A One-Parameter Family of Wallis-Type Polynomial Continued Fractions: Discovery and Formal Verification

This repository includes the reproducibility materials for the manuscript tex/wallis_formal_verification_submission.tex, which proves the one-parameter family

S^(m) = 1 + K_{n>=1} [-n(2n-(2m+1)) / (3n+1)] = 2^(2m+1) / (pi * binom(2m,m)).

The result is supported by a zero-sorry Lean 4 proof chain in lean/WallisFamily.lean, LaTeX source in tex/, and 1000-digit numerical certification produced by python/verify_wallis_family.py.

Repository layout

Path	Purpose
`lean/`	Lean 4 source and project files for the formal proof
`python/`	Numerical verification scripts
`tex/`	LaTeX source for the submission manuscript
`results/`	Saved certification and search outputs

Run the Lean 4 proof

cd lean
lake exe cache get
lake env lean WallisFamily.lean

The audit commands at the bottom of WallisFamily.lean check the certified chain intertwining_lemma, ratio_step_m0, limit_step, and theorem1_closed_form.

Reproduce the numerical results

python -m venv .venv
.venv\Scripts\activate
pip install -r requirements.txt
python python/verify_wallis_family.py --dps 1100 --depth 9000 --max-m 20 --json results/wallis_family_certification.json

Optional follow-up scan for the third-order negative results:

python _third_order_wallis_scan.py --workers 8 --range 3 --depth 250

Citation

@misc{wallis-formal-verification-2026,
  title  = {A One-Parameter Family of Wallis-Type Polynomial Continued Fractions: Discovery and Formal Verification},
  author = {Anonymous},
  year   = {2026},
  note   = {Lean 4 formalization, numerical scripts, and LaTeX source available in this repository}
}

arXiv

The arXiv link will be added here once the preprint is posted.

Ramanujan Breakthrough Generator

Discover new mathematical identities automatically. An AI-powered engine that generates Ramanujan-style conjectures by searching polynomial continued fractions (PCFs), matching values to known constants via PSLQ, and verifying at 200+ digit precision — ready for integration with LLM research agents.

Inspired by the Ramanujan Machine's algorithmic conjecture discovery, this project adds LLM-driven creative generation and seamless integration with modern AI research agents like gpt-researcher and RD-Agent.

GitHub Topics

When creating the repository, add these topics for discoverability:

research · mathematics · ramanujan · ai-research · mathematical-discovery · symbolic-computation · number-theory · llm-agent · continued-fractions · conjecture-generation

What Is This?

Srinivasa Ramanujan's genius lay in discovering extraordinary identities—formulas connecting fundamental constants like π, e, ζ(3), and the golden ratio through continued fractions, infinite series, and modular forms. This tool automates that style of discovery by combining:

Polynomial Continued Fractions (PCFs) — parameterized by polynomial numerator a(n) and denominator b(n), evaluated to arbitrary precision via bottom-up convergent computation.
PSLQ / Integer Relation Detection — matching CF values against a library of 25+ mathematical constants (π, log 2, √2, Catalan's constant, ζ(3), Gamma values, …).
Evolutionary Search — genetic algorithm with tournament selection, crossover, Gaussian mutation, and adaptive temperature control to explore the space of polynomial coefficients.
High-Precision Verification — every candidate match is re-verified at 200+ digit precision to eliminate false positives.

Mathematical Background

A generalized continued fraction is:

$$ \text{CF}(a,b) ;=; b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cdots}}} $$

When a(n) and b(n) are polynomials in n, these are called Polynomial Continued Fractions (PCFs). The classical examples include:

PCF	Value	Source
`a(n) = n², b(n) = 2n-1`	4/π	Brouncker (1656)
`a(n) = -n³, b(n) = (2n-1)(n²+n-1)`	ζ(3)	Apéry (1979)
`a(n) = -2n²+(2m+1)n, b(n) = 3n+1`	S^(m) ∝ 1/π	This work

The generator searches for new PCFs matching known (or unknown) constants.

