Theory Modeling

Stage-Scoped References

Load per stage, not all at once.

The four gates below are the creation-time correctness floor — walked at every implementation dispatch, including rough exploratory work. references/integration.md is the readability layer, loaded when the document needs polishing for a human reader.

The Iron Law

NO MANIPULATION WITHOUT DEFINED OBJECTS, INTERPRETABLE ASSUMPTIONS, AND STATED INTUITION

Every symbol has a meaning. Every assumption has a plain-language interpretation a researcher can defend. Every non-trivial move has a one-sentence reason.

Symbol with no stated meaning, assumption with no economic reading, derivation step invoked mechanically: back up and write the missing meaning, interpretation, or reason first.

Non-default constraints (traps the gates don't catch):

• Restrictions live on primitives. Do not silently move one onto an endogenous variable because the latter is shorter to write.
• Notation is fixed. Do not rename objects mid-derivation without an explicit old-to-new mapping.
• Restrictions are stated up front. Do not back-fill a key restriction (e.g., "assume interior solution") after the algebra is already written.

The Four Gates

Ordered by the reader's trust chain: Objects & Notation → Assumptions → Derivations → Verification & Rendering. Each gate has an artifact the implementer produces and a checklist walked while producing it. The gates are concurrent — every modeling step exercises all four, with documentation built into the artifact.

[BLOCKING] items must be fixed for APPROVE; [ADVISORY] items are recorded and do not block. Verdict mechanics live in superRA:review-task.

Falsification tests (per ledger entry in Gates 1 and 2)

Reviewer diagnostics run against a slot suspected of not pulling its weight — text that looks like a justification but would survive any small change to the object it claims to justify.

• Substitution test. Read the entry's "What the name carries" / "What this assumption carries" slot, then mentally swap in a hypothetical sibling — a different symbol $\mathbf{z}_q$ from the same proof, or a different assumption on the same primitive. Re-read the slot under the swap. Still true for the substituted object → generic, pins nothing down → BLOCKING. The slot must contain something false of any other object: a specific sign meaning, a specific structural role, a named scalar cited at a specific site.

Worked example. "Plays a role in the proof of Lemma 3.1" stays true under a swap to $\mathbf{z}q$ — vacuous → BLOCKING. "Sign-bearing scalar — positive iff the comparative static of price w.r.t. dividend $k$ is positive, cited at eq. (12)" goes false under the swap unless $\mathbf{z}q$ carries the same sign meaning at the same site → passes.

• Proof-deletion test (Meaning slot). Cover the surrounding proof and re-read only the Meaning slot. Can it still tell a reader what the object is — type, denotation in already-introduced terms, construction? A slot that evaporates without the proof was stating what the symbol is used for, not what it is → BLOCKING.

Worked example. "Used to verify $hk = mD \beta{E,k}$" has nothing left once the proof is gone — usage, not meaning → BLOCKING. "Column-$k$ object of the loading matrix $H \in \mathbb{R}^{N\times K}$, with $\mathbf{c}k := H^\top e_k$" still carries the type ($K$-vector), the parent object ($H$), and the construction → passes.

Gate 1 — Objects & Notation

Pin down the objects and their names before manipulating them.

At write-up, review the notation you reasoned with and replace each shorthand rather than justify keeping it — substitute its expression inline, or map it to an existing symbol. Shorthand left standing in the output is a defect.

A new symbol earns its place two ways: standard notation an economist writes without prompting ($r$, $w$, $\beta$), defined at first use; or writing its expression out each time would lose real meaning — the one-site test in the ledger's What the name carries slot, applied to every new output symbol. A symbol meeting neither is inlined — introducing notation that fails this bar counts against the work like an algebra error.

Artifact: notation ledger table at the end of the task's ## Results. One row per object. An indexed family ($x_k$ for $k=1,\dots,K$) is one row; distinct symbols never share a row. Tasks introducing no new symbols record "None."

Symbols in the governing ancestor's canonical Notation Conventions table are reused with their canonical meaning and need no row.

