Formalize Problem

Formalize Problem

Transform a research goal into precise, testable hypotheses with explicit success/failure conditions.

Usage

Invoke this skill by name with the research goal as a quoted string. On slash-command hosts, prefix with / (e.g. /formalize-problem "<goal>").

formalize-problem "determine if the security proof in Section 4 holds under asynchrony"

Instructions

1. Read Inputs

Always read:

• reaper-workspace/notes/clarified-goal.md — refined research goal, scope, assumptions, and success criteria
• reaper-workspace/notes/paper-summary.md — what the paper claims and how. If this file does not exist (paper-less mode), proceed without it — use the literature review and clarified goal as primary inputs instead.
• The research goal from the argument (use the refined goal from clarified-goal.md if available)

Read selectively from:

• reaper-workspace/notes/literature.md — focus on the "Key Prior Results" and "Gaps Identified" sections, not the full paper tables. Only deep-read individual paper entries if a specific prior result is needed for hypothesis screening. In paper-less mode, this becomes the primary technical context — read it more thoroughly.

2. Identify the Core Question

What exactly needs to be resolved? Be specific. "Is this protocol secure?" is too vague. "Does the safety proof of Protocol X hold when the adversary is adaptive rather than static?" is testable.

3. Pin Down Model Assumptions (must be unambiguous before hypotheses)

Pin down every dimension of the system/trust model. If any dimension is left vague, the investigation will produce ambiguous results. Consult ../reaper/references/model.md for the domain-appropriate checklist of dimensions that must be specified. Every field must have a concrete answer, not "TBD".

Every hypothesis must reference these model assumptions by specifying which parameters it holds under. A hypothesis that states a claim without pinning every relevant dimension is rejected.

4. Apply Importance Filter (Hamming)

Not all questions are equally worth investigating. Prioritize by consequence — ask: "If we resolved this question, who would care and why?" Consult ../reaper/references/model.md for domain-specific examples of how to rank by importance.

5. Apply Gap-Finding (Qian)

Map the dimensions of existing work and find unexplored combinations. Consult ../reaper/references/model.md for the domain-appropriate gap-finding matrix dimensions. Which cells in this matrix are empty? Those are candidate hypotheses.

6. Screen Against Known Impossibilities

For each hypothesis, check whether it contradicts a known impossibility or lower bound. Consult ../reaper/references/impossibility-results.md for the domain-relevant impossibility results and lower bounds.

If a hypothesis asks to prove something that an impossibility result rules out:

1. Flag it explicitly in the hypothesis with a warning
2. Reformulate: "Under what additional assumptions does this become possible?" or "What weaker property can be achieved?"
3. Do NOT leave it as a hypothesis for the investigate skill to waste cycles on

7. Enforce Definitional Hygiene

Each core property listed in the output must be stated precisely. Consult ../reaper/references/definitional-standards.md for the domain-appropriate acceptable definition forms. Informal or ambiguous terms without formal definitions are NOT acceptable — different papers define the same terms differently, so pin it down. If the paper under analysis uses informal definitions, formalize them explicitly and note that you are doing so.

8. Write Output

Write to reaper-workspace/notes/problem-statement.md:

# Problem Statement

## Research Goal
[Restate the goal precisely]

## Model Assumptions
[One field per dimension from `../reaper/references/model.md`. Every dimension must have a concrete answer.]

## Security Properties Under Investigation
[What must hold? List each property with its formal definition or reference.]
1. **[Property name]**: [formal definition or citation]
2. ...

## Performance Metrics
[Concrete targets, if applicable]
- Communication complexity:
- Round complexity:
- Other:

Write ideas to a separate file reaper-workspace/notes/ideas.md:

# Ideas

## Initial Ideas

### H1: [Short descriptive title]
- **Statement**: [Precise, falsifiable claim — specific enough that a reader could disagree]
- **Success condition**: [What evidence would confirm this?]
- **Failure condition**: [What evidence would refute this?]
- **Priority**: High / Medium / Low
- **Rationale**: [Why this priority? What's the consequence of resolving it?]

### H2: [Short descriptive title]
...

Relationship to Brainstorm

/formalize-problem handles initial formalization: pinning down trust assumptions, the core question, definitional hygiene, and the first set of hypotheses. It runs once at the start of the pipeline. The /brainstorm skill handles recurring ideation — generating additional ideas as the investigation progresses, applying Hamming/Qian heuristics to current state, and proposing follow-up hypotheses based on patterns in results. /brainstorm updates the ideas.md that this skill creates (adding new ideas and editing existing ones inline).

Quality Criteria

• Every hypothesis is falsifiable — a reader could disagree with it
• Every hypothesis has explicit success and failure conditions — not "we'll know it when we see it"
• Priorities are justified by consequence, not convenience
• Trust assumptions are complete — no missing dimensions
• The hypotheses collectively cover the research goal — resolving all of them would answer the original question
• At most 5-7 hypotheses — focus beats breadth. If you have more, merge or drop the low-priority ones
• clarified-goal.md is always required — return an error if missing. paper-summary.md is optional (absent in paper-less mode) — if missing, rely on literature.md and clarified-goal.md instead