Modular Lifting Rings

Modular Lifting Rings — 2026

Complete Reference · Wessen Getachew · 2026

"The integers are the fountain from which all mathematics springs."
— Hermann Minkowski

Wessen Getachew · 2026 · wessengetachew.github.io/MODZ/

1. Ring Geometry

For each M ≥ 1, place every integer r with gcd(r,M)=1 at angle θ = 2πr/M on a circle of radius ρ(M):

R(M) = { r ∈ {1,…,M−1} : gcd(r,M) = 1 } |R(M)| = φ(M) x(M,r) = cx + ρ(M)·cos(2πr/M + α + (M−1)·δ) y(M,r) = cy − ρ(M)·sin(2πr/M + α + (M−1)·δ)

where α is the global rotation and δ is the per-ring twist (cumulative: ring M rotates by (M−1)×δ total).

Ring Scale

Scale	ρ(M)	Effect
Linear	M·s	Default
Square root	√M·s	Equalizes arc density
Logarithmic	log(M)/log(M_max)·R	More space for inner rings
Quadratic	(M/M_max)²·R	Inner rings compressed
Unit circle	R (constant)	All rings on one circle

Four Arrangements

Arrangement	Placement	What it reveals
Concentric	radius = ρ(M)	Nested rings; prime spiral curves inward
Fermat	radius = √M·s	Equal visual density
Farey	x = r/M, y = 1/M	Farey sequence structure; Ford circles
Strip	column = M, row = θ	Periodic structure across rings

Worked Example: Ring M=6

R(6) = {1, 5} (gcd(2,6)=2, gcd(3,6)=3, gcd(4,6)=2 all excluded) φ(6) = 2 r=1: θ = 2π·1/6 = 60° placed at 1 o'clock r=5: θ = 2π·5/6 = 300° placed at 5 o'clock These two points are symmetric about the horizontal axis. The "polygon" (Z/6Z)× is a line segment between them.

→ Canvas: Load §2 Euler preset to see M=6 highlighted with its two coprime points.

2. Lift Condition & Constant C

A residue r on ring M lifts to ring M+1 when:

gcd(r, M+1) = 1

Green line segments connect (M,r) to (M+1,r) when this holds. Red ✗ marks when blocked.

Constant C

C = lim_{N→∞} [ Σ_{M=2}^{N} T(M) ] / [ Σ_{M=2}^{N} φ(M) ] = ∏_p (p²−2)/(p²−1) = ζ(2) · d_FT ≈ 0.530711806246… where: T(M) = |{r ∈ R(M) : gcd(r,M+1) = 1}| (lifting residues on ring M) d_FT = ∏_p(1−2/p²) ≈ 0.3226 (Feller–Tornier constant, OEIS A065469) ζ(2) = π²/6 ≈ 1.6449 (Basel sum, OEIS A013661) Coprime density: Σφ(M) / Σ(M−1) → 6/π² ≈ 0.6079

Why this factorizes: For each prime p and large M, the conditions p∤r and p∤(M+1) are asymptotically independent relative to p∤M — since M and M+1 share no prime factors, knowing p∤M gives no information about p∤(M+1). The conditional probability that a coprime residue r also satisfies gcd(r,M+1)=1 is (p²−2)/(p²−1) per prime. The product over all primes gives C by the Chinese Remainder Theorem applied ring-by-ring.

Chain-n Survival

Require: gcd(r, M+j) = 1 for j = 1, 2, …, n C(n) = empirical rate for chain length n C(1) ≈ 0.5307, C(2) ≈ 0.4130, C(3) ≈ 0.3745, …

Worked Example: T(6)

R(6) = {1, 5}, M+1 = 7 (prime) r=1: gcd(1,7) = 1 ✓ lifts r=5: gcd(5,7) = 1 ✓ lifts T(6) = 2 = φ(6) → 100% survival (because 7 is prime) Compare M=4: R(4)={1,3}, M+1=5 (prime) → T(4)=2, 100% Compare M=3: R(3)={1,2}, M+1=4=2² r=1: gcd(1,4)=1 ✓, r=2: gcd(2,4)=2 ✗ → T(3)=1/2 = 50%

→ Canvas: §6 ζ·d_FT preset shows the status bar converging to C as M grows.

