70,0.80,0.85,0.90,0.95\× _d∈\0.05,0.10,0.20,0.30,0.50\ changes the charted set on 55 to 1919 of the 2525 cells, depending on the dataset, and retraining every distinct alternative set the grid produces (ten seeds each, Table 15) changes no conclusion: every configuration keeps exact validity, alternates move margin KS by at most ≈0.03≈0.03 on the low-R datasets, and every cell near the paper’s setting keeps every win and tie. The one qualification is wids, where single extreme cells that chart seven or more constraints move to KS 0.1430.143 to 0.1520.152, at or slightly above the constraint layer’s 0.1400.140. The one large change shows where the sensitivity lies. On faults, the extreme τd=0.05 _d=0.05 row stops charting the two bounding-box margins (including the high-R one), and the main win shrinks (KS 0.117→0.5480.117→ 0.548, still below the constraint layer’s 0.7450.745). The outcome is sensitive not to the threshold values but to whether the high-R margins are charted, which is what R predicts before training. Table 15: Threshold sensitivity on the RQ3 benchmark (margin KS, n=10n=10, mean ± std). “cells” is how many of the 2525 grid cells select each charted set. The paper’s cell is (τs,τd)=(0.9,0.2)( _s, _d)=(0.9,0.2). Dataset Paper’s charted set (cells) KS Alternative sets (cells) KS faults 4 charted (20) 0.117±0.0080.117± 0.008 2 charted (5) 0.548±0.0060.548± 0.006 heloc none (15) 0.158±0.0140.158± 0.014 1 charted (10) 0.132±0.0080.132± 0.008 lcld none (20) 0.133±0.0070.133± 0.007 1 charted (5) 0.134±0.0070.134± 0.007 url 1 charted (9) 0.144±0.0210.144± 0.021 none (16) 0.150±0.0110.150± 0.011 news 2 charted (6) 0.297±0.0070.297± 0.007 none or 1 charted (19) 0.2920.292 to 0.2960.296 wids 1 charted (6) 0.133±0.0080.133± 0.008 11 distinct sets (19) 0.1280.128 to 0.1520.152 RQ4 in table form. Table 16 restates the RQ4 numbers of Section 5.4. Table 16: RQ4: predicate placements on the high-R orderings (ordering satisfaction, the per-ordering satisfaction fraction averaged over the dataset’s orderings, n=10n=10, mean ± std). “chance” is a free generator with no constraint mechanism. “G-LTN-GAN” places the predicate in the generator loss. “D re-weight” re-weights each sample’s discriminator loss by its predicate satisfaction. “D feature aug.” appends the predicate value to the discriminator’s input. FSG-LTN-GAN is function-symbol grounding. No placement of the predicate moves the ordering more than 0.090.09 above chance, and the discriminator-feature placement falls well below it. Function-symbol grounding satisfies the ordering for every sample. Dataset chance G-LTN-GAN D re-weight D feature aug. FSG-LTN-GAN Alchemy 0.49±0.030.49± 0.03 0.58±0.020.58± 0.02 0.51±0.030.51± 0.03 0.14±0.080.14± 0.08 1.000±0.0001.000± 0.000 tmQM 0.53±0.080.53± 0.08 0.59±0.030.59± 0.03 0.55±0.080.55± 0.08 0.29±0.070.29± 0.07 1.000±0.0001.000± 0.000 Appendix K Code and Data Availability All datasets are publicly available (Appendix F); code, experiment scripts, and the prediction record of Appendix J are available at FSG-LTN-GAN.