Feedback

ChatGPT referee report — Feb 12 2026

Context: Uploaded paper.tex (with new related literature section) for referee-style feedback. Prompted for JLS/JLEO-targeted assessment.

Overall verdict

• Core idea (holdings as linear constraints on feasible set) is promising and novel as an economics model
• Revise-and-rebuild: keep the feasible-set + linear-constraint core, fix the holding definition, deliver 3–6 propositions
• Currently imbalanced: polished motivation + skeleton model + two long illustrations + agenda, but no theorems

Critical flaw: entailment/no-dicta constraint is backwards

• Current formalization requires $\mathcal{F}_t(d_t; z_t) \subseteq H_t$ — holding must contain all outcome-consistent rules
• This means the holding cannot exclude any outcome-consistent rule → breadth/citation channel collapses
• Legal practice is the opposite: holdings are selected rationales that narrow the feasible set beyond the bare outcome
• Fix: holdings should be subsets of $\mathcal{F}_t(d_t; z_t)$, or judge chooses an intended rule $(w,c)$ and the holding must be consistent with that chosen rule
• This is the #1 priority — everything else depends on it

Formal results a referee expects (minimum for JLS)

1. Geometry lemmas (mandatory): plausibility characterization via $m(z) = \min, M(z) = \max$ over $\mathcal{F}_t$; penumbra measure $P_t$ monotone in set inclusion; effect of halfspace holding on $m, M$
2. Equilibrium holding choice: myopic equilibrium — judge chooses outcome among plausible set, then holding within allowed language to maximize $(\gamma - \rho)B(H_t)$; optimal holding is extreme element of allowed language
3. One drift/path dependence proposition: law-following decisions produce monotone movement in feasible set; under $\gamma > \rho$ diameter shrinks faster but can tilt ideologically
4. Overruling threshold: characterize when judge overrules — $\alpha$ gain exceeds $Cr$
5. Total: ~5 propositions + corollaries. Do NOT need VC dimension unless it produces a clean comparative static

Structural advice

• Two applications is worse than one without formal results — reads like compensating for missing theory
• Keep EP (easier to map to constrained language / tiers of scrutiny); cut or drastically shorten DP
• Related literature is unusually good but "sets a very high bar" — referee will ask where the results using those tools are
• Introduction: pick ONE flagship claim ("law binds without being determinate" or "slippery slopes without lawlessness"), not three
• Sanctions too automatic — need at least probabilistic detection or reduced-form audience cost

Other weaknesses flagged

• Holding language is too unconstrained — "any finite set of linear inequalities" is enormous; restrict to a small menu matching doctrinal forms (tiers, factors, thresholds)
• Overruling cost linear in number is too naive — should depend on age/citations/reliance
• Affine rules for constitutional doctrine may feel ad hoc — present as local approximation / tractable reduced form

Strategic priorities (in order)

1. Fix the holding/no-dicta definition
2. Pick a single holding language and prove results for it
3. Deliver one flagship drift/path dependence result
4. Collapse to one tight, model-anchored application (EP)
5. Optimize for JLS/JLEO first

Offered to provide

• Explicit alternative formalization of holdings (two variants: "chosen rationale" vs "minimal doctrinal rule")
• Concrete list of 5–6 propositions with proof roadmaps in current notation

ChatGPT holding formalizations + proposition roadmaps — Feb 12 2026

Context: Follow-up to referee report. Asked for two alternative holding formalizations and 5–6 propositions with proof roadmaps.

Three holding formalizations

Variant A: "Chosen rationale" — judge selects intended rule $(\hat{w}_t, \hat{c}_t) \in \mathcal{F}_t$, outcome follows from that rule, holding is a constraint set containing the chosen rule

• A1 (ball): $H_t = \{(w,c) : \|(w,c) - (\hat{w}_t, \hat{c}_t)\| \leq \varepsilon\}$ — breadth knob is $\varepsilon$
• A2 (halfspaces): $H_t = \bigcap_{j=1}^{m} \{(w,c) : a_{t,j}^\top (w,c) \geq b_{t,j}\}$ with normals generated from $(z_t, \hat{w}_t, \hat{c}_t)$
• No-dicta: consistency $(\hat{w}_t, \hat{c}_t) \in H_t$ + case-anchoring (holding generated from case facts and chosen rationale)
• Assessment: very tractable (especially A1); medium legal realism; excellent for equilibrium + drift results

Variant B: "Minimal doctrinal rule" — holding is the least constraining element of the holding language $\mathcal{H}$ that supports the outcome

• $H_t \in \arg\min_{H \in \mathcal{H}} B(H; \mathcal{F}_t)$ subject to $\mathcal{F}_t(d_t; z_t) \subseteq H$
• No-dicta is the minimization itself — prevents adding unrelated constraints
• Assessment: medium tractability (depends on $\mathcal{H}$); high legal realism ("narrowest grounds"); great for geometry lemmas, weaker for strategic holding manipulation

Option C: "Single case-normal halfspace" — each case induces a canonical cut

• C1: $H_t = \{(w,c) : (-1)^{1-d_t}(w^\top z_t - c) \geq 0\}$ — no discretion in holding, only in outcome
• C2: C1 plus one additional halfspace with normal tied to $z_t$ and a chosen threshold $b_t$ — gives one breadth knob
• Assessment: maximally tractable; cleanest geometry; propositions 1–3 and corollary fall out beautifully; but C1 has no strategic holding-writing

