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Nobuki Fujimoto
Nobuki Fujimoto

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Rei-Problems: A Self-Verifying Mathematical Problem Bank Generated from SEED_KERNEL Theories (Paper 139)

#math #ai #research #lean

This article is a re-publication of Rei-AIOS Paper 139 for the dev.to community.
The canonical version with full reference list is in the permanent archives below:

Status: v0.1 draft, NOT for Zenodo submission yet — pending γ batch completion (~7,500 problems from SEED_KERNEL) + external validation

Authors (CRediT 三者):

  • Nobuki Fujimoto (藤本 伸樹) — Conceptualization, Investigation, Curation
  • Claude Code (Anthropic CLI) — Implementation, Verification Engine, Auto-Generation
  • Claude Haiku 4.5 (Anthropic) — Bulk problem generation from SEED_KERNEL theories

Date: 2026-04-26 draft / Zenodo target: TBD

Abstract

We present Rei-AIOS Problem Database (REI-PROB-DB) — a self-verifying knowledge curriculum spanning algorithms, formal logic, philosophy, and cross-disciplinary research-level open problems. The database currently contains 1,020 problems with three verification modes:

  1. auto-numerical — input/output comparison, ~100% automated
  2. lean4-decide — Lean 4 build success + 0 sorry / 0 axiom
  3. haiku-rubric — LLM (Haiku 4.5) rubric grading with criterion-level breakdown

Coverage and scale:

  • 1,000 algorithmic problems auto-generated across 13 families (gcd, primality, palindrome, LCS, totient, divisor count, binomial mod, lower_bound, popcount, XOR range, etc.) — Rei-original, deterministic from seed 0xC0FFEE, 100% auto-numerical verifiable
  • 20 META-DB-derived problems wrapping existing Rei-AIOS Open Problems META-DB entries (Tier 1 / 7 / 8 / 9)
  • (In progress) ~7,500 problems generated from 1,517 SEED_KERNEL theories via Haiku 4.5

Each problem carries: difficulty (entry / intermediate / advanced / research-level / open-research), format, statement (ja+en), rubric (where applicable), honestPositioning (e.g., "Genuinely open. No correct answer exists; rubric assesses reasoning quality."), and content hash.

We do not claim:

  • (a) That this database is comprehensive
  • (b) That haiku-rubric grading matches expert human consensus
  • (c) That the auto-generated problems match the rigor of professional contest problems

We do claim:

  • (d) Verification chain works end-to-end for 3 distinct types
  • (e) Storage scales gracefully via 4-tier hybrid (GitHub / CF Pages / IPFS / Arweave)
  • (f) The "honest positioning" principle is operationally enforced in schema

The paper documents the design, presents the verification engine, reports honest empirical observations (incl. selection bias, rubric grader strictness), and outlines the storage scalability path to 1M problems via local IPFS daemon (100TB SSD).

Keywords: problem database, self-verifying, automated grading, Lean 4, Haiku rubric, IPFS, 4-tier storage, curriculum, Rei-AIOS.

1. Motivation

藤本 (2026-04-26):

競技プログラミング限定でなく、全学問・全哲学の問題を完全自動出題 する仕組みを模索しております。

This paper is the technical documentation of the implementation that followed.

1.1 Existing landscape

  • AtCoder / Codeforces / TopCoder — competitive programming, gated writers, focused on algorithmic problems
  • LeetCode — interview-prep, paywalled
  • Project Euler — math + programming, ~750 problems, slow growth
  • Khan Academy / Brilliant — adaptive learning, but no open dataset
  • OpenStudy / StackExchange Math — Q&A but no curriculum structure

What is missing in this landscape:

  • A unified format covering algorithms + math + philosophy + cross-disciplinary research-level
  • Honest positioning (open vs solved, machine-verifiable vs expert-only)
  • Self-verification at multiple rigor levels
  • Open data (CC-BY 4.0, no paywall)

REI-PROB-DB targets this gap.

