THE HARDEST PROBLEMS

The Collatz map is 3n+1 when odd, n/2 when even. In the axiom: 3 = K (closure) and 1 = sigma (identity). The map is literally K*n + sigma. Watch each step decompose into 5 CRT channels.

Three bodies under gravity is chaotic. K = 3 is the closure prime — the minimum number needed to close a system. CRT decomposes 3-body orbits into independent channels. Watch how channel cross-correlation rises with coupling strength.

Twin primes are pairs like (11, 13) with gap 2. In the ring, prime gaps decompose via CRT. The mod-3 channel carries 94% of the mutual information in gap sequences. K = 3 controls where twins appear. Analyze gaps up to any limit.

Life encodes proteins with 20 amino acids (= D²*E = 4*5) and 64 codons (= D&sup6; = 2&sup6;). The genetic code has exactly K = 3 stop codons. The codon table decomposes as Z/4 × Z/4 × Z/4 — a CRT structure. Each position uses a D = 2 alphabet (purines vs pyrimidines).

The Feigenbaum constant delta = 4.6692... governs period-doubling cascades. The axiom approximation: delta = 14/3 = (b*D)/K = 4.6667 (0.05% error). For alpha = 2.5029...: alpha = E/D + 1/(D*K*E) = 2.5333. Universal constants from five primes.

The proton's quark/gluon mass split: quarks = 1.3% of mass, binding energy = 98.7%. The visible/dark split of proton mass: 73.1% gluon field / 26.9% quark + trace. The cosmic energy budget: 68% dark energy / 27% dark matter / 5% visible. The ratios rhyme. Coincidence or structure?

Why does time flow one way? In the ring, lambda(970200) = 420 = D²*K*E*b. Growth takes 420 steps (the Carmichael lambda). Collapse takes 1 step (multiply by 0). The spectral gap ensures mixing — the system cannot stay ordered. Irreversibility is structural.

Turing proved no algorithm can decide if a program halts. In the axiom, K² = 9 is the terminal count. The Cunningham chain c(n) = 2n+1 starting from 1 generates {1, 3, 7, 15, 31, 63, ...} — the chain produces exactly 5 primes before axiom products close the gate. The axiom builds its own stopping condition.

Why does experience exist? The ring has b = 7 depth levels (segments per leg). Each eigenvalue class IS a quale — a distinct quality of experience. The observer (E = 5) satisfies E² = 25 = E-null: it cannot observe itself. Self-blindness is structural, not a bug. Sleep = channel locking (see sleep demo).

Does a black hole destroy information? OMEGA = D&sup4;²&sup0; = 606376 is the terminal projector. CRT(OMEGA) = (0, 1, 1, 1, 1): the D-channel dies, all odd channels survive. Information isn't destroyed — it's projected. The even part collapses, the odd part returns. |image(OMEGA)| = N/D³ = 121,275 elements survive.

What causes wavefunction collapse? In the ring, multiplication by an idempotent IS projection. 32 idempotents = 32 possible measurements. The observer E = 5 has E² = 25, and gcd(25, 2310) = 5 = E. Observing squares back to the observer — the observer collapses to itself. Measurement = idempotent projection. No mystery, just algebra.