The State of the Union
Saturday, August 22, 2026
Friday, August 21, 2026
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
| ||||||||||||||||||||||
Wednesday, August 19, 2026
Question for Google Gemini: 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 $n = 6$:
6 is even $\rightarrow$ $6 / 2 = 3$
3 is odd $\rightarrow$ $3(3) + 1 = 10$
10 is even $\rightarrow$ $10 / 2 = 5$
5 is odd $\rightarrow$ $3(5) + 1 = 16$
16 is even $\rightarrow$ $16 / 2 = 8$
8 is even $\rightarrow$ $8 / 2 = 4$
4 is even $\rightarrow$ $4 / 2 = 2$
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
The Hodge Conjecture is one of the seven Millennium Prize Problems designated by the Clay Mathematics Institute and remains unsolved.
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.
## 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.
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).
To claim the prize, a mathematician must prove one of two outcomes for the 3D incompressible Navier-Stokes equations (either in all of
Global Regularity (Proof): Given smooth, finite initial energy conditions and smooth external forces, a smooth, physically sensible solution exists for all time $t > 0$.
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?
Subscribe to:
Posts (Atom)