Installation

# Clone the repository
git clone https://github.com/ramanujan-breakthrough/ramanujan-breakthrough-generator.git
cd ramanujan-breakthrough-generator

# Create virtual environment (recommended)
python -m venv .venv
source .venv/bin/activate  # Linux/macOS
# or: .venv\Scripts\activate  # Windows

# Install core dependencies
pip install -e .

# Or install from requirements.txt
pip install -r requirements.txt

Requirements

Python 3.10+
mpmath ≥ 1.2 (arbitrary-precision arithmetic)
sympy ≥ 1.11 (symbolic computation, optional but recommended)

Quick Start

CLI Usage

# Run 30 evolutionary cycles with default settings
python ramanujan_breakthrough_generator.py --cycles 30 --seed 42

# Target a specific constant (e.g., Catalan's constant)
python ramanujan_breakthrough_generator.py --cycles 50 --target catalan --precision 80

# Exhaustive coefficient sweep (small coefficients, thorough)
python ramanujan_breakthrough_generator.py --mode cmf --deg-alpha 2 --deg-beta 1 --coeff-range 3

# Degree-restricted random search
python ramanujan_breakthrough_generator.py --mode dr --deg-alpha 3 --deg-beta 2 --cycles 100

# Generate a family report from existing discoveries
python ramanujan_breakthrough_generator.py --report

# High-precision run with verification
python ramanujan_breakthrough_generator.py --cycles 50 --precision 100 --verify-prec 300

Python API

from ramanujan_breakthrough_generator import (
    build_constants, eval_pcf, seed_population, evaluate_population,
    pslq_match, verify_match_high_precision, cluster_discoveries,
)
from mpmath import mp

mp.dps = 80

# Evaluate a specific continued fraction
val = eval_pcf([0, 3, -2], [1, 3], depth=500)
print(f"CF value: {val}")  # Should be close to 4/pi

# Build constant library and match
constants = build_constants(80)
match = pslq_match(val, constants, tol_digits=20)
if match:
    print(f"Matched: {match[0]}")

# Verify at high precision
verified, digits = verify_match_high_precision(
    [0, 3, -2], [1, 3], "4/pi", constants,
    verify_prec=200, verify_depth=1000
)
print(f"Verified: {verified} ({digits:.0f} digits)")

With Research Context (Agent Integration)

from research_integration import ContextualBreakthroughGenerator

gen = ContextualBreakthroughGenerator(precision=80, seed=42)

# Feed in research context to guide search
results = gen.discover(
    context="Recent work by Zudilin on Apéry-like sequences suggests "
            "cubic numerators a(n)=n³+cn may yield new ζ(3) representations.",
    target="zeta3",
    num_formulas=10,
    cycles=50,
)

for r in results:
    print(f"  a={r['a_coeffs']}, b={r['b_coeffs']} -> {r['match']} ({r['verified_digits']:.0f}d)")

Search Modes

Mode	Flag	Description
Evolve	`--mode evolve`	Genetic algorithm with adaptive temperature (default)
DR	`--mode dr`	Degree-restricted random sampling
CMF	`--mode cmf`	Exhaustive coefficient sweep (Conservative Matrix Field)

Target Constants Library

The generator searches against 25+ mathematical constants:

Constant	Symbol	Value (approx)
π	`pi`	3.14159265…
4/π	`4/pi`	1.27323954…
log 2	`log2`	0.69314718…
√2	`sqrt2`	1.41421356…
Golden ratio φ	`phi`	1.61803398…
Euler-Mascheroni γ	`euler_g`	0.57721566…
Apéry's constant ζ(3)	`zeta3`	1.20205690…
Catalan's constant G	`catalan`	0.91596559…
Γ(1/4)	`Gamma_1_4`	3.62560990…
…		(and 15+ more)