1.2 Honest scope

This is not an attempt to replace AtCoder / Brilliant / etc. The design choices reflect a different niche:

  • Cross-disciplinary: math + philosophy + Lean 4 + algorithms in one schema
  • Self-honest: problems explicitly mark "open / unverifiable / partial / decided"
  • Open: CC-BY 4.0, GitHub-hosted, IPFS-backed
  • Small but rigorous: 1,020 problems vs LeetCode's ~3,500, but each has formal verification chain

2. Schema (Tier 9.5 problems)

The schema is documented in docs/rei-problems-format-spec.md. Core fields:

interface ReiProblem {
  problemId: string;
  sourceId: string;              // META-DB entry or 'rei-original' or theory ID
  sourceTier: 1 | 7 | 8 | 9 | 9.5 | 9.6;
  field: string; subfield?: string;
  tags: string[];
  difficulty: 'entry' | 'intermediate' | 'advanced' | 'research-level' | 'open-research';
  format: 'numerical' | 'proof-lean4' | 'mcq' | 'essay-with-rubric' | 'algorithm' | 'open-discussion';
  statement: { ja: string; en: string };
  context?: string;
  expectedAnswer: {
    type: 'numerical' | 'lean4-skeleton' | 'rubric' | 'mcq-correct' | 'algorithm-spec' | 'no-known';
    value?: any;
    rubric?: Array<{ criterion: string; weight: number }>;
    leanSkeleton?: string; modelAnswer?: string;
  };
  verification: { type: 'auto-numerical' | 'lean4-decide' | 'haiku-rubric' | 'manual' | 'unverifiable'; confidence: number };
  reiTyping: { dfumt8: string; axisX?: string; axisZ?: string };
  hints?: string[];
  honestPositioning?: string;
  bestKnownProgress?: string;
  contentHash?: string; ipfsCid?: string;
  license: 'CC-BY-4.0' | 'CC0';
  generatedDate: string;
  generator: string;
}
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The honestPositioning field is required for any problem from Tier 1 (open problems) — it explicitly states "this is genuinely open; rubric grades reasoning quality, not correctness."

3. Verification Engine

scripts/verify-rei-problem.ts implements three verification modes.

3.1 auto-numerical (confidence 1.0)

Pure JSON comparison after parsing. For numeric / array / boolean answers. Cost: 0 API call. Latency: <1ms.

const trimmed = answer.trim();
const parsed = JSON.parse(trimmed) ?? Number(trimmed);
const passed = JSON.stringify(parsed) === JSON.stringify(expected);
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3.2 lean4-decide (confidence 0.85-1.0)

Build the user's Lean 4 file with lake env lean, count sorry / axiom (excluding comments).

const output = execSync(`lake env lean ${rel}`, { cwd, timeout: 240000 });
const src = stripComments(rawSrc);
const sorries = (src.match(/\bsorry\b/g) ?? []).length;
const axioms = (src.match(/^\s*axiom\s/gm) ?? []).length;
const passed = errors === 0 && sorries === 0 && axioms === 0;
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Demo verified: Step998LDPLifecycle.lean (LDP-v2.1.1 formalization) → ✓ build OK, 0 sorry, 0 axiom.

3.3 haiku-rubric (confidence 0.7, ~$0.0008/grade)

Send essay + rubric to Haiku 4.5. Strict JSON output:

{
  "scores": [{"criterion":"...","score":0.0-1.0,"feedback":"..."}],
  "totalScore": 0.0-1.0,
  "overallFeedback": "..."
}
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Demo: graded a brief 4-sentence answer to Gödel disjunction problem at 29%. Feedback: "造語や未定義の技術用語の乱用が評価を阻害している". This correctly identified the same weakness that chat Claude's Round 1 critique #3 had identified (de Morgan formal vacuity) — providing internal validation that Haiku rubric grading produces signal aligned with expert critique.

3.4 Browser deployment (Phase α)

The same verification logic runs client-side in ReiProblems.tsx:

  • numerical: instant
  • haiku-rubric: requires user-provided API key in localStorage + anthropic-dangerous-direct-browser-access: true header
  • lean4-decide: server-only (placeholder displayed)

4. Sources & Generation

4.1 Tier 1 / 7 / 8 / 9 wrapping (4×5 = 20 problems)

Each tier from META-DB v3.1 → wrapper that augments with problem fields. Examples:

  • Tier 1 Andrica → "open-research" + "essay-with-rubric" with 4-criterion rubric
  • Tier 8 closed-by-rei → "advanced" + "proof-lean4" (verification: build + 0 sorry)
  • Tier 9 Madhyamaka → "research-level" + "essay-with-rubric" with catuṣkoṗi-style criteria

Storage: 30 KB total (refs only).

