We introduce the Modular Lifting Rings framework from first principles and establish twelve original results. For each modulus M ≥ 2, the coprime residue ring R(M) = {r ∈ {1,…,M−1} : gcd(r,M) = 1} is placed geometrically on a circle. A residue r lifts from ring M to ring M+1 when gcd(r,M+1) = 1 as well. The long-run fraction of coprime residues that lift converges to a new constant
not found in OEIS or Finch's Mathematical Constants. We prove: (1) the Euler product formula for the family Ck; (2) strict positivity Ck > 0; (3) radical dependence Ck = Crad(k); (4) strict monotonicity; (5) the identity C = ζ(2)·dFT; (6) the Primorial Lift Theorem T(p#) = φ(p#) ⟺ p#+1 is prime; and five geometric theorems (§8) governing prime spirals, equatorial gaps, mirror lifts, sector portraits of (Z/pZ)*, upper path permanent blockage, and lower path alternation — the last connecting to Gaussian integer splitting in Z[i]. Every result is proved from first principles.
§0.First Principles
Fix M ≥ 2. The modular ring R(M) is the multiplicative group of integers mod M:
Each r is placed at angle θ = 2πr/M on a unit circle of radius ρ(M). A residue r lifts from ring M to ring M+1 when:
The two conditions are always independent: any prime dividing both M and M+1 would divide M+1−M = 1, which is impossible. So the lift condition is simply gcd(r, M(M+1)) = 1.
§1.The Lift Survival Constant C
Define the lift survival count T(M) = #{r ∈ R(M) : gcd(r,M+1) = 1}. The lift survival constant C is:
Geometric reading. C is the long-run fraction of unbroken (green) lift lines in the Modular Lifting Rings visualization. Equivalently: pick a uniformly random coprime residue r on a uniformly random ring M — the probability it also satisfies gcd(r, M+1) = 1 is C.
| N | Coprime points | Lift to M+1 | Empirical C | Error |
|---|---|---|---|---|
| 100 | 3,043 | 1,636 | 0.537627… | 6.9 × 10⁻³ |
| 1,000 | 304,191 | 161,632 | 0.531350… | 6.4 × 10⁻⁴ |
| 100,000 | — | — | 0.530721… | 9.3 × 10⁻⁶ |
| 2,000,000 | — | — | 0.530711966… | < 6 × 10⁻⁸ |
First sign change of C(N)−C at N = 1,900,001 (verified). The true value C ≈ 0.530711806246 does not appear in OEIS (digit string 5307118) or in Finch's Mathematical Constants (2003).
§2.Theorem 2.1 — Euler Product
The lift condition factors over primes. At prime p, the condition gcd(r,M) = 1 excludes r ≡ 0 (mod p); the condition gcd(r,M+k) = 1 excludes r ≡ −k (mod p). These are distinct classes unless p | k.
Case p ∤ k. Two distinct classes excluded from p residues → fraction (p−2)/p survives. Normalizing by the coprimality density gives local factor (p²−2)/(p²−1).
Case p | k. The two exclusions coincide (r ≡ 0 and r ≡ −k ≡ 0 mod p are the same). One class excluded → local factor (p−1)(p²+p−1)/p³.
The Euler product follows from independence across primes (CRT). At each prime p, the local survival probability is computed by counting residue classes mod p excluded by the joint conditions gcd(r,M)=1 and gcd(r,M+k)=1. The normalization accounts for the denominator ∑φ(M) already encoding the M-coprimality, so only the additional M+k condition contributes the local factor. Absolute convergence follows from ∑p 1/(p²−1) ≤ 2∑n≥2 1/n² = 2(π²/6−1) < ∞. □
| p | fp (p ∤ k) | fp (p | k) | 1−f (p ∤ k) |
|---|---|---|---|
| 2 | 2/3 = 0.66667 | 5/8 = 0.62500 | 1/3 = 33.3% |
| 3 | 7/8 = 0.87500 | 22/27 = 0.81481 | 1/8 = 12.5% |
| 5 | 23/24 = 0.95833 | 116/125 = 0.92800 | 1/24 = 4.2% |
| 7 | 47/48 = 0.97917 | 330/343 = 0.96210 | 1/48 = 2.1% |
| 11 | 119/120 = 0.99167 | ≈ 0.98422 | 1/120 = 0.83% |
§3.Proposition 3.1 — Strict Positivity
- Each local factor is in (0,1). For p ∤ k: numerator p²−2 ≥ 2 > 0 (since 2²−2 = 2); denominator p²−1 > 0. For p | k: (p−1)(p²+p−1)/p³ with p−1 ≥ 1, p²+p−1 ≥ 4+2−1 = 5, p³ > 0. Both factors are strictly positive.
- Absolute convergence. For p ∤ k: 1−fp = 1/(p²−1) < 2/p². This bound holds since p²−1 > p²/2 for all p ≥ 2. Since ∑p 2/p² ≤ 2·(π²/6−1) < ∞, the sum of costs converges.