Writing the Meaning slot. Three components, all required:

1. Type / space. Scalar in $(0,1)$, $K$-vector in $\mathbb{R}^K$, $N\times N$ symmetric matrix, function $X \to Y$, random variable on $(\Omega,\mathcal{F},\mathbb{P})$ — dimension and domain wherever applicable.
2. Denotation in the model's vocabulary. What the object represents in already-introduced terms: "the coefficient on $X_t$ in eq. (12)", "the Lagrange multiplier on the resource constraint", "row $k$ of the dividend-loading matrix $H$". References only objects already defined.
3. Origin if derived. Construction from prior symbols, e.g. $\mathbf{c}k := H^\top ek$ — the construction, not a step that uses it later.

The Proof-deletion test diagnoses whether the slot satisfies this recipe.

Anti-patterns for Meaning:

Checklist:

• [BLOCKING] Ledger entry written before the symbol appears in proof text. Other documentation runs concurrently with the math; symbol introduction does not — post-hoc entries reverse-engineer justifications for choices already made.
• [BLOCKING] Notation is interpretable or genuinely conventional. Placeholder labels (A/B/C/D, T1/T2, eq1, var2) are not acceptable; conventional notation (r for an interest rate, w for a wage) is, defined at first use.
• [BLOCKING] Meaning slot satisfies the type / denotation / origin recipe and survives the proof-deletion test.
• [BLOCKING] Reuse-sites slot cites every additional appearance with line or equation refs, or states "none." Claims of reuse without refs are REVISE.
• [BLOCKING] Inline-alternative slot shows the actual substituted expression, not a description of it ("would be unwieldy" is not an inline alternative).
• [BLOCKING] One-site symbols (Reuse sites: none) justify via concrete content in "What the name carries" — sign meaning, structural role, named scalar cited elsewhere. One-time abbreviation fails regardless of length.
• [BLOCKING] Every new non-standard symbol in the shipped output clears the bar — standard notation, or meaning that inlining would lose. A symbol clearing neither is inlined or mapped to an existing one. "Nearest existing symbol considered: none in scope" is falsifiable: a candidate in the canonical Notation Conventions table, the active lemma, or an upstream derivation is REVISE.
• [BLOCKING] One entry per object. Indexed families ($x_k$ for $k=1,\dots,K$) count as one; bundling distinct objects under a shared justification is REVISE.
• [BLOCKING] Domains, units, and sign restrictions stated whenever they matter for the algebra, comparative statics, or numerical checks.
• [ADVISORY] Multiple reasonable notation choices: prefer the one matching the literature or existing project docs; note the mapping on any deviation.

Gate 2 — Assumptions

Assumptions carry the economic content. Each is attached to a primitive object, readable as economics, and no weaker than it needs to be — one interpretable primitive over a scattering of weak technical restrictions.

Artifact: assumption ledger table at the end of the task's ## Results. One row per assumption. Tasks introducing no new assumptions record "None."

Checklist:

• [BLOCKING] Assumption attached to a primitive (preferences, technology, endowments, information, timing, distributions, parameter domains, boundary conditions, normalizations). Assumptions stated as desired properties of endogenous objects are REVISE unless those properties are later proved.
• [BLOCKING] Interpretation slot carries a one-sentence plain-language reading a researcher can defend ("risk aversion bounded so the value function is finite"). Math-only restrictions with no economic interpretation are REVISE.
• [BLOCKING] "Without this assumption" slot names a specific conclusion that changes. Vague claims ("the result would be weaker") are REVISE.
• [BLOCKING] "What this assumption carries" slot names the additional restriction on the named primitive that existing assumptions do not impose. Generic claims ("makes the proof cleaner") are REVISE.
• [BLOCKING] Multiple scattered assumptions replaceable by one stronger primitive assumption with a cleaner interpretation: prefer the synthesis, record the trade in "Why state it this way." Reviewer applies a judgement margin — flag only a clearly cleaner synthesis.

Gate 3 — Derivations

Derivations must be auditable — a correct result that cannot be checked is not an acceptable handoff artifact. Every non-trivial move needs both the technical rule and a reason for invoking it here.

Artifact: the proof / derivation body in the task's ## Results.

