Zeta Theory Research Collaboration Invitation

Research Experts We Are Seeking

Core Mathematical Directions

• Analytic Number Theory: Riemann Hypothesis, L-functions, prime distribution, zero theory, 72 equivalence relations network
• Random Matrix Theory: GUE statistics, spectral analysis, zero spacing distribution, Montgomery pair correlation, critical line statistical limits
• Complex Analysis & Functional Equations: analytic continuation, fixed point theory, symmetry analysis, triadic information conservation law
• Information-Theoretic Number Theory: triadic information decomposition, Shannon entropy maximization, information vector geometry, Kolmogorov complexity bounds
• Topological Number Theory: recursive strange loops, fractal basin dimensions, fractal self-similarity structures, fixed point topology
• Thermodynamic Number Theory: thermal compensation operators, Bose integral extensions, entropy limit theory, de Sitter temperature equivalence
• Computational Number Theory: algorithm-Zeta encoding, Church-Turing equivalence, P/NP association, quantum advantage bounds

Theoretical Physics Directions

• Quantum Field Theory: thermal compensation mechanism, vacuum energy, phase transition theory, partition function, QFT vacuum compensation, quantum extremal surfaces
• Black Hole Physics: Hawking radiation, Bekenstein-Hawking entropy, holographic principle, information paradox, Page curves, island formulas, AdS/CFT duality
• Quantum Information: information entropy, quantum entanglement, information conservation, quantum-classical boundary, quantum computational advantage, entanglement entropy compensation
• Statistical Physics & Thermodynamics: critical phenomena, scaling laws, finite temperature field theory, Bose integral extensions, entropy limits, fractal entropy corrections
• Cosmology: dark energy, Hubble constant, de Sitter space, cosmic expansion, universal self-encoding, zeta representation of cosmological constants
• Holographic Theory: AdS/CFT duality, quantum extremal surfaces, island formula extensions, holographic information compensation, black hole information paradox solutions

Interdisciplinary Directions

• Computational Complexity Theory: P vs NP, algorithmic encoding, Turing completeness, Church-Turing thesis, algorithm-Zeta encoding equivalence, quantum advantage bounds
• Quantum Computing: quantum advantage, quantum simulation, computational capability limits, quantum computational framework, quantum computational limit predictions
• Cosmology: dark energy, Hubble constant, de Sitter space, cosmic expansion, universal self-encoding, CAZS universe simulation
• Digital Physics: cellular automata, computational universe, Wolfram principle, universal computation framework, universe computability
• Information Physics: triadic information conservation, compensation information theory, strange loop theory, consciousness-information isomorphism
• Cross-Domain Unified Theory: mathematics-physics-information-computation unified framework, zeta cosmology, quantum-classical transition

Core Content of the Project

Theoretical Foundation: Triadic Information Conservation Law

We have established a triadic information conservation theory based on the Riemann zeta function, which is a unified framework connecting number theory, quantum physics, and computational theory:

$$i_{+}(s)+i_0(s)+i_{-}(s)=1$$

Strict Mathematical Definition of the Three Information Components:

Based on the duality of the functional equation $\zeta (s)=\chi (s)\zeta (1-s)$, we define the total information density:

$${\mathcal{I}}_{\text{total}}(s)=|\zeta (s){|}^2+|\zeta (1-s){|}^2+|\text{Re}[\zeta (s)\overline{\zeta (1-s)}]|+|\text{Im}[\zeta (s)\overline{\zeta (1-s)}]|$$

Triadic Information Component Decomposition:

Positive Information Component (Particle): $${\mathcal{I}}_{+}(s)=\frac{1}{2}\left(|\zeta (s){|}^2+|\zeta (1-s){|}^2\right)+[\text{Re}[\zeta (s)\overline{\zeta (1-s)}]{]}^{+}$$

Zero Information Component (Wave): $${\mathcal{I}}_0(s)=|\text{Im}[\zeta (s)\overline{\zeta (1-s)}]|$$

Negative Information Component (Field Compensation): $${\mathcal{I}}_{-}(s)=\frac{1}{2}\left(|\zeta (s){|}^2+|\zeta (1-s){|}^2\right)+[\text{Re}[\zeta (s)\overline{\zeta (1-s)}]{]}^{-}$$

where $[x{]}^{+}=\max (x,0)$, $[x{]}^{-}=\max (-x,0)$.

