Tusk Series — Definitions and Spike Theorem
Tusk Series — Definitions and Spike Theorem
Definitions
sopfr (sum of prime factors with repetition):
- sopfr(1) = 0
- For n > 1: sopfr(n) = Σ_{p^a ‖ n} a·p
θ (prime-power detector):
- θ(n) = p if n = p^k for some prime p and k ≥ 1
- θ(n) = 0 otherwise
Möbius pair:
- sopfr(n) = Σ_{d|n} θ(d)
- θ(n) = Σ_{d|n} μ(d)·sopfr(n/d)
The Tusk series (for n ≥ 2):
T(n) = sopfr(n) − sopfr(n−1) = Σ_{d|n} θ(d) − Σ_{d|n−1} θ(d)
Key inequality
sopfr(m) ≤ m for all m ≥ 1, with equality if and only if m = 4 or m is prime.
(sopfr(1) = 0 < 1.)
Spike Theorem
For every prime p:
T(p) = p − sopfr(p−1) ≥ 1.
T(p) = 1 if and only if p ∈ {3, 5}. For every other prime, T(p) ≥ 2.
Small primes on the lattice
| p | p−1 | Right-hand lattice | T(p) |
|---|---|---|---|
| 2 | 1 | empty (θ ≡ 0) | 2 |
| 3 | 2 | single prime point {2} | 1 |
| 5 | 4=2² | single prime-power chain | 1 |
| ≥7 | composite, ≥6 | product of prime-power chains | large spike |
Detector-not-characteriser corollary
T(n) > 0 whenever n is prime. The converse is false: T(4) = 4 − 3 = 1.
Note the rhyme: 4 is the unique composite with sopfr(n) = n, so it is the unique composite that can mimic the minimal prime spike. (This echoes Nagaπ’s Law: composites must mimic primes to cross any boundary.)
Analytic hook
Dirichlet series:
Σ sopfr(n)/n^s = ζ(s) · Σ_p p/(p^s − 1)
OGF of T:
Σ T(n)·x^n = (1−x) · Σ_p Σ_{k≥1} p·x^{p^k}/(1−x^{p^k})
Cyclotomic remark (analogy, not theorem)
Cyclotomic polynomials factor x^n − 1 over the divisor lattice of n. θ factors sopfr(n) over the same lattice. T is the difference of those two factorisations at neighbouring integers. Same lattice, different monoid.
This is structural analogy, not identification.
Non-claims (explicit)
- No product formula for T(n). The homomorphism lands in an additive group.
- No identification of Φ_p with T(p). They rhyme (both see “p has no proper multiplicative structure”) but are not the same theorem.
- No T(kn) = T(n). Scale invariance does not hold for T.
- The inequality T(p) ≥ 1 is a one-line corollary of a known fact about sopfr. The contribution is the framing.
Relationship to spectral claims
The original Tusk Series paper’s load-bearing claims are spectral (FFT peaks at 1/p, lock to prime-gap spectrum). Those survive or die independently of this section. This theorem makes them interpretable: a peak at 1/p is the periodic return of a left lattice that is a single point. It does not prove the peaks.
What We Don’t Know
- Q-TST-01: Can the spike amplitude T(p) = p − sopfr(p−1) be bounded more tightly for large p?
- Q-TST-02: Is there a characterisation of composites n where T(n) > 0? (4 is one witness; are there infinitely many?)
- Q-TST-03: Does the Dirichlet series of sopfr connect to the spectral claims via the prime zeta function P(s)?