Generate in the Chart, Not on the Boundary: Function-Symbol Grounding for Hard Constraints in LTN-GANs

# Generate in the Chart, Not on the Boundary: Function-Symbol Grounding for Hard Constraints in LTN-GANs
Source: [https://arxiv.org/html/2608.21605](https://arxiv.org/html/2608.21605)
Vaishak BelleEmail:[vbelle@ed\.ac\.uk](mailto:[email protected])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 ratioRR, the data’s scale over the margin’s spread, is a diagnostic, computable before training, of which constraints a chosen grounding can learn\. WhenRRis 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 to25×25\\timessmaller than the constraint layer’s, and a hybrid of charting and clamping matches or exceeds the constraint layer on its own benchmark\.

## 1Introduction

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\([39](https://arxiv.org/html/2608.21605#bib.bib13)\), score\-based and diffusion models\([17](https://arxiv.org/html/2608.21605#bib.bib40);[18](https://arxiv.org/html/2608.21605#bib.bib39)\), and relational\-structure models\([22](https://arxiv.org/html/2608.21605#bib.bib14)\)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 from00K \(U0U\_\{0\}\) to room temperature \(UU\) and remain below its enthalpy \(HH\), so thatU0<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 \(LTN\-GAN\)\([37](https://arxiv.org/html/2608.21605#bib.bib2)\)*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\([3](https://arxiv.org/html/2608.21605#bib.bib1)\)\. 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](https://arxiv.org/html/2608.21605#bib.bib4), 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 guarantees100%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 stepU−U0U\-U\_\{0\}can fail the same way\. We call each such amount \(drop\-off minus pick\-up, orU−U0U\-U\_\{0\}\) 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*RR, the data’s scale over the margin’s spread, computed before training\. WhenRRis 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\. WhenRRis large, the margin is invisible at the data’s scale\. For thermochemical energies the stepU−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 reads100%100\\%and every per\-feature statistic still matches the real data \(margin Kolmogorov–Smirnov \(KS\) distance near11; Figure[2](https://arxiv.org/html/2608.21605#S5.F2)\)\. High\-RRconstraints 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 orderingb\>ab\>a, the generator emits a free value foraaand assemblesbbfromaaby 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\-RRcontinuous constraints while low\-RRand discrete margins are better clamped, we use a short pre\-run to decide per constrained variable, yielding ahybridmodel\. We show thatRRacts as a condition number for the constrained density\-estimation problem \(Section[4](https://arxiv.org/html/2608.21605#S4)\)\.

#### 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 ratioRRacts as a condition number that predicts this failure before training\. \(ii\) 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*\. \(iii\) We demonstrate that function\-symbol grounding cuts the margin KS by up to25×25\\timesat equal validity on four real high\-resolution datasets, that the hybrid matches or exceeds the constraint layer on the benchmark of[35](https://arxiv.org/html/2608.21605#bib.bib4), that theRR\-based prediction holds out of sample \(nycflights13\), and that the method transfers unchanged to CTGAN and TVAE backbones\.

## 2Background: 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\([3](https://arxiv.org/html/2608.21605#bib.bib1)\)\. A*grounding*𝒢\\mathcal\{G\}assigns meaning to symbols, and every term becomes a tensor\. Each*predicate*PPbecomes a map𝒢⁡\(P\)\\mathcal\{G\}\(P\)into the truth interval\[0,1\]\[0,1\]that*scores*its arguments\. Each*function symbol*ffof aritykkbecomes a real map𝒢⁡\(f\):ℝD​k→ℝD\\mathcal\{G\}\(f\):\\mathbb\{R\}^\{Dk\}\\\!\\to\\\!\\mathbb\{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 \(a t\-norm for∧\\wedge, its dual for∨\\vee, a fuzzy implication\), and quantifiers become aggregations \(∀\\forallas a generalized mean over errors\)\. The truth value of a closed formula under𝒢\\mathcal\{G\}is its*satisfaction*Sat∈\[0,1\]\\mathrm\{Sat\}\\in\[0,1\], with11fully true\. An*axiom*is a closed formula asserted to hold of the domain, a knowledge baseKB\\mathrm\{KB\}is a finite set of axioms, and learning maximizes their aggregated satisfactionSat⁡\(KB\)\\mathrm\{Sat\}\(\\mathrm\{KB\}\)\.

#### The LTN\-GAN objective\.

A GAN couples a generatorGθ:𝒵→ℝDG\_\{\\theta\}:\\mathcal\{Z\}\\\!\\to\\\!\\mathbb\{R\}^\{D\}, mapping latent noiseζ∼pζ\\zeta\\sim p\_\{\\zeta\}\(a standard Gaussian\) to a sample, and a discriminatorDψ:ℝD→\[0,1\]D\_\{\\psi\}:\\mathbb\{R\}^\{D\}\\\!\\to\\\!\[0,1\], trained to score real samples near11and generated ones near00, whileGθG\_\{\\theta\}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 objectiveℒG=ℒadv​\(Gθ,Dψ\)\+λ⁡\(1−Sat⁡\(KB\)\)\\mathcal\{L\}\_\{G\}=\\mathcal\{L\}\_\{\\mathrm\{adv\}\}\(G\_\{\\theta\},D\_\{\\psi\}\)\+\\lambda\\,\(1\-\\mathrm\{Sat\}\(\\mathrm\{KB\}\)\), whose axioms encode the constraints\.

#### Predicate grounding of a constraint\.

The standard generator\-side LTN\-GAN \(G\-LTN\-GAN\)\([37](https://arxiv.org/html/2608.21605#bib.bib2)\)grounds an ordering axiomb\>ab\>aas a predicate,𝒢⁡\(P\>\)​\(a,b\)=σ⁡\(\(b−a\)/s\)\\mathcal\{G\}\(P\_\{\>\}\)\(a,b\)=\\sigma\\\!\\bigl\(\(b\-a\)/s\\bigr\)withσ\\sigmathe logistic, at a bandss\(a width hyperparameter\), and the termλ⁡\(1−Sat\)\\lambda\(1\-\\mathrm\{Sat\}\)in the objective pushes samples towardb\>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](https://arxiv.org/html/2608.21605#S4)shows this band covers a fractionΘ⁡\(1/R\)\\Theta\(1/R\)of samples, so the usable gradient vanishes asRRgrows \(RQ4\)\. Our method keeps the LTN\-GAN objective but regrounds each structural axiom through a function symbol\. The constrained variable becomes a term that𝒢⁡\(f\)\\mathcal\{G\}\(f\)computes rather than a free output that a predicate scores, soSat=1\\mathrm\{Sat\}=1holds by construction\.*Soft*thus refers to the satisfaction signal, since function\-symbol grounding also uses smooth links yet satisfies the axiom for every output\.

## 3Problem Statement

GeneratorGθG\_\{\\theta\}ζ\\zetaconstraintsΠ\\PiFunction\-symbol groundingφ\\varphiFourier\-Motzkin bounds\[ℓi,ui\]\[\\ell\_\{i\},u\_\{i\}\]e\.g\.xi=ℓi\+softplus⁡\(zi\)x\_\{i\}=\\ell\_\{i\}\+\\mathrm\{softplus\}\(z\_\{i\}\)valid samplexx,Sat⁡\(KB\)=1\\mathrm\{Sat\(KB\)\}=1DiscriminatorDψD\_\{\\psi\}real / fakereal dataxxzzzzφ−1​\(x\)\\varphi^\{\-1\}\(x\)Figure 1:Overview of the FSG\-LTN\-GAN\. The grounding mapφ\\varphiassembles valid samples from the generator’s free coordinateszzwithin the Fourier\-Motzkin bounds ofΠ\\Pi, and the discriminator operates in the chart coordinates\. The hybrid instead clamps low\-RRand discrete variables in the loop \(Appendix[B](https://arxiv.org/html/2608.21605#A2)\)\.LetpXp\_\{X\}be an unknown distribution overX∈ℝDX\\in\\mathbb\{R\}^\{D\}and𝒟\\mathcal\{D\}a dataset ofNNi\.i\.d\. samples\. A*sample*is a vectorx=\(x1,…,xD\)x=\(x\_\{1\},\\dots,x\_\{D\}\)whose scalar componentsxk∈ℝx\_\{k\}\\in\\mathbb\{R\}are its*features*\. A generative model with parametersθ\\theta\(for us the generatorGθG\_\{\\theta\}\) induces the distributionpθp\_\{\\theta\}of its outputGθ​\(ζ\)G\_\{\\theta\}\(\\zeta\),ζ∼pζ\\zeta\\sim p\_\{\\zeta\}, and learning choosesθ\\thetaso thatpθ≈pXp\_\{\\theta\}\\approx p\_\{X\}\. The background knowledge is a finite setΠ\\Piof linear\-inequality axioms over the features\{x1,…,xD\}\\\{x\_\{1\},\\dots,x\_\{D\}\\\}, each of the form∑kwk​xk\+b​⊵​0\\sum\_\{k\}w\_\{k\}x\_\{k\}\+b\\trianglerighteq 0with⊵∈\{≥,\>\}\\trianglerighteq\\in\\\{\\geq,\>\\\}, real coefficientswkw\_\{k\}, and offsetbb, following the formulation of[35](https://arxiv.org/html/2608.21605#bib.bib4)\. A samplex~\\tilde\{x\}*satisfies*ϕ∈Π\\phi\\in\\Piif∑kwk​x~k\+b​⊵​0\\sum\_\{k\}w\_\{k\}\\tilde\{x\}\_\{k\}\+b\\trianglerighteq 0\. A generator is*compliant*\(valid\) if all its samples satisfy all ofΠ\\Pi\. Orderings \(xi\>xjx\_\{i\}\>x\_\{j\}\), positivity \(xi\>0x\_\{i\}\>0\), and definitional identities \(xi=∑jwj​xjx\_\{i\}=\\sum\_\{j\}w\_\{j\}x\_\{j\}, encoded as two inequalities\) are the*structural*fragment we study\.

#### The margin and its distribution\.

Forϕ:∑kwk​xk\+b≥0\\phi:\\sum\_\{k\}w\_\{k\}x\_\{k\}\+b\\geq 0, its*margin*on a sample ismϕ​\(x\)=∑kwk​xk\+bm\_\{\\phi\}\(x\)=\\sum\_\{k\}w\_\{k\}x\_\{k\}\+b, withmϕ≥0m\_\{\\phi\}\\geq 0exactly whenϕ\\phiholds\. 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\}\(m\_\{\\phi\}\\mid m\_\{\\phi\}\\geq 0\)\. 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=std𝒟​\(mϕ\)\\sigma\_\{m\}=\\mathrm\{std\}\_\{\\mathcal\{D\}\}\(m\_\{\\phi\}\)is the standard deviation ofϕ\\phi’s margin over the dataset\. The*scale*σs=maxk:wk≠0std𝒟\(xk\)\\sigma\_\{s\}=\\max\_\{k:\\,w\_\{k\}\\neq 0\}\\mathrm\{std\}\_\{\\mathcal\{D\}\}\(x\_\{k\}\)is the largest standard deviation among the featuresϕ\\phirelates: forϕ:U−U0≥0\\phi:U\-U\_\{0\}\\geq 0it isstd⁡\(U\)\\mathrm\{std\}\(U\)\. Both are computed on the raw, unstandardized data, before training\. The*resolution ratio*ofϕ\\phiisRϕ=σs/σmR\_\{\\phi\}=\\sigma\_\{s\}/\\sigma\_\{m\}\. WhenRϕR\_\{\\phi\}is large the margin is a tiny difference of large near\-equal quantities\. In standardized coordinates it occupies a band of relative width about1/Rϕ1/R\_\{\\phi\}, sub\-resolution to a Lipschitz discriminator, so a*free*generator \(one trained with no constraint mechanism\) that has matched the marginals still violatesϕ\\phion a constant fraction of samples \(RQ4\)\. Section[4](https://arxiv.org/html/2608.21605#S4)castsRϕR\_\{\\phi\}as a scaling condition number for recovering the distribution of the constraint margin\.

