A Large Open Multi-Energy Corpus of Soil Compaction Tests, with Machine-Learning Baselines

# A Large Open Multi-Energy Corpus of Soil Compaction Tests, with Machine-Learning Baselines
Source: [https://arxiv.org/html/2609.03337](https://arxiv.org/html/2609.03337)
Sompote Youwai1, Chana Phutthananon, and Warat KongkitkulAffiliation:AI Research Group, Department of Civil Engineering King Mongkut’s University of Technology Thonburi 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand 1Corresponding author: sompote\.you@kmutt\.ac\.th

###### Abstract

Every engineered fill is specified by a maximum dry density and an optimum moisture content\. Each determination needs a full Proctor test\. Published correlations rest on one to four hundred specimens, usually from one laboratory at one compactive energy, and are seldom released\. This paper releases a corpus without those limits\. It holds 2,854 laboratory compaction tests from six public sources, across 162 provenance groups and four Proctor energy levels, with fines from 1\.5 to 100 %\. Every record is audited to the Proctor method its source names, and no energy is inferred\. Screening on the zero\-air\-voids condition removed 11\.8 % of harmonised records, and 5\.7 % of those with a measured specific gravity\. A material share of published compaction data is physically impossible\. The optimum degree of saturation over the corpus is 0\.815 at a coefficient of variation of 11 %\. That is a baseline, not a constant\. Both parameters are then estimated from one classification suite and the compaction standard\. A tabular foundation model reaches R20\.824 for density and 0\.784 for water content under random folds\. It reaches 0\.727 and 0\.696 with folds drawn around provenance, and 0\.520 and 0\.614 with a whole source held out\. Compactive energy is negligible marginally yet decisive conditionally\. Density on the 66 modified\-Proctor records is predicted at R20\.740 with it and−\-0\.651 without\. Symbolic regression yields closed forms coupled through a phase relation\. No predicted pair can then exceed the zero\-air\-voids line\. The predictions are for screening, not acceptance\.

###### Index Terms:

machine learning, symbolic regression, tabular foundation models, in\-context learning, physics\-constrained prediction, out\-of\-distribution generalisation, maximum dry density, optimum moisture content, compaction energy level

## IIntroduction

### I\-ACompaction parameters and why they are expensive

Compaction control governs the strength, stiffness, compressibility and permeability of every engineered fill\. Two parameters specify it\. The maximum dry density \(MDD\) is the highest dry density reached under a stated compactive effort\. The optimum moisture content \(OMC\) is the water content at which that maximum occurs\. Both come from the test of Proctor \[1\], standardised as ASTM D698 \[2\] and D1557 \[3\] and as AASHTO T99 \[4\] and T180 \[5\]\. The test compacts four to six specimens at different water contents\. Each is oven\-dried for 24 h, so one determination takes about a working day\. Specifications require a determination for every change of material\. The number of tests therefore follows the variability of the deposits, not the volume of fill placed\. This burden motivates the present study\.

The optimum matters in its own right, not only as a companion to the density\. It governs soil fabric\. Lambe \[6\] showed that a clay compacted dry of optimum develops a flocculated particle arrangement, and the same soil compacted wet of optimum develops a dispersed one\. Seed and Chan \[7\] showed that the resulting differences in strength, volume change and permeability persist at equal dry density\. The compaction water content also sets the soil\-water potential of the fill, and with it the susceptibility to swelling or to collapse on wetting \[8\]\. Daniel and Benson \[9\] proposed that acceptance be defined as a zone in the dry density–water content plane rather than as independent bounds on each quantity\. That definition makes the position of the optimum a design quantity\. Continuous\- and intelligent\-compaction systems report a mechanistic response across an entire lift \[10\], and the laboratory optimum remains the reference against which those measurements are interpreted\.

The optimum is bounded\. The bound is the zero\-air\-voids \(ZAV\) line, along which the remaining voids are entirely water\-filled\. No compactive effort can produce a state above that line\. The condition is therefore a criterion of physical admissibility, and every record used here is screened against it \(Section 2\)\. Raising the compactive effort raises MDD and reduces OMC\. The peaks obtained at different energy levels trace a locus approximately parallel to the ZAV line\. Analytical descriptions of that family of curves \[11–13\] start from one observation\. For a given soil, the degree of saturation at the optimum is nearly invariant\.

Tatsuoka and Gomes Correia \[14\] developed that observation into a framework for compaction control\. They define the optimum degree of saturation\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}as the saturation at which the maximum dry density is reached for a given energy level\. From a large body of laboratory and field data they show that\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}and the normalised compaction curve change little with limited variation in energy level and soil type\. They argue that placement should be specified in terms of degree of saturation rather than water content\. Tatsuoka et al\. \[15\] carry the same normalisation into field control, and the nine soils tabulated there have\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}between 64\.1 % and 92\.3 %\. Both frameworks need\(ρd\)m​a​x\(\\rho\_\{d\}\)\_\{max\}and\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}for the material before any field measurement can be interpreted\. They therefore sit downstream of the laboratory compaction test rather than in place of it\. The present study addresses the upstream problem of estimating those quantities for a material that has not yet been compacted\. The evidence behind those frameworks combines laboratory and field data\. The corpus assembled here is laboratory\-only\. The two are not interchangeable, and Section 6 states the resulting restriction\.

### I\-BEstimating the parameters from index properties

Where the test is expensive and the parameters are needed for every material, the established approach is to estimate them from properties already measured\. Correlations of that kind fall into two groups\.

The first regresses on a single consistency limit\. Sridharan and Nagaraj \[16\] argued that the plastic limit governs the position of the peak\. Gurtug and Sridharan \[17\], Nagaraj et al\. \[18\], Sharma and Sridharan \[19\] and Vinod and Sridharan \[20\] developed the resulting forms\. Pandian et al\. \[21\] had earlier re\-examined the compaction characteristics of fine\-grained soils on a similar basis\. Blotz et al\. \[22\] remain the standard reference for estimating both parameters across compactive efforts from the liquid limit alone\.

The second group applies regression or machine learning to a wider set of index properties\. It begins with the artificial neural networks of Günaydın \[23\] and extends to recent tree\-ensemble, deep\-learning and hybrid formulations \[24,25\]\. Verma and Sivapullaiah \[26\] review the field\. Reported coefficients of determination are frequently above 0\.90\.

Five features of the underlying datasets limit what those figures show\. Table 1 lists them beside what this study does about each\. The fifth is the feature this paper is organised around\. Almost every study in the second group scores its model on a random hold\-out or a shuffled k\-fold partition of its own dataset\. Where that dataset comes from one laboratory, the resulting figure means what it states\. Where a dataset is assembled from several sources and then pooled and shuffled, the meaning changes\. A random fold can place records sharing a soil, a pavement layer, a laboratory protocol or a publication on both sides of the boundary\. The score then measures interpolation within the assembled corpus\.

Table 1 — Five features of the datasets behind published compaction correlations, and what this study does about each\.
### I\-CWhat public compaction data exists

Two kinds of dataset underlie the correlations above\. Neither is a database in the sense used here\.

The first kind is the set a study assembles for its own paper\. These are the one to four hundred specimens already described, and they are rarely deposited\. A correlation keyed on a consistency limit is also undefined for a soil that has none, so that branch of the literature is restricted to plastic material by construction\. Its reference points are accordingly clays and fine\-grained soils \[17,21,22\]\. The largest compilation in this literature is that of Rehman et al\. \[25\]\. They assemble 1,001 observations and model them across compactive energy levels with gradient\-boosted trees, reporting the liquid limit and the energy as the two leading predictors\. That work is the closest antecedent of the present study and the point from which it departs\. Its scope is cohesive soils, as its title states\. A granular fill has no consistency limits and therefore no place in it\. The compilation is not among the deposits retained in Section 2\.

