TODOs

Model — critical path (per ChatGPT referee report, Feb 2026)

• Fix the holding/entailment definition — replaced backwards $\mathcal{F}_t(d_t; z_t) \subseteq H_t$ with canonical holdings (one halfspace per case, mechanically determined by outcome)

  • notes: Option C (single case-normal halfspace) adopted as base model; richer holding language (Variant A, chosen rationale) noted as extension for Holger to decide on
  • completed: 2026-02-12

• Pick a single holding language and restrict $\mathcal{H}$ to a small menu

  • notes: Base model uses canonical holdings: one halfspace per case. Extension to richer language deferred pending decision on Variant A.
  • completed: 2026-02-12

• Prove geometry lemmas: plausibility characterization ($m(z), M(z)$), penumbra measure $P_t$, monotonicity under set inclusion, effect of halfspace holding

  • notes: Props 1–3 added to paper.tex with proof sketches
  • completed: 2026-02-12

• Characterize myopic equilibrium under canonical holdings: judge chooses ideal outcome if plausible, else unique plausible outcome (Prop 4)

  • notes: $K > \alpha$ assumption added; simple but sets up drift result
  • completed: 2026-02-12

• Prove drift/path dependence proposition (Prop 5): monotone shrinkage, path dependence via constructive 2D example, ideological drift under asymmetric judge distribution

  • notes: flagship "slippery slopes without lawlessness" result; works purely through geometry under canonical holdings
  • completed: 2026-02-12

• Prove overruling threshold result (Prop 6): overrule iff $\alpha > C \cdot W^*$ where $W^*$ is minimum hitting set of blocking constraints

  • completed: 2026-02-12

• VC dimension corollary: $d_{VC} = k+1$ for affine classifiers; sample complexity $O(k/\varepsilon)$ for legal determinacy; multidimensionality result

  • completed: 2026-02-12

Model — extensions

• Make sanctions less automatic: probabilistic detection or reduced-form audience cost depending on distance from $\mathcal{F}_t$

  • created: 2026-02-12

• Make overruling cost depend on age/citations/reliance rather than linear in number of holdings

  • created: 2026-02-12

• Explore neighborhood / decaying binding force (Holger idea: cases only bind in neighbourhood, or with decaying force)

  • created: 2026-02-12

Paper

• Restructure paper: model → results → one application (EP) → empirical implications

  • notes: current structure (motivation → model → two illustrations → agenda) is imbalanced; need theorems earlier and applications shorter
  • created: 2026-02-12

• Trim to one application (EP); cut or drastically shorten DP section

  • notes: DP can be preserved as extension or brief illustration if it shows a genuinely different holding language
  • created: 2026-02-12

• Sharpen intro to one flagship claim: "law binds without being determinate" or "slippery slopes without lawlessness"

  • created: 2026-02-12

• Integrate intro.tex framing (focal-point motivation) into paper/paper.tex

  • created: 2026-02-12

• Add head-off paragraph for "affine rules are ad hoc" critique — present as local approximation / tractable reduced form for convex constraints on parameterized family

  • notes: added "Why affine rules?" paragraph after decision rules subsection
  • completed: 2026-02-12

• Write full proofs appendix for all 6 propositions + corollary

  • notes: Appendix A with formal proofs for Props 1–6 and Corollary. Prop 1 uses support function theory (Rockafellar 1970). Prop 3 monotonicity tightened to set-inclusion ordering. Prop 5(iii) drift formalized via one-step barycenter shift. Prop 6(iii) proved via removal-set mapping across time periods. VC dimension proof uses Radon's theorem for upper bound.
  • completed: 2026-02-12

• Fix stale Testable Predictions and Empirical Strategies in Next Steps

  • notes: replaced strategic-breadth/citation-incentive predictions with overruling and Prop 6 predictions; anchored all three predictions and empirical strategies to specific propositions
  • completed: 2026-02-12

• Add TikZ figure for Prop 5 path dependence example

  • notes: single-panel figure showing both F_3 (Order 1, w≤0, blue) and F_3' (Order 2, w≥0, red) as disjoint triangles within the initial square; cross-referenced from Prop 5 proof sketch
  • completed: 2026-02-12

• Add comparative statics subsection

  • notes: four paragraphs between VC corollary and Discussion: dimension & determinacy, doctrinal entrenchment (W* increasing), bench composition & drift direction, sanctions & binding-force threshold
  • completed: 2026-02-12

• Tighten Prop 3 statement to match appendix proof

  • notes: replaced vague "monotone in volume removed" with precise set-inclusion ordering; aligned main text with appendix proof
  • completed: 2026-02-12

External input

• Ask ChatGPT for explicit alternative holding formalizations and 5–6 proposition proof roadmaps
  • notes: done; three variants (C, A, B) and six propositions with proof roadmaps recorded in feedback.md
  • completed: 2026-02-12

Decisions pending (require Holger)

• Decide whether Variant A (strategic holding-writing / chosen rationale) goes in the main model or as an extension

  • notes: see docs/briefs/holding-formalization.md for full analysis; determines whether $\gamma/\rho$ are in the base model
  • created: 2026-02-12

• Decide on paper structure: one unified model vs. two-layer (Option C spine + Variant A extension)

  • created: 2026-02-12