#### The constraint layer \(clamping\)\.

C\-DGM\([35](https://arxiv.org/html/2608.21605#bib.bib4)\)appends a differentiable*constraint layer*\(CL\): given a variable ordering, it computes for each variable an admissible interval\[ℓi,ui\]\[\\ell\_\{i\},u\_\{i\}\]\(piecewise\-linear in the already\-set variables, by Fourier\-Motzkin reduction\) and*clamps*the generated value into it,CL​\(x~\)i=min⁡\(max⁡\(x~i,ℓi\),ui\)\\mathrm\{CL\}\(\\tilde\{x\}\)\_\{i\}=\\min\(\\max\(\\tilde\{x\}\_\{i\},\\ell\_\{i\}\),u\_\{i\}\), guaranteeing validity\. A clamp moves a violating sample to the nearest boundary, where the margin is near zero\. Section[4\.1](https://arxiv.org/html/2608.21605#S4.SS1)shows this is exactly where the distribution of the constraint margin is lost, at a rate governed byRR\.

## 4Function\-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 setℳ=\{x:⋀ϕϕ⁡\(x\)\}\\mathcal\{M\}=\\\{x:\\bigwedge\_\{\\phi\}\\phi\(x\)\\\}of a satisfiable linear system is a relatively open convex polytope, diffeomorphic toℝd\\mathbb\{R\}^\{d\}and covered by a single global chart\. The mapφ\\varphibelow is such a chart, with one coordinate per free variable\.

We ground each structural axiom through a function symbol \(Figure[1](https://arxiv.org/html/2608.21605#S3.F1)\)\. The generator emits free termsz∈ℝdz\\in\\mathbb\{R\}^\{d\}\(one per variable not fixed by an identity\), and a grounded mapφ\\varphiassembles the constrained sample by processing the variables in the Fourier\-Motzkin order\. WithΠi\+\\Pi\_\{i\}^\{\+\}\(Πi−\\Pi\_\{i\}^\{\-\}\) the reduced constraints in whichxix\_\{i\}has positive \(negative\) coefficient\([35](https://arxiv.org/html/2608.21605#bib.bib4)\), the admissible interval ofxix\_\{i\}is piecewise\-linear in the already\-assembledx<ix\_\{<i\},

ℓi\(x<i\)=maxϕ∈Πi\+εiϕ\(x<i\),ui\(x<i\)=minϕ∈Πi−εiϕ\(x<i\),εiϕ=−\(∑k<iwkxk\+b\)/wi,\\ell\_\{i\}\(x\_\{<i\}\)=\\max\_\{\\phi\\in\\Pi\_\{i\}^\{\+\}\}\\varepsilon\_\{i\}^\{\\phi\}\(x\_\{<i\}\),\\quad u\_\{i\}\(x\_\{<i\}\)=\\min\_\{\\phi\\in\\Pi\_\{i\}^\{\-\}\}\\varepsilon\_\{i\}^\{\\phi\}\(x\_\{<i\}\),\\quad\\varepsilon\_\{i\}^\{\\phi\}=\-\\Bigl\(\\textstyle\\sum\_\{k<i\}w\_\{k\}x\_\{k\}\+b\\Bigr\)/w\_\{i\},\(1\)the bounds the constraint layer clamps into \(Section[3](https://arxiv.org/html/2608.21605#S3)\)\. Per variable,φ\\varphiapplies the grounding that fits the interval:

one\-sided:xi\\displaystyle\\text\{one\-sided: \}x\_\{i\}=ℓi\+softplus⁡\(zi\)​or​ui−softplus⁡\(zi\),\\displaystyle=\\ell\_\{i\}\+\\mathrm\{softplus\}\(z\_\{i\}\)\\ \\text\{ or \}\\ u\_\{i\}\-\\mathrm\{softplus\}\(z\_\{i\}\),box:xi\\displaystyle\\text\{box: \}x\_\{i\}=ℓi\+\(ui−ℓi\)​σ​\(zi\),\\displaystyle=\\ell\_\{i\}\+\(u\_\{i\}\-\\ell\_\{i\}\)\\,\\sigma\(z\_\{i\}\),free:xi\\displaystyle\\text\{free: \}x\_\{i\}=zi,\\displaystyle=z\_\{i\},identity:xi\\displaystyle\\text\{identity: \}x\_\{i\}=∑j<iwi​j​xj\+wi​0\.\\displaystyle=\\textstyle\\sum\_\{j<i\}w\_\{ij\}\\,x\_\{j\}\+w\_\{i0\}\.\(For Alchemy’sU0<U<HU\_\{0\}<U<H:U0=z1U\_\{0\}=z\_\{1\},U=U0\+softplus⁡\(z2\)U=U\_\{0\}\+\\mathrm\{softplus\}\(z\_\{2\}\),H=U\+softplus⁡\(z3\)H=U\+\\mathrm\{softplus\}\(z\_\{3\}\)\.\) Each bounded variable is a smooth, monotone function of a unit\-scale coordinateziz\_\{i\}that stays within its admissible interval, and each identity derives its dependent variable, so the axiom holds by construction \(Sat⁡\(KB\)=1\\mathrm\{Sat\}\(\\mathrm\{KB\}\)=1, hence the logical term in the objective vanishes for these axioms\)\. The decoderφ:z↦x\\varphi:z\\mapsto xis the*chart*ofℳ\\mathcal\{M\}defined above, turning unconstrained coordinateszzinto feasible samplesxx\. It inverts in closed form \(for a one\-sided variable,zi=softplus−1​\(xi−ℓi\)z\_\{i\}=\\mathrm\{softplus\}^\{\-1\}\(x\_\{i\}\-\\ell\_\{i\}\)\), which gives the encoderφ−1\\varphi^\{\-1\}applied to real data\. The discriminator operates on the chart coordinateszz, where every margin is unit scale, receivingzzfor generated samples andφ−1​\(x\)\\varphi^\{\-1\}\(x\)for real ones, so a standard discriminator can learn the distribution of the constraint margin directly\. For heavy\-tailed margins we ground throughexp\\exprather than softplus \(a multiplicative increment\), set by a fixed dynamic\-range rule \(Appendix[G](https://arxiv.org/html/2608.21605#A7)\)\. Monotone softplus links enforcing order and positivity go back to[9](https://arxiv.org/html/2608.21605#bib.bib42)\. Function\-symbol grounding deploys them as the groundings of structural axioms inside adversarial training\.

###### Proposition 1\(Validity by construction\)\.

For any satisfiable finite setΠ\\Piof linear inequalities and any generator outputzz, the assembled sampleφ⁡\(z\)\\varphi\(z\)satisfiesΠ\\Pi\.

The proof \(Appendix[C](https://arxiv.org/html/2608.21605#A3)\) 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 andσ\\sigmamapℝ\\mathbb\{R\}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\.1Why the chart can preserve the distribution of the constraint margin and the clamp cannot

Forϕ:b\>a\\phi:b\>a, the clamp sends every violator tob=ab=a, while function\-symbol grounding emits the margin as a unit\-scale coordinate,mϕ=softplus⁡\(z\)m\_\{\\phi\}=\\mathrm\{softplus\}\(z\)\. Remark[2](https://arxiv.org/html/2608.21605#Thmtheorem2)castsRRas a condition number, and Proposition[5](https://arxiv.org/html/2608.21605#Thmtheorem5)quantifies both mechanisms\.

###### Corollary 3\(Definitional identities are measure\-zero\)\.

Letℳ==\{x∈ℝD:c⁡\(x\)=0\}\\mathcal\{M\}\_\{=\}=\\\{x\\in\\mathbb\{R\}^\{D\}:c\(x\)=0\\\}be the zero set of the identities, withc:ℝD→ℝpc:\\mathbb\{R\}^\{D\}\\to\\mathbb\{R\}^\{p\}aC1C^\{1\}map whose Jacobian has full rankpponℳ=\\mathcal\{M\}\_\{=\}\. Thenℳ=\\mathcal\{M\}\_\{=\}is Lebesgue\-null, so any generated distributionν\\nuthat is absolutely continuous hasPrx∼ν\[c\(x\)=0\]=0\\Pr\_\{x\\sim\\nu\}\[c\(x\)=0\]=0, and no satisfaction loss can raise exact satisfaction above probability00\. Function\-symbol grounding derives the dependent variables from their parents, soc≡0c\\equiv 0holds with probability11\.

###### Corollary 4\(Predicate\-grounded ordering:Θ⁡\(1/R\)\\Theta\(1/R\)gradient\)\.

Ground an orderingϕ:b\>a\\phi:b\>aas the predicatePs=σ⁡\(\(b−a\)/s\)P\_\{s\}=\\sigma\\\!\\big\(\(b\-a\)/s\\big\)with bands=Θ⁡\(σm\)s=\\Theta\(\\sigma\_\{m\}\)\. If the generator has matched the marginals ofaaandbbbut not their dependence \(margin correlation bounded away from11\), its margin has spreadΘ⁡\(R​σm\)\\Theta\(R\\,\\sigma\_\{m\}\)and densityΘ⁡\(1/σs\)\\Theta\(1/\\sigma\_\{s\}\)nearb=ab=a\(no anomalous concentration\)\. The predicate gradientPs​\(1−Ps\)/sP\_\{s\}\(1\-P\_\{s\}\)/sisΘ⁡\(1/s\)\\Theta\(1/s\)on the band\|b−a\|≲s\|b\-a\|\\lesssim sand exponentially small outside it, so the fraction of generated samples with usable gradient isΘ⁡\(1/R\)\\Theta\(1/R\), vanishing asR→∞R\\to\\infty\.

###### Proposition 5\(Margin laws\)\.

Let the real margin ofϕ:b\>a\\phi:b\>ahave CDFFFsupported on\(0,∞\)\(0,\\infty\), and clamp a free generator with violation ratevvpost hoc, with offsetε\\varepsilonbelow the real support\. \(i\) The clamped margin distribution isv​δε\+\(1−v\)​ν\+v\\,\\delta\_\{\\varepsilon\}\+\(1\-v\)\\,\\nu^\{\+\}, withν\+\\nu^\{\+\}the generator’s satisfying\-margin distribution\. Under the hypotheses of Corollary[4](https://arxiv.org/html/2608.21605#Thmtheorem4), its KS distance toFFis at leastmax⁡\(v,1−v\)−Θ⁡\(1/R\)≥12−Θ⁡\(1/R\)\\max\(v,1\-v\)\-\\Theta\(1/R\)\\geq\\tfrac\{1\}\{2\}\-\\Theta\(1/R\), tending to11asv→1v\\to 1\(observed in\-loop; Section[5](https://arxiv.org/html/2608.21605#S5)\)\. \(ii\) The chartφ\\varphiis a bijection fromℝd\\mathbb\{R\}^\{d\}onto the relative interior ofℳ\\mathcal\{M\}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\(0,∞\)\(0,\\infty\)exactly \(proofs in Appendix[D](https://arxiv.org/html/2608.21605#A4)\)\.

### 4\.2Which axioms to ground through function symbols: a hybrid

Function\-symbol grounding is most effective where clamping is most costly, on high\-RRconstraints 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 orσ\\sigmaincrement 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 fractionsis\_\{i\}of its samples that already lie in variablexix\_\{i\}’s admissible interval, and the largest single\-value frequency ofxix\_\{i\}’s binding margin in𝒟\\mathcal\{D\}measures the discretenessdid\_\{i\}\. We chartxix\_\{i\}ifsi<0\.9s\_\{i\}<0\.9anddi≤0\.2d\_\{i\}\\leq 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\-RRcontinuous variables and clamps the rest\. Validity remains exact \(Proposition[1](https://arxiv.org/html/2608.21605#Thmtheorem1)\) since every bounded variable is either charted or clamped\. Algorithm[B](https://arxiv.org/html/2608.21605#A2)\(Appendix[B](https://arxiv.org/html/2608.21605#A2)\) summarizes the procedure\.

