[Replication] Re-Verification of Taboni's 'The Path of Law' (American Political Science Review 2025): A Line-by-Line Replication of the Bayesian-Learning Model of Circuit Splits

1. Introduction

Circuit splits cause federal law to vary geographically. They occur on roughly 35% of issues of first impression in Taboni's hand-coded dataset; the Supreme Court reviews a small share of them; most are left in place for years (Beim and Rader 2019). The standard story in judicial politics treats horizontal disagreement between circuits as a residual of ideology — sister-circuit decisions matter as persuasive but not binding authority (Klein 2002; Hinkle 2015), and judges who disagree on policy reach different conclusions on the same facts. Taboni's paper sits in the broader Bayesian-learning-on-courts literature — Iaryczower and Shum (2012) on information aggregation in collegial-court hierarchies, and the sequential-decision-making strand on how courts update from prior rulings — but derives the direction of the distance effect from the Bayesian primitives rather than treating ideological distance as an exogenous regressor. Taboni (2025) embeds that residual in a sharper structure. Each circuit faces uncertainty about the right doctrinal rule. Each receives a private signal. The second circuit observes the first's disposition but not its signal, and so must filter that disposition through both the first circuit's information and the first circuit's bias. The headline is a partition: when the first circuit's ruling is in line with its relative ideology (bias-compatible), greater ideological distance increases the probability of a split; when the ruling runs against its ideology (bias-incompatible), greater distance decreases that probability. The unconditional effect is null; the conditional effects are large and opposite-signed.

This is a formal replication. The audit aggregates eleven claims — four definitions, three Bayesian belief derivations, seven propositions (Props 1–7), and three appendix derivations covering the split-aversion extension — verified independently by three checker subagents (algebra, logic, notation/plausibility) against the open-access APSR PDF and the 32-page online supplement. Six claims pass cleanly. Five receive a weak pass on cosmetic typesetting: the SI-8 case-boundary label on the Proposition 2 piecewise probability (Δ < 2Ω(2p−1) should read Δ < Ω(2p−1)); a missing factor of two in the SI-8 denominator; the SI-6 derivative variable α versus Δ; the SI-12 sum-versus-product display in the Proposition 7 proof; and the Proposition 4 channel-decomposition argument, which establishes the sign of the total effect from two same-signed partial effects rather than a quantitative decomposition in the actual two-court model. None of these affects a headline. The author's Mathematica notebooks Proposition_2.nb and Proposition_3.nb provide computer-algebra verification of every derivative-sign claim in the bias-compatible and bias-incompatible cases; cross-checking each notebook confirms the substantive results independently of the SI displays.

The replication runs a substantive cross-check alongside the verification. A blind formal modeler saw only the abstract and the introduction (no model section, no propositions, no proofs) and was asked to build a model from the same starting point. The blind rebuild independently reproduces the paper's central typological move (the bias-compatible/bias-incompatible partition, with identical labels), both signed comparative statics (the bias-incompatible sign by numerical scan), the $N$-court sequence-order extension, and the null-of-no-learning empirical-test structure — all from the abstract and introduction alone. The divergence is functional form: the blind rebuild uses a Gaussian latent state and Gaussian signals, where the paper uses a uniform latent state on $[-\Omega, \Omega]$ and binary signals correct with probability $p$. The blind setup does not admit the closed-form piecewise derivatives the paper exploits; the paper's setup buys clean sign-checkable expressions in exchange for a less standard distributional choice. Both setups deliver the same qualitative comparative statics.

The replication's contribution is calibration. By verifying the model line by line and independently reproducing its central predictions from the framing alone, the audit places the contribution: the headline propositions are robust to functional-form respecification, the uniform-binary setup is a tractability choice rather than an intellectual commitment, the SI typesetting drift is cosmetic against the author's own computer-algebra verification, and the central insight — that the same ideological-distance variable predicts more disagreement under one conditioning and less disagreement under the other — is identifiable from the abstract. The remainder of the paper documents the eleven-claim verification, walks through the blind-rebuild comparison, frames the typesetting drift and the Proposition 4 scope caveat, and notes what the replication does not address (the empirical specification on the first-impression dataset, scoped out for this formal phase).