The second kind is the deposited dataset\. Section 2 reports a census of these rather than a sample, and Table 2 gives the result\. The retained deposits are small, between 7 and 395 records\. Each comes from one laboratory or one compilation at one compactive energy\. Beside them sits the Long\-Term Pavement Performance release, which is large at 1,976 records but is a single testing programme following one protocol at standard effort\. No public source holds the individual points of a compaction curve, and none holds a field density\.

What is assembled here is therefore not a larger version of any of them\. It is the six sources pooled under one set of harmonisation conventions\. The provenance of every record is kept, so a fold can be drawn around it\. Four compactive energies are present rather than one\. 134 non\-plastic soils are carried alongside clays of liquid limit above 100 %\. The median fines content is 48\.6 % against a range of 1\.5 to 100 %\. The corpus therefore sits across the coarse–fine boundary rather than on the cohesive side of it\.

### I\-DScope and objectives

The consequence for practice is direct\. An engineer meeting a new borrow material cannot tell whether a published correlation was fitted on soils like it\. The engineer can neither refit it nor know how it will behave on material from a laboratory the correlation never saw\. The gap is therefore not a better regression on the datasets that already exist\. It is the absence of the data those regressions need\.

Pooling the sources does three things that none of them does alone\. It spans the soils a project actually meets rather than one deposit\. It carries more than one compactive energy, so effort becomes a variable rather than a constant\. And it keeps the provenance of every record, so a model can be scored on a laboratory it has never seen\. That last point is the question an engineer is asking, and it governs the evaluation throughout\.

Two estimators are then given, because they are used differently\. A fitted ensemble is the more accurate\. It has to be run as software, its coefficients cannot be inspected, and nothing in it prevents a physically impossible prediction\. A closed\-form expression is less accurate\. It can be written into a specification, checked by hand, and built so that the pair it returns cannot lie above the zero\-air\-voids line\. Both are reported, and the cost of the closed form is measured rather than assumed\.

The estimation problem is stated in full in Section 4\.1 and summarised in Table 8\. One routine classification suite on a disturbed sample gives the plastic limit, the plasticity index, the fines content, the sand content and the specific gravity\. The compaction standard is also known\. The task is to estimate MDD and OMC from those inputs\. No quantity derived from the compaction curve itself is used as a predictor\.

This study has three objectives\.

1. 1\.To assemble a corpus of laboratory compaction tests from public sources\. It holds 2,854 tests from six sources, spans four Proctor energy levels and crosses the coarse–fine boundary\. It is an order of magnitude larger than the datasets on which the published correlations rest\. Every record carries the compactive energy its source states, and records failing the zero\-air\-voids condition are reported rather than removed silently\.
2. 2\.To evaluate a model that predicts MDD and OMC from a single classification suite\. Three classes of learner are carried throughout, so the results reflect the database rather than one modelling choice\. The evaluation uses fold designs that respect the provenance of each record\.
3. 3\.To propose closed\-form expressions for both parameters\. The optimum moisture content is recovered from the estimated density through the phase relation at a fixed optimum degree of saturation\. That constraint keeps every estimated pair below the zero\-air\-voids line\.

Three further results follow\. Drawing folds around the 162 provenance groups costs about 0\.10 in R2at an identical training\-set size\. Compactive energy appears negligible marginally, yet it is decisive once plasticity and gradation are held constant\. The optimum degree of saturation of Tatsuoka and Gomes Correia \[14\], measured over 2,854 tests, has a mean of 0\.815 and a coefficient of variation of 11 %\.

Section 2 assembles and screens the corpus\. Section 3 describes the surviving records\. Section 4 defines the inputs, the models and the three fold designs\. Section 5 reports the results\. Section 6 states what the corpus cannot support, and Section 7 concludes\.

## IIAssembling and screening the corpus

### II\-ASearch and retention

Two channels were searched in August 2026, following PRISMA 2020 \[27\] and PRISMA\-S \[28\]\. Table 2 gives the result of each\.

Deposited datasets were sought through the DataCite application programming interface across Zenodo, figshare, Dryad, Mendeley Data and the Open Science Framework\. DataCite registers most research datasets, so this search approximates a census rather than a sample\. Open\-access articles were located through OpenAlex\. Both channels reach only open\-access material and query in English, so the corpus samples the openly available literature rather than compaction testing at large\.

A deposited dataset was retained only if it met five conditions: an open licence permitting redistribution, both MDD and OMC on every record, at least one index property on every record, provenance identifiable at record level, and the two targets being the peak of a laboratory compaction test\. The last condition sets the scope of everything that follows\. Field densities, nuclear\-gauge readings and optima back\-calculated from field data are all outside it\.

Table 2 — Search and retention\. Both channels were run in August 2026\. The article stream was excluded on completeness, not on quality\.Two exclusions deserve comment because they are decisions rather than filters\. Two figshare deposits of 502 records met all five conditions but were excluded on integrity\. Each follows every apparently genuine specimen with about nine near\-copies perturbed in the fourth to sixth decimal place\. The article and thesis stream was excluded in full on completeness\. Extraction recovers both targets on all of it, but fines on 3\.7 % of records, sand on 4\.8 % and compactive energy on none\. Admitting it is the most costly choice reported anywhere in this paper\. It takes the gradient\-boosted model from R20\.819 to 0\.536 for density, and from 0\.783 to 0\.467 for water content\.

### II\-BThe six retained sources

Table 3 describes the six sources\. Every retained record is the peak of a laboratory compaction curve, and the origin of each was audited source by source\.

For LTPP the evidence is documentary\. The programme’s own documentation describes the source table as holding standard Proctor results \[29\], which makes every LTPP record AASHTO T99 at 592\.5 kJ/m3\. The density field in that table carries no unit, and only one of the two candidate readings returns physically sensible saturations, which settles it\. For the five deposits the evidence is the deposit itself\. No source reports a field density, so the dataset is laboratory\-only\. Every statement in this paper about MDD, OMC and\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}is a statement about laboratory compaction\.

Table 3 — The six retained sources\. Deposits are named by their identifier: LTPP SDR 39 is Federal Highway Administration \[30\], figshare 28681187 is Günaydın \[23\], Zenodo 20737270 is the Türkiye deposit and Zenodo 14251190 is Soranzo \[31\]\.nnis records retained after screening\. The fines and OMC columns are medians over those records\.GsG\_\{s\}meas\. counts the records reporting a measured specific gravity rather than the substituted median\.Two things follow from Table 3\. LTPP carries 69 % of the records, and its material is coarser and drier than four of the five deposits\. A pooled accuracy figure is therefore weighted towards LTPP soils\. And the multi\-energy deposit, figshare 32955851, is a single source\. Every non\-standard test in the corpus comes from it, which constrains what the source\-held\-out design can test \(Sections 5\.4 and 6\)\.

### II\-CHarmonisation

Three columns need not mean the same thing in every source\. Each was audited record by record, and Table 4 states the convention adopted and its measured consequence\.

Table 4 — The three harmonisation audits\. The effect column reports a measurement, not an assumption\.The energy convention is worth stating twice, because it is what makes the corpus multi\-energy\. No compactive energy is inferred from the data\. A record whose source does not identify its Proctor method is excluded rather than assigned a value\. The four levels are reduced standard, standard, reduced modified and modified Proctor\.

### II\-DPhysical admissibility

Admissibility was checked on every harmonised record, using quantities the record already contains\. Withρw\\rho\_\{w\}taken as 1 Mg/m3,

e\\displaystyle e=GsMDD−1\\displaystyle=\\frac\{G\_\{s\}\}\{\\mathrm\{MDD\}\}\-1\(1\)So​p​t\\displaystyle S\_\{opt\}=OMC⋅Gse\\displaystyle=\\frac\{\\mathrm\{OMC\}\\cdot G\_\{s\}\}\{e\}\(2\)
So​p​tS\_\{opt\}is the optimum degree of saturation\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}of Tatsuoka and Gomes Correia \[14\], computed here per record rather than per soil\. It gives a direct test\. A value above unity puts the record above the zero\-air\-voids line, which is impossible as published\. A very low value points to a quantity that has been scaled wrongly\. Records were kept whereSo​p​tS\_\{opt\}lies between 0\.20 and 1\.0, MDD between 0\.8 and 2\.6 Mg/m3and OMC between 1 and 70 %\. The window onSo​p​tS\_\{opt\}is deliberately wide, because the quantity is derived and inherits any error in an assumed specific gravity\.