## 5Experimental 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](https://arxiv.org/html/2608.21605#bib.bib4)?RQ4:can predicate grounding \(the standard LTN\-GAN satisfaction loss\) reach high\-RRconstraints instead?

Datasets\.We use four real high\-resolution datasets, each carrying an ordering whose margin is a small difference of large quantities:Alchemy\([7](https://arxiv.org/html/2608.21605#bib.bib11)\)\(U0<U<HU\_\{0\}<U<H\),tmQM\([4](https://arxiv.org/html/2608.21605#bib.bib15)\),Transition1x\([32](https://arxiv.org/html/2608.21605#bib.bib16)\)\(reaction barriers\), andTaxi\([28](https://arxiv.org/html/2608.21605#bib.bib17)\)\(trip duration\)\. Constraints and per\-marginRR\(3\.63\.6to7×1067\{\\times\}10^\{6\}\) are in Table[4](https://arxiv.org/html/2608.21605#A6.T4), Appendix[F](https://arxiv.org/html/2608.21605#A6)\. For RQ3 we use the six\-dataset benchmark of[35](https://arxiv.org/html/2608.21605#bib.bib4)\.Baselines\.The*constraint layer*\(C\-DGM\) on the same MLP\-GAN as FSG\-LTN\-GAN;*CTGAN*and*TVAE*\([39](https://arxiv.org/html/2608.21605#bib.bib13)\)with and without it; and post\-hoc*projection*\.Metrics\.Validity, the*margin KS*against a fixed reference sample of the real data \(Appendix[E](https://arxiv.org/html/2608.21605#A5)\), and per\-property moment error, as means±\\pmstd overn=10n\{=\}10seeds with paired Wilcoxonpp\-values\. Full tables and the constraint\-layer corrections are in Appendices[E](https://arxiv.org/html/2608.21605#A5),[H](https://arxiv.org/html/2608.21605#A8),[I](https://arxiv.org/html/2608.21605#A9), and[J](https://arxiv.org/html/2608.21605#A10)\.

### 5\.1RQ1: 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±\\pmstd\. Both methods are100%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 Wilcoxonp<0\.01p<0\.01\.![Refer to caption](https://arxiv.org/html/2608.21605v1/figures/constraint_margins.png)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 are100%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\-RRmargin\. Table[1](https://arxiv.org/html/2608.21605#S5.T1)averages each dataset’s constraints, so its Taxi value also counts the low\-RRtotal\>fare\\mathrm\{total\}\>\\mathrm\{fare\}margin, which the clamp handles well\.Throughout, the constraint layer runs from the code released by[35](https://arxiv.org/html/2608.21605#bib.bib4), with corrections that only help it \(including one for a latent bug in its Fourier\-Motzkin reduction; Appendix[E](https://arxiv.org/html/2608.21605#A5)\)\. Table[1](https://arxiv.org/html/2608.21605#S5.T1)shows that on all four datasets the constraint layer and FSG\-LTN\-GAN are both100%100\\%valid, yet the constraint layer’s margin KS is0\.570\.57to1\.001\.00while FSG\-LTN\-GAN’s is0\.040\.04to0\.100\.10\(p<0\.01p<0\.01, all ten seeds\)\. Inspecting the raw margins \(Figure[2](https://arxiv.org/html/2608.21605#S5.F2)\) explains the numbers\. The clamp places every sample’s margin at the boundary \(about10−610^\{\-6\}at every quantile\), a point mass disjoint from the real distribution \(the collapse of Proposition[5](https://arxiv.org/html/2608.21605#Thmtheorem5)\(i\)\)\. 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 tracksRRper margin \(KS near11on the high\-RRmargins, e\.g\. AlchemyU−U0U\-U\_\{0\}atR≈105R\\approx 10^\{5\}\)\. Taxi’s lower table value averages in a low\-RRmargin the clamp handles well, so the effect tracksRRrather than the domain\. A final, pre\-registered test on a fifth dataset outside both papers’ suites \(nycflights13\) confirmed every prediction \(Appendix[J](https://arxiv.org/html/2608.21605#A10), Table[11](https://arxiv.org/html/2608.21605#A10.T11)\)\.

#### Overall sample quality is preserved\.

On the constraint layer’s own density and coverage metrics\([27](https://arxiv.org/html/2608.21605#bib.bib12)\), 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 \(coverage0\.200\.20and0\.700\.70against0\.050\.05; Appendix[J](https://arxiv.org/html/2608.21605#A10)\)\.

### 5\.2RQ2: the effect persists across the architectures we test

Table 2:Architecture generality \(RQ2\)\.Margin KS↓\\downarrow\(n=10n\{=\}10, mean±\\pmstd\) for CTGAN\+\+CL, TVAE\+\+CL, the constraint layer on our MLP generator \(C\-DGM\), and FSG\-LTN\-GAN\. All are100%100\\%valid\. The constraint layer misses the distribution of the constraint margin on all three generator families\.Table[2](https://arxiv.org/html/2608.21605#S5.T2)applies the constraint layer to the two most\-cited tabular generators, CTGAN and TVAE\. Both reach100%100\\%validity but leave margin KS at0\.350\.35to0\.670\.67, against0\.040\.04to0\.100\.10for FSG\-LTN\-GAN\. The in\-loop constraint layer on our MLP generator is worse still \(0\.570\.57to1\.001\.00\), 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 throughφ\\varphiis exactly valid and recovers the margin \(Appendix[J](https://arxiv.org/html/2608.21605#A10), Table[10](https://arxiv.org/html/2608.21605#A10.T10)\)\.

### 5\.3RQ3: the hybrid matches or exceeds the constraint layer on the Stoian et al\. \(2024\) benchmark

Table 3:The constraint\-layer benchmark of[35](https://arxiv.org/html/2608.21605#bib.bib4)\(RQ3\)\.Margin KS↓\\downarrow,n=10n\{=\}10, mean±\\pmstd, all methods100%100\\%valid\.RmaxR\_\{\\max\}is the dataset’s largest per\-constraintRR\. The hybrid outperforms the constraint layer \(paired Wilcoxon\) on faults \(p=0\.002p\{=\}0\.002, the high\-RRcase\), 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\.*Does the method help on the constraint layer’s own benchmark? Yes, whereverRRis high\.*Table[3](https://arxiv.org/html/2608.21605#S5.T3)reports all six datasets of[35](https://arxiv.org/html/2608.21605#bib.bib4), predominantly low\-RRtabular\-ML tasks where clamping is already adequate\. On the one high\-RRdataset, faults \(bounding\-box orderings on large sensor coordinates,Rmax≈1\.8×103R\_\{\\max\}\\approx 1\.8\\times 10^\{3\}\), the hybrid charts every bounded variable, coinciding with FSG\-all, and improves \(p=0\.002p\{=\}0\.002, all ten seeds\)\. On the low\-RRremainder the hybrid clamps and matches the constraint layer, for three wins, three ties, and zero losses across the benchmark, all at100%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 \(Section[4\.2](https://arxiv.org/html/2608.21605#S4.SS2)\) is therefore necessary, andRRpredicts 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 \(Appendix[J](https://arxiv.org/html/2608.21605#A10), Table[15](https://arxiv.org/html/2608.21605#A10.T15)\)\.

### 5\.4RQ4: predicate grounding does not reach high\-RRconstraints

Corollary[4](https://arxiv.org/html/2608.21605#Thmtheorem4)bounds a predicate grounding’s usable\-gradient fraction atΘ⁡\(1/R\)\\Theta\(1/R\)\. 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 \(n=10n\{=\}10, mean±\\pmstd\):0\.58±0\.020\.58\\pm 0\.02,0\.51±0\.030\.51\\pm 0\.03,0\.14±0\.080\.14\\pm 0\.08on Alchemy, where a free generator with no mechanism scores0\.49±0\.030\.49\\pm 0\.03, and0\.59±0\.030\.59\\pm 0\.03,0\.55±0\.080\.55\\pm 0\.08,0\.29±0\.070\.29\\pm 0\.07on tmQM \(free generator0\.53±0\.080\.53\\pm 0\.08\), against1\.000±0\.0001\.000\\pm 0\.000for function\-symbol grounding\. No placement helps by more than0\.090\.09\. The failure tracks the coordinates, not the placement \(Table[16](https://arxiv.org/html/2608.21605#A10.T16)\)\.

## 6Related Work

Neuro\-symbolic generation and LTNs\.Logic Tensor Networks train by maximizing grounded satisfaction\([3](https://arxiv.org/html/2608.21605#bib.bib1)\), and related work realises logic as a differentiable loss\([38](https://arxiv.org/html/2608.21605#bib.bib3);[11](https://arxiv.org/html/2608.21605#bib.bib6)\)\. All are*predicate\-style*soft constraints \(RQ4\)\. Hard\-constraint output layers guarantee specific fragments\([2](https://arxiv.org/html/2608.21605#bib.bib5);[15](https://arxiv.org/html/2608.21605#bib.bib7);[12](https://arxiv.org/html/2608.21605#bib.bib8)\)\.Constrained tabular generation\.The constraint layer of[35](https://arxiv.org/html/2608.21605#bib.bib4), our primary baseline, guarantees linear inequalities by clamping\. GOGGLE\([22](https://arxiv.org/html/2608.21605#bib.bib14)\)injects only simple correlations, and CTGAN and TVAE\([39](https://arxiv.org/html/2608.21605#bib.bib13)\)give no guarantees\. Closest are constrained adversarial networks\([8](https://arxiv.org/html/2608.21605#bib.bib19)\)and a non\-convex constraint layer\([36](https://arxiv.org/html/2608.21605#bib.bib21)\); Appendix[A](https://arxiv.org/html/2608.21605#A1)surveys the rest and our function symbols’ antecedents\([9](https://arxiv.org/html/2608.21605#bib.bib42);[5](https://arxiv.org/html/2608.21605#bib.bib43)\)\.

## 7Discussion 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 fromRR\. 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](https://arxiv.org/html/2608.21605#bib.bib4), with the same advantage in conditional inverse design \(Appendix[J](https://arxiv.org/html/2608.21605#A10)\)\.

Scope and limitations\.Our charts cover the linear fragment, where Fourier\-Motzkin gives the admissible intervals\. Non\-convex feasible sets from nonlinear or disjunctive constraints are left to future work, and discrete margins are boundary point masses the hybrid correctly clamps\.

