proof-checker

Proof Checker: Rigorous Mathematical Verification & Fixing

Systematically verify a mathematical proof via cross-model adversarial review, fix identified gaps, re-review until convergence, and generate a detailed audit report with proof-obligation accounting.

Context: $ARGUMENTS

Constants

• MAX_REVIEW_ROUNDS = 3
• REVIEWER_MODEL = gpt-5.5 via Codex reviewer agent, reasoning effort always xhigh
• REVIEWER_BACKEND = codex — Default: Codex reviewer agent (spawn_agent, xhigh). Override with — reviewer: oracle-pro for GPT-5.4 Pro via Oracle MCP. See shared-references/reviewer-routing.md.
• AUDIT_DOC: PROOF_AUDIT.md at the paper directory root, alongside main.tex (cumulative log; when invoked via /paper-writing, this is paper/PROOF_AUDIT.md)
• REPORT_TEX: proof_audit_report.tex (formal before/after PDF)
• STATE_FILE: PROOF_CHECK_STATE.json (for recovery)
• SKELETON_DOC: PROOF_SKELETON.md (micro-claim inventory)
• RENDER_HTML = true — When true (default), auto-render PROOF_AUDIT.md to HTML at workflow end via /render-html. Uses full review gate (audit-class, math-heavy — render-fidelity check protects against MathJax breakage). Set false to skip, or pass — render html: false. Non-blocking: failures don't invalidate the proof audit.

Acceptance Gate (objective, replaces subjective scoring)

The proof passes when ALL of the following hold:

1. Zero open FATAL or CRITICAL issues
2. Every theorem/lemma has: (i) explicit hypotheses, (ii) proof with all interchanges justified, (iii) every application discharges hypotheses in the ledger
3. All big-O/Θ/o statements have declared parameter dependence and uniformity scope
4. Counterexample pass executed on all key lemmas (log candidates even if none found)

Issue Taxonomy (20 categories, 4 groups)

Group A: Logic & Proof Structure

| Category | Description | Example | |----------|-------------|---------| | UNJUSTIFIED_ASSERTION | Claim stated without proof or reference | "The Hessian splits into Gram blocks" | | UNPROVEN_SUBCLAIM | "Clearly" / "it follows" hides a nontrivial lemma | "By symmetry, the cross-terms vanish" without checking | | QUANTIFIER_ERROR | Wrong order ∀/∃, missing "for sufficiently small κ" | "For all π, there exists ε" vs "there exists ε for all π" | | IMPLICATION_REVERSAL | Uses (A⇒B) as (B⇒A), or claims equivalence with only one direction | | | CASE_INCOMPLETE | Misses boundary/degenerate cases | Singular covariance, zero weight, non-unique argmin | | CIRCULAR_DEPENDENCY | Lemma uses theorem that depends on it | | | LOGICAL_GAP | A step is not justified by what precedes it | B=Θ(1) → β_K=0 without analyzing W |

Group B: Analysis & Measure Theory

| Category | Description | Example | |----------|-------------|---------| | ILLEGAL_INTERCHANGE | Swaps limit/expectation/derivative/integral without DCT/MCT/Fubini | Differentiating under E without domination | | NONUNIFORM_CONVERGENCE | Pointwise convergence used as uniform | sup and limit swapped | | MISSING_DOMINATION | DCT cited but no dominating function given | | | INTEGRABILITY_GAP | Uses E|X|^p without proving/assuming finite moments | | | REGULARITY_GAP | Differentiability/Lipschitz/convexity used but not established | | | STOCHASTIC_MODE_CONFUSION | Mixes a.s./in prob./in L²/in expectation | |

Group C: Model & Parameter Tracking

| Category | Description | Example | |----------|-------------|---------| | MISSING_DERIVATION | A quantity is used but never derived from the model | Risk functional with undefined B, W | | HIDDEN_ASSUMPTION | Proof silently uses a condition not in the theorem | Gaussianity assumed but not stated | | INSUFFICIENT_ASSUMPTION | Hypotheses too weak for proof (counterexample exists) | Moment conditions admitting 2-point distributions | | DIMENSION_TRACKING | Parameter dependence (d, n, K, ...) not explicit | d enters only through κ | | NORMALIZATION_MISMATCH | Coordinate/scaling conventions inconsistent | Rescaled vs raw coordinates | | CONSTANT_DEPENDENCE_HIDDEN | "C" depends on d,n,K but treated as universal | |

