Intermodulation Collision Mass

How nonlinear frequency mixing explains why primes outperform composites — and why the Tusk-resonant set wins.

What We Know

The Two-Term Model

Kimi K3 (Moonshot AI) independently performed combinatorial intermodulation product enumeration on V3 experimental data (Aug 10-11, 2026) and proposed a two-term model that may unify the Four-Factor Theory:

Ŷ = α(1 − e^(−δn)) − β·(clutter_ratio)^γ · LCM⁻¹

Parameter	Fitted Value	Meaning
α	0.866	Asymptotic scaffold coherence
β	−0.235	Clutter penalty coefficient
γ	0.743	Sub-linear clutter scaling
δ	0.509	Scaffold saturation rate (~2 tones for half-max)
R²	0.459	Moderate (15 captures; single-tone outlier drags fit)
RMSE	0.072	±7% prediction error

The model decomposes spectral quality into two mechanistic terms:

1. Scaffold coherence — integer harmonic ratios creating standing-wave modes (the {1,2,3} backbone)
2. Clutter penalty — non-harmonic intermodulation collisions weighted by amplitude and in-band fraction

Collision Mass: O(1) vs O(n) — The Core Finding

When tones are added progressively, prime collision mass stays flat while composite collision mass grows monotonically:

Tones	Prime Collision Mass	Composite Collision Mass
1	0.000	0.000
2	0.074	0.401
3	0.074	0.564
4	0.074	1.387
5	0.074	1.591
6	0.230	1.716

Primes at 0.074 from 2→5 tones — each new prime finds empty eigenstate space. The jump at tone 6 (/13) may reflect hardware or measurement-window boundary.

Composites grow 0→1.72 — every new composite creates collisions with existing structure because shared divisors force intermod products onto occupied frequencies.

This is the spectral manifestation of the prime–composite duality: primes are orthogonal modes; composites are linear combinations of existing modes.

Harmonic Collisions Anti-Correlate with Coherence

ρ = −0.756, p = 0.011 — the strongest single mechanistic finding across V3 data. Where n·fᵢ = m·fⱼ (harmonic collisions), cross-correlation drops. Mechanism: collisions inject energy back as phase-modulating sidebands → decorrelates torsion channels.

LCM Confinement Explains Peak Counts (ρ = −1.000)

Primes generate more intermod products (558 vs 194 for 6-tone sets) but spread across a 42× finer LCM grid. The V3 “14 vs 27 peaks” result is an FFT resolution artifact: prime products blend below scope resolution into a smooth envelope, while composite products form resolvable clusters.

This doesn’t diminish the prime advantage — it reveals the mechanism. Primes don’t suppress intermod; they disperse it.

Clutter Ratio Rankings (V3 Data, V2 Exact Fractions)

Set	Clutter Ratio	N Products	Collision Mass	Rank
Tusk resonant {1,2,3,5,6,7}	12.53 ✨	221	4.364	1st
Harmonic {1,2,3,4,5,6}	12.97	130	6.331	2nd
Prime baseline {1,2,3,5,7,11,13}	13.13	558	0.411	3rd
Composite narrow {2,4,6,8,9,10}	14.77	263	1.993	4th
Composite wide {2,4,6,8,10,12}	15.86	130	6.331	5th

Key insight: The Tusk-resonant set has the lowest clutter ratio despite not having the fewest products or lowest collision mass. It achieves this through scaffold coherence: the {1,2,3,6} backbone creates coherent standing-wave modes, while coprime extensions {5,7} add spectral richness without clutter.

The {3,5,7} vs {4,9,7} Resolution — Square-Number Pathology

Both sets are 100% pairwise coprime. Identical collision counts. Nearly identical confinement ratios. Yet flatness differs by ~10%.

Resolved: Not a collision effect — both have zero true collisions. The difference is in-band concentration: {4,9,7} crams 100% of product weight in-band vs 99.3% for {3,5,7}. The square numbers (4=2², 9=3²) prevent spectral escape by creating phantom carriers via harmonic reinforcement.

V5 design rule: Avoid square-number ratios even if pairwise coprime.

Relationship to Four-Factor Theory

The original four factors:

1. Structural resonance (dominant)
2. Anchor frequency
3. Coprimality
4. Factor depth

The two-term model potentially subsumes all four:

• Scaffold term captures structural resonance + anchor frequency (harmonic backbone effects)
• Clutter term captures coprimality (via LCM dispersion) + factor depth (via collision amplitude weighting)

If V5 validates the model at R² > 0.75 on sine-wave data, the four factors may reduce to two mechanistic primitives with a clear physical interpretation.

Connection to Prime Eigenstates

Kimi’s key insight connects directly to the prime-eigenstates topic:

“Primes don’t just avoid collisions — they avoid them regardless of how many you add. The prime-numbered submultiples aren’t merely non-interfering; they’re orthogonal modes in the spectral basis. Each new prime tone finds empty eigenstate space. The composite monotonic accumulation is exactly what you’d expect from a system with shared divisors: every new tone is a linear combination of old ones in the nonlinear basis.”

This reframes the prime–composite distinction from number theory into linear algebra: primes span new dimensions; composites project onto existing ones.

What We Don’t Know

1. Does the two-term model hold for sine sources? R²=0.459 on PWM data. V5 sine-wave tests should clarify whether the scaffold/clutter decomposition is robust or partially a harmonic-distortion artifact.
2. Can the model predict untested sets? The real test: fit on V3, predict V5 results for novel frequency combinations.
3. Why does collision mass jump at prime tone 6 (/13)? Hardware limitation, measurement-window boundary, or genuine saturation?
4. Is the O(1) growth truly asymptotic? Need more tones (8, 10, 12+) to test whether prime collision mass remains flat indefinitely or eventually climbs.
5. What is the η term? Single-tone self-coherence (xcorr=0.852) isn’t captured by the scaffold model. Adding η·e^(−δn) should improve fit.

V5 Falsifiable Predictions

1. {3,5,7} vs {4,9,7} gap collapses with sine waves → confirms square-number effect is harmonic distortion (if not → real primality effect beyond coprimality)
2. High-res FFT reveals “missing” prime intermod products at extended bandwidth
3. Phase noise increases when colliding harmonics are active (selective tone enable/disable)
4. Progressive collision mass: prime FLAT, composite MONOTONIC reproduces with DDS sine
5. Two-term model predicts V5 results within ±5% for untested frequency sets
6. Composite dip shrinks >30% with sine sources → current effect is partly harmonic distortion
7. Resolution sweep inverts prime/composite peak-count ratio at high nperseg
8. 7-tone prime set ({1,3,5,7,11,13,17}) gains +3-5% xcorr, flatness unchanged

Source Data

• V3 raw captures: github.com/nagapi2357-ui/pwt-v3-data (53 NPZ + CSV, CC BY-SA 4.0)
• Kimi analysis scripts: projects/Prime_Maxel-v5/KIMI_K3_ANALYSIS_NOTES.md
• Kimi chat logs: projects/Adrian's Stuff/Background for Nagaπ/AI Chat Logs/Kimi K3/
• Four-panel dashboard figures: projects/Prime_Maxel-v5/kimi-figures/

Influences

• Kimi K3 (Moonshot AI) — independent intermod enumeration, two-term model, O(1) vs O(n) framing
• V3 torsion ring experiments (May 2026) — source data
• Four-Factor Theory — theoretical predecessor