Recommended packaging for JLS/JLEO:

• Main model: Option C (single halfspace) + sanctions $K$ — tight geometric spine
• Extension: Variant A with disciplined language (C2 or A2) — adds strategic holding-writing
• This gives (i) clean theorems referees can't poke holes in + (ii) strategic behavior where it adds value

Six propositions with proof roadmaps

Prop 1: Plausibility and the penumbra (geometry lemma)

• Define $m_t(z) = \min_{\mathcal{F}_t} (w^\top z - c)$, $M_t(z) = \max_{\mathcal{F}_t} (w^\top z - c)$
• Outcome 1 plausible iff $M_t(z) \geq 0$; outcome 0 plausible iff $m_t(z) < 0$; penumbra iff both
• $m_t$ is concave, $M_t$ is convex (support function properties)
• Penumbra probability $P_t = \Pr_G(m_t(z) < 0 \leq M_t(z))$ is well-defined
• Proof: linear optimization over compact convex set + support function properties
• Works under any holding formalization

Prop 2: Monotone determinacy (geometry lemma)

• $\mathcal{F}_{t+1} \subseteq \mathcal{F}_t$ implies $m_{t+1}(z) \geq m_t(z)$, $M_{t+1}(z) \leq M_t(z)$
• Therefore penumbra probability $P_t$ is weakly decreasing in $t$
• Proof: min over smaller set rises, max over smaller set falls
• Works under any holding formalization
• Buys: core "stare decisis makes law more determinate" theorem

Prop 3: Breadth = shrinkage in determinacy (geometry lemma)

• Define $B(H; \mathcal{F}) = P(\mathcal{F}) - P(\mathcal{F} \cap H)$ — penumbra reduction
• $B \geq 0$ and monotone: $H_1 \subseteq H_2 \implies B(H_1; \mathcal{F}) \geq B(H_2; \mathcal{F})$
• Proof: immediate from Prop 2 + set inclusion monotonicity
• Buys: pins "breadth" directly to the jurisprudential story

Prop 4: Equilibrium outcome + holding choice (requires Variant A)

• Under A1 (ball holdings): judge chooses $(\hat{w}_t, \hat{c}_t)$ then $\varepsilon$
• Optimal $\varepsilon$ solves FOC $\gamma B'(\varepsilon) = \rho R'(\varepsilon)$; corner solutions when $\gamma \leq \rho$ (narrowest) or $\gamma$ large (broadest)
• Outcome choice: ideal outcome if plausible (Prop 1), otherwise only plausible outcome; violation iff $\alpha > K$
• Proof: 1D optimization given rationale; backward induction within period
• Buys: microfoundation of breadth — $\gamma, \rho$ map to holding breadth; $K$ disciplines deviation

Prop 5: Drift / path dependence (flagship result)

• Under Option C + ideal-outcome-when-plausible: every decision is lawful, but $\{\mathcal{F}_t\}$ is path dependent
• Same multiset of outcomes with different fact orderings yields different $\mathcal{F}_T$
• If judge ideal-type distribution is asymmetric, expected barycenter of $\mathcal{F}_t$ drifts toward dominant ideology
• Proof: (i) path dependence via 2D counterexample; (ii) drift via correlation between chosen outcomes and case normals $z_t$ under asymmetric ideals
• Buys: "slippery slope without lawlessness" — the flagship story

Prop 6: Overruling threshold

• If ideal outcome not plausible under $\mathcal{F}_t$ but plausible after removing constraints of total weight $W^*$: overrule iff $\alpha > C \cdot W^*$
• $W^*$ characterized as minimum hitting set over blocking constraints
• Proof: feasibility restoration is combinatorial optimization; threshold immediate
• Cleanest under Option C (each holding = one halfspace = one facet)
• Buys: overruling tied to geometry of $\mathcal{F}_t$, not just exogenous reset

Corollary: VC dimension / sample complexity (non-decorative)

• Under Option C, $\mathcal{F}_t$ is exactly the version space of consistent affine classifiers
• VC dimension of affine separators in $\mathbb{R}^k$ is $k+1$
• After $t = O((k + \log(1/\delta))/\varepsilon)$ cases, any rule in $\mathcal{F}_t$ has generalization error $\leq \varepsilon$ w.p. $1 - \delta$
• Buys: higher-dimensional fact spaces require more precedent to determine law — maps to multidimensionality claim

---

Bård Harstad — Nov 26 2025

• One way to microfound the coordination game: cost of complaining depends on number who complain. E.g., 1/n probability of having to testify. The cost is divided between those who complain.
• Theory should have predictions about which institutions should be correlated with less corruption in the judiciary
  • We should also see consequences of complaints in certain institutional environments but not others
• Cool if we can come up with a theoretical explanation for dissent
• Don't need a dynamic game if it is not necessary for the conclusions. Can keep the coordination game shorthand.
• Possible institutions: ordering of complaints — who has the responsibility to call out wrongdoing. Example: in judicial panels the other members are expected to do so; if they don't, they will be associated with the corruption since they sign.
• Any countries without public reasoned dissent?
• Ways to go about equilibrium selection:
  1. Refinements (for instance Markov perfect are considered particularly robust)
  2. Sometimes one can assume that the most powerful party (e.g., the principal) selects the equilibrium