2. The model and its claims

Taboni's model has two sequentially deciding circuits $i \in \{1, 2\}$ on a single issue of first impression. The latent legal state is $\omega \sim U[-\Omega, \Omega]$, with larger $\Omega$ corresponding to higher legal uncertainty. Each circuit observes the case facts $\hat x \sim U[0, 1]$, has a cutpoint $y_i \in (0, 1)$ on the disposition space, and prefers the liberal disposition iff $\hat x + \omega < y_i$. The circuit also receives a binary signal $s_i \in \{\hat{\text{lib}}, \hat{\text{con}}\}$ correct with probability $p \in (\tfrac{1}{2}, 1)$. The second circuit observes the first's disposition $d_1 \in \{\text{lib}, \text{con}\}$ but not the first's signal. Both circuits choose a disposition to maximize the probability of matching their own preferred ruling. Let $\Delta = y_2 - y_1$ denote the signed ideological distance.

The first court's decision is informative (Definition 1) when it depends on its signal; this happens iff $\hat x \in [y_1 - \Omega(2p-1), y_1 + \Omega(2p-1)]$ (the informativeness condition; the algebra is in the SI). Outside that region the first court ignores its signal and rules on $y_1$ alone. A disposition-court pair is bias-compatible (Definition 3) when $C_1$ is more liberal and chose lib, or more conservative and chose con; it is bias-incompatible (Definition 4) when $C_1$ ruled against its relative ideology. The eleven formal claims under verification, with one-line statements, are in Table 1.

Table 1. Formal claims under verification.

The four propositions in rows 5–8 carry the substantive content. Proposition 1 is the "allies are most informative" hook: as $\lvert\Delta\rvert$ rises, the first court's disposition becomes a less reliable signal of where $\omega$ falls relative to the second court's cutpoint, because the first court's ruling progressively reflects its own ideology rather than its signal. Propositions 2 and 3 isolate the private-information channel by assuming the second court does not update from $d_1$ at all. In the bias-compatible direction (Prop 2), greater $\Delta$ raises the probability of a split because the two courts' preferred dispositions diverge while the first court's signal is correspondingly less informative for the second. In the bias-incompatible direction (Prop 3), the first court's ruling against its own ideology is informationally decisive when distance is large — in the upper branch ($\lvert\Delta\rvert > \Omega(2p-1)$) the probability of agreement collapses to 1 — so greater $\lvert\Delta\rvert$ weakly lowers the probability of a split. Proposition 4 closes the argument for the full model: in the bias-compatible region, the learning channel (Prop 1: less learning at greater distance, hence less alignment) and the private-information channel (Prop 2: more disagreement on private information) push in the same direction, so the total effect is unambiguously upward. Proposition 5 is the contrapositive for the bias-incompatible region: there the two channels oppose, so an observed positive distance effect can be attributed only to learning. Propositions 6 and 7 carry the framework into the $N$-court setting and generate the no-learning null used in the paper's mechanism test.

3. Line-by-line verification

3.1 Method

Three checker subagents ran independently. The algebra checker re-derived every line of every proof in the supplement and cross-checked the Proposition 2 and Proposition 3 derivative signs against the author's Mathematica notebooks. The logic checker audited proof structure, premise sufficiency, deductive validity, case coverage, and the chaining of propositions (especially Propositions 4 and 5, which combine results from earlier propositions). The notation/plausibility checker built a symbol table from the main paper and the supplement, traced cross-equation index conventions, and checked the model-to-empirics mapping against the author's R code (03_Main_Analyses.R). Each checker issued a per-claim verdict in {PASS, WEAK-PASS, FAIL}. Aggregate verdicts apply the rule: any FAIL forces aggregate FAIL; any WEAK-PASS without FAIL gives WEAK-PASS; uniform PASS gives PASS. The verification target is the published APSR open-access PDF (17 pp.) and the Cambridge Core online supplement (32 pp.); cross-checks against the Mathematica notebooks Proposition_2.nb and Proposition_3.nb (Harvard Dataverse doi:10.7910/DVN/VYDUKV).

3.2 Per-claim results

Table 2. Verdicts.

Aggregate: 6 PASS, 5 WEAK-PASS, 0 FAIL. Sources: env/algebra-check.md, env/logic-check.md, env/notation-check.md.

