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SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization

Wondesen Teshome Bekele, Marco D'Oria

arXiv:2609.33515v1cs.LGstat.ML

TL;DR

Probabilistic predictors may need to satisfy exact structure in every stochastic realization, while common approaches predict in ambient coordinates and then correct or project. SLP-ProbHard defines the predictive law directly through a structural latent law and feasible map. Across its evaluations, exact feasibility does not determine a unique predictive law, and representation choices bring different predictive trade-offs.

  • Problem

    Probabilistic prediction requires feasibility in every stochastic realization, while common approaches predict in ambient coordinates and then enforce feasibility through penalties, post-processing, optimization, or projection.

    Source: §1 · p. 1

  • Method

    SLP-ProbHard pairs a structural latent predictive law with an explicit feasible map to define a sample-wise hard-feasible pushforward law.

    Source: §1 · p. 2

  • Results

    Both SLP-ProbHard and ProbHardE2E/DPPL achieve practical hard feasibility on the compatible affine benchmark, while their stochastic representations yield different predictive trade-offs.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Exact feasibility alone does not specify a predictive law: the chosen structural representation shapes support, boundary probability, stochastic coordinates, attainable covariance, and calibration–sharpness trade-offs.

    Source: §6 · p. 9

    Positive-part maps are nonsmooth, and the normalized positive-part map adds a discontinuous argmax-vertex fallback on the all-inactive region, requiring separate analysis of expected-score gradient estimation.

    Source: L LIMITATIONS · p. 40

Abstract

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Many probabilistic predictors must satisfy exact structure in every stochastic realization, yet common hard-constraint approaches form predictions in ambient coordinates and then correct or project them. We introduce SLP-ProbHard, a cross-family, representation-centered framework for probabilistic hard-constrained learning when explicit structural parameterizations are available. Its core object, a Structural Feasible Latent Parameterization (SFLP), combines a structural latent law $Z \sim P^Z_\theta(\cdot\mid x)$ with a feasible map $Y=h_\phi(x,Z)$ that satisfies the constraint for every latent realization. Together these components define the predictive law itself, including its support and boundary probabilities, rather than serving as a final feasibility wrapper. We study how feasible coordinates and maps affect stochastic dimension, dependence, calibration, expressiveness, and computation. Experiments use Gaussian latent laws and fixed geometry-derived maps across affine equalities, ordering and simplex constraints, nonlinear manifolds, and three structural representations of seven-basin hydrological flow-duration-curve (FDC) data. In an official-source affine comparison with ProbHardE2E/DPPL, both methods achieve zero practical constraint violations. SLP-ProbHard uses 8 instead of 11 stochastic coordinates and improves MSE/MAE, while DPPL yields better marginal CRPS and closer-to-nominal coverage; a paired test detects no Energy Score difference across ten seeds. Real-world affine and nonlinear FDC representations reduce 13 to 7 and 14 to 8 ambient versus computational coordinates, respectively. Exact feasibility alone thus does not determine a predictive law, motivating direct structural generation when meaningful feasible coordinates are available.

1 INTRODUCTION

Probabilistic predictors need to satisfy structural constraints in every realization, but common approaches predict in ambient coordinates before enforcing feasibility. SLP-ProbHard instead defines the predictive law through a structural latent law and an explicit feasible map, then studies how that representation shapes the law and its computation.