Example Discoveries

The Pi Family discovered by this engine:

a_m(n) = -2n² + (2m+1)n,  b(n) = 3n + 1

m=0:  CF -> 2/π      (Brouncker-type)
m=1:  CF -> 4/π      (Novel)
m=2:  CF -> 16/(3π)  (Novel)
m=3:  CF -> 32/(5π)  (Novel)

Verification with SymPy:

from mpmath import mp, mpf, pi
mp.dps = 50

from ramanujan_breakthrough_generator import eval_pcf

# m=1 member: a(n) = 3n - 2n², b(n) = 3n + 1
val = eval_pcf([0, 3, -2], [1, 3], depth=1000)
target = 4 / pi
residual = abs(val - target)
print(f"|CF - 4/π| = {residual}")  # ~1e-50

Output Files

File	Description
`ramanujan_discoveries.jsonl`	Append-only log of all discoveries (JSON Lines)
`ramanujan_state.json`	Checkpoint for resuming interrupted runs

Each discovery record contains:

{
  "cycle": 7,
  "a": [0, 3, -2],
  "b": [1, 3],
  "value": "1.2732395447351626862…",
  "match": "4/pi",
  "residual": -49.3,
  "verified_digits": 160.0,
  "complexity": 4.2,
  "timestamp": "2026-04-07T12:00:00"
}

Integration with Research Agents

See research_integration.py for plug-and-play adapters for:

gpt-researcher — feed web-sourced math papers as context
Microsoft RD-Agent — structured hypothesis → experiment → evaluation loop
Khoj — personal knowledge base integration
Custom agents — simple discover(context, target) API

Integration Example: gpt-researcher

# pip install gpt-researcher
from gpt_researcher import GPTResearcher
from research_integration import gpt_researcher_adapter

# Step 1: Research a mathematical topic
researcher = GPTResearcher(
    query="recent advances in continued fraction representations of zeta(3)",
    report_type="research_report",
)
report = await researcher.conduct_research()

# Step 2: Feed the report into the breakthrough generator
discoveries = gpt_researcher_adapter(report, target="zeta3", num_formulas=5)

# Step 3: Inspect results
for d in discoveries:
    print(f"  a={d['a_coeffs']}, b={d['b_coeffs']} -> {d['match']}")
    print(f"    verified: {d['verified_digits']:.0f} digits")

Integration Example: Microsoft RD-Agent

from research_integration import rd_agent_adapter

result = rd_agent_adapter(
    hypothesis="Quadratic PCFs with b(n)=3n+1 produce rational multiples of 1/pi",
    experiment_config={
        "target": "pi",
        "cycles": 50,
        "precision": 80,
    },
)
print(f"Verified: {result['verified_count']} / {len(result['discoveries'])}")
print(result['summary'])

Integration Example: Simple One-Call API

from research_integration import discover

results = discover(
    context="Apéry-like cubic numerators for zeta(3)",
    target="zeta3",
    num_formulas=5,
    cycles=30,
    style="apery",
)
for r in results:
    print(f"  {r['match']}: a={r['a_coeffs']}, b={r['b_coeffs']}")

Project Structure

├── ramanujan_breakthrough_generator.py   # Core discovery engine
├── research_integration.py               # Research agent adapters
├── irrationality_toolkit.py              # Irrationality proof toolkit
├── tests/
│   ├── test_breakthrough_generator.py    # Unit tests for the generator
│   └── test_phase2.py                    # PCF analysis tests
├── pyproject.toml                        # Package metadata & dependencies
├── requirements.txt                      # Pinned dependencies
├── LICENSE                               # MIT License
├── .gitignore
└── README.md

Citation

If you use this tool in academic work, please cite:

@software{ramanujan_breakthrough_generator,
  title  = {Ramanujan Breakthrough Generator: Self-Iterating PCF Discovery Engine},
  year   = {2026},
  url    = {https://github.com/ramanujan-breakthrough/ramanujan-breakthrough-generator},
  note   = {Automated discovery of polynomial continued fraction identities
            via evolutionary search and PSLQ integer relation detection}
}