Table 5 gives the result\. Screening removed 11\.8 % of the 8,242 harmonised records, and 94 % of those removals were above the zero\-air\-voids line\. Taken at face value, about one published compaction record in nine describes a soil that cannot exist\.

Table 5 — The zero\-air\-voids screen\. The upper block splits the harmonised records by how specific gravity was obtained\. The lower block re\-runs the screen on the same records with the substituted value changed\.The two blocks of Table 5 answer different questions\. The upper block shows that the rejection rate is more than twice as high where specific gravity had to be substituted\. The lower block is the more reliable comparison, because the records do not change between runs\. Moving the substituted value from 2\.60 to 2\.75 moves the rate from 19\.8 % to 8\.2 %\. The finding that a material share of published compaction data is physically impossible is robust\. The figure of one in nine is not\.

The full protocol behind this section is released with the code, including the query strings, the per\-candidate screening decisions, the source adapters and the three harmonisation audits\.

## IIIThe dataset

### III\-AComposition and the two deliberate gaps

The records surviving the screening of Section 2 form the dataset modelled here\. It holds 2,854 laboratory compaction tests across 162 provenance groups\. A record is included when it carries every field the model requires, which is a test of model\-readiness rather than of completeness\. Table 6 gives the composition by compactive effort and the state of every column\.

Table 6 — Composition of the modelling set,nn= 2,854\. The upper block is by compactive effort\. The lower block reports the columns that are not complete, and what an absent entry means in each\.
The two gaps are kept rather than filled, and the reason differs for each\. Requiring sand would delete every record from figshare 32955851 and so fix energy at a single level, which removes the variable of principal interest\. The consistency limits are missing where the tests do not apply, so an absent value is the correct entry rather than a number\. Most of those 134 records also report a plastic limit above the liquid limit, which cannot be a measurement\. XGBoost and TabPFN take both gaps directly\. The perceptron cannot, so it receives the training\-fold median together with a was\-missing indicator, and the indicator carries the information\.

### III\-BProvenance and what a random fold leaks

Every record carries the identifier of the publication, site or deposit it came from\. That gives 162 groups, and Table 7 describes them\. The groups are very uneven: 94 hold a single record and the largest holds 269\.

Table 7 — The grouping variable, and what a random five\-fold partition places on both sides of the boundary\.Two properties of this variable govern the evaluation\. First, records sharing a group share much more than a label\. They share a soil deposit, a laboratory, a sampling campaign and a reporting convention, and they are often repeat tests on one pavement layer\. Under a random partition the model is therefore handed a near\-copy, and in 40 % of cases an exact copy, of the record it is about to be scored on\. Second, the grouping is coarse where the corpus is heaviest\. An LTPP group is a state rather than a laboratory, and all 54 belong to one testing programme following one protocol\. Holding out a state tests transfer to a new material, not to a new laboratory or a new reporting convention\.

### III\-CThe dataset in the compaction plane

Figure 1 shows the dataset\. Panel \(a\) places every record on the MDD–OMC plane, marked by the compactive effort its source states, with the zero\-air\-voids line and two constant\-saturation contours atGsG\_\{s\}= 2\.68\. Panel \(b\) gives the six sources side by side on five quantities\.

![Refer to caption](https://arxiv.org/html/2609.03337v1/figures/Fig1.png)Fig\. 1:The dataset: \(a\) the MDD–OMC plane by compactive effort, with envelopes atGsG\_\{s\}= 2\.68; \(b\) the six sources of Table 3 on five quantities\.Three features of panel \(a\) matter later\. The corpus occupies a band beneath the ZAV line rather than a cloud\. A soil reaches its peak at an approximately fixed degree of saturation, so MDD and OMC are two coordinates of one point on a saturation curve\. The band is closed above by the screening rule of Section 2 and thins belowSrS\_\{r\}= 0\.6\. The modified\-effort records do not sit above the standard ones\. Between 5 and 10 % water content their median density is 0\.192 Mg/m3below the standard records at the same water content, because the soils compacted harder are also the more plastic\. The vertical striping reflects LTPP reporting the optimum to the nearest whole per cent, which accounts for many of the repeated input vectors of Table 7\.

Three consequences follow for the rest of the paper\. The band structure means the two targets are not independent quantities to be estimated separately, which is why Section 5\.5 recovers the optimum from the estimated density through the phase relation\. The overlap of the effort classes means compactive energy cannot be read from the plane alone, so Section 5\.4 scores each effort level separately rather than pooling them\. And the weight of LTPP within the corpus means Section 4\.3 must draw folds around provenance\.

## IVMethodology

### IV\-AThe estimation problem

Section 3 established two properties the methodology must respect\. The corpus carries a group structure, and two of its columns are deliberately incomplete\. The estimation problem is stated once against that background, and Table 8 sets it out\.

The inputs are five index properties from a single routine classification suite on a disturbed sample, together with the compaction standard to which the material is to be compacted\. From those the task is to estimate the two coordinates of the laboratory compaction\-curve peak\. No input is required beyond what a classification and a specific\-gravity determination already produce\. No quantity taken from the compaction curve itself enters the predictors\.

Table 8 — Inputs, targets, and the quantities the corpus holds but the mapping never sees\. Index properties are reported as percentages\. OMC is modelled as a decimal fraction and reported as a percentage\.
Figure 2 sets the problem out as five stages: what a laboratory measures, the arithmetic on those measurements, the mapping itself, the pair of targets, and the intended use\.

![Refer to caption](https://arxiv.org/html/2609.03337v1/figures/Fig2.png)Fig\. 2:The estimation problem: six inputs, two targets, and the quantities held out of the mapping\.Formally, writing𝐱=\(wP,IP,F,s,ln⁡E¯,Gs\)∈ℝ6\\mathbf\{x\}=\(w\_\{P\},I\_\{P\},F,s,\\ln\\bar\{E\},G\_\{s\}\)\\in\\mathbb\{R\}^\{6\}, two regressors are fitted independently,f1​\(𝐱\)→MDDf\_\{1\}\(\\mathbf\{x\}\)\\rightarrow\\mathrm\{MDD\}andf2​\(𝐱\)→OMCf\_\{2\}\(\\mathbf\{x\}\)\\rightarrow\\mathrm\{OMC\}, on identical inputs and identical partitions\. The two targets are physically coupled, being the coordinates of a single curve peak, but no coupling is imposed on the estimators\. Whether they nevertheless respect it is measured in Section 5\.1, and a structurally coupled alternative is given in Section 5\.5\.

Intended use\.Three uses are intended: ranking borrow sources before any is tested, deciding which material warrants a Proctor determination first, and setting the water content of its first specimen\. They are not a basis for accepting placed fill\. A mean absolute error of 0\.066 Mg/m3is about 3\.5 % of the median maximum dry density\. That is the same order as the margin a specification leaves when it requires 95 % of MDD in the field, so an acceptance decision would spend that entire margin on the prediction\. The estimate is inexpensive not because the six inputs are free, but because they are made in any case and made once per material, whereas a Proctor test is made once per material and compactive standard\.

### IV\-BModels

Three learners carry the principal results\. Eight further architectures drawn from the compaction literature are refitted on identical folds in Appendix A, which tests at breadth the proposition that the accuracy reached is a property of the data rather than of the model class\. Section 5\.5 additionally searches for closed\-form expressions by symbolic regression using PySR \[35,36\]\.