1 INTRODUCTION

  • Predicting in ambient space and then applying penalties, post-processing, optimization, or projection is a common route to feasibility.ProbHardE2E is identified as the closest dedicated probabilistic projection framework; projection is useful when the feasible set is implicit or difficult to parameterize.Source: §1 · p. 1
  • The paper asks whether a predictive distribution can be defined directly in feasible coordinates when a scientifically meaningful generative representation is available.The motivation is that feasibility must hold for every stochastic realization, not only for a predictive mean.Source: §1 · pp. 1–2
  • A structural latent law and explicit feasible map jointly define SLP-ProbHard’s sample-wise hard-feasible pushforward predictive law.The map satisfies the constraint for every latent realization; together, the components determine which feasible outputs and boundary events can receive probability and the number of modeled stochastic coordinates.Source: §1 · p. 2
  • The framework audits how parameterization affects support, boundary probability, stochastic dimension, uncertainty propagation, dependence, calibration, expressiveness, and computation.The audit is validated across representative constraint geometries.Source: §1 · p. 2
  • The experiments span affine equality, ordering, simplex, composed nonnegative ordering, nonlinear manifold and scale–shape problems, and three hydrological FDC representations.The compatible affine benchmark includes an official-source comparison with ProbHardE2E/DPPL: both achieve practical hard feasibility, with representation-dependent predictive trade-offs.Source: §1 · p. 2

2 RELATED WORK

Related work includes methods that enforce feasibility after ambient prediction and methods that construct outputs directly in feasible coordinates. Several probabilistic structural approaches address particular settings, while SLP-ProbHard frames representation choice as a cross-family modeling question.

2 RELATED WORK

  • Ambient prediction followed by optimization, correction, radial mapping, or projection is a major branch; ProbHardE2E constructs hard-constrained predictive laws through differentiable probabilistic projection.The cited approaches include Amos & Kolter (2017), Agrawal et al. (2019), Donti et al. (2021), Liang et al. (2023; 2024), Schneider & Kuhn (2026), and Utkarsh et al. (2026).Source: §2 · p. 2
  • Direct feasible parameterization includes null-space and inequality parameterizations, convex feasible layers, and hard-constrained network architectures.SMT-HCSD generates stochastically ordered temperature outputs using a nonnegative increment.Source: §2 · p. 2
  • In the probabilistic structural viewpoint, CLOVER constructs coherent probabilistic forecasts by construction for hierarchical forecasting through a differentiable structural factor representation.Source: §2 · p. 2

A. Where hard feasibility enters the predictive model

The section contrasts the modeling object used by SLP-ProbHard with a hard-feasibility requirement: its structural latent law is mapped directly into feasible outputs, rather than treating feasibility as the only defining feature.

A. Where hard feasibility enters the predictive model

  • SLP-ProbHard’s structural latent law is paired with a structural feasible pushforward.The section contrasts this modeling object with the shared requirement of hard feasibility, which can still induce different predictive laws.Source: A. Where hard feasibility enters the… · p. 3

B. Constraint structures and real-world representations evaluated here

The paper positions structural generation alongside projection-based feasibility methods and evaluates representative constraint geometries, including multiple structural views of hydrological FDC data. Its related-work framing situates this cross-family comparison among prior probabilistic structural approaches.

B. Constraint structures and real-world representations evaluated here

  • Figure 1 contrasts probabilistic projection of an ambient predictive law with an SFLP that defines the feasible law as a structural-latent pushforward.The figure’s empirical comparison is with ProbHardE2E/DPPL on the compatible affine benchmark; DC3 and Πnet provide conceptual positioning only.Source: Figure 1 · p. 3
  • The related-work passage describes a formulation separating free and constrained variables for linearly constrained multiple time series and extending it to probabilistic forecast reconciliation.It also identifies probabilistically sound deep generative learning under hard linear equality constraints.Source: B. Constraint structures and… · p. 3
  • The study brings structural-probabilistic perspectives from hierarchical forecasting, general linear time-series constraints, hard linear generative learning, and stochastic temperature ordering into a common representation audit.The audit spans affine, ordering, simplex, and explicit nonlinear geometries and asks how coordinate choice affects support, boundary probability, stochastic coordinates, uncertainty, expressiveness, calibration, and computation.Source: B. Constraint structures and… · p. 3

3 SLP-PR O BHA R D

An SFLP pairs a structural latent predictive law with a feasible structural map, jointly defining the family of feasible predictive distributions. The paper also gives affine equalities a null-space parameterization whose latent dimension equals the number of free coordinates.

