axiomize · diff
git:20260824.ac527d4 to git:20260903.93d7e34
42 added, 42 removed. Audit A to A.
---
name: axiomize
description: Transforms any idea, problem, or real-world phenomenon into a rigorous mathematical model. Decomposes the idea into sub-problems, matches known modeling archetypes, extracts active parameters into structured tables, models the system from multiple mathematical perspectives (deterministic, stochastic, optimization, agent-based, network, control, game theory, causal inference, information theory, reliability, SPC, thermodynamic analogies, decision theory, demographic, spatial), compares the resulting models, and delivers a standardized report with runnable Python code and falsifiability criteria. Use when the user wants to formalize, model, simulate, or mathematically analyze a concept, process, system, or hypothesis.
---
# Axiomize: Idea → Rigorous Mathematical Model
Turn vague ideas into formal mathematical models through disciplined decomposition and multi-perspective analysis.
## Core Workflow
Follow ALL phases in order. Never skip a phase. Show intermediate outputs to the user.
- ### Phase 0 — Set Rigor Level
+ ### Phase 0: Set Rigor Level
- Pick **basic**, **standard** (default), or **research** using the signals table in [rigor.md](rigor.md), announce it in your first line, and offer: *"say 'deeper' or 'quicker' anytime."* The ladder defines what each phase requires at each level — basic stays light but honest, research adds model criticism, dimensionless reduction, uncertainty quantification and reproducibility. Whatever the tier, the final answer opens with a plain-language summary (≤ 5 sentences).
+ Pick **basic**, **standard** (default), or **research** using the signals table in [rigor.md](rigor.md), announce it in your first line, and offer: *"say 'deeper' or 'quicker' anytime."* The ladder defines what each phase requires at each level, basic stays light but honest, research adds model criticism, dimensionless reduction, uncertainty quantification and reproducibility. Whatever the tier, the final answer opens with a plain-language summary (≤ 5 sentences).
- ### Phase 1 — Parse the Idea
+ ### Phase 1: Parse the Idea
Restate the idea in one sentence. Then extract:
- **System**: What is the thing being modeled? What are its boundaries?
- **State**: What quantities change over time / space / across entities?
- **Inputs**: What drives or perturbs the system?
- **Goal**: What question must the model answer? (prediction? explanation? optimization? control?)
- **Horizon**: Time scale and spatial scale that matter.
- If any of these is unclear, ask the user BEFORE proceeding — but ask like a modeler, not generically:
+ If any of these is unclear, ask the user BEFORE proceeding, but ask like a modeler, not generically:
- Vague goal → "Do you want to PREDICT what happens, DECIDE what to do, or CONTROL it to a target?" (this single question routes the whole session)
- No quantities named → "What would you MEASURE to know this is working?"
- No scale given → "Over what time period? At what size?"
Never proceed on an idea where neither system boundary nor goal question can be stated.
- ### Phase 2 — Decompose
+ ### Phase 2: Decompose
Break the idea into independent sub-problems:
- 1. List 3–7 sub-problems (e.g., "spread of influence" → population dynamics + interaction rules + external shocks).
+ 1. List 3-7 sub-problems (e.g., "spread of influence" → population dynamics + interaction rules + external shocks).
2. For each sub-problem classify its nature:
- - `flow` — quantity accumulating over time (→ differential/difference equations)
- - `interaction` — entities affecting each other (→ networks / agent-based)
- - `decision` — choices under constraints (→ optimization / game theory)
- - `uncertainty` — randomness dominates (→ probability / stochastic processes)
+ - `flow`, quantity accumulating over time (→ differential/difference equations)
+ - `interaction`, entities affecting each other (→ networks / agent-based)
+ - `decision`, choices under constraints (→ optimization / game theory)
+ - `uncertainty`, randomness dominates (→ probability / stochastic processes)
3. State which sub-problems couple to which.
4. Draw the coupling map as a Mermaid graph (renders on GitHub):
````
```mermaid
graph LR
Demand[uncertainty] --> Stock[flow]
Reorder[decision] --> Stock
Stock --> Answer[goal question]
```
````
- ### Phase 3 — Parameter Table
+ ### Phase 3: Parameter Table
Extract every active parameter using the template in [templates/parameters.md](templates/parameters.md):
| Symbol | Name | Unit | Type | Range | Source | Sensitivity |
|--------|------|------|------|-------|--------|-------------|
Rules:
- Mark each parameter as **exogenous** (given) or **endogenous** (model output).
