Coprimality

Status: validated Domain: math Source: → projects/Prime_Maxel-v3/research/v3_board3_coprimality_dimensions_22may2026.md

What We Know

Coprimality is necessary but not sufficient for optimal resonance
Discovered through v3 experiments: coprime sets don’t automatically outperform; structural resonance matters more
Odd divisors ≈ primes (93% coprime) — both produce 1140 mV Vpp with integer f₀ divisors
Even divisors produce cluttered harmonics due to shared factor 2
Coprimality explains WHY prime frequencies don’t interfere destructively, but doesn’t explain WHY they resonate constructively
This is Factor 3 in Four-Factor Theory — important but subordinate to structural resonance

Zeta Zeros — Coprimality Deficit (2026-06-04)

Zeta zeros rounded to nearest integers {14, 21, 25, 30, 33, 38} achieve only 47% pairwise coprimality (7/15 pairs) vs 100% for primes. Shared factors include: gcd(14,21)=7, gcd(21,33)=3, gcd(25,30)=5, gcd(30,38)=2. This contributes ~1-2% of the 10.5% performance gap — significant but smaller than structural resonance (~5-6%) or missing anchor (~3-4%). The penalty is muted because as irrational values, zeros don’t truly “share factors” — incommensurable frequencies can’t constructively interfere the way shared-factor integers do.

Phyllotaxis Coprimality — Φ Connection (2026-06-04)

Plants using Fibonacci spirals achieve coprimality rate 0.650 vs 0.608 expected for random integers (= 6/π²) — 7% higher than random. Phyllotaxis IS coprimality optimisation, and the golden ratio Φ is the mechanism through which biology implements it. This connects coprimality to the 2,3-scaffold via Φ’s Pisano period mod 6 = 24.

Maxwell Cavity Coprime Modes (2026-06-04)

Prime-ratio rectangular EM cavities (2,3,5) produce 24% more unique frequencies (297 vs 240) than composite-ratio cavities (4,6,8). The mechanism is purely coprimality: incommensurable prime dimensions prevent frequency degeneracies. Coprimality of mode indices is identical (41.0% for both cavities) — the difference is entirely in how coprime dimensions map indices to frequencies. This is coprimality operating in classical electrodynamics, independent of the torsion ring.

What We Don’t Know

Q-CP-01: Is there a “coprimality dimension” — a measure of how coprime a set is beyond binary yes/no?
Q-CP-02: How does partial coprimality (e.g., {6,10,15} — pairwise coprime but sharing prime factors across the set) affect resonance?
Q-CP-03: At what network size does coprimality become the dominant factor over structural resonance?

Relationships

[[four-factor-theory]] — nature: depends-on — coprimality is Factor 3
[[tusk-resonant-set]] — nature: supports — {1,2,3,5,6,7} is NOT fully coprime yet outperforms fully coprime sets
[[prime-composite-duality]] — nature: extends — coprimality is the mathematical expression of why primes “don’t interfere”
[[ras]] — nature: analogous-to — RAS “resist” term captures the same non-interference property geometrically
[[prime-resonance-computing]] — nature: supports — coprimality ensures orthogonal channels in Prime-OFDM
[[golden-ratio-scaffold]] — nature: extends — phyllotaxis coprimality rate 0.650 (7% above random) via Φ
[[maxwell-prime-cavities]] — nature: extends — 24% more unique EM modes from coprime dimensions

Bridging Potential

If combined with [[zeta-zeros-physical]], could explain the 85-90% performance of zeros (they’re not coprime by construction)
Coprimality dimension concept could connect to coding theory (orthogonal codes)

Key Evidence

Coprimality dimensions experiments: v3_board3_coprimality_dimensions_22may2026.md
Odd vs even experiments: v3_board3_odd_even_22may2026.md