Decision brief: Paper status and path to submission

Date: Feb 12, 2026 Context: Paper revised through six propositions, VC corollary, figure, comparative statics, related literature, and full proofs appendix. ChatGPT provided detailed referee-style feedback on the current draft. This brief summarizes the paper's state, identifies technical fixes already implemented, and lays out the 5 decisions that require co-author judgment.

Where the paper stands

What's done and solid:

Model core: Canonical holdings (one halfspace per case), feasible-set dynamics, plausibility, sanctions, overruling, breadth — all defined and internally consistent.
Six propositions + corollary: Plausibility characterization (Prop 1), monotone determinacy (Prop 2), breadth/penumbra shrinkage (Prop 3), myopic outcome choice (Prop 4), drift and path dependence (Prop 5), overruling threshold (Prop 6), VC dimension (Corollary 1).
Full proofs appendix: All six propositions and the corollary have complete proofs.
Figure: TikZ diagram for the path-dependence constructive example.
Comparative statics: Four paragraphs (dimension, entrenchment, bench composition, sanctions).
Related literature: Four paragraphs positioning against economics/polisci models, legal indeterminacy, AI & Law, and learning theory. ~30 bibliography entries.
Two applications: Equal Protection (~4 pages, anchored to propositions) and Due Process (~4 pages, not anchored).
Length: 24 pages including appendix.

What was just fixed (technical cleanup from ChatGPT feedback):

Proposition numbering: Remark shared the proposition counter, making "Prop 6" (overruling) render as "Prop 7." Fixed: remark and corollary now have separate counters. Numbers match abstract.
Strict/weak inequality inconsistency: Text described $d_t=0$ holding as $w^\top z_t < c$ (strict) while the formula gives $w^\top z_t \le c$ (weak, closed halfspace). Fixed text + added footnote explaining the measure-zero boundary. Rewrote the sanctions "collapse" claim to avoid the problematic "iff" and instead state that plausible outcomes always generate feasible holdings.
Prop 5(i) — open penumbra: Volume decrease now requires $m_t(z_t) < 0 < M_t(z_t)$ (strict on both sides), not just the weak penumbra. Avoids the tangent-case fragility.
Prop 5(iii) — symmetry assumption for drift: Added explicit condition that $\mathcal{F}_t$ must be symmetric about the cutting hyperplane (e.g., $\mathcal{F}_1$ centrally symmetric, $G$ symmetric). The previous proof relied on unstated symmetry.
Prop 6(ii) — "facets" → "constraints": Removed holdings may be redundant/inactive (not facets of the polytope).
Prop 6(iii) — no-overruling qualifier: $W^*$ monotonicity now explicitly requires "along histories without overruling," since overruling can expand $\mathcal{F}_t$.
VC sample complexity: Changed from "agrees with any other surviving rule on $\varepsilon$-fraction" (wrong — pairwise disagreement is $\le 2\varepsilon$) to the correct PAC statement: "classifies $(1-\varepsilon)$-fraction identically to the generating rule."
"Why affine rules?": Added sentence noting that version-space results generalize to any parametric class with convex constraints; affine is the simplest case.
Set-inclusion note after Prop 3: Surfaced the appendix observation that set-inclusion ordering is strictly stronger than volume ordering.

What ChatGPT identified as the three biggest referee risks

"Promised extension" problem. The paper repeatedly mentions a richer holding language (breadth choice, $\gamma$, $\rho$, strategic rationale selection) that is never developed. Multiple "to be developed" and "deferred to future draft" references + a whole "Next Steps" section signal "not ready" to a JLS/JLEO referee.
Canonical model–application mismatch. The EP and DP sections talk about sign restrictions on race weights, conditional constraints, relative-weight inequalities — none of which are canonical holdings. Under canonical holdings, a holding is just "this point must be labeled this way." A technically literate referee will say the applications don't map to the model.
Technical slippage in proof sketches. The fixes above address these, but the paper should be read carefully one more time for any remaining gaps (especially the drift proof under non-symmetric $\mathcal{F}_t$).

ChatGPT's assessment of strengths:

Clean geometric spine ("law as shrinking feasible set") with explicit penumbra notion.
Path-dependence example + figure are genuinely clarifying.
Overruling as "remove blocking constraints at cost" is tractable and novel.
Learning-theory bridge (version space, VC dimension, cutting planes) is a credible interdisciplinary hook.
"Slippery slopes without lawlessness" is a sticky, distinctive flagship claim.

Decisions requiring co-author judgment

Decision 1: Develop Variant A (strategic holding-writing) or keep canonical-only?

This is the highest-stakes structural decision.

Option A: Keep canonical-only paper (current state)

Pro: Clean, minimal, fast to submit. One clear message: "even the simplest holdings generate binding force + drift."
Con: The paper repeatedly promises a richer model it doesn't deliver. Breadth and $\gamma/\rho$ are defined but inert — a reviewer will ask why they're there. The EP application overstates what canonical holdings imply (talking about "race weights" and "conditional constraints" as holdings, which the model doesn't support).
If chosen: Must purge most extension talk. Delete or drastically shorten the "Citation incentives and risk" paragraph, the "Remark (richer holding languages)" paragraph, and all "to be developed" language. Rewrite EP application to be faithful to canonical holdings (outcomes at points, not weight restrictions).

