The Cross-Alignment Matrix
Compare every fraction to every other, digit by digit. The result is a symmetric matrix whose eigenvalues encode the internal structure of the fractional field.
The Coherence Decomposition
Alignment splits into two parts: a focused component from the repetend orbit, and a pairwise component from cross-matches. The decomposition explains why some integers are more coherent than others.
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.
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.
Long Division and Euclid's Lemma
The digit function is the floor quotient from Euclidean division. It maps remainders to digit bins. Everything in the collision program starts here.
The Effect of Base on Numeric Fields
Change the base and the field changes with it. But the complement symmetry does not. Something underneath is base-independent.
Geometries Hidden in the Number System
Field glyphs sort the integers into three visible categories. The geometry is base-independent. The structure was there before the formalism.
The Golden Ratio
The simplest self-referential equation produces the slowest-converging continued fraction and the threshold that separates the prime 3 from all others.
Arithmetic Foundations
The integers expand outward through each range and contract back along the mirror path. The breathing pattern repeats at every magnification. The complement map is already here.