## References

- Ahmedet al\.\(2023\)K\. Ahmed, K\. Chang, and G\. Van den BroeckA pseudo\-semantic loss for autoregressive models with logical constraints\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.36\.External Links:[Link](http://papers.nips.cc/paper_files/paper/2023/hash/3accfe8332366a6f740d8740cd4cd653-Abstract-Conference.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px1.p1.1)\.
- Ahmedet al\.\(2022\)K\. Ahmed, S\. Teso, K\. Chang, G\. Van den Broeck, and A\. VergariSemantic probabilistic layers for neuro\-symbolic learning\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.35\.External Links:[Link](https://proceedings.neurips.cc/paper_files/paper/2022/hash/c182ec594f38926b7fcb827635b9a8f4-Abstract-Conference.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px2.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Badreddineet al\.\(2022\)S\. Badreddine, A\. d’Avila Garcez, L\. Serafini, and M\. SprangerLogic Tensor Networks\.Artificial Intelligence303,pp\. 103649\.External Links:[Document](https://dx.doi.org/10.1016/j.artint.2021.103649),[Link](https://doi.org/10.1016/j.artint.2021.103649)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.21605#S1.p2.1),[§2](https://arxiv.org/html/2608.21605#S2.SS0.SSS0.Px1.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Balcells and Skjelstad \(2020\)D\. Balcells and B\. B\. SkjelstadtmQM dataset—quantum geometries and properties of 86k transition metal complexes\.Journal of Chemical Information and Modeling60\(12\),pp\. 6135–6146\.External Links:[Document](https://dx.doi.org/10.1021/acs.jcim.0c01041),[Link](https://doi.org/10.1021/acs.jcim.0c01041)Cited by:[Appendix F](https://arxiv.org/html/2608.21605#A6.p2.1),[§5](https://arxiv.org/html/2608.21605#S5.p2.1)\.
- Carpenteret al\.\(2017\)B\. Carpenter, A\. Gelman, M\. D\. Hoffman, D\. Lee, B\. Goodrich, M\. Betancourt, M\. Brubaker, J\. Guo, P\. Li, and A\. RiddellStan: a probabilistic programming language\.Journal of Statistical Software76\(1\),pp\. 1–32\.External Links:[Document](https://dx.doi.org/10.18637/jss.v076.i01)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Chaoet al\.\(2021\)X\. Chao, J\. Cao, Y\. Lu, Q\. Dai, and S\. LiangConstrained generative adversarial networks\.IEEE Access9,pp\. 19208–19218\.External Links:[Document](https://dx.doi.org/10.1109/ACCESS.2021.3054822),[Link](https://doi.org/10.1109/ACCESS.2021.3054822)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Chenet al\.\(2019\)G\. Chen, P\. Chen, C\. Hsieh, C\. Lee, B\. Liao, R\. Liao, W\. Liu, J\. Qiu, Q\. Sun, J\. Tang, R\. Zemel, and S\. ZhangAlchemy: a quantum chemistry dataset for benchmarking AI models\.arXiv preprint arXiv:1906\.09427\.External Links:[Document](https://dx.doi.org/10.48550/arXiv.1906.09427),[Link](https://arxiv.org/abs/1906.09427)Cited by:[Appendix F](https://arxiv.org/html/2608.21605#A6.p2.1),[§5](https://arxiv.org/html/2608.21605#S5.p2.1)\.
- Di Lielloet al\.\(2020\)L\. Di Liello, P\. Ardino, J\. Gobbi, P\. Morettin, S\. Teso, and A\. PasseriniEfficient generation of structured objects with constrained adversarial networks\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.33\.External Links:[Link](https://proceedings.neurips.cc/paper/2020/hash/a87c11b9100c608b7f8e98cfa316ff7b-Abstract.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Dugaset al\.\(2009\)C\. Dugas, Y\. Bengio, F\. Bélisle, C\. Nadeau, and R\. GarciaIncorporating functional knowledge in neural networks\.Journal of Machine Learning Research10\(42\),pp\. 1239–1262\.Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1),[§4](https://arxiv.org/html/2608.21605#S4.p2.3),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Ferberet al\.\(2024\)A\. M\. Ferber, A\. Zharmagambetov, T\. Huang, B\. Dilkina, and Y\. TianGenCO: generating diverse designs with combinatorial constraints\.InProceedings of the 41st International Conference on Machine Learning \(ICML\),Proceedings of Machine Learning Research, Vol\.235,pp\. 13445–13459\.External Links:[Link](https://proceedings.mlr.press/v235/ferber24a.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Fischeret al\.\(2019\)M\. Fischer, M\. Balunović, D\. Drachsler\-Cohen, T\. Gehr, C\. Zhang, and M\. VechevDL2: training and querying neural networks with logic\.InProceedings of the 36th International Conference on Machine Learning \(ICML\),Proceedings of Machine Learning Research, Vol\.97,pp\. 1931–1941\.External Links:[Link](https://proceedings.mlr.press/v97/fischer19a.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px1.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Giunchiglia and Lukasiewicz \(2020\)E\. Giunchiglia and T\. LukasiewiczCoherent hierarchical multi\-label classification networks\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.33\.External Links:[Link](https://proceedings.neurips.cc/paper/2020/hash/6dd4e10e3296fa63738371ec0d5df818-Abstract.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px2.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Heim \(2019\)E\. HeimConstrained generative adversarial networks for interactive image generation\.InIEEE/CVF Conference on Computer Vision and Pattern Recognition \(CVPR\),pp\. 10753–10761\.External Links:[Document](https://dx.doi.org/10.1109/CVPR.2019.01101),[Link](https://doi.org/10.1109/CVPR.2019.01101)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Hesset al\.\(2022\)P\. Hess, M\. Drüke, S\. Petri, F\. M\. Strnad, and N\. BoersPhysically constrained generative adversarial networks for improving precipitation fields from Earth system models\.Nature Machine Intelligence4\(10\),pp\. 828–839\.External Links:[Document](https://dx.doi.org/10.1038/s42256-022-00540-1),[Link](https://doi.org/10.1038/s42256-022-00540-1)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Hoernleet al\.\(2022\)N\. Hoernle, R\. Karampatsis, V\. Belle, and K\. GalMultiplexNet: towards fully satisfied logical constraints in neural networks\.InProceedings of the 36th AAAI Conference on Artificial Intelligence,pp\. 5700–5709\.External Links:[Document](https://dx.doi.org/10.1609/aaai.v36i5.20512),[Link](https://doi.org/10.1609/aaai.v36i5.20512)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px2.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Huet al\.\(2018\)Z\. Hu, Z\. Yang, R\. Salakhutdinov, L\. Qin, X\. Liang, H\. Dong, and E\. P\. XingDeep generative models with learnable knowledge constraints\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.31,pp\. 10522–10533\.External Links:[Link](https://proceedings.neurips.cc/paper/2018/hash/d7e77c835af3d2a803c1cf28d60575bc-Abstract.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Kimet al\.\(2023\)J\. Kim, C\. Lee, and N\. ParkSTaSy: score\-based tabular data synthesis\.InProceedings of the 11th International Conference on Learning Representations \(ICLR\),External Links:[Link](https://openreview.net/forum?id=1mNssCWt_v)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px5.p1.1),[§1](https://arxiv.org/html/2608.21605#S1.p1.1)\.
- Kotelnikovet al\.\(2023\)A\. Kotelnikov, D\. Baranchuk, I\. Rubachev, and A\. BabenkoTabDDPM: modelling tabular data with diffusion models\.InProceedings of the 40th International Conference on Machine Learning \(ICML\),Proceedings of Machine Learning Research, Vol\.202,pp\. 17564–17579\.External Links:[Link](https://proceedings.mlr.press/v202/kotelnikov23a.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px5.p1.1),[§1](https://arxiv.org/html/2608.21605#S1.p1.1)\.
- Lagariset al\.\(1998\)I\. E\. Lagaris, A\. Likas, and D\. I\. FotiadisArtificial neural networks for solving ordinary and partial differential equations\.IEEE Transactions on Neural Networks9\(5\),pp\. 987–1000\.External Links:[Document](https://dx.doi.org/10.1109/72.712178),[Link](https://doi.org/10.1109/72.712178)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px2.p1.1)\.
- Li \(2020\)W\. LiSupporting database constraints in synthetic data generation based on generative adversarial networks\.InProceedings of the 2020 ACM SIGMOD International Conference on Management of Data,pp\. 2875–2877\.External Links:[Document](https://dx.doi.org/10.1145/3318464.3384414),[Link](https://doi.org/10.1145/3318464.3384414)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px5.p1.1)\.
- Liet al\.\(2023\)Z\. Li, Y\. Huang, Z\. Li, Y\. Yao, J\. Xu, T\. Chen, X\. Ma, and J\. LuNeuro\-symbolic learning yielding logical constraints\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.36\.External Links:[Link](http://papers.nips.cc/paper_files/paper/2023/hash/4459c3c143db74ee52afebdf56836375-Abstract-Conference.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px1.p1.1)\.
- Liuet al\.\(2023\)T\. Liu, Z\. Qian, J\. Berrevoets, and M\. van der SchaarGOGGLE: generative modelling for tabular data by learning relational structure\.InProceedings of the 11th International Conference on Learning Representations \(ICLR\),External Links:[Link](https://openreview.net/forum?id=fPVRcJqspu)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px5.p1.1),[§1](https://arxiv.org/html/2608.21605#S1.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Liuet al\.\(2020\)X\. Liu, Q\. Liu, S\. Song, and J\. PengA chance\-constrained generative framework for sequence optimization\.InProceedings of the 37th International Conference on Machine Learning \(ICML\),Proceedings of Machine Learning Research, Vol\.119,pp\. 6271–6281\.External Links:[Link](http://proceedings.mlr.press/v119/liu20i.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1),[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px4.p1.1)\.
- Luet al\.\(2021\)L\. Lu, R\. Pestourie, W\. Yao, Z\. Wang, F\. Verdugo, and S\. G\. JohnsonPhysics\-informed neural networks with hard constraints for inverse design\.SIAM Journal on Scientific Computing43\(6\),pp\. B1105–B1132\.External Links:[Document](https://dx.doi.org/10.1137/21M1397908),[Link](https://doi.org/10.1137/21M1397908)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px2.p1.1)\.
- Méndez\-Luceroet al\.\(2025\)M\. Á\. Méndez\-Lucero, E\. Bojorquez Gallardo, and V\. BelleSemantic objective functions: a distribution\-aware method for adding logical constraints in deep learning\.InProceedings of the 17th International Conference on Agents and Artificial Intelligence \(ICAART\),pp\. 909–917\.External Links:[Document](https://dx.doi.org/10.5220/0013229200003890),[Link](https://doi.org/10.5220/0013229200003890)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px1.p1.1)\.
- Misinoet al\.\(2022\)E\. Misino, G\. Marra, and E\. SansoneVAEL: bridging variational autoencoders and probabilistic logic programming\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.35\.External Links:[Link](http://papers.nips.cc/paper_files/paper/2022/hash/1e38b2a0b77541b14a3315c99697b835-Abstract-Conference.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px1.p1.1)\.
- Naeemet al\.\(2020\)M\. F\. Naeem, S\. J\. Oh, Y\. Uh, Y\. Choi, and J\. YooReliable fidelity and diversity metrics for generative models\.InProceedings of the 37th International Conference on Machine Learning \(ICML\),Proceedings of Machine Learning Research, Vol\.119,pp\. 7176–7185\.External Links:[Link](https://proceedings.mlr.press/v119/naeem20a.html)Cited by:[Appendix J](https://arxiv.org/html/2608.21605#A10.SS0.SSS0.Px1.p1.1),[Table 8](https://arxiv.org/html/2608.21605#A10.T8),[§5\.1](https://arxiv.org/html/2608.21605#S5.SS1.SSS0.Px1.p1.1)\.
- New York City Taxi and Limousine Commission \(2024\)New York City Taxi and Limousine CommissionTLC trip record data\.Note:[https://www\.nyc\.gov/site/tlc/about/tlc\-trip\-record\-data\.page](https://www.nyc.gov/site/tlc/about/tlc-trip-record-data.page)Accessed: 2026\-06\-16Cited by:[Appendix F](https://arxiv.org/html/2608.21605#A6.p2.1),[§5](https://arxiv.org/html/2608.21605#S5.p2.1)\.
- Paraet al\.\(2021\)W\. R\. Para, S\. F\. Bhat, P\. Guerrero, T\. Kelly, N\. J\. Mitra, L\. J\. Guibas, and P\. WonkaSketchGen: generating constrained CAD sketches\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.34,pp\. 5077–5088\.External Links:[Link](https://proceedings.neurips.cc/paper/2021/hash/28891cb4ab421830acc36b1f5fd6c91e-Abstract.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Penget al\.\(2025\)Y\. Peng, Z\. Zha, Y\. Jin, Z\. Luo, W\. Dai, Z\. Ren, Y\. Ding, and K\. ZhouGenerating by understanding: neural visual generation with logical symbol groundings\.InProceedings of the 31st ACM SIGKDD Conference on Knowledge Discovery and Data Mining,pp\. 2291–2302\.External Links:[Document](https://dx.doi.org/10.1145/3711896.3736978),[Link](https://doi.org/10.1145/3711896.3736978)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Ramakrishnanet al\.\(2015\)R\. Ramakrishnan, P\. O\. Dral, M\. Rupp, and O\. A\. von LilienfeldBig data meets quantum chemistry approximations: theΔ\\Delta\-machine learning approach\.Journal of Chemical Theory and Computation11\(5\),pp\. 2087–2096\.External Links:[Document](https://dx.doi.org/10.1021/acs.jctc.5b00099)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Schreineret al\.\(2022\)M\. Schreiner, A\. Bhowmik, T\. Vegge, J\. Busk, and O\. WintherTransition1x—a dataset for building generalizable reactive machine learning potentials\.Scientific Data9,pp\. 779\.External Links:[Document](https://dx.doi.org/10.1038/s41597-022-01870-w),[Link](https://doi.org/10.1038/s41597-022-01870-w)Cited by:[Appendix F](https://arxiv.org/html/2608.21605#A6.p2.1),[§5](https://arxiv.org/html/2608.21605#S5.p2.1)\.
- Seffet al\.\(2022\)A\. Seff, W\. Zhou, N\. Richardson, and R\. P\. AdamsVitruvion: a generative model of parametric CAD sketches\.InProceedings of the 10th International Conference on Learning Representations \(ICLR\),External Links:[Link](https://openreview.net/forum?id=Ow1C7s3UcY)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Serafini and d’Avila Garcez \(2016\)L\. Serafini and A\. d’Avila GarcezLearning and reasoning with Logic Tensor Networks\.InAI\*IA 2016: Advances in Artificial Intelligence,Lecture Notes in Computer Science, Vol\.10037,pp\. 334–348\.External Links:[Link](https://doi.org/10.1007/978-3-319-49130-1_25)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px1.p1.1)\.
- Stoianet al\.\(2024\)M\. C\. Stoian, S\. Dyrmishi, M\. Cordy, T\. Lukasiewicz, and E\. GiunchigliaHow realistic is your synthetic data? constraining deep generative models for tabular data\.InProceedings of the 12th International Conference on Learning Representations \(ICLR\),External Links:[Link](https://openreview.net/forum?id=tBROYsEz9G)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px2.p1.1),[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px5.p1.1),[Appendix E](https://arxiv.org/html/2608.21605#A5.p1.1),[Appendix F](https://arxiv.org/html/2608.21605#A6.p1.1),[Table 7](https://arxiv.org/html/2608.21605#A9.T7),[§1](https://arxiv.org/html/2608.21605#S1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.21605#S1.p2.1),[§3](https://arxiv.org/html/2608.21605#S3.SS0.SSS0.Px3.p1.1),[§3](https://arxiv.org/html/2608.21605#S3.p1.1),[§4](https://arxiv.org/html/2608.21605#S4.p2.1),[§5\.1](https://arxiv.org/html/2608.21605#S5.SS1.p1.1),[§5\.3](https://arxiv.org/html/2608.21605#S5.SS3.p1.1),[Table 3](https://arxiv.org/html/2608.21605#S5.T3.2),[Table 3](https://arxiv.org/html/2608.21605#S5.T3.3),[§5](https://arxiv.org/html/2608.21605#S5.p1.1),[§5](https://arxiv.org/html/2608.21605#S5.p2.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1),[§7](https://arxiv.org/html/2608.21605#S7.p1.1)\.
- Stoian and Giunchiglia \(2025\)M\. C\. Stoian and E\. GiunchigliaBeyond the convexity assumption: realistic tabular data generation under quantifier\-free real linear constraints\.InProceedings of the 13th International Conference on Learning Representations \(ICLR\),External Links:[Link](https://openreview.net/forum?id=rx0TCew0Lj)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px2.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Upreti and Belle \(2026\)N\. Upreti and V\. BelleLogic Tensor Network\-Enhanced Generative Adversarial Network\.Electronic Proceedings in Theoretical Computer Science439,pp\. 89–113\.External Links:[Document](https://dx.doi.org/10.4204/EPTCS.439.8),[Link](https://doi.org/10.4204/EPTCS.439.8)Cited by:[§1](https://arxiv.org/html/2608.21605#S1.p2.1),[§2](https://arxiv.org/html/2608.21605#S2.SS0.SSS0.Px3.p1.1)\.
- Xuet al\.\(2018\)J\. Xu, Z\. Zhang, T\. Friedman, Y\. Liang, and G\. Van den BroeckA semantic loss function for deep learning with symbolic knowledge\.InProceedings of the 35th International Conference on Machine Learning \(ICML\),Proceedings of Machine Learning Research, Vol\.80,pp\. 5502–5511\.External Links:[Link](https://proceedings.mlr.press/v80/xu18h.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px1.p1.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Xuet al\.\(2019\)L\. Xu, M\. Skoularidou, A\. Cuesta\-Infante, and K\. VeeramachaneniModeling tabular data using conditional GAN\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.32,pp\. 7333–7343\.External Links:[Link](https://proceedings.neurips.cc/paper/2019/hash/254ed7d2de3b23ab10936522dd547b78-Abstract.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px5.p1.1),[§1](https://arxiv.org/html/2608.21605#S1.p1.1),[§5](https://arxiv.org/html/2608.21605#S5.p2.1),[§6](https://arxiv.org/html/2608.21605#S6.p1.1)\.
- Xue and van Hoeve \(2019\)Y\. Xue and W\. van HoeveEmbedding decision diagrams into generative adversarial networks\.InIntegration of Constraint Programming, Artificial Intelligence, and Operations Research \(CPAIOR\),Lecture Notes in Computer Science, Vol\.11494,pp\. 616–632\.External Links:[Link](https://doi.org/10.1007/978-3-030-19212-9_41)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Yanget al\.\(2022\)Z\. Yang, J\. Lee, and C\. ParkInjecting logical constraints into neural networks via straight\-through estimators\.InProceedings of the 39th International Conference on Machine Learning \(ICML\),Proceedings of Machine Learning Research, Vol\.162,pp\. 25096–25122\.External Links:[Link](https://proceedings.mlr.press/v162/yang22h.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px1.p1.1)\.
- Younget al\.\(2022\)H\. Young, M\. Du, and O\. BastaniNeurosymbolic deep generative models for sequence data with relational constraints\.InAdvances in Neural Information Processing Systems \(NeurIPS\),Vol\.35\.External Links:[Link](http://papers.nips.cc/paper_files/paper/2022/hash/f13ceb1b94145aad0e54186373cc86d7-Abstract-Conference.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Zenget al\.\(2021\)Y\. Zeng, J\. Wu, and H\. XiaoEnforcing imprecise constraints on generative adversarial networks for emulating physical systems\.Communications in Computational Physics30\(3\),pp\. 635–665\.External Links:[Document](https://dx.doi.org/10.4208/cicp.OA-2020-0106),[Link](https://doi.org/10.4208/cicp.OA-2020-0106)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px3.p1.1)\.
- Zhaoet al\.\(2021\)Z\. Zhao, A\. Kunar, R\. Birke, and L\. Y\. ChenCTAB\-GAN: effective table data synthesizing\.InProceedings of the 13th Asian Conference on Machine Learning \(ACML\),Proceedings of Machine Learning Research, Vol\.157,pp\. 97–112\.External Links:[Link](https://proceedings.mlr.press/v157/zhao21a.html)Cited by:[Appendix A](https://arxiv.org/html/2608.21605#A1.SS0.SSS0.Px5.p1.1)\.