- Estimate sensitivity qualitatively (low/med/high): "if this doubles, does the answer change wildly?"
- Explicitly list parameters you deliberately EXCLUDE and why (dimension reduction).
- ### Phase 4 — Assumptions
+ ### Phase 4: Assumptions
- Write assumptions using [templates/assumptions.md](templates/assumptions.md). Every assumption must have a **violation consequence** — what breaks in the model if reality violates it.
+ Write assumptions using [templates/assumptions.md](templates/assumptions.md). Every assumption must have a **violation consequence**, what breaks in the model if reality violates it.
- ### Phase 5 — Multi-Perspective Modeling
+ ### Phase 5: Multi-Perspective Modeling
- **First, check [archetypes.md](archetypes.md):** scan the catalog against each sub-problem from Phase 2. If two or more core features match an archetype (SIR, Bass diffusion, newsvendor, M/M/c, logistic, Lotka–Volterra...), START from that canonical model and adapt — declare the match and what you changed. Inherited closed forms become Phase 7 validation targets.
+ **First, check [archetypes.md](archetypes.md):** scan the catalog against each sub-problem from Phase 2. If two or more core features match an archetype (SIR, Bass diffusion, newsvendor, M/M/c, logistic, Lotka, Volterra...), START from that canonical model and adapt, declare the match and what you changed. Inherited closed forms become Phase 7 validation targets.
- **If nothing matches:** announce 'novel territory' and execute the seven-step protocol in [first-principles.md](first-principles.md) — analogy mining, conservation skeleton, mechanism-driven rate laws, dimensional scaffold (Buckingham π), minimal viable model, falsifier-first design, triangulated validation. Novel territory RAISES the bar: research tier demands ≥ 3 lenses from unrelated mathematical families.
+ **If nothing matches:** announce 'novel territory' and execute the seven-step protocol in [first-principles.md](first-principles.md), analogy mining, conservation skeleton, mechanism-driven rate laws, dimensional scaffold (Buckingham π), minimal viable model, falsifier-first design, triangulated validation. Novel territory RAISES the bar: research tier demands ≥ 3 lenses from unrelated mathematical families.
- Then read EVERY perspective file in `perspectives/`. For each applicable perspective, build an actual model — not just "this could apply":
+ Then read EVERY perspective file in `perspectives/`. For each applicable perspective, build an actual model, not just "this could apply":
- 1. [perspectives/deterministic.md](perspectives/deterministic.md) — ODEs, difference equations, compartmental models
- 2. [perspectives/stochastic.md](perspectives/stochastic.md) — random variables, Markov chains, Monte Carlo
- 3. [perspectives/optimization.md](perspectives/optimization.md) — objective functions, constraints, equilibria
- 4. [perspectives/agent-based.md](perspectives/agent-based.md) — local rules → global behavior
- 5. [perspectives/network.md](perspectives/network.md) — graph structure, centrality, dynamics on networks
- 6. [perspectives/control.md](perspectives/control.md) — feedback, regulation, steering to setpoint
- 7. [perspectives/game-theory.md](perspectives/game-theory.md) — strategic interaction, equilibria, mechanisms
- 8. [perspectives/causal-inference.md](perspectives/causal-inference.md) — intervention claims from observational data
- 9. [perspectives/information-theory.md](perspectives/information-theory.md) — what can be known, transmitted, compressed
- 10. [perspectives/reliability.md](perspectives/reliability.md) — failure times, maintenance economics, availability
- 11. [perspectives/spc.md](perspectives/spc.md) — detecting process change vs common-cause noise
- 12. [perspectives/thermodynamic.md](perspectives/thermodynamic.md) — stock-flow analogies with explicit break points
- 13. [perspectives/decision-theory.md](perspectives/decision-theory.md) — choosing under deep uncertainty, EVPI
- 14. [perspectives/demographic.md](perspectives/demographic.md) — aging populations, actuarial liabilities
- 15. [perspectives/spatial.md](perspectives/spatial.md) — hotspots, clustering, spatial interpolation