TabPFN \[37\] predicts by in\-context learning\. The training records are supplied as context and the test records evaluated in a single forward pass, with no gradient step on the present data\. The network is pre\-trained once, on synthetic datasets drawn from structural causal models, to approximate the posterior predictive distribution under that prior \[38\]\. It therefore introduces no hyperparameter and no search conducted on these records, so no comparison below can be confounded by unequal search effort\. The whole training fold is re\-supplied for every prediction, which is why the cost is 522 s against 2 s for the ensemble\.

The gradient\-boosted model is anXGBRegressorwith values fixed in advance\. A nested random search over nine hyperparameters, selected on a 3\-fold inner cross\-validation within each training fold, returned R20\.816 and 0\.779 against 0\.819 and 0\.783 for the fixed values\. Tuning therefore yields no improvement\. The perceptron and the two transformer baselines are described with the released code\.

### IV\-CThe three fold designs

The boundary around which a fold is drawn determines what the resulting score means\. Where records carry a group structure, a fold that ignores it reports a score for interpolation within the groups rather than prediction for a new one\. The remedy is to block the folds on the grouping variable \[39\]\. Three designs are used here, on the same 2,854 records, the same six inputs and the same learners\. Only the boundary changes between them\. Table 9 defines them\.

Table 9 — The three fold designs\. The random and grouped designs are matched in training\-set size and differ only in composition\.The random design is the convention of the comparative literature\. It is retained rather than replaced, because it is the design under which the figures below may be set beside published ones, and because the difference between it and the grouped design is itself a result\. It is not the basis of any statement about a new laboratory, publication or data source\.

The source\-held\-out design is the harshest of the three\. Holding out LTPP leaves 878 training records, and holding out any other source leaves a training set at least 69 % LTPP\. A fourth design, leave\-one\-group\-out over all 162 groups, was also run for the gradient\-boosted model and is released with the code\. Its error distribution across groups is skewed: the median group is predicted at 0\.066 Mg/m3and the ninetieth percentile at 0\.110, but the tail runs to 0\.268\.

Performance is reported as the coefficient of determination, the mean absolute error and the root\-mean\-square error, against the floor set by a mean predictor computed on the training fold and scored on the identical held\-out records\. Each is computed once over the pooled out\-of\-fold predictions rather than as the mean of per\-fold scores, because the admissibility check requires one prediction attached to every record\. Fold\-to\-fold dispersion is reported separately as the±\\pmin Table 10\. Under the source\-held\-out design the coefficient of determination for a source is taken against that source’s own mean, which is a harder reference than the corpus mean where a source is narrow, so the mean absolute error is set beside it\. Because MDD and OMC are predicted separately, a predicted pair may be physically impossible, and the proportion above the zero\-air\-voids line is reported as a consistency check\. Bold in a results table marks the best value in the comparison that table makes, and a note beneath each table states which comparison that is\.

Three qualifications apply\. First, the symbolic search of Section 5\.5 runs on a single 80/20 random partition\. The evolutionary search sees only the 2,176 training records, but two later steps are not held to the same standard: the entry taken off the Pareto front was chosen with held\-out accuracy in view, and the constant\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}is a mean over all 2,720 records\. Both are model selection performed with sight of the 544, so the accuracies quoted for that partition are selection scores rather than test scores\. The expressions are refitted and rescored under all three designs in Section 5\.5\. Second, no untouched test set exists, and Section 6 states what that costs\. Third, the 1,405 repeated input vectors of Table 7 cannot leak under the grouped or source\-held\-out designs, since every repetition lies inside one group\. Under the random design they can, and the consequence was measured\. Collapsing every repeated vector to a single record carrying the median target leaves 2,126 records and changes the random result by−\-0\.005 in R2for MDD and \+0\.002 for OMC\. That effect is small while the grouped design costs an order of magnitude more, so duplication is not the mechanism\. Shared provenance is\.

## VResults

### V\-APredictive accuracy

Table 10 reports the three learners on the six inputs under all three fold designs, so that the leakage the random design admits can be read against the figure it produces\. Within a design, every learner is trained and scored on one fold assignment\.

TabPFN attains R20\.824 for maximum dry density at a mean absolute error of 0\.066 Mg/m3, and 0\.784 for optimum moisture content at 1\.87 % water content\. These are reductions in error of 63 % and 60 % against the mean predictor\. Both leading learners lie within the range at which the prediction is of practical use\. An error of 1\.87 % in optimum moisture content is comparable with the between\-laboratory reproducibility of the Proctor test, and 0\.066 Mg/m3is below 4 % of the median density\.

Table 10 — Out\-of\-fold performance, six inputs, 2,854 records\. The last column is a mean predictor computed on the training fold and scored on the identical held\-out records\. Water\-content errors are percentage points\. The±\\pmis the standard deviation across the five random folds\. The last block counts predicted pairs implying a degree of saturation above unity\.*Bold marks the best value in each row\. The mean\-predictor column is a floor, not a competitor, and is excluded from the comparison\.*

Three readings follow from Table 10, and they are separate\.

*The random figures are interpolation figures\.*They are the only figures in this paper that can be set beside published ones\. They measure how well the corpus interpolates within itself, not how well the model transfers to a source it has not seen\. Drawing the fold boundary around the provenance group, at an identical training\-set size, costs about 0\.10 in R2on both targets and on every learner\. The mean absolute error in water content rises from 1\.87 to 2\.28 %\. Nothing about the model, the inputs or the amount of training data changes between those two blocks\. What changes is whether the model had already seen the publication on which it was scored\.

*The learners agree under two designs and separate under the third\.*Under random and grouped folds the three span 0\.03 in density\. With a whole source held out they span 0\.19, and the ordering reverses\. Appendix A shows that the cause is the missing\-data encoding rather than the architecture\.

*Accuracy is not the only criterion\.*Because the two targets are predicted independently, a predicted pair may imply a degree of saturation above unity\. The learners separate more sharply on that criterion than on accuracy\. Predicting each target by its training\-fold mean already places 28 pairs above the line, so the perceptron does no better than a constant under random folds\. The rate cannot be driven to zero by improved fitting, because nothing in the formulation forbids a violation\. Building the phase relation into the formulation does forbid it, which is the approach of Section 5\.5\.

Sections 5\.4 and 5\.5 are reported with TabPFN and the gradient\-boosted model\. On this evidence neither is more accurate than the other, so the choice rests on two other grounds\. TabPFN puts 1 of its 2,854 random\-fold pairs above the ZAV line against 7 for the ensemble, a sevenfold difference on a quantity that is not measurement noise\. And TabPFN has no hyperparameters, so no result below can be attributed to a search conducted on these records\.

### V\-BTransfer to an unseen source

Table 11 opens the source\-held\-out block of Table 10 source by source\. Each source in turn is predicted by a model trained on the other five\. The coefficient of determination is taken against that source’s own mean, so a narrow source is scored against a hard reference\. The standard deviation of the target within each source is given for that reason\.

Table 11 — Leave\-one\-source\-out, source by source\. R2is taken against the held\-out source’s own mean\. sd is the standard deviation of the target within that source, given because it sets how hard that reference is\.*Bold marks the better of the two learners in each row\. On Zenodo 20737270 the two water\-content errors differ only in the third decimal\.*

Three things are visible in Table 11 that the pooled figure hides\. The hardest fold for the gradient\-boosted model is figshare 32955851 at R2−\-0\.490 for density\. That is the multi\-energy deposit, and removing it removes every non\-standard record from training, leaving the energy variable present but untrained at three of its four levels\. The smallest source, at 7 records, is predicted worse than its own mean by both learners, so no per\-source conclusion should be drawn from it\. And the two learners diverge most on the OMC column of the two largest folds, which is where the missing\-data encoding of Appendix A does its damage\.

Leave\-one\-group\-out over all 162 groups gives a complementary view of the same question at finer resolution\. The median group is predicted at a mean absolute error of 0\.066 Mg/m3in density and 1\.73 % in water content\. The ninetieth percentile is 0\.110 Mg/m3and 3\.49 %, and the worst group reaches 0\.268 Mg/m3and 11\.0 %\. The error distribution across groups is therefore strongly skewed, and a pooled mean absolute error understates the risk on an unlucky material\.

