The Tusk Series

Status: validated Domain: math Source: → projects/Prime_Maxel-v4/research/tusk_series.md, tusk_series_fft.py

What We Know

• The discovery: Δ(Σ Pf) — first differences of sum-of-prime-factors
• Primes = positive spikes, composites = negative dips — a binary-like signature emerging from arithmetic
• Scale-invariant “DNA” pattern — like musical scales that stretch logarithmically
• FFT proof: Tusk Series and prime gaps produce harmonically locked spectra (3:2, 2:1, 3:1 ratios)
• 3 direct frequency matches + 16 harmonic relationships between Tusk Series and prime gap spectra
• Tusk peaks at reciprocals of small primes: 1/2, 1/3, 2/5, 3/7 — exactly the v4 board’s divisor ratios
• Works for any nonce but breaks on odd-only sequences
• Published on Zenodo: DOI 10.5281/zenodo.19852116 (Apr 28, 2026)

FFT Peak Analysis (Q-TRS-03 research, 2026-06-04)

Full FFT of Δ(sopfr(n)) for n=2..20000 reveals dominant peaks at reciprocals of small integers:

Rank	Frequency	Period	Amplitude	Interpretation
1	0.5000	2.00	34,224,796	Parity (every-other-number)
2	0.3333	3.00	20,122,549	Divisibility by 3
3	0.4000	2.50	12,771,673	Period-5 alias (5/2)
8	0.2000	5.00	6,986,070	Divisibility by 5
15	0.1667	6.00	5,352,426	6k scaffold — anomalously strong for a composite
23	0.1429	7.00	4,311,972	Divisibility by 7

Key finding: Amplitudes fall roughly as 1/d for primes. Composite 6 has anomalously high amplitude (5.35M) — comparable to prime 5 (6.99M) — because 6 = lcm(2,3) captures the combined 2×3 sieve structure.

Spectral energy rankings directly predict which frequency divisors contribute most to resonance, and correctly rank {1,2,3,5,6,7} above pure primes {1,2,3,5,7,11} by ~28% in spectral score.

What We Don’t Know

• Q-TS-01: Why does the Tusk Series break on odd-only sequences? What does this tell us about even numbers’ structural role?
• Q-TS-02: Is there a closed-form expression for the Tusk Series envelope?
• Q-TS-03: Can the Tusk Series predict optimal frequency sets for arbitrary network sizes (beyond 6)?
• Q-TS-04: What is the relationship between Tusk Series scale-invariance and the Riemann zeta function’s functional equation?

Prime Harmonic Transform Analysis (2026-06-04)

The PHT applied to the Tusk Series reveals it is overwhelmingly a prime signal:

• PRS = 0.967 — 96.7% of spectral energy at prime indices
• Dominant PHT peaks: |P(2)| = 249.1, |P(3)| = 97.3, |P(5)| = 24.7, |P(7)| = 9.1
• Amplitude decay ∝ 1/n — primes dominate every amplitude rank in their neighbourhood
• Composite indices carry residual energy: |P(4)| = 47.5, |P(6)| = 12.9
• Möbius deconvolution separates unique vs inherited energy at each index

Amputation Experiment (6 Jun 2026)

The Tusk Series Amputation Experiment systematically removed multiples of each alphabet element, revealing:

• Prime spike invariance: 100% spike preservation across all 10 amputations — a mathematical invariant
• Distinct failure modes: each element removal produces a unique FFT/variance/AC signature
• Sparse primes as keystones: removing mult(5) or mult(7) more destructive than mult(2)
• Composites are the medium: primes-only series collapses to near-Gaussian (variance 34 vs 7.4M)
• f=1/2 is Nyquist: the 1/2 frequency belongs to consecutive integer sampling, not Binary

See [[tusk-series-amputation]] for full results.

Relationships

• [[tusk-series-amputation]] — nature: extends — amputation experiment reveals structural invariants and failure modes
• [[four-factor-theory]] — nature: depends-on — Four-Factor Theory explains WHY Tusk-derived sets work physically
• [[tusk-resonant-set]] — nature: extends — {1,2,3,5,6,7} was discovered through Tusk Series analysis
• [[zeta-zeros-physical]] — nature: bridges — both encode prime structure in frequency domain; Tusk peaks at reciprocals of small primes
• [[v3-experimental-proof]] — nature: supports — v3 physically confirmed the frequency relationships Tusk Series predicts
• [[prime-composite-duality]] — nature: depends-on — the positive/negative spike pattern IS the give/resist duality
• [[ras]] — nature: analogous-to — both map number theory to geometry/shape
• [[prime-harmonic-transform]] — nature: supports — PHT proves Tusk is 96.7% prime-structured (PRS = 0.967)
• [[prime-resonance-computing]] — nature: informs (strong) — Tusk FFT peaks match optimal encoding frequencies for Prime-OFDM Layer 1
• [[levin-bioelectricity-prime-resonance]] — nature: supports (moderate) — Tusk Δ(Σ Pf) models the nonce→address→broadcast architecture Levin describes for bioelectric signalling
• [[six-dimensional-scaffold]] — nature: depends-on (strong) — Tusk FFT amplitude at 1/6 reveals the scaffold’s spectral importance; peaks cluster around 6k±1 structure
• [[onm-set-architecture]] — nature: bridges (moderate) — Tusk Series encodes the same structural relationships spectroscopically that set architecture formalises set-theoretically
• [[prime-tree-architecture]] — nature: supports (moderate) — Tusk Δ(Σ Pf) encodes sopfr structure that drives tree branching decisions and give/resist ratios
• [[waveform-torsion-division]] — nature: bridges (moderate) — positive/negative spike pattern IS the give/resist duality; torsion waveform character at the series level

Bridging Potential

• If combined with [[zeta-zeros-physical]], could establish whether Tusk Series IS a discretized form of the zeta function
• If combined with [[cymatics]], Tusk peaks could predict which Chladni patterns emerge at prime-ratio frequencies
• Triangulation: Tusk FFT peaks + v3 measured resonances + zeta zero spacings → three independent confirmations of the same harmonic structure

Key Evidence

• Zenodo publication: https://zenodo.org/records/19852116
• FFT analysis: projects/Prime_Maxel-v4/research/tusk_series_fft.py
• Research doc: projects/Prime_Maxel-v4/research/tusk_series.md