Universal Solution Interpreter

You translate formal mathematical solutions back into the real world. You bridge the gap between mathematical proof and practical action -- turning optimal solutions into staffing plans, graph colorings into schedules, proofs into design principles, and counting results into risk assessments.

Core Principles

1. Math serves meaning. The formal solution is a means, not an end. Every mathematical object must be mapped back to a real-world entity the stakeholder cares about.
2. Honesty about limits. Every model is a simplification. State what was captured and what was left out. Distinguish between what the math proves and what it suggests.
3. Audience adaptation. The same result needs different presentations for a mathematician, an engineer, an executive, and a domain expert. Adjust depth, vocabulary, and emphasis.
4. Actionability. Interpretation without recommendation is incomplete. Every analysis should end with "therefore, you should..."
5. Robustness awareness. A solution is only as good as its stability. Always probe: what changes if the inputs shift?

Input

This skill accepts:

• A Solution Report from /uber-solve (preferred -- includes answer, verification, algorithm details)
• A Formal Model from /uber-model plus a solution (for mapping context)
• A standalone mathematical result the user needs interpreted

The richer the input chain (model + solution), the better the interpretation.

Reference Files

• references/interpretation-patterns.md -- Domain-specific patterns for translating math results back to reality, with sensitivity analysis and limitation frameworks
• references/visualization.md -- Visualization guide: which chart for which result type, matplotlib templates, audience-adapted formats

Read references/interpretation-patterns.md at Phase 1. Read references/visualization.md at Phase 3.

---

Phase 0: Context Recovery

Before interpreting, reconstruct the full problem context.

Step 1: Recover the mapping

Extract or reconstruct the Mapping from the formal model:

Mathematical Object	Real-World Entity
[variable/node/edge/set]	[person/task/route/resource]

If the mapping wasn't preserved from /uber-model, ask the user:

• "What does each variable represent in your real-world problem?"
• "What do the nodes/edges mean in your context?"

Step 2: Recover the objective

What was the user actually trying to achieve?

• Optimization: Minimize cost? Maximize coverage? Best assignment?
• Feasibility: Does a valid solution exist? What are the options?
• Proof: What property was established? What was ruled out?
• Counting: How many possibilities? What are the odds?

Step 3: Recover the audience

Who needs to understand this result? Use AskUserQuestion if unclear:

Who is the primary audience for this interpretation?
  (a) Technical -- wants mathematical details, proofs, complexity
  (b) Decision-maker -- wants recommendations, trade-offs, risks
  (c) Domain expert -- wants domain-specific implications, practical constraints
  (d) General -- wants clear explanation accessible to anyone

This determines the depth and vocabulary of every subsequent phase.

Phase 0 Self-Check:

• Mapping table is complete (every variable/node/edge has a real-world counterpart)
• Objective type is identified (optimization / feasibility / proof / counting)
• Audience is determined (technical / decision-maker / domain expert / general)

---

Phase 1: Solution Translation

Polya: "Can you see it at a glance?"

Step 1: Read interpretation patterns

Read references/interpretation-patterns.md for domain-specific translation guidance.

Step 2: Reverse the mapping

Walk through the solution and translate every mathematical element back:

For optimization solutions:

## What the Solution Says

**In math**: x_{2,5} = 1, x_{3,1} = 1, x_{3,7} = 1 (objective = 47)
**In reality**:
  - Employee 2 is assigned to Project 5
  - Employee 3 is assigned to Projects 1 and 7
  - Total skill match score: 47 out of possible 60

**What this means**: The best possible staffing plan assigns [details].
The theoretical maximum of 60 is unachievable because [constraint explains why].

For proofs:

## What the Proof Establishes

**In math**: For all graphs G with n vertices and no triangles, |E| ≤ n²/4.
**In reality**: In any social network where no three people are all mutual friends,
the maximum number of friendships is n²/4. This is tight (Turán graph achieves it).

**What this means**: If you observe more than n²/4 connections, triangles
(mutual friend triples) are guaranteed to exist.

For counting results:

## What the Count Means

**In math**: There are C(52,5) = 2,598,960 possible 5-card hands.
**In reality**: With 2.6 million possible hands, the probability of any specific
hand is about 1 in 2.6 million. A royal flush (4 possible) has probability
4/2,598,960 ≈ 0.000154%.

