proof-checker

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:

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
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 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:

1. Minimal set of blocking FATAL/CRITICAL issues that could not be resolved
2. Salvage options ranked: (a) weaken claim, (b) strengthen assumptions, (c) add missing lemmas, (d) restructure argument
3. Which parts of the proof are likely still reusable
4. 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:

1. Overview table: All issues with two-axis severity, category, fix strategy, status
2. Before/After logic chain: Red (BEFORE) → Green (AFTER) comparison
3. For each fix: original proof → why wrong → counterexample (if any) → complete derivation → remaining subtleties
4. Proof-obligation diff: What was unverified before, what is verified now
5. Summary: Now proven / still assumed / open problems
6. 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"