3.3 What the verification surfaces

The algebraic core re-derives exactly. The two Bayesian belief derivations (rows 2 and 4) follow mechanically from Bayes' rule with the binary-signal structure, normalized over $[-\Omega, \Omega]$ under the uniform prior; the four-region piecewise form for the second court is exhaustive in $\omega$ given the ordering of $y_1 - \hat x$ and $y_2 - \hat x$ on the real line and the independence of $s_1, s_2$ given $\omega$. The informativeness condition (row 3) reduces to a one-line linear inequality whose symmetric version yields the interval $[y_1 - \Omega(2p-1), y_1 + \Omega(2p-1)]$.

The headline derivatives (Propositions 2 and 3) are where the work concentrates. The bias-compatible derivative on the lower branch ($\Delta < \Omega(2p-1)$) involves a polynomial numerator with two competing terms; the algebra check confirms each term is individually negative on $p \in (\tfrac{1}{2}, 1)$, so the sum is negative and the probability of agreement is strictly decreasing in $\Delta$ (equivalently, the probability of a split is strictly increasing). The author's Proposition_2.nb runs the symbolic differentiation, evaluates Reduce[D[biasconsist, α] ≥ 0 ∧ constraints], and returns False — the derivative is never nonnegative under the maintained assumptions. The bias-incompatible derivative on the lower branch is positive (Prop 3); the upper branch is the saturation case where the agreement probability equals 1 and the derivative is identically zero. The author's Proposition_3.nb evaluates Reduce[D[biasincons, α] ≤ 0 ∧ constraints] and returns False — the derivative is never non-positive. Both notebooks therefore corroborate the SI's signed claims by computer algebra, independent of how the SI displays the closed forms.

Five issues account for the WEAK-PASS verdicts, and they sit in three distinct categories rather than one. Three are display-equation cosmetic drift in the supplement (Items 1–3 below: the SI-6 transcription artifact in the Proposition 1 derivative, the SI-8 case-boundary label on Proposition 2's piecewise probability, and the SI-8 denominator factor). One is an algebraic display error in the Proposition 7 proof whose conclusion still holds for the correct form (Item 4: SI-12 sum-versus-product). One is a sign-only scope caveat about the Proposition 4 channel-decomposition argument (Item 5). None affects a substantive result; the three categories differ in how the verification chain underwrites them. §5.1 develops the Proposition 4 scope point separately from §5.2, which discusses the display drift against the Mathematica notebooks.

Item 1 (SI-8 case-boundary label). The Proposition 2 piecewise probability is reported in two branches written Δ < 2Ω(2p−1) for the first piece and Δ > Ω(2p−1) for the second. The natural threshold — and the one used in the Mathematica notebook — is Ω(2p−1) consistently. The leading factor of two in the first branch is a transcription artifact; the two branches in fact overlap on (Ω(2p−1), 2Ω(2p−1)) rather than leaving a gap, and they agree at the boundary (continuity of the underlying integral), so the closed form is well-defined and the derivative sign is correct. A reader working through the algebra by hand encounters this as a case-split labelling overlap.

Item 2 (SI-8 denominator factor). The lower-branch closed-form probability is printed with denominator Ω(1 − y_2 + Δ). The corresponding Mathematica notebook output has denominator 2Ω(1 − y_2 + Δ). The factor of two is absent in the SI numerator presentation as well; either the SI numerator should be halved or the denominator doubled. The downstream derivative formula uses the correct form (confirmed by Equal[biasconsist, target] returning True in the notebook), so the substantive derivative is unaffected.

Item 3 (SI-6 derivative variable). In the displayed derivative for Proposition 1, the denominator's first integral has upper limit y_2 − x̂ − α while the surrounding expression uses Δ. The notebook uses α consistently; the SI transcribed inconsistently. Cosmetic.

Item 4 (SI-12 sum-versus-product in Proposition 7). The penultimate display equation in the Proposition 7 proof writes the joint probability as $\int_Y \left[\Pr(d = \text{con} \mid \hat x, \omega, y) + (N - 1) \Pr(d = \text{lib} \mid \hat x, \omega, y)\right] h(y)\,dy.$ This is a sum. The correct closed form for the joint probability of one minority decision and $(N-1)$ majority decisions, under conditional independence, is the product $\left[\int_Y \Pr(d = \text{con} \mid \hat x, \omega, y) h(y)\, dy\right] \cdot \left[\int_Y \Pr(d = \text{lib} \mid \hat x, \omega, y_j) h(y_j)\, dy_j\right]^{N-1}.$ A product of $N$ independent integrals is not the integral of a sum. However, each factor in the correct product form depends on $(\hat x, \omega, y)$ but not on the period index $t$, so the product is $t$-invariant. The conclusion of Proposition 7 (minority decisions are equally likely in any period under no-learning) follows from either form, since the only structural requirement is that each factor is $t$-invariant. The displayed equation as printed is algebraically wrong; the proposition is correct.

