01 · population
Totient shell count
#Sₙ = φ(n)
Shell n contains exactly the reduced residue classes modulo n.
reduced fraction
selected / prime-denominator overlay
hover
θ = 0 east · positive θ counterclockwise
point color mode
Four classical structures, one view
the map displays, it does not invent
02 · angle
Primitive roots
exp(2πik/n), gcd(k,n)=1
The angular positions are precisely the primitive n-th roots of unity.
03 · radius
Equal-area annuli
Area(Aₙ) = π
The law r = √n assigns every denominator the same annular area budget.
04 · cumulative count
Farey / summatory totient
Φ(N) = Σₙ≤N φ(n)
The total point count is the summatory totient; 1 + Φ(N) is the Farey length.
Density hygiene
same word, different denominator
Four quantities that must stay separate
shell population#Sₙ = φ(n)
annular normalizationDₙᵃⁿⁿ = φ(n)/π
circumferential densityDₙᶜⁱʳᶜ = φ(n)/(2π√n)
average disk densityΦ(R²)/(πR²) ~ 3R²/π³
The measure fence
The points lie on circles. The equal-area budget belongs to the assigned annuli. Therefore φ(n)/π is an annular normalization, not a literal pointwise density over the disk.
None of these quantities is the number-theoretic visible-lattice density 1/ζ(2).
Lock: before saying density, name the population, denominator, measure, and ambient space.
The TFG collapse
many cells can share one reduced polar address
(N,K)→
g = gcd(N,K)→
(s,t) = (N/g,K/g)→
t/s→
(√s, 2πt/s)
(12,8) collapses to 2/3
g=4 → (s,t)=(3,2) → (√3, 4π/3)
(6,4) lands on the same point
g=2 → (s,t)=(3,2) → (√3, 4π/3)
The square grid supplies the cells, reduction supplies the denominator shell, the polar map supplies the coordinates, and the totient supplies the population.
Same shell metaphor, different shell law
corpus distinction
Polar denominator shells
rₙ = √n · n↑ ⇒ r↑
Reduced fractions expand outward by denominator. Shell n carries exactly φ(n) discrete points, and its assigned annulus has area π.
Zeta isolation shells
rₘ(ρ) ≈ [m|ζ′(ρ)|]⁻¹ · m↑ ⇒ r↓
Local analytic level sets contract inward around a known simple zero and support a finite-range isolation diagnostic.
Same shell metaphor. Different generator, geometry, and theorem family.
Claims and nonclaims
the fence travels with the figure
What the page claims
- The stated polar map is exact.
- Shell n has φ(n) points.
- The angular positions are primitive n-th roots of unity.
- Each assigned denominator annulus has area π.
- The TFG denominator-collapse map is exact.
What the page does not claim
- No new totient or cyclotomic theorem.
- No constant-density filling of the disk.
- No transfer of visible-lattice density into this geometry.
- No shared shell law with the zeta annular-shell diagnostic.
- No claim that absence of found prior art proves novelty.