Four visualizations probing the framework's limits: lift-constant friction, ghost prime residues in the primorial sieve, logarithmic sign-change rarity at scale, and the parity obstruction that prevents standard sieve methods from counting twin primes exactly.
Each integer M contributes a local lift factor L(M) = φ(M+1)/(M+1). The theoretical constant C ≈ 0.5307 is the limit R(N) converges to. Friction measures the signed relative deviation: f(M) = L(M)/C − 1. Smooth numbers (products of small primes) create deep negative friction; primes spike sharply positive. The histogram shows the full deviation distribution — a fat-tailed landscape that reveals where the convergence R(N) → C is most turbulent, and where potential counter-examples to the density claims would most naturally hide.
At primorial level M = p₁·…·pₖ, most coprime residues r in [1, M) are genuine primes — but some are composites of primes > pₖ that only appear prime-like. These are the ghost composites: they pass the gcd(r, M) = 1 test but will be eliminated when M advances to M·p_{k+1}. The transient residues are those that fail the very next sieve step. Twin-prime pair candidates where at least one member is a ghost are the "composite interference paths" — pairs that look like twin primes at this level but are not.
Chart 06 shows the raw gap between consecutive sign changes of E(N). Here the gaps are plotted on a logarithmic scale — log₁₀(gap_k) vs sign-change index k — exposing any long-term growth trend in waiting time. A rising OLS regression line indicates sign changes grow rarer at larger N, connecting the small-N behaviour of E(N) to Littlewood's theorem about π(x) − li(x): the changes must occur infinitely often, but with astronomically increasing spacing as N → ∞.
The parity problem in sieve theory: standard multiplicative sieves cannot distinguish integers with an even number of prime factors (Liouville λ = +1) from those with an odd number (λ = −1). The top chart shows L(N) = Σ λ(n) — the cumulative Liouville sum — whose smallness relative to N reflects near-perfect parity balance. The bottom chart tracks, in real time, the fraction of coprimes to M = 30 with odd Ω. If this fraction deviated detectably from ½, a sieve could exploit the imbalance to count twin primes exactly. It does not — and the chart shows why.
The log-derivative D(x) = d/dx log C(x) is smooth everywhere except at the primes. At x = p−1, D(x) jumps by exactly 1/(p(p−1)). Composites pass in silence. Given any jump size s, the prime is recovered exactly: p = (1 + √(1 + 4/s)) / 2. The discriminant 1 + 4p(p−1) = (2p−1)² is always a perfect square for integer primes — the algebraic fingerprint of the theorem. The golden ratio φ is the supremum of the jump spectrum: s = 1 is the only value no integer prime reaches. Enter any integer to verify.