+ 1. [perspectives/deterministic.md](perspectives/deterministic.md). ODEs, difference equations, compartmental models
+ 2. [perspectives/stochastic.md](perspectives/stochastic.md), random variables, Markov chains, Monte Carlo
+ 3. [perspectives/optimization.md](perspectives/optimization.md), objective functions, constraints, equilibria
+ 4. [perspectives/agent-based.md](perspectives/agent-based.md), local rules → global behavior
+ 5. [perspectives/network.md](perspectives/network.md), graph structure, centrality, dynamics on networks
+ 6. [perspectives/control.md](perspectives/control.md), feedback, regulation, steering to setpoint
+ 7. [perspectives/game-theory.md](perspectives/game-theory.md), strategic interaction, equilibria, mechanisms
+ 8. [perspectives/causal-inference.md](perspectives/causal-inference.md), intervention claims from observational data
+ 9. [perspectives/information-theory.md](perspectives/information-theory.md), what can be known, transmitted, compressed
+ 10. [perspectives/reliability.md](perspectives/reliability.md), failure times, maintenance economics, availability
+ 11. [perspectives/spc.md](perspectives/spc.md), detecting process change vs common-cause noise
+ 12. [perspectives/thermodynamic.md](perspectives/thermodynamic.md), stock-flow analogies with explicit break points
+ 13. [perspectives/decision-theory.md](perspectives/decision-theory.md), choosing under deep uncertainty, EVPI
+ 14. [perspectives/demographic.md](perspectives/demographic.md), aging populations, actuarial liabilities
+ 15. [perspectives/spatial.md](perspectives/spatial.md), hotspots, clustering, spatial interpolation
Applicability rule: model from at least **two** perspectives whenever possible. A single-perspective analysis is acceptable only if the user explicitly asks for speed.
### Parallel Dispatch Protocol
If the runtime supports subagents (Claude Code `Task` tool, opencode `task` tool, equivalent), Phase 5 MUST run in parallel:
- 1. **Freeze first.** Phases 1–4 are completed and FROZEN before any dispatch: idea statement, goal question, decomposition, parameter table, assumptions. Frozen inputs go into every brief verbatim.
- 2. **Select lenses** (≥ 2 applicable). Sub-problems marked *coupled* in Phase 2 stay inside one brief — never split coupled dynamics across agents.
+ 1. **Freeze first.** Phases 1-4 are completed and FROZEN before any dispatch: idea statement, goal question, decomposition, parameter table, assumptions. Frozen inputs go into every brief verbatim.
+ 2. **Select lenses** (≥ 2 applicable). Sub-problems marked *coupled* in Phase 2 stay inside one brief, never split coupled dynamics across agents.
3. **Dispatch simultaneously**: fill [templates/subagent-brief.md](templates/subagent-brief.md) once per lens and send ALL briefs in a single message so they execute concurrently. Each brief is self-contained; subagents see no other lens's output (independence prevents anchoring bias between lenses).
4. **Merge**: collect blocks verbatim → resolve every `ASSUMPTION CONFLICT` yourself and document resolutions → deduplicate overlapping insights → proceed to Phase 6 with the collected scores.
Fallback (no subagent support): run the same briefs sequentially through your own context, in the order listed above, and note in the report that independence was sequential rather than parallel.
For each perspective output:
```
### Perspective: <name>
Model: <equations / rules, written formally in LaTeX>
Why it fits: <one sentence tied to Phase 2 classification>
Key insight this lens reveals: <what ONLY this view shows>
Blind spots: <what this view cannot see>
```
- ### Phase 6 — Compare & Recommend
+ ### Phase 6: Compare & Recommend
Build a comparison table:
| Criterion | Det | Stoch | Opt | ABM | Net | Ctrl |
|-----------|-----|-------|-----|-----|-----|------|
- Criteria (score 1–5): fidelity to reality, data requirements, computational cost, analytical tractability, answerability of the user's goal question. Only include columns for perspectives you actually built; mark rejected ones with a one-line rejection reason under the table.