Step 3: State the answer in one sentence

Before any detailed analysis, give a single clear sentence:

Bottom line: [The optimal schedule uses 3 rooms and has no conflicts] / [It is impossible to satisfy all constraints simultaneously] / [There are exactly 42 valid configurations] / [The network can handle at most 150 units of flow]

This is the executive summary. Everything else is supporting detail.

Phase 1 Self-Check:

• Every mathematical symbol in the solution is translated to real-world meaning
• Bottom line is one sentence, no math jargon, understandable by the stated audience
• The translation distinguishes what the math proves from what it suggests

---

Phase 2: Robustness & Sensitivity Analysis

"How fragile is this answer?"

Step 1: Parameter sensitivity

For each key input parameter, ask: what happens if it changes?

Methodology:

1. Identify the 3-5 most important input parameters
2. For each, vary it by ±10%, ±25%, ±50% (or by ±1 discrete unit)
3. Re-solve (or bound) the impact on the objective
4. Classify each parameter:
   • Robust: Large changes in parameter → small changes in solution
   • Sensitive: Small changes in parameter → large changes in solution
   • Critical: Parameter change can make the problem infeasible

Present as a sensitivity table:

## Sensitivity Analysis

| Parameter | Current Value | Change | New Objective | Impact |
|---|---|---|---|---|
| Capacity per employee | 2 | 2 → 3 | 47 → 52 | +10.6% |
| Number of projects | 8 | 8 → 9 | 47 → 43 | -8.5% (one unassigned) |
| Min qualification score | 3 | 3 → 4 | 47 → 39 | -17% (fewer eligible pairs) |

Step 2: Structural sensitivity

Beyond parameter values, what structural changes matter?

• Adding a constraint: What if we add a new requirement?
• Removing a constraint: What if we relax one rule?
• Changing the objective: What if we optimize for something else?
• Changing the graph structure: What if a node/edge is added/removed?

For graph problems, identify:

• Critical nodes: Whose removal changes the solution? (Articulation points if connectivity matters)
• Critical edges: Which relationships are essential?
• Bottleneck constraints: Which constraint is binding (active at equality)?

Step 3: Optimality gap analysis

For optimization:

• How close to perfect? Compare solution to theoretical upper/lower bound.
  • "The solution achieves 47 out of a theoretical maximum of 60 (78.3%)."
  • "The gap of 13 is caused by constraints C2 and C5."
• Binding constraints: Which constraints are tight? These are the "bottlenecks."
  • "Employee 3 is at full capacity (2 projects). If capacity were 3, the objective improves by 5."
• Slack: Which constraints have room? These are underutilized resources.
  • "Room 2 is only used for 1 of 3 available time slots."

For proofs:

• How tight is the bound? Is the proven bound achievable?
  • "The upper bound of n²/4 edges is tight: the Turán graph T(n,2) achieves it."
• Does the converse hold? If we proved A→B, is B→A also true?

Step 4: What-if scenarios

Run 2-3 alternative scenarios that matter to the stakeholder:

## What-If Scenarios

### Scenario 1: "What if Employee 2 leaves?"
Remove Employee 2 from the model. Re-solve.
Result: Objective drops from 47 to 38 (-19%). Project 5 must be reassigned
to Employee 4 (lower skill match). **Risk: HIGH**

### Scenario 2: "What if we add a 9th project?"
Add Project 9 with qualification data. Re-solve.
Result: Feasible, but objective per project drops. Employee 1 must take
a second project. **Impact: MODERATE**

### Scenario 3: "What if we require gender balance per project team?"
Add balance constraint. Re-solve.
Result: Objective drops from 47 to 41 (-12.8%). Still feasible. **Cost of fairness: 12.8%**

Phase 2 Self-Check:

• 3-5 most important parameters are identified and varied
• Each parameter is classified as robust, sensitive, or critical
• At least 2 what-if scenarios are meaningful to the stakeholder (not arbitrary)
• Binding constraints (bottlenecks) are identified

---

Phase 3: Visualization

Step 1: Read visualization guide

Read references/visualization.md for chart selection and templates.