3 SLP-PR O BHA R D

  • An SFLP pairs a structural latent predictive law with a feasible map that returns an output in the sample-level feasible set for every latent input.Together, the pair determines the family of feasible predictive distributions that can be represented.Source: §3.1 · p. 3
  • Table 1 covers empirical constraint coverage across representative geometry groups, with the direct official-source DPPL comparison limited to the synthetic affine benchmark.The caption does not provide cell values.Source: Table 1 · p. 4
  • For affine equalities, Y = y0 + NZ uses q = d − m free latent coordinates, which are both intrinsic and computational stochastic dimensions.The construction assumes a feasible point y0 and a full null-space basis N.Source: §3.1 · p. 4

A. SFLP design/audit:

The SFLP audit checks feasibility and records how the chosen representation shapes coverage, boundary probability, dimensions, dependence, tractability, and generation cost.

A. SFLP design/audit:

  • The audit verifies samplewise feasibility and records map coverage, boundary mass, intrinsic and computational dimensions, sensitivity, dependence restrictions, density tractability, and whether generation needs a separate correction stage.It makes representation choice operational by checking consequences beyond feasibility.Source: A. SFLP design/audit · p. 4

B. Probabilistic training:

Training samples latent variables, maps them to feasible outputs, and scores the resulting predictive law; the representation audit shows that these choices affect probability allocation and computation. Structural generation and projection can share feasible support while inducing different predictive laws.

B. Probabilistic training:

  • Inference samples Z and applies hϕ to return feasible predictive samples without a separate correction stage.The workflow does not assume structural parameterization is preferable when no useful explicit SFLP is available.Source: B. Probabilistic training · p. 4

3.2 REPRESENTATION-INDUCED PROBABILISTIC CONSEQUENCES

  • Stochastic-dimension savings depend on the map: intrinsic and computational dimensions coincide for affine null-space and circle-angle coordinates, but differ for boundary-capable simplex and nonlinear FDC implementations.Ordering need not reduce computational dimension.Source: §3.2 · p. 5

3.3 STRUCTURAL-MAP DESIGN CONSIDERATIONS

  • For the positive-part map, expected active-gradient probability equals the probability of a positive output, so the same latent parameters control boundary mass and this sensitivity diagnostic.The identity is not an optimization guarantee.Source: §3.3 · p. 5

3.4 RELATIONSHIP TO PROJECTION AND PROBHARDE2E

  • Equal feasible support does not ensure equal covariance families or probability allocation; projection suits implicit or difficult feasible sets, while structural generation is attractive when meaningful feasible coordinates or boundary behavior matter.An SFLP defines the law through a structural latent law and feasible map rather than projecting an ambient predictive law.Source: §3.4 · p. 5

4 EXPERIMENTS

Experiments evaluate SLP-ProbHard on frozen held-out synthetic and seven-basin FDC benchmarks against unconstrained, soft, post-hoc, and matched end-to-end hard baselines. The FDC study reuses the same underlying data and split across three target representations, while comparisons also reflect preprocessing choices.

4 EXPERIMENTS

  • The benchmark suite uses frozen held-out synthetic and seven-basin FDC data, comparing unconstrained, soft, post-hoc, and matched end-to-end hard baselines.The affine benchmark adds an official-repository ProbHardE2E/DPPL comparison using a verified adapter; existing SLP-ProbHard seed-20-29 results were reused unchanged.Source: §4 · p. 6
  • The seven-basin FDC dataset has 756 samples from 1996–2022, split temporally into 532 training, 112 validation, and 112 test samples.It uses 22 predictors for seven Emilia-Romagna basins.Source: §4 · p. 6
  • The three targets represent FDC levels, adjacent increments, and scale–shape allocation, reusing the same underlying FDCs and split to examine representation choices.The affine target exposes increment coherence, while the nonlinear target separates low-flow baseline, discharge range, and relative increment allocation.Source: §4 · p. 6
  • Cross-representation results combine representation and preprocessing choices, rather than isolating feasible-map changes under identical target preprocessing.The affine and nonlinear extensions reuse the frozen predictors, raw FDCs, identities, and split, with a train-only multiplicative RMS scale of 43.7191.Source: §4 · p. 6

5 RESULTS

Across synthetic and hydrological benchmarks, structural maps provide hard feasibility while coordinate choices shape predictive trade-offs. Results include fewer stochastic coordinates, representation-dependent calibration and scores, and finite-precision violations in some scaling configurations.