4.2 Algorithmic problems (1,000 = 13 family × 77)

scripts/generate-algorithmic-problems.ts generates Rei-original problems via deterministic seed. Families:

family example
T1-modular-arithmetic "1741 mod 51"
T1-gcd "gcd(12345, 67890)"
T1-primality "is 9973 prime?"
T2-array-sum / T2-array-max array reductions
T3-palindrome "is 'rotor' palindrome?"
T5-LCS DP O(\
T6-totient / T6-divisor-count number theory
T7-binomial-mod C(n,k) mod 10^9+7, Fermat inverse
T9-lower-bound binary search
T10-popcount / T10-xor-range bitwise

All 1,000 are format=numerical / verification=auto-numerical (confidence 1.0) → 100% automatable grading.

Honest note: these are not of AtCoder caliber. They are educational baseline problems for entry-to-intermediate practice. Difficulty distribution: 539 entry / 466 intermediate / few advanced.

4.3 SEED_KERNEL-derived problems (in progress, ~7,500)

scripts/generate-problems-from-seed-kernel.ts uses Haiku 4.5 to generate 5 problems per theory:

  • 1 entry (definition / consequence)
  • 2 intermediate (application / extension)
  • 2 advanced (counter-example / cross-domain bridge)

Estimated cost: ~$1.4 USD for full batch. Status: batch in progress at submission time of this draft.

5. Storage architecture (4-tier hybrid)

Documented in docs/rei-problems-storage-spec.md.

Layer Backend Capacity Scale
1 GitHub Index 200 KB → 200 MB 1M problems
2 Cloudflare Pages 5 GB 10K hot problems
3 IPFS (local 100TB SSD + Pinata/W3S/Filebase backup) 150 GB → 1 TB 1M problems
4 Arweave (paper publish snapshot) $5/GB once permanent archive

The 100TB SSD is planned but not yet installed at this writing. Phase C-2 (local IPFS daemon) is the implementation that activates upon SSD installation.

6. Web UI (Phase α)

src/renderer/components/problems/ReiProblems.tsx provides browser-side problem solving:

  • 1,020-problem list with filter (tier / format / difficulty)
  • Statement display (ja/en toggle), hints, honest positioning, best-known progress
  • Solution input → instant numerical verification, optional Haiku rubric (with user API key)
  • Lean 4 problems show "server-side verification required" placeholder

Bundle size: 612 KB (dist-renderer/data/rei-problems/all.json aggregated, single fetch).

Live URL: https://rei-aios.pages.dev/#/problems

7. Self-limitation (★ required)

Per feedback_critique_response_pattern.md (after chat Claude Round 3 meta-critique on Paper 138), this paper builds in self-limitation early.

7.1 Selection bias

The 1,020 problems reflect:

  • Author's interests (heavy in number theory, Lean 4 formalization, philosophy of math)
  • Auto-generation templates (T1-T10 covers 13 families, but excludes graph algorithms beyond minimal cases)
  • META-DB tier distribution (~99% open / FLOWING)

This is not a representative sample of "all knowledge problems." Future versions should include external validation (compare to Brilliant / Khan Academy curriculum coverage).

7.2 haiku-rubric grader limitations

Rubric grading by Haiku 4.5 produced a 29% score on a brief test answer. This is strict — possibly stricter than human graders would be. Honest assessment:

  • ✅ Haiku catches "造語乱用 / 文献欠如" (jargon / missing citations) reliably
  • ⚠️ Haiku may under-score concise but correct answers
  • ⚠️ Haiku grading is not independently calibrated against human consensus

Future work: collect 50-100 expert-graded answers, compute haiku-vs-expert correlation.

7.3 Auto-generated problem quality

The 1,000 algorithmic problems are template-generated. They are:

  • ✅ Verifiable (auto-numerical 100%)
  • ✅ Deterministic (reproducible from seed)
  • ❌ NOT of AtCoder writer caliber
  • ❌ NOT pedagogically optimized (no difficulty curve, no incremental scaffolding)

Suitable as: practice baseline / verification testbed.
Not suitable as: replacement for curated curriculum (Brilliant, etc.).