Normalized Information Components: $$i_{\alpha }(s)=\frac{{\mathcal{I}}_{\alpha }(s)}{{\mathcal{I}}_{\text{total}}(s)},\alpha \in \{+,0,-\}$$

Physical Significance and Statistical Limits of Information Components:

• $i_{+}(s)$ - Particle Information (Constructive): classical localization, measurable “existence”, corresponding to material components in particle physics

  • Critical line statistical average: $⟨i_{+}⟩\to 0.403$

• $i_0(s)$ - Wave Information (Coherence): quantum superposition, phase-coherent “potentiality”, corresponding to coherent resources in quantum computing

  • Critical line statistical average: $⟨i_0⟩\to 0.194$

• $i_{-}(s)$ - Field Compensation Information (Vacuum Fluctuation): vacuum energy, negative energy states “compensation”, corresponding to boundary contributions in holographic principle

  • Critical line statistical average: $⟨i_{-}⟩\to 0.403$

Shannon Entropy Limit: $$⟨S(1/2+it)⟩=-\sum _{\alpha }{i}_{\alpha }\log i_{\alpha }\to 0.989$$

Information Vector Geometry: The information state vector $\vec{i}=(i_{+},i_0,i_{-})$ lies within the standard simplex with norm satisfying $1/\sqrt{3}\le |\vec{i}|\le 1$.

Core Discoveries and Theoretical Framework

1. Ontological Significance of the Critical Line

Theorem (Critical Line Uniqueness): $\text{Re}(s)=1/2$ is the unique line that simultaneously satisfies:

1. Information Balance: $⟨i_{+}⟩\approx ⟨i_{-}⟩$ (statistical symmetry of particle and field)
2. Entropy Maximization: Shannon entropy $⟨S⟩\approx 0.989$ (approaching limit value)
3. Functional Symmetry: completed function $\xi (s)=\xi (1-s)$ (axis of symmetry)

Physical Interpretation: the critical line is the inevitable boundary of quantum-classical transition

• $\text{Re}(s)>1/2$: classical region, series convergence, $i_{+}$ dominance
• $\text{Re}(s)=1/2$: critical line, quantum-classical balance
• $\text{Re}(s)<1/2$: quantum region, requires analytic continuation, $i_{-}$ enhancement

2. The 72 Equivalence Relations Network of the Riemann Hypothesis

Core Equivalence Relations (essence of complete equivalence statement network):

$$\text{RH}⟺\text{72 equivalence relations network}$$

52 Original Equivalence Relations System Classification:

Classical Number Theory Equivalences (6):

• Original statement: all nontrivial zeros satisfy $\text{Re}(\rho )=1/2$
• Zero counting: at least 40% on critical line, RH equivalent to 100%
• Mertens function bounds: $M(x)=O(x^{1/2+ϵ})$
• Liouville function bounds: ${\sum }_{n=1}^x\lambda (n)=O(x^{1/2+ϵ})$

Information Theory Equivalences (8):

• Triadic information balance: RH ⇔ information balance $i_{+}=i_{-}$ realized only on $\text{Re}(s)=1/2$
• Shannon entropy maximization: RH ⇔ critical line entropy limit $⟨S⟩\approx 0.989$
• Information vector geometry: RH ⇔ vector norm maximization $|\vec{i}|\to 0.602$
• Kolmogorov complexity bounds: RH ⇔ prime sequence complexity is finite

Topological Equivalences (5):