5.1 PRIMARY SYNTHETIC BENCHMARKS

  • SLP-ProbHard lowers MSE by 1.30% and MAE by 0.43%, while DPPL lowers marginal CRPS by 1.18% and gives closer-to-nominal coverage; neither has observed affine violations.The paired test detects no Energy Score difference across ten seeds.Source: §5.1 · p. 6, Table 2 · p. 7
  • On the unit circle, the angle map uses one stochastic coordinate instead of two and has zero observed violations, with no detected MSE, CRPS, or Energy Score differences across ten seeds.For ordering, SLP’s exact-tie rate is 0.208 versus 0.396 for isotonic projection, against an observed 0.255; on the d = 5 simplex, projection’s zero rate is 0.246 versus SLP’s 0.180, against an observed 0.250.Source: §5.1 · p. 7

5.2 COMPOSED AND REAL-WORLD STRUCTURAL REPRESENTATIONS

  • In the controlled nonnegative-ordering task, both methods satisfy the constraints exactly, and SLP-ProbHard is numerically lower on CRPS and Energy Score.The targets use cumulative positive-part, so this is mechanism validation rather than independent evidence of general superiority.Source: §5.2 · p. 7
  • For affine FDC coherence, SLP-ProbHard uses 7 rather than 13 stochastic coordinates, with zero violations for both methods and no detected score differences.SLP intervals are narrower (10.94 vs. 13.57) but coverage is lower (0.712 vs. 0.746); this branch enforces increment coherence only, not nonnegativity or FDC ordering.Source: §5.2 · p. 7, Table 3 · p. 8
  • For nonlinear FDC scale-shape coherence, SLP-ProbHard uses 8 rather than 14 computational coordinates, while both hard methods have zero violations.No paired Raw-Q score differences were statistically detected; post-hoc feasible correction has lower mean raw-Q CRPS (3.568 vs. 4.047) and higher 90% coverage (0.823 vs. 0.649).Source: §5.2 · p. 8, Table 3 · p. 8

5.3 COVARIANCE AND COMPUTATION

  • Rank-4 or full covariance improves SLP-ProbHard CRPS from 0.2889 to about 0.2837 and Energy Score from 1.2171 to 1.2071 while retaining zero observed violations.Median inference latency is 1.65 ms for SLP-ProbHard versus 3.22 ms for the DPPL affine adapter under matched CPU instrumentation.Source: §5.3 · p. 8

5.4 SCALING, FINITE PRECISION, AND STATISTICAL INTERPRETATION

  • Across twelve affine configurations, SLP-ProbHard uses q = d −m coordinates and reduces stochastic dimension by 12.3–38.5%, while point accuracy remains close to Projection-Or.Float64 reduces primary affine residuals from roughly 10−7 to 10−15; in one float32 scaling configuration, violations at 10−5 are 0.0193 for SLP and 0.3958 for Projection-Or.Source: §5.4 · p. 8

6 DISCUSSION AND LIMITATIONS

The discussion emphasizes that feasibility alone does not determine predictive uncertainty: representations affect boundary probability, dimension, covariance, and calibration. It also bounds the framework’s scope and distinguishes architectural coordinate savings from implementation-dependent runtime.