+ Criteria (score 1-5): fidelity to reality, data requirements, computational cost, analytical tractability, answerability of the user's goal question. Only include columns for perspectives you actually built; mark rejected ones with a one-line rejection reason under the table.
Recommend ONE primary model (+ optionally one secondary for validation). Justify with the table, not vibes.
- ### Phase 7 — Implement & Validate
+ ### Phase 7: Implement & Validate
Generate runnable Python for the recommended model:
- Use `numpy`/`scipy.integrate.solve_ivp` for ODEs, `numpy` RNG for stochastic, `scipy.optimize` for optimization, plain loops/dataclasses for agent-based.
- Parameter values: use literature-typical defaults, clearly marked as placeholders.
- **If the user has real data** (CSV of observations), calibrate instead of guessing: `tools/fit.py --model <sir|logistic> --data <file>` returns fitted parameters with confidence intervals and derived quantities (R₀, K...). Report both fit quality (RMSE) and parameter uncertainty.
- Then validate with `tools/validate.py` (dimensional checks, sanity bounds, conservation laws).
- Run a sensitivity sweep on the 2 highest-sensitivity parameters from Phase 3.
- When matplotlib is available, produce a plot of the model behavior (`--plot`) and reference it in the report.
- Present results as: predicted behavior summary + plot description + limitations.
- ### Phase 8 — Deliverable Format
+ ### Phase 8: Deliverable Format
Assemble the final answer using [templates/report.md](templates/report.md). Its non-negotiable elements:
1. **One-line model statement** ("The idea reduces to a SIR-type system with reinfection")
2. Assumptions & parameter tables
3. All perspective models (formal notation)
4. Comparison table + recommendation
5. Code + validation results
- 6. **What would falsify this model** — observable predictions that, if wrong, kill the model
- 7. **Confidence ledger** — every major claim tagged as established / assumption / speculation
+ 6. **What would falsify this model**, observable predictions that, if wrong, kill the model
+ 7. **Confidence ledger**, every major claim tagged as established / assumption / speculation
**LaTeX export (research tier):** when the user wants thesis/paper-ready output, convert the final report with `tools/report_to_latex.py --input <report.md> --output <report.tex>` (add `--pdf` if a LaTeX toolchain is installed). The converter handles tables, verbatim code, math passthrough and symbol transliteration; always compile-check before delivering.
- **Archive rule:** after delivering the report, save it to `reports/YYYY-MM-DD-<short-slug>.md` in the working directory (create the folder if needed) using the template header verbatim (Date / Rigor level / Model in one sentence), then run `tools/index_reports.py` to rebuild `reports/INDEX.md`, and tell the user the path. Reference earlier indexed sessions when relevant ("this extends your 2026-08-24 barista model") — modeling sessions should accumulate into a searchable personal archive, not evaporate into chat scrollback.
+ **Archive rule:** after delivering the report, save it to `reports/YYYY-MM-DD-<short-slug>.md` in the working directory (create the folder if needed) using the template header verbatim (Date / Rigor level / Model in one sentence), then run `tools/index_reports.py` to rebuild `reports/INDEX.md`, and tell the user the path. Reference earlier indexed sessions when relevant ("this extends your 2026-08-24 barista model"), modeling sessions should accumulate into a searchable personal archive, not evaporate into chat scrollback.
## Hard Rules
- NEVER present an equation without defining every symbol.
- - NEVER skip Phase 5 multi-perspective analysis silently — if reducing scope, say so explicitly.
+ - NEVER skip Phase 5 multi-perspective analysis silently, if reducing scope, say so explicitly.
- ALWAYS state units for parameters.
- ALWAYS open the final deliverable with a plain-language summary, regardless of tier.
- If the idea is purely qualitative (no measurable quantity exists), say so and propose the closest measurable proxy instead of inventing fake precision.
- Distinguish clearly between: established science, reasonable assumption, pure speculation.