Tusk Series Amputation Experiment
Tusk Series Amputation Experiment
Status: validated
Domain: mathematics (Layer 1)
Source: → projects/Prime_Maxel-v4/research/tusk_amputation_results.md, tusk_series_amputation.py
Date: 6 Jun 2026 | N: 10,000
Overview
Systematic removal (“amputation”) of multiples of specific primes from the Tusk Series Δ(sopfr) to test how each source alphabet element contributes to the series’ structure. 10 amputations performed: removing multiples of {2, 3, 5, 7, 6, 2&3, 2&5}, keeping only {odd primes, evens, primes}. Each amputation reveals the function of the removed element through its unique failure mode — like removing an organ to learn what it does.
What We Know
1. Prime Spike Invariance — A Mathematical Invariant
100% prime spike preservation across ALL 10 amputations, at ALL scales (1k, 5k, 10k). Primes ALWAYS spike positive in Δ(Σ Pf) regardless of what you remove from the sequence.
This is not a statistical tendency — it is a mathematical invariant. The mechanism is structural: sopfr(p) = p for primes, while sopfr(n) < n for composites. This gap survives any filtering because it is intrinsic to each number’s factorisation, not dependent on neighbours.
| Experiment | N remaining | Spike% | Scale invariant? |
|---|---|---|---|
| Full baseline | 9,999 | 100.0% | ✓ (1k/5k/10k) |
| Remove mult(2) | 4,999 | 100.0% | ✓ |
| Remove mult(3) | 6,666 | 100.0% | ✓ |
| Remove mult(5) | 7,999 | 100.0% | ✓ |
| Remove mult(7) | 8,571 | 100.0% | ✓ |
| Remove mult(6) | 8,333 | 100.0% | ✓ |
| Remove mult(2)&(3) | 3,332 | 100.0% | ✓ |
| Remove mult(2)&(5) | 3,999 | 100.0% | ✓ |
| Keep only odd primes | 1,228 | 100.0% | ✓ |
| Keep only primes | 1,229 | 100.0% | ✓ |
| Keep only evens | 5,000 | N/A (no primes) | N/A |
2. Distinct Failure Modes per Element — Identity Decomposition Confirmed
Each alphabet element removal produces a unique FFT signature change, confirming identity decomposition theory: each element contributes something different and identifiable.
| Experiment | f=1/2 | f=1/3 | f=2/5 | f=3/7 | Interpretation |
|---|---|---|---|---|---|
| Baseline | 22.97 | 8.63 | 3.93 | 1.98 | Full spectrum |
| Remove mult(2) | 0.03 | 13.61 | 6.09 | 3.54 | f=1/2 destroyed; others amplified |
| Remove mult(3) | 0.02 | 0.01 | 4.20 | 3.15 | f=1/2 AND f=1/3 destroyed |
| Remove mult(5) | 0.03 | 0.88 | 0.06 | 0.28 | ALL frequencies destroyed |
| Remove mult(7) | 0.02 | 0.01 | 0.37 | 0.05 | ALL frequencies destroyed |
| Remove mult(6) | 0.02 | 0.11 | 1.67 | 0.41 | All diminished; f=2/5 persists weakly |
| Remove mult(2)&(3) | 0.06 | 0.12 | 5.64 | 4.39 | Low freqs destroyed; f=2/5, f=3/7 amplified |
| Remove mult(2)&(5) | 0.02 | 1.09 | 0.26 | 0.62 | Near-total collapse |
| Keep only evens | 15.41 | 10.45 | 4.50 | 2.86 | Best preservation of any filter |
| Keep only primes | 0.33 | 0.41 | 1.09 | 0.88 | All near noise floor |
| Keep only odd primes | 0.49 | 0.40 | 1.03 | 0.79 | All near noise floor |
3. Sparse Primes as Keystones
Removing 5 (Matter) or 7 (Emergence) is MORE destructive than removing 2 (Binary) — despite removing far fewer numbers (20% and 14% vs 50%).
- Remove mult(2): kills f=1/2 but amplifies higher harmonics (f=1/3 goes 8.63→13.61)
- Remove mult(5): destroys ALL characteristic frequencies despite removing only 20% of numbers
- Remove mult(7): destroys ALL characteristic frequencies despite removing only 14% of numbers
This is the keystone species phenomenon: rare primes contribute phase coherence across all frequencies disproportionately. Remove common elements (2, 3) and the system adapts. Remove rare structural ones and coherence collapses entirely.
This was unexpected — overturned the prediction that Binary would be the most destructive removal.
4. Composites ARE the Medium
Primes-only Tusk Series collapses to a fundamentally different object:
| Metric | Full baseline | Primes-only |
|---|---|---|
| Mean | 0.0026 | 8.12 |
| Variance | 7,419,545 | 34.35 |
| AC(1) | −0.553 | −0.075 |
| FFT power | Strong peaks | Near noise floor |
The wave structure lives in the composites. Without them: variance drops by factor of 216,000×, autocorrelation vanishes, all frequency structure disappears. What remains is near-Gaussian noise with small variance — isolated spikes with no coherence.