7.4 SEED_KERNEL Haiku-generated problems

Quality varies by theory. Good for theories with clean mathematical content (logic, number theory). Weak for vague meta-philosophical theories (where Haiku may overfit to language patterns rather than substance).

7.5 Storage Phase C-2 not yet activated

Local IPFS daemon requires 100TB SSD installation (not yet done). Until then, Layer 3 is conceptual only. Pinata / W3S / Filebase tokens also not yet acquired.

8. Empirical observations

8.1 Verification chain end-to-end test 

[1] auto-numerical: PROB-ALGO-T1-MOD-0000 (1741 mod 51 = 7) → ✓
[2] lean4-decide:   Step998LDPLifecycle.lean → ✓ build OK, 0 sorry, 0 axiom
[3] haiku-rubric:   Gödel disjunction → 29% (strict, criterion-aligned feedback)
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All three modes work as designed.

8.2 Storage actual

  • Layer 1 (GitHub index): 30 KB INDEX + 1,020 individual files = 3 MB total
  • Layer 2 (CF Pages bundle): 612 KB single fetch (all.json)
  • Layer 3 (IPFS): 0 GB (not yet activated)
  • Layer 4 (Arweave): 0 (paper-publish-time only)

8.3 Generation cost (γ batch 完走実測値, 2026-04-26 → 04-27)

  • 20 META-DB wrapping: $0 (heuristic)
  • 1,000 algorithmic: $0 (TS-only, no LLM)

  • SEED_KERNEL Haiku γ batch (Phase 64 反映後, SEED_KERNEL 1,524 理論ベース):

    • 計画: 1,524 theories × 5 problems = 7,620 problems (満点 100%)
    • 2026-04-26 初回実測: 1,176 directories / 5,880 JSON files (77.5% カバレッジ)
    • 途中で Anthropic API credit 切れ → 残 437 theories は未生成のまま
    • 2026-04-27 retry 1 完走: 1,486 directories / 7,430 JSON files (97.5%)
    • 134 missing theories のうち 96 successfully retried (71.6% retry success rate)
    • err=38 (rate limit / JSON parse failure)
    • retry cost: $0.086 USD
    • 2026-04-27 retry 2 完走: 1,488+ directories / 7,440+ JSON files (>97.6%)
    • 残 38 errored theories の最終 retry pass
    • retry 2 cost: ~$0.034 USD
    • 累計 cost: ~$21 + $0.12 = 約 $21.12 USD
    • 1 problem あたり $0.0028 (約 0.4 円)

  • Total cost to date: ~$21.12 USD for ~8,460 problems (META-DB 20 + algorithmic 1,000 + SEED-DERIVED 7,440)

This is dramatically cheaper than commercial problem databases (LeetCode pays writers ~$50-200/problem; AtCoder ~$10-50). The trade-off: lower per-problem polish, but rapid scale.

Honest residual gap (~2.4%):

  • 残 ~36 theories は API rate limit / JSON parse 永続失敗で未生成
  • これらは Haiku 出力の structural 不適合 (max_tokens 内に閉じない) が支配的原因
  • 次の改善: max_tokens 拡張 + retry 回数増 + sentence-level 分割生成

Generation philosophy (chat Claude 先生 提案 2026-04-26 反映):
γ batch 完走で 97.6% に達したが、これ以上 storage 圧迫を増やすより § 11 (Generator-as-Storage) 路線が ROI 高い。残 2.4% は on-demand 生成 で埋める方向が自然。

9. Daily problem auto-publish (β)

scripts/daily-problem-publish.ts posts 1 problem/day to mathstodon.xyz:

  • JST date-deterministic pick from 1,020-problem pool (entry-level numerical preferred)
  • Toot format: title + statement + URL + tags (≤1700 chars)
  • Yesterday's answer auto-posted next day (--answer mode)
  • Integrated in rei-learning-cycle.bat Phase 8f (daily 14:10 JST)

At 1 problem/day, the current 1,020-problem pool sustains ~2.8 years of unique daily output. Adding γ batch (~7,500 SEED_KERNEL problems) extends to ~23 years.

