Research
5 published papers ยท Open access ยท CC BY-SA 4.0
๐ Published Papers
The Tusk Series: Cumulative Prime Factor Differences as a Structural Signature
28 April 2026 ยท DOI: 10.5281/zenodo.19852116
Introduces the Tusk Series ฮ(ฮฃ Pf) โ a deterministic sequence where primes produce positive spikes and composites produce negative dips. FFT analysis reveals peaks at reciprocals of small primes. Scale-invariant across all tested ranges.
Resonant Amplitude Spectrum: Prime Structure in Frequency Space
2 June 2026 ยท DOI: 10.5281/zenodo.20512346
Develops the Resonant Amplitude Spectrum (RAS) โ a framework mapping prime factorisation to amplitude in a frequency-ratio system. Establishes the Four-Factor Theory: structural resonance, anchor frequency, coprimality, and factor depth.
Prime Resonance Theory: A Structural Framework
4 June 2026 ยท DOI: 10.5281/zenodo.20541350
The unifying theoretical paper. Connects the Tusk Series, RAS, and ONM into a single framework โ Prime Resonance Theory. Proposes that prime-ratio frequency networks exhibit measurably different resonance behaviour than composite-ratio networks.
Prime Tree Architecture: Self-Evolving Structure from the Resonant Amplitude Spectrum
9 June 2026 ยท DOI: 10.5281/zenodo.20609886
Demonstrates that the RAS generates a self-evolving tree structure. The spiral equals the coupling constant ฮฑ. Composites nest 2.05ร deeper than primes (p = 3ร10โปยนยน). The 45ยฐ angle = 360ยฐ/ฯ(24) = Re(s) = ยฝ.
Experimental Validation: Prime-Ratio Resonance in Electronic Torsion Rings
11 June 2026 ยท DOI: 10.5281/zenodo.20637347
First experimental proof that prime-ratio frequency sets outperform composite-ratio sets: +28% amplitude, +18% sharpness, +22% coherence. Zeta zeros resonate at 85โ90% of prime performance. Conducted on the V3 torsion ring board.
๐ฌ How We Test
The principles that govern what we publish and what we discard.
Null Models First
Every claim is tested against Monte Carlo null models or random baselines. If a pattern survives randomisation, it may be structural. If not, it is discarded.
Multiplicative, Not Smooth
Smooth functions of n cannot encode prime structure โ primality is a multiplicative property. We build prime-ratio systems and measure their behaviour, rather than hunting for primes in existing physics.
Honest About Failures
Null results are published, not hidden. The hydrogen analysis (Aug 2025) was debunked by our own methodology โ it taught us more than the results that worked.
Quantitative Over Qualitative
Claims carry p-values, correlation coefficients, or structural proofs. "This number appears therefore primes" is explicitly rejected โ the Fundamental Theorem of Arithmetic makes such observations trivially true for any number.
Reproducible
All experiments use documented hardware, published code, and described methodology. The V3 board design, measurement scripts, and raw data are available.
โ What We Got Wrong
Null results, debunked claims, and the lessons they taught. Published here because honesty is methodology.
Hydrogen energy level differences encode prime structure
Null โ debunked April 2026
Smooth functions evaluated at prime indices produce artifacts of smoothness, not prime structure. Properly tested with null models. Led to the critical insight: don't search for primes in existing physics โ build prime-ratio physics and measure what happens.
Flatness as a discriminator between prime and composite ratios
Null โ May 2026
Flatness does not distinguish prime from composite frequency ratios. The real discriminators are peak-to-peak amplitude (Vpp) and cross-correlation coherence. Redirected experimental focus to the metrics that matter.
Unit-dependent factorisation claims
Rejected as methodology
Factorising a measurement that depends on arbitrary units (metres, seconds) produces artifacts of the unit system, not physical structure. Only dimensionless ratios and structural properties are valid inputs.
๐ Citation
If referencing this work:
Tusk Innovations. (2026). Prime Resonance Theory: A Structural Framework.
Zenodo. https://doi.org/10.5281/zenodo.20541350