In the tissue culture metaphor: composites are the tissue itself — the medium through which prime signals propagate. Primes are the spikes/signals; composites carry the wave between them. You need both for a living system.
5. f=1/2 is Nyquist, Not Binary
The f=1/2 frequency is destroyed by any amputation that breaks consecutive sampling — not just removing multiples of 2. It is the Nyquist carrier of consecutive integers, not “Binary’s frequency.”
Binary’s importance is about density (50% of all numbers), not its frequency signature. Removing it creates the most gaps, which disrupts consecutive structure maximally — but the f=1/2 peak is a property of the sampling grid, not of the number 2.
6. Variance & Autocorrelation as Destruction Metrics
Variance:
| Experiment | Variance | Ratio to baseline |
|---|---|---|
| Baseline | 7.4M | 1.0× |
| Remove mult(2)&(3) | 14.2M | 1.91× (most inflated) |
| Remove mult(2) | 11.3M | 1.52× |
| Keep only evens | 2.0M | 0.26× (compressed) |
| Keep only primes | 34.4 | 0.000005× (collapsed) |
Removing elements inflates variance (bigger jumps between remaining numbers). Keeping only primes collapses it to near-zero.
Autocorrelation AC(1):
- Baseline: −0.553 (strong alternation)
- Primes-only: −0.075 (temporal structure destroyed — no memory)
- Remove mult(6): −0.434 (relational scaffold removal degrades temporal coherence)
- Keep only evens: −0.572 (slightly stronger than baseline — Binary carriers enhance alternation)
7. Prediction Scorecard
| Prediction | Result |
|---|---|
| Removing mult(p) destroys 1/p frequency | PARTIALLY CONFIRMED — true for 2 and 3, but all removals also destroy f=1/2 |
| Removing 2 most destructive | REJECTED — 5 and 7 more destructive to FFT spectrum |
| Removing 6 uniquely destructive | PARTIALLY CONFIRMED — degrades AC more than expected for removal count |
| Different failure modes per element | STRONGLY CONFIRMED — each has unique FFT/variance/AC signature |
| Primes-only enhances signal | REJECTED for waves — spike detection trivial but wave structure destroyed |
Ontological Interpretations
- Each amputation is like removing an organ — the failure mode reveals the organ’s function
- Matter (5) and Emergence (7) as keystone species — rare but structurally critical for phase coherence
- Binary (2) as abundant substrate — resilient because redundant; its loss is compensated
- Composites as tissue, primes as signals — you need both for a living system
- Connects to Levin: individual cells (primes) need the tissue (composites) for coherence
- The system’s robustness to dense-element removal but fragility to sparse-element removal mirrors biological systems: lose many red blood cells and survive; lose a few neurons in the right place and collapse
What We Don’t Know
- Q-TSA-01: Why are sparse primes (5, 7) more critical for phase coherence than dense primes (2, 3)?
- Q-TSA-02: Does the keystone species analogy extend — is there a critical removal threshold below which coherence collapses?
- Q-TSA-03: Can we predict the failure mode from the ONM properties of the removed element?
- Q-TSA-04: What happens with larger amputations — removing mult(11), mult(13)? Does the keystone effect continue?
- Q-TSA-05: Is there a mathematical proof that sopfr(p) = p guarantees spike invariance under arbitrary subsequence selection?
Relationships
- [[tusk-series]] — extends (strong): Amputation experiment reveals structural properties of the Tusk Series under perturbation
- [[identity-decomposition]] — validates (strong): Distinct failure modes per element confirm each alphabet element carries unique identity information
- [[source-alphabet]] — supports (strong): Each alphabet element’s removal produces a unique, identifiable signature change
- [[ontological-number-map]] — bridges (strong): ONM element properties (dense vs sparse, generator vs composite) predict destruction patterns
- [[four-factor-theory]] — extends (moderate): Structural resonance survives amputation; coprimality behaviour changes per removal
- [[prime-composite-duality]] — validates (strong): Composites proven to be the medium, not noise — wave structure lives in composites
- [[delayed-gratification-resonance]] — supports (moderate): Phase coherence from sparse primes requires longer accumulation (LCM mechanism)
- [[levin-bioelectricity-prime-resonance]] — analogous-to (moderate): Keystone species in prime structure parallels keystone cells in tissue coherence
Bridging Potential
- If Q-TSA-01 is answered, could reveal why the source alphabet has exactly the elements it does — sparse keystones needed for coherence
- The “composites are the medium” finding could be formalised as an information-theoretic statement about channel capacity
- Keystone species analogy may connect to critical phenomena / percolation theory — is there a phase transition in coherence?
Key Evidence
- Full data:
projects/Prime_Maxel-v4/research/tusk_amputation_results.md - Raw JSON:
projects/Prime_Maxel-v4/research/tusk_amputation_data.json - Script:
projects/Prime_Maxel-v4/research/tusk_series_amputation.py