Statement of Arthur Schaper's Departure from MassResistance

Hey, everyone!

Many supporters and fellow activists have reached out to me over the last three weeks about MassResistance’s ongoing efforts. Local and international activists have been in touch with me about projects as well.

Today, it is time for me to make the announcement official:

July 31, 2026, was my last day working for MassResistance.

I stayed in touch with the organization day by day for the next few weeks to assist the main office with transition efforts.

My work with the international pro-family group that makes the difference has come to a close.

Their mission has not ceased. My mission has changed.

I am more committed than ever to standing for truth, and I believe—no, I know!—that MassResistance is the only pro-family organization that has maintained a steadfast commitment to telling the whole truth about the LGBT agenda, unafraid to confront cowardly conservatives as well as lame liberals and poisonous progressives intent on accommodating or imposing the LGBT agenda on others.

I have a growing interest in intellectual activism, not just organizing for wins in legislatures or government entities. There is a need to confront the lies and promote the truth in more ways, and I have enjoyed writing and commenting on these issues at great length.

It’s my hope that I can continue along those lines to fight the good fight and stand for truth.

MassResistance still needs your help and your activism!

Please let President Brian Camenker know if you want to stay in the fight!

From this point forward, if there are any MassResistance-related questions or ideas, please send them to Brian directly or visit the website.

Email him at brian@massresistance.org

Website: www.massresistance.org

Thank you for all your support and ongoing commitment to stand for truth and fight the lies of the LGBT agenda.

Sincerely,

Arthur Schaper

John McKinney for LA City Attorney: Marissa Roy Endorsed George Gascon

 

Marissa Roy Endorsed George Gascón Dear Friend, This is John McKinney, candidate for LA City Attorney.    Last week, we released our first ad highlighting Marissa Roy’s dangerous position allowing homeless encampments to exist near on our sidewalks and near our schools and parks.    Today, we are releasing another ad exposing the fact that Marissa Roy endorsed George Gascón—even as his policies wreaked havoc on LA:

As a prosecutor, I saw firsthand how George Gascón let dangerous criminals off the hook and  into our communities. Our neighborhoods suffered, our streets became less safe, and victims were robbed of justice. Instead of doing his job and keeping Los Angeles safe, Gascón pushed a radical agenda that put politics over people. With the same extreme philosophy and platform, Marissa Roy would take LA back to the dangerous days of George Gascón. For years, Marissa has advocated against prosecuting misdemeanors. She has said that the City Attorney should "decline to prosecute” and "decriminalize” crimes like theft, trespassing, public drunkenness, vandalism, and resisting arrest. With a plan that dangerous and reckless, it is no surprise that she endorsed George Gascón.   We can’t let LA return to Gascón-style lawlessness. Help us fight back by donating to help us air this ad:  Donate to John McKinney for City Attorney 2026 Thank you for your ongoing support. Sincerely, John McKinney Candidate for LA City Attorney

Question for Google Gemini: Solve the Collatz Conjecture

 

Solve the Collatz Conjecture

The Collatz Conjecture remains unsolved and has not yet been proven for all integers, so I cannot provide a complete, universally accepted mathematical proof for it.

However, we can look at what the conjecture is, why it is notoriously difficult, and what modern mathematics has achieved so far.

What is the Collatz Conjecture?

Proposed by Lothar Collatz in 1937, the conjecture applies to any positive integer $n$ and follows a remarkably simple rule:

• If $n$ is even, divide it by 2 ($n / 2$).

• If $n$ is odd, multiply it by 3 and add 1 ($3n + 1$).

The conjecture states that no matter what positive integer you start with, repeating this process will always eventually reach the number 1 (entering the infinite loop of $4 \rightarrow 2 \rightarrow 1 \rightarrow 4$).

The Rules in Action (Example)

Let's take $n = 6$:

1. 6 is even $\rightarrow$ $6 / 2 = 3$

2. 3 is odd $\rightarrow$ $3(3) + 1 = 10$

3. 10 is even $\rightarrow$ $10 / 2 = 5$

4. 5 is odd $\rightarrow$ $3(5) + 1 = 16$

5. 16 is even $\rightarrow$ $16 / 2 = 8$

6. 8 is even $\rightarrow$ $8 / 2 = 4$

7. 4 is even $\rightarrow$ $4 / 2 = 2$

8. 2 is even $\rightarrow$ $2 / 2 = 1$ (Reached!)

Why hasn't it been solved?

Despite its simple rules, the sequence trajectories can behave chaotically, shooting up and down unpredictably before finding their way to 1.

• Empirical Verification: Supercomputers have tested and verified the conjecture for all starting values up to astronomical numbers (roughly $2^{69}$ or higher), finding no counterexamples. But computer verification can never cover an infinite set of numbers.

• The "Almost All" Breakthrough: In 2019, Fields Medalist Terence Tao made the most significant modern advancement on the problem. He used a probabilistic framework to prove that almost all Collatz orbits attain almost bounded values. While a monumental step forward, it still falls short of a universal proof for every single integer.

Mathematicians often joke that "mathematics is not yet ready for such problems," as resolving it likely requires entirely new mathematical tools.

Would you like to explore how computer algorithms test the conjecture, or look into Terence Tao's partial approach?

Question to Google Gemini: Resolve The Hodge Conjecture

 

Resolve The Hodge Conjecture

The Hodge Conjecture is one of the seven Millennium Prize Problems designated by the Clay Mathematics Institute and remains unsolved. Because it is an open problem in modern algebraic geometry, I cannot resolve or provide a completed proof for it.

However, we can look at what the conjecture states, why it is important, and how far mathematicians have come in understanding it.