Option B: Add minimal "Variant A-lite" (2–3 pages, 1 proposition)

Pro: Cures the "incomplete" feel. Makes EP application legitimate (judges can impose sign restrictions / relative-weight constraints as holdings). Increases novelty over Callander & Clark. The building blocks ($B$, $\gamma$, $\rho$) are already defined — just need a threshold result: "judges go broad when $\gamma > \rho$, subject to entailment."
Con: Risk of bloating or derailing if overbuild. Adds ~2–3 pages, potentially 1–2 weeks of work.
If chosen: Keep it brutally simple. Judge picks target rule $\hat\theta_t \in \mathcal{F}_t$ and breadth parameter. One proposition about optimal breadth. Then EP can legitimately discuss holding content.

Option C: Save Variant A entirely for follow-up paper

Pro: Fastest path to submission. The canonical-only paper has a complete set of results.
Con: Same as Option A cons, plus you lose the opportunity to show Variant A is tractable.
If chosen: Same purging as Option A, but frame the paper's contribution as "we isolate the geometric channel" and note Variant A as "the natural next step."

ChatGPT recommendation: Option B (Variant A-lite). My assessment: Options A and B are both defensible. The question is whether Holger wants to spend the time. If Option A, the purging must be thorough.

Decision 2: Keep two applications or trim to one?

Option A: Keep EP + DP (current state)

Pro: Shows breadth of the framework across two major doctrinal areas.
Con: DP (~4 pages) does not anchor to specific propositions, largely repeats the "multidimensional + balancing test" storyline, and exposes the paper to doctrinal nitpicks in two huge areas. DP also weakens the paper by diluting focus.

Option B: EP only; cut DP entirely

Pro: Frees ~4 pages for either tightening proofs, adding Variant A-lite, or strengthening EP anchoring. One crisp application is better than two mushy ones.
Con: Loses breadth.

Option C: EP as main application; DP as 1-paragraph second illustration

Pro: Best of both worlds — keeps the breadth signal in a single paragraph while focusing effort on EP.
Con: Requires judgment about what one DP paragraph would say.

ChatGPT recommendation: Option C (EP main, DP as short second illustration). My assessment: Option B or C. DP in its current form is clearly the weakest section. If Holger values showing both doctrinal areas, Option C; otherwise Option B saves the most space.

Decision 3: Restructure "Next Steps" → "Implications and testable predictions"?

The problem: The current "Next Steps: Closing the Model, Empirical Tests, and Policy Implications" section reads like a grant proposal — it lists what the authors plan to do rather than what the paper establishes. ChatGPT flagged this as the single most impactful structural change for submission readiness.

Proposed change: Delete "Next Steps" as a section. Move the useful content (testable predictions, empirical strategies) into a shorter "Implications" section written as claims, not plans. Fold "Closing the Model" items into a brief "Extensions" paragraph in the Conclusion.

Pro: Immediately makes the paper feel finished rather than in-progress. The predictions are already anchored to propositions — they just need declarative framing.
Con: Loses the "Closing the Model" discussion, which signals awareness of limitations.

My assessment: Strong yes on this one. The predictions and empirical strategies are real contributions worth keeping. The "Next Steps" framing is the problem, not the content.

Decision 4: Integrate Schelling/focal-point framing from intro.tex?

The alternative intro (paper/intro.tex) motivates the admissible set as a focal-point equilibrium using Schelling's convention theory. The key passage: "A convention is a coordination equilibrium that acquires binding force through shared expectations."

Option A: Integrate as 1 paragraph in the current intro

Pro: Provides microfoundation for why peer sanctions ($K$) exist and why the admissible set is a shared coordination object. Speaks to JLS/JLEO readers who care about institutions.
Con: Risk of a second competing motivation alongside the constraint-accumulation story.

Option B: Don't integrate

Pro: Keeps the intro focused on one narrative.
Con: The sanctions assumption ($K > \alpha$) currently has no institutional justification beyond "reputational/institutional consequences."

ChatGPT recommendation: Option A, but keep it to one paragraph tied directly to the plausibility/sanction mechanism. My assessment: Mild yes. One paragraph won't hurt and adds depth. But it's low priority relative to Decisions 1–3.

Decision 5: Flagship claim framing

The intro currently gestures at two framings:

"Law binds without being determinate" (jurisprudence canon — true but not distinctive)
"Slippery slopes without lawlessness" (distinctive, memorable, maps to Prop 5 + Remark 1)

ChatGPT's suggested one-sentence contribution:

Horizontal stare decisis binds judges by shrinking an admissible set of decision rules, which can generate path-dependent doctrinal drift ("slippery slopes") even when every decision is legally plausible.

My assessment: "Slippery slopes without lawlessness" is the stronger hook. It's specific to this paper's contribution (Prop 5 + the constructive example), whereas "law binds without being determinate" describes the framework generally. The suggested one-sentence version could work as the intro's topic sentence.

Recommended priority order

Decision 3 (restructure Next Steps → Implications): Highest impact, lowest risk, can be done independently.
Decision 2 (trim DP): Clear improvement, frees space.
Decision 1 (Variant A-lite vs purge): Highest stakes, determines what EP application can say.
Decision 5 (flagship claim): Quick to decide, sharpens the intro.
Decision 4 (Schelling framing): Nice to have, low priority.