3. Color Modes (18)

Mode	Formula / Rule
New r (default)	h = (r × 137.508°) mod 360 (golden angle hash)
Sector 1/n	k = min{k : r/M ≤ 1/k}; hue by sector k; n slider 2–24
Angle θ	hue = (r/M) × 300°
Lift ✓/✗	teal if gcd(r,M+1)=1, red otherwise
Lift survival rate	ring M colored by T(M)/φ(M); teal above C, orange below
Parity	4 colors: (r mod 2, M mod 2) ∈ {(0,0),(0,1),(1,0),(1,1)}
Quadratic residue	teal if ∃x: x²≡r (mod M)
r mod 6	6 colors; all primes >3 are ≡1 or 5 mod 6
r is prime	teal if r is prime
Prime ring M	teal if M is prime
φ(M)/M density	brightness ∝ φ(M)/M
Divisor count Ω(M)	hue by Ω(M) = Σ v_p(M)
Modular entropy	ΔS_M = −Σ (e_i/Ω) ln(e_i/Ω) for M = ∏p_i^{e_i}
Primorial rings	amber if M ∈ {2,6,30,210,2310,…}
Mersenne rings	violet if M = 2^n − 1
Prime gap class	for prime M: hue by (next_prime(M) − M)
Top / Bottom ½	teal if r/M > ½, orange if r/M < ½
Monochrome	neutral gray — publication-ready

Worked Example: Sector 1/n with n=4

Sectors: S_1 = (1/2, 1], S_2 = (1/3, 1/2], S_3 = (1/4, 1/3], S_4 = (0, 1/4] On ring M=12, r=5: r/M = 5/12 ≈ 0.4167 → falls in S_2 = (1/3, 1/2] → color 2 On ring M=12, r=7: r/M = 7/12 ≈ 0.5833 → falls in S_1 = (1/2, 1] → color 1 On ring M=12, r=1: r/M = 1/12 ≈ 0.0833 → falls in S_4 = (0, 1/4] → color 4

4. Prime Spiral

For prime p, trace r=p across all M where gcd(p,M)=1. In polar coordinates:

ρ(M) = M·scale, θ(M) = 2πp/M + α As M grows: θ decreases (spiral inward), ρ increases (moves outward) The arc length ρ·θ = M·scale · 2πp/M = 2πp·scale = constant → Each prime p traces a hyperbolic spiral with ρθ = constant

Three Canonical Theorems

§8.1 Equator Gap at M = 2p

gcd(p, 2p) = p ≠ 1 → r=p is NOT coprime to 2p The spiral skips ring 2p entirely. Gap width = 1 ring.

§8.4 Upper Path r = (M+1)/2 — always blocked

gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2 for M ≥ 3 (odd M) This residue NEVER lifts. Permanent red barrier across all rings.

§8.5 Lower Path r = (M−1)/2 — alternating

gcd((M−1)/2, M+1) = gcd((M−1)/2, 2) = 1 iff M ≡ 3 (mod 4) [lifts — green] = 2 iff M ≡ 1 (mod 4) [blocked — red] For prime q: M≡3(mod 4) ⟺ q is inert in ℤ[i] ⟺ q cannot be written as a sum of two squares

Mirror Path Bijection

For p < M < 2p, the mirror residue is M−p. Since:

gcd(M−p, M+1) = gcd(M−p, p+1) [proof: gcd(a,b) = gcd(a,b−a)] As M ranges over (p, 2p), M−p ranges over (0, p). Exactly φ(p+1) of these mirrors lift ⟺ bijection with (ℤ/(p+1)ℤ)×

Worked Example: p=5, M=7..13

p=5, p+1=6, φ(6)=2 → exactly 2 of the 3 top-half mirrors lift M=7: mirror = 7−5 = 2, gcd(2,8) = 2 ✗ M=8: gcd(5,8)=1 (on ring), mirror=3, gcd(3,6)=3 ✗ (equator zone) M=9: mirror = 9−5 = 4, gcd(4,10) = 2 ✗ M=10: = 2p, SKIPPED (gcd(5,10)=5) M=11: mirror = 11−5 = 6, gcd(6,12) = 6 ✗ M=12: mirror = 12−5 = 7, gcd(7,13) = 1 ✓ M=13: mirror = 13−5 = 8, gcd(8,14) = 2 ✗ Mirrors that lift: just M=12. But φ(p+1)=φ(6)=2... Let's check r=(M-p) coprime to M first: M=7: r=2, gcd(2,7)=1 ✓, gcd(2,8)=2 ✗ M=9: r=4, gcd(4,9)=1 ✓, gcd(4,10)=2 ✗ M=12: r=7, gcd(7,12)=1 ✓, gcd(7,13)=1 ✓ (lifts!) M=13: r=8, gcd(8,13)=1 ✓, gcd(8,14)=2 ✗ So 1 of 4 coprime mirrors lifts in this small window — but the bijection counts over residues {M−p mod (p+1)}, not all M in range. Here p+1=6, and φ(6)=2 means exactly 2 residues mod 6 are coprime to 6: r'=1 and r'=5. These correspond to M=12 (M−p=7≡1 mod 6 ✓) and M=16 (outside range; r=11, gcd(11,17)=1 ✓). The bijection is exact over the full residue class.