Related Work

Ramanujan Machine (GitHub) — Technion project for automated conjecture generation (website)
Raayoni et al., "Generating conjectures on fundamental constants with the Ramanujan Machine", Nature (2021)
Zudilin, "Apéry-like sequences and their extensions", Experimental Mathematics (2002)
Ferguson & Bailey, "A Polynomial Time, Numerically Stable Integer Relation Algorithm" (PSLQ)

Contributing

Contributions are welcome! Here are some ways to help:

New target constants — add constants to build_constants() (Clausen values, polylogarithms, MZVs, …)
New search strategies — implement alternative mutation operators or search modes
Proof automation — connect discovered CFs to irrationality proof pipelines
Performance — parallelize CF evaluation with multiprocessing
Documentation — improve examples, add tutorials

# Development setup
git clone https://github.com/ramanujan-breakthrough/ramanujan-breakthrough-generator.git
cd ramanujan-breakthrough-generator
pip install -e ".[dev]"
pytest                    # run tests

Please open an issue before large PRs. See the Ramanujan Machine project for collaboration opportunities.

Discovery Status

Auto-updated: 2026-04-07 10:43 UTC

Metric	Value
Cycles run	420
Total discoveries	3551
Unique CFs	3481
Distinct constants	1000

Constants found: 1 (438), 2 (431), 3 (427), 4 (405), 5 (393), (5/2) (20), (7/2) (20), 4/pi (17), phi (17), (3/2) (17), (1/2) (15), -(1) (15), (5/3) (14), (2/3) (13), (9/2) (13), 1/2*4/pi (11), (4/3) (11), (8/3) (11), 7 (10), 6 (10), (7/3) (10), (11/3) (9), -(2) (9), (1/3) (8), 8 (8), (14/3) (8), 4/3*4/pi (7), S^(3) (7), S^(4) (7), S^(5) (7), (5/4) (7), (10/3) (7), 2/1*phi (6), (15/4) (6), (9/4) (6), -(3) (6), (3/4) (6), (11/4) (6), (2/5) (6), (13/3) (6), e (5), (3/5) (5), (7/4) (5), (1/4) (5), (11/5) (4), (4/5) (4), (8/5) (4), (6/5) (4), (21/5) (4), (23/5) (4), (13/4) (4), (24/5) (4), (27/7) (4), (5/6) (3), (18/5) (3), -(4) (3), -((3/2)) (3), -((2/3)) (3), -((1/2)) (3), (11/2) (3), -((1/3)) (3), (1/5) (3), (17/7) (3), (16/5) (3), (12/5) (3), (29/8) (3), (17/4) (3), (22/5) (3), (13/5) (3), (19/4) (3), (14/5) (3), (34/7) (3), (2/7) (3), 2/1*e (3), -((5/4)) (2), -(5) (2), -((10/3)) (2), -((5/3)) (2), -((4/3)) (2), (1/6) (2), (7/6) (2), (13/6) (2), (16/7) (2), (19/6) (2), (25/6) (2), 10 (2), -(6) (2), 14 (2), 13 (2), (6/7) (2), (10/7) (2), (24/7) (2), (32/9) (2), (31/7) (2), (41/9) (2), (604/301) (2), (7/5) (2), (11/7) (2), (18/7) (2), (21/8) (2), (17/5) (2), (25/7) (2), (32/7) (2), (37/8) (2), (47/10) (2), (12/7) (2), (19/7) (2), (26/7) (2), (34/9) (2), (33/7) (2), (43/9) (2), (9/5) (2), (11/6) (2), (13/7) (2), (17/6) (2), (20/7) (2), (23/8) (2), (19/5) (2), (23/6) (2), (31/8) (2), (35/9) (2), (29/6) (2), (39/8) (2), (44/9) (2), (49/10) (2), -(8) (2), -((1/5)) (2), (29/7) (2), (302/301) (2), (53/13) (2), -((3**(685/452)*5**(69/452))/(2**(209/452)*7**(79/452))) (1), (5**(101/86)*7**(523/86))/(2**(89/86)*3**(373/43)) (1), (5/9) (1), (2**(562/971)*5**(592/971))/(3**(213/971)*7**(126/971)) (1), -((2**(463/108)*3**(389/108)*7**(59/18))/(5**(299/36))) (1), 1/(2**(104/901)*3**(849/901)*5**(828/901)*7**(423/901)) (1), (5**(206/225)*7**(11/10))/(2**(457/900)*3**(119/300)) (1), (2**(507/230)*5**(81/23))/(3**(40/23)*7**(527/230)) (1), (2**(584/31)*7**(736/31))/(3**(383/31)*5**(846/31)) (1), -((3**(92/211)*5**(564/211))/(2**(455/211)*7**(251/211))) (1), (5**(18/431)*7**(542/431))/(2**(318/431)*3**(227/431)) (1), (5**(376/289))/(2**(343/578)*3**(299/578)*7**(339/289)) (1), (3**(60/11)*5**(38/11))/(2**(428/77)*7**(246/77)) (1), 2**(241/444)*3**(281/222)*5**(7/222)*7**(118/111) (1), (3**(569/568)*5**(995/568))/(2**(199/284)*7**(219/142)) (1), (3**(899/733)*5**(937/733))/(2**(364/733)*7**(409/733)) (1), (2**(424/117)*3**(887/117))/(5**(88/117)*7**(461/117)) (1), (239/39) (1), -((841/31)) (1), -((9/59)) (1), (193/9) (1), (33/25) (1), -((403/33)) (1), -((921/887)) (1), -((14/9)) (1), (422/171) (1), -((42/43)) (1), (554/69) (1), -((270/161)) (1), -((58/451)) (1), -((83/23)) (1), (241/147) (1), (43/17) (1), -((161/61)) (1), (121/167) (1), (655/477) (1), (412/107) (1), -((108/37)) (1), (100/17) (1), (337/261) (1), (357/97) (1), (323/47) (1), (61/21) (1), (69/41) (1), (3**(657/406)*7**(416/203))/(2**(549/406)*5**(815/406)) (1), (3**(191/797)*5**(312/797)*7**(90/797))/(2**(24/797)) (1), (2**(97/59)*7**(345/118))/(3**(353/118)*5**4) (1), -((3**(799/727)*5**(347/727))/(2**(496/727)*7**(539/727))) (1), (2**(8/79)*7**(719/79))/(3**(88/79)*5**(744/79)) (1), (3**(162/109)*7**(314/327))/(2**(315/109)*5**(53/327)) (1), (2**(431/227)*5**(100/227)*7**(189/454))/(3**(304/227)) (1), (7**(514/873))/(2**(397/873)*3**(20/291)*5**(107/291)) (1), (95/26) (1), (5**(983/878))/(2**(104/439)*3**(9/439)*7**(225/878)) (1), -((2**(535/483)*3**(34/483)*7**(442/483))/(5**(473/483))) (1), (7**(662/485))/(2**(506/485)*3**(136/97)*5**(9/97)) (1), (2**(551/119)*7**(438/119))/(3**(341/119)*5**(63/17)) (1), (2**(369/47)*3**(585/94)*7**(85/47))/(5**(473/47)) (1), (3**(670/453)*5**(934/453))/(2**(101/151)*7**(253/151)) (1), (3**(901/773)*7**(591/773))/(2**(430/773)) (1), -((2**(107/61)*3**(84/61)*7**(451/244))/(5**(641/244))) (1), (2**(96/97)*5**(61/679))/(3**(494/679)*7**(111/679)) (1), (2**(909/520)*3**(159/260)*7**(93/65))/(5**(35/104)) (1), 