For each modulus M ≥ 2, consider the multiplicative group (ℤ/Mℤ)* — the integers r in [1, M−1] with gcd(r, M) = 1. There are φ(M) such residues, where φ is Euler's totient function. Each residue r is placed on a circle of radius proportional to M at angular position θ = 2πr/M. This defines the modular ring for M, and the collection of all such rings for M = 1, 2, … , N forms the modular lifting ring system.
A direct lift connects residue r on ring M to the same integer r on ring M+1 when r remains coprime to M+1. The existence condition is:
The lift graph — rings of coprime residues joined by these conditional edges — is the primary object of study.
Every coprime residue r on ring M encodes a reduced rational fraction r/M in (0, 1). The angular position θ = 2πr/M is simply the fraction r/M scaled to a full turn. This means any reduced fraction a/b with 0 < a < b can be located immediately: it appears on ring M = b at the angular position corresponding to a/b.
Use the Fraction Inspector in the controls panel to enter any a/b and highlight it directly on the visualization, along with its Farey neighbors and Stern-Brocot path.
In polar coordinates with ρ = M and θ = 2πr/M, the trajectory of residue r across rings satisfies:
This is the hyperbolic spiral (reciprocal spiral) ρ = C/θ, studied since Pierre Varignon (1704) and Newton. Each residue r traces a distinct hyperbolic spiral with parameter C = 2πr. The family of all such spirals constitutes the geometric backbone of the visualization.
The normalized angular positions θ/(2π) = r/M on ring M are precisely the elements of the Farey sequence FM — all fractions p/q in (0,1) with q = M in lowest terms. The lift structure connects Farey fractions across consecutive sequences:
When M+1 = p is prime, the p−1 new fractions r/p (for r = 1, …, p−1) appear simultaneously. Each is the mediant of its two Farey neighbors in Fp−1, connecting this structure to the Stern-Brocot tree. The angular "explosion" at prime rings is visible in the visualization as a sudden dense ring of new points.
When M+1 = p is prime (so M = p−1), every coprime residue on ring M survives to ring M+1. Specifically, lifts(p−1 → p) = φ(p−1).
Proof. For any r with gcd(r, p−1) = 1, we have 1 ≤ r ≤ p−2 < p, so gcd(r, p) = 1 automatically (p is prime and r < p). Thus every element of (ℤ/(p−1)ℤ)* lifts successfully. □
Visually: prime rings are transparent to lift lines. The thinning of lift bundles occurs only when M+1 is composite, as composite structure introduces new prime divisors that can trap residues.
When M = 2k − 1 for k ≥ 1, exactly φ(M)/2 residues on ring M survive the lift. The survival ratio equals 1/2 for all such M.
Proof. Since M = 2k−1 is odd, M+1 = 2k. A residue r survives iff gcd(r, 2k) = 1, i.e., r is odd. The map r ↦ M−r is an involution on (ℤ/Mℤ)* (since gcd(r, M) = 1 ⇔ gcd(M−r, M) = 1) that swaps odd and even elements. Hence exactly half the φ(M) units are odd. □
Verified for M = 1, 3, 7, 15, 31, 63, 127, 255, … (all 2k−1). The Mersenne halving produces a characteristic visual signature: exactly half the lift lines are cut at each Mersenne ring.
For general M, the number of surviving lifts is well-approximated by:
This approximation holds because among the φ(M) units mod M, the fraction that avoid M+1's prime factors is approximately φ(M+1)/(M+1) (the density of integers coprime to M+1), with error from boundary effects. The aggregate relative error is under 0.1% for N > 100.
Define the lift survival constant C as the asymptotic weighted fraction of residues that survive lifting:
This is the probability that a uniformly random coprime residue r mod M also satisfies gcd(r, M+1) = 1, averaged with weight φ(M) over all M.
The lift survival constant admits the closed-form Euler product:
The derivation proceeds in two steps. First, partial summation gives:
where AHS = limN→∞ (1/N3) ∑n≤N φ(n)φ(n+1) is the Hausman-Shapiro consecutive totient correlation. Second, local density analysis at each prime p establishes:
The local factor (1−2/p²) at prime p arises because exactly two residue classes mod p fail the lift condition — r≡0 mod p (excluded from the source ring M) and r≡−1 mod p (not coprime to M+1 when p|M+1) — each with density 1/p, giving exclusion 2/p². Combining: C = (π²/2)⋅AHS = (π²/6)⋅∏(1−2/p²).
This parallels the Hardy-Littlewood twin prime constant C2 = ∏p≥3 p(p−2)/(p−1)² ≈ 0.6601…, with quadratic factors replacing linear ones. The constant C is not tabulated in the OEIS (digit string 5307116 returns no results, searched March 2026) and does not reduce to standard zeta values or known constants.