6 DISCUSSION AND LIMITATIONS

  • Exact feasibility does not uniquely specify a predictive law; the selected representation shapes boundary probability, stochastic coordinates, attainable covariance, and calibration-sharpness trade-offs.Coordinate savings depend on the map: affine and circle constructions use intrinsic coordinates, while normalized simplex and nonlinear scale-shape retain computational redundancy.Source: §6 · p. 9
  • The three FDC views test physical ordering, increment coherence, and scale-shape coherence on one fixed dataset and split, with distinct preprocessing choices affecting probabilistic predictions.Source: §6 · p. 9
  • Across the three FDC representations, SLP’s reported 90% marginal coverage is 0.722, 0.712, and 0.649, indicating undercoverage; interval width alone does not establish better uncertainty quantification.Source: §6 · p. 9
  • Explicit nonlinear-equality demonstrations cover the unit circle and FDC scale-shape map, not arbitrary nonlinear constraints; projection or correction remains more general for implicit or difficult feasible sets.Source: §6 · p. 9
  • Stochastic-coordinate savings are architectural and hardware independent, but wall-clock improvements depend on workload and implementation; optimized affine conditioning can itself be inexpensive.Source: §6 · p. 9
  • SLP-ProbHard requires a useful explicit map, and non-surjective maps introduce representation bias; nonlinear pushforwards may lack tractable densities or moments.The hydrological target is GloFAS v5.0 LISFLOOD-based reanalysis discharge rather than observed-gauge discharge; the nonlinear FDC comparator is an alternating feasible correction, not an exact joint Euclidean projection.Source: §6 · p. 9

7 CONCLUSION

SLP-ProbHard defines hard-feasible predictive laws through a structural latent law and feasible map, with the representation shaping support, boundary probability, and attainable dependence. Experiments span several constraint families and FDC representations; the affine DPPL comparison shows shared practical feasibility but different predictive trade-offs.

7 CONCLUSION

  • SLP-ProbHard’s structural latent law and feasible map jointly define a hard-feasible predictive law, shaping support, boundary probability, stochastic representation, and attainable dependence, calibration, expressiveness, and computation.Experiments span affine equalities, ordering, simplex, explicit nonlinear maps, and three representations of the same hydrological FDC data; projection and correction remain useful for implicit or difficult feasible sets.Source: §7 · p. 9

FUNDING

This section presents SLP-ProbHard as a structural-latent approach to feasible probabilistic prediction, distinguishing it from prior methods and explaining how structural maps shape predictive distributions, boundary mass, computational dimension, and training. It also discusses mathematical guarantees alongside limitations involving gradients, numerical precision, and map choice.

FUNDING

  • The study received no dedicated project funding; doctoral and computational support are acknowledged elsewhere.Source: FUNDING · p. 10

A.4 CONSTRUCTIVE FEASIBILITY LINEAGE

  • SLP-ProbHard centers a full predictive distribution by mapping stochastic latent samples to feasible outputs, rather than targeting only deterministic feasibility or optimization solutions.Its methodological contribution is the coupled analysis of structural latent generation, map-induced support and boundary atoms, stochastic coordinates, approximation bias, and scoring and computational consequences; it does not claim feasible-by-construction transforms are new, alongside related work such as Li et al. (2026).Source: §A.4 · p. 17

B.1 STRUCTURAL LATENT DISTRIBUTION

  • The framework permits latent families beyond Gaussians, and training can use tractable likelihoods or sample-based proper scores while applying the feasibility map during training and inference.Examples of alternative latent families include Student-t, Gamma, LogNormal, zero-inflated, mixture, autoregressive, and flow-based distributions.Source: §B.1 · p. 18, §B.4 · p. 18, §B.5 · p. 18

C STRUCTURAL FEASIBILITY MAP TAXONOMY

  • The taxonomy groups maps by structural representation, including affine, ordered, normalized, nonlinear-manifold, composed, and functional maps; ordering implementations can induce exact ties or only reach them.The primary ordering experiment uses positive-part increments, for which latent values on a half-line map to zero increments and can create tie atoms; boundary conditions are included in the taxonomy but are not empirically benchmarked.Source: C STRUCTURAL FEASIBILITY MAP TAXONOMY · p. 19, §C.1 · p. 20, §D.1 · p. 21

