omega
plain-language theorem explainer
ω(n) counts the distinct prime factors of a natural number n without multiplicity. Researchers extending the J-cost to completely additive arithmetic functions on integers cite this when decomposing costs into irreducible quanta from the prime spectrum. The definition is a direct one-line extraction of the cardinality of the support of the factorization map.
Claim. For a natural number $n$, let $ω(n)$ denote the number of distinct prime factors of $n$.
background
The Prime Cost Spectrum module extends the Recognition Science J-cost from positive reals to natural numbers via prime factorization. For each $n ≥ 1$ the cost $c(n)$ is defined as $Σ_p v_p(n) · J(p)$, where $J(x) = (x + x^{-1})/2 - 1$, making $c$ completely additive: $c(m·n) = c(m) + c(n)$. This rests on upstream ledger factorization of $(ℝ₊, ×)$ and phi-forcing derived calibration of J, together with spectral emergence structures that fix the gauge content and particle generations from the Q₃ face-pair count.
proof idea
The definition is a one-line wrapper that applies the support cardinality operation directly to the prime factorization of $n$.
why it matters
This definition supplies the count of distinct irreducible cost quanta in the prime spectrum, supporting the module's reformulation of classical prime-counting functions in terms of $c$. It feeds the additivity and nonnegativity results for costSpectrumValue and connects to T5 J-uniqueness plus the eight-tick octave in the forcing chain. No open scaffolding remains in the module.
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