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OSCC: Certified Observation-Safe Coupling Optimization for Gradient-Noise Control in Imperfect-Information Learning
Miaobo Hu, Shuhao Hu, Xiaobo Guo, Xin Wang, Bokun Wang, Rui Chen, Daren Zha
TL;DR
In imperfect-information games, common-random-number rollouts can reduce comparison noise, but unsafe sharing and scalar contrast objectives limit their use for multi-action policy updates. OSCC defines admissible couplings and a gradient-aware selector; in fixed-root Leduc comparisons, fully coupled CP-GRPO reduces return-contrast variance by 55.87%.
Problem
Common-random-number rollouts can reduce counterfactual comparison noise, but unsafe sharing and return-contrast variance alone do not address multi-action policy-gradient noise.
Source: §1 · p. 1Method
OSCC defines observation-safe, marginal-preserving couplings, while OSCC-Select uses separate safety and cost-adjusted gain certificates to choose a coupling or fall back to independent sampling.
Source: §1 · p. 2Results
55.87%: CP-GRPO reduced return-contrast variance from 41.1158 to 18.1441 on 100,000 Leduc pairs, with no failed groups.
Source: §6 · p. 9Takeaways & Limitations
For multi-action policy optimization, coupling selection should account for policy-gradient-weighted off-diagonal return covariance rather than return-contrast variance alone.
Source: §1 · p. 2The certificate assumes independent bounded calibration blocks, so a changed checkpoint or root law requires recalibration.
Source: §5 · p. 9
Abstract
from arXiv · showhide
Coupled rollouts can reduce the noise of counterfactual action comparisons, but two issues prevent standard common-random-number constructions from serving as a general learning primitive in imperfect-information environments. First, an invalid coupling may expose hidden state, synchronize endogenous policy randomness, or misalign chance events after counterfactual histories diverge. Second, in multi-action policy optimization, lower return-contrast variance is not by itself the relevant objective: the optimizer depends on the return covariance matrix after projection through the local policy-gradient geometry. We introduce observation-safe counterfactual coupling (OSCC), a framework that defines an admissible class through marginal preservation, information-state safety, branch-local policy randomness, semantic event alignment, and trace-before-oracle replay. We derive a gradient-aware coupling criterion showing that, for marginal-preserving couplings, policy-gradient noise changes are determined by policy-Jacobian-weighted off-diagonal return covariance. This motivates OSCC-Select, a calibration-only selector that chooses among independent, root-only, continuation-only, and fully coupled rollouts using separate safety and gain certificates. Its gain target combines projected gradient noise with measured physical sampling cost and falls back to independent sampling whenever a simultaneous lower confidence bound does not certify improvement. On 100,000 fixed-root Leduc comparisons, the fully coupled CP-GRPO instantiation reduces return-contrast variance from 41.1158 to 18.1441, a 55.87% reduction, while preserving the declared branch marginals. With three actions, OSCC-Select chooses continuation coupling and attains gradient-noise trace 0.0783 versus 0.0917 for return-variance selection. Increasing calibration from 64 to 2,048 groups raises certification from 0.327 to 0.995.
1 INTRODUCTION
OSCC frames counterfactual rollout coupling as a constrained problem under partial observation, then uses OSCC-Select to choose among rollout designs. Its gradient-aware criterion and evaluation distinguish return-contrast precision from policy-update noise and downstream learning outcomes.
1 INTRODUCTION
- 55.87% lower return-contrast variance over 100,000 Leduc comparisons is the fixed-root audit’s main result.The evaluation keeps estimator claims distinct from learning claims and reports lower final diagnostic exploitability for independent sampling on every seed.Source: §1 · p. 2
- OSCC defines admissible couplings through marginal preservation, information-state safety, branch-local policy randomness, semantic event alignment, and trace-before-oracle ordering.Source: §1 · p. 2
- For multi-action policies, gradient noise depends on policy-gradient-weighted off-diagonal return covariance, while the binary-action case reduces to return-contrast variance.This makes the covariance after projection through local policy-gradient geometry—not scalar contrast variance alone—the relevant object.Source: §1 · p. 2
- OSCC-Select chooses among independent, root-only, continuation-only, and fully coupled rollouts using separate safety and simultaneous cost-adjusted gain certificates, with an independent fallback.It estimates an update-relevant noise functional and physical sampling cost on a calibration split; the mapping is frozen before held-out learning.Source: §1 · pp. 1–2
- The evaluation connects marginal validity and covariance to projected gradient noise, realized update variance, physical cost, held-out quality, solver evaluation, and cross-environment transfer.Source: §1 · p. 2
2 RELATED WORK
The related work establishes common scenarios as a comparison-noise tool and positions OSCC as adding information-safety and policy-randomness constraints for imperfect-information games. It complements policy-optimization objectives and differs from control-variate evaluation by coupling action alternatives before their return difference is formed.