C.2 PRIMARY SIMPLEX MAP

  • The simplex map uses d computational latent coordinates for d − 1 intrinsic degrees of freedom, and its all-inactive fallback assigns samples to an argmax vertex.The fallback event’s mean rate was 3.81 × 10^-4 in the final d = 5 primary experiment; this frequency alone does not establish negligible gradient bias.Source: §B.2 · p. 18, §C.2 · p. 20

C.3 EXPLICIT NONLINEAR-EQUALITY MAPS

  • Computational dimension depends on the chosen map: the FDC scale-shape representation has 7 intrinsic degrees of freedom but uses 8 computational latent coordinates.A convenient nonlinear map may be redundant, so intrinsic and computational dimensions are distinguished.Source: §B.2 · p. 18, §C.3 · p. 21

E.1 PUSHFORWARD AND SAMPLE-LEVEL FEASIBILITY

  • For measurable maps whose outputs lie in the feasible set, the pushforward predictive law is sample-level feasible, and map geometry determines where it can place probability.For nonlinear maps, samples can be transformed directly without approximating the full predictive law as Gaussian; affine maps also admit exact moment propagation.Source: §E.1 · p. 22, §E.3 · p. 22

E.3 DISTRIBUTIONAL PROPAGATION THROUGH STRUCTURAL MAPS

  • The positive-part map assigns boundary probability Pr(R = 0) = Φ(−α), with latent location and scale determining both predictive moments and exact boundary mass.For weak ordering, adjacent outputs can also have exact tie probabilities determined by predicted latent parameters.Source: §E.3 · p. 23
  • Structural maps determine boundary probabilities as well as feasibility: ties and exact zeros can receive positive mass, while a continuous Gaussian latent law can reach a boundary without assigning it positive probability.The circle pushforward has analytically available first moments, but when boundary atoms or face-supported mass matter, a Gaussian moment approximation should not replace the full pushforward law.Source: §E.3 · pp. 24–25

E.4 SAMPLEWISE DIFFERENTIATION AND EXPECTED-SCORE GRADIENTS

  • Almost-everywhere samplewise differentiation does not establish unbiased expected-score gradients when the structural map has discontinuities, including the normalized positive-part map’s fallback boundaries.The expected-score gradient needs separate justification or an explicitly reported approximation; reparameterizing the latent sampler does not resolve map discontinuities.Source: §E.4 · p. 25

E.6 CONSTRAINED UNIVERSAL APPROXIMATION

  • A continuous feasible target is uniformly approximable through a continuous structural map when it has a continuous latent lifting and the latent network family universally approximates continuous maps.The circle map is continuous and surjective but has no global continuous section, so the continuous-lifting theorem does not apply globally to it.Source: §E.6 · p. 26

E.7 DISTRIBUTIONAL APPROXIMATION AND PARAMETERIZATION BIAS

  • A measurable surjective map with a measurable section transfers total-variation density of latent laws to pushforward laws, but this conditional result is not a universal-expressiveness guarantee for the Gaussian heads used in experiments.A chosen map also restricts representable laws to its image, so targets assigning mass outside that image cannot be represented exactly.Source: §E.7 · p. 27

E.8 CLASSICAL IDENTITY USED AS A REACHABILITY AND BOUNDARY-MASS TAXONOMY

  • Boundary reachability and positive boundary probability are distinct: normalized positive-part maps can assign positive mass to simplex faces under continuous latents, whereas normalized squares require a measure-zero event for exact zeros and softmax excludes zero.These image and boundary-mass statements concern real arithmetic; floating-point rounding and underflow can change exact-zero or tie diagnostics.Source: §E.8 · pp. 27–28

E.9 FINITE PRECISION

  • Feasibility theorems hold in real arithmetic, while floating-point operations can introduce residuals; continuous residuals are primary numerical evidence and thresholded violation rates are secondary diagnostics.A numerical threshold is an evaluation convention, not a relaxation of the mathematical constraint.Source: §E.9 · p. 28

F.2 DETAILED COMPARISON WITH PROBHARDE2E

  • Projection suits implicit or complex feasible sets, while structural generation is attractive when an explicit, scientifically acceptable representation is available.The mechanism comparison distinguishes SLP-ProbHard by making a constraint-specific structural map part of its probabilistic pushforward model.Source: §F.2 · p. 28, Table 7 · p. 28

G.1 FROZEN BENCHMARK FAMILIES AND HELD-OUT SEEDS

The experiments use frozen benchmark families and held-out seeds to compare structural maps with correction, projection, and soft-constraint baselines. Across scaling and real-world FDC tests, both approaches can be hard-feasible while differing in stochastic dimension and predictive trade-offs.

