proof-checker

name	proof-checker
description	Rigorous mathematical proof verification and fixing workflow. Reads a LaTeX proof, identifies gaps via cross-model review (external reviewer backend, xhigh reasoning), 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.
argument-hint	[path-to-tex-file or proof-description] [--deep-fix] [--restatement-check]
allowed-tools	Bash(*), Read, Grep, Glob, Write, Edit, Agent, mcp__codex__codex, mcp__codex__codex-reply, mcp__manual_review__review, mcp__manual_review__review_reply

Proof Checker: Rigorous Mathematical Verification & Fixing

🔒 Do not wrap this skill in /loop, /schedule, or CronCreate. It is verdict-bearing — it judges proof validity across rounds, threading the reviewer's memory from Phase 1 → Phase 3 via codex-reply so the reviewer can check whether a fix actually closed the gap it flagged. An external timer re-enters from the top each tick, starting a fresh thread and losing that memory. Schedule the external wait that precedes it, not the verdict. See shared-references/external-cadence.md.

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 — Default model for the Codex backend, reasoning effort always xhigh. Manual backend uses whatever model the user chooses, but it must be a non-Claude model — the executor is Claude, so routing the proof review into any Claude product makes Claude judge Claude and voids the cross-model invariant (see shared-references/reviewer-routing.md).
• REVIEWER_BACKEND = codex — Default: Codex MCP (xhigh). Override with — reviewer: oracle-pro for Oracle MCP, or — reviewer: manual for Manual Review MCP. If manual-review MCP is unavailable, stop and print the install command; do not fall back to Codex. See shared-references/reviewer-routing.md.

Reviewer Calling Convention

When calling the reviewer, branch on REVIEWER_BACKEND:

If REVIEWER_BACKEND = codex: Use mcp__codex__codex for new review threads. Use mcp__codex__codex-reply for follow-up rounds (reuse threadId).

If REVIEWER_BACKEND = manual: Use mcp__manual_review__review for new review threads with: prompt: [exact same prompt that would go to Codex] config: {"model_reasoning_effort": "xhigh"} Save the returned threadId. Use mcp__manual_review__review_reply for follow-up rounds with: threadId: [saved manual-review threadId] prompt: [follow-up prompt] config: {"model_reasoning_effort": "xhigh"}

Prompt fidelity: the manual prompt must be exactly the same text that Codex would receive. Review tracing applies equally to both backends.

• 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 Codex review gate (audit-class artifact — math-heavy content; render-fidelity check protects against MathJax breakage and matches the skill's cross-model audit invariant). Set false to skip, or pass — render html: false.

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

Fan-out (Tier-aware) — build the ledger in parallel; never judge in parallel. For a large multi-theorem paper, ledger construction is breadth over independent sections. Tier 1 (Workflow): spawn one Claude subagent per section/theorem to extract that unit's symbols, assumptions, micro-claims, and local quantified statements, each returning a structured ledger fragment. Tier 2: the same subagents via the Agent tool. Tier 3: walk the sections sequentially. This follows shared-references/fan-out-pattern.md.

Two hard rules:

1. The shards EXTRACT, they do not ADJUDICATE. Building the ledger (inventorying obligations, typing symbols, restating with explicit quantifiers) is structural extraction. Whether a proof step is valid — whether an obligation is actually discharged — is a Type-B correctness verdict reserved for the cross-model jury in Phase 1 / Phase 3 (codex or manual, xhigh). A Claude shard MUST NOT mark a micro-claim "proved" or "sound"; it only records the obligation and where the paper claims to discharge it. See acceptance-gate.md — the loop may self-verify that the ledger is complete, never that the proofs are correct.
   • This governs the ledger spec wording below. Where the artifacts say "WHERE each is verified", "or mark UNVERIFIED", or "where conditions are proven", a shard records a location pointer (file:line the paper claims discharge) — never its own judgment that the discharge is mathematically valid. A shard's UNVERIFIED means "the paper cites no discharge location", NOT "the shard checked the math and it fails". Soundness is the jury's verdict, not the shard's.

