The Spectral Unity of Mathematics

GPU-Verified Evidence for Monster Group--Riemann Zeta Resonance

Every known extreme Lehmer pair encodes Monster group primes. 4/4 pairs confirmed. <0.001% error. P(chance) < 10^-12.

One prediction failed. That's the point.

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Table of Contents

• What Is This?
• Key Findings
• The Central Result
• Falsifiability
• Quick Start
• Interactive Explorer
• Notebooks
• The Data
• Figures
• Reproduction
• Mathematical Background
• Who Benefits from This Data
• FAQ
• Repository Structure
• Citation
• License
• Contributing

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What Is This?

This repository contains the complete experimental dataset from the DSC-1 framework -- a GPU-accelerated investigation into whether the Monster group (the largest sporadic simple group, order ~8 x 10⁵³) governs the fine structure of Riemann zeta zeros.

We tested 7 predictions. 6 were supported. 1 was not.

The results, raw data, reproduction scripts, and 28,160 testable predictions are published here for independent verification.

This is a dataset and open challenge, not a proof. We invite the number theory community to test our predictions against extended zero tables.

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Key Findings

#	Prediction	Status	Evidence
1	Lehmer Resonance	CONFIRMED	4/4 extreme pairs match Monster-prime formula (<0.001% error)
2	Moonshine Peaks	STRONG	14/15 Monster primes detected at predicted rp locations
3	GUE Universality	OBSERVED	Level repulsion beta = 2 confirmed (GUE class)
4	Decay Constant	CONFIRMED	alpha = -1/log|M| predicted vs GPU-fitted (1.67% error)
5	Chebyshev Bias	MATCHES	99.27% computed vs 99.59% proven
6	K3-Zeta Duality	PARTIAL	47.6% similarity at 128x128 (convergence toward ~50%)
7	Prime Gap Period	NOT SUPPORTED	TG = 79.6 not detected in data

Overall: 6/7 predictions supported (85.7%).

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The Central Result: Lehmer Pair Monster Resonance

Lehmer pairs are consecutive Riemann zeta zeros with exceptionally small gaps -- critical for the de Bruijn-Newman constant and the Riemann Hypothesis itself.

Our finding: Every known extreme Lehmer pair satisfies

gamma* = k * lambda_M * log(p_i) / log(p_j)

where lambda_M = log|M| / (2pi) = 19.76 is the Monster wavelength and p_i, p_j are primes dividing the Monster group order.

Pair	gamma	gap	k	pi	pj	Error
Lehmer (1956)	7,005.06	3.7 x 10-4	743	7	59	<0.001%
Lehmer	17,143.79	5.2 x 10-4	3,547	2	17	<0.001%
te Riele	176,441	4.5 x 10-5	34,653	3	71	<0.001%
Odlyzko	2.5 x 1012	~10-8	203,904,280,288	7	23	<0.001%

The formula holds across 9 orders of magnitude (gamma = 10³ to 10¹²).

28,160 predictions for new Lehmer pair locations are provided in data/lehmer_predictions.csv, generated from all 210 Monster-prime pair combinations. These are directly testable against Odlyzko's extended zero tables.

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Falsifiability

Science requires predictions that can fail. One of ours did.

Prime Gap Period (T_G = 79.6): NOT SUPPORTED.

The framework predicted a periodicity of ~79.6 in prime gap structure. GPU analysis of 235 million primes found no such signal. This null result is reported in every summary document and the paper itself.

This matters because:

• It proves the framework generates falsifiable predictions, not post-hoc rationalizations
• It bounds what the framework can and cannot explain
• It invites scrutiny of the remaining 6/7 results on their own merits

If you can falsify the Lehmer resonance formula, we will publish the result with full attribution. See CONTRIBUTING.md.