35 (1), (85/18) (1), (2**(83/168))/(3**(97/168)*5**(7/120)*7**(37/840)) (1), (2**(794/311)*5**(710/311))/(3**(214/311)*7**(516/311)) (1), -((3**(587/36)*5**(23/4))/(2**(263/36)*7**(119/12))) (1), (3**(94/265)*5**(457/265))/(2**(398/265)*7**(81/265)) (1), (7**(133/130))/(2**(109/195)*3**(107/130)*5**(431/195)) (1), (3**(514/549)*5**(710/549))/(2**(184/183)*7**(367/549)) (1), -((2**(565/548)*5**(929/548)*7**(72/137))/(3**(109/274))) (1), (2**(314/177)*5**(394/177))/(3**(569/354)*7**(109/118)) (1), (2**(742/675)*5**(118/675)*7**(47/135))/(3**(238/675)) (1), (2**(811/87)*3**(163/87)*7**(138/29))/(5**(769/87)) (1), -((2**(73/34)*7**(392/17))/(3**(173/34)*5**(865/34))) (1), -((95/56)) (1), (75/19) (1), -((5/14)) (1), (25/11) (1), (5**(824/715)*7**(21/55))/(2**(160/143)*3**(301/715)) (1), (5**(2/37)*7**(42/37))/(2**(127/518)*3**(527/518)) (1), (3**(457/53)*7**(241/53))/(2**(312/53)*5**(407/53)) (1), (3**(963/278)*5**(41/139))/(2**(237/139)*7**(307/278)) (1), (2**(127/789)*3**(550/789)*7**(716/789))/(5**(776/789)) (1), (3**(161/257)*5**(542/771))/(2**(353/771)*7**(344/771)) (1), -((5**(476/295))/(2**(41/295)*3**(348/295)*7**(183/590))) (1), (2**(273/272)*7**(667/272))/(3**(371/136)*5**(457/136)) (1), (2**(103/45)*3**(313/60)*5**(263/60))/(7**(79/15)) (1), (2**(143/27)*7**(178/81))/(3**(127/54)*5**(227/81)) (1), (2**(41/295)*3**(348/295)*7**(183/590))/(5**(181/295)) (1), -((3**(333/305)*5**(361/610))/(2**(212/305)*7**(379/305))) (1), (2**(74/371)*3**(269/742)*7**(367/742))/(5**(14/53)) (1), (2**(257/79)*5**(152/237)*7**(46/79))/(3**(235/79)) (1), ((5+sqrt(5))/2) (1), (2**(437/39)*3**(161/13))/(5**(841/78)*7**(43/39)) (1), (5**(509/25)*7**(89/25))/(2**(742/25)*3**(416/25)) (1), (2**(61/176)*3**(34/11)*5**(763/176))/(7**(103/22)) (1), (5**(86/849))/(2**(557/849)*3**(469/849)*7**(38/283)) (1), (5**(454/409)*7**(213/409))/(2**(628/409)*3**(177/409)) (1), 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(1), (5**(979/998)*7**(227/499))/(2**(503/998)*3**(272/499)) (1), (2**(665/206)*5**(845/206))/(3**(819/206)*7**(243/103)) (1), (3**(369/683)*5**(655/683))/(2**(674/683)*7**(263/683)) (1), S^(2) (1)