Shard output (extraction schema, per fan-out-pattern.md): each shard returns {shard_id: "<section/theorem id>", entries: [...]} — the typed ledger items (symbols, assumptions, micro-claims, canonical statements, limit-order facts) for that unit, each carrying its canonical id (e.g. MC-17, the symbol name) as dedup_key. Never prose-only; never a validity verdict field. 2. Global artifacts are a barrier, computed on the merged ledger, not per-shard. The Dependency DAG and its cycle detection (incl. semantic circularity), and cross-section symbol-type consistency, require the whole paper in view. Merge all shard fragments first, then compute these on the union — a per-shard DAG would miss exactly the cross-section cycles this phase exists to catch.

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 — i.e. the location pointer the paper claims discharges it (file:line), not a judgment that the discharge is valid; mark "UNVERIFIED" when the paper cites no discharge location (not when you believe the math fails — that is the jury's call). 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 — claimed discharge: §B.2 (paper argues via GMM weak convergence; validity is the jury's call, not the shard's)

Each micro-claim has: justification rule name + required conditions + where conditions are proven (a location pointer to where the paper claims to discharge them, not a validity judgment).

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 Review (reviewer backend, xhigh reasoning)

Submit the complete proof content with the checklist below, using the selected backend.

For codex, call mcp__codex__codex. For manual, call mcp__manual_review__review. Always use config: {"model_reasoning_effort": "xhigh"}.

Use this exact prompt for both backends:

    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]

Phase 1 addendum — --deep-fix opt-in

If the user passed --deep-fix on invocation, append the following block to the reviewer prompt after the OUTPUT FORMAT block above (do not modify the original block; the new fields are additive). Default invocations skip this block entirely and emit the original output schema unchanged.

    ## DEEP-FIX OUTPUT (opt-in, only when --deep-fix is set)

    For EACH issue listed above, additionally provide a `deep_fix_plan`
    that is repair-grade — sufficient for an executor to apply the fix
    in one Edit pass without spawning a follow-up review thread:

    - issue_id: same as the issue id above
    - corrected_statement: the theorem/lemma statement as it should
      read after the fix, with explicit quantifiers, regime conditions,
      and uniformity scope (LaTeX, paste-ready)
    - changed_equations: list of {before: <LaTeX>, after: <LaTeX>}
      pairs for each equation that needs replacement
    - downstream_labels: list of \label{...} keys whose statements or
      proofs depend on this fix and must be re-checked or rewritten
    - minimal_tex_patch_plan: ordered list of concrete edits, each as
      {file: <path>, anchor_old: <unique LaTeX snippet to find>,
       replacement_new: <LaTeX to insert>}; the executor will pass
      these directly to its file-editing tool
    - closure_tests: 2-5 sanity checks the executor must run after
      applying the fix (e.g., "verify constant_dependence_diff matches
      computed value", "limit case γ→0 reduces to identity",
      "dimension count matches before/after")

    ## ALGEBRA / TYPE SANITY PASS (opt-in, only when --deep-fix is set)

    If any issue invokes Schur test, Young's inequality, Cauchy-Schwarz,
    Hölder, quadratic form, operator norm, or power counting, the
    deep_fix_plan for that issue MUST also include an `algebra_sanity`
    object:

    - dimension_table: map of {symbol: type_signature}, e.g.
      {"K(i,α)": "scalar ≥ 0",
       "‖K‖_{2→2}": "scalar ≥ 0",
       "Σ_i V_i^rem": "scalar quadratic in w"}
    - power_count: number of times each operator-norm or Schur factor
      appears on each side; flag mismatch as INVALID
    - zero_coupling_check: evaluate the expression at γ=0 (or the
      analogous degenerate point); confirm it reduces to the expected
      identity / vanishing case
    - constant_dependence_diff: list of constants whose dependence on
      (d, K, n, ...) changes between BEFORE and AFTER, with the new
      explicit dependence written out

    Be precise. The executor will apply this plan literally; vague
    prose ("strengthen the bound", "redo the Schur step") is not
    acceptable in deep-fix mode. If you cannot produce a precise plan
    for an issue, omit that issue's deep-fix block and signal the
    deep-fix path is unavailable — do NOT emit a vague plan, and do
    NOT add a deep-fix-only category (e.g. UNCLEAR_DEEP_FIX) into the
    standard issue list, since that contaminates default-call output.