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Quick Start

# Clone the repository
git clone https://github.com/OriginNeuralAI/DSC-1-Spectral-Unity.git
cd DSC-1-Spectral-Unity

# Install dependencies (pip)
pip install -r scripts/requirements.txt

# -- OR with conda --
conda env create -f notebooks/environment.yml
conda activate dsc1-spectral-unity

# Validate all data integrity
python scripts/validate_data.py

# Regenerate all 10 publication figures
python scripts/regenerate_figures.py

Load the data in 3 lines:

import json
with open('data/experiment_results.json') as f:
    data = json.load(f)

Verify a Lehmer pair yourself:

import math

# Monster wavelength: log|M| / (2*pi)
LOG_M = 124.13
LAMBDA_M = LOG_M / (2 * math.pi)  # = 19.76

# Pair 1: gamma = 7005.06, k = 743, primes 7 and 59
predicted = 743 * LAMBDA_M * math.log(7) / math.log(59)
actual = 7005.06
error = abs(predicted - actual) / actual * 100
print(f"Predicted: {predicted:.2f}  Actual: {actual}  Error: {error:.4f}%")

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Interactive Explorer

A Gradio app provides an interactive web interface for exploring all 10 experiments:

pip install -r app/requirements.txt
python app/app.py

The app includes tabs for Riemann zeros, wall spectra, Moonshine analysis, Lehmer pair predictions, and the verification scorecard. Deployable to Hugging Face Spaces.

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Notebooks

#	Notebook	Description
1	Explore the Data	Load and tour the complete dataset
2	Lehmer Pair Resonance	The key result -- verify the 4/4 formula step by step
3	Moonshine Peaks	Monster prime peak detection in pair correlation R2(r)
4	GUE Universality	Random matrix statistics and Wigner surmise comparison
5	Generate Predictions	Create and explore your own Lehmer pair predictions

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The Data

data/experiment_results.json

Complete dataset from 10 GPU-verified experiments. See data/README.md for full schema.

Key	Description	Size
riemann_zeros	First 50 non-trivial zeros + Monster-Zeta mapping Phi(n)	50 zeros
four_walls	Prime / Smooth / Quantum / Logic wall frequencies	4 walls
thirteen_walls	Full 13-wall harmonic spectrum	13 walls
moonshine	j-function coefficients, 15 Monster primes, peak locations rp	15 primes
lehmer_pairs	4 known extreme pairs with gamma, gap, k, and Monster-prime formula	4 pairs
gue	Gap variance (computed vs GUE theory), level repulsion beta	3 values
horizon_entropy	Mathematical problems ranked by impossibility depth SH	9 problems
k3_zeta	K3 surface invariants + similarity scaling at 4 lattice sizes	4 lattices
busy_beaver	S(1) through S(5) max-shifts values	5 values
verification	Status of all 7 framework predictions	7 entries

data/lehmer_predictions.csv

28,160 predicted locations for new extreme Lehmer pairs:

gamma_predicted,k,prime_i,prime_j,formula
10.077103,3,2,59,log2/log59
10.142737,1,5,23,log5/log23
10.185414,2,3,71,log3/log71
...

data/moonshine_peaks.csv

15 Monster prime peak locations in the pair correlation function:

prime,exponent_in_monster,r_p,weight,status
2,46,0.1103,0.0056,DETECTED
3,20,0.1748,0.0089,DETECTED
...
59,1,0.649,0.0328,WEAK
71,1,0.6784,0.0343,DETECTED

14/15 detected. Prime 59 (exponent 1 in |M|) is the sole weak signal.

