Proof Checker

Proof Checker: Rigorous Mathematical Verification & Fixing

Codex assurance: base Codex proof judgments are review_independence: same-family and acceptance_status: provisional. Deterministic compilation/algebra checks may be accepted; a semantic proof acceptance requires a cross-family overlay. Reviewer failure emits BLOCKED.

Systematically verify a mathematical proof via fresh-agent 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-6-astra via Codex reviewer agent, reasoning effort ultra for this deep-audit skill (capability fallback never below xhigh)
• REVIEWER_BACKEND = codex — Default: Codex reviewer agent (spawn_agent, ultra — deep-audit tier). Override with — reviewer: oracle-pro for GPT-5.5 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

Group B: Analysis & Measure Theory

Group C: Model & Parameter Tracking

Group D: Scope & Claims

Two-Axis Severity System

Axis A — Proof Status (what is wrong)

Axis B — Impact (how much breaks)

Severity Labels (derived)

Side-Condition Checklists for Common Theorems

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

Workflow

Proof-obligation fan-out

Independent sections/theorems may be extracted by fresh read-only spawn_agent shards, with a sequential fresh-context fallback. Each shard returns {"shard_id": ..., "entries": [{"payload": ..., "dedup_key": "<theorem-or-obligation-id>"}]} and must not declare the proof valid. The parent mechanically merges obligations; the fresh Codex review that evaluates them records review_independence: same-family and acceptance_status: provisional. See fan-out-pattern.md.

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.

Optional Lean verification

When the user requests Lean or a specific obligation warrants formal checking, delegate that obligation and its original hypotheses to /lean-formalize. Record the checked scope, exported declaration, statement-alignment evidence, build output and transitive-axiom audit, then resume this audit. A proved sublemma closes only the corresponding obligation. Keep the complete proof audit, acceptance decision and canonical wiki artifacts here; the nested Lean task returns evidence without starting another proof-checker run. Routine proof checks do not require Lean. Reviewer agreement and unsuccessful counterexample searches do not discharge a proof obligation.

Phase 1: First Review (Codex GPT-6-Astra ultra)

Submit the complete proof content with the following mandatory reviewer checklist in the prompt:

spawn_agent:
  model: gpt-6-astra
  reasoning_effort: ultra
  message: |
    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 reviewer agent_id. 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:

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 — it is a scope-changing decision when it alters theorem statements.

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: Re-Review (Codex GPT-6-Astra ultra)

Launch a fresh reviewer agent for the next review round. Do not use send_input here; 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 Codex thread:

spawn_agent:
  model: gpt-6-astra
  reasoning_effort: ultra
  message: |
    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.