Prime Period Theory

Path Equivalence and Confluence

The limits of path uniqueness

In the Prime Lattice, every point is defined by an ordered sequence of steps along prime axes, descending recursively into smaller subdivisions of a period.

As stated in the prime lattice definition, within a single prime family, the subdivision grid is regular and every path maps to a unique position. The mathematics of balanced base-p signed-digit systems guarantees that no two different paths of pure Tri digits, or pure Qui digits, can arrive at the same fractional position.

However, when paths mix different prime families (e.g., interleaving Tri and Qui steps), this uniqueness property breaks down.

Commutativity collision

This lack of uniqueness across mixed primes is not an edge case or a bug — it is a mathematically guaranteed structural feature of the system.

The physical position of any point in the lattice is determined by the formula:

position = Σᵢ aᵢ / Pᵢ, where Pᵢ = ∏ⱼ₌₁ⁱ pⱼ

Because the denominator Pᵢ relies on the product of all primes used so far, and multiplication commutes (i.e., 3 × 5 = 5 × 3), different sequences of prime choices can yield the same denominator, and ultimately the same position.

A concrete example

Consider a depth-2 path aiming for the fractional position 4/15:

Path 1 (Tri then Qui):

  • Step 1: Tri, digit +1 (position 1/3)
  • Step 2: Qui, digit −1 (position 1/3 - 1/15 = 4/15)

Path 2 (Qui then Tri):

  • Step 1: Qui, digit +1 (position 1/5)
  • Step 2: Tri, digit +1 (position 1/5 + 1/15 = 4/15)

These are two structurally distinct decision sequences, in different prime orders, that resolve to the exact same physical address in the period space.

Confluence

We do not frame this commutativity collision as a problem to be solved. Rather, it is an observation of how the lattice behaves. In PPT, this relationship is known as Confluence.

Confluence is the comma-space analogue to enharmonic equivalence in pitch space (e.g., G♯ and A♭ being different spellings of the same pitch). It documents an equivalence class between distinct decision-paths — whether they use the same or different prime orders, at the same or different depths — that arrive at the same location.

Recognising Confluence allows a composer or theorist to treat the path taken as a meaningful choice (a compositional decision about how a period is recursively subdivided) even when the final destination is identical to another route. Furthermore, because there is no longer a separate Boundary family, a coarsest-frame Du digit (licensed pivot) and an interior Du digit now participate in Confluence relations on the exact same footing as any other prime’s digits.

See also

Knowledge Graph
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