### V\-CWhat the estimate rests on

Three questions are asked of the input set\. Which input groups carry the prediction, which of the three consistency limits should be used, and what the continuous prediction adds to a soil\-classification label\.

Input groups\.Table 12 removes each input group in turn from the gradient\-boosted model and rescores under the random and grouped designs\.

Table 12 — Input ablation, gradient\-boosted model\.Δ\\Deltais the loss in R2against the full six\-input model in the first row\. Gradation is fines and sand together\.Table 12 separates the inputs into two tiers\. The consistency limits and gradation each cost between 0\.096 and 0\.213 in R2\. Sand alone, compactive energy and specific gravity never cost more than 0\.030, so the cheapest entry in the leading tier is still more than three times the most expensive in the second\. Which of the two leading groups leads depends on the target and the design, so they are best read as jointly load\-bearing rather than ranked\. Gradation earns its place through the fines content rather than through sand: removing both grading terms costs seven times what removing sand alone does under grouped folds\. The two leading costs both grow when the fold boundary respects provenance, so neither is inflated by shared provenance\. The three small costs do not change materially between designs, and the pooled energy figure is qualified in Section 5\.4\.

Consistency limits\.Of the three limits only two are algebraically independent, sinceIP=wL−wPI\_\{P\}=w\_\{L\}\-w\_\{P\}\. Table 13 scores all four sets under the three designs\.

Table 13 — The four consistency\-limit sets, gradient\-boosted model, each with fines, sand, energy and specific gravity\. Each cell gives R2for density then water content\.*Bold marks the highest value in each column\. It is the set used throughout, and it is best under all three designs and on both targets\.*

The pairwPw\_\{P\}withIPI\_\{P\}equals or exceeds all three limits together in every design\. The margin is negligible under random folds and 0\.041 in density with a source held out\. The three\-limit set is not rank\-deficient, but the redundant column costs transfer\. The liquid limit is therefore released with the dataset and not carried into the mapping\. Section 6 records that this choice was made on the learners and charged to the closed forms, which preferwLw\_\{L\}\.

Value beyond a classification label\.Gradation and the consistency limits already place a soil in a classification group, and a group carries a qualitative expectation of how it will compact\. Table 14 asks what the continuous prediction adds to that label\. Every class mean is computed inside the training fold and scored on the held\-out records, so the label is evaluated on the same terms as the model\.

Table 14 — What the prediction adds to a classification label\. Each cell gives R2for density then water content\. The USCS block covers all 2,854 records\. The AASHTO block covers the 1,976 LTPP records, the only ones carrying an AASHTO symbol\.The USCS symbol alone explains 0\.563 of the variance in density and 0\.561 in water content under random folds\. The six inputs used linearly add 0\.116 and 0\.134 on top of that\. Used by an ensemble they add a further 0\.139 and 0\.086\. Under grouped folds the margin narrows without closing\. A class mean is a low\-capacity predictor that was never exploiting shared provenance, so it falls only to 0\.543 while the ensemble falls to 0\.722\. The gain over the label is then 0\.179 rather than 0\.256 in density, and 0\.157 rather than 0\.222 in water content\. The practical advantage survives the change of design at about two\-thirds of its apparent size\.

Two causes separate the label from the prediction\. A classification group is a broad interval\. CL spans liquid limits from 20 to 50 % and fines from 50 to 100 %, so soils that compact differently share a symbol\. And the residual gap between the linear model and the ensemble is the cost of assuming additivity\. Neither is removed by classifying more finely\. The AASHTO block of Table 14 uses a finer scheme on the LTPP records and both gaps persist\.

### V\-DEffect of compactive energy

Table 12 ranked compactive energy near the bottom of the six inputs, at 0\.015 to 0\.025 in R2\. That figure substantially understates its influence, because standard Proctor accounts for 96\.5 % of the dataset\. The pooled value is an average over a corpus in which the variable barely varies\. Table 15 splits the same predictions by the effort actually applied, and Figure 3 draws them\.

Table 15 — R2and MAE for density within each effort stratum, with and without the energy input, gradient\-boosted model\. The strata are those of Table 6\.![Refer to caption](https://arxiv.org/html/2609.03337v1/figures/Fig3.png)Fig\. 3:R2for density by compactive effort, with and without the energy input, gradient\-boosted model, under \(a\) random and \(b\) grouped folds\.The modified\-Proctor row is the relevant one\. Given the energy variable, the model predicts those 66 records at R20\.740 under random folds\. Deprived of it, the prediction falls below the mean of the records it is scored on, to−\-0\.651, and the mean absolute error rises from 0\.058 to 0\.162 Mg/m3\. The model has no means of knowing that those soils were compacted at 4\.5 times the standard effort, so it defaults to the standard\-Proctor relationship that describes almost all of its training data\. The same happens less sharply under grouped folds\. The standard\-Proctor row, 96\.5 % of the corpus, barely moves\. That contrast is the substance of the result\. A variable can be worthless on average and decisive on the stratum that requires it\. The reduced\-modified row rests on 7 records and carries no weight\.

Table 15 cannot be extended to the source\-held\-out design, for a reason that is a property of the corpus rather than of the method\. All 101 non\-standard records come from figshare 32955851, spread over 51 of its 83 groups\. A grouped fold therefore still contains non\-standard records in training\. A source\-held\-out fold that removes that deposit removes every non\-standard record with it, leaving the energy variable present but untrained at three of its four levels\. That is one reason it is the hardest source of the six in Table 11\.

The result does not depend on the precision with which those energies are recorded, and cannot, because the variable takes four distinct values across 2,854 records\. For the gradient\-boosted model the exact conversion, the value recomputed from the governing standard and the rounded SI convention all return R20\.819 and 0\.783, as does an ordered category carrying no magnitude\. Four indicator variables return 0\.819 and 0\.780\. Compactive energy is therefore a categorical variable carried as a number, and the numerical value functions as a label\.

### V\-EClosed\-form expressions by symbolic regression

Gradient boosting provides the accuracy of Table 10 but not an expression a specification can quote\. Symbolic regression was run to establish the cost of a closed form\. MDD and OMC are searched independently, as they are modelled independently throughout\. In the recommended form the two are then coupled through the phase relation, so that admissibility is structural\.

PySR \[35,36\] implements a multi\-population evolutionary search over expression trees, minimising mean\-square loss against a complexity, and returning the most accurate expression at every complexity rather than a single model\. The operator set is\{\+,−,×,÷,⋅,log,\(⋅\)2,\(⋅\)−1\}\\\{\+,\-,\\times,\\div,\\sqrt\{\\cdot\},\\log,\(\\cdot\)^\{2\},\(\\cdot\)^\{\-1\}\\\}\. The reciprocal is included as a primitive because1/wL1/w\_\{L\}is the motif every density search rediscovers\. Nested unary operators are forbidden\. The reported front comes from 250 iterations over 36 populations at maximum size 40 with parsimony2×10−42\\times 10^\{\-4\}\. PySR cannot accept a missing value, so for this search alone sand is dropped and the 134 non\-plastic soils excluded\. That leaves 2,720 records and six inputs with all four compactive energies intact\. The search runs on a random 80 % of those records, 2,176 in total, and the front is scored on the remaining 544\.

The protection that partition provides is limited, as Section 4\.3 states\. The 0\.692 and 0\.658 quoted below are selection scores rather than scores on records untouched by every step\. The transfer claim of this paper does not rest on them\. It rests on the refitted figures below, where the structure is held fixed and every constant is re\-estimated inside each training fold\.