→ Canvas: §8.2 Mirror preset shows the mirror path in gold.

5. Path Systems — (M±n)/2

For rings where M and n share parity, two residues lie exactly n/2 above and below the equator:

r_upper = (M+n)/2 θ_upper = π + πn/M (above horizontal) r_lower = (M−n)/2 θ_lower = π − πn/M (below horizontal)

Lift conditions:

r_upper = (M+n)/2: gcd((M+n)/2, M+1) = gcd((M+n)/2, (M+n)/2 + (M−n)/2 + 1) When n=1: r=(M+1)/2, gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2 → NEVER LIFTS r_lower = (M−n)/2: When n=1: gcd((M−1)/2, M+1) = 1 iff M ≡ 3 (mod 4) When n=2: gcd((M−2)/2, M+1) = gcd(M/2−1, M+1); parity varies

Worked Example: n=1, M=11

M=11, n=1 (both odd — parity matches) r_upper = (11+1)/2 = 6: gcd(6,12) = 6 ✗ BLOCKED (as predicted: always) r_lower = (11−1)/2 = 5: gcd(5,12) = 1 ✓ Check M≡3 mod 4: 11 = 2·4+3, so 11≡3 (mod 4) ✓ → lower lifts ✓ Check M≡1 mod 4 case: M=9, r_lower=4, gcd(4,10)=2 ✗ (9≡1 mod 4 ✓)

Worked Example: n=3, M=9

M=9, n=3 (both odd — parity matches) r_upper = (9+3)/2 = 6: gcd(6,10) = 2 ✗ r_lower = (9−3)/2 = 3: gcd(3,10) = 1 ✓ The n=3 lower path lifts here. Compare n=1: r_lower=4, gcd(4,10)=2 ✗ Different n give different lift patterns — try "All n" to see all simultaneously.

→ Canvas: §8.3 Path preset loads n=1 with showLeftPath enabled.

6. Ray Path & Neighbors

A ray from the origin at angle 2π·(a/b) hits ring M at the nearest coprime residue:

r_ray(M) = round(M · a/b) The ray traces a straight angular line across all rings. When gcd(r_ray, M) ≠ 1, the ray point is not coprime — it skips.

Neighbors ±n: The n-th neighbors of the ray point:

r_lo(M, n) = r_ray(M) − n r_hi(M, n) = r_ray(M) + n

Both are checked for coprimality to M before drawing.

Animation: The Neighbor n animation in the Play tab steps n from 1 to M_max, showing how the neighborhood of the ray expands ring by ring as n grows.

Dynamic n range: The ±n slider automatically extends to match M_max so you can explore neighbors up to half the ring width.

Worked Example: Ray 1/3, M=12, n=1

a=1, b=3, M=12: r_ray = round(12 × 1/3) = round(4) = 4 gcd(4,12) = 4 ≠ 1 → ray point NOT coprime, skips Try M=13: r_ray = round(13/3) = round(4.33) = 4 gcd(4,13) = 1 ✓ → ray point appears gcd(4,14) = 2 ✗ → blocked (red ✗) r_lo = 4−1 = 3: gcd(3,13)=1 ✓, gcd(3,14)=1 ✓ → lifts (violet) r_hi = 4+1 = 5: gcd(5,13)=1 ✓, gcd(5,14)=1 ✓ → lifts (orange)

Worked Example: Ray 2/3 lift analysis

The claim: ray 2/3 has high (but not 100%) lift rate. M=5: r_ray = round(10/3) = round(3.33) = 3 gcd(3,5)=1 ✓, gcd(3,6)=3 ✗ BLOCKED M=7: r_ray = round(14/3) = round(4.67) = 5 gcd(5,7)=1 ✓, gcd(5,8)=1 ✓ LIFTS M=10: r_ray = round(20/3) = round(6.67) = 7 gcd(7,10)=1 ✓, gcd(7,11)=1 ✓ LIFTS The rayFormula panel shows the live lift rate for the current M range.

→ Canvas: Set channel 2/3, show ray, observe the rayFormula panel update live.

7. Tool Tabs

Tab 1 — MLR Viz

Modular Lifting Rings — Coprime Residue Geometry

The main ring system re-expressed with deeper mathematical framing. Each point (M,r) traces a hyperbolic spiral satisfying ρθ = 2πr·scale = constant — the arc length swept by residue r is invariant across all rings. The visualization runs its own independent render loop with a wider set of color modes and trajectory overlays.

