| name | proof-checker |
| description | Rigorous mathematical proof verification and fixing workflow. Reads a LaTeX proof, identifies gaps via AGENTS.md paper architect/reviewer audit, fixes each gap with full derivations, re-reviews, and generates an audit report. Use when user says "检查证明", "verify proof", "proof check", "审证明", "check this proof", or wants rigorous mathematical verification of a theory paper. |
Proof Checker: Rigorous Mathematical Verification & Fixing
Systematically verify a mathematical proof via independent paper architect/reviewer audit, fix identified gaps, re-review until convergence, and generate a detailed audit report with proof-obligation accounting.
Context: $ARGUMENTS
Constants
- MAX_REVIEW_ROUNDS = 3
- REVIEW_ROLE = paper architect/reviewer via the AGENTS.md paper workflow. Do not hard-code a model or backend.
- 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)
Acceptance Gate (objective, replaces subjective scoring)
The proof passes when ALL of the following hold:
- Zero open FATAL or CRITICAL issues
- Every theorem/lemma has: (i) explicit hypotheses, (ii) proof with all interchanges justified, (iii) every application discharges hypotheses in the ledger
- All big-O/Θ/o statements have declared parameter dependence and uniformity scope
- 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 |
| 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
- Locate the proof: Find the main
.tex file(s).
- Read the entire proof: Extract list of all theorems/lemmas/propositions/corollaries/definitions/assumptions.
- Read reference materials: Reference papers, prior results.
- Build a section map: Structured list with line numbers and key claims.
- 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 type is inconsistent across uses.
4. Canonical Quantified Statements
For each theorem/lemma, rewrite the statement with explicit quantifiers, domains, and limit order:
∀K ≥ 3, ∀π ∈ Π_K^{ms,∘} \ E_K, ∃κ_0 > 0 such that ∀κ ∈ (0, κ_0):
h_act^{(K,π)} = Θ(κ^{α_K^act}) [uniform in π on compact subsets]
If you cannot restate a theorem this precisely, mark it UNCLEAR — needs disambiguation.
5. Micro-Claim Inventory
Every nontrivial step becomes a numbered micro-claim in sequent form:
MC-17: Context: [Lemma 3.1, κ < κ_0, Z_κ has bounded moments up to order 2m+2]
⊢ Goal: P̂_0 is positive definite
Rule: monomials linearly independent on support of continuous distribution
Side-conditions: positive density near origin ✓ (by GMM weak convergence)
Each micro-claim has: justification rule name + required conditions + where conditions are proven.
6. Limit-Order Map
Track every asymptotic statement's limit order and uniformity scope:
h_act = Θ(κ^α) [as κ→0, uniform in π on compact subsets of Π_K, for fixed K]
τ_act ~ (b/a)n [as n→∞, for fixed κ,K,π with x_K ≪ 1]
Flag any statement where limit order is ambiguous or uniformity is unclear.
Phase 1: First Paper Architect/Reviewer Audit
Submit the complete proof content with the following mandatory reviewer checklist in the prompt:
Paper architect/reviewer task:
You are performing a rigorous mathematical proof review. For EVERY theorem,
lemma, and proposition, check ALL of the following:
## MANDATORY CHECKS
A. DEFINITIONS: List any symbol whose meaning is ambiguous or changes.
B. HYPOTHESIS DISCHARGE: For each lemma/theorem APPLICATION (not statement),
list each hypothesis and whether it was verified, with location.
C. INEQUALITY AUDIT: For each inequality chain, verify direction, missing
absolute values, missing conditions (convexity, PSD, integrability).
D. INTERCHANGE AUDIT: Flag every limit/derivative/expectation/integral
interchange. State which theorem justifies it (DCT/MCT/Fubini/Leibniz)
and which conditions are verified/missing.
E. PROBABILITY MODE: Track whether claims are a.s./in prob./in expectation/
w.h.p. Ensure transitions are justified.
F. UNIFORMITY & CONSTANTS: For every O(·), o(·), Θ(·), ≲, state whether
it is uniform over all parameters. List hidden parameter dependence.
G. EDGE/DEGENERATE CASES: Attempt to break each key lemma with a 1D,
low-rank, or extreme-parameter construction.
H. DEPENDENCY CONSISTENCY: Detect cycles or forward references to unproven
results.