2 RELATED WORK
- The paper’s comparison unit is a legal root-action group with explicitly shared exogenous draws; prior game systems motivate exploitability-oriented evaluation.Source: §2 · p. 2
- OSCC complements clipped and trust-region policy updates and group-relative objectives by changing trajectory coupling before a contrast is formed, without changing the action space or policy parameterization.Source: §2 · p. 2
- Unlike AIVAT’s control variates and known strategies, CP-GRPO couples action alternatives before forming their return difference.Source: §2 · p. 2
- The classical covariance identity says shared exogenous draws can reduce difference variance when induced covariance is positive without changing either marginal target.Source: §2 · p. 2
- OSCC operationalizes the covariance identity with root, continuation, and policy keys, observation-only inputs, event handling, and replay checks, separating estimator claims from learning claims.Source: §2 · p. 3
- Classical paired scenarios and common-state action comparisons reduce comparison noise, but OSCC additionally constrains shared events after histories diverge and keeps policy draws branch-local.The paper identifies PEGASUS, paired comparisons, TRPO’s vine procedure, and simulation-based planning as precedents.Source: §2 · pp. 2–3
3 METHOD
OSCC defines admissible couplings for imperfect-information action comparisons and selects among them using gradient-aware noise and physical sampling cost. Its method preserves branch marginals while constraining information access, policy randomness, event alignment, and replay order.
3.1 TASK DEFINITION
- A comparison group evaluates each legal root action under a fixed continuation policy, with training, validation, and test roots assigned separate stream namespaces.Source: §3.1 · p. 3
3.2 OBSERVATION-SAFE COUPLING
- Policy lookups use the acting player’s information state, excluding opponent cards, terminal-oracle labels, and branch identity from policy-visible inputs.Replay checks the observation invariant across branch swaps; the reported 774-state audit found no opponent-card exposure.Source: §3.2 · p. 3
3.3 ADMISSIBLE OBSERVATION-SAFE COUPLINGS
- Admissible couplings preserve each branch marginal, restrict policy inputs to legal information states, keep policy draws branch-local, align chance draws by semantic event, and freeze traces before oracle joins.Source: §3.3 · p. 4
3.4 TARGET PRESERVATION AND COUPLING VARIANCE
- Branch marginals preserve the expected action-value difference, while return covariance separates contributions from shared hidden roots and shared continuation randomness.The interaction between these sources need not be additive when interventions change which semantic chance events branches reach.Source: §3.4 · pp. 4–5
3.5 FROM RETURN COVARIANCE TO GRADIENT NOISE
- With fixed branch marginals, gradient-noise changes depend on off-diagonal return covariance weighted by local policy-gradient geometry, not on scalar contrast variance alone.For two actions, the ranking matches contrast variance at fixed information state and equal cost; with three or more actions, scalar contrasts can rank candidates differently.Source: §3.5 · p. 5
3.6 CERTIFIED GRADIENT-AWARE COUPLING SELECTION
- OSCC-Select chooses only among safe candidates whose simultaneous calibration lower bound certifies gain, minimizing projected gradient noise plus measured sampling cost and otherwise falling back to independent sampling.Its guarantee applies to the frozen checkpoint and calibration law; checkpoint, root-law, or event-interface changes require recertification.Source: §3.6 · pp. 5–6
3.7 CP-GRPO AS A FULL-COUPLING INSTANTIATION
- CP-GRPO is OSCC’s full-coupling rollout instantiation, changing rollout construction rather than the policy objective or parameterization.It enumerates legal root actions, shares compatible hidden roots and semantic continuation streams, and keeps policy-side action draws branch-local.Source: §3.7 · p. 6
3.8 TRAINING AND REPLAY INTERFACE
- Figure 1 places trace freezing before replay validation and policy updates, with an audit path that supplies no oracle feedback to the online policy.Source: Figure 1 · p. 7
4 EXPERIMENTS
The experiments assess coupling validity, estimator precision, downstream learning, robustness, and selector calibration across declared Leduc and Kuhn settings. The fixed-root audit finds lower return-contrast variance with shared streams, while later results show why policy geometry and calibration matter.