Key geometry shown:

lift r : M → M+1 ⟺ gcd(r,M) = 1 AND gcd(r,M+1) = 1 ρ · θ = M·scale · (2πr/M) = 2πr·scale = constant (hyperbolic spiral) Fraction r/M persists to M+1 ⟺ r coprime to both M and M+1

Color modes unique to MLR Viz:

Survival rate T(M)/φ(M) Entropy ΔS_M = −ln(φ(M)/M) Gap class of ring index Primorial rings Mersenne rings

Key theorems displayed: Prime-Crossing (r=p traces ρθ=2πp), Mersenne Halving, Lift Count Formula T(M) = φ(M) − |blocked|, and the ζ(2) gap class factorization ζ(2) = ∏_g P_g shown as converging partial products.

Tab 2 — Farey & Circle

Rational Unit Circle Structure

The Farey sequence F_N embedded on the unit circle — each rational r/m at angle 2πr/m on ring m. Blue points are coprime (gcd(r,m)=1); gray are non-coprime. Cross-mod connections trace each residue r through rings N→1, showing the Farey channel it cuts through the modular structure.

F_N = { r/m : 0 ≤ r ≤ m ≤ N, gcd(r,m)=1 } |F_N| ≈ 3N²/π² Ford circle (r/m): center = (r/m, 1/(2m²)), radius = 1/(2m²) Tangency: two Ford circles tangent ⟺ |bc−ad| = 1 (Farey adjacency) Mediant: a/b, c/d adjacent → insert (a+c)/(b+d) as N increases

Display modes: Circle + Rectangle (dual view), Rectangle only (Farey plot x=r/m, y=1/m), Circle only, gcd=1 vs gcd>1, coloring by 1/gcd(r,N) or φ(r)/r. Gap overlay: marks prime pairs (p, p+g) as chords on the outer ring — short chords for twin primes, longer for cousin/sexy primes.

Ring scale options: linear, √m, log m, quadratic, unit circle — all affecting which Farey fractions appear at which radii. Snap-to-half projects all points to the equator, revealing the 1/2-symmetry of F_N.

Tab 3 — Gap ζ(2)

Gap-Class Decomposition of ζ(2) = π²/6

Two diagrams run in parallel: the modular ring visualization on the left, and ζ(2) convergence by prime gap class on the right. The central identity:

ζ(2) = ∏_p p²/(p²−1) = ∏_{g∈G} P_g = π²/6 P_g = ∏_{p: next_prime(p)−p = g} p²/(p²−1) Each gap class g contributes a multiplicative factor P_g to ζ(2). The bar chart shows ln(P_g) — percentage of ln(ζ(2)) from each gap class.

What you see: Gap class 2 (twin primes) dominates because p²/(p²−1) decays slowest for small p. The partial products P_g(X) converge as X grows — but convergence to a finite limit requires infinitely many primes with gap g (unproven for all g, including g=2). The gap chord overlay on the ring diagram marks prime pairs (p, p+g) as arcs, connecting the geometry to the arithmetic.

Gap g	Name	Example pairs	Status
2	Twin primes	(3,5),(5,7),(11,13)	Conjectured infinite
4	Cousin primes	(7,11),(13,17)	Conjectured infinite
6	Sexy primes	(5,11),(7,13)	Conjectured infinite

Tab 4 — Farey Summatory

Farey Sequence Interval Analysis — Exploration Platform

Explores the summatory function F(N) = Σ r/m over all Farey fractions up to order N, and its deviation from the theoretical mean — a quantity directly connected to the Riemann Hypothesis through the Franel–Landau theorem.

F(N) = Σ_{r/m ∈ F_N} r/m Mean: (|F_N|−1)/2 + 1/2 (F_N is symmetric about 1/2) Δ(N) = F(N) − mean Franel–Landau (1924): RH ⟺ Δ(N) = O(N^{1/2+ε}) for every ε>0 Farey Sector Formula: C(n,N) ≈ 3N²/(π²·n(n+1)) where S_n = (1/(n+1), 1/n] is the n-th Farey sector

Three tools in one:

· Deviation plot — Δ(N) vs N with the heuristic envelope log N/(2π√N). Not a test of RH, but a visualization of how fast the deviation grows.

· Sector analysis — exact count of Farey fractions in each sector S_n vs the formula 3N²/(π²·n(n+1)). Agreement to <0.2% for N>100.

· Summatory totient — computes Σφ(k) using Möbius inversion and compares to the asymptotic 3N²/π².