Hyperbolic spirals are classical curves (Varignon 1704, Newton c. 1680). The identification of coprime residue trajectories with this family is immediate from the polar parametrization (ρ, θ) = (M, 2πr/M) but does not appear to be stated in the literature.
Farey sequences (Farey 1816) and the Stern-Brocot tree (Stern 1858, Brocot 1861) govern the angular distribution of residues. Every reduced fraction a/b with b = M corresponds to a point on ring M. This visualization is therefore a complete geometric representation of all rational numbers in (0,1) with bounded denominator.
Consecutive totient correlations ∑φ(n)φ(n+1) appear in Hausman & Shapiro (1984). The connection to the lift survival rate and its geometric interpretation via the ring system is new.
Prime-crossing and Mersenne theorems follow from elementary properties of φ and divisibility. Their formulation in the lifting context — transparent prime rings and Mersenne half-silvering — provides a new geometric language for these classical facts.
Each ring m carries a local entropy measuring how much coprimality information it destroys relative to a uniform residue system:
Primorials 2, 6, 30, 210, 2310, 30030, … are the entropy maxima of modular arithmetic. Their coprime density φ(P#)/P# = ∏p|P#(1−1/p) approaches e−γ/ln(ln(P#)) by Mertens' Third Theorem and is the minimum among all integers of comparable size. Between primorials, prime powers are local entropy minima. The fundamental limit:
Select Modular entropy ΔS_m in the coloring menu: prime rings (low entropy) appear cool blue; primorial rings (maximum entropy) glow amber. The entropy and statistics panels in the sidebar show ΔS_m rankings live for the current range.
The Euler product ζ(2) = ∏p p²/(p²−1) can be reorganized by the gap class of each prime. Define gap(p) = next_prime(p) − p, and:
The rearrangement is valid by absolute convergence. Gap classes contribute a rapidly converging hierarchy:
The Gap Decomposition panel below the canvas shows these partial products live for the current range.
The Hardy-Littlewood singular series predicts the asymptotic density of prime gap classes:
On the primitive polygon of mod 30, the ratio S(6)/S(2) = 2 is not merely asymptotic — it is an exact integer ratio of chord counts. At M = 30: gap-2 chord pairs = 3, gap-4 chord pairs = 3, gap-6 chord pairs = 6. Ratio = 2.0000 exactly. The Hardy-Littlewood prediction is geometrically encoded in the finite mod-30 structure.
Enable Gap Chords in the controls and select mod 30 (set M range to 1–30) to see the pairwise chord connections. Gap 6 chords (violet) will appear exactly twice as numerous as gap 2 (green) and gap 4 (blue).
The gap decomposition motivates three open problems. These are honest open conjectures, not claimed results. None has a proof path bypassing the parity problem in sieve theory.
The gap class {p : gap(p) = g} is infinite ⇔ the remainder Rg(X) = ln Pg − ln Pg(X) satisfies Rg(X) > 0 for all finite X.
The "if" direction is immediate: infinitely many primes keep contributing factors, so Pg(X) never stabilizes. The non-trivial claim is proving Rg(X) > 0 for all X. If the class is infinite, Hardy-Littlewood predicts Rg(X) ~ Cg·S(g)/ln(X) → 0 slowly. If finite with last prime pmax: Rg(X) = 0 for all X > pmax. The function R2(X) remains positive and smoothly decaying at X = 2,000,000 — strong computational evidence for infinitely many twin primes, but not a proof.
For any two even gap classes g, g′ with S(g) = S(g′): the ratio ρ(g,g′) = Pg/Pg′ satisfies 0 < ρ(g,g′) < ∞.
Since S(2) = S(4), proving 0 < P2/P4 < ∞ would show twin and cousin primes have the same finiteness status. Combined with known results on gap 6 (infinite from Bombieri-Vinogradov + Dirichlet), this would propagate infiniteness to gaps 2 and 4.
No gap class can be finite while preserving the exact infinite product identity ζ(2) = π²/6. The transcendental value π²/6 rigidly determines the gap structure of the primes.
If gap class g0 were finite, the remaining primes would redistribute into other gap classes, requiring ∏g ≠ g0 Pg = π²/(6·Pg0fin). The conjecture is that no such redistribution is consistent with the exact value π²/6. This is analogous in spirit to the algebraic independence results of Nesterenko (1996) which force certain products to be transcendental in specific ways. Formalizing "redistribution" precisely is the main open challenge.
Farey sector formula: C(n, N) ≈ 3N²/(π²n(n+1)) coprime pairs in sector n of ℤ/Nℤ.
Primorial lifting recursion: T(p#) = T((p−1)#) × (p−2), verified through 510510.