4.1 SETUP AND RESEARCH QUESTIONS
- The evaluation uses fixed-root Leduc groups and declared tabular schedules, separating marginal/replay validity, covariance and contrast variance, gradient/update noise, calibration and cost, and downstream decision quality.The setup specifies 100,000 paired comparisons for the main audit and 5,120 logical groups per seed for the authoritative tabular screen.Source: §4.1 · p. 7
4.2.1 RETURN-CONTRAST PRECISION
- 18.1441 versus 41.1158: shared root and continuation streams reduce return-contrast variance by 55.87% over 100,000 Leduc comparisons.Both arms use the same root groups and budgets; the difference is exogenous coupling.Source: Table 2 · p. 7, §4.2.1 · p. 7, Figure 2 · p. 8
4.2.2 BOUNDED UPDATE DIAGNOSTICS
- The bounded tabular diagnostic has lower final diagnostic exploitability for independent sampling on all three seeds, while a separate five-seed policy screen favors CP-GRPO on return, update variance, and sealed Leduc exploitability.The reported policy-screen values are 0.2847 ± 0.0196 versus 0.2369 ± 0.0284 for return, 0.0917 versus 0.1835 for update variance, and 0.1184 versus 0.1715 for sealed Leduc exploitability.Source: §4.2.2 · p. 8
4.3.1 CONTROLS AND ROBUSTNESS
- 25.4317 for continuation-only, 32.7641 for root-only, and 18.3269 for full CP-GRPO versus 41.0876 for independent sampling show that both partial couplings reduce contrast variance.The ablation uses identical logical root/action budgets and five seeds; the combined coupling has the lowest reported value.Source: §4.3.1 · p. 9
4.3.3 ROOT–BUDGET AND UPDATE DIAGNOSTICS
- With 61,440 physical calls per arm, OSCC-Select has gradient-noise trace 0.0837 at 107.09 s versus 0.1396 at 105.21 s for independent sampling.Across private-card, public-round, and continuation-budget strata, paired variance remains below independent variance.Source: §4.3.3 · p. 9
4.3.4 ACTION SUPPORT AND CALIBRATION SIZE
- For three actions, OSCC-Select chooses continuation-only coupling with gradient-noise trace 0.0783, while return-variance selection chooses full coupling with trace 0.0917.The oracle trace is 0.0778; across 64–2,048 calibration groups, certification rises from 0.327 to 0.995 and false selection falls from 0.031 to 0.001.Source: §4.3.4 · p. 9
5 DISCUSSION AND LIMITATIONS
The discussion frames OSCC as covariance control under partial observation and distinguishes estimator precision from downstream learning evidence.
5 DISCUSSION AND LIMITATIONS
- The certificate assumes independent bounded calibration blocks, so a changed checkpoint or root law requires recalibration.The reported experiments cover finite Leduc and Kuhn games, frozen-checkpoint calibration, and the stated tabular schedules; the fixed-root audit improves precision while the bounded diagnostic favors independent sampling on all three seeds.Source: §5 · p. 9
- The three-action result illustrates why selection must account for policy geometry rather than contrast variance alone.The discussion connects estimator precision to downstream policy, solver, transfer, and matched-cost comparisons.Source: §5 · p. 9
6 CONCLUSION
The conclusion summarizes OSCC, OSCC-Select, and CP-GRPO, emphasizing a certified covariance-based approach and a fixed-root variance reduction in Leduc.
6 CONCLUSION
- On 100,000 Leduc pairs, CP-GRPO cuts variance 55.87%, from 41.1158 to 18.1441, with no failed groups.The conclusion also notes target preservation, gradient-aware covariance weighting, and a finite-sample fallback.Source: §6 · p. 9
- The paper introduces OSCC, OSCC-Select, and CP-GRPO as its central contributions.Source: §6 · p. 9
AI-USE STATEMENT
The authors disclose using generative AI tools to polish the text and summarize reference literature, while stating that they were not used to generate experimental results or formulate mathematical claims.
