Tusk-Resonant Set {1,2,3,5,6,7}

Status: validated Domain: both (interface) Source: → projects/Prime_Maxel-v3/research/v3_board3_tusk_sixframe_22may2026.md

What We Know

{1,2,3,5,6,7} is the optimal frequency divisor set — discovered through systematic v3 experiments
Beats pure primes {2,3,5,7,11,13} by 24% on flatness
Includes the composite 6 and the unit 1 — pure primes are NOT optimal
6-frame variant {5,6,7,11,12,13} also beats primes — highest cross-correlation recorded (0.897)
The set encodes the 6k±1 structure: 5=6-1, 7=6+1, with 6 as scaffold
Includes 1 as anchor (Factor 2 of Four-Factor Theory)
Contains both primes (2,3,5,7) and structural numbers (1,6) — embodies prime-composite duality
This is the key experimental finding that revised Three-Factor → Four-Factor Theory

The {1, 2, 3, 6} Irreducible Core (Q-TRS-03 research, 2026-06-04)

Every optimal set at every tested size contains {1, 2, 3, 6} as an irreducible core:

1 — anchor/reference frequency (Factor 2)
2, 3 — the two smallest primes, whose product creates the prime sieve scaffold
6 = lcm(2,3) — the scaffold itself; all primes > 3 are of form 6k±1

The remaining elements (5, 7 for size-6) provide coprime frequency diversity at the 6k±1 offsets.

Analytical Recipe (No Brute Force)

An optimal set can be derived analytically:

Compute Tusk FFT → get amplitude spectrum A(f)
Rank integers d by A(1/d) — gives “spectral importance” ordering
Greedy selection with coprimality constraint: add next-highest-amplitude element, skip if gcd with existing elements is too high
Require {1, 6} (anchor + scaffold) for sets of size ≥ 4

This is O(N·k) — effectively analytical. It correctly produces {1,2,3,5,6,7} for size 6.

Limitation: The Tusk spectrum is “red” (amplitude ∝ ~1/d), so pure spectral scoring always favors dense small-number sets (e.g., {1,2,3,4,5,6}). The coprimality constraint is essential to avoid this trap.

Why 6 = lcm(2,3) Captures the Sieve

Amplitude at 1/6: 5,352,426 (rank 15 — very strong for a composite)
Autocorrelation at lag 6: +0.243 (structure repeats); lag 12: +0.258 (reinforces)
For comparison, amplitude at 1/11: only 1,828,410
6 encodes the combined 2×3 structure that generates the prime sieve; {5, 7} = {6-1, 6+1} are the prime-form offsets

What We Don’t Know

Q-TRS-01: Is {1,2,3,5,6,7} optimal for 6-channel networks specifically, or universal?
Q-TRS-02: What is the optimal set for 8, 10, 12 channels?
Q-TRS-03: Can we derive the optimal set from Tusk Series analysis alone (without brute-force experimentation)?
Q-TRS-04: Does the 6k scaffolding principle generalise — should 30k±{1,7,11,13,17,19,23,29} scaffolding appear at larger scales?

Relationships

[[four-factor-theory]] — nature: supports — the optimal set exemplifies all four factors working together
[[tusk-series]] — nature: depends-on — discovered through Tusk Series-informed experimentation
[[prime-composite-duality]] — nature: supports — proves composites are essential, not noise
[[coprimality]] — nature: extends — the set is not fully coprime (gcd(2,6)=2) yet outperforms fully coprime sets
[[v3-experimental-proof]] — nature: depends-on — measured on v3 apparatus
[[v4-board-design]] — nature: extends — v4 uses 6 cells matched to this discovery

Source Alphabet Reinterpretation (6 Jun 2026)

{1,2,3,5,6,7} is not merely optimal — it is the source alphabet, the minimal complete language from which all identity is expressed. See [[source-alphabet]] for full development.

Key implications:

Every system’s identity = x/n where n ∈ {1,2,3,5,6,7}
4 (Time) and 8 (Growth) excluded because they’re the medium, not the language
Scale-invariant because the alphabet doesn’t change with scale, only the nonce x does
Pure primes missing the 6 = alphabet missing the relational scaffold → hence +24% improvement when 6 is included
[[six-dimensional-scaffold]] — nature: depends-on (strong) — {1,2,3,6} core IS the 2×3 scaffold; 6k±1 structure defines the set’s architecture
[[prime-resonance-computing]] — nature: extends (strong) — defines the optimal encoding basis for Layer 1 Prime-OFDM
[[levin-bioelectricity-prime-resonance]] — nature: analogous-to (strong) — Tusk set reinterpreted as source alphabet; Δ(Σ Pf) = biological system address
[[prime-harmonic-transform]] — nature: depends-on (strong) — Tusk FFT + PHT scoring validates the set’s spectral optimality
[[ontological-number-map]] — nature: maps-to (strong) — set elements map to ONM positions: 1=Source, 2=State, 3=Dimension, 5=Matter, 6=Relationship, 7=Emergence
[[golden-ratio-scaffold]] — nature: supports (moderate) — Pisano period π(6)=24 connects Fibonacci structure to the scaffold the set orbits
[[zeta-zeros-physical]] — nature: bridges (moderate) — zeros encode the prime distribution that the Tusk set makes explicit; die→Tusk = hidden→revealed
[[maxwell-prime-cavities]] — nature: corroborates (moderate) — 24% mode advantage from prime ratios independently matches Tusk set’s 24% flatness advantage
[[cymatics]] — nature: analogous-to (speculative) — hexagonal Chladni patterns may preferentially emerge at Tusk-set frequency ratios
[[delayed-gratification-resonance]] — nature: supports (moderate) — coprime elements in set force longer LCM realignment, building deeper resonance
[[biological-resonance]] — nature: extends (moderate) — set’s coprimality structure parallels biological frequency selection (phyllotaxis, circadian)
[[neural-resonance]] — nature: extends (speculative) — neural oscillation bands may map onto Tusk-set ratios
[[three-tiers-of-primes]] — nature: formalises (strong) — {1,2,3} = source tier, {5,7} = scaffold tier, {6} = composite tier — all three tiers represented in one set
[[relative-time]] — nature: supports (moderate) — set excludes 4 (state space) and 8 (time) — the Tusk set IS the timeless alphabet
[[cosmological-onm]] — nature: extends (moderate) — Tusk set governs developmental milestones at both lunar and solar timescales
[[truth-as-prime-information]] — nature: bridges (speculative) — Tusk set as the irreducible “truth alphabet” from which all composite knowledge is built
[[onm-set-architecture]] — nature: depends-on (strong) — set architecture formalises the dimensional emergence that explains WHY {1,2,3,5,6,7} and not other combinations
[[prime-tree-architecture]] — nature: extends (moderate) — Tusk set as optimal set architecture drives branching geometry; tree growth uses sopfr/Ω from these elements
[[waveform-torsion-division]] — nature: supports (moderate) — Tusk-set frequency ratios produce informationally novel zero-crossing patterns; coprime torsion signatures

Bridging Potential

If combined with [[prime-resonance-computing]], defines the optimal encoding basis for Layer 1
If combined with [[ras]], the geometric shape of {1,2,3,5,6,7} factorisation could explain WHY it’s optimal
Triangulation: optimal set + Tusk FFT peaks + v4 DDS sweeps → validate or refine the set with higher precision

Key Evidence

Tusk/6-frame experiments: v3_board3_tusk_sixframe_22may2026.md
24% improvement over pure primes (flatness metric)
6-frame xcorr: 0.897 (highest recorded in all experiments)
Full experiment suite context: v3_board3_experiment_suite_22may2026.md