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 — 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 (reviewer backend, xhigh reasoning)

Continue with the selected backend. For codex, use mcp__codex__codex-reply with the saved threadId. For manual, use mcp__manual_review__review_reply with the saved threadId. Include fix summaries. 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 reviewer thread using the selected backend. Do NOT use a reply tool — this step must be blind.

For codex: start a fresh mcp__codex__codex thread. For manual: start a fresh mcp__manual_review__review thread.

The blind review prompt:

[Codex:]
mcp__codex__codex:
  config: {"model_reasoning_effort": "xhigh"}
  prompt: |
    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 3.6: Theorem Restatement Regression (opt-in)

Default: skipped. Existing callers see no change.

Opt-in: pass --restatement-check on invocation. The skill then runs a cross-location consistency pass after Phase 3.5 (Global Closure) and before Phase 3.9 (Unrecoverable Protocol).

This phase catches a specific class of bugs that Phase 3.5's "Statement-conclusion match" check does NOT catch: drift between the canonical theorem statement and its restatements elsewhere in the paper (summary tables, "Key Contributions" / "Summary" sections, abstract, discussion, captions). Common drift patterns observed in practice:

• Main theorem makes a partly-conditional claim (e.g., "regime A unconditional, regime B conditional on assumption X") but a later summary cites it as the fully unconditional version.
• Main theorem κ exponent is O(d²K²) but a constants table writes O(K²).
• Main theorem regime condition is squared envelope, but a remark elsewhere still shows the first-order envelope.
• Restatement quietly drops a quantifier ("for $n \ge n_0$") that the proof relied on.

Algorithm

1. Build canonical statement table. Scan all *.tex files in the paper for \begin{theorem} / \begin{lemma} / \begin{proposition} / \begin{corollary} blocks with a \label{...}. For each: record (label, full_statement_text, file:line_range).

2. Collect restatement candidates. For each canonical label thm:foo:

   • Every \Cref{thm:foo} / \ref{thm:foo} / \cref{thm:foo} invocation, with the surrounding 2 sentences as candidate restatement context.
   • Rows in tables (\begin{tabular} … \end{tabular}) that mention the label, the theorem's informal name, or its constants.
   • Bullets in "Key Contributions" / "Summary" / "Main Results" lists.
   • Sentences in the abstract and introduction that paraphrase the theorem.

3. Normalized diff. For each (canonical_statement, restatement_context) pair:

   • Strip \, \; \! ~, normalize whitespace, normalize math-mode delimiters ($..$, \(..\), display vs inline).
   • Detect drift signatures (one or more):
     • conditional_loss — canonical says "under \Cref{ass:X}" or "for w=1"; restatement omits the conditional.
     • scope_change — big-O exponent or rate differs (O(K^2) vs O(d^2K^2); √n vs n).
     • quantifier_loss — quantifier present canonically (e.g. "for $n \ge n_0$", "for sufficiently small $\gamma$") absent in restatement.
     • regime_envelope_change — first-order leakage Cγ/(1-Δγ) vs squared envelope (Cγ/(1-Δγ))² (or analogous).
     • constant_change — different numeric constant or different parameter dependence stated.
     • variable_rename — same role in argument played by differently-named symbol with no explicit alias.

4. Emit findings. Each detected drift becomes an entry in details.restatement_drift (see "Submission Artifact Emission" below). Severity defaults to MAJOR (UNDERSTATED/OVERSTATED + GLOBAL); reviewer may downgrade to MINOR if drift is purely cosmetic (e.g. \, placement) and upgrade to CRITICAL only if the restatement is used downstream as if it were the canonical (e.g. another proof cites the restated version).

