Geometric Canvas — (ℤ/Mℤ)* Explorer

ℤ[i] · Gaussian Integer Structure

The 90°/270° axis encodes ℤ[i]. Only M=4, r=1 sits exactly at i. The mirror σ(r)=M−r is complex conjugation. Primes split by r mod 4.

χ₄ Coloring

r≡1 mod 4 · i · splits r≡3 mod 4 · −i · inert r≡1 mod 2 · −1 · real axis r≡0 mod 2 · non-coprime

Imaginary Axis Overlay

Draw imaginary axis (90°/270° line)

Quadrant labels Highlight M=4 ring

L(s, χ₄) — Dirichlet L-function

L(s,χ₄) = 1 − 1/3ˢ + 1/5ˢ − 1/7ˢ + … separates the two channels. At s=1: π/4 (Leibniz). The two channels must balance — this is Dirichlet's theorem.

s = 1.0 terms

Click compute below

Prime Channel Balance

Dirichlet: #{p≤N : p≡1 mod 4} ≈ #{p≡3 mod 4} asymptotically. But the race fluctuates — Chebyshev's bias: ≡3 leads for most N.

Up to M =

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Presets

PRESETS

Theorems §8

Chain Survival C(n)

Dynamic Presets

Rate Correlation

          Colors each ring by its E(N) contribution — whether it pushes R(N) above or below C.
          Gold = prime ring (spiral dropout). Bar chart shows the full per-ring sequence.
          Correlation between spiral dropouts and large swings is the geometric signature
          of the Möbius cancellation driving O(1/N) convergence.
        

C(n) Quick

INFO

Legend

Coprime r

Lift ✓

Lift ✗

Non-coprime

Prime ring

Lift line

Broken line

Spiral

Gap chord

Theorems §8 — Reference

§8.1 Lift Survival

C = ∏_p (1−1/p(p−1))
C ≈ 0.530711806246

§8.2 Jump Theorem

D(x) jumps by 1/(p(p−1)) at each prime p

§8.3 Farey Sector

C(n,N) ~ 3N²/(π²n(n+1))

§8.4 J = D₀ + T

J ≈ 0.773155 · D₀ ≈ 0.452247 · T ≈ 0.221463

§8.5 Wessen Identity

S(p#) = ∏_{q≤p}(1−1/q)·∏_{q≤p}(q−1)/(q−2)
numerical conjecture

Arrangement

Concentric: ρ = M

Clicked Point

Click a point

p Spiral Table

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ANALYSIS

Canvas Analysis — select a preset to see description.

φ(M): —
T(M): —
C(n): —
R(N): —
E(N): —
|E·N|: —
pts: —
M max: —

Lift Survival Framework · 2026

C = ∏_p(p²−2)/(p²−1) ≈ 0.530711806246

R(N) → C · E(N) = O(1/N) · Wessen Getachew

§1 — The Lift Ratio R(N)

L(M) = φ(M)·φ(M+1)/(M+1) [weight at ring M]
R(N) = Σ L(M) / Σ φ(M) [φ-weighted average]

φ(M)/M is the density of coprimes on ring M.
Large when M prime (all residues coprime).
Small when M highly composite (few survive).

§2 — Theorem: R(N) → C (Proved)

THEOREM 2.1 (PROVED)

R(N) → C = ∏_p(p²−2)/(p²−1)
via Möbius inversion + equidistribution of φ in APs.

§3 — Conjecture: E(N) = O(1/N)

CONJECTURE 3.1 (OPEN · VERIFIED N=10M)

E(N) = R(N) − C = O(1/N)
max|E(N)·N| < 0.929 to N = 10,000,000
Naive bound: O(log N / N) · Observed: O(1/N)

VERIFICATION DATA

N	E(N)	\|E(N)\|·N
1,000	+1.73×10⁻⁴	0.173
10,000	+3.92×10⁻⁵	0.392
100,000	+3.03×10⁻⁶	0.303
1,000,000	+3.87×10⁻⁷	0.387

Cancellation: prime-M terms (−) vs M+1=prime terms (+) nearly cancel. Residual = O(N). Elliott–Halberstam connection.

§4 — Wessen Identity (Proved)

THEOREM 4.1 — EXACT FINITE HARDY–LITTLEWOOD

A(H, p#) = ∏_q≤p (q − ν_H(q))
Error = 0 exactly at every primorial. CRT proof.

Gap g =

p	p#	A(g,p#)	Formula	Error

§5 — Gaussian ℤ[i] Theorems (Proved)

THM 5.1 — UNIQUENESS OF i

Only M=4, r=1 sits at exactly 90°=i.
gcd(M/4,M)=1 only when M=4.

THM 5.2 — MIRROR = CONJUGATION

σ(r)=M−r is complex conjugation on primitive roots. σ(i) = −i on ring 4.

THM 5.3 — INERTNESS CRITERION

r=(q−1)/2 lifts from ring q iff q≡3(mod 4) iff q inert in ℤ[i].
Proof: gcd((q−1)/2, q+1) = gcd((q−1)/2, 2). Two lines.

PRIME SPLITTING vs INERTNESS

q	q mod 4	ℤ[i] behavior	r=(q−1)/2 lifts?	Sum of squares?
3	3	Inert	✓ Yes	No
5	1	Splits	✗ No	Yes 1²+2²
7	3	Inert	✓ Yes	No
11	3	Inert	✓ Yes	No
13	1	Splits	✗ No	Yes 2²+3²
17	1	Splits	✗ No	Yes 1²+4²

§6 — Cyclotomic Tower Identification

Ring M = primitive M-th roots of unity = generators of Φ_M(x).
Full diagram = cyclotomic tower ∪_M Φ_M.
χ_n(r) and angle 2πr/n encode the same information.

n	Algebra	Axis	L(1,χ)
3	ℤ[ω] Eisenstein	120°/240°	π/3√3
4	ℤ[i] Gaussian	90°/270°=i,−i	π/4
6	same as χ₃	60°/300°	π/3√3
8	Klein 4-group	45° octagonal	π/4√2
30	All above ×	All axes at once	product

TECHNICAL REVIEW COMMENTARY

O(1/N) Conjecture:

Safe prime connection: |E(N)·N| peaks at N=p−1 for safe primes p=2q+1. Suggests "resistance" driven by structural rigidity of primes with fewest factors in preceding neighbors. K=ζ(2)·C≈0.872986 is a concrete target for future analytic proof.

Wessen Identity — most publishable insight:

A(6,p#)/A(2,p#)=2 exactly provides finite anchor for HL conjectures. CRT table invites immediate self-verification by the reader.

ℤ[i] inertness:

Tethers modular ring diagram to quadratic residues and Gaussian primes. Geometric reason for why jumps survive or die by orientation in complex plane.

Constant L ≈ 0.300437:

Investigate connection to Feller–Tornier constant or higher-order zeta combinations. If C=ζ(2)·d_FT, L is likely a related derivative.

§8 — Open Problems

OPEN 1 — MAIN CONJECTURE

Prove Σ φ(M)·[φ(M+1)/(M+1)−C] = O(N).
→ Implies E(N)=O(1/N). Likely from Elliott–Halberstam.

OPEN 2 — CONSTANT L ≈ 0.300437

Identify L geometrically. J_twin/J ≠ C₂/C. What does L count?

OPEN 3 — CONSTANT K ≈ 0.872986

Prove K=ζ(2)·C. Explain maxima at N=p−1 for safe primes analytically.

SUBMIT & SHARE

arXiv math.NT submission ↗ Submit C to OEIS ↗ MathOverflow — post the O(1/N) conjecture ↗