#### V\-E1The recommended pair

The search returns a front rather than a single expression, so one entry has to be selected\. Density is taken at complexity 20, on a front that is flat over a wide range of length\. Three symbols already reach R20\.613 and complexity 11 reaches 0\.669\. The entry at complexity 18 scores marginally higher, 0\.693 against 0\.692, but it carries no energy term, which Section 5\.4 shows would be an unsound omission\. Complexity 20 retainsln⁡E¯\\ln\\bar\{E\}at a cost of 0\.001\. The optimum is then recovered from the predicted density through the phase relation rather than from a direct expression:

MDDρw\\displaystyle\\frac\{\\mathrm\{MDD\}\}\{\\rho\_\{w\}\}=1\.969\+0\.0100​Gs2wP\+IP−0\.3285​F\\displaystyle=1\.969\+0\.0100\\,\\frac\{G\_\{s\}^\{\\,2\}\}\{w\_\{P\}\+I\_\{P\}\}\-0\.3285\\,F−0\.8985​wP\+0\.003285​ln⁡E¯\\displaystyle\\quad\-0\.8985\\,w\_\{P\}\+0\.003285\\,\\ln\\bar\{E\}\(3\)OMC\\displaystyle\\mathrm\{OMC\}=0\.820Gs​\(Gs​ρwMDD−1\)\\displaystyle=\\frac\{0\.820\}\{G\_\{s\}\}\\left\(\\frac\{G\_\{s\}\\,\\rho\_\{w\}\}\{\\mathrm\{MDD\}\}\-1\\right\)\(4\)
Every term is a pure number\. The plastic limit, the plasticity index and the fines content enter as the reported percentages divided by 100 %\. A soil atwPw\_\{P\}= 18 % andIPI\_\{P\}= 15 % with 45 % fines therefore giveswPw\_\{P\}= 0\.18,IPI\_\{P\}= 0\.15 andFF= 0\.45, and OMC likewise returns a decimal fraction\. Specific gravity is already a ratio, density is divided byρw\\rho\_\{w\}= 1 Mg/m3, andE¯=E/E0\\bar\{E\}=E/E\_\{0\}withE0E\_\{0\}= 592\.5 kJ/m3the standard\-Proctor reference, soln⁡E¯\\ln\\bar\{E\}is zero at standard effort\.

Equation \([4](https://arxiv.org/html/2609.03337#S5.E4)\) is the phase relation of Section 2\.4 read in reverse, and Figure 4 shows it\. It is an exact identity, so the only estimated quantity in it is the constant 0\.820\. At a fixed saturation the equation is a curve in the MDD–OMC plane, one per specific gravity, lying under the zero\-air\-voids bound by construction\. The pair is therefore this paper’s density model composed with an identity rather than a second fit\.

![Refer to caption](https://arxiv.org/html/2609.03337v1/figures/Fig4.png)Fig\. 4:Equation \([4](https://arxiv.org/html/2609.03337#S5.E4)\): \(a\) the identity at\(Sr\)o​p​t=0\.820\(S\_\{r\}\)\_\{opt\}=0\.820for three specific gravities, with the measured records behind it; \(b\) the fold\-to\-fold range of the constant against the span a different saturation would imply\.The formulation is motivated by the optimum\-saturation framework of Tatsuoka and Gomes Correia \[14\] and is not taken from it\. That framework relates density, water content and saturation to one another\. It does not predict MDD and OMC from index properties, so Equation \([4](https://arxiv.org/html/2609.03337#S5.E4)\) is not a Tatsuoka predictor and is not scored as one anywhere in this paper\. The constant is estimated here as the mean measured saturation over the 2,720 records of the symbolic subset\. That it falls close to the average of about 0\.82 reported for the standard\-Proctor compilation of Tatsuoka and Gomes Correia \[14\] is external corroboration rather than the origin of the number\. Re\-estimated inside every fold it spans 0\.814 to 0\.832, worth 0\.30 % in water content at the median density, whereas a different saturation would give a different curve entirely\.

Two features of the density expression merit comment\. Every sign is the physical one and none was prompted\. Density falls with fines, with the plastic limit and with the plasticity index, and rises with specific gravity and with compactive effort\. The recurring groupingGs2/\(wP\+IP\)G\_\{s\}^\{2\}/\(w\_\{P\}\+I\_\{P\}\)is nonetheless empirical\. It is returned because density falls with both limits and one reciprocal costs fewer symbols than two linear terms, and it should be read as a fitted grouping rather than as a mechanism\. The expressions also hold over the corpus on which they were fitted and not beyond it\. The 2,720 records span a plastic limit of 1\.0 to 49\.3 %, a plasticity index of 1\.0 to 126\.2 %, a fines content of 1\.5 to 100 %, a specific gravity of 2\.30 to 2\.94 and the four Proctor energies\. Outside those ranges the equations extrapolate, and the released predictor flags such an input\.

Figure 5 shows the pair applied to the 544 held\-out records\. Panel \(c\) addresses the claim in the title\. The predicted pairs lie on a single constant\-saturation contour because Equation \([4](https://arxiv.org/html/2609.03337#S5.E4)\) places them there, and none of the 544 can cross the zero\-air\-voids bound for any input\.

![Refer to caption](https://arxiv.org/html/2609.03337v1/figures/Fig5.png)Fig\. 5:The closed\-form pair on the 544 held\-out records: \(a\) density, Equation \([3](https://arxiv.org/html/2609.03337#S5.E3)\); \(b\) the optimum recovered through Equation \([4](https://arxiv.org/html/2609.03337#S5.E4)\); \(c\) the predicted pairs against the zero\-air\-voids bound\.
#### V\-E2Why the coupling, and what it costs

A second search was given the void ratio implied by Equation \([3](https://arxiv.org/html/2609.03337#S5.E3)\) alongside every soil variable\. It returned a single term proportional to the void ratio and discarded the rest\. That is worth recording, because the search was free to introduce plasticity, gradation or effort terms and did not\. It returnedOMC\(%\)=λe\\mathrm\{OMC\}\\,\(\\%\)=\\lambda ewithλ≈30\.70\\lambda\\approx 30\.70, a constantOMC/e\\mathrm\{OMC\}/eand not a constant\-saturation form\. Substituting it into the phase relation gives\(Sr\)o​p​t=0\.307​Gs\(S\_\{r\}\)\_\{opt\}=0\.307\\,G\_\{s\}, so the implied saturation rises with specific gravity from 0\.706 at the corpus minimum to 0\.903 at its maximum\.

Table 16 compares four ways of obtaining OMC from the same predicted density\. Every coefficient in it, including\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}, is estimated within the training fold\. The density expression is common to all four rows, so the rows differ only in the coupling\.

Table 16 — Four ways of obtaining OMC from the same predicted density, over the 2,720 records of the symbolic subset\.nnis the count of predicted pairs above the zero\-air\-voids line, out of 2,720\. The common density expression returns R20\.710, 0\.676 and 0\.579 under the three designs\.*Bold marks the recommended formulation, not the highest R2\. Form C scores 0\.012 higher in every design and is rejected on the grounds given below\.*

Coupling secures admissibility, and it secures it completely\. The independent expression places 75 of 2,720 pairs above the zero\-air\-voids line under random folds, 98 under grouped folds and 81 with a source held out\. One of them reaches a saturation of 132\.7 in the last design, which is not a small error\. All three coupled forms produce none, under all three designs, at an R2spread of 0\.022 across the four rows\.

Form B is recommended\. It is bounded for anyGsG\_\{s\}, whereasλ​e\\lambda eis admissible only becauseλ<Gs/100\\lambda<G\_\{s\}/100happens to hold for every soil present, and it would fail aboveGs≈3\.26G\_\{s\}\\approx 3\.26\. Letting the saturation itself be predicted, form C, adds 0\.012 at the cost of a second fitted model\. Because Equation \([4](https://arxiv.org/html/2609.03337#S5.E4)\) requires only the predicted density andGsG\_\{s\}, it composes with any density source, including the ensemble of Section 5\.1\.

Table 17 sets the closed form against the gradient\-boosted model on the same records and the same folds, so that the price of the closed form is a measurement rather than an assertion\.