## Appendix AExtended Related Work

This appendix expands Section[6](https://arxiv.org/html/2608.21605#S6)\. The body cites only the most directly related work\.

#### Logic as a soft training signal\.

Beyond Logic Tensor Networks\([3](https://arxiv.org/html/2608.21605#bib.bib1);[34](https://arxiv.org/html/2608.21605#bib.bib18)\), many methods inject logic as a differentiable loss\. The semantic loss\([38](https://arxiv.org/html/2608.21605#bib.bib3)\)and its pseudo\-semantic extension for autoregressive models\([1](https://arxiv.org/html/2608.21605#bib.bib25)\)penalise probability mass on violating assignments\. DL2\([11](https://arxiv.org/html/2608.21605#bib.bib6)\), straight\-through estimators\([41](https://arxiv.org/html/2608.21605#bib.bib26)\), distribution\-aware objectives\([25](https://arxiv.org/html/2608.21605#bib.bib27)\), and bilevel formulations\([21](https://arxiv.org/html/2608.21605#bib.bib28)\)optimise logical losses directly, and VAEL\([26](https://arxiv.org/html/2608.21605#bib.bib24)\)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 aΘ⁡\(1/R\)\\Theta\(1/R\)fraction of samples \(Corollary[4](https://arxiv.org/html/2608.21605#Thmtheorem4)\), 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\([2](https://arxiv.org/html/2608.21605#bib.bib5)\), MultiplexNet\([15](https://arxiv.org/html/2608.21605#bib.bib7)\), and coherent hierarchical classifiers\([12](https://arxiv.org/html/2608.21605#bib.bib8)\)cover classes of logical constraints, and physics\-informed networks enforce differential constraints\([19](https://arxiv.org/html/2608.21605#bib.bib9);[24](https://arxiv.org/html/2608.21605#bib.bib10)\)\. For linear constraints on tabular data, the constraint layer of[35](https://arxiv.org/html/2608.21605#bib.bib4)clamps onto the feasible polytope, recently extended to quantifier\-free non\-convex constraints\([36](https://arxiv.org/html/2608.21605#bib.bib21)\)\. 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\([8](https://arxiv.org/html/2608.21605#bib.bib19);[16](https://arxiv.org/html/2608.21605#bib.bib20);[6](https://arxiv.org/html/2608.21605#bib.bib29)\), and related mechanisms appear in interactive image editing\([13](https://arxiv.org/html/2608.21605#bib.bib30)\), physical fields\([14](https://arxiv.org/html/2608.21605#bib.bib31);[43](https://arxiv.org/html/2608.21605#bib.bib32)\), decision diagrams\([40](https://arxiv.org/html/2608.21605#bib.bib33)\), parametric CAD sketches\([33](https://arxiv.org/html/2608.21605#bib.bib34);[29](https://arxiv.org/html/2608.21605#bib.bib35)\), combinatorial design\([10](https://arxiv.org/html/2608.21605#bib.bib36)\), sequence optimisation\([23](https://arxiv.org/html/2608.21605#bib.bib37);[42](https://arxiv.org/html/2608.21605#bib.bib23)\), and visual scene generation\([30](https://arxiv.org/html/2608.21605#bib.bib22)\)\. 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](https://arxiv.org/html/2608.21605#bib.bib42), and probabilistic\-programming samplers routinely transform positive, interval, and ordered parameters to unconstrained space before sampling\([5](https://arxiv.org/html/2608.21605#bib.bib43)\)\. 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 echoesΔ\\Delta\-machine learning for chemical energies\([31](https://arxiv.org/html/2608.21605#bib.bib44)\), a supervised antecedent of the margin\-form baseline of Appendix[J](https://arxiv.org/html/2608.21605#A10)\.