## OUTPUT FORMAT (per issue)
For each issue found, provide:
- id: sequential number
- status: INVALID / UNJUSTIFIED / UNDERSTATED / OVERSTATED / UNCLEAR
- impact: GLOBAL / LOCAL / COSMETIC
- category: [from taxonomy]
- location: section/equation/line
- statement: what the proof claims
- why_invalid: why this is wrong or unjustified
- counterexample: YES (describe) / NO / CANDIDATE (describe attempt)
- affects: which downstream results break if this is wrong
- minimal_fix: how to fix it
[FULL PROOF CONTENT HERE]
Save the threadId. Parse into structured issue list. Write to PROOF_AUDIT.md.
Phase 1.5: Counterexample Red Team
For each CRITICAL or MAJOR issue, and for every key lemma that introduces:
- a new inequality bound
- an identifiability/uniqueness claim
- a curvature/PSD/strong convexity assertion
- a uniform-in-parameter claim
- a convergence mode upgrade (pointwise → uniform, in prob → w.h.p.)
Systematically attempt to construct counterexamples using:
| Strategy | Description |
|---|
| Dimensional collapse | Set d=1 or 2, K=2, n small |
| Degeneracy | Singular covariance, tiny weight, overlapping means, identical components |
| Extremal distributions | Two-point ±a, bounded non-subGaussian, heavy tails |
| Adversarial parameter scaling | Pick parameters making neglected terms dominate |
| Numeric falsification | Translate lemma to a function, brute-force optimize over small domain |
Rule: Label "counterexample found" ONLY if algebraically verified. Otherwise log as "candidate counterexample — needs verification."
Record all attempts (successful or not) in PROOF_AUDIT.md.
Phase 2: Fix Implementation
For each issue, ordered by severity (FATAL → CRITICAL → MAJOR → MINOR):
Step 2a: Choose fix strategy
For each issue, explicitly choose one of:
- ADD_DERIVATION: Write missing proof steps
- STRENGTHEN_ASSUMPTION: Add conditions to theorem statement
- WEAKEN_CLAIM: Reduce conclusion scope
- ADD_REFERENCE: Cite known result + verify its conditions apply
Log this choice. STRENGTHEN_ASSUMPTION and WEAKEN_CLAIM are scope-changing decisions: propose the exact theorem/claim delta and get explicit user approval before editing the statement, theorem scope, or paper-level claim. Without approval, record the proposed fix in PROOF_AUDIT.md and only apply derivation or exposition fixes that preserve the original claim.
Step 2b: Derive the fix mathematically
- Complete mathematical derivation, not just a claim
- If new proposition/lemma needed, write in full theorem-proof style
Step 2c: Implement in LaTeX
- Edit the
.tex file
- Preserve existing
\label references where possible
Step 2d: Record the fix
### Fix N: [SHORT TITLE]
**Issue**: [id] [CATEGORY] — [description]
**Severity**: FATAL / CRITICAL / MAJOR / MINOR
**Status**: INVALID / UNJUSTIFIED / UNDERSTATED / OVERSTATED
**Impact**: GLOBAL / LOCAL / COSMETIC
**Fix strategy**: ADD_DERIVATION / STRENGTHEN_ASSUMPTION / WEAKEN_CLAIM / ADD_REFERENCE
**Location**: Section X, Lines Y-Z
**BEFORE**: [what the proof originally did]
**WHY WRONG**: [mathematical problem, with counterexample if applicable]
**AFTER**: [what the fix does]
**KEY EQUATION**: [central new equation]
**PROOF OBLIGATIONS ADDED**: [new conditions/lemmas introduced]
**DOWNSTREAM EFFECTS**: [which results now need re-checking]
Step 2e: Compile check
pdflatex -interaction=nonstopmode <file>.tex 2>&1 | grep -E "Error|Warning|undefined"
Phase 3: Paper Architect/Reviewer Re-Review
Use a fresh reviewer context for the next review round; proof-checker keeps each round independent. Request the same mandatory checklist.
Check acceptance gate. If not met, repeat Phases 2-3 (up to MAX_REVIEW_ROUNDS).
Phase 3.5: Global Closure & Independent Verification
Global closure checks
After all fixes, verify the proof as a whole:
- Statement–conclusion match: Does the proof end with EXACTLY what the theorem claims (quantifiers, constants, uniformity)?
- All obligations discharged: Every node in the obligation DAG is proven or explicitly assumed (and the theorem statement includes it).
- Case analysis coverage: Cases partition the domain AND include boundary/degenerate cases.