AI-USE STATEMENT
- The authors report that methodology, experimental setup, analysis, and interpretation were conducted and verified by the authors.They take responsibility for the final content, claims, analyses, and artifacts.Source: AI-USE STATEMENT · p. 10
ETHICS STATEMENT
The work evaluates variance reduction in finite, simulated Leduc and Kuhn poker environments, without human participants or personal data.
ETHICS STATEMENT
- The study concerns finite, simulated Leduc and Kuhn poker environments and uses no human participants or personal data.The observation contract excludes opponent private cards and post-hoc solver labels from online policy inputs.Source: ETHICS STATEMENT · p. 10
- The reported results concern controlled game benchmarks; applying automated policies to gambling settings raises risks of financial harm and unfair play.The code and artifacts support research evaluation of the stated protocols.Source: ETHICS STATEMENT · p. 10
- Opponent private cards and post-hoc solver labels are excluded from online policy inputs.Source: ETHICS STATEMENT · p. 10
REPRODUCIBILITY STATEMENT
The supplementary materials provide code, numerical records, replay fixtures, and manifests linking reported measurements to their experimental protocols.
REPRODUCIBILITY STATEMENT
- The Supplementary Material includes code and artifacts supporting research evaluation of the stated protocols.Source: REPRODUCIBILITY STATEMENT · p. 10
- Appendices A–E cover coupling construction and proofs, experiment protocols and accounting, complete result tables, and artifact mappings.Appendix A specifies the coupling construction, update equations, and proofs; Appendices B and C describe experimental procedures and accounting.Source: REPRODUCIBILITY STATEMENT · p. 10
- The 100,000-group audit is represented by hash-bound sufficient statistics, and the replay fixture checks the observation and sampling contract.Source: REPRODUCIBILITY STATEMENT · p. 10
- Training seeds, evaluation roots, and physical sampling calls are reported with their respective experiments.Source: REPRODUCIBILITY STATEMENT · p. 10
A TASK AND METHOD DETAILS
The method couples counterfactual rollouts while preserving branch marginals and information boundaries, then evaluates covariance effects through policy-gradient noise and auditable replay. Its selector uses calibration evidence and simultaneous gain bounds to choose a coupling or fall back to independent sampling.
A.1 ESTIMATOR DERIVATION AND ASSUMPTIONS
- A marginal-preserving coupling can change return covariance without changing the marginal target, but finite-sample marginal comparisons remain a separate check.The contrast variance identity separates changes in marginal sample-variance sums from changes in covariance.Source: §A.1 · p. 13
- The policy observation is limited to public history, the acting player's private card, round, and legal mask; hidden cards and oracle fields are excluded.Unused continuation counters remain reserved after early termination, preventing reuse under a different semantic event.Source: §A.1 · p. 13
A.3 FULL COUPLING AND REPLAY CONTRACT
- The replay contract validates root, observation, counter, terminal-return, and branch-swap predicates before accepting a quantitative contrast.The ledger retains explicit terminal reasons and consumed counters.Source: §A.3 · p. 13, Table 6 · p. 14
A.5 PRIOR-ART DESIGN RECORD
- CP-GRPO is positioned against common-random-number precedents by adding observation-state invariance, branch-local policy randomness, and an artifact ledger binding event order to variance.The contract makes estimator claims separable from learning claims.Source: §A.5 · pp. 14–15
A.6 CONDITIONAL COVARIANCE AND THE UPDATE DIRECTION
- Root sharing primarily changes covariance across hidden deals, while continuation sharing changes covariance conditional on a deal; their effects can interact.For multiple actions, policy-gradient noise depends on policy-derivative-weighted off-diagonal return covariance, so contrast variance alone is insufficient.Source: §A.6 · p. 15, §A.8 · p. 17
A.7 COLLECTOR PROCEDURE AND SUPPORT CONDITIONS
- The collector freezes the checkpoint and sampling contract, executes branch-local policy draws and semantically addressed chance draws, then freezes the trace before joining evaluator fields.Complete-action enumeration requires finite legal support; zero-probability actions require an explicit complete-action objective or a positive-support proposal.Source: §A.7 · pp. 15–16, Figure 3 · p. 16
A.8 PROOFS FOR GRADIENT-AWARE SELECTION
- The selector compares candidate couplings using calibration scores and simultaneous confidence bounds, selecting a non-independent arm only when its lower bound certifies gain.If no bound is positive, it falls back to the independent objective; a changed calibration law or checkpoint requires a new certificate.Source: §A.8 · pp. 16–17
B.1 OBSERVATION AND STREAM INVARIANCE AUDIT
- The event-level audit derives policy randomness only from run, group, branch, and action counter, and checks replay observations, stream monotonicity, and branch-swap behavior.The validator checks root equality, observations, streams, legal actions, and terminal codes before joining returns or solver fields.Source: §B.1 · p. 17, §B.2 · p. 18
B.4 IMPLEMENTATION CHECKS AND UPDATE ELIGIBILITY
The implementation and evaluation sections distinguish replay eligibility, estimator precision, physical cost, and learned-policy quality, using matched budgets and explicit statistical units. Across the reported screens, coupling improves precision, but learned-policy outcomes vary by evaluation and are not inferred from variance alone.