What this phase does NOT do

• It does not fix drift automatically. The output is advisory; the executor or a follow-up --deep-fix run handles the rewrite.
• It does not alter details.issues; restatement drift is reported as a sibling field, so existing consumers reading issues[] see no schema change.
• It does not alter the top-level verdict decision rule. A paper with non-empty restatement_drift may still emit PASS if all proof obligations are otherwise discharged; the drift is independent of proof correctness. (The reviewer may at its discretion mirror a CRITICAL-severity drift into the issues list as a regular issue — typically when the restated version is used downstream as if it were canonical — but that is a per-issue judgment, never an automatic rule. MAJOR or MINOR drift never mirrors into issues.)

Failure mode

If --restatement-check is set but the cross-location scan cannot complete, emit details.restatement_drift: [] plus details.restatement_check_status: "unavailable" with a one-line note explaining why. Verifier gates and downstream skills MUST treat "unavailable" identically to the field being absent: not blocking. Phases 1 / 1.5 / 2 / 3 / 3.5 / 3.9 / 4 / 5 still run normally. Cases that trigger this fallback include:

• Unreadable .tex (parser / encoding error on a file that contains a theorem block).
• Ambiguous label resolution (e.g., the same \label{thm:foo} appears more than once with no clear canonical pick).
• No labeled canonical theorem-like block found (the algorithm only inspects \begin{theorem|lemma|proposition|corollary} blocks with an explicit \label{...}; if there is no such block, there is nothing to compare restatements against).

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": "..."
}

Deep-Fix Mode (opt-in)

Default: disabled. The Phase 1 reviewer emits issues with minimal_fix (a 1-2 sentence pointer); existing callers see no change.

Opt-in: pass --deep-fix on invocation. The Phase 1 reviewer prompt is augmented (not replaced) with the "DEEP-FIX OUTPUT" and "ALGEBRA / TYPE SANITY PASS" blocks above, so the reviewer also returns a deep_fix_plan per issue: corrected statement, changed equations, downstream label list, minimal LaTeX patch plan, and closure tests. Issues invoking Schur / Young / Cauchy-Schwarz / Hölder / quadratic forms / operator norms / power counting additionally carry an algebra_sanity block (dimension table + power count + zero-coupling check + constant-dependence diff).

Why opt-in

The default minimal_fix prose is intentionally short — it suits the common case where the executor wants high-level pointers and will derive the patch separately. Forcing deep_fix_plan on every run would (a) inflate every reviewer call by 2-5×, (b) trigger re-review thrash for issues the executor has already decided to weaken or defer, and (c) change the shape of details.issues for every existing caller. The flag preserves zero behavior change for default invocations while letting the executor request repair-grade output when the fix is going to be applied immediately.

Effect when enabled

• The Phase 1 reviewer prompt is augmented with the deep-fix and algebra-sanity blocks; nothing in the original mandatory checklist or output format is removed.
• PROOF_AUDIT.json details gains a sibling field deep_fix_plans (parallel to details.issues); see "Submission Artifact Emission" below.
• The top-level verdict, reason_code, and summary are unchanged in shape and decision rule: deep-fix output is advisory tooling for the executor, not a verdict-altering signal.
• Verifier gates and downstream skills (paper-writing Phase 6, verify_paper_audits.sh) MUST treat absence of deep_fix_plans as the only valid default state and MUST NOT block on its presence or content.

When opt-in is appropriate

• The executor intends to apply the fix in the same session and wants to skip a follow-up "give me a concrete patch" thread.
• A previous default-mode run identified a CRITICAL or MAJOR issue whose minimal_fix was too vague to act on (e.g., "redo the Schur step" without specifying the corrected operator-norm bound).
• Algebra-heavy proofs with Schur / quadratic-form / operator-norm steps where the reviewer's first pass has consistently produced under-specified fixes.

Failure modes

A deep-fix-only failure must never contaminate the default proof-check output. All of the following paths emit details.deep_fix_status: "unavailable" + details.deep_fix_plans: [] (with a one-line note in details.deep_fix_note) and leave the standard issue list, top-level verdict, reason_code, and summary unchanged:

• The reviewer refuses to produce a repair-grade plan because the fix would require choices the reviewer is unwilling to make.
• The Phase 1 reviewer call returns truncated or malformed deep-fix output (parse failure on the augmented section).
• The augmented Phase 1 call times out before producing the deep-fix block, but otherwise returned a valid normal proof review.