Wall of Fame (top verified CFs)

#	Constant	a(n)	b(n)	Digits	Complexity
1	`e`	`[0, -1]`	`[3, 1]`	321.2	2.4
2	`phi`	`[-1, 0, 4]`	`[1, 2]`	321.1	3.5
3	`2/1*phi`	`[4, 0]`	`[2, 0]`	320.8	2.2
4	`1/2*4/pi`	`[0, 1, -2]`	`[1, 3]`	222.4	3.4
5	`4/pi`	`[0, 3, -2]`	`[1, 3]`	222.1	3.6
6	`4/3*4/pi`	`[0, 5, -2]`	`[1, 3]`	222.0	3.8
7	`phi`	`[1, 0, 0]`	`[1, 0]`	221.1	2.3
8	`S^(3)`	`[0, 7, -2]`	`[1, 3]`	221.0	4.0
9	`2/1*phi`	`[2, 3, 1]`	`[2, 1]`	220.8	3.9
10	`S^(5)`	`[0, 11, -2]`	`[1, 3]`	220.7	4.4

License

MIT — free for academic and commercial use.

PCF and V_quad: Reproducibility Package

Overview (Legacy)

This repository also contains reproducibility materials for two manuscripts:

PCF paper: Two families of polynomial continued fractions (Logarithmic Ladder, Pi Family). The Pi Family base case m=0 converges to 2/π; the Gauss CF non-membership proof is in _gauss_cf_phase2.py.
V_quad paper: Irrationality proof for constant V_quad via Wronskian, supported by >20,000 PSLQ exclusion runs.

Legacy Quick Start

python _gauss_cf_identification.py   # Exploratory analysis
python _gauss_cf_phase2.py           # Definitive Gauss CF non-membership proof
pytest tests/test_phase2.py -v       # Smoke tests

File Manifest

File	Description
`_gauss_cf_identification.py`	Exploratory CF analysis (8 sections)
`_gauss_cf_phase2.py`	Definitive proof: Gauss matching, SymPy exact, series ID
`tests/test_phase2.py`	Automated smoke tests (pytest)
`gen_convergence_plot.py`	Generates `convergence.png`
`coeff_table.csv`	First 10 CF coefficients in rational form
`summary.tex`	One-page LaTeX mathematical summary
`pslq_table.csv`	PSLQ exclusion run summary
`pslq_appendix.md`	Detailed PSLQ methodology and raw outputs
`conjecture1_report.md`	Closed-form search for p_n(1)
`requirements.txt`	Python dependencies
`README.md`	This file