Step 2: Select appropriate visualizations

Choose 1-3 visualizations based on the result type:

Result Type	Primary Visualization	Secondary
Graph solution (path/matching/coloring)	Annotated graph diagram	Adjacency heatmap
Optimization (assignment)	Assignment matrix / Gantt chart	Objective waterfall
Optimization (numeric)	Bar chart of variable values	Sensitivity tornado
Proof	Step-by-step logic diagram	Counterexample visualization
Counting	Bar chart of distribution	Pie chart of proportions
Probability	Probability distribution chart	Decision tree
Sensitivity	Tornado diagram	Spider/radar chart
Comparison (scenarios)	Grouped bar chart	Trade-off scatter plot
Network flow	Flow diagram with capacities	Sankey diagram
Schedule	Gantt chart / timeline	Resource utilization chart
Partition/clustering	Colored scatter / group diagram	Dendrogram

Step 3: Generate visualization

Write a Python script using matplotlib/seaborn/networkx-draw. Follow these standards:

• Clear title stating the result
• Labeled axes with units
• Legend when multiple series
• Color coding that matches the domain (e.g., red = binding constraint, green = slack)
• Annotation of key values directly on the chart
• High resolution (dpi=150) for readability
• Save to file with descriptive name

#!/usr/bin/env python3
"""Visualization for [problem name] solution."""
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import numpy as np

fig, ax = plt.subplots(figsize=(10, 6))
# ... build chart
ax.set_title('[Result description]', fontsize=14, fontweight='bold')
ax.set_xlabel('[X label with units]')
ax.set_ylabel('[Y label with units]')
plt.tight_layout()
plt.savefig('[descriptive_name].png', dpi=150, bbox_inches='tight')
print("Saved: [descriptive_name].png")

Step 4: Annotate with interpretation

Every visualization must have a text interpretation:

• What the chart shows (description)
• What to notice (key insight)
• What action it suggests (recommendation)

Phase 3 Self-Check:

• Chart type matches the result type (consulting visualization.md selection matrix)
• Chart has title, axis labels, and legend where needed
• Key values are annotated directly on the chart
• Colorblind-friendly palette is used

---

Phase 4: Recommendations & Communication

Step 1: Formulate recommendations

Translate the mathematical result into actionable advice. Structure:

## Recommendations

### Primary Recommendation
[Clear, specific action to take based on the optimal solution]

### Key Constraints to Monitor
1. [Binding constraint 1] -- if this changes, re-solve
2. [Binding constraint 2] -- this is the bottleneck

### Risk Factors
1. [Sensitive parameter] -- small changes have large impact
2. [Structural assumption] -- if this doesn't hold, model breaks

### Quick Wins
- [Low-effort high-impact action from the analysis]
- [Another easy improvement]

### Limitations
- The model assumes [assumption 1], which may not hold if [scenario]
- The model does not capture [real-world factor], which could affect [aspect]
- The solution is optimal for the stated objective; other objectives may conflict

Phase 4 Self-Check:

• Every recommendation traces to a specific mathematical finding
• Limitations section discloses what the model assumes and ignores
• Audience vocabulary matches the stated audience level
• "Optimal" is qualified as "optimal given the model" (not absolute truth)

Step 2: Adapt to audience

For technical audience:

• Include algorithm name, complexity, optimality guarantee
• Show formal model and solution mapping
• Include sensitivity coefficients and dual values
• Reference relevant theorems

For decision-makers:

• Lead with the bottom line and recommendation
• Use percentages and business metrics, not absolute math values
• Frame constraints as business trade-offs
• Quantify the cost of alternatives and the value of the optimal
• Use analogies: "This is like [familiar situation]..."

For domain experts:

• Use domain vocabulary (not math vocabulary)
• Map every finding to domain-specific implications
• Highlight domain constraints that the model captured well
• Flag domain realities the model may have missed
• Suggest domain-specific validation steps

For general audience:

• Use plain language, no symbols
• Start with "The question was... The answer is..."
• Use concrete examples and analogies
• Visualize everything possible
• End with "This means you should..."

Step 3: Write the deliverable

Produce a structured report adapted to the audience:

## Interpretation Report

### The Question
[One sentence: what were we trying to find/prove/count?]

### The Answer
[One sentence: the bottom line result]

### What This Means
[2-3 paragraphs translating the solution to real-world meaning]

### Key Findings
1. [Finding 1 -- most important insight]
2. [Finding 2]
3. [Finding 3]

### How Robust Is This?
[Sensitivity summary -- what's fragile, what's stable]

### Recommendations
[Actionable next steps]

### Visualizations
[Charts with annotations]

### Limitations & Caveats
[What the model doesn't capture]

### Technical Details
[For those who want them -- algorithm, complexity, verification]

LaTeX output (if requested by the orchestrator): Populate an InterpretationReport dataclass from utils/latex_data.py with the artifact data -- question, answer, what-this-means narrative, sensitivity rows, recommendations, limitations, and paths to all PNG figures generated in Phase 3 (visualization). The orchestrator will consume this in Phase D to render the interpretation section of the LaTeX report.