B.3 LEARNING-BOUNDARY AND STATISTICAL-UNIT AUDIT
- In the seed-resolved comparison, each of the five seeds has 5,120 valid groups, and paired variance is below independent variance in every seed, supporting a repeatable precision effect at this budget.The observed ratios range from 0.4393 to 0.4451.Source: §B.6 · p. 19
D.3.1 NON-PAIRED INDEPENDENT ROLLOUT BASELINE
- The primary five-seed comparison reports CP-GRPO with the highest learned-arm return and success and the lowest update variance and exploitability.The learned arms use the same five-seed protocol and physical-call ledger.Source: §D.3.1 · p. 23, Table 21 · p. 23, Table 22 · p. 23
D.3.2 SOLVER-BACKED VALUE AND EXPLOITABILITY
- The solver screen reports CP-GRPO value 0.2136 and exploitability 0.1184, with exact CFR+ as the reference anchor.The comparison uses sealed Leduc test states and a common held-out root convention.Source: §D.3.2 · p. 24, Table 24 · p. 24
D.3.3 CROSS-ENVIRONMENT TRANSFER
- 0.0109 versus 0.0211 exploitability on sealed Kuhn test states is reported for CP-GRPO and the PPO control, respectively.The paired-coupling variance ratios are 0.4996 versus 1.0324 for the PPO control under the same observation schema and held-out evaluation convention.Source: §D.3.3 · p. 24, Table 25 · p. 24
D.4.1 ROOT-BUDGET FACTORIZATION
- The root-budget screen reports paired variance below independent variance in every completed row, with the same four-row comparison shown in its plot.The plot is a visual aid to the table, not an additional estimate.Source: §D.4.1 · p. 25, Table 27 · p. 25, Figure 5 · p. 26
D.5 COUPLING-FACTOR AND CLIPPING STUDIES
- The coupling-factor study separates root reuse from continuation reuse and reports logical coupling keys separately from physical sampler calls.A lower-variance contrast is not automatically a matched-cost comparison; promotion also requires marginal agreement and disjoint held-out roots.Source: §D.5 · p. 26, Table 28 · p. 26
D.6 EXPANDED COUPLING MECHANISM MATRIX
- The expanded mechanism matrix varies shared root, shared continuation, and policy namespace one component at a time, treating the arms as a mechanism decomposition rather than a leaderboard.All arms use matched root IDs, legal masks, horizon, optimizer schedule, and seed names.Source: §D.6 · p. 27, Table 30 · p. 27
D.7 SELECTOR STRESS AND SAFETY CONTROLS
- The selector uses full coupling only when a simultaneous lower confidence bound certifies positive variance gain, otherwise falling back to a weaker coupling or independent sampling.Source: §D.7 · p. 27
D.8 ACTION-SUPPORT AND CALIBRATION-SIZE STRESS
The selector stress tests show that the preferred coupling depends on action support and that certification improves with calibration size. The later protocols keep noise, cost, learning, solver, transfer, and artifact evidence tied to distinct measured endpoints and validation records.