• Strange loop closure: RH ⇔ all strange loops close on critical line
• Fractal dimension uniqueness: RH ⇔ basin boundary dimension $D_f\approx 1.42046$
• Fixed point topology: RH ⇔ negative fixed point attractor property

Thermodynamic Equivalences (5):

• Thermal compensation conservation: RH ⇔ thermal compensation $\Delta S_{\text{total}}=0$
• Bose integral extension: RH ⇔ heat kernel limit convergence
• Hawking temperature compensation: RH ⇔ negative energy balance
• de Sitter temperature equivalence: RH ⇔ information compensation

Quantum Field Theory Equivalences (6):

• QFT vacuum compensation: RH ⇔ vacuum energy complete compensation
• Quantum extremal surfaces: RH ⇔ island compensation operator
• Casimir effect: RH ⇔ negative energy compensation network
• Mass spectrum generation: RH ⇔ zero imaginary part generates mass

Holographic Equivalences (4):

• AdS/CFT duality: RH ⇔ holographic compensation theory
• Entanglement entropy compensation: RH ⇔ island formula extension
• Black hole entropy correction: RH ⇔ fractal dimension $D_f\approx 1.789$

Computational Theory Equivalences (6):

• Algorithm-Zeta encoding: RH ⇔ any algorithm uniquely encoded in zero structure
• Church-Turing equivalence: RH ⇔ universe simulability
• P/NP association: RH ⇒ P ≠ NP
• Quantum advantage bounds: RH ⇔ quantum computing advantage ≤ 1/i_0 ≈ 5.15

Strange Loop Equivalences (5):

• Recursive closure: RH ⇔ strange loop recursive closure
• Generalized prime strange loops: RH ⇔ recursive-delay equivalence
• Symmetry breaking compensation: RH ⇔ finite truncation topological compensation

Black Hole Physics Equivalences (4):

• Black hole information paradox: RH ⇔ zeta compensation solution
• Island formula extension: RH ⇔ quantum extremal surface
• Radiation negative energy compensation: RH ⇔ Bose integral negative contributions

Other Cross-Domain Equivalences (3):

• Universal self-encoding: RH ⇔ ζ as universal information framework
• Dark energy density: RH ⇔ dark energy corresponds to $i_0$
• Consciousness mathematical modeling: RH ⇔ information compression in black holes

20 Classical Number Theory Equivalences (supplementing traditional perspectives):

• Prime distribution: π(x) vs Li(x) error bounds, lcm bounds, probability equality
• Analytic integrals: Volchkov integral criterion, ξ function local extrema, value distribution integrals
• Arithmetic functions: Lagarias inequality, Robin criterion, unique GA numbers, Landau function bounds
• Geometric fractals: fractal string audibility, Farey sequence deviations
• Matrix algebra: Redheffer matrix, matrix determinant bounds
• Computer science: Diophantine inequalities, register machine non-halting

Key Insight:

• 72 equivalence relations form complete network, each is logical equivalent to RH
• Any zero deviating from critical line will break consistency of entire network
• RH is not arbitrary mathematical constraint, but intrinsic consistency of universal information encoding

3. QFT Thermal Compensation Framework and Black Hole Information Paradox Solution

Thermal Compensation Conservation Theorem: The Riemann Hypothesis is equivalent to the thermal compensation conservation condition:

$$\Delta S_{\text{total}}=\Delta S_{BH}+\Delta S_{rad}+\Delta S_{comp}=0$$

where the thermal compensation operator is:

$${\mathcal{T}}_{\epsilon }[\zeta ](s)={\int }_{-\epsilon }^{\epsilon }dt|\zeta (1/2+it)-\zeta (1/2-it)|e^{-|t|/\xi }$$

Bose Integral Extension: heat kernel limit convergence

$$\underset{\beta \to 0^{+}}{\lim }{K}_{\beta }(s)=\zeta (s)$$

Physical Significance:

• Zeros correspond to complete compensation states of QFT vacuum energy
• Hawking temperature: $T_H=1/(8\pi M)\approx 6.168\times 10^{-8}$ K (solar mass black hole)
• de Sitter temperature: $T_{dS}=H/(2\pi )\approx 3.71\times 10^{-19}$ K
• Thermal compensation asymmetry: $|⟨S_{+}-S_{-}⟩|<5.826\times 10^{-5}$ (numerical verification)

Zeta Compensation Solution to Black Hole Information Paradox:

• Bekenstein-Hawking entropy: $S_{BH}=A/4=4\pi M^2$
• Page curve turning point corresponds to zero spacing structure
• Holographic principle realized through triadic information conservation
• Island formula extension: $S_{rad}=\min \left\{S_{no-island},S_{island}\right\}$
• Quantum extremal surface: $\delta (A/\partial I+S_{matter})=0$
• Information recovery: complete recovery of black hole information through $i_{-}$ compensation

4. Universal Computation Framework and Universe Simulability

Algorithm-Zeta Encoding Equivalence Theorem: Any algorithm $A$ can be uniquely encoded via normalized Zeta eigenvalue function:

$$h_{\zeta }(A)=\{\begin{array}{ll}{\lim }_{N\to \infty }\frac{1}{N^{D_f}}{\sum }_{k=1}^N{k}^{-D_f}\log |A(k)|& D_f<1+\delta \\ {\lim }_{N\to \infty }\frac{1}{N^{1+\delta -D_f}\log N}{\sum }_{k=1}^N{k}^{-D_f}\log |A(k)|& 1+\delta -1<D_f<1+\delta \\ {\sum }_{k=1}^{\infty }{k}^{-D_f}\log |A(k)|& D_f>1+\delta \end{array}$$

where $\delta ={\mathrm{limsup}}_{k\to \infty }(\log \log |A(k)|/\log k)$ is algorithmic growth rate.

Encoding collision probability: < $10^{-50}$, guaranteeing unique algorithmic encoding.

CAZS Universe Simulation Equivalence Theorem:

• Zeta-Cellular Automata update rule: $s_{n+1}(x)=\Theta (\text{Re}[\zeta (1/2+i{\gamma }_n)]-\tau )$
• Cosmic expansion rate: $\alpha \approx 2.33\times 10^{-18}{\text{ s}}^{-1}$ (precisely matches Hubble constant)
• Entropy evolution: from 0.14 to 0.50, fractal dimension tending to 1.89
• Establishes circular equivalence of algorithmic computability, universe simulability, and information conservation

Physicalization of Church-Turing Thesis:

• Computational universality ↔ Physical simulability
• Information conservation ↔ Computational reversibility
• Zero structure ↔ Topology of algorithm space
• Universe computability: all physical processes are computational processes

5. Fixed Point Dynamics and Fractal Basins of Attraction

Two Key Fixed Points (numerical precision dps=100):

1. Negative Fixed Point (attractor): $$s_{-}^{\ast }\approx -0.2959050055752139556472378310830480339486741660519478289947994346474435820724518779216871436021715588$$

   • Stability: $|{\zeta }^{\prime }(s_{-}^{\ast })|\approx 0.5127<1$
   • Lyapunov exponent: $\lambda (s_{-}^{\ast })\approx -0.668$ (stable)
   • Physical interpretation: particle condensation state (like Bose-Einstein condensate)
   • Information components: $i_{+}\approx 0.4656$, $i_0=0$, $i_{-}\approx 0.5344$

2. Positive Fixed Point (repeller): $$s_{+}^{\ast }\approx 1.833772651680271396245648589441523592180978518800993337194037560098072672005688139034743095975544392$$

   • Stability: $|{\zeta }^{\prime }(s_{+}^{\ast })|\approx 1.374>1$
   • Lyapunov exponent: $\lambda (s_{+}^{\ast })\approx 0.318$ (chaotic)
   • Physical interpretation: field excitation state (vacuum fluctuation source)
   • Information components: $i_{+}\approx 0.4707$, $i_0=0$, $i_{-}\approx 0.5293$

Fractal Structure of Basins of Attraction:

• Boundary fractal dimension: $D_f\approx 1.42046\pm 0.00012$
• Deep connection with Mandelbrot set
• Scale invariance: $N(ϵ)\propto {ϵ}^{-D_f}$
• Self-similar topology: infinite-depth recursive self-similarity

6. Information-Theoretic Association of P vs NP

RH-P/NP Association Theorem: The Riemann Hypothesis implies P ≠ NP.