The Farey sector prefactor 3/π² = (1/2)·(6/π²) = 1/(2ζ(2)) arises directly from the coprime pair density and connects the Farey sector geometry to the ζ(2) normalization. The primorial lifting recursion encodes the Sieve of Eratosthenes in the primorial tower: each new prime p contributes a factor of (p−2) admissible residues for twin prime patterns.
For each modulus M ≥ 2, place every r ∈ [1, M−1] with gcd(r,M)=1 on a circle of radius ∝ M at angle θ = 2πr/M. This is the modular ring for M. The points are exactly the elements of (ℤ/Mℤ)*, with |Ring M| = φ(M).
Residue r on ring M lifts to ring M+1 if and only if r is coprime to both M and M+1.
The second equivalence holds because gcd(M, M+1) = 1 always — consecutive integers share no prime factors. This is the most fundamental constraint of the system.
Count all surviving lifts across all rings up to M = N, divided by all coprime residues:
Probabilistic reading: C is the probability that a uniformly random coprime residue r mod M also satisfies gcd(r, M+1) = 1, averaged over all M weighted by φ(M).
| N | Empirical C | Digits |
|---|---|---|
| 1,000 | 0.530638… | 3 |
| 10,000 | 0.530692… | 4 |
| 100,000 | 0.530711… | 6 |
| 2,000,000 | 0.530711611… | 8+ |
| ∞ (theory) | 0.530711806… | exact |
We analyze the lift condition one prime at a time. For prime p, exactly two residue classes fail the joint condition:
These two classes never coincide: since 0 ≠ −1 (mod p) for any prime p, the exclusions are always independent. Over the p² pairs (r mod p, M mod p), exactly 2 fail. The local survival factor at prime p is:
The denominator p²−1 = (p−1)(p+1) is the product of the two integers neighboring p².
| p | p²−1 | Factor (exact) | Running ∏ |
|---|---|---|---|
| 2 | 3 | 2/3 | 0.66667 |
| 3 | 8 | 7/8 | 0.58333 |
| 5 | 24 | 23/24 | 0.55903 |
| 7 | 48 | 47/48 | 0.54738 |
| 11 | 120 | 119/120 | 0.54282 |
| 13 | 168 | 167/168 | 0.53959 |
| 17 | 288 | 287/288 | 0.53772 |
| 19 | 360 | 359/360 | 0.53622 |
| 23 | 528 | 527/528 | 0.53521 |
| all p | — | converges… | 0.53071… |
The product converges absolutely because the costs 1/(p²−1) decay as 1/p²:
C converges slightly faster than C2: the cost 1/(p²−1) is smaller than C2's cost 1/(p−1)² for every prime p, since p²−1 > (p−1)² for all p ≥ 2. Both converge absolutely. The partial products of C start at 2/3 and decrease monotonically from above toward the limit.
We need to connect the local product to the geometric limit. The bridge is a 1984 result on consecutive totient correlations:
By partial summation, the numerator of the C definition satisfies:
And since ∑M ≤ N φ(M) ~ (3/π²) N² (Mertens 1874), the limit is:
Use the identity π²/6 = ∏p p²/(p²−1) (Euler's product for ζ(2)) to collapse the expression:
One more algebraic step gives the cleanest form:
All three expressions are identical:
The lift survival constant. New. Absolutely convergent. Not previously named or tabulated.
C is structurally parallel to the Hardy-Littlewood twin prime constant C⊂2;, but at the modular level with quadratic rather than linear local factors:
| Constant | Value | Local factor | Convergence |
|---|---|---|---|
| 6/π² (Mertens) | 0.607927… | — | exact |
| C (new) | 0.530712… | (p²−2)/(p²−1) | absolute |
| C⊂2; (twin prime) | 0.660162… | p(p−2)/(p−1)² | absolute |
| A (Artin) | 0.373956… | 1−1/(p(p−1)) | absolute |
The Feller-Tornier constant dFT (OEIS A065474 ≈ 0.3226) shares the inner product ∏(1−2/p²) with C, but uses a different prefactor:
These are distinct constants. The identity connecting them is:
Define Ck as the lift survival rate for ring gap k (between ring M and ring M+k). The Euler product formula generalizes:
When p divides k, the two exclusion classes become correlated, increasing the local factor. Extra factor when p|k:
Ck depends only on rad(k) — the product of distinct primes dividing k. Integers with the same prime radical have identical lift survival rates:
Therefore: C2 = C4 = C8 = C16 = … and C6 = C12 = C18 = C30 = …
The equality C2 = C4 mirrors the Hardy-Littlewood result S(2) = S(4) exactly — the HL symmetry lifts to the modular level. The HL ratio S(6)/S(4) = 2 does not lift, because the C-family extra factors grow as O(1/p²) while HL's grow as O(1/p).