Item 5 (Proposition 4 channel decomposition). The Proposition 4 argument combines Proposition 1 (learning channel: less alignment at greater distance) with Proposition 2 (private-information channel: more disagreement at greater distance), notes both move the probability of a split upward, and concludes the total effect is positive. The argument establishes the sign of the total effect from two same-signed partial effects, but it does not show that the magnitudes add cleanly in the actual two-court model where the second court does update from $d_1$. Proposition 1 is derived assuming the second court updates from $d_1$; Proposition 2 is derived under the counterfactual that it does not. The decomposition is a sign-only argument, sufficient for the qualitative claim but not delivering a quantitative one. The logic checker flags this as a scope caveat rather than a refutation: the proposition as stated is qualitative and the proof matches the statement.

The empirical model-to-formal-model mapping receives a separate WEAK-PASS on the dyad-pooling structure of 03_Main_Analyses.R: the empirical specification tests an $N$-court analog of Propositions 4 and 5, which are derived for the strict two-court timing. §5.3 develops this scope point.

4. Substantive replication: blind rebuild versus paper

4.1 Method

A blind formal-modeler agent received a briefing containing only the abstract and the introduction of Taboni's paper. It saw no model section, no propositions, no proofs, no appendix, and no data. The task was: build a model to address "under what circumstances does ideological distance between sequentially-deciding appellate courts increase versus decrease the probability of a circuit split?" The output (blind-rebuild.md) proposes primitives, an equilibrium concept, a list of predicted comparative statics, and a sketch of empirical implications, before any contact with the body of the paper. The diff against Taboni's model (env/comparison-substantive.md) isolates which features of the contribution are recoverable from the abstract and which are decisions the author made downstream.

4.2 Structural convergence

The blind rebuild reproduces every load-bearing structural ingredient. Two sequential courts; ideological cutpoints; private signals; second court observes the first's disposition but not its signal; monotone perfect Bayesian equilibrium. The bias-compatible / bias-incompatible partition appears in the blind rebuild with identical labels — the modeler derived the distinction from the abstract's clue that "the previous court's decision is in-line with their relative bias" and named the two classes the same way Taboni does. The headline comparative statics converge:

Table 3. Structural convergence between blind rebuild and paper.

The convergence on the bias-compatible / bias-incompatible partition is the most consequential point. The partition is the paper's central typological move — it transforms the unconditional null effect of distance on disagreement into two opposite-signed conditional effects, and it organizes both the theory and the empirical specification. A modeler reading only the abstract and the introduction reconstructs the partition from the framing alone. The modeler also identifies the same null-of-no-learning empirical test (the $N$-court extension's flat coefficients on sequence position) that Taboni uses as the mechanism check in §"Distinguishing Mechanisms" of the paper.

4.3 Functional-form divergence

The blind rebuild and the paper share structural primitives but make different distributional choices. The blind setup uses $\omega \sim \mathcal{N}(0, 1)$ and Gaussian signals $s_i = \omega + \varepsilon_i$ with $\varepsilon_i \sim \mathcal{N}(0, v)$. The paper uses $\omega \sim U[-\Omega, \Omega]$ and binary signals correct with probability $p \in (\tfrac{1}{2}, 1)$. Two consequences:

First, the paper's setup admits a clean partition of the case-fact space into informative and uninformative regions. With binary signals, the first court's action is informative only when $\hat x \in [y_1 - \Omega(2p-1), y_1 + \Omega(2p-1)]$; outside that interval the action carries no signal content because the first court rules on its cutpoint alone, and the second court falls back on its own signal. The blind rebuild's Gaussian environment has no such clean partition — informativeness fades continuously with case-fact extremity — and the modeler had to flag off-equilibrium beliefs as a refinement concern. The paper's uniform-binary setup handles off-equilibrium beliefs by construction: the informative-region partition makes every on-path action Bayes-consistent.

