Imagine we are standing at the edge of an infinitely wide, bottomless chasm. We want to build a bridge across it.
We have only three types of building blocks at our disposal: Wood (a), Stone (b), and Iron (c).
However, the laws of physics in this universe are extremely strict. The bridge is highly sensitive to harmonic resonance.
The Fundamental Rule: The bridge must never contain two consecutive sections that are "structurally identical". If any two adjacent sections of the bridge contain the exact same amounts of materials, a resonance will build up and the bridge will instantly collapse.
Before we begin, we test the strictest possible rule: never two consecutive identical sections, not even a single block long.
This fails almost immediately — after 7 blocks, every subsequent choice creates a repetition. It is an old, indisputable fact: a bridge of three materials can never be completely abelian square-free (anagram repetition-free).
Mäkelä proposed a more subtle question in 2002: what if we allow a single block repetition (aa, bb, cc are OK), but forbid anything longer?
This is a much more promising starting point — but for over twenty years, no one has been able to prove either way whether an infinite bridge can succeed under this rule. This unresolved question is the true starting point of the B16 research.
Years later, master architects proved that an infinite bridge over the chasm is possible! But their blueprint is incredibly complex.
They realized that just counting the raw materials isn't enough. The structural integrity also depends on the joints—how the blocks are connected to each other.
The master architects introduced a stricter rule: The 2-abelian rule. Now, two adjacent sections are considered identical only if they have the exact same amounts of raw materials AND the exact same amounts of every single one of the 9 joint types.
By monitoring and restricting all 9 joint types, the bridge can be extended infinitely!
The master blueprint works perfectly, but monitoring all 9 joint types simultaneously is an absolute nightmare. The builders ask a natural question: "Do we really need all 9 joints?"
What if we only monitor a few, and let nature handle the rest? We run massive simulations up to 16 blocks to measure how many valid paths the bridge can take. We call this its "strength".
There is no magic switch. The strength of the bridge grows in a grueling, slow staircase. If you leave even a single joint type unregulated, nature forces the construction into a state where it overuses that crack. The bridge becomes unbalanced.
Just as the builders are about to lose hope, they test the combinations of 6 joints.
Suddenly, the strength of the bridge skyrockets! It leaps straight to 99.92% of the full 9-joint strength. But there is a catch: not just any 6 joints will work. Exactly one specific combination unlocks this massive leap.
The grand secret of the architecture is revealed: Nature allows you to be careless with identical materials. You don't need to monitor how often Wood connects to Wood, or Iron connects to Iron.
But the moment you transition from one material to another, the structure is vulnerable. You must strictly monitor every single mixed joint. If you do that, the bridge achieves almost the exact same strength as the master blueprint.
"There are no shortcuts in the deep structure of sequences. But if you listen closely, nature will tell you exactly where the true structural load lies."
Sixteen blocks is a fine place to stop and celebrate. But the builders, curious, keep watching the bridge as it stretches further — from 16 blocks out to 22. Two things happen, and they are opposites of each other.
One mystery solved for good. One mystery deepened, and still wide open. That is what real research looks like.
This is not a mockup. Every number below is the exact, exhaustively-computed count of valid length-16 bridge sections for whichever joints you select — all 512 possible combinations were computed and verified (MATH_CLAIMS.md rows 90–99).
Click joints on or off. Watch the strength meter. See if you can find the Golden Six yourself — and notice that adding a joint can never make the bridge weaker, only ever the same or stronger.
Fun fact, found the same day this matrix was built: every single "8 of 9 joints" combination equals the full 9-joint strength exactly — try unchecking just one box and see for yourself.