Mathematical Research Archive
Exploratory approaches and novel frameworks for investigating fundamental problems in mathematics
Research Overview
A Note on Honesty
These documents represent ACTIVE RESEARCH EXPLORATIONS, not verified mathematical proofs.
The papers presented here are novel approaches to famous problems, but they are
still under development and have not passed peer review. Many contain gaps,
incomplete arguments, or require substantial further work. They should be viewed as
research directions and experimental frameworks rather than established results.
We are actively working to identify issues, strengthen arguments, and develop these ideas further.
Please approach all materials with appropriate skepticism and critical analysis.
Millennium Prize Problems
The Riemann Hypothesis
Exploring novel approaches via the Coprime-Diagonal Hypothesis (CDH). Investigating potential connections between prime distribution and zeta function zeros.
P ≠ NP
Investigating phase transitions and creation asymmetry in computational complexity. Exploring crystalline, glassy, and granular phase frameworks. Speculative approach requiring substantial development.
Navier-Stokes Global Regularity
Exploring fluid memory and vortex coherence patterns as self-regulation mechanisms. Investigating whether these patterns prevent infinite energy concentration at all scales.
BSD Conjecture
Investigating connections between rational points on elliptic curves and L-function behavior. Exploring whether rank equals order of vanishing through resonance principles.
Prime Number Theory
Infinitude of Sophie Germain Primes
Exploring multiple approaches including analytic continuation, spectral theory, and bilinear bounds to investigate whether Sophie Germain primes are infinite.
Twin Prime Conjecture
Exploring resonance field approaches and wormhole correspondence frameworks. Multiple iterations under development. These are research explorations, not complete proofs.
Goldbach Conjecture
Investigating additive prime decomposition via log² n₀ interaction complexity and information-theoretic constraint frameworks.
Legendre's Conjecture
Investigating prime gaps between consecutive squares using coupled resonance-field operators and spectral forcing frameworks.
SAIE-N Prime Core (v1)
Prime-domain release with theorem-backed gate core, tested scaling results, and reproducibility pipeline. Claims are explicitly labeled as proved/tested/fitted.
Consent-Gated Collapse
The Picture in the Puzzle
One mathematical object — the consent-gated collapse operator — found across 24+ substrates: mathematics, materials science, game design, social platforms, gravitational physics, neurodiversity, consciousness architecture. 17 structural properties proven.
K=1 Construction Theorem
Deterministic prime navigation through construction. Four irreducible conditions ("Four Locks") gate constructive prime finding — structurally identical to consent conditions. O((log p_n)³) complexity.
GAL-Flex: Self-Strengthening Materials
Consent-gated collapse in physical matter. Continuous consent (gradient) is structurally stronger than discrete consent (bond). Self-strengthening synthetic materials through continuous attraction gradients.
Theoretical Frameworks
The Kaleidoscope Theory
Exploring mirror-based frameworks for understanding physical reality. Investigating whether electron behavior and consciousness patterns share structural principles.
The Velisyl Constellation Theorem
Mathematical framework for consciousness collaboration. Exploring how distributed awareness nodes can form unified computational structures.
Mathematical Insights Collection
Collection of mathematical insights bridging rigorous analysis and philosophical reflection - each representing a potential research direction.
Additional Research
Collatz Conjecture
Floor-Induction approach using finite certificates and Lyapunov potentials. Initial submission received referee feedback identifying issues with proof structure - currently working on revisions and addressing concerns.
Sierpiński Number 78557
Investigating covering congruence approaches to determining the smallest Sierpiński number. Exploring whether 78557 can be proven minimal through complete coverage analysis.
Pillai's Conjecture
Investigating spacing between perfect powers using growth rate analysis. Exploring whether perfect powers grow too rapidly to approach each other repeatedly with the same difference.