#### 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\-RRregime \(that validity is0\.050\.05on Alchemy, Table[5](https://arxiv.org/html/2608.21605#A8.T5), and decreases asRRgrows\), and for definitional identities it is zero outright\. By Corollary[3](https://arxiv.org/html/2608.21605#Thmtheorem3)an absolutely continuous generator satisfies an identity with probability00, so no rejection budget suffices\.*Chance\-constrained*formulations\([23](https://arxiv.org/html/2608.21605#bib.bib37)\)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\-RRgradient obstruction \(Corollary[4](https://arxiv.org/html/2608.21605#Thmtheorem4)\)\. 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\([39](https://arxiv.org/html/2608.21605#bib.bib13)\), CTAB\-GAN\([44](https://arxiv.org/html/2608.21605#bib.bib41)\)\), score\-based and diffusion models \(STaSy\([17](https://arxiv.org/html/2608.21605#bib.bib40)\), TabDDPM\([18](https://arxiv.org/html/2608.21605#bib.bib39)\)\), and relational\-structure models \(GOGGLE\([22](https://arxiv.org/html/2608.21605#bib.bib14)\)\) improve marginal and dependency fidelity but provide no validity guarantees\. Database\-constraint enforcement has also been added to GAN\-based synthesis\([20](https://arxiv.org/html/2608.21605#bib.bib38)\)\. The constraint layer\([35](https://arxiv.org/html/2608.21605#bib.bib4)\)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 BThe Hybrid Procedure

\\SetKwComment

tcc⊳\\triangleright\\SetKwCommenttcp⊳\\triangleright\\SetCommentStytextnormal\{algorithm2e\}\[H\]\\DontPrintSemicolon\\SetKwInOutInputInput\\SetKwInOutOutputOutput\\Inputlinear constraintsΠ\\Pi; data𝒟\\mathcal\{D\}; thresholdsτs=0\.9\\tau\_\{s\}\{=\}0\.9\(satisfaction\),τd=0\.2\\tau\_\{d\}\{=\}0\.2\(discreteness\)\\Outputan LTN\-GAN generator with exact validity onΠ\\Pi\\BlankLine\\tccPhase 1: reduceΠ\\Pito per\-variable interval bounds Fix a variable order; by Fourier\-Motzkin elimination, write eachxix\_\{i\}’s boundsℓi,ui\\ell\_\{i\},u\_\{i\}as maxima and minima of linear functions of the strictly earlier variables\\BlankLine\\tccPhase 2: choose a grounding per bounded variable \(one short pre\-run\) Train a free generator briefly; measure each bounded variablexix\_\{i\}’s interval satisfactionsis\_\{i\}on the generator’s samples and its binding\-margin discretenessdid\_\{i\}on𝒟\\mathcal\{D\}\\Foreach bounded variablexix\_\{i\}\\eIfsi<τss\_\{i\}<\\tau\_\{s\}anddi≤τdd\_\{i\}\\leq\\tau\_\{d\}modei←chart\\mathrm\{mode\}\_\{i\}\\leftarrow\\textsc\{chart\}\\tcp\*high\-RR, continuousmodei←clamp\\mathrm\{mode\}\_\{i\}\\leftarrow\\textsc\{clamp\}\\tcp\*low\-RRor discrete\\BlankLine\\tccPhase 3: train; charted variables’ axioms hold by construction \(no satisfaction loss\)\\Foreach training step draw noiseζ\\zetaand set the free coordinatesz←Gθ​\(ζ\)z\\leftarrow G\_\{\\theta\}\(\\zeta\)\\Foreach variableiiin Fourier\-Motzkin order\\uIfxix\_\{i\}is fixed by an identityxi←∑j<iwi​j​xj\+wi​0x\_\{i\}\\leftarrow\\textstyle\\sum\_\{j<i\}w\_\{ij\}x\_\{j\}\+w\_\{i0\}\\uElseIfxix\_\{i\}is free \(highest\-order in no axiom\)xi←zix\_\{i\}\\leftarrow z\_\{i\}\\uElseIfmodei=chart\\mathrm\{mode\}\_\{i\}=\\textsc\{chart\}xi←x\_\{i\}\\leftarrowthe link fitting\[ℓi,ui\]\[\\ell\_\{i\},u\_\{i\}\]\(softplus orσ\\sigma, Section[4](https://arxiv.org/html/2608.21605#S4);exp\\exp, Appendix[G](https://arxiv.org/html/2608.21605#A7)\)\\tcp\*Sat=1\\mathrm\{Sat\}\{=\}1\\Elsexi←min⁡\(max⁡\(zi,ℓi\),ui\)x\_\{i\}\\leftarrow\\min\(\\max\(z\_\{i\},\\ell\_\{i\}\),\\,u\_\{i\}\)\\tcp\*in\-loop clamp updateGθ,DψG\_\{\\theta\},D\_\{\\psi\}adversarially on the chart features: each charted variable enters as its coordinateziz\_\{i\}and each clamped variable as its standardized clamped value, with real samples encoded the same way throughφ−1\\varphi^\{\-1\}Hybrid function\-symbol grounding in an LTN\-GAN\.

## Appendix CProof of Proposition[1](https://arxiv.org/html/2608.21605#Thmtheorem1)

Fourier\-Motzkin elimination overΠ\\Piin the variable order gives, for eachxix\_\{i\}, its lower and upper boundsℓi,ui\\ell\_\{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; ifΠ\\Piis satisfiable the interval\[ℓi,ui\]\[\\ell\_\{i\},u\_\{i\}\]is non\-empty \(Fourier\-Motzkin completeness\)\. The interior placement also requires the*open*interval to be non\-empty,ℓi<ui\\ell\_\{i\}<u\_\{i\}: ifΠ\\Pipinchesℓi=ui\\ell\_\{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 andσ\\sigmamapℝ\\mathbb\{R\}into the interior of its admissible interval, so each strict inequality in whichxix\_\{i\}is highest\-order holds with positive margin; a dependent variable is set by its identity to the single admissible valueℓi=ui\\ell\_\{i\}=u\_\{i\}, meeting its two non\-strict encoding inequalities with margin00; and a free variable is highest\-order in no constraint, so it imposes nothing\. Induction over the order givesφ⁡\(z\)⊧Π\\varphi\(z\)\\models\\Pi\.□\\square

## Appendix DThe Resolution Ratio as Condition Number

Figure[3](https://arxiv.org/html/2608.21605#A4.F3)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’sU−U0U\-U\_\{0\}hasR≈9×104R\\approx 9\\times 10^\{4\}\. The margin’s spreadσm\\sigma\_\{m\}is about10510^\{5\}times smaller than the scaleσs\\sigma\_\{s\}, so after standardization the margin occupies a band of relative width∼1/R≈10−5\{\\sim\}1/R\\approx 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 ontoU=U0U\{=\}U\_\{0\}, and the distribution of the constraint margin collapses to a point mass at00\. Taxi’stotal−fare\\mathrm\{total\}\-\\mathrm\{fare\}hasR≈3\.6R\\approx 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 \(highRR\) and clamps the latter \(lowRR\)\.

![Refer to caption](https://arxiv.org/html/2608.21605v1/figures/resolution_regimes.png)Figure 3:High versus low resolution ratioRR\(schematic\)\.The same orderingb\>ab\>ain two regimes\.*Top:*the data\(a,b\)\(a,b\)and the boundaryb=ab=a\. At lowRR\(left\) the feasible marginb−ab\-ais a visible fraction of the scale, so a free generator resolves it and the constraint layer’s clamp rarely acts\. At highRR\(right\) 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 lowRRthe clamp \(orange\) preserves the distribution of the real constraint margin \(grey\); at highRRthe free generator cannot place the sub\-resolution margin, so the clamp collapses every sample onto the boundary, a point mass at00disjoint from the real distribution\. Function\-symbol grounding repairs the high\-RRcase by making the margin a unit\-scale coordinate \(Figure[4](https://arxiv.org/html/2608.21605#A4.F4)\)\.We now prove Corollaries[3](https://arxiv.org/html/2608.21605#Thmtheorem3)and[4](https://arxiv.org/html/2608.21605#Thmtheorem4)and Proposition[5](https://arxiv.org/html/2608.21605#Thmtheorem5)\.

*Proof of Corollary[3](https://arxiv.org/html/2608.21605#Thmtheorem3)\.*Full rank of∇c\\nabla conℳ=\\mathcal\{M\}\_\{=\}makes00a regular value, so by the regular\-value theoremℳ==c−1​\(0\)\\mathcal\{M\}\_\{=\}=c^\{\-1\}\(0\)is an embeddedC1C^\{1\}submanifold of dimensionD−pD\-p; a submanifold of dimension below the ambientDDhas Lebesgue measure zero\. Ifν≪Leb\\nu\\ll\\mathrm\{Leb\}with densityff, thenν⁡\(ℳ=\)=∫ℳ=f​𝑑Leb=0\\nu\(\\mathcal\{M\}\_\{=\}\)=\\int\_\{\\mathcal\{M\}\_\{=\}\}f\\,\\mathrm\{d\}\\,\\mathrm\{Leb\}=0\. Grounding sets the dependent coordinate to the exact value its identity prescribes, so every generatedxxlies inℳ=\\mathcal\{M\}\_\{=\}by construction; the generated distribution is then carried by the null setℳ=\\mathcal\{M\}\_\{=\}, soν≪̸Leb\\nu\\not\\ll\\mathrm\{Leb\}, escaping the hypothesis, andc⁡\(x\)=0c\(x\)=0for every sample\.□\\square

*Proof of Corollary[4](https://arxiv.org/html/2608.21605#Thmtheorem4)\.*Differentiating,∂\(b−a\)σ⁡\(\(b−a\)/s\)=σ′​\(\(b−a\)/s\)/s=Ps​\(1−Ps\)/s\\partial\_\{\(b\-a\)\}\\sigma\\\!\\big\(\(b\-a\)/s\\big\)=\\sigma^\{\\prime\}\\\!\\big\(\(b\-a\)/s\\big\)/s=P\_\{s\}\(1\-P\_\{s\}\)/s\. The logistic derivativeσ′​\(t\)=σ⁡\(t\)​\(1−σ⁡\(t\)\)\\sigma^\{\\prime\}\(t\)=\\sigma\(t\)\\,\(1\-\\sigma\(t\)\)isΘ⁡\(1\)\\Theta\(1\)on any fixed band\|t\|≤c\|t\|\\leq c\(peakσ′​\(0\)=14\\sigma^\{\\prime\}\(0\)=\\tfrac\{1\}\{4\}\) and decays ase−\|t\|e^\{\-\|t\|\}for\|t\|≫1\|t\|\\gg 1, so the gradient isΘ⁡\(1/s\)\\Theta\(1/s\)exactly for\|b−a\|≲s\|b\-a\|\\lesssim sand exponentially suppressed otherwise\. Withb−ab\-aat scaleσs=R​σm\\sigma\_\{s\}=R\\,\\sigma\_\{m\}and densityΘ⁡\(1/σs\)\\Theta\(1/\\sigma\_\{s\}\)near00, the probability mass on the width\-Θ⁡\(s\)=Θ⁡\(σm\)\\Theta\(s\)=\\Theta\(\\sigma\_\{m\}\)active band isΘ⁡\(σm/σs\)=Θ⁡\(1/R\)\\Theta\(\\sigma\_\{m\}/\\sigma\_\{s\}\)=\\Theta\(1/R\)\.□\\square