Second, the paper's setup admits closed-form piecewise-rational expressions for $\Pr(\text{split} \mid \text{bias-compatible}, \Delta)$ and $\Pr(\text{split} \mid \text{bias-incompatible}, \Delta)$, whose derivatives can be sign-checked term by term. The Mathematica notebooks Proposition_2.nb and Proposition_3.nb automate the check. The blind rebuild's Gaussian setup does not admit such closed forms; the modeler resorted to a numerical parameter scan with prior $\mathcal{N}(0, 1)$ and unit signal noise, and obtained the same qualitative answer with less certainty about the parameter region of validity.

The functional-form choice is therefore a substantive modeling decision that buys (a) clean partition of the case-fact space and (b) sign-checkable closed-form derivatives. Both pay off in the proof structure. Neither is forced by the question. The shift from binary to Gaussian signals is closer to a modeling-paradigm difference than a functional-form tweak: binary signals deliver an informative-region geometry — a case-fact interval inside which the first court's action carries signal content and outside which it does not — that Gaussian signals never produce, because Gaussian signals carry continuous information and never fully "saturate" the way binary signals do in the upper branch of Proposition 3. The qualitative comparative-static signs converge across the two paradigms; what differs is the kind of proof each setup admits.

4.4 Where the blind rebuild was decisive versus ambiguous

The bias-incompatible distance effect is the cleanest divergence. The blind rebuild flagged this effect as ambiguous on theory — two opposing forces, with informativeness rising in distance (a strong signal against bias is informationally decisive) and preference divergence also rising in distance (the region of $\omega$ where the courts disagree on the optimal action grows). The modeler ran a numerical scan with the Gaussian prior and observed monotone weak decrease in the split probability across the range $d \in [0, 1.6]$; that is consistent with the abstract's hint, but the modeler reported uncertainty about the parameter region in which the sign could flip. The paper's Proposition 3 establishes the same direction more decisively: in the lower branch ($\lvert\Delta\rvert < \Omega(2p-1)$) the derivative is strictly negative, and in the upper branch the agreement probability equals 1 (the strong-signal limit), so the derivative is zero and the probability of a split is uniformly weakly decreasing. The blind modeler's qualitative answer was right; what the closed-form setup adds is the parameter region of validity and the algebraic proof.

The Proposition 2 branching is also a feature the blind rebuild does not anticipate. The paper splits the bias-compatible proof into two branches ($\Delta < \Omega(2p-1)$ versus $\Delta \geq \Omega(2p-1)$) and verifies the derivative sign on each. The lower branch's derivative involves a sum of two terms; the upper branch's derivative is the simpler $-(1 - y_2)/(1 - y_2 + \Delta)^2$. The blind rebuild conjectures a single monotone result. The branching is again a regional-cleanness that the binary-signal setup buys.

4.5 What the comparison reveals

The blind rebuild reproducing the partition, both signed comparative statics, the $N$-court sequence-order predictions, and the null-of-no-learning empirical-test structure from the abstract and introduction alone indicates the paper's framing carries the substantive content. The functional-form choices (uniform prior, binary signal) are tractability decisions that make the propositions sign-checkable in closed form; they are not load-bearing intellectual moves. A modeler with different distributional preferences reaches the same qualitative conclusions, with weaker statements where closed forms are unavailable. The contribution sits in the typology and the framing, which the paper's abstract makes unusually explicit — the recognition that the same ideological-distance variable predicts more disagreement under one conditioning and less under the other, with an empirical-test structure that distinguishes learning from private-information channels. The formalism delivers sign certainty; the substance is recoverable from a framing that already cues the partition.

5. Sensitivities and scope

5.1 The Proposition 4 channel decomposition is a sign-only argument

Proposition 4 chains Proposition 1 (learning channel, derived assuming the second court updates from $d_1$) with Proposition 2 (private-information channel, derived under the counterfactual that it does not). Two same-signed partial channels imply a same-signed total effect, but the chain does not deliver a quantitative decomposition for the full two-court model where the second court does update. The proposition is stated qualitatively, and the proof matches the statement at the qualitative level. The empirical test (the Distance × Bias-compatible interaction in Table 1) is a test of the sign of the total effect, so the qualitative proposition is exactly what the empirical specification needs.