Group D: Scope & Claims

| Category | Description | Example | |----------|-------------|---------| | SCOPE_OVERCLAIM | Conclusion stated more broadly than proof supports | "β_K=0" with only generic overlap | | REFERENCE_MISMATCH | Cited theorem's hypotheses not verified at point of use | |

Two-Axis Severity System

Axis A — Proof Status (what is wrong)

| Status | Meaning | |--------|---------| | INVALID | Statement false as written (counterexample exists or contradiction) | | UNJUSTIFIED | Could be true, but current proof does not establish it | | UNDERSTATED | True only after strengthening assumptions | | OVERSTATED | True only after weakening conclusion / adding qualifiers | | UNCLEAR | Ambiguous notation / definition drift (not wrong per se) |

Axis B — Impact (how much breaks)

| Impact | Meaning | |--------|---------| | GLOBAL | Breaks main theorem or core dependency chain | | LOCAL | Affects a side result but not the main theorem | | COSMETIC | Exposition only |

Severity Labels (derived)

| Label | Definition | |-------|------------| | FATAL | INVALID + GLOBAL | | CRITICAL | (INVALID + LOCAL) or (UNJUSTIFIED + GLOBAL) | | MAJOR | (UNJUSTIFIED + LOCAL) or (UNDERSTATED/OVERSTATED + GLOBAL) | | MINOR | Clarity / notation / dimension bookkeeping that doesn't change claims |

Side-Condition Checklists for Common Theorems

When the proof invokes any of the following, require explicit verification of ALL listed conditions:

| Theorem | Required Conditions | |---------|-------------------| | DCT (Dominated Convergence) | Pointwise a.e. convergence + integrable dominating function | | MCT (Monotone Convergence) | Monotone increasing + non-negative | | Fubini/Tonelli | Product measurability + integrability (Fubini) or non-negative (Tonelli) | | Leibniz integral rule | Continuity of integrand + dominating function for derivative | | Implicit Function Theorem | Continuous differentiability + non-singular Jacobian | | Taylor with remainder | Sufficient differentiability + remainder form (Lagrange/integral) | | Jensen's inequality | Convexity of function + integrability | | Cauchy-Schwarz | Correct inner product space + integrability of both factors | | Weyl/Davis-Kahan | Symmetry/Hermiticity + perturbation bound conditions | | Analytic continuation | Domain connectivity + identity theorem conditions | | WLOG reduction | Invariance under claimed symmetry + reduction is reversible |

Workflow

Phase 0: Preparation

1. Locate the proof: Find the main .tex file(s).
2. Read the entire proof: Extract list of all theorems/lemmas/propositions/corollaries/definitions/assumptions.
3. Read reference materials: Reference papers, prior results.
4. Build a section map: Structured list with line numbers and key claims.
5. Identify the main theorem: Central result, assumptions, claims.

Phase 0.5: Proof-Obligation Ledger

Build formal accounting artifacts. Save to PROOF_SKELETON.md:

1. Dependency DAG

Nodes = Definitions / Assumptions / Lemmas / Theorems. Edges = "uses". Detect cycles (including semantic circularity where Lemma A uses a corollary that quietly depends on A).

2. Assumption Ledger

For each theorem/lemma, list every hypothesis with WHERE each is verified (or mark "UNVERIFIED"). Track usage-minimal assumption sets — which assumptions were actually used vs merely stated.

3. Typed Symbol Table

Each symbol must have a type signature:

κ : scalar ∈ (0,1), depends on (d, α_t, Σ, μ)
u* : vector ∈ ℝ^d, u* = C^{-1}m
B^even : matrix ∈ ℝ^{(L+1)×(L+1)}, symmetric PSD
Ψ_v : function ℝ → ℝ, analytic in (ζ,κ), parity determined by v

Flag any symbol whose meaning changes or whose typ