Checklist:

• [BLOCKING] The active solution concept named before derivation starts: planner problem, competitive equilibrium, recursive equilibrium, steady state, fixed point, or other.
• [BLOCKING] Top-level proof goal stated in one sentence before the first displayed equation. Derivations opening with algebra and no stated target are REVISE. (Reader-facing recursive signposting — sub-arguments at every level, transition prose — is references/integration.md Section A.)
• [BLOCKING] A derivation step depending on a previously established equation, lemma, or proposition cites it by name or equation number. Asserted equations with no path to a named source are REVISE. (Cite-with-operative-form-recall for distant sources, and prose-level precision — math symbol vs. English description, equation reference vs. positional pointer — are Section B.)
• [BLOCKING] One logical algebraic move per displayed step. No collapsing multiple substitutions, cancellations, and sign changes into "therefore". (Section B's half-page mask test is the integration-stage detection layer.)
• [BLOCKING] Each non-obvious step names the rule it invokes — substitute a constraint, differentiate, apply the envelope theorem, impose market clearing, linearize — and a one-sentence reason for invoking it. A rule-label without a reason is REVISE.
• [BLOCKING] Results depending on case splits or domains (interior vs corner, positive vs negative branch, existence/uniqueness conditions): the active case stated, excluded cases checked or explicitly deferred.
• [BLOCKING] Comparative statics state what is held fixed, which object moves, and what sign or ranking is claimed.
• [BLOCKING] Reused symbols keep the same meaning throughout the task; notation changes carry an explicit old-to-new mapping.
• [BLOCKING] Claims of existence, uniqueness, monotonicity, or concavity carry a stated argument, not assertion by inspection.
• [BLOCKING] New equations, named statements (lemmas, propositions, definitions, corollaries), and derivation steps face the same necessity lens as ledger entries: removal leaving the reasoning intact means remove it. They need no individual ledger entries — the one-move-per-step and reason-per-move items above enforce the lens.

Gate 4 — Verification & Rendering

A derivation is complete only after surviving at least one independent check and reading cleanly for a human audience.

Artifact: verification work + rendered output. Report symbolic checks in the human-readable markdown / LaTeX shipping in ## Results. Run numerical checks when needed to verify algebra; keep their parameters, evaluations, and outcomes internal.

Checklist:

• [BLOCKING] Every headline symbolic result checked against at least one independent verification mode: substitute back into the original conditions, test a limiting or special case, or evaluate a simple numerical example.
• [BLOCKING] Numerical verification, when needed, uses explicit parameter values and a pass condition: residual near zero, sign, monotonicity, feasibility, branch selection, or fixed-point convergence. It is not reported.
• [BLOCKING] Special and limiting cases interpreted economically, not just numerically confirmed ("at $\beta \to 0$ the policy reduces to the myopic rule, matching the one-period benchmark"), and compared against intuition and any hypotheses stated in the task objective; divergences flagged before proceeding.
• [BLOCKING] Results checked back against the assumption map. A step quietly needing a stronger sign, domain, or regularity restriction than the map states: update the map before using the result.
• [BLOCKING] With code, CAS output, or a solver, the human-readable result matches the computed object exactly. No manual transcription drift.
• [ADVISORY] Numerically delicate objects: verify more than one parameter set, or a small perturbation around the baseline.

Implementation standards

• [BLOCKING] Evidence showing an extra lemma, case split, derivation step, or verification pass is required to trust the result: add it inside the current task and rewrite the step text to match.
• [BLOCKING] Solver scripts, symbolic code, and model notes organized so a reviewer can trace the chain from primitives and assumptions to the reported result.
• [BLOCKING] Major modeling decisions (normalization, timing, equilibrium selection, parameter baseline, approximation point) carry a markdown explanation or nearby comment.

Documentation and handoff

Beyond the per-symbol / per-assumption ledgers:

• [BLOCKING] The canonical Notation Conventions table is user-gated — implementers do NOT inline-edit it. A symbol is promoted from the task-level ledger only when the user confirms it should become canonical project-wide; until then the ledger entry is that task's source of truth.
• [BLOCKING] Definitions, assumptions, and the reason for major derivation choices written alongside the math or code, not left only in chat.
• [BLOCKING] Human-readable equations, tables, and figures routed through superRA:communicate.
• [BLOCKING] Rendered math, prose, and supporting code use consistent notation for the same object.

Common Rationalizations

Excuses the gate checklists alone do not catch.

See also

• superRA:communicate — format discipline for equations, tables, figures, and LaTeX in Markdown.