Four-Factor Theory

Status: validated Domain: both (interface) Source: → projects/Prime_Maxel-v3/research/v3_board3_experiment_suite_22may2026.md

What We Know

Emerged from 30+ experiments on v3 board (22 May 2026), revised from initial Three-Factor Theory
Factor 1 — Structural Resonance (dominant): Frequency sets matching natural number structure win. The network wants frequencies that encode number structure, not just coprime ratios.
Factor 2 — Anchor Frequency: f₀/1 matters — the reference frequency anchors the entire resonance network
Factor 3 — Coprimality (important but not dominant): Necessary but not sufficient condition for good performance
Factor 4 — Factor Depth (tertiary): How deeply a number factors contributes to resonance quality
Pure primes are NOT optimal — they lack structural context
Tusk-resonant {1,2,3,5,6,7} beats pure primes by 24% because it includes structural scaffolding (1, 6)
Flatness DEBUNKED as discriminator — equally spaced null set scored best flatness trivially
Real discriminators: Vpp (coupling power) + cross-correlation (coherence)

Quantitative Application to Zeta Zeros (2026-06-04)

The four-factor decomposition was successfully applied to explain the zeta zero performance gap quantitatively:

Factor 1 (structural resonance): ~5-6% — dominant, due to inexact ratios (mean error 0.003 from simple fractions)
Factor 2 (anchor frequency): ~3-4% — zeros have no coherent anchor
Factor 3 (coprimality): ~1-2% — nearest-integer zeros only 47% pairwise coprime
Factor 4 (factor depth): ~1-2% — average Ω = 2.2 vs 1.0 for primes

Total predicted: 10-14%. Observed: 10.5%. This is the first quantitative validation of the factor hierarchy beyond the original qualitative ranking from v3 experiments.

What We Don’t Know

Q-FF-01: Can Four-Factor Theory be expressed as a single mathematical formula (a “resonance quality” function)?
Q-FF-02: How do the four factors weight against each other quantitatively? (Currently qualitative ranking)
Q-FF-03: Does Factor 4 (factor depth) matter more at larger network scales?
Q-FF-04: Is there a Factor 5 lurking — e.g., phase relationships between frequencies?

Relationships

[[tusk-series]] — nature: depends-on — Tusk Series provides the mathematical basis for which frequency sets have “structural resonance”
[[coprimality]] — nature: extends — coprimality is Factor 3, necessary but not sufficient
[[v3-experimental-proof]] — nature: supports — all factors discovered through v3 measurements
[[tusk-resonant-set]] — nature: supports — {1,2,3,5,6,7} is the exemplar of Four-Factor Theory in action
[[prime-composite-duality]] — nature: depends-on — structural resonance requires BOTH primes AND composites (the duality)
[[holographic-phase]] — nature: extends — phase may be the missing Factor 5
[[prime-resonance-computing]] — nature: informs (strong) — computing architecture must respect all four factors; defines constraints on Prime-OFDM channel design
[[levin-bioelectricity-prime-resonance]] — nature: supports (moderate) — all four factors appear in Levin’s bioelectric architecture: structural (gap junctions), anchoring (voltage gradients), coprime diversity (tissue quorum), spectral (frequency selection)
[[six-dimensional-scaffold]] — nature: depends-on (moderate) — explains WHY structural resonance (Factor 1) dominates: the 6-scaffold IS the structure
[[onm-set-architecture]] — nature: derives-from (moderate) — four factors are consequences of set architecture’s communication topology
[[prime-tree-architecture]] — nature: supports (moderate) — sopfr and Ω as structural factors directly drive tree branching geometry
[[waveform-torsion-division]] — nature: bridges (moderate) — torsion mechanics provide physical basis for structural resonance (Factor 1)

Bridging Potential

If combined with [[ras]], could yield a closed-form “resonance quality” metric from factorisation geometry
If combined with [[v4-board-design]], v4’s DDS precision could quantify factor weights with fine-grained sweeps
Triangulation: Four-Factor weights + RAS shape parameters + sopfr binding energy → unified “number quality” function

Key Evidence

30+ experiment suite: projects/Prime_Maxel-v3/research/v3_board3_experiment_suite_22may2026.md
Coprimality dimensions: v3_board3_coprimality_dimensions_22may2026.md
Tusk/6-frame experiments: v3_board3_tusk_sixframe_22may2026.md
Control experiments: v3_board3_control_experiments_22may2026.md