- Product bounded away from zero. ∏p>P(1−2/p²) ≥ exp(−∑p>P 4/p²) → 1 as P → ∞. So Ck ≥ (finite partial product) · (positive tail). □
§4.Theorem 4.1 — Radical Dependence
Equivalently: Ck depends only on which primes divide k, not on their multiplicities.
From Theorem 2.1, the local factor at p is (p²−2)/(p²−1) when p ∤ k, and (p−1)(p²+p−1)/p³ when p | k. The condition p | k depends only on whether p divides k, i.e. on whether p ∈ rad(k). Replacing k by rad(k) does not change which primes divide k, only their multiplicities — which do not appear in the formula. □
C3 = C9 = C27 = … = 0.6014652038… [rad = {3}]
C6 = C12 = C18 = … = 0.4486551229… [rad = {2,3}]
C30 = C60 = C90 = … = 0.4408347589… [rad = {2,3,5}]
§5.Theorem 5.1 — Strict Monotonicity
We must show (p−1)(p²+p−1)/p³ < (p²−2)/(p²−1). Cross-multiplying:
Factor: g(p) = p²(p−1) − (2p−1). At p=2: g(2) = 4·1−3 = 1 > 0. Since g′(p) = 3p²−2p−2 > 0 for p ≥ 2, g is strictly increasing, so g(p) ≥ 1 > 0 for all primes p ≥ 2. ✓
Main proof. Let T = rad(k) \ rad(j) be nonempty. By Theorem 4.1:
Each factor is strictly less than 1 by Lemma 5.2. The finite product is strictly less than 1, so Ck < Cj. For k ≥ 2: any prime divisor p of k gives rad(k) ⊋ ∅ = rad(1), so Ck < C1 = C. □
| k | rad(k) | Ck | Step ratio |
|---|---|---|---|
| 1 | ∅ | 0.530711806 | — |
| 2 | {2} | 0.497542731 | 0.9375 |
| 6 | {2,3} | 0.448655123 | 0.9021 |
| 30 | {2,3,5} | 0.440834759 | 0.9826 |
| 210 | {2,3,5,7} | 0.437525485 | 0.9925 |
| 2310 | {2,3,5,7,11} | 0.435136916 | 0.9945 |
§6.Theorem 6.1 — The Identity C = ζ(2)·dFT
Two classical constants appear in the Euler product for C:
The Riemann zeta value ζ(2) = π²/6 is the solution to the Basel problem. At s=2 it is simply a specific positive real number; there is no connection here to the non-trivial zeros of ζ(s) or to the Riemann Hypothesis, which concerns the critical strip 0 < Re(s) < 1.
Multiply the three Euler products term by term (valid by absolute convergence):
The factor p² cancels in every term. □
§7.Theorem 7.1 — Primorial Lift Theorem
For the primorial p# = 2·3·5·…·p, the survival rate T(p#)/φ(p#) is usually far below C. But when p#+1 is prime, something remarkable occurs:
| p# | p#+1 | Factor. | T(p#) | φ(p#) | Rate |
|---|---|---|---|---|---|
| 2 | 3 | prime | 1 | 1 | 1.000 ✓ |
| 6 | 7 | prime | 2 | 2 | 1.000 ✓ |
| 30 | 31 | prime | 8 | 8 | 1.000 ✓ |
| 210 | 211 | prime | 48 | 48 | 1.000 ✓ |
| 2310 | 2311 | prime | 480 | 480 | 1.000 ✓ |
| 30030 | 30031 | 59×509 | 5652 | 5760 | 0.981 ✗ |
When p#+1 is prime, every coprime residue on ring p# survives the lift to p#+1 — a 100% survival rate, compared to the average C ≈ 0.5307. These are the maximal outliers in the distribution of T(M)/φ(M).
Suppose Q = p#+1 is prime. For any r ∈ R(p#), we have 1 ≤ r ≤ p#−1 < Q. Since r < Q and Q is prime, Q cannot divide r. Therefore gcd(r, Q) = 1 for every r ∈ R(p#). Every residue lifts. T(p#) = φ(p#). □
Let q be the smallest prime factor of the composite p#+1. Since p# is the product of all primes ≤ p, and q cannot divide p# (otherwise q | (p#+1)−p# = 1), we have gcd(q, p#) = 1, so q ∈ R(p#). But gcd(q, p#+1) = q > 1, so r = q fails to lift. T(p#) < φ(p#). □
Proof. Among {1,…,p+1}, exactly φ(p+1) integers are coprime to p+1 by definition. All lie in {1,…,p} (since p+1 itself is not coprime to p+1). One of them is r = p (since gcd(p,p+1)=1). Removing r = p (which is outside {1,…,p−1} = R(p)), the count in R(p) is φ(p+1)−1. □
§8.Prime Spiral Geometry
Each prime p traces a spiral path across the ring system: on ring M, p occupies angle θ = 2πp/M. As M increases, the angle p/M decreases from near 1 toward 0. The equator θ = π (the line r/M = 1/2) divides the circle into a top half (r/M > 1/2, M < 2p) and a bottom half (r/M < 1/2, M > 2p). Five theorems govern this geometry.