Table 17 — The closed\-form pair against the ensemble, on the 2,720 records of the symbolic subset\. Each cell gives R2for density then water content\.*Bold marks the highest value in each column and target\.*

The closed form costs about 0\.11 in R2under random folds\. That gap closes almost entirely under grouped folds and reverses for water content with a source held out, where the coupled expression beats both ensembles\. An expression with five coefficients has far less capacity to memorise a source than an ensemble, which is what Table 17 shows\.

Two limits apply to this section\. The figures are confined to the 2,720 records of the symbolic subset, so they are not those of Table 10\. And the functional form was selected on these data by a search that saw every source\. What is established is that the fitted constants transfer, not that the structure was found without sight of the corpus\. On these six inputs the closed forms are not recommended over the ensemble for a soil outside the corpus\. What holds without qualification is that they cannot violate the zero\-air\-voids condition\.

### V\-FThe optimum degree of saturation

A third symbolic search, targeting the degree of saturation at the compaction peak, returned a constant\. No expression up to size 26 exceeded R20\.05, because little variation remains once the mean is removed\. Figure 6\(a\) shows the distribution\. Over the 2,854 recordsSo​p​tS\_\{opt\}has a mean of 0\.815 and a coefficient of variation of 11 %\. The distribution is tight and bounded above by the zero\-air\-voids line\. The quantity is that of Tatsuoka and Gomes Correia \[14\], and the contribution here is a measurement of it at scale rather than its definition\.

![Refer to caption](https://arxiv.org/html/2609.03337v1/figures/Fig6.png)Fig\. 6:The degree of saturation at the compaction peak: \(a\) its distribution over the 2,854 records, against the zero\-air\-voids bound; \(b\) its mean within each stratum, against the corpus mean\.Table 18 gives the same quantity by stratum, and Figure 6\(b\) draws it\. The measurement does not support treating 0\.815 as a constant of the soil\.

Table 18 — The optimum degree of saturation by stratum, 2,854 records\. CV is the coefficient of variation\. The corpus mean has a 95 % interval of 0\.811 to 0\.818, or 0\.802 to 0\.824 when clustered on the 162 provenance groups\. A fifth soil group, fine\-grained non\-plastic, holds 4 records and is omitted\.*Bold marks the whole\-corpus reference against which each stratum is read\. No value in this table is a performance figure\.*

Soil group moves the mean systematically, from 0\.706 for coarse\-grained non\-plastic soils to 0\.853 for fine\-grained soils of high plasticity\. That range of 0\.147 is more than six times the width of the cluster\-robust interval on the corpus mean\. Source means and the treatment of specific gravity also move it, the latter by 0\.027 in the direction the substitution predicts\. Compactive effort moves it least, which is the one result that agrees with the earlier work without qualification, although at 3\.5 % non\-standard records these data do not test it severely\. Within\-group variance accounts for 74 % of the total and between\-group variance for 26 %, so most of the scatter is not explained by provenance either\.

What the measurement supports is a baseline rather than a constant\. The corpus figure of 0\.815, with a fifth\-to\-ninety\-fifth percentile spread of 0\.655 to 0\.951, is of the same order as the 64\.1 % to 92\.3 % reported across the nine soils of Tatsuoka et al\. \[15\], but is obtained from 2,854 laboratory records rather than nine\. The corpus mean is a defensible first estimate where no test exists\. It is not a substitute for determining\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}for the material at hand, which is what a control procedure normalised onΔ​Sr\\Delta S\_\{r\}requires\.

## VILimitations

Table 19 lists what the corpus cannot support\. Each row states the limitation, the evidence for it, and the claim it restricts\. The paragraphs below expand the three that most affect how the results should be used\.

Table 19 — What the corpus cannot support\.The corpus is laboratory\-only\.The accuracies of Section 5, the closed forms and the mean\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}of 0\.815 all describe laboratory compaction\. Whether\(Sr\)o​p​t\(S\_\{r\}\)\_\{opt\}near 0\.82 also describes field\-compacted lifts is not tested here and cannot be tested on a corpus of this kind\. The comparison with Tatsuoka and Gomes Correia \[14\] is therefore one across different evidence bases\.

The grouping variable is as coarse as the sources allow\.An LTPP group is a state rather than a laboratory\. No source publishes the laboratory that performed each test, so a strictly laboratory\-held\-out evaluation cannot be run on any public corpus known to the authors\. That is the single change to reporting practice that would most improve work of this kind\. The source\-held\-out design is also coarse and cannot be otherwise\. A single programme contributes 69 % of the records and the corpus has six sources, so each fold changes the training set substantially\. One per\-source figure in Table 11 rests on 7 records\.

No independent test set exists\.All results are cross\-validated within the same 2,854 records, and the input set, the ablations and the constant of Section 5\.5 were chosen after inspecting them\. The source\-held\-out design is the closest substitute available without new laboratory work, and it is not equivalent\. A held\-out source is still one the authors selected, harmonised and screened under the conventions of Section 2, whereas a genuinely new laboratory would also differ in the mould, the oversize correction and the rammer\. The degradation reported in Table 10 is a lower bound on the cost of a new laboratory rather than an estimate of it\.

Two further rows deserve a number\. Dropping the liquid limit was decided on the learners and charged to the closed forms\. It gains the gradient\-boosted model 0\.041 in density with a source held out \(Table 13\), but the symbolic search performs worse without it\. An earlier version whose search could usewLw\_\{L\}returned a density expression reaching R20\.687 with a source held out, against 0\.579 for Equation \([3](https://arxiv.org/html/2609.03337#S5.E3)\)\. A variable that adds nothing to a model with capacity to spare can still be load\-bearing for a simple expression, so where the closed form is the priority it should be fitted on all three limits\. Specific gravity is nearly inert as an input but not as the denominator of the screening rule\. It also costs accuracy in the expected direction: against the 1,148 records reporting a measured value, the 1,706 substituted records lose 0\.018 in density and 0\.044 in water content, sinceGsG\_\{s\}enters the optimum through the phase relation and the density only weakly\.

## VIIConclusions

The primary contribution of this study is a corpus of laboratory compaction tests\. On it a model was built for maximum dry density and optimum moisture content, and closed\-form expressions were fitted for the same two parameters\.

*The corpus\.*

1. 1\.2,854 laboratory compaction tests were assembled from six public sources, five of them deposits meeting five stated retention conditions out of 618 candidates \(Tables 2 and 3\)\. The corpus is an order of magnitude larger than the datasets behind the published correlations\. It spans four Proctor energy levels and both sides of the coarse–fine boundary\. Every record is the peak of a moisture–density curve determined to a Proctor method its source names, and the compactive energy is the standard the source states rather than a value inferred here\. The corpus holds no field density, so every result reported here concerns laboratory compaction\. It is released with per\-record provenance under CC BY 4\.0, together with the query strings, the screening decisions and the source adapters\.
2. 2\.Provenance is what the corpus adds beyond size\. Every record carries an identifier for the publication, site or deposit it came from, giving 162 groups \(Table 7\)\. The two transfer\-facing fold designs of Table 9 are built on that field, and neither can be run without it\.
3. 3\.A material share of published compaction data is physically impossible, and the share depends on how specific gravity is treated\. Screening removed 11\.8 % of harmonised records overall and 5\.7 % of those reporting a measured value\. The rate runs from 19\.8 % to 8\.2 % as the substituted value moves from 2\.60 to 2\.75 \(Table 5\)\. The zero\-air\-voids condition is recommended as a filter rather than as a diagnostic\.
4. 4\.The optimum degree of saturation of Tatsuoka and Gomes Correia \[14\] is quantified across 2,854 laboratory tests at a mean of 0\.815 and a coefficient of variation of 11 %, with a fifth\-to\-ninety\-fifth percentile range of 0\.655 to 0\.951 \(Table 18\)\. It is a corpus\-level baseline rather than a soil\-independent constant, and a laboratory value rather than evidence of field invariance\.

*The model\.*Three learner classes were carried throughout and eleven architectures evaluated, on identical folds and inputs\.