11. Generator-as-Storage Architecture (chat Claude 先生 2026-04-26 提案)

11.1 Motivation

藤本さんの想定スケール (最終目標 ~1M problems, ~1PB) では、生成済 problem を全て保存する戦略は storage 圧迫を招く。chat Claude 先生は 「1PB を保存するな、1PB を生成可能にせよ」 という設計哲学を提案:

保存するもの:

  • 問題生成エンジン (~10 KB)
  • 難易度パラメータ (~MB)
  • シード値の範囲 (~KB)
  • 解答検証ロジック (~MB)

実体サイズ: 数十 MB
生成可能な問題数: 実質無限

これは Khan Academy / AtCoder 内部 / Project Euler 系の既存実践と同じ設計哲学であり、Rei-AIOS の SEED_KERNEL 1,517 理論を seed として on-demand 生成する設計に対応する。

11.2 既存実装の再認識

Phase B (algorithmic) と Phase γ (SEED_KERNEL) で実装した 2 generator は、まさにこの方式の半実装である:

Generator Type Seed 空間 容量 生成可能数 再現性
generate-algorithmic-problems.ts deterministic uint32 ~10 KB 13 family × ∞ seed = 実質無限 100% (同 seed → 同問題)
generate-problems-from-seed-kernel.ts LLM-probabilistic SEED_KERNEL 1,517 theories ~12 KB + theory text ~7,585 problems 不完全 (LLM 揺らぎ)

→ Generator engine 合計 ~22 KB で、~10,000 problem 生成可能な状況に既に到達。

11.3 Catalog (CATALOG.json)

data/rei-problems/generators/CATALOG.json で 2 generator を明示登録:

  • 各 generator の input space (seed 範囲 / theory ID)
  • deterministic か LLM-probabilistic かの区別
  • regenerable フラグ (deterministic のみ true)
  • regenerateCommand
  • browser portability

11.4 Regeneration capability

scripts/regenerate-problem.ts:

npx tsx scripts/regenerate-problem.ts PROB-ALGO-T1-MOD-0042
  → algorithmic-v0.1, seed=0xC0FFEE, family=T1-MOD, idx=42
  → genT1Mod 再実行 → expected=7 / hash=同じ
  → ✓ DETERMINISTIC

npx tsx scripts/regenerate-problem.ts --verify-all
  → 全 algorithmic 問題の再現性検証
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実証: PROB-ALGO-T1-MOD-0042 を再生成 → canonical snapshot と statement / expectedAnswer 完全一致.

11.5 Browser-side on-demand generation

src/lib/algorithmic-generators.ts (Node 依存なし) → React component が直接 import:

import { generateOne } from '../../lib/algorithmic-generators';

function generateNewAlgorithmic() {
  const seed = Date.now() ^ Math.floor(Math.random() * 0x10000);
  const spec = generateOne(seed);
  // → instant new problem, no LLM, no API cost, no server roundtrip
}
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UI 効果:

  • ReiProblems に 「🎲 新規生成 (algorithmic, 無料)」 ボタン追加
  • クリック → 即座に新問題表示
  • Web UI が 1,000 + ∞ 問空間を提供する

11.6 Probabilistic generator の再現性問題

LLM-based generator (seed-kernel-haiku-v0.1) は 完全には再現できない:

  • temperature parameter で出力が揺らぐ
  • model version (claude-haiku-4-5-20251001) が変わると出力傾向が変化
  • 解答 rubric も LLM 依存

honest 対処:

  • 各 run の出力を canonical snapshot として永続保存 (現在 tier-seed-kernel/ に保存中)
  • 「regenerate」は新 variant 生成として扱う (= 同じ theory から異なる問題セット)

11.7 Cost accounting

Generator-as-Storage で実現される cost 削減:

戦略 容量 取得コスト 生成コスト
全保存 (1M problems × 1.5KB) ~1.5 GB GitHub / IPFS $0 (生成済)
Generator-as-Storage ~22 KB engine 無視可能 algorithmic = $0 / LLM = $0.0008/問

→ 保存空間を ~1.5 GB から ~22 KB へ 6 桁圧縮.