D.8 ACTION-SUPPORT AND CALIBRATION-SIZE STRESS
- With three actions, OSCC-Select chooses continuation-only coupling at gradient-noise trace 0.0783, versus 0.0917 for RV-Select’s full coupling, despite higher contrast variance.The oracle reference trace is 0.0778; binary groups select full coupling under both criteria.Source: Table 32 · p. 28, §D.8 · p. 28, Figure 6 · p. 29
- At 2,048 groups, certification reaches 0.995, false selection is 0.001, and gradient-noise regret is 0.0009, at a cost of 16,419 physical calls.At 64 groups, the median gain lower bound is −4.6831 and 514 calls are used; at 128 groups the bound is 3.1478 and the selected arm changes to full coupling.Source: Table 33 · p. 28, §D.8 · p. 28, Figure 7 · p. 29
D.9 RETURN-TO-UPDATE NOISE TRANSFER; D.10 PHYSICAL-COST FRONTIER
- Noise-transfer evaluation tracks return, projected-gradient, and realized-update ratios, clipping, and finite-gradient counts at a frozen checkpoint; the cost comparison matches physical calls across admissible arms.The matched-cost setup equalizes physical-call or transition budgets, with 61,440 calls per arm in Figure 8.Source: §D.9 · p. 29, §D.10 · p. 30, Figure 8 · p. 30
D.11 SOLVER-BACKED EXPLOITABILITY PROTOCOL; D.12 CROSS-ENVIRONMENT TRANSFER PROTOCOL
- The solver protocol evaluates policies on the same held-out roots as policy evaluation, while transfer tests a frozen public schema on Kuhn after training on Leduc.Solver queries are post-hoc; transfer changes deck multiplicities, horizon, and terminal history while retaining explicit legal-action representation.Source: §D.11 · pp. 30–31, §D.12 · p. 31, Table 38 · p. 31
D.13 QUALITATIVE FAILURE CASES AND ARTIFACT PROTOCOL; E.2 IMPLEMENTATION ARTIFACTS IN THE SUPPLEMENTARY MATERIAL
- The failure ledger links each qualitative case to its group key, terminal reason, and consumed cost, while supplementary artifacts preserve configuration and reproduction records.The listed cases include observation-hash drift, reused chance counters, illegal actions, early termination, and solver timeouts.Source: §D.13 · pp. 31–32, Table 40 · p. 32, §E.2 · p. 33
D.14 THREE-SEED UPDATE DIAGNOSTICS
- Independent sampling has lower final diagnostic exploitability in all three seeds of the bounded-update screen, despite the separate focus on reducing contrast noise.The independent-minus-paired differences are −0.0207, −0.0704, and −0.0709 for seeds 13, 17, and 23; the plotted mean is −0.0540.Source: §D.14 · p. 32, Figure 10 · p. 32
E.1 EVIDENCE ACCEPTANCE MATRIX
- The evidence matrix promotes variance, learning, and transfer claims only after their respective observation/replay, optimizer/cost, and test-environment/serializer requirements pass.It keeps statistical denominators distinct and connects each claim to a measured endpoint and supporting artifact.Source: §E.1 · pp. 32–33, Table 41 · p. 33
E.3 EVIDENCE-TO-CLAIM MAP
- The evidence-to-claim map separates estimator evidence based on return covariance and the variance identity from observation-safety evidence based on hashes, masks, and stream counters.Learning and transfer rows retain explicit seed, split, and artifact requirements.Source: §E.3 · p. 33, Table 43 · p. 33
E.4 FIGURE DATA AND STATISTICAL INTERPRETATION
The figure-data discussion identifies the statistical units and comparison conventions used across the result plots. It distinguishes fixed-root variance estimates, matched-cost noise measurements, solver intervals, and sealed transfer comparisons.
E.4 FIGURE DATA AND STATISTICAL INTERPRETATION
- The fixed-root variance bars use the same 100,000-group estimates as Table 2, with uncertainty reported in Table 3.The root-budget plot retains each stratum’s sample count, and the three-seed diagnostic preserves within-seed pairing.Source: §E.4 · p. 34
- The noise-transfer plot reports ratios to independent sampling at a fixed checkpoint, while the matched-cost scatter compares wall time and gradient noise at 61,440 physical calls per arm.The selector plots show measured action-support and calibration-size conditions without interpolating unmeasured conditions.Source: §E.4 · p. 34
- Solver intervals summarize five-seed player-value estimates, whereas the Kuhn bars represent a separate sealed transfer comparison.Table 44 frames the result figures in terms of their interpretation and statistical units.Source: §E.4 · p. 34, Table 44 · p. 34