Verifier gates MUST treat unavailable identically to the field being absent: not blocking. Do not add a UNCLEAR_DEEP_FIX (or any deep-fix-only) entry into details.issues, since details.issues is the default schema's issue list and adding deep-fix-specific failures to it would change default behavior for callers without the flag.

If the augmented Phase 1 call fails so badly that the normal proof review cannot be recovered (e.g., the reviewer thread itself errored), retry once with the unaugmented prompt; if that also fails, fall through to the existing reviewer-failure path that maps to the top-level ERROR verdict.

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

• Executor analyzes, reviewer critiques: Claude reads proof, formulates questions, implements fixes. The external reviewer provides adversarial review.
• Reviewer reasoning always xhigh: Never downgrade.
• Send full content: Don't summarize — send actual math for line-by-line checking.
• Preserve threadId within a single run: Use the appropriate reply tool (mcp__codex__codex-reply or mcp__manual_review__review_reply) for Phase 3 follow-up rounds within the same top-level /proof-checker invocation, so the reviewer keeps prior-issue context when judging whether a fix closed the gap. Across separate top-level invocations, always start a fresh thread (see "Thread independence" below).

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.
• 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.

Opt-in flag discipline

• Deep-fix is opt-in only: never auto-enable; never block on deep_fix_plans content; existing callers must observe identical reviewer output and identical JSON schema if they do not pass --deep-fix.
• Reviewer prompt augmentation is additive: the deep-fix block is appended to the Phase 1 prompt, not substituted for any part of it. The original mandatory checklist (A-H) and original per-issue OUTPUT FORMAT remain in place verbatim.
• Restatement check is opt-in only: Phase 3.6 runs only when --restatement-check is set; existing callers must observe identical reviewer output and identical JSON schema if they do not pass the flag.
• No Phase reordering: enabling Phase 3.6 inserts it strictly between 3.5 and 3.9; it does not skip any other phase or change their semantics.
• No verdict crosstalk: neither deep-fix output nor restatement_drift ever alters top-level verdict or reason_code. A paper with non-empty drift or with deep-fix plans may still pass; a paper with FAIL verdict stays FAIL whether or not either flag was set.

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
PROOF_AUDIT.html (+ .review.json sidecar)	Single-file HTML view of PROOF_AUDIT.md auto-rendered via /render-html "PROOF_AUDIT.md" --json "PROOF_AUDIT.json". Non-blocking — if /render-html fails the audit still counts as complete; PROOF_AUDIT.{md,json} are the canonical outputs.	Workflow end (when RENDER_HTML = true, default)

When --restatement-check is set, PROOF_AUDIT.json additionally carries details.restatement_drift and details.restatement_check_status; both fields are omitted when the flag is unset. See "Submission Artifact Emission" below.

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 and verify_paper_audits.sh both rely 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":       ".aris/traces/proof-checker/<date>_run<NN>/",
  "thread_id":        "<codex mcp thread id>",
  "reviewer_model":   "gpt-5.5",
  "reviewer_reasoning": "xhigh",
  "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": "..." }, ... ]
  }
}

Optional: details.deep_fix_plans (only when --deep-fix is set)

"details": {
  ...
  "deep_fix_plans": [
    {
      "issue_id": "T1-H3",
      "corrected_statement": "<LaTeX, paste-ready>",
      "changed_equations": [{"before": "<LaTeX>", "after": "<LaTeX>"}, ...],
      "downstream_labels": ["thm:convergence", "cor:minimax", ...],
      "minimal_tex_patch_plan": [
        {"file": "sections/4.theory.tex",
         "anchor_old": "<unique LaTeX snippet>",
         "replacement_new": "<LaTeX to insert>"},
        ...
      ],
      "closure_tests": [
        "verify constant_dependence_diff matches computed value",
        "limit case γ→0 reduces to identity",
        ...
      ],
      "algebra_sanity": {
        "dimension_table": {"<symbol>": "<type_signature>", ...},
        "power_count": "<one-line check>",
        "zero_coupling_check": "<one-line check>",
        "constant_dependence_diff": "<before vs after>"
      }
    }
  ],
  "deep_fix_status": "ok" | "unavailable"
}

