Updated: 2026-09-07
The population ranking and welfare results are checked with the source clarifications below, including the source’s standing strict candidate-value order in Theorem 1.
formalized.The source is arXiv:2101.05853. The selected surface includes its named definitions, Theorems 1–9, and lemmas, plus the selected Appendix B and Plackett–Luce claims in Section 4. Computational illustrations, numerical and figure observations, and the source’s stated open observation are not claimed as proved theorems. The scope note also separates the finite-support and simulation remarks from the checked results.
| Result | Comparison with source |
|---|---|
| Theorem 1 | Exact. The strict full-set comparison is evaluated on the source’s standing strictly ordered candidate-value domain. Source reading. |
| Theorem 2 | Formalized under explicit outer- regularity: the candidate-value law has ordered support, finite first moments, measurable ranking probabilities, and well-defined conditional and payoff expectations. Conditions. |
| Theorem 3 | Corrected domain: at least three candidates; with two, total hiring welfare is constant. Example. |
| Theorem 4 | Exact. |
| Theorem 5 | Density condition clarified: global absolute continuity and an integrable derivative suffice; smooth full support alone does not. Necessity of this sufficient condition is unresolved. Counterexample and condition. |
| Theorem 6 | Exact. |
| Theorem 7 (Laplace) | Exact. |
| Theorem 8 (Gaussian) | Exact. |
| Theorem 9 | Exact. |
| Appendix C Lemma 1 | Changed inequality: weak globally, strict on overlap. Exact comparison. |
| Lemma 4 | Strictness clarified: a strict likelihood-ratio witness for the strict expectation comparison. Condition. |
| Lemmas 2–3, 5–8 | Exact. |
| Appendix B counterexamples and smoothing | Exact. |
| Gumbel RUM / Plackett–Luce identification | Scale clarified: unit-variance noise rescales accuracy by . The specialization uses the standard Gumbel variance identity as an analytic premise. Details. |
| Plackett–Luce strategy comparison and zero monoculture effect | Exact. |
No selected named theorem remains uncredited. The source’s broad finite-support and simulation remarks are not stated as unrestricted theorem claims; their scope is described in the source note.
Appendix C.1’s derivative-to-limit step is replaced by direct strict comparisons for Laplace and Gaussian noise. The Gaussian and Mallows proofs also use local algebra fixes; see the precise changes.
None.
No additional generalization is established here.
The table links the density, cardinality, strictness, Gumbel-scale, and Theorem 1 source-domain readings. The sequential-experiment note states Theorem 4’s source-domain reading. Proof-only changes are in Section 7.
See the result-specific clarifications in Section 4 and the explicit Equation (6) regularity extension in Section 6.
The source map retains 49 selected result routes and nine source-model rows, including Definitions 1–4, Theorems 1–9, Lemmas 1–8, and the corrected Appendix-C comparison. The current source-to-Spec ledger reuses the unchanged routes and includes the direct outer-law Theorem 1 endpoint. Supporting formulas and experiments are not counted as extra named theorems.
Section 6 records the total-family Equation (6) extension and Theorem 2’s outer- regularity. The density and candidate-count clarifications are distinguished from ordinary source-model conditions in Section 4. The current scoped graph records its paper-local and library prerequisites in the paper and library ledgers.
The source map binds the ranking laws, expected welfare comparisons, sequential experiments, Laplace and Gaussian likelihood calculations, and Mallows normalization. The source memo and Laplace note give the corrected local formulas and inequality domains.
The selected reusable surface consists of finite ranking, probability, expectation, and order-comparison infrastructure. Paper-specific hiring experiments, noise families, theorem numbering, and corrected formula bundles remain paper-local.
The dependency DAG groups the population-ranking, welfare, sequential-experiment, Laplace, Gaussian, and Mallows routes. Theorem 1 is a formalized consequence of the source model and Definitions 1–3. The current one-page rendering has readable labels and arrows and no page clipping.
An independent source-to-Spec review checks the direct Theorem 1 route together with the unchanged selected routes. The focused build and final closure receipt record the corresponding proof and import-closure checks.
Checked definitions include the source ranking systems and quality/noise models, candidate pools, fresh-ranking hiring experiment, welfare and selection probabilities, and the Laplace, Gaussian, and Mallows families.
Theorems 1–6: the monoculture paradox, welfare examples, strict loss for at least three candidates, sequential experiments, and the density-regularity result.
Theorems 7–9: Laplace, Gaussian, and Mallows comparisons with the exact formula readings in Section 4.
Lemmas 1–8 and Appendix C Lemma 1: the supporting probability, derivative, strictness, and weak-global/strict-overlap comparisons.
Selected unnumbered claims: Appendix B counterexamples and smoothing, Gumbel–Plackett–Luce identification, and the Plackett–Luce strategic comparisons.
Direct comparisons are in the source-to-Spec ledger, row-level checks in the statement ledger, and correction provenance in source-proof fidelity.
The source map retains 49 selected result routes and nine source-model rows, including the direct outer-law Theorem 1 route. The source-to-Spec ledger binds that current surface. These are structural scope records; the closure receipt supplies the separate accepted proof basis.