1. 5\.Under the random five\-fold design used in this literature, a tabular foundation model predicts maximum dry density at R20\.824 and a mean absolute error of 0\.066 Mg/m3, and optimum moisture content at 0\.784 and 1\.87 %\. Gradient\-boosted trees return 0\.819 and 0\.783 \(Table 10\)\.
2. 6\.Those figures measure interpolation within the corpus\. Under that design 96\.6 % of records share a provenance group with their training fold, and 40\.1 % share an exact input vector \(Table 7\)\. Grouped folds cost about 0\.10 in R2on both targets at an unchanged training\-set size, and holding out a whole source costs more again\. Every claim made here about a new laboratory rests on those two designs\.
3. 7\.Under random folds the data rather than the capacity of the model limits accuracy\. Eleven architectures agree closely \(Table A1\), and admitting records with largely missing gradation and energy costs 0\.283 in density\. Under source\-held\-out folds the learners cease to agree, and much of that spread is the missing\-data encoding rather than the architecture\. One gradient\-boosted model moves from 0\.392 to 0\.597 in density when gaps are median\-imputed with an indicator instead of passed to the learner\. A comparison run on random folds therefore identifies neither the model nor the protocol that transfers\.
4. 8\.The consistency limits and gradation are the two load\-bearing input groups, each worth 0\.096 to 0\.213 in R2\. Sand alone, compactive energy and specific gravity never reach 0\.030 \(Table 12\)\. Of the three consistency limits only two are independent, andwPw\_\{P\}withIPI\_\{P\}matches or exceeds all three in every design, by 0\.041 in density with a source held out \(Table 13\)\.
5. 9\.Compactive energy is marginally negligible yet conditionally decisive\. Omitting it drives modified\-Proctor density prediction to R2−\-0\.651, far below the mean of those records \(Table 15\)\. Its effect is categorical, five encodings differing by at most 0\.003\. Only a multi\-energy corpus can show this, which is why the four levels were kept at the cost of an incomplete sand column\.

*The closed forms\.*

1. 10\.With the structure fixed and every constant re\-estimated inside the training fold, the recommended pair reaches R20\.710 and 0\.688 under random folds, 0\.676 and 0\.666 under grouped folds, and 0\.579 and 0\.606 with a whole source held out \(Table 17\)\. Coupling separates the formulations on admissibility rather than on accuracy\. An independent OMC expression places 75 to 98 predicted pairs above the zero\-air\-voids line depending on the design, and every coupled form places none \(Table 16\)\. The closed forms are not recommended over the ensemble for a soil outside this corpus\. What holds without qualification is that they cannot violate the zero\-air\-voids condition\.

*What the corpus asks of the next dataset\.*

1. 11\.The practical recommendation concerns compaction databases rather than models\. The compaction standard should be recorded\. Specific gravity should be reported as measured rather than assumed\. An identifier should be carried for the laboratory and study each test came from\. Without the last, no evaluation can test transfer to a new laboratory\. The corpus released here carries all three fields, and its acquisition protocol and screening rule are released with it\.

## CRediT authorship contribution statement

Sompote Youwai:Conceptualization, Methodology, Software, Formal analysis, Data curation, Writing – original draft, Writing – review & editing, Visualization, Supervision, Funding acquisition\.Chana Phutthananon:Methodology, Validation, Writing – review & editing\.Warat Kongkitkul:Conceptualization, Methodology, Writing – review & editing, Supervision\.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper\.

## Declaration of generative AI and AI\-assisted technologies in the writing process

During the preparation of this work the authors used a large language model \(Claude, Anthropic\) to assist with editing prose for concision, with restructuring results into tabular form, and with routine analysis scripting\. After using this tool the authors reviewed and edited the content as needed and take full responsibility for the content of the published article\. No text, figure, table or numerical result was accepted without verification against the released data and analysis code\. Generative AI was not used to generate, interpret or alter any measurement, and is not credited as an author\.

## Acknowledgements

The authors thank the depositors of the six datasets listed in Table 3, whose decision to publish their records under open licences made this corpus possible\.

## Data availability

The corpus, the fitted models, the command\-line predictor and all analysis code are released at[https://github\.com/Sompote/open\_compaction](https://github.com/Sompote/open_compaction)under CC BY 4\.0, at version v2\.2\.0, with a checksum manifest, a data dictionary and a validation script\. The repository is public and requires no account to read\. The dataset is one flat file, one row per laboratory compaction test, carrying the six inputs, the two targets, the provenance identifier and the compaction standard\. The liquid limit and a flag separating measured from substituted specific gravity are held in a wider file beside it\. The code reproduces every quantity reported here, including the record\-level out\-of\-fold predictions under all three fold designs, so any stratification can be recomputed without refitting\. Analyses used Python 3\.11 with xgboost 2\.1\.4, scikit\-learn 1\.5\.1, PySR 0\.19\.4 and version 6\.4\.1 of the PyPI packagetabpfn\. All seeds were fixed\. The symbolic search is multi\-threaded and is therefore not bit\-reproducible\. \[DOI TO BE INSERTED ON DEPOSIT\.\]

## References

\[1\] Proctor RR\. Fundamental principles of soil compaction\. Eng News\-Rec 1933;111:245–8\.

\[2\] ASTM D698\-12\(2021\)\. Standard test methods for laboratory compaction characteristics of soil using standard effort \(12,400 ft\-lbf/ft3\(600 kN⋅\\cdotm/m3\)\)\. West Conshohocken, PA: ASTM International; 2021\.

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## Appendix AArchitectures from the compaction literature

The learners of Table 10 are not those most used in this literature\. Neural networks, support vector machines, random forests and stacked hybrids recur across it \[23–26\]\. None of those studies releases its fitted parameters, so none can be applied to these records as published\. The architectures can be refitted on identical folds and inputs, which Table A1 reports\. None accepts a missing value, so all share one preprocessing: training\-fold median, a was\-missing indicator per gapped column, and standardisation on training\-fold statistics\. None is tuned on these records, so the table compares model families at fixed effort rather than at fixed tuning budget\.

Table A1 — Architectures from the compaction literature, refitted on identical folds and inputs\. Each cell gives R2for density then water content\. The second column names a study that applied that family to this task, not the source of any hyperparameter\. Two rows are variants added here\. The last three rows are the learners of Table 10\.*Bold marks the highest value in each design column, separately for density and water content\.*

Under random folds the table confirms the reading of Section 5\.1\. The seven tree and ensemble architectures span 0\.021 in density, and eight of the eleven learners fall within 0\.036 of the best\. The exceptions are the linear model, the shallow network and the support vector machine\. The conclusion that the data rather than the learner limits what is reached under this design therefore rests on eleven architectures rather than three\. Under grouped folds the pattern is the same at a lower level\.

The source\-held\-out column is the informative one\. Scikit\-learn’s gradient boosting reaches 0\.606 for density and the stacked hybrid 0\.587, against 0\.392 for the gradient\-boosted model of Table 10 and 0\.520 for the foundation model\. The cause is not the architecture, and the anchor row establishes this\. The sameXGBRegressor, with the same hyperparameters and the same folds, reaches 0\.597 and 0\.597 when the gaps are median\-imputed with a was\-missing indicator, and 0\.392 and 0\.225 when they are passed to the learner directly\. That is a difference of 0\.205 in R2for density from the encoding alone\. A gradient\-boosted tree learns a default branch for a missing value from the sources it trains on, and that branch does not transfer to a source it has not seen, whereas an explicit indicator does\. Under random folds the two encodings are indistinguishable, at 0\.819 and 0\.817, so the design this literature reports would not reveal the difference\. A random split therefore identifies neither the model that transfers nor the missing\-data protocol that transfers\. The linear model no longer collapses either, returning 0\.513 and 0\.588 on the six inputs where on all three limits it returned−\-51 and−\-101, so removing one near\-collinear column accounts for the whole of that difference\.