- Induction correctness (if applicable): Base case, inductive step, correct use of IH, induction measure strictly decreases.
- WLOG reductions: Each "without loss of generality" spawns a micro-claim proving the reduction is lossless.
- No silent assumption strengthening: Any fix that strengthened assumptions has propagated to the main theorem statement.
Independent second review for FATAL/CRITICAL fixes
For any fix that resolved a FATAL or CRITICAL issue, submit the fixed section alone (without showing the previous critique) to a fresh paper architect/reviewer context:
Paper architect/reviewer task:
Blind review of the following proof section. You have NOT seen any prior
review or discussion. Check every step for correctness, hidden assumptions,
illegal interchanges, and counterexamples.
[FIXED SECTION ONLY]
If the blind reviewer finds new issues, re-enter Phase 2.
Regression proof-audit
After fixes, re-run:
- DAG acyclicity check (no new cycles introduced)
- Counterexample suite on all DOWNSTREAM lemmas of modified results
- Assumption-delta report: what became stronger/weaker due to fixes?
Phase 3.9: Unrecoverable Proof Protocol
If acceptance gate is not met after MAX_REVIEW_ROUNDS, output a Proof Unrecoverable Report:
- Minimal set of blocking FATAL/CRITICAL issues that could not be resolved
- Salvage options ranked: (a) weaken claim, (b) strengthen assumptions, (c) add missing lemmas, (d) restructure argument
- Which parts of the proof are likely still reusable
- Recommended next steps for the author
Do NOT silently declare success. The report must be honest.
Phase 4: Audit Report Generation
Generate proof_audit_report.tex with:
- Overview table: All issues with two-axis severity, category, fix strategy, status
- Before/After logic chain: Red (BEFORE) → Green (AFTER) comparison
- For each fix: original proof → why wrong → counterexample (if any) → complete derivation → remaining subtleties
- Proof-obligation diff: What was unverified before, what is verified now
- Summary: Now proven / still assumed / open problems
- Colored boxes: BEFORE (red), AFTER (green), WHY WRONG (orange), KEY INSIGHT (blue), WARNING (yellow)
Compile: pdflatex proof_audit_report.tex && pdflatex proof_audit_report.tex
Phase 5: State Persistence
Write PROOF_CHECK_STATE.json:
{
"status": "completed",
"rounds": 2,
"threadId": "...",
"fatal_fixed": 0,
"critical_fixed": 3,
"major_fixed": 2,
"minor_fixed": 1,
"counterexamples_found": 1,
"counterexample_candidates": 2,
"acceptance_gate": "PASS",
"timestamp": "..."
}
Key Rules
Mathematical rigor
- Never accept a proof step on faith. "Clearly" / "it follows" / "by standard arguments" are red flags — each must spawn a micro-claim.
- Hypothesis discharge: Every time a lemma is APPLIED, verify EACH of its hypotheses at that point. Use the side-condition checklists above.
- Interchange discipline: Every swap of limit/expectation/derivative/integral must cite a theorem (DCT/MCT/Fubini/Leibniz) and verify its conditions with explicit dominating function or integrability proof.
- Uniformity discipline: Every O(·)/Θ(·) must declare what parameters it is uniform over. "O(1)" that secretly depends on d,n,K is a CONSTANT_DEPENDENCE_HIDDEN issue.
- Quantifier discipline: Check ∀/∃ order. "For sufficiently small κ" must specify: does κ₀ depend on K? On π? On d?
- Counterexample-first: Before trying to fix a gap, first try to break it.
- WLOG prohibition: Every "without loss of generality" must have an explicit micro-claim proving the reduction. No free WLOGs.
- No silent assumption strengthening: Any fix that adds conditions must propagate to the theorem statement.
Cross-model protocol
- Role separation: the paper executor reads the proof, formulates questions, and implements fixes; the paper architect/reviewer provides adversarial review.
- Use sufficient reasoning for proof risk: allocate enough review depth for theorem-level correctness without hard-coding a backend or model setting.
- Send full content: Don't summarize — send actual math for line-by-line checking.
- Preserve review continuity only when appropriate: for independent proof rounds, prefer fresh reviewer context; for narrow follow-ups, pass only the revised artifact and targeted question.
Fix quality
- Minimal fixes: Fix exactly what's broken, nothing more.
- Full derivation: Every fix includes complete mathematical argument.