G.1 FROZEN BENCHMARK FAMILIES AND HELD-OUT SEEDS; G.4 SHARED HYDROLOGICAL FDC SOURCE AND PREPROCESSING

  • The frozen experiments span affine, ordering, simplex, composed, and circle benchmarks, alongside seven-basin FDC data with physically ordered quantile targets and held-out temporal samples.Primary evaluations use 100 predictive samples; FDC comparisons condition on a fixed dataset and temporal split, not new years, basins, or datasets.Source: §G.1 · pp. 29–30, §G.4 · p. 31, §G.8 · p. 33

G.6 REAL-WORLD AFFINE FDC COHERENCE; G.7 REAL-WORLD NONLINEAR SCALE-SHAPE COHERENCE

  • For real-world FDCs, the affine map uses 7 rather than 13 stochastic coordinates, while the nonlinear scale-shape map uses 8 rather than 14; the nonlinear comparator uses five alternating correction steps.The affine map enforces equalities only; the nonlinear finite alternating procedure is not claimed to be the exact joint Euclidean projection.Source: §G.6 · p. 32, §G.7 · pp. 32–33

H.1 FLOAT64 AFFINE DIAGNOSTIC

  • Float64 reduced mean affine residuals from approximately 10^-7 to 10^-15 for all three hard methods, supporting numerical realization effects as the source of small base-case residuals.The diagnostic alone does not attribute every high-dimensional scaling exceedance to the same source.Source: Table 11 · p. 33, §H.1 · p. 34

H.2 LINEAR SCALING

  • Across twelve affine configurations, SLP-ProbHard used q = d − m coordinates, reducing stochastic dimension by 12.3% to 38.5%; point accuracy stayed close, while proper scores usually slightly favored Projection-Or.At (d, m, q) = (97, 33, 64), VR10^-5 was 0.3958 for Projection-Or versus 0.0193 for SLP-ProbHard.Source: §H.2 · p. 34

H.3 ORDERING SCALING

  • From d = 3 to d = 40, positive-part SLP-ProbHard had lower CRPS and Energy Score at every evaluated dimension, while isotonic projection’s tie rate rose from 0.300 to 0.891.Across five scaling dimensions, absolute tie-rate error averaged 0.030 for SLP-ProbHard versus 0.283 for end-to-end isotonic projection.Source: §H.3 · p. 34

H.4 SYNTHETIC COMPOSED NONNEGATIVE ORDERING: MECHANISM VALIDATION

  • The deliberately mechanism-aligned composed benchmark gave both hard methods zero feasibility violations; SLP-ProbHard had lower Energy Score and pair-tie calibration error, while projection better matched anchor-zero frequency.The five-seed result supports the boundary-preimage mechanism, not universal boundary-calibration superiority.Source: §H.4 · p. 34, Table 12 · p. 34

H.5 NONLINEAR CIRCLE PRIMARY RESULTS

  • At ten seeds, circle paired tests detected no differences in MSE, CRPS, Energy Score, or circular MAE; SLP-ProbHard MAE was 1.17% lower.Both hard maps had zero 10^-5 violations, and SLP-ProbHard used one stochastic coordinate versus two for radial projection.Source: §H.5 · p. 35, Table 13 · p. 35

H.6 REAL-WORLD AFFINE FDC RESULTS

  • Both affine FDC hard methods have zero 10^-5 violations; SLP-ProbHard uses 7 rather than 13 stochastic coordinates, while paired tests found no principal-score differences and a sharpness-calibration trade-off.The reported inference latency was 1.148 ms for SLP-ProbHard versus 1.623 ms for end-to-end projection; the reduction is implementation-specific.Source: §H.6 · p. 35, Table 15 · p. 35

H.7 REAL-WORLD NONLINEAR SCALE-SHAPE RESULTS

The real-world nonlinear scale-shape FDC experiments compare feasible representations and enforcement methods, while the broader analyses show that predictive behavior and computational gains depend on the chosen map, latent law, and implementation.