*Proof of Proposition[5](https://arxiv.org/html/2608.21605#Thmtheorem5)\.**\(i\)*On the orderingϕ\\phi, the clamp leaves a satisfying sample’s margin unchanged and moves each violator to the boundary valueε\\varepsilon, giving the mixture CDFFCL\(t\)=v1\[t≥ε\]\+\(1−v\)F\+\(t\)F\_\{\\mathrm\{CL\}\}\(t\)=v\\,\\mathbf\{1\}\[t\\geq\\varepsilon\]\+\(1\-v\)\\,F^\{\+\}\(t\), withF\+F^\{\+\}the CDF ofν\+\\nu^\{\+\}\. Two evaluations boundKS=supt\|FCL​\(t\)−F⁡\(t\)\|\\mathrm\{KS\}=\\sup\_\{t\}\|F\_\{\\mathrm\{CL\}\}\(t\)\-F\(t\)\|\. Att=εt=\\varepsilon, below the real support,FCL​\(ε\)−F⁡\(ε\)≥vF\_\{\\mathrm\{CL\}\}\(\\varepsilon\)\-F\(\\varepsilon\)\\geq v\. Att=qαt=q\_\{\\alpha\}, the real margin’sα\\alpha\-quantile \(soqα=Θ⁡\(σm\)q\_\{\\alpha\}=\\Theta\(\\sigma\_\{m\}\)\),F⁡\(qα\)−FCL​\(qα\)≥α−v−\(1−v\)​F\+​\(qα\)F\(q\_\{\\alpha\}\)\-F\_\{\\mathrm\{CL\}\}\(q\_\{\\alpha\}\)\\geq\\alpha\-v\-\(1\-v\)\\,F^\{\+\}\(q\_\{\\alpha\}\), and under Corollary[4](https://arxiv.org/html/2608.21605#Thmtheorem4)’s density hypothesis the generator places massΘ⁡\(qα/\(R​σm\)\)=Θ⁡\(1/R\)\\Theta\\bigl\(q\_\{\\alpha\}/\(R\\,\\sigma\_\{m\}\)\\bigr\)=\\Theta\(1/R\)in\(0,qα\]\(0,q\_\{\\alpha\}\], so\(1−v\)​F\+​\(qα\)=Θ⁡\(1/R\)\(1\-v\)\\,F^\{\+\}\(q\_\{\\alpha\}\)=\\Theta\(1/R\)\. Takingα→1\\alpha\\to 1,KS≥max⁡\(v,1−v−Θ⁡\(1/R\)\)≥max⁡\(v,1−v\)−Θ⁡\(1/R\)≥12−Θ⁡\(1/R\)\\mathrm\{KS\}\\geq\\max\\bigl\(v,\\,1\-v\-\\Theta\(1/R\)\\bigr\)\\geq\\max\(v,1\-v\)\-\\Theta\(1/R\)\\geq\\tfrac\{1\}\{2\}\-\\Theta\(1/R\), and the first evaluation alone givesKS→1\\mathrm\{KS\}\\to 1asv→1v\\to 1\. With several constraints the statement reads per ordering through its own clamp step; empirically the atom sits at the boundary on every dataset \(Figure[2](https://arxiv.org/html/2608.21605#S5.F2)\)\.*\(ii\)*Each link is a smooth bijection onto its admissible interval with smooth inverse: softplus mapsℝ\\mathbb\{R\}onto\(0,∞\)\(0,\\infty\)with inversesoftplus−1​\(y\)=log⁡\(ey−1\)\\mathrm\{softplus\}^\{\-1\}\(y\)=\\log\(e^\{y\}\-1\); the affine logistic mapℓ\+\(u−ℓ\)​σ​\(⋅\)\\ell\+\(u\-\\ell\)\\,\\sigma\(\\cdot\)mapsℝ\\mathbb\{R\}onto\(ℓ,u\)\(\\ell,u\)with inverse the scaled logitz=log⁡t1−tz=\\log\\frac\{t\}\{1\-t\}att=\(x−ℓ\)/\(u−ℓ\)t=\(x\-\\ell\)/\(u\-\\ell\); the heavy\-tailedexp\\explink mapsℝ\\mathbb\{R\}onto\(ℓ,∞\)\(\\ell,\\infty\)with inverselog⁡\(x−ℓ\)\\log\(x\-\\ell\); a free coordinate is the identity\. Assembling in the Fourier\-Motzkin order, the bounds ofxix\_\{i\}read onlyx<ix\_\{<i\}\(Appendix[C](https://arxiv.org/html/2608.21605#A3)\), so the inverse is computed coordinate\-wise\. Eachziz\_\{i\}is the link inverse ofxix\_\{i\}at the bounds set byx<ix\_\{<i\}, defined and smooth exactly when every constrainedxix\_\{i\}is strictly interior, i\.e\. on the relative interior ofℳ\\mathcal\{M\}\(dependent coordinates are recovered by their identities and contribute no coordinate\)\. Induction along the order givesφ∘φ−1=id\\varphi\\circ\\varphi^\{\-1\}=\\mathrm\{id\}onrelint​ℳ\\mathrm\{relint\}\\,\\mathcal\{M\}andφ−1∘φ=id\\varphi^\{\-1\}\\circ\\varphi=\\mathrm\{id\}onℝd\\mathbb\{R\}^\{d\}, and interior placement \(Appendix[C](https://arxiv.org/html/2608.21605#A3)\) meansφ\\varphiplaces no probability on the boundary\. Realizability: given any margin distributionμ\\muon\(0,∞\)\(0,\\infty\), the coordinate distributionsoftplus\#−1​μ\\mathrm\{softplus\}^\{\-1\}\_\{\\\#\}\\mupushes forward under softplus to exactlyμ\\mu\. When several reduced constraints bound the same variable,ℓi\\ell\_\{i\}anduiu\_\{i\}are maxima and minima of linear forms, hence piecewise\-linear\.φ\\varphiandφ−1\\varphi^\{\-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\.□\\square

Figure[4](https://arxiv.org/html/2608.21605#A4.F4)shows the mechanism of Remark[2](https://arxiv.org/html/2608.21605#Thmtheorem2)in pictures\. The change of coordinates that givesΘ⁡\(1\)\\Theta\(1\)conditioning turns the sub\-resolution margin of Figure[3](https://arxiv.org/html/2608.21605#A4.F3)into a unit\-scale coordinate the GAN can learn, then decodes it back to valid samples with the distribution of the constraint margin recovered\.

![Refer to caption](https://arxiv.org/html/2608.21605v1/figures/chart_mechanism.png)Figure 4:How function\-symbol grounding repairs the high\-RRcollapse \(schematic\)\.The same high\-RRordering margin as in Figure[3](https://arxiv.org/html/2608.21605#A4.F3); top row shows the margin*distribution*at each stage, bottom row the corresponding\(a,b\)\(a,b\)*geometry*\.*\(1\)*In ambient space the marginb−ab\-ais sub\-resolution, so the constraint layer clamps it to a point mass at00\.*\(2\)*The chart coordinatez=φ−1​\(x\)z=\\varphi^\{\-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 \(generator in blue matches real in grey\)\.*\(3\)*Decodingx=φ⁡\(z\)=ℓ\+softplus⁡\(z\)x=\\varphi\(z\)=\\ell\+\\mathrm\{softplus\}\(z\)carries that distribution back to the ambient margin\.*\(4\)*Geometrically the clamp moves every violating sample onto the boundaryb=ab=a\(margin00, the point mass of panel 1\)\.*\(5\)*The function symbolb=a\+softplus⁡\(z\)b=a\+\\mathrm\{softplus\}\(z\)is positive for every generator outputzz, sob\>ab\>aholds by construction \(Sat⁡\(KB\)=1\\mathrm\{Sat\(KB\)\}=1\); it is smooth and monotone, so matching the distribution ofzzmatches the margin’s distribution, and asymptotically linear, so generator tails do not diverge\.*\(6\)*Our decoded samples therefore land inside the feasible regionb\>ab\>a, matching the real distribution with exact validity\.

## Appendix EConstraint\-Layer Code Corrections and Evaluation Protocol

We run the constraint layer from the code released by[35](https://arxiv.org/html/2608.21605#bib.bib4)\. For genuine100%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\-inequalityε\\varepsilonset above one unit in the last place \(its defaultε=10−12\\varepsilon=10^\{\-12\}underflows here, falsely reporting0%0\\%valid\)\. 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 inevalmode\. 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 a20,00020\{,\}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 \(softplus orexp\\exp\) is set by the fixed dynamic\-range rule of Appendix[G](https://arxiv.org/html/2608.21605#A7)\. A definitional identity is scored satisfied when its residual is within a per\-dataset tolerance in raw units \(0\.050\.05on Alchemy,10−310^\{\-3\}on tmQM,0\.50\.5on nycflights13\)\. Corollary[3](https://arxiv.org/html/2608.21605#Thmtheorem3)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 is0\.0000\.000\. Per\-datasetRR, sizes, constraints, and full per\-method results with standard deviations are given in Appendices[F](https://arxiv.org/html/2608.21605#A6)to[J](https://arxiv.org/html/2608.21605#A10)\.

## Appendix FDatasets, Constraints, and Resolution Ratios

Table[4](https://arxiv.org/html/2608.21605#A6.T4)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 ratioR=σs/σmR=\\sigma\_\{s\}/\\sigma\_\{m\}\(Section[3](https://arxiv.org/html/2608.21605#S3)\) is computed on the raw data before training\. The constraint\-layer benchmark of[35](https://arxiv.org/html/2608.21605#bib.bib4)used in RQ3 is described in Appendix[I](https://arxiv.org/html/2608.21605#A9)\.

Table 4:The four real high\-resolution datasets\.NNrows,DDproperties;RRis the resolution ratio of each evaluated ordering margin\.The full structural constraint set per dataset is as follows\.Alchemy\([7](https://arxiv.org/html/2608.21605#bib.bib11)\), 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 identitylumo=homo\+gap\\mathrm\{lumo\}=\\mathrm\{homo\}\+\\mathrm\{gap\}over the frontier\-orbital energies; the thermochemical orderingsU0<U<HU\_\{0\}<U<HandG<HG<H, whereU0U\_\{0\}andUUare the internal energies at00K and298298K,HHthe enthalpy, andGGthe free energy\.tmQM\([4](https://arxiv.org/html/2608.21605#bib.bib15)\), 8 properties of transition\-metal complexes: the high\-RRdispersion orderingEtot<EelecE\_\{\\mathrm\{tot\}\}<E\_\{\\mathrm\{elec\}\}; positivity of dipole, gap, and polarizability; the low\-RRidentityLUMO=HOMO\+gap\\mathrm\{LUMO\}=\\mathrm\{HOMO\}\+\\mathrm\{gap\}\.Transition1x\([32](https://arxiv.org/html/2608.21605#bib.bib16)\), reaction energy profiles: the transition state is the maximum,ETS\>EreactE\_\{\\mathrm\{TS\}\}\>E\_\{\\mathrm\{react\}\}andETS\>EprodE\_\{\\mathrm\{TS\}\}\>E\_\{\\mathrm\{prod\}\}\. \(0\.08%0\.08\\%of raw rows violateETS\>EprodE\_\{\\mathrm\{TS\}\}\>E\_\{\\mathrm\{prod\}\}, near\-barrierless reverse reactions, and are retained, so the real reference margin carries that small negative mass\.\)Taxi\([28](https://arxiv.org/html/2608.21605#bib.bib17)\), NYC yellow\-cab records \(2024\-01\): the duration orderingdropoff\>pickup\\mathrm\{dropoff\}\>\\mathrm\{pickup\}on absolute second\-scale timestamps \(highRR\);total\>fare\\mathrm\{total\}\>\\mathrm\{fare\}\(lowRR\); positivity of distance and fare\.

## Appendix GArchitecture 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 dropout0\.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 perceptronD→256→128→1D\\to 256\\to 128\\to 1with LeakyReLU\(0\.20\.2\), dropout0\.10\.1, and a sigmoid output\. Both train with Adam \(learning rate2×10−42\{\\times\}10^\{\-4\},β=\(0\.5,0\.999\)\\beta=\(0\.5,0\.999\)\), batch size 256, binary cross\-entropy adversarial loss with label smoothing \(targets0\.90\.9real,0\.10\.1generated\), for 1000 steps\. Two numerical safeguards apply throughout\. The decoder floors each charted margin at10−610^\{\-6\}in raw units, and the encoder clips the real chart coordinates at their0\.50\.5and99\.599\.5percentiles before standardization\. Both act only in a thin boundary layer, and neither affects validity\. The analysis of Section[4](https://arxiv.org/html/2608.21605#S4)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 coordinateszzand the grounding mapφ\\varphiassembles the sample \(Section[4](https://arxiv.org/html/2608.21605#S4)\)\. No satisfaction loss is added because charted axioms hold by construction\. The constraint layer runs on the identical architecture \(Appendix[E](https://arxiv.org/html/2608.21605#A5)\)\. The grounding link is softplus by default andexp\\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 ratioq99/q50q\_\{99\}/q\_\{50\}of its positive part on the training data, and selects theexp\\explink when any charted margin reachesq99/q50≥10q\_\{99\}/q\_\{50\}\\geq 10\. Across the eleven datasets in this paper it selectsexp\\expexactly once, on faults, whose bounding\-box margins haveq99/q50=18\.0q\_\{99\}/q\_\{50\}=18\.0and13\.013\.0\. It selects softplus everywhere else \(every charted margin outside faults hasq99/q50≤5\.1q\_\{99\}/q\_\{50\}\\leq 5\.1\)\. The released code records the resulting per\-dataset choice\.

