Formula Intake

논문 수식은 notation이 아니라 claim의 계약입니다. 수식을 읽을 때는 모양을 외우기보다 어떤 object, distribution, parameter, objective, metric을 고정하는지 확인합니다.

$$\text{formula}=(\text{objects},\text{indices},\text{distribution},\text{operation},\text{claim})$$

Intake Fields

Field	Question	Route
Explanation depth	Should the note use name-only, canonical formula, operational form, claim contract, or derivation depth?	Formula explanation ladder
Pattern	Does the equation match a known loss, likelihood, contrastive, generative, graph, coordinate, or metric pattern?	Formula pattern catalog
Object type	Is the quantity a scalar, vector, matrix, tensor, graph, coordinate, distribution, or trajectory?	Linear algebra, Tensor shape notation, Geometry
Index set	Does the index run over batch, token, node, edge, residue, atom, time, sample, or candidate?	Discrete math and graphs
Distribution	Is the expectation over data, model samples, noise, policy rollouts, or test population?	Probability and statistics
Objective	Is the formula a loss, likelihood, bound, score, reward, constraint, or metric?	Information and likelihood
Constraint	Is the formula a hard feasible set, penalty, barrier, projection, or repair rule?	Constrained optimization
Bayesian quantity	Is the formula a prior, likelihood, posterior, MAP estimate, or posterior predictive distribution?	Bayesian inference
Objective-metric link	Does the optimized loss support the reported metric and claimed utility?	Objective-metric alignment
Derivative	Is the derivative with respect to parameters, inputs, coordinates, time, or latent variables?	Calculus and gradients
Curvature	Does the method use Hessian, Fisher, Newton step, curvature, or local quadratic approximation?	Second-order optimization
Symmetry	Should the output be invariant or equivariant under permutation, translation, rotation, or reflection?	Geometry and symmetry
Numerics	Does implementation require stable softmax, log-sum-exp, precision control, or conditioning checks?	Numerical computing
Evidence	Is the reported number a point estimate, confidence interval, paired comparison, or selected checkpoint?	Evaluation math
Benchmark score	What aggregation, split, selection rule, and denominator define the reported number?	Benchmark intake

Rewrite Pattern

For any objective, rewrite the paper equation into this shape when possible:

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

Then define:

• $u$: sampled unit, such as example, token, node, pair, time, noise, or trajectory.
• $q(u)$: sampling distribution used by the objective.
• $w(u)$: weighting, mask, importance weight, or class balance term.
• ${\ell }_{\theta }(u)$: per-unit loss, score, residual, or reward term.
• $\theta$: optimized parameter set.

This makes hidden assumptions visible: sampling policy, masking policy, class imbalance, time weighting, and whether the objective matches the evaluation metric.

Common Formula Families

Family	Canonical Question
Linear map	What are the input/output shapes, axes, rank, and projection matrices?
Attention	How are $Q$, $K$, $V$, mask, softmax, head concatenation, and output projection defined?
Message passing	What node, edge, neighbor, message, aggregation, and update functions are used?
Likelihood	What distribution is modeled, and what observation is conditioned on?
Bayesian inference	What is prior, likelihood, posterior, evidence, MAP estimate, or predictive distribution?
Contrastive loss	What is the anchor, positive, negative set, similarity, and temperature?
Diffusion or flow	What is the state, time variable, noise/score/velocity target, and sampling path?
Energy model	What is the energy, score, force, partition term, sampler, or negative process?
Constraint	What is hard, soft, projected, repaired, filtered, and counted as invalid?
Equivariance	What group acts on the input, and how should the output transform?
Metric	What is averaged, over which examples, after which model-selection rule?

For worked canonical patterns, use Formula pattern catalog.

Explanation Depth

Use Formula explanation ladder before deciding how much math to place in the current note.

Target	Minimum
Korean post	canonical formula with symbols, or operational form for the main method
Concept note	operational form with shapes, indices, distribution, and checks
Paper note	claim contract connecting objective, estimator, benchmark, and evidence
Math note	derivation or assumptions when the formula itself is the topic

Red Flags

• A metric is treated as a loss without explaining the surrogate.
• A loss is assumed to optimize the reported metric without an alignment argument.
• An expectation omits the sampling distribution.
• A constraint is written as a penalty but described as guaranteed validity.
• Invalid generated samples are filtered before reporting without a denominator.
• A graph or set equation does not state permutation behavior.
• A coordinate model does not state translation or rotation handling.
• A benchmark average hides per-target, per-scaffold, or per-source variation.
• A selected checkpoint is reported as if it were an untouched test estimate.

Related

• Math
• Formula pattern catalog
• Formula explanation ladder
• AI paper intake
• Computational Biology paper intake
• Claim routing
• Objective-metric alignment
• Bayesian inference
• Second-order optimization
• Constrained optimization
• Energy-based model
• Benchmark intake
• Paper review workflow