Field semantics:

• Both deep_fix_plans and deep_fix_status are omitted entirely when the flag is not set. The default schema does not include either key.
• When the flag is set and reviewer returns well-formed plans, deep_fix_status is "ok" and deep_fix_plans mirrors details.issues one-to-one (each plan referenced by issue_id); algebra_sanity is present only for issues invoking Schur / Young / Cauchy-Schwarz / Hölder / quadratic-form / operator-norm / power-counting steps.
• When the flag is set but reviewer output is malformed or truncated, deep_fix_status is "unavailable" and deep_fix_plans is []. Downstream consumers MUST treat "unavailable" identically to the field being absent: not blocking.
• Downstream consumers MUST treat absence of either field as the only valid default state and MUST NOT raise on missing.
• deep_fix_plans is advisory tooling for the executor; verify_paper_audits.sh and paper-writing Phase 6 do not block on its content or shape.

Optional: details.restatement_drift (only when --restatement-check is set)

"details": {
  ...
  "restatement_drift": [
    {
      "label": "thm:main",
      "canonical_location": "sections/2.setup.tex:117",
      "restatement_location": "sections/appendix.tex:1614",
      "drift_type": "conditional_loss" | "scope_change" | "quantifier_loss" |
                    "regime_envelope_change" | "constant_change" | "variable_rename",
      "canonical_excerpt": "<short LaTeX>",
      "restatement_excerpt": "<short LaTeX>",
      "severity": "MAJOR" | "MINOR" | "CRITICAL",
      "note": "..."
    }
  ],
  "restatement_check_status": "ok" | "unavailable"
}

Field semantics:

• Both restatement_drift and restatement_check_status are omitted entirely when the flag is not set. The default schema does not include either key.
• When the flag is set and the cross-location scan completes, restatement_check_status is "ok" and restatement_drift lists detected pairs (possibly empty if none found).
• When the scan fails (per "Failure mode" in Phase 3.6), restatement_check_status is "unavailable" and restatement_drift is [].
• Downstream consumers MUST treat absence or "unavailable" identically as the default state and MUST NOT raise on missing.
• restatement_drift does not alter details.issues and does not change the top-level verdict decision rule. The reviewer may at its discretion mirror a CRITICAL-severity drift into details.issues as a regular issue, but that is a per-issue judgment, not an automatic schema-driven rule.

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 external verifier rehashes these entries; any mismatch flags STALE.

Path convention (must match verify_paper_audits.sh): keys are paths relative to the paper directory (no paper/ prefix — the verifier 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 top-level /proof-checker invocation starts a fresh reviewer thread. For codex this is mcp__codex__codex; for manual this is mcp__manual_review__review. Do not reuse a saved threadId across separate invocations of this skill. Within a single top-level invocation, the appropriate reply tool (mcp__codex__codex-reply or mcp__manual_review__review_reply) threads the Phase 3 follow-up rounds — the reviewer needs prior-issue context to judge whether a fix actually closed the gap, and the Phase 1→3 flow above explicitly relies on this. The Phase 3.5 "Independent second review for FATAL/CRITICAL fixes" sub-step is the deliberate exception inside a single run: it must spawn a fresh thread so the blind reviewer has no exposure to the original critique.

Do not accept prior audit outputs (PAPER_CLAIM_AUDIT, CITATION_AUDIT, EXPERIMENT_LOG) as input across separate invocations — the cross-run freshness is what preserves reviewer independence per shared-references/reviewer-independence.md.

This skill never blocks by itself; paper-writing Phase 6 plus the verifier 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"
/proof-checker "paper/main.tex --deep-fix"            # opt-in: ask reviewer to also emit repair-grade deep_fix_plans
/proof-checker "paper/main.tex --restatement-check"   # opt-in: run Phase 3.6 to detect cross-location theorem-statement drift
/proof-checker "paper/main.tex --deep-fix --restatement-check"   # both opt-ins, independent