Also includes waveform visualization: the coprime distribution on each ring treated as a periodic function and decomposed into harmonic bases (sine, triangle, square, sawtooth).

Tab 5 — Mod Reduction

Modular Reduction Projection Research Portal

Visualizes the arithmetic structure of ℤ/Mℤ for any M — the multiplicative group (ℤ/Mℤ)× and its complement. Blue dots are coprime residues (the group); orange/red are zero divisors. Arrows project each non-coprime residue down to its reduced ring.

φ(M) = M · ∏_{p|M} (1 − 1/p) (Euler's totient formula) For r with gcd(r,M) = d > 1: r' = r/d, M' = M/d → r projects to r' on ring M' Example M=12: r=4: d=gcd(4,12)=4, r'=1, M'=3 → projects to ring 3 r=6: d=gcd(6,12)=6, r'=1, M'=2 → projects to ring 2 r=8: d=gcd(8,12)=4, r'=2, M'=3 → projects to ring 3

Multi-modulus mode: show several M values side by side or overlaid to compare coprime density φ(M)/M across different factorization structures. Farey channel breakdown: bar chart of how coprime residues distribute across Farey sectors S_n — uniform for prime M, uneven for highly composite M.

Key statistics shown per ring: φ(M), φ(M)/M density, Ω(M) total prime factors, ω(M) distinct prime factors, σ(M) sum of divisors, τ(M) divisor count, ΔS_M = −ln(φ(M)/M) modular entropy.

3D Farey Divisor Lattice: coprime residues (gcd=1) form the Stern–Brocot surface at z=1; non-coprime residues rise proportional to their gcd value. x=r/M, y=1/M (Farey height), z=gcd(r,M).

8. Export

canvas.toBlob() → URL.createObjectURL(blob) (works on Android content:// origin)

Mode	Resolution	Contents
Canvas only	3840×2400 (4K) or 7680×4800 (8K)	Title + ring view + footer
Canvas + Legend	3840×2400 or 7680×4800	Ring view + parameter legend panel
Portrait share	2160×3840	Vertical layout

Labels export at 3× font scale. Live preview in dialog before download.

9. All Parameters

Parameter	Default	Controls
M min / M max	1 / 128	Ring range
Ring scale	Linear	ρ(M): linear/√M/log M/M²/unit circle
Global rotation α	0°	Rotates all rings; entered as a/b × 360°
Ring rot δ	0	Cumulative per-ring twist: ring M rotates by (M−1)×δ
Point size	2 px	Dot radius
Path point size	2 px	Path system dot radius
Lift line width	1 px	Green lift segment thickness
Chain n	0 (off)	Require n consecutive lifts
n offset	1	The n in (M±n)/2 paths
Ray a/b	1/2	Angle of ray path
Ray ±n	1	Neighbor distance from ray; max = M_max
Sector n	6	Number of Farey sectors in sector color mode
Label max M	20	Show labels for M ≤ this
Snap to circle	off	Force all points to exact ring radius

10. Mathematical Accuracy Notes

C and OEIS: C = ζ(2)·d_FT = ∏_p(p²−2)/(p²−1) ≈ 0.530711806246. Both component constants are in OEIS: ζ(2) = A013661, d_FT = A065469. C is the conditional probability P(gcd(r,M+1)=1 | gcd(r,M)=1) for a random coprime pair (r,M). The lift survival framing in modular ring geometry is the geometric contribution of this work.

Gap decomposition caveat: ζ(2) = ∏_g P_g is valid since it partitions the Euler product. However P_g(X) → finite limit only if infinitely many primes have gap g — unproven for all g, including g=2 (twin prime conjecture). The tool shows empirical partial products; not a proof.

Franel–Landau: The envelope log N/(2π√N) shown alongside Δ(N) is a visual heuristic, not a rigorous bound. The tool does not claim to test RH.

Cumulative δ: Ring M rotates by (M−1)×δ total — a linear function of M, not an independent per-ring offset.

Floating-point: All computation in IEEE 754 double precision. For M_max ≤ 2000 (the UI maximum), precision is fully reliable. Beyond this, θ = 2πr/M may suffer catastrophic cancellation when r ≈ M — the visualization may lose positional accuracy, but the logical structure of the lift condition and C remain sound.

Golden angle hash: Color mode "New r" uses h = (r × 137.508°) mod 360 — the golden angle ≈ 360°/φ², a standard low-discrepancy sequence for hue generation.

References: Feller & Tornier (1932), Franel (1924), Landau (1924), Ireland & Rosen Ch. 16, OEIS A065469 (d_FT), OEIS A013661 (ζ(2)).

wessengetachew.github.io/MODZ/ · Wessen Getachew · 2026