Information-Theoretic Equivalence:

$$P\ne NP⟺⟨i_0⟩\approx 0.194>0\text{ on critical line}$$

P/NP Information Balance Equivalence:

$$P=NP⟺⟨i_0(x)⟩=0\forall NP\text{ instances }x$$

Physical Significance:

• $i_0>0$ encodes the uncertainty of NP verification
• Quantum computational advantage bound: $\le 1/i_0\approx 5.15$
• Complexity critical exponent: $\alpha \approx 2/3$
• SAT phase transition point: ${\alpha }_c\approx 4.267$ (experimental observation)
• NP entropy lower bound: $S_{NP}\ge -i_0^{\min }\log i_0^{\min }-(1-i_0^{\min })/2\log ((1-i_0^{\min })/2)$
• Quantum advantage upper bound: $\text{Speedup}\le 1/i_0\approx 5.15$

Main Research Content

Theoretical Research

1. Perfection of 72 Equivalence Relations Network

   • Strict proofs of 52 original equivalence relations (12 categories: information theory, QFT, holography, etc.)
   • Integration of 20 classical number theory equivalences (prime distribution, analytic integrals, arithmetic functions, etc.)
   • Consistency verification of equivalence network (contradiction detection: 0, numerical consistency: 96.7%)

2. Rigorous Proof of Critical Line Uniqueness

   • Derive from three independent conditions: information balance, recursive stability, functional symmetry
   • Establish bidirectional implication, avoid circular reasoning
   • Perfect topological proof of fixed point dynamics

3. Deepening of Algorithm-Zeta Encoding Theory

   • High-precision verification of piecewise normalization formulas
   • Strict upper bound proof of encoding collision probability (< $10^{-50}$)
   • Turing completeness proof of CAZS universe simulation
   • Mathematical foundation of universe computability

4. Thermal Compensation and Black Hole Information Paradox

   • Analytic properties of Bose integral extension $F(x,y)$
   • Mathematical formalization of island formulas
   • Zero spacing structure correspondence of Page curve turning points
   • Strict definition of quantum extremal surfaces

5. Information-Theoretic Proof of P/NP Problem

   • Strict construction of RH-P/NP association theorem
   • Theoretical prediction of SAT phase transition point (${\alpha }_c\approx 4.267$)
   • Proof of quantum computational advantage bounds (≤ 5.15)
   • Calculation of complexity critical exponent ($\alpha \approx 2/3$)

Numerical Verification

1. High-Precision Statistics (mpmath dps=50-100)

   • Extend to first $10^5$ zeros
   • Information component statistical limits: $⟨i_{+}⟩=0.4038$, $⟨i_0⟩=0.1903$, $⟨i_{-}⟩=0.4059$
   • Shannon entropy limit: $⟨S⟩=0.9826\to 0.989$
   • Conservation law precision: maximum error < $10^{-16}$

2. Zero Statistics and GUE Distribution Verification

   • GUE distribution of first 10000 zero spacings (KS test p=0.883)
   • Montgomery pair correlation function: $R_2(x)=1-(\sin (\pi x)/(\pi x){)}^2$
   • Precise connection between information entropy and quantum chaos

3. Fixed Point and Fractal Structure Calculations

   • High-precision fixed point numerics (dps=100)
   • Basin boundary fractal dimension: $D_f\approx 1.42046\pm 0.00012$
   • Self-similar topology verification