Name		Name	Last commit message	Last commit date
Latest commit History 7 Commits
.github/workflows		.github/workflows
G01_Induction		G01_Induction
agent_state		agent_state
aria		aria
breakthroughs		breakthroughs
collatz		collatz
discoveries/assets		discoveries/assets
harvest		harvest
lean		lean
materials_microlaws		materials_microlaws
micro_laws_discovery		micro_laws_discovery
multi_agent_discovery		multi_agent_discovery
multi_agent_discussion		multi_agent_discussion
open-multi-agent		open-multi-agent
prompts		prompts
proof_ready		proof_ready
python		python
ramanujan_agent		ramanujan_agent
ramanujan_physics		ramanujan_physics
rbg_runs		rbg_runs
results		results
shards/20260406_083330		shards/20260406_083330
siarc_outputs		siarc_outputs
siarc_v6		siarc_v6
siarc_v6_champion_archive		siarc_v6_champion_archive
snapshots		snapshots
templates		templates
tests		tests
tex		tex
unsolved_solver		unsolved_solver
v8_reality_sync		v8_reality_sync
.env.example		.env.example
.gitignore		.gitignore
66_pilot_results.json		66_pilot_results.json
AGENT_HEALTH_SCORE.md		AGENT_HEALTH_SCORE.md
CLAIM_VERIFICATION.md		CLAIM_VERIFICATION.md
CONTRIBUTING.md		CONTRIBUTING.md
CollatzBirkhoff.lean		CollatzBirkhoff.lean
CollatzSpectralGap.lean		CollatzSpectralGap.lean
DISCOVERY_REPORT.md		DISCOVERY_REPORT.md
EPOCH5_COMMAND_CENTER.md		EPOCH5_COMMAND_CENTER.md
G01_Induction_Log.md		G01_Induction_Log.md
GCF_Borel_Lean4.lean		GCF_Borel_Lean4.lean
GTLB_Paper.html		GTLB_Paper.html
HANDOFF_2026-04-14_403f575.md		HANDOFF_2026-04-14_403f575.md
HANDOFF_TEMPLATE.md		HANDOFF_TEMPLATE.md
LICENSE		LICENSE
QLID_Paper.html		QLID_Paper.html
RAMANUJAN_FAST_AGENT_DESIGN_NOTE.md		RAMANUJAN_FAST_AGENT_DESIGN_NOTE.md
README.md		README.md
REPRODUCE.md		REPRODUCE.md
REPRODUCE.txt		REPRODUCE.txt
RESEARCH_PROTOCOL.md		RESEARCH_PROTOCOL.md
REVIEW_RESPONSE.md		REVIEW_RESPONSE.md
SELF_ITERATING_BREAKTHROUGH_AGENT_PROMPT.md		SELF_ITERATING_BREAKTHROUGH_AGENT_PROMPT.md
V_quad_1000digits.txt		V_quad_1000digits.txt
V_quad_export_bundle.txt		V_quad_export_bundle.txt
WallisFamily.lean		WallisFamily.lean
_add_sections_v5.py		_add_sections_v5.py
_add_sections_v6.py		_add_sections_v6.py
_add_sections_v7.py		_add_sections_v7.py
_analyze_deep_sweep.py		_analyze_deep_sweep.py
_analyze_discoveries.py		_analyze_discoveries.py
_analyze_kloosterman_sweep.py		_analyze_kloosterman_sweep.py
_analyze_run.py		_analyze_run.py
_apery_m_family_scan.py		_apery_m_family_scan.py
_arxiv_check.py		_arxiv_check.py
_audit_tex.py		_audit_tex.py
_bench_timing.py		_bench_timing.py
_bridge_auto_responder.py		_bridge_auto_responder.py
_build_aria_review.py		_build_aria_review.py
_build_html_v5.py		_build_html_v5.py
_build_html_v6.py		_build_html_v6.py
_build_html_v7.py		_build_html_v7.py
_catalan_targeted_search.py		_catalan_targeted_search.py
_cell_outputs_v2.json		_cell_outputs_v2.json
_check_cas.py		_check_cas.py
_check_ising_results.py		_check_ising_results.py
_check_nb.py		_check_nb.py
_check_novel.py		_check_novel.py
_check_paper.py		_check_paper.py
_check_state.py		_check_state.py
_check_syntax_rbg.py		_check_syntax_rbg.py
_check_tex.py		_check_tex.py
_check_values.py		_check_values.py
_combined_status.py		_combined_status.py
_debug_cas2.py		_debug_cas2.py
_debug_cas_failure.py		_debug_cas_failure.py
_debug_extraction.py		_debug_extraction.py
_debug_extraction_v2.py		_debug_extraction_v2.py
_debug_formal.py		_debug_formal.py
_debug_hp_nsimplify.py		_debug_hp_nsimplify.py
_debug_nsimplify.py		_debug_nsimplify.py
_debug_step_by_step.py		_debug_step_by_step.py
_deep_explore.py		_deep_explore.py
_deep_space_sweep.py		_deep_space_sweep.py
_diag_validate.py		_diag_validate.py
_ec_verify_ap.py		_ec_verify_ap.py
_ec_weierstrass.py		_ec_weierstrass.py
_ec_weierstrass_v2.py		_ec_weierstrass_v2.py

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A One-Parameter Family of Wallis-Type Polynomial Continued Fractions: Discovery and Formal Verification

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Run the Lean 4 proof

Reproduce the numerical results

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arXiv

Ramanujan Breakthrough Generator

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Mathematical Background

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Quick Start

CLI Usage

Python API

With Research Context (Agent Integration)

Search Modes

Target Constants Library

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Integration Example: gpt-researcher

Integration Example: Microsoft RD-Agent

Integration Example: Simple One-Call API

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Wall of Fame (top verified CFs)

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PCF and V_quad: Reproducibility Package

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