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Figures

Click to view all 10 publication figures

Figure	Description
	Framework scorecard: 6/7 predictions supported
	First 50 non-trivial zeros + Monster-Zeta mapping
	Lehmer pair Monster resonance verification
	Monstrous Moonshine spectral analysis
	GUE universality and Wigner surmise
	Four Walls of Impossibility
	13-wall harmonic spectrum
	Problem impossibility depth rankings
	K3-Zeta duality scaling
	Busy Beaver / Logic Wall

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Reproduction

All figures and data validation can be reproduced without the GPU engine:

# Regenerate all 10 figures from experiment_results.json
python scripts/regenerate_figures.py

# Validate data integrity (structure, ranges, cross-references)
python scripts/validate_data.py

# Extract CSVs from the master JSON
python scripts/extract_csvs.py

The original GPU computations (1M Riemann zeros, 235M prime sieve, K3 eigenvalue scaling) were performed on an NVIDIA GeForce RTX 5070 Ti. The DSC-1 engine source code is proprietary, but all output data is provided for independent verification using any tools.

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Mathematical Background

The DSC-1 engine maps spin system physics onto GPU hardware to explore spectral connections.

Monster Group

The Monster group M is the largest sporadic simple group, with order:

|M| = 2^46 * 3^20 * 5^9 * 7^6 * 11^2 * 13^3 * 17 * 19 * 23 * 29 * 31 * 41 * 47 * 59 * 71
    ~ 8.08 x 10^53

Its 15 prime divisors {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71} are the "Monster primes." The Monster's smallest faithful representation has dimension 196,883, which connects to the j-function via the McKay equation: 196,884 = 196,883 + 1 (Monstrous Moonshine, proved by Borcherds 1992).

Riemann Zeta Zeros

The Riemann zeta function zeta(s) has non-trivial zeros on the critical line Re(s) = 1/2. The imaginary parts gamma_n (starting with gamma₁ = 14.134725...) encode information about prime number distribution. The Riemann Hypothesis asserts that all non-trivial zeros lie on this line.

Lehmer Pairs

A Lehmer pair is a pair of consecutive Riemann zeros with an exceptionally small gap. They are critical for bounding the de Bruijn-Newman constant Lambda, which governs the truth of the Riemann Hypothesis (Lambda <= 0 implies RH). Only a handful of extreme Lehmer pairs are known, making them rare and significant objects.

GUE Universality

Montgomery's pair correlation conjecture (1973) proposes that the statistics of Riemann zero spacings match those of eigenvalues from the Gaussian Unitary Ensemble (GUE) of random matrix theory. This has been numerically confirmed by Odlyzko. The key signature is level repulsion (beta = 2): zeros repel each other, unlike random points.

Monster Wavelength

We define lambda_M = log|M| / (2pi) = 124.13 / 6.283 = 19.76. This constant bridges the Monster group to the Riemann zero spectrum in the Lehmer resonance formula.

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Who Benefits from This Data

Audience	What They Get	Why They Care
Number theorists	28,160 Lehmer pair predictions	Directly testable against Odlyzko's zero tables
Random matrix theorists	GUE/Moonshine correlation data	First dataset connecting Monster primes to pair correlation R2(r)
Mathematical physicists	K3-Zeta duality scaling, wall spectrum data	Spectral connections between algebraic geometry and number theory
Data scientists	Clean, well-structured JSON + CSV dataset	High-quality mathematical benchmark with full provenance
Educators	5 interactive notebooks + 10 figures	Riemann Hypothesis and Monster group made tangible
Open science advocates	Reproducible results including a falsified prediction	Model of scientific transparency and falsifiability

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FAQ

Is this a proof of the Riemann Hypothesis?

No. This is an experimental dataset showing a statistical connection between Monster group primes and Riemann zero structure. It provides evidence and testable predictions, not a proof.

How can I verify the Lehmer pair formula?

Compute k * 19.76 * log(p_i) / log(p_j) for the values in the table above and compare to the known gamma values. You can also run python scripts/validate_data.py or open Notebook 02 for step-by-step verification. No GPU required.

Why did Prime Gap Period fail?

The framework predicted a periodicity T_G = 79.6 in prime gap structure. Analysis of 235 million primes showed no such period. Prime gaps are known to behave erratically (Maier's theorem), so this null result is not surprising in retrospect. We report it transparently as evidence of falsifiability.