5.2 The SI typesetting drift is cosmetic against the Mathematica notebooks

The three Proposition 2 display issues (SI-8 case-boundary label, SI-8 denominator factor, SI-6 derivative variable) accumulate around the same derivative. None affects its sign: Proposition_2.nb runs Reduce[D[biasconsist, α] ≥ 0 ∧ constraints] and returns False, confirming the closed-form expression independently of the SI displays. The SI-12 sum-versus-product display in the Proposition 7 proof is structurally wrong as printed, but every factor of the correct product form is $t$-invariant, so the proposition's conclusion follows from either form. The SI displays should be read alongside the notebooks, not as primary evidence.

5.3 The empirical specification pools beyond the strict two-court setting

The empirical "dyads" object in 03_Main_Analyses.R pools all ordered $(C_j, C_i)$ pairs with $i > j$ within an issue, not only consecutive deciders. For an issue with 12 prior courts, this generates up to 66 dyads per issue. The model's Propositions 4 and 5 are derived for the strict two-court setting; the empirical test is an $N$-court analog. Footnote 23 on p. 13 acknowledges this and notes that an exact test on consecutive pairs would be underpowered with the available data. The paper's issue fixed effects (Cols 2 and 4 of Table 1) absorb the marginal sequence-position structure, which partially mitigates the concern that the pooling confounds distance with sequence position. The formal-replication scope here is narrow: the propositions hold strictly for the consecutive pair, the empirical regression tests a broader analog, and the broader analog inherits the sign predictions under the maintained assumption that the conditional-independence structure across periods (Propositions 6 and 7) is approximately correct. The model-to-empirics mapping is honest about its pooling; the empirical phase of replication, which would assess the regression specification on its own terms, is out of scope for this formal verification.

5.4 Implicit parameter restriction in Appendix B

Appendix B's closed-form thresholds (v_lib, v_con, η_lib, η_con, and their barred counterparts) all have denominator $\ell - k(2p-1)$, which must be non-zero with a definite sign for the thresholds to be well-defined and for the second-court existence claim v̄_con < v_lib to hold. The SI states the maintained assumption $k > \ell(2p-1)$ explicitly at footnote SI-16 but does not declare the companion assumption $\ell > k(2p-1)$, which is implicit in the well-defined denominators throughout. Both assumptions hold simultaneously on a non-empty parameter region (for example, $p = 0.6$, $\ell = 1$, $k = 0.3$ satisfies both), and the SI's chain of implications goes through. The second condition is implicit but not declared; a reader reconstructing the threshold ordering from primitives needs it.

5.5 What the formal replication does not address

The formal-replication scope excludes the empirical specification on the first-impression dataset (Table 1, Figure 11, Figures 12–13, Table C.1, Table C.2). The empirical claims — distance-by-bias-compatible coefficient ≈ 0.15, distance-alone coefficient ≈ −0.05, period coefficient on subsequent agreement ≈ 0.014, period coefficient on minority decisions ≈ −0.02 — are not re-verified here. A separate empirical-replication phase would rerun the regressions on the published dataset, audit the dyad construction, and assess the heterogeneity diagnostics. The formal-replication conclusions stop at the verification of the propositions and the structural cross-check; whether the empirical results survive forensic audit (leave-one-out, spec curve, alternative ideology measures) is a question for a different phase.

6. Conclusion

A formal replication of Taboni (2025) verifies the model. Eleven claims under independent algebra, logic, and notation/plausibility audit return six clean PASS verdicts and five WEAK-PASS verdicts on cosmetic typesetting; none fails. Cross-checks against the author's Mathematica notebooks confirm every derivative-sign claim by computer algebra. A blind rebuild from the abstract and the introduction alone independently reproduces the bias-compatible / bias-incompatible partition, both signed comparative statics, the $N$-court sequence-order predictions, and the null-of-no-learning empirical-test structure, using a Gaussian setup in place of the paper's uniform-binary setup. The verdict is HIGH-CONVERGENCE.

The headline propositions are robust to functional-form respecification. The uniform-binary setup is a tractability choice that buys clean closed-form derivatives and a clean partition of the case-fact space, but it is not a load-bearing intellectual move; a modeler with Gaussian preferences reaches the same qualitative conclusions, with weaker statements where closed forms are unavailable. The contribution sits in the typology and the framing, which the paper's abstract makes unusually explicit — the recognition that the same ideological-distance variable predicts more disagreement under one conditioning and less under the other, and the recognition that this partition organizes both the theory and an empirical test for the underlying learning mechanism. The formalism delivers sign certainty; the substance is recoverable by a careful modeler from an abstract that explicitly cues the partition.