- Explicit scope decisions: Each fix is tagged ADD_DERIVATION / STRENGTHEN_ASSUMPTION / WEAKEN_CLAIM / ADD_REFERENCE. Scope-changing fixes require user approval before implementation.
- Compile after each fix: LaTeX must compile cleanly.
Scope honesty
- Don't overclaim: If a fix makes a result conditional, say so.
- Separate "proven" from "assumed": The audit report has an explicit section for this.
- Log open problems: Issues requiring future work are listed, not hidden.
Output Files
| File | Content | When |
|---|
PROOF_SKELETON.md | Dependency DAG + assumption ledger + micro-claims | Phase 0.5 |
PROOF_AUDIT.md | Cumulative round-by-round audit log | Updated each round |
PROOF_AUDIT.json | Machine-readable submission verdict (see below) | Always emitted |
proof_audit_report.tex/.pdf | Formal before/after report | Phase 4 |
PROOF_CHECK_STATE.json | State for recovery | Phase 5 |
Submission Artifact Emission
This skill always writes PROOF_AUDIT.json at the paper directory
root (i.e. paper/PROOF_AUDIT.json when invoked from /paper-writing
with paper-dir paper/; <your-paper-dir>/PROOF_AUDIT.json when invoked
standalone), regardless of caller or whether the paper contains theorems.
A paper with no \begin{theorem} / \begin{lemma} / \begin{proof} emits
verdict NOT_APPLICABLE; silent skip is forbidden. paper-writing
Phase 6 relies on this artifact existing at <paper-dir>/PROOF_AUDIT.json.
The artifact conforms to the schema in ../shared-references/assurance-contract.md:
{
"audit_skill": "proof-checker",
"verdict": "PASS | WARN | FAIL | NOT_APPLICABLE | BLOCKED | ERROR",
"reason_code": "all_proofs_complete | minor_gaps | critical_gap | no_theorems | ...",
"summary": "One-line human-readable verdict summary.",
"audited_input_hashes": {
"main.tex": "sha256:...",
"sections/4.theory.tex": "sha256:..."
},
"trace_path": "paper/review-traces/proof-checker/<date>_run<NN>/",
"review_trace": "<fresh paper architect/reviewer trace id>",
"review_role": "paper architect/reviewer",
"generated_at": "<UTC ISO-8601>",
"details": {
"theorems_audited": <int>,
"issues": [ { "id": "T1-H3", "severity": "FATAL|CRITICAL|MAJOR|MINOR",
"category": "quantifier|domination|...",
"location": "sections/4.theory.tex:L182",
"note": "..." }, ... ]
}
}
audited_input_hashes scope
Hash the declared input set actually reviewed — the theorem-bearing
.tex files passed into this invocation — not a repo-wide union and not
the reviewer's self-reported opened subset. The local artifact gate rehashes
these entries; any mismatch flags STALE.
Path convention: keys are
paths relative to the paper directory (no paper/ prefix — the
local artifact gate resolves relative to the paper dir; prefixing produces
paper/paper/... and false-fails as STALE). Use absolute paths for
files outside the paper dir.
Verdict decision table
| Input state | Verdict | reason_code example |
|---|
| No theorems / lemmas / proofs in paper | NOT_APPLICABLE | no_theorems |
| Theorems present but referenced files unreadable | BLOCKED | source_unreadable |
| All proof obligations discharged, no gaps | PASS | all_proofs_complete |
| Only MINOR issues (notation / exposition) | WARN | minor_gaps |
| Any FATAL or CRITICAL issue (logic gap, wrong claim) | FAIL | critical_gap |
| Reviewer invocation failed (network / malformed) | ERROR | reviewer_error |
MAJOR issues alone map to WARN or FAIL at the reviewer's discretion and
must carry an explicit justification in summary + details.issues.
Thread independence
Every invocation uses a fresh reviewer context. Never continue a prior
proof-checker run. Do not accept prior audit outputs
(PAPER_CLAIM_AUDIT, CITATION_AUDIT, EXPERIMENT_LOG) as input — the fresh
thread preserves reviewer independence per
../shared-references/reviewer-independence.md.
This skill never blocks by itself; paper-writing Phase 6 plus the
local artifact gate decide whether the verdict blocks finalization based on the
assurance level.
Example Invocations
/proof-checker "neurips_2025.tex"
/proof-checker "check the GMM generalization proof, focus on dimension dependence"
/proof-checker "verify proof in paper.tex — difficulty: nightmare"