## What is the Hodge Conjecture?

Proposed by Scottish mathematician William Vallance Douglas Hodge in 1950, the conjecture bridges two different fields of mathematics:

• Algebraic Topology: Studies the global shape of spaces using holes, loops, and topological invariants (via singular homology or de Rham cohomology).

• Algebraic Geometry: Studies geometric spaces defined by systems of polynomial equations (algebraic varieties).

The Core Premise:

The Hodge conjecture asserts that for projective algebraic varieties, the basic topological information—specifically, the classes of cycles that correspond to "Hodge classes"—can be completely understood as rational linear combinations of geometric subvarieties cut out by polynomial equations. In plain terms: every "hole" in a complex algebraic variety that can be detected by algebraic differential forms is actually built out of the zero sets of polynomials.

## Current Status and Known Cases

While a general proof for all dimensions has eluded mathematicians, several partial results and special cases have been established:

• Low Dimensions: The conjecture is known to be true when the projective variety has a dimension of less than four.

• Divisors (Lefschetz Theorem): It was proven early on (Lefschetz (1,1)-theorem) that the conjecture holds true for Hodge classes of degree 2 (the simplest non-trivial case).

• Specific Families: Certain higher-dimensional families, such as specific types of abelian varieties and certain smooth cubic fourfolds, have been verified, but a universal framework remains out of reach.

## Why Is It So Difficult?

As fields medalist Pierre Deligne once remarked regarding attempts to bypass or solve the problem, the sheer conceptual gap between analytic/transcendental methods (used to study Hodge structures) and purely algebraic methods (polynomial cycles) makes the translation exceptionally rigid and deep.

Would you like to explore the mathematical background of de Rham cohomology, or look closer at the specific cases where the conjecture has been proven?

Question to Google Gemini: Resolve the Navier-Stokes Existence and Smoothness Problem

 

Resolve the Navier-Stokes Existence and Smoothness problem

The Navier-Stokes Existence and Smoothness problem is one of the seven Millennium Prize Problems designated by the Clay Mathematics Institute, carrying a $1 million reward.

As of 2026, this problem remains officially unsolved. Nobody has successfully proven that smooth, physically reasonable solutions always exist for all time in three dimensions, nor has anyone constructed an accepted mathematical counterexample showing that they break down into singularities (infinite velocities or pressures).

Why the Problem is Unsolved

While engineers and scientists routinely use the Navier-Stokes equations to simulate aircraft design, weather patterns, and ocean currents, these practical uses rely on numerical approximations (discretization on computers). Mathematically proving what happens in the continuous, exact equations is an entirely different challenge.

To claim the prize, a mathematician must prove one of two outcomes for the 3D incompressible Navier-Stokes equations (either in all of $\mathbb{R}^3$ or on a periodic torus $\mathbb{T}^3$):

1. Global Regularity (Proof): Given smooth, finite initial energy conditions and smooth external forces, a smooth, physically sensible solution exists for all time $t > 0$.

2. Finite-Time Blowup (Disproof): There exist smooth initial conditions and external forces for which a smooth solution develops a singularity (breaks down) in finite time.

What is Currently Known

Although the full 3D problem is open, mathematicians have established several major partial results:

• The 2D Case is Solved: In two dimensions, O.A. Ladyzhenskaya proved in 1969 that smooth solutions always exist globally for all time. The difficulty is uniquely tied to three dimensions, specifically due to vortex stretching (the mechanism where stretching fluid vortex tubes can rapidly concentrate energy and vorticity).

• Weak Solutions Exist: Jean Leray proved in 1934 that "weak solutions" (solutions that satisfy the equations in an averaged, integral sense) exist globally in 3D. However, it remains unknown whether these weak solutions are smooth or unique.

• Singularities are Extremely Rare: Caffarelli, Kohn, and Nirenberg (1982) proved a partial regularity theorem showing that even if singularities could form in 3D, the set of singular points has a one-dimensional parabolic Hausdorff measure of zero (meaning they are extraordinarily sparse).

If you are working on a specific approach or proof attempt for this problem, what particular angle (such as energy estimates or vorticity bounds) are you exploring?

Question to Google Gemini: Prove the Riemann Hypothesis

Prove the Riemann Hypothesis

While I would love to help you claim the $1,000,000 Millennium Prize from the Clay Mathematics Institute, the Riemann Hypothesis remains one of the greatest unsolved problems in mathematics!

Because a formal proof has not yet been discovered, I can't write out the solution for you. However, here is a quick breakdown of what the hypothesis is and why it has baffled mathematicians for over 160 years.

What is the Riemann Hypothesis?

Proposed by Bernhard Riemann in 1859, the hypothesis concerns the Riemann zeta function, denoted as $\zeta(s)$, where $s$ is a complex number.

• The Equation: The function has "trivial" zeros at the negative even integers ($-2, -4, -6, \dots$).

• The Core Claim: Riemann hypothesized that all the other non-trivial zeros lie on a specific vertical critical line in the complex plane, defined by the equation $\text{Re}(s) = \frac{1}{2}$.

Why Does It Matter?

• Prime Numbers: The zeros of the zeta function dictate the exact distribution of prime numbers. Proving the hypothesis would unlock profound secrets about prime behavior and the gaps between them.

• Modern Cryptography: Much of our digital security relies on the assumption that prime numbers behave in certain ways. While a proof wouldn't instantly break encryption, it would fundamentally revolutionize number theory.

Current Status

• Numerical Verification: Supercomputers have checked and verified that the first 10 trillion non-trivial zeros all lie precisely on the critical line ($\text{Re}(s) = \frac{1}{2}$).

• The Missing Link: Despite this overwhelming numerical evidence, mathematics requires a general proof that holds true for infinite zeros, which no one has been able to construct yet.

Would you like to explore how the zeta function connects to the distribution of prime numbers, or look into some of the near-misses and approaches mathematicians have tried over the decades?