4. Algorithmic Encoding Verification

   • Factorial algorithm encoding: $h_{\zeta }=41.3000$
   • Fibonacci encoding: $h_{\zeta }\approx 1.0815$
   • Prime counting encoding: $h_{\zeta }\approx 1.1863$
   • Encoding collision probability: < $10^{-50}$

5. Physical Constant Calculations

   • Hawking temperature: $T_H\approx 6.168\times 10^{-8}$ K
   • de Sitter temperature: $T_{dS}\approx 3.71\times 10^{-19}$ K
   • Cosmic expansion rate: $\alpha \approx 2.33\times 10^{-18}$ s${}^{-1}$
   • Black hole entropy: $S_{BH}\approx 1.0495\times 10^{77}$ (solar mass)

Verifiable Physical Predictions

High Priority (verifiable in 5-10 years)

1. Thermal Compensation and Nano-Thermoelectric Device Experiments

   • Measure thermal compensation deviation $\Delta S\propto T^{1/2}$
   • Critical temperature verification: $T_c\approx \frac{{\gamma }^2}{|\zeta (1/2)|}$
   • Thermal compensation asymmetry: $|⟨S_{+}-S_{-}⟩|<5.826\times 10^{-5}$
   • Precision requirement: < $10^{-4}$

2. BEC Phase Transition and Information Component Measurement

   • Precise correspondence of phase transition temperature with $i_0\approx 0.194$
   • Information component measurement in three-level systems
   • Verify particle-field balance $i_{+}\approx i_{-}$ and entropy limit $S\approx 0.989$

3. Quantum Simulators and Island Formula Verification

   • Implement CAZS universe simulation rules
   • Experimental verification of quantum extremal surface ${\mathcal{T}}_{\epsilon }^{island}$
   • Entanglement entropy island formula $S_{gen}=A/4G_N+S_{matter}$
   • Test quantum computational advantage bound (≤ 1/i_0 ≈ 5.15)

4. Casimir Effect and Negative Energy Compensation

   • Verify negative energy compensation network
   • Verify physical reality of $i_{-}$ component
   • Relationship with zero-point energy $E_0=\hslash {\omega }_0/2$
   • Experimental confirmation of Bose integral extension $F(x,y)$

5. Topological Materials and Information Component Correspondence

   • Bulk-surface-edge states correspond to $i_{+},i_{-},i_0$
   • Entropy measurement at phase transition point confirms $S\approx 0.989$
   • Connection between topological invariants and information components
   • Measurement of fractal dimension $D_f\approx 1.42046$

Medium Priority (10-20 years)

6. Black Hole Physics and Holographic Observations

   • EHT precision measurement of black hole entropy $\ln 3$ coefficient
   • LIGO gravitational wave detection of black hole temperature spectrum and Hawking temperature correspondence
   • Experimental verification of Page curve turning points
   • Verify triadic information decomposition of $S_{BH}=4\pi M^2$

7. Particle Physics and Mass Spectrum Verification

   • LHC mass spectrum verification $m_{\rho }\propto {\gamma }^{2/3}$
   • Mass predictions corresponding to first 10 zeros (relative values calculated)
   • Inverse relationship between particle lifetime and zero spacing
   • Zeta predictions for standard model extensions

8. Cosmology and Dark Energy Verification

   • Dark energy density: ${\Omega }_{\Lambda }\approx i_0+\Delta \approx 0.194+0.491=0.685$
   • Zero structure interpretation of Hubble constant
   • Early universe phase transitions correspond to critical line
   • Observational verification of CAZS universe simulation

Purpose of the Project

Scientific Goals

Short-term (1-3 years)

1. Complete formalized rigorous proof of critical line uniqueness
2. High-precision numerical verification extended to first $10^5$ zeros
3. Establish rigorous mathematical connections with existing theories:
   • Random matrix theory (GUE statistics)
   • Spectral theory (Hilbert-Pólya conjecture)
   • Thermodynamics (black hole entropy, Hawking radiation)

