Primes and the Major Scale
The primes that organize long division are the same primes that organize musical pitch. In base 30, the alignment deficit lattice produces the just major scale. In base 12, the Pythagorean comma.
alignment
The Three-Tier Theorem
Every positive integer falls into one of three tiers. The mirror bound from the nines complement forces everything with rough part past 7 below the golden line. The classification is unconditional.
The Alignment Limit for All Primes
Past the digit-partitioning boundary, the alignment splits into lanes. Different smooth factors choose different lanes, and the limit may not exist as a single number. But no prime past 3 reaches the golden threshold.
Digit-Partitioning Primes and the Alignment Formula
In base 10, exactly three primes produce total digit separation: 3, 7, and 11. Three different mechanisms, one shared condition, one universal formula.
Why the Golden Ratio Selects the Prime Three
A cubic equation has a root in (0,1) for every prime. The golden ratio's minimal polynomial divides it at exactly one. The remainder names the prime 3.