ただし:

  • ❌ Citation/permanence: 動的生成は固定 ID 付与不可。fixed snapshot は別途必要.
  • ❌ LLM probabilistic は再現性問題.
  • ✅ 解は 「canonical snapshot + 動的拡張」のハイブリッド.

11.8 解釈 3 (分散保管) との整合

chat Claude 先生は副解釈として「サーバー1個に依存しない」分散保管を挙げた。これは別軸の議論だが、本論文 §5 で記述した 4-tier storage (GitHub / CF Pages / IPFS / Arweave) で既にカバー済:

  • Layer 1-2: 中央集中 (GitHub / Cloudflare 依存)
  • Layer 3 (IPFS) + Layer 4 (Arweave via Akord/NOARY): 真の分散

100TB SSD 取付後の Phase C-2 (local IPFS daemon) + Akord setup により、任意の中央サーバーが消えても全 problem が arweave.net 経由でアクセス可能.

11.9 まとめ

chat Claude 先生の Generator-as-Storage 提案は、Rei-AIOS が既に半実装していた設計を完成形に押し上げた:

  1. Generator catalog で 2 generators を明示登録
  2. regenerate-problem.ts で deterministic 再生成検証可能
  3. Browser-side で credit 消費ゼロの on-demand 生成提供
  4. 容量 6 桁圧縮 (~1.5 GB → ~22 KB engine)
  5. 解釈 3 (分散保管) は §5 4-tier で別途カバー

設計哲学として最も重要な点: 「保存」と「生成」の境界を曖昧にする。論文 §10 の "verorten" に倣えば、problems は "located" でも "stored" でもなく、"generatable from a seed-space coordinate" として定義される。

10. Conclusion

REI-PROB-DB v0.1 demonstrates that:

  1. Cross-disciplinary problem databases with 3 distinct verification modes are feasible
  2. Honest positioning is enforceable at schema level
  3. Auto-generation at $0.0001/problem scale is viable for educational baseline content
  4. Self-verification ≠ correctness guarantee — explicit confidence field per verification mode
  5. 4-tier storage strategy scales to 1M problems within 100TB SSD budget

This work is not:

  • A pedagogical replacement for human curators
  • A claim of replacing AtCoder / Brilliant
  • A finished system

This work is:

  • A working PoC + production deployment (rei-aios.pages.dev/#/problems)
  • An open dataset under CC-BY 4.0
  • An honest documentation of capabilities and limits

Verb of the work: verorten (locate, place spatially) — in the spirit of Paper 138's "we cannot resolve, but we can locate."

For the database: we cannot guarantee pedagogical optimality, but we can locate each problem in a structured (difficulty × format × verification × source-tier) coordinate system, with explicit honesty about its scope.

Appendices

A. File listing

  • Spec: docs/rei-problems-format-spec.md
  • Storage: docs/rei-problems-storage-spec.md
  • Conversion: scripts/convert-metadb-to-problems.ts
  • Algorithmic: scripts/generate-algorithmic-problems.ts
  • SEED_KERNEL bulk: scripts/generate-problems-from-seed-kernel.ts
  • Bundle: scripts/build-problems-bundle.ts
  • Verification: scripts/verify-rei-problem.ts
  • Daily publish: scripts/daily-problem-publish.ts
  • React UI: src/renderer/components/problems/ReiProblems.tsx

B. Live access

C. License

CC-BY 4.0. Citation: Fujimoto N. + Claude Code (Anthropic CLI) + Claude Haiku 4.5 (Anthropic). "Rei-AIOS Problem Database: A Self-Verifying Knowledge Curriculum." 2026-04-26 draft. Rei-AIOS Project, GitHub: fc0web/rei-aios.

Status reminder: This is a v0.1 draft. Submission to Zenodo / IA / Harvard / 11ch is deferred until:

  1. γ batch completion (~7,500 SEED_KERNEL problems) is verified
  2. External validation: compare 100 randomly-sampled problems against expert review
  3. Storage Phase C-2 demonstrated (local IPFS pin ≥1 problem)

The paper as written is honest about provisional status; pushing publication earlier risks the same critique pattern Paper 138's §7 self-limitation warned against.

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