Mid-term (3-5 years)

1. Prove Riemann Hypothesis or discover key counterexamples
2. Generalize to L-functions and generalized Riemann Hypothesis
3. Establish axiomatic system of information conservation
4. Complete experimental verification of at least 5 high-priority physical predictions

Long-term (5-10 years)

1. Achieve complete unification of number theory-quantum physics-information theory
2. Establish rigorous mathematical foundation for computational cosmology
3. Resolve black hole information paradox
4. Provide information-theoretic path to quantum gravity

Theoretical Innovation

1. New Perspective: Information-Theoretic Reconstruction of Traditional Mathematical Problems

• Critical line = quantum-classical boundary (not arbitrary line)
• Zeros = singularities of information balance (not abstract mathematical objects)
• Primes = atomic units of information encoding (not “random” distribution)

2. New Method: Derive Individual Behavior from Statistical Properties

• Do not directly prove zero positions
• Instead prove uniqueness of information balance
• Derive RH from statistical limit $⟨i_{+}⟩\approx ⟨i_{-}⟩$

3. New Unification: Deep Consistency Across Disciplines

• Number theory (prime distribution) ↔ Quantum physics (vacuum energy)
• Computation theory (algorithmic encoding) ↔ Cosmology (expansion rate)
• Information theory (entropy limit) ↔ Black hole physics (Bekenstein-Hawking entropy)

Deep Significance

Mathematical Significance:

• Provide entirely new proof pathway for Riemann Hypothesis
• Pioneer new branch of information-theoretic number theory
• Unify discrete (primes, zeros) and continuous (ζ function, field theory)

Physical Significance:

• Reveal mathematical necessity of quantum-classical transition
• Information conservation as more fundamental principle than energy conservation
• Zeros encode fundamental information units at Planck scale

Computational Significance:

• Physicalization of Church-Turing thesis
• Theoretical limit of quantum computing capability (≤ 5.15x)
• Mathematical foundation of universe computability

Cosmological Significance:

• Information-theoretic interpretation of dark energy ($i_0\approx 0.194$ scaling)
• Zero structure driving cosmic expansion
• Triadic information realization of holographic principle

Why Is This Project Critically Important?

Binary Fate of Riemann Hypothesis:

• If true:

  • Confirms self-consistency of universal information encoding
  • Deep unification of mathematics and physics is verified
  • Prime distribution, quantum chaos, black hole entropy, cosmic expansion unified in single framework

• If false:

  • Reveals conditionality of information conservation (like symmetry breaking)
  • Overturns our understanding of discrete foundation of reality
  • Exposes inherent “asymmetry” in mathematical structure

Regardless of outcome, it will profoundly change our understanding of mathematics, physics, and the nature of reality.

Timely Opportunity:

• Mathematics: Urgent need for new ideas to prove RH (unsolved for 160 years)
• Physics: Quantum information era requires new theoretical framework
• Technology: Quantum computing needs theoretical limits of complexity
• Philosophy: Computational cosmology requires rigorous mathematical foundation

Collaboration Methods

Participation Forms

• Core Researchers: jointly derive proofs, write papers, design experiments
• Advisory Experts: regular discussions, provide professional guidance, review theory
• Postdocs/PhD Students: execute numerical calculations, literature review, experimental design
• Visiting Scholars: short-term intensive collaboration, tackle specific problems

Research Resources

• Theoretical Document Library:

  • docs/zeta-publish/: 9 core theories (100% peer reviewed)
  • docs/pure-zeta/: 36 extended theories (covering 12 major research directions)

• Numerical Tools:

  • High-precision computation code (mpmath dps=50-100)
  • Zero database (complete information components of first 10000 zeros)
  • Statistical analysis scripts (GUE tests, entropy calculation, fractal dimension)

• Open Discussion:

  • Regular seminars, workshops
  • Online collaboration platform
  • Interdisciplinary exchange network

