reason

structure

graph LR
    A[parse] --> B[branch]
    B --> C[reduce]
    C --> D[ground]
    D --> E[emit]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffe1f5
    style D fill:#e1ffe1
    style E fill:#f5e1ff

[[#parse]] → [[#branch]] → [[#reduce]] → [[#ground]] → [[#emit]] ^structure

---

parse

1. receive input ^input
2. extract¹:
   • #component/parameter — what varies ^parameters
   • #component/constant — what is fixed ^constants
   • #component/operation — what transforms ^operations
   • #component/unknown — what is sought^unknowns
3. A question is a function awaiting its arguments². Identify what arguments it needs. ^function-analogy

---

branch

Generate multiple resolution paths before committing³:

Path Type	Tag	Purpose	Anchor
Typical	#path/typical	The conventional approach	^path-typical
Alternative	#path/alternative	A valid but less obvious route	^path-alternative
Boundary	#path/boundary	The edge case or inversion	^path-boundary

Assign rough likelihood. Retain all paths until [[#reduce|reduction]] forces convergence or reveals genuine divergence worth surfacing.

[!warning] Distribution Sampling ^distribution-sampling This is not optional. Single-path reasoning collapses to the mode. The goal is to sample from the full distribution of valid approaches.

---

reduce

Transform step by step⁴. Each step follows the primitive: ^transformation-primitive

$$\text{input} \xrightarrow{\text{operation}} \text{output}$$

[!caution] Skip Nothing ^skip-nothing An obvious step is still a step. A hidden step is where errors hide.

five primitive operations

Operation	Tag	Definition	Anchor
select	#primitive/select	condition → one of two branches	^op-select
sequence	#primitive/sequence	output of one feeds [[SKILL#^input|input]] of next	^op-sequence
iterate	#primitive/iterate	apply operation n times	^op-iterate
recurse	#primitive/recurse	define in terms of self, with base case	^op-recurse
search	#primitive/search	find first element satisfying constraint	^op-search

Name which [[SKILL#five primitive operations|primitive]] you are applying.

Substitution is explicit:

Given X, applying rule R yields X transformed by R

Never say "therefore" without showing the step that licenses it. ^explicit-substitution

---

ground

Before [[#emit|emitting]], verify:

• Each conclusion traces to stated premises only^premise-tracing
• Terms from separate contexts are not conflated^context-separation
• The result is terminal (no further reduction possible) or explicitly partial (further [[SKILL#^input|input]] required) ^termination-check

convergence logic

State	Tag	Meaning	Anchor
Converged	#state/converged	confidence is high	^state-converged
Diverged	#state/diverged	present the divergence—it is the answer	^state-diverged
Continuing	#state/continuing	[[#reduce|reduction]] could continue	^state-continuing
Blocked	#state/blocked	state what is missing	^state-blocked

---

emit

Output the synthesis.

Context	Tag	Strategy	Anchor
Simple query	#query/simple	direct answer	^emit-simple
Complex query	#query/complex	show the [[#^structure|structure]] that produced the answer	^emit-complex
Creative task	#task/creative	weight toward [[#branch|boundary paths]]5	^emit-creative
Analytical task	#task/analytical	weight toward [[#ground|convergent paths]]6	^emit-analytical

[!note] Trace Principle ^trace-principle The trace is part of the answer when the trace is informative. The trace is omitted when it would obscure.

---

Footnotes

1. Component extraction as defined in [[#parse|parse phase]] identifies [[SKILL#^parameters|parameters]], [[SKILL#^constants|constants]], [[SKILL#^operations|operations]], and [[SKILL#^unknowns|unknowns]]. ↩

2. See [[SKILL#^function-analogy|function analogy]] for the functional interpretation of queries. ↩

3. Multi-path generation prevents [[SKILL#^distribution-sampling|mode collapse]] by exploring [[SKILL#^path-typical|typical path]], [[SKILL#^path-alternative|alternative path]], and [[SKILL#^path-boundary|path boundary]] simultaneously. ↩

4. Stepwise transformation using [[SKILL#^op-select|op-select]], [[SKILL#^op-sequence|op-sequence]], [[SKILL#^op-iterate|op-iterate]], [[SKILL#^op-recurse|op-recurse]], or [[SKILL#^op-search|op-search]]. ↩

5. Boundary paths (see [[SKILL#^path-boundary|path-boundary]]) explore inversions and edge cases suitable for creative divergence. ↩

6. Convergent paths (see [[SKILL#^state-converged|state-converged]]) prioritize analytical rigor through [[SKILL#^premise-tracing|premise-tracing]] and [[SKILL#^context-separation|context-separation]]. ↩