## Appendix HFull Per\-Method Results \(RQ1 and RQ2\)

Table[5](https://arxiv.org/html/2608.21605#A8.T5)extends Table[1](https://arxiv.org/html/2608.21605#S5.T1)with all baselines and their validities; Table[6](https://arxiv.org/html/2608.21605#A8.T6)extends Table[2](https://arxiv.org/html/2608.21605#S5.T2)with the raw unconstrained architectures, whose validities are given in its caption\. The paired Wilcoxon test for FSG\-LTN\-GAN versus C\-DGM returnsp=0\.00195p=0\.00195on every dataset, the minimum attainable atn=10n=10\. In Table[5](https://arxiv.org/html/2608.21605#A8.T5), the projection row coincides with the unconstrained row on tmQM for the reason given in Table[6](https://arxiv.org/html/2608.21605#A8.T6)’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=10seeds, mean±\\pmstd\. 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\.Table 6:Full RQ2 \(margin KS by architecture,n=10n=10, mean±\\pmstd\)\. Raw CTGAN/TVAE validity is below3%3\\%on the chemistry sets and3030to46%46\\%on Transition1x and Taxi\. The\+\+CL rows, C\-DGM, and FSG\-LTN\-GAN all reach100%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 atε\\varepsilonwhile the reference margin lies aboveε\\varepsilon, so the KS supremum, attained atε\\varepsilon, 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\.
## Appendix IThe Constraint\-Layer Benchmark, Full \(RQ3\)

Table[7](https://arxiv.org/html/2608.21605#A9.T7)extends Table[3](https://arxiv.org/html/2608.21605#S5.T3)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 reach100%100\\%validity; unconstrained validity ranges from0\.0050\.005\(wids\) to0\.7360\.736\(url\)\. The hybrid charts the constraints its pre\-run selects \(free\-generator satisfaction below0\.90\.9, margin not discrete\) 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 \(p=0\.03p=0\.03and0\.0040\.004\) and a statistical tie on news \(heloc and lcld are ties as well\)\. On faults \(whose high\-RRmargin is the bounding\-box orderingYmax\>YminY\_\{\\max\}\>Y\_\{\\min\}\) it charts every variable and improves substantially \(paired Wilcoxonp=0\.002p=0\.002, all ten seeds\)\. 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](https://arxiv.org/html/2608.21605#bib.bib4),n=10n=10seeds, mean±\\pmstd, margin KS\. “charted” is the number of variables the hybrid charts as function symbols;boldmarks where the hybrid significantly outperforms the constraint layer \(paired Wilcoxonp<0\.05p<0\.05\)\.
## Appendix JAdditional Analyses

#### Density and coverage\.

Table[8](https://arxiv.org/html/2608.21605#A10.T8)reports density and coverage\([27](https://arxiv.org/html/2608.21605#bib.bib12)\), the constraint layer’s own realism metrics, computed on standardized features and shown alongside margin KS\.

Table 8:Density and coverage\([27](https://arxiv.org/html/2608.21605#bib.bib12)\)\(n=10n=10, mean±\\pmstd\), with margin KS for reference\. Higher is better for density and coverage, lower for KS\. The better method per column is in bold\.
#### The failure is not a variable\-ordering artifact\.

A constraint layer must fix a Fourier\-Motzkin elimination order\. Table[9](https://arxiv.org/html/2608.21605#A10.T9)runs the constraint layer on Transition1x under every dependency\-valid ordering: all stay near1\.01\.0margin KS, so this failure is intrinsic to clamping\. Function\-symbol grounding, which fixes one ordering, is at0\.0650\.065\. All configurations are100%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 \(Transition1x,n=10n=10, mean±\\pmstd, margin KS\)\. Bracketed lists are the Fourier\-Motzkin variable elimination orders\.
#### Other backbones: CTGAN and TVAE in the chart\.

Table[2](https://arxiv.org/html/2608.21605#S5.T2)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 byφ−1\\varphi^\{\-1\}, fit the unmodified backbone with its package\-default hyperparameters at100100epochs \(the RQ2 protocol\), and decode its samples throughφ\\varphi\. Table[10](https://arxiv.org/html/2608.21605#A10.T10)applies this recipe to CTGAN and TVAE with no architectural change: both become exactly valid and recover the margin, with margin KS3\.63\.6to8\.6×8\.6\\timesbelow their constraint\-layer counterparts \(paired Wilcoxonp=0\.002p\{=\}0\.002on every dataset–backbone pair\)\. 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\(margin KS↓\\downarrow,n=10n\{=\}10, mean±\\pmstd\)\. “in chart” trains the unmodified backbone onφ−1\\varphi^\{\-1\}\-encoded data and decodes throughφ\\varphi\. Both chart columns are exactly100%100\\%valid\. “\+\+CL” columns from Table[2](https://arxiv.org/html/2608.21605#S5.T2)\.
#### 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 \(327,346327\{,\}346rows after cleaning, subsampled to50,00050\{,\}000; six properties\), with the orderingarrival\>departure\\mathrm\{arrival\}\>\\mathrm\{departure\}on absolute timestamps spanning a year, the exact identitydep=sched\+delay\\mathrm\{dep\}=\\mathrm\{sched\}\+\\mathrm\{delay\}, and positivity of air time and distance\.*Before training*we computedR=3\.0×103R=3\.0\\times 10^\{3\}for the ordering from the raw data and recorded the predicted outcome \(constraint\-layer margin KS near11, FSG\-LTN\-GAN below0\.150\.15, both exactly valid; unconstrained validity near00\), together with explicit falsification criteria, in a file included in the code release\. The sweep then ran once, with ten seeds\. Every prediction held \(Table[11](https://arxiv.org/html/2608.21605#A10.T11)\)\. 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 \(nycflights13\)\(n=10n\{=\}10, mean±\\pmstd\)\.RRand the predicted outcome were recorded before training\.
#### 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 coordinateszz, 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 chartφ\\varphiare unchanged, so validity remains exact, but the discriminator receives the decoded, standardized ambient sample instead ofzz\. Table[12](https://arxiv.org/html/2608.21605#A10.T12)shows margin KS degrades on every dataset \(paired Wilcoxonp=0\.002p\{=\}0\.002each\), and the degradation tracksRR\(largest on the chemistry sets\), 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\. \(Validity is1\.0001\.000in every run except a single tmQM seed at0\.9990\.999, a float32 identity\-reconstruction round\-off in the ambient arm’s training\-time decode\.\)

Table 12:Validity by construction versus discriminating in the chart\.Margin KS \(n=10n\{=\}10, mean±\\pmstd\)\. FSG\-ambient\-D keeps the chart generator \(validity exact\) but shows the discriminator the decoded ambient sample; C\-DGM from Table[5](https://arxiv.org/html/2608.21605#A8.T5)for reference\.
#### Margin\-form preprocessing is a manual chart\.

For a single ordering one can instead transform the*data*: on Taxi, replace dropoff withduration=dropoff−pickup\\mathrm\{duration\}=\\mathrm\{dropoff\}\-\\mathrm\{pickup\}, standardize, train the constraint layer withduration\>0\\mathrm\{duration\}\>0, and decode afterwards\. Because the margin becomes its own column, its resolution ratio drops toR=1R\{=\}1\(from1\.1×1031\.1\\times 10^\{3\}\), and the clamp then largely preserves the margin distribution\. Margin KS falls0\.574→0\.134±0\.0200\.574\\to 0\.134\\pm 0\.020\(duration KS1\.000→0\.118±0\.0331\.000\\to 0\.118\\pm 0\.033\) at exact validity \(Table[13](https://arxiv.org/html/2608.21605#A10.T13)\)\. 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 is52\.1%±5\.552\.1\\%\\pm 5\.5valid \(duration can still be generated negative, and the untouched constraints are violated\), so a clamp or a chart is still required\. Second, rewriting a*conjunction*into consistent margin form \(Alchemy’sU0<U<HU\_\{0\}<U<Hchain together with its lumo identity\) 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 \(0\.099±0\.0100\.099\\pm 0\.010; paired Wilcoxonp=0\.002p\{=\}0\.002against the preprocessed constraint layer, all ten seeds\)\.

Table 13:Margin\-form preprocessing on Taxi\(n=10n\{=\}10, mean±\\pmstd\)\. 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\-RRduration margin alone\.
#### 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 valuettfor 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, \(ii\) on target, and \(iii\) 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](https://arxiv.org/html/2608.21605#A10.T14), function\-symbol grounding is the only configuration that delivers all three at once: exact validity \(the tmQM chart value is0\.99990\.9999before rounding, a float32 identity round\-off\), 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 highRR\. Pushed toward satisfaction, it misses the target \(target error22to5×5\\timesworse\) yet still fails validity\. Projection attains validity but inherits the unconstrained margins\.

Table 14:Conditional inverse design\(n=10n\{=\}10seeds, mean±\\pmstd, 1500 steps\)\. 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\.
#### Sensitivity of the hybrid’s thresholds\.

The hybrid’s two thresholds are fixed once \(τs=0\.9\\tau\_\{s\}\{=\}0\.9,τd=0\.2\\tau\_\{d\}\{=\}0\.2\) and shared by every dataset in the paper\. Sweeping the gridτs∈\{0\.70,0\.80,0\.85,0\.90,0\.95\}×τd∈\{0\.05,0\.10,0\.20,0\.30,0\.50\}\\tau\_\{s\}\\in\\\{0\.70,0\.80,0\.85,0\.90,0\.95\\\}\\times\\tau\_\{d\}\\in\\\{0\.05,0\.10,0\.20,0\.30,0\.50\\\}changes the charted set on55to1919of the2525cells, depending on the dataset, and retraining every distinct alternative set the grid produces \(ten seeds each, Table[15](https://arxiv.org/html/2608.21605#A10.T15)\) changes no conclusion: every configuration keeps exact validity, alternates move margin KS by at most≈0\.03\{\\approx\}0\.03on the low\-RRdatasets, 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 KS0\.1430\.143to0\.1520\.152, at or slightly above the constraint layer’s0\.1400\.140\. The one large change shows where the sensitivity lies\. On faults, the extremeτd=0\.05\\tau\_\{d\}\{=\}0\.05row stops charting the two bounding\-box margins \(including the high\-RRone\), and the main win shrinks \(KS0\.117→0\.5480\.117\\to 0\.548, still below the constraint layer’s0\.7450\.745\)\. The outcome is sensitive not to the threshold values but to whether the high\-RRmargins are charted, which is whatRRpredicts before training\.

Table 15:Threshold sensitivity on the RQ3 benchmark\(margin KS,n=10n\{=\}10, mean±\\pmstd\)\. “cells” is how many of the2525grid cells select each charted set\. The paper’s cell is\(τs,τd\)=\(0\.9,0\.2\)\(\\tau\_\{s\},\\tau\_\{d\}\)\{=\}\(0\.9,0\.2\)\.
#### RQ4 in table form\.

Table[16](https://arxiv.org/html/2608.21605#A10.T16)restates the RQ4 numbers of Section[5\.4](https://arxiv.org/html/2608.21605#S5.SS4)\.

Table 16:RQ4: predicate placements on the high\-RRorderings\(ordering satisfaction, the per\-ordering satisfaction fraction averaged over the dataset’s orderings,n=10n\{=\}10, mean±\\pmstd\)\. “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 than0\.090\.09above chance, and the discriminator\-feature placement falls well below it\. Function\-symbol grounding satisfies the ordering for every sample\.

## Appendix KCode and Data Availability

All datasets are publicly available \(Appendix[F](https://arxiv.org/html/2608.21605#A6)\); code, experiment scripts, and the prediction record of Appendix[J](https://arxiv.org/html/2608.21605#A10)are available at[FSG\-LTN\-GAN](https://github.com/nuuoe/FSG-LTN-GAN)\.