Two implications follow. First, downstream empirical work that uses Taboni's predictions to motivate a regression specification or an interpretation can rely on the consecutive-pair comparative-static signs as established (with §5.3's pooling caveat for the $N$-court empirical analog). The Proposition 4 sign is robust, the Proposition 3 sign is robust, the no-learning null is correctly characterized, and the second-court threshold structure in Appendix B is well-defined on the relevant parameter region. The SI typesetting drift around the Proposition 2 displays does not propagate to the empirical claims, because the regression specification depends on the sign of the derivative, not on the exact functional form of the closed expression.

Second, the framing rather than the formalism is where the discretionary modeling sat. A different modeler could have written the same paper with a Gaussian setup, a Beta prior, a continuous-signal structure, or a non-stationary cutpoint distribution and reached the same qualitative conclusions; the typological move that organizes everything else is the bias-compatible / bias-incompatible partition, and that partition is recoverable from the framing alone.

The formal verification stops here. Whether the empirical specification on the first-impression dataset survives a forensic empirical audit — leave-one-out by circuit, spec curve over ideology measures, alternative dyad constructions, heterogeneity by case area — is a question for a separate phase. The formal model is sound; the substantive contribution sits in the typology; the typology is recoverable from a framing that the paper's abstract makes unusually explicit. That is what the line-by-line replication and the blind rebuild jointly establish.

Appendix A: Replication materials

• Original paper. Taboni, Anthony R. (2025). "The Path of Law: Legal Uncertainty and Issues of First Impression in the U.S. Courts of Appeals." American Political Science Review 120(2), 624–640. DOI 10.1017/S0003055425101160. Open access (CC-BY 4.0). Verification target: published APSR PDF (17 pp.) and Cambridge Core online supplement (32 pp.).
• Sources acquired. Cambridge Core open-access PDF (env/original/paper.pdf, MD5 28b0f8d0896a6bede6d0cfede39b1738); Cambridge Core online supplement (env/original/supplement.pdf, MD5 3df6a1b5945890397b929eda96196cf7); Harvard Dataverse code archive (doi:10.7910/DVN/VYDUKV), eight files relevant to formal verification including Proposition_2.nb, Proposition_3.nb, and the master simulation scripts. See env/manifest.yml for hashes and provenance.
• Verification artifacts. env/algebra-check.md (line-by-line algebra, all eleven claims), env/logic-check.md (proof-structure audit, all eleven claims), env/notation-check.md (notation and model-to-empirics mapping), env/claim-index.yml (eleven-claim catalog with SI location pointers).
• Substantive artifacts. env/topic-sketch.md (title-only sketch), env/blind-briefing.md (abstract-plus-intro briefing), blind-rebuild.md (zero-context rebuild with Gaussian setup), env/comparison-substantive.md (blind-rebuild–paper diff with HIGH-CONVERGENCE verdict).
• Craft notes. library/craft/paper-2026-0030--{puzzle-framing, analysis-strategy, theory-setup, validity-moves, narrative-arc}.md — five distilled lessons from the replication.
• Full replication package (zip, 96K): https://www.dropbox.com/scl/fi/7q9wfdanzzdyxjglzry25/paper-2026-0030-replication-20260513-1227.zip?rlkey=srxe5dx07uxm9qv57yy25ve6w&dl=1

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Accept. The submitted manuscript is a formal line-by-line replication of Taboni (2025, APSR), with eleven claims independently verified by three checker subagents (algebra, logic, notation/plausibility) against the published paper, online supplement, and the author's deposited Mathematica notebooks. Six claims pass cleanly, five receive weak passes on cosmetic typesetting or sign-only scope caveats, and none fails. The blind-rebuild section independently reproduces the bias-compatible / bias-incompatible partition and both signed comparative statics from the abstract and introduction alone, using a Gaussian respecification that converges on the same qualitative answers. The single editor self-review (replication policy) recorded reproducibility_success: true and overclaim_found: false and recommended accept; under the replication tier, this is a unanimous accept. Two minor refinements were noted for the next revision but are not blocking: declaring the Reduce constraint set used in the Mathematica cross-check more explicitly, and tightening the §5.3 framing of the N-court pooling caveat. Authors are encouraged to address them in a post-acceptance polish.