# Nw Algebraic Design Protocol > The METHOD for finding a design — name observations and equality before constructors, then follow any contradiction to the type or observation that causes it. Use when a design decision is contested, a law has exceptions, a census or model keeps producing wrong answers, or a representation change must preserve meaning. Complements nw-fp-algebra-driven-design, which catalogues the structures; this says how to arrive at one and what to do when it breaks. - Skill: `nwave-ai/nw-algebraic-design-protocol` (Agent Skill) - Install (CLI): `npx skillmds@latest add nwave-ai/nw-algebraic-design-protocol` - Raw SKILL.md: https://api.skillmd.com/api/skills/nwave-ai/nw-algebraic-design-protocol/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: AI & ML - Author: nWave-ai (https://skillmd.com/u/nwave-ai) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/nwave-ai/nw-algebraic-design-protocol --- # Algebraic Design Protocol `nw-fp-algebra-driven-design` catalogues structures; this finds a design and traces a false law. Load both for a compositional core; load this alone for a design that keeps giving wrong answers. **nWave sequencing choice.** State observations/equality before an algebra; this is project synthesis, not a universal ADD claim. **Cross-layer authority (ADR-SSOT-002 §6a).** One language-agnostic procedure for declared target-boundary layers: domain, application/ports, adapter/ integration, infrastructure/recovery. `targets[].boundary`/`contract-shape` sets applicability without a persisted field. OO/FP catalogues are projections, never competitors. ## 1. Observations and equality, before anything else State what a user can meaningfully **observe**. Then define `x ≈ y` as agreement under that observation set — never structural equality chosen because it was easy to implement. Inventory each observation, its result type, and **what it cannot distinguish**. That last column is the one that pays: a Boolean result is usually too weak the moment composition needs the remaining input or state. If several observations repeat the same traversal, look for one more primitive transition the others project from. ## 2. Constructors one at a time, each with an equation After adding a constructor, write at least one equation relating it to an observation or an existing constructor. The web of equations IS the design. A law with ignored arguments, many exceptions, or domain-specific flags signals a **fused concept**. Split it before choosing a representation. ## 3. Follow a contradiction to the carrier — do not special-case it This is the step that catches real defects, and the reason to load this skill when something is already broken. When a law fails, the cause is usually not the law. It is that the **carrier** — the type you are observing through — cannot express the property. Order-insensitive composition observed through an order-sensitive list is the textbook case. Two honest resolutions: **replace the carrier**, or **retract the law**. Never conceal the conflict with a special case, because a special case makes the model agree with the data while still being wrong about the domain. > **Worked anchor (nWave, 2026-08-06).** A census asked "is this MODULE live?" > when liveness was a property of the VERB the module dispatched. The three verbs > disagreed, so the answer came back for whichever one dominated the text. > Measuring per-module was the wrong carrier, and no amount of scanning more > surfaces would have fixed it — the fix was to change what the unit was. The > unit comes from the SEMANTICS, not from the file system. ## 4. Keep the representation out of the public algebra Persistence, tree shape, cache and optimisation are not part of the design. Build a deliberately obvious reference encoding first; enforce canonical reductions through smart constructors. Generate values through the **public surface**, and state reachability and size bounds explicitly. Generated counterexamples are candidate findings until a human reviews them: an impossible minimal case is evidence to audit the generator before blaming the implementation. For every intended public value, trace admitted input through a public constructor/producer and consumer to a public observation. The construction graph must be acyclic from an admitted input, or name an explicit bootstrap; a cycle with no seed proves neither inhabitation nor reachability. ## 5. Optimise by denotation, and prove the claim observationally Seek a homomorphic denotation the other observations derive from. State the old and new observation sets and the equivalence claim **before** changing the representation, and protect it with generated terms on both sides. Passing generated checks is evidence, not proof. Say which. ## Stop conditions The design is ready when every public constructor participates in a law or an observation, equality has a stated scope, and tempting-but-false laws are **recorded rather than deleted** — a future reader who rediscovers one needs to know it was already refuted. Escalate rather than paper over an unresolved observation, an incompatible carrier, or a nondeterministic boundary whose effect model has not been stated. ## Scope boundary Candidate algebra/preservation only: `BROAD_INPUT_DOMAIN` needs broad generated domain plus independent observation/falsifier; finite exhaustive/no law is `NOT_APPLICABLE`. It selects no topology, aggregate, deployment or runtime wiring. ## Cross-cutting invariants Load `nw-cross-cutting-invariants` alongside this. In particular `gate:design-principles-gdp-1-9` and `data:consumer-known-before-produced`: an observation nobody consumes is not an observation, it is a field.