H.7 REAL-WORLD NONLINEAR SCALE-SHAPE RESULTS

  • Both primary end-to-end hard methods had zero bilinear violations at 10^-5, near-float32-roundoff simplex residuals, and zero negative reconstructed discharge and FDC ordering violations.SLP-ProbHard used 8 computational stochastic coordinates versus 14 ambient coordinates for correction, with median inference latency of 1.478 ms versus 7.369 ms for the five-iteration correction.Source: Table 16 · p. 36, §H.7 · p. 36

H.9 LATENT COVARIANCE EXPRESSIVENESS

  • The covariance ablation improved CRPS from 0.28887 to about 0.2837 and Energy Score from 1.21714 to about 1.20710, while MSE/MAE remained essentially unchanged.The low-rank and full-covariance variants were numerically close on this q = 8 benchmark.Source: §H.9 · p. 37, Table 19 · p. 37

I.1 CONTROLLED COMPUTATIONAL PROFILING

  • Controlled profiling found computational advantages that vary by constraint family and implementation, rather than a universal speedup over projection.For example, the circle was 1.22× faster end to end with one structural coordinate versus two ambient coordinates; ordering gains depend on the project’s PAVA implementation.Source: §I.1 · p. 37, §K.5 · p. 39

K.1 THE FEASIBLE REPRESENTATION IS PART OF THE PROBABILISTIC MODEL

  • The maps and latent law jointly define the predictive distribution: weak-order cumulative positive-part permits adjacent ties but not a positive anchor-zero atom under a continuous anchor, unlike the nonnegative map.The hydrological representations hold the underlying dataset and split fixed while testing different structural coherence conditions.Source: §K.1 · pp. 38–39

K.2 DIMENSIONALITY IS FAMILY- AND MAP-DEPENDENT

  • The affine, circle, and nonlinear FDC examples use fewer computational than ambient coordinates, but the dimensional reduction is map-dependent rather than universal.The real-world affine representation uses 7 rather than 13 coordinates; the nonlinear scale-shape map uses 8 for a 7-DOF object, and the weak-order map does not reduce dimension.Source: §K.2 · p. 39

K.3 BOUNDARY CAPABILITY IS NOT BOUNDARY CALIBRATION

  • Boundary reachability does not ensure boundary calibration, and richer latent covariance can improve probabilistic scores without changing the structural feasibility guarantee.The study does not resolve the source of observed undercoverage or establish nominal calibration.Source: §K.3 · p. 39, §K.4 · p. 39, L LIMITATIONS · p. 40

K.6 RELATIONSHIP TO PROBHARDE2E/DPPL

  • On the matched affine benchmark, both methods were feasible, but SLP-ProbHard used 27.27% fewer stochastic coordinates and had lower MSE/MAE, while DPPL had lower marginal CRPS and better 90% coverage.The paired test did not detect an Energy Score difference; the comparison uses a verified official-source affine adapter and does not establish dominance on other constraint classes.Source: §K.6 · p. 40, §O.8 · p. 44

L LIMITATIONS

  • The scope is limited to suitable explicit maps: arbitrary nonlinear-equality parameterizations and higher-dimensional manifold scaling are not established, and timing comparisons remain implementation-specific.The seven-basin benchmark uses reanalysis rather than observed-gauge discharge, so it evaluates constraint structure and distributional behavior rather than observational hydrological skill.Source: L LIMITATIONS · p. 40

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