Formula Explanation Ladder

Formula explanation ladder defines how much math to include when turning a paper equation into a wiki note or Korean synthesis post. The goal is not to prove everything; it is to make the claim readable and checkable.

$$\text{formula explanation}=\text{symbols}+\text{shapes}+\text{distribution}+\text{operation}+\text{claim}$$

Levels

Level	Include	Use When
0. Name	Concept name and link only	the equation is not central to the claim
1. Canonical formula	one displayed equation plus symbol definitions	a reader needs the formula to follow the post
2. Operational form	intermediate variables, shapes, masks, aggregation, update rule	the method is defined by the computation
3. Claim contract	objective, estimator, metric, split, uncertainty, and failure mode	the formula supports a paper or benchmark claim
4. Derivation note	assumptions, approximation, proof sketch, edge cases	the math itself is the topic

Most Korean posts should use level 1 or 2 and link to wiki notes for level 3 or 4. Paper notes should reach level 3 when a reported result depends on the equation.

Required Pieces

Piece	Question
Symbols	What does every variable mean?
Shape	Is each quantity scalar, vector, matrix, tensor, graph, coordinate set, or distribution?
Index	What does the index range over: token, node, edge, residue, atom, time, sample, or candidate?
Distribution	Where does each expectation or sample come from?
Operation	Is the formula projecting, normalizing, aggregating, differentiating, sampling, or scoring?
Parameter	What is optimized, estimated, conditioned on, or held fixed?
Evidence	How does this formula connect to the metric, split, benchmark, or claim?

Example Pattern

Use this shape for objectives when possible:

$$\mathcal{J}(\theta )={\mathbb{E}}_{u\sim q(u)}\left[w(u){\ell }_{\theta }(u)\right]$$

• $u$: sampled unit.
• $q(u)$: sampling distribution.
• $w(u)$: optional weight, mask, or importance term.
• ${\ell }_{\theta }(u)$: per-unit loss, score, reward, residual, or surrogate.
• $\theta$: optimized parameters.

Then state whether $\mathcal{J}$ is aligned with the reported metric or only a training surrogate.

Formula Families

Family	Minimum Level	Extra Detail
Attention	2	$Q$, $K$, $V$, scaling, mask, softmax, head concatenation, output projection
GNN message passing	2	node state, edge state, neighbor set, message, aggregation, update
Diffusion or flow	2	state, time, noise/score/velocity target, conditioning, sampler
Contrastive learning	2	anchor, positive, negative set, similarity, temperature, denominator
Coordinate model	3	frame, symmetry, alignment, atom/residue mapping, coordinate metric
Benchmark metric	3	denominator, aggregation, selection rule, uncertainty, invalid-output policy
Likelihood or ELBO	3	modeled distribution, latent variables, approximation, bound direction

Post vs Wiki Boundary

Surface	Good Depth
Korean post	level 1 or 2, enough to explain the reader-facing claim
Concept wiki note	level 2 or 3, with reusable symbol definitions and checks
Paper note	level 3, because the equation is evidence for a specific claim
Math note	level 3 or 4 when assumptions or derivation are the topic

Stop Conditions

• A formula is copied without local notation.
• Symbols are defined by memory instead of the source.
• A shorthand hides important intermediate variables.
• A loss is presented as if it were the final metric.
• A finite-sample metric is presented as a population guarantee.
• A coordinate equation omits frame, symmetry, or mapping assumptions.

Related

• Formula intake
• Math
• Math foundations
• Objective-metric alignment
• Benchmark claim contract
• Claim-evidence boundary