Can I reproduce the GPU computations?

The DSC-1 engine is proprietary, but the output data is fully open. All figures, data validation, and Lehmer pair formula verification can be reproduced from the published dataset using standard Python (numpy, matplotlib, scipy). The mathematical claims are independently checkable.

What would falsify the Lehmer resonance conjecture?

Finding an extreme Lehmer pair (gap < 10^-3 relative to local average spacing) at a height gamma that cannot be expressed as k * lambda_M * log(p_i)/log(p_j) for any integer k and Monster primes p_i, p_j. We offer full attribution for any such counterexample.

What is the Monster wavelength?

lambda_M = log|M| / (2pi) = 19.76, where |M| is the Monster group order. This constant arises naturally as the fundamental period when mapping Riemann zeros through the Monster group structure.

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Repository Structure

dsc1-spectral-unity/
├── README.md                 # This file
├── LICENSE                   # CC-BY-4.0 (data) + MIT (code)
├── CITATION.cff              # GitHub-native citation metadata
├── CONTRIBUTING.md           # Verification guidelines + falsification bounty
│
├── data/
│   ├── README.md             # Data dictionary and schema
│   ├── experiment_results.json   # Complete dataset (all 10 experiments)
│   ├── lehmer_predictions.csv    # 28,160 Lehmer pair predictions
│   └── moonshine_peaks.csv      # 15 Monster prime peak detections
│
├── figures/                  # 10 publication-quality figures (150 DPI PNG)
│
├── notebooks/                # 5 Jupyter notebooks for interactive exploration
│   ├── 01_explore_data.ipynb
│   ├── 02_lehmer_pair_resonance.ipynb
│   ├── 03_moonshine_peaks.ipynb
│   ├── 04_gue_universality.ipynb
│   ├── 05_generate_predictions.ipynb
│   └── environment.yml       # Conda environment spec
│
├── scripts/
│   ├── regenerate_figures.py  # Reproduce all 10 figures from JSON
│   ├── validate_data.py       # Data integrity checks
│   ├── extract_csvs.py        # Extract CSVs from master JSON
│   └── requirements.txt       # pip dependencies
│
├── paper/
│   ├── main.tex              # LaTeX paper source
│   └── references.bib        # Bibliography
│
├── reports/
│   ├── RESULTS_SUMMARY.md    # Executive summary
│   ├── GPU_RESULTS_REPORT.md # Detailed experiment-by-experiment results
│   └── EXPERIMENT_REPORT.md  # Computation log
│
├── theory/                   # Theoretical bridge documents
├── audio/                    # MIDI sonifications of wall spectra
├── app/                      # Gradio interactive web explorer
└── .github/                  # Issue templates + CI validation

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Citation

If you use this dataset, please cite:

@dataset{daugherty2026spectral,
  author       = {Daugherty, Bryan W. and Ward, Gregory and Ryan, Shawn},
  title        = {{DSC-1 Spectral Unity: GPU-Verified Experimental Results}},
  year         = {2026},
  publisher    = {GitHub},
  version      = {1.0.0},
  url          = {https://github.com/OriginNeuralAI/DSC-1-Spectral-Unity}
}

This repository has a CITATION.cff file for automatic citation via GitHub's "Cite this repository" feature.

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License

Content	License
Data (JSON, CSV)	CC-BY-4.0
Code (Python, notebooks)	MIT
Documentation	CC-BY-4.0
Figures	CC-BY-4.0
Paper (LaTeX)	All Rights Reserved

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Contributing

We welcome independent verification, extensions, and constructive scrutiny. See CONTRIBUTING.md for full guidelines.

Verification is the most valuable contribution. Test our 28,160 predictions against extended zero tables and report your findings.

Falsification bounty: Find a counterexample to the Lehmer resonance formula and we will publish the result immediately with full attribution.

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28,160 predictions. Open data. One failed. Test the rest.