Ecta theory model

Theorem and Proof Strategy (ecta-theory-model)

When to trigger

• The central result is stated informally or its conditions are scattered through the text
• The proof has gaps: "it is easy to see," "by a standard argument," or a missing step
• Regularity conditions are invoked mid-proof but never stated as assumptions
• You are unsure whether the result is as general as claimed, or whether the proof secretly uses more

This is the heart of an Econometrica paper. The product is a complete, correct proof of a general, cleanly stated theorem. Econometrica is the field's theorem-proof journal: the referee pool (routed by the handling co-editor) reads proofs line by line, and a single genuine gap can sink an otherwise strong paper — this is the most common rejection cause here, in contrast to applied siblings (AER / QJE / JPE / REStud) where a credible empirical narrative can carry the day. Plan around the 45-page main-text limit (incl. references and appendices): the body conveys the theorem, its interpretation, and the architecture of the proof, while the complete, formal proofs live in the Supplemental Material (the ≤25-page Supplemental Appendix or the unrestricted online Supplemental Material), which does not count against the body's page budget. "Proof omitted to save space" is not acceptable — the proof must exist somewhere complete.

Structure: definitions → assumptions → theorems → proofs

Organize the formal content in this order, with consistent theorem numbering:

1. Definitions — every object used in the theorem (spaces, operators, parameter set, solution concept) defined before it appears.
2. Assumptions — numbered, each stated once, each used. Group standing assumptions separately from those invoked only for specific results.
3. Theorems / Propositions / Lemmas — the headline result first or clearly flagged; supporting lemmas factored out so the main proof reads cleanly.
4. Proofs — in-text for short proofs; the full proof in the Supplemental Material (Econometrica's term for the online appendix; ≤25-page Supplemental Appendix or unrestricted online Supplemental Material) when length would break the exposition or exceed the 45-page body cap. Either way the proof must be complete and self-contained.

Stating the theorem

• State the theorem so it is self-contained: a reader should be able to verify the hypotheses and conclusion without hunting through the text.
• Reference assumptions by number in the hypothesis ("Under Assumptions 1–3, ...").
• Make the conclusion a precise mathematical statement (existence, the limiting law, the representation, the bound), not a verbal summary.
• If the result is sharp (assumptions are necessary, the rate is optimal), say so and prove it or cite the matching lower bound.

Proof strategy toolkit

Pick the architecture before writing line-by-line:

Result type	Typical machinery
Existence of equilibrium / solution	Fixed-point (Brouwer / Kakutani / Banach / Schauder), Berge maximum theorem
Representation theorem	Separation / Hahn–Banach, mixture-space, biseparable arguments
Consistency	Uniform law of large numbers, argmax / M-estimation continuity, identification + compactness
Limiting distribution	CLT (for arrays / dependent data), delta method, empirical-process / Donsker arguments, stochastic equicontinuity
Uniqueness / comparative statics	Contraction, monotone-comparative-statics (lattice / single-crossing), index theory
Bounds / minimax	Le Cam two-point / Fano, coupling

Write the proof sketch first (the architecture and the one or two hard steps), then expand every step. Flag the genuinely novel step — referees want to see where the work is.

Generality audit

• For each assumption: is it used? Locate the line of the proof that needs it. Unused assumptions must go.
• Is each assumption as weak as the proof allows? If the proof only needs continuity, do not assume differentiability.
• Does the result hold on the largest natural class, or is it tied to a parametric example? If the latter, either generalize or reframe honestly as an example.
• Are there edge cases (boundary of the parameter space, degenerate distributions, ties, non-uniqueness) the statement quietly excludes? State the exclusions.

Checklist

• Every object in the theorem is defined before use
• Assumptions numbered; each is used at an identifiable step; none redundant
• Theorem statement is self-contained and precisely mathematical
• Proof complete — no "easy to see," no skipped step; full version in Supplemental Material if long
• The novel / hard step is flagged and given full detail
• Regularity conditions invoked in the proof all appear as stated assumptions
• Generality audited: assumptions minimal; result not secretly an example
• Sharpness addressed (necessity / optimal rate) where claimed

Anti-patterns

• "It is easy to see that ..." standing in for the load-bearing step
• "By standard arguments" where the argument is not standard in this setting
• A regularity condition that appears only inside the proof, never in the assumptions
• Differentiability / compactness assumed for convenience but never genuinely needed
• A theorem that holds only for one functional form, presented as general
• Existence proven, uniqueness silently assumed
• Citing a CLT / fixed-point theorem whose hypotheses your setting does not actually satisfy

Output format

【Central theorem】... (one-line statement + assumption numbers)
【Proof architecture】fixed-point / ULLN+M-estimation / empirical process / ...
【Hard step】...
【Assumptions audit】all used? [yes/no — list unused]; minimal? [...]
【Generality】largest class: ...; excluded edge cases: ...
【Full proof location】in-text / Supplemental Material §...
【Next step】ecta-robustness