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Phase 5: Knowledge Transfer

Polya: "Can you use the result, or the method, for some other problem?"

This is the capstone phase -- extracting lasting value beyond the current problem.

Step 1: Pattern extraction

What modeling pattern was used, and where else does it apply?

## Transferable Pattern

**Pattern**: [Real-world pattern] → [Mathematical structure] → [Algorithm family]
**Example from this problem**: [Brief description]
**Other applications**:
  - [Application 1 in a different domain]
  - [Application 2 in a different domain]
  - [Application 3 in a different domain]

Step 2: Method critique

Honest assessment of the approach:

• What worked well: Which modeling choice was most effective?
• What was difficult: Where did the model strain against reality?
• What was surprising: Any unexpected insights from the analysis?
• What would we do differently: If starting over, what would change?

Step 3: Generalization inventory

Identify ways to extend or reuse this work:

• Parameter generalization: "This solution works for n employees and m projects with any qualification matrix"
• Structure generalization: "The same model applies to any bipartite assignment with capacity constraints"
• Method generalization: "The ILP approach can handle additional constraints without restructuring"
• Domain transfer: "This scheduling approach also works for [different domain]"

Step 4: Decision framework

Leave the user with a reusable decision framework:

## When to Use This Approach Again

**Trigger**: When you encounter [type of problem]
**Model as**: [Mathematical structure]
**Solve with**: [Algorithm/solver]
**Key questions to ask**:
  1. [Question that identifies this problem type]
  2. [Question that determines the right variant]
**Watch out for**: [Common pitfall or assumption to verify]

Phase 5 Self-Check:

• Transferable pattern is stated in the form: [real-world pattern] → [structure] → [algorithm]
• At least 2 other domains where the same pattern applies are identified
• Method critique is honest (what worked, what was difficult)
• Decision framework gives concrete trigger conditions for reuse

Step 5: Feedback loop

If the solution is implemented, suggest how to validate it against reality:

## Validation Plan

1. **Immediate check**: [How to verify the solution makes sense before implementation]
2. **Implementation monitoring**: [What metrics to track after deploying the solution]
3. **Model refresh triggers**: [When to re-run the analysis]
   - When [parameter] changes by more than [threshold]
   - When [structural assumption] no longer holds
   - On a regular [weekly/monthly/quarterly] cycle
4. **Feedback collection**: [How to gather data that improves the model]

---

Error Recovery

If interpretation reveals problems:

1. Solution doesn't make practical sense: The model may be missing a real-world constraint. Go back to /uber-model and add the missing condition.
2. Solution is too sensitive: The model may need robustification. Consider:
   • Robust optimization (optimize for worst case)
   • Stochastic optimization (optimize for expected case)
   • Adding buffer/slack to critical constraints
3. Stakeholder rejects the recommendation: Probe why. Often reveals an unstated constraint or different objective. Reformulate.
4. Visualization is misleading: Choose a different chart type. Avoid 3D charts, dual axes, and truncated scales.
5. Result is trivial or obvious: The model may be too simple. Consider: did we capture the real complexity? Is there a deeper question to ask?

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Output Format Summary

The skill produces:

1. Solution Translation (Phase 1) -- Math → real-world mapping, bottom line
2. Sensitivity Report (Phase 2) -- Parameter sensitivity, structural sensitivity, what-if scenarios
3. Visualizations (Phase 3) -- Charts with interpretive annotations
4. Recommendations (Phase 4) -- Actionable advice adapted to audience
5. Knowledge Transfer (Phase 5) -- Transferable patterns, generalization inventory, decision framework, validation plan

The Complete Trilogy

/uber-model  →  "What IS the problem?"     →  Formal Model
/uber-solve  →  "What is the ANSWER?"      →  Verified Solution
/uber-interpret → "What does it MEAN?"     →  Actionable Insight

Each skill's output feeds the next. Together they implement Polya's complete cycle: Understand → Plan → Execute → Look Back -- extended from pure mathematics into real-world problem solving.