SKILL

State the problem and assumptions up front, using explicit hypotheses and named conventions.
Use a sparse, formal tone: definitions and propositions first, then proofs.
Keep proofs linear: each step should cite the exact definition or lemma used.
Use minimal narrative; prefer symbolic precision and structured claims.
Separate “interpretation” from “proof” to avoid mixing explanation into the argument.
Apply math-symbol-discipline to ensure domain-appropriate notation.

Proof voice (dense, formal)

High density of definitions, lemmas, and constructions.
Hypotheses and notation are enumerated explicitly before use (“Fix”, “Let”, “Assume”).
Proofs are short, direct, and avoid rhetorical flourish.
Use named structures and cross-references instead of restating informal intuition.
Prefer “Construction”, “Remark”, “Warning”, and “Example” blocks for auxiliary material.

Use short, declarative sentences.
Explain what a definition is for, not just what it is.
Avoid metaphor; use operational descriptions and measurable quantities.
When introducing a model, name the inputs/outputs and what is being optimized or preserved.

“Assume $T$ is a small site and $H_{t}$ is finite for all $t \in T$ .”
“Define $U G$ as the least closure operator satisfying (i)-(iii).”
“Proof. By Definition 2.1 and Lemma 2.3, we obtain…”
“Construction. For each $t$ define a fiber $H_{t}$ and reindex along $T$ .”
“Interpretation. This definition isolates the minimal objects forced by gluing.”
“Interpretation. The bound quantifies how much structure is lost under compression.”