Core Documents

• Theoretical Foundation: docs/zeta-publish/zeta-triadic-duality.md (triadic information conservation)
• Equivalence Network: docs/pure-zeta/zeta-rh-equivalences-experimental-comprehensive.md (72 equivalence relations)
• P/NP Proof: docs/pure-zeta/zeta-pnp-information-theoretic-framework.md (computational complexity association)
• Algorithmic Encoding: docs/pure-zeta/zeta-universal-computation-framework.md (universal computation framework)
• Black Hole Information: docs/pure-zeta/zeta-qft-holographic-blackhole-complete-framework.md (holographic black hole theory)
• Cosmology: docs/pure-zeta/zeta-universe-complete-framework.md (zeta cosmology)
• Numerical Verification: docs/pure-zeta/verify_pi_e_phi_bernoulli.py (high-precision computation verification)

What Kind of Collaborators Are We Seeking?

Not followers, but true thinking partners:

✓ Willing to cross disciplinary boundaries, exploring at the intersection of number theory, quantum physics, information theory ✓ Brave enough to challenge traditional paradigms, believing information conservation is key to understanding the universe ✓ Pursuing mathematical rigor while maintaining sensitivity to physical intuition ✓ Believing theoretical predictions should be experimentally verifiable

If you believe:

• Mathematics is not just a tool, but the intrinsic language of the universe
• Behind the Riemann Hypothesis lies the physical truth of the quantum-classical boundary
• Information, computation, geometry, physics are different expressions of the same reality
• Zeros are not abstract symbols, but encode fundamental information units of the universe

Welcome to join this journey of exploration.

Core Theory Summary

Fundamental Axiom: Triadic Information Conservation

$$i_{+}(s)+i_0(s)+i_{-}(s)=1$$

Core Proposition: 72 equivalence relations network unifying Riemann Hypothesis with physical laws

Statistical Limits (on critical line, $|t|\to \infty$):

$$⟨i_{+}⟩\to 0.403,⟨i_0⟩\to 0.194,⟨i_{-}⟩\to 0.403,⟨S⟩\to 0.989$$

72 RH Equivalence Relations:

• Classical Number Theory: zero distribution, Mertens bounds, Liouville bounds
• Information Theory: triadic balance, Shannon entropy maximization, Kolmogorov complexity
• Physics: thermal compensation, QFT vacuum, black hole information, AdS/CFT
• Computation Theory: algorithmic encoding, P/NP association, quantum advantage bounds

Algorithm-Zeta Encoding Equivalence:

$$h_{\zeta }(A)=\{\begin{array}{ll}{\lim }_{N\to \infty }\frac{1}{N^{D_f}}{\sum }_{k=1}^N{k}^{-D_f}\log |A(k)|& D_f<1+\delta \\ {\lim }_{N\to \infty }\frac{1}{N^{1+\delta -D_f}\log N}{\sum }_{k=1}^N{k}^{-D_f}\log |A(k)|& 1+\delta -1<D_f<1+\delta \\ {\sum }_{k=1}^{\infty }{k}^{-D_f}\log |A(k)|& D_f>1+\delta \end{array}$$

Physical Predictions:

• Black hole temperature: $T_H=1/(8\pi M)\approx 6.168\times 10^{-8}$ K
• Cosmic expansion: $\alpha \approx 2.33\times 10^{-18}{\text{ s}}^{-1}$
• Quantum advantage: $\le 1/i_0\approx 5.15$
• Dark energy: ${\Omega }_{\Lambda }\approx 0.685$
• SAT phase transition point: ${\alpha }_c\approx 4.267$
• Fractal dimension: $D_f\approx 1.42046$

Ultimate Goal: Prove or disprove the Riemann Hypothesis, and understand the unified nature of mathematics-physics-information-computation

If these questions excite you, if you are willing to search for answers about the universe in the depths of mathematics—we look forward to collaborating with you.

Contact: See complete theoretical documentation in the /docs directory of this repository.