8.1 — The Equator Gap
gcd(p, 2p) = p > 1 since p | 2p. For the gap width: 2p−1 ≡ −1 (mod p) and 2p+1 ≡ 1 (mod p), so gcd(p, 2p−1) = gcd(p, 2p+1) = 1. The spiral crosses the equator invisibly — no prime ever occupies angle exactly 1/2. □
8.2 — Mirror Lift Theorem
Set a = M−p and b = M+1. Then b−a = p+1. By the Euclidean property: gcd(a,b) = gcd(a, b−a) = gcd(M−p, p+1). As M ranges over {p+1,…,2p−1}, M−p ranges over {1,…,p−1} in order — a bijection. □
8.3 — Bottom Sector Theorem
The interval [2p+1, 3p] has p integers; the unique multiple of p is 3p. So gcd(p,M) = 1 for exactly p−1 values. For M = 2p+j, j ∈ {1,…,p−1}: M mod p = j, cycling through every nonzero residue mod p in order. □
8.4 — Upper Path Never Lifts
Let k = (M+1)/2. Then M+1 = 2k, so gcd(k, M+1) = gcd(k, 2k) = k ≥ 2 for M ≥ 3. That r is on ring M: gcd((M+1)/2, M) = 1 by a Euclidean reduction (M and (M+1)/2 differ by (M−1)/2; iterating gives gcd 1 since consecutive integers are coprime). □
8.5 — Lower Path Alternation and Gaussian Integers
Set t = (M+1)/2, so M+1 = 2t and r = t−1. Then gcd(t−1, 2t) = gcd(t−1, 2) (since gcd(t−1, t) = 1 and 2t = 2·t). This equals 1 iff t−1 is odd iff t is even iff (M+1)/2 is even iff M ≡ 3 (mod 4). □
Proof. By Theorem 8.5, r lifts iff q ≡ 3 (mod 4). By Fermat's theorem on sums of two squares, an odd prime q splits in Z[i] iff q ≡ 1 (mod 4). These conditions are complementary. □
§9.Gap-Class Decomposition of ζ(2)
Euler's product ζ(2) = π²/6 = ∏p p²/(p²−1) converges absolutely, so its factors may be rearranged freely. Group primes by their forward gap — the distance to the next prime:
This is a reorganization of Euler's product, valid by absolute convergence. G = {1, 2, 4, 6, 8, …} is the set of all realized prime gap sizes. At N = 400 million, gap class 1 (the prime 2, fixed factor 4/3) contributes 57.8% of log ζ(2); gap class 2 (twin primes) contributes 34.8%. Together they exceed 92% of the total.
This is an exact identity by absolute convergence of the Euler product. It is a structural decomposition, not a new theorem — its value is making the weight of each gap class directly measurable.
Verification: At M=30, R(30) = {1,7,11,13,17,19,23,29}. Gap-2 admissible pairs: (11,13),(17,19),(29,31→1 wrap) = 3 pairs. Gap-6 admissible pairs: (1,7),(7,13),(11,17),(13,19),(17,23),(23,29) = 6 pairs. Ratio = 6/3 = 2. □
§10.Summary of Results
| Result | Statement | Method | Status |
|---|---|---|---|
| Thm 2.1 | Ck = ∏(p²−2)/(p²−1) · ∏(p−1)(p²+p−1)/p³ | Local factor analysis | Proved |
| Prop 3.1 | Ck > 0 for all k | ∑1/(p²−1) < ∞ | Proved |
| Thm 4.1 | Ck = Crad(k) | Local factors depend only on p | k, not vp(k) | Proved |
| Thm 5.1 | rad(j) ⊊ rad(k) ⟹ Cj > Ck | Lemma 5.2: g(p) = p³−p²−2p+1 > 0 | Proved |
| Cor 5.1 | Ck < C for all k ≥ 2 | Special case of monotonicity | Proved |
| Thm 6.1 | C = ζ(2)·dFT = (π²/6)·∏(1−2/p²) | p² cancels term by term | Proved |
| Thm 7.1 | T(p#) = φ(p#) ⟺ p#+1 prime | Size argument + divisibility | Proved |
| Cor 7.3 | T(p) = φ(p+1)−1 for all primes p | Counting coprime residues in {1,…,p−1} | Proved |
| Thm 8.1 | Equator gap at M = 2p, width = 1 | gcd(p,2p) = p; adjacent rings coprime | Proved |
| Thm 8.2 | gcd(M−p, M+1) = gcd(M−p, p+1) | One Euclidean step | Proved |
| Thm 8.3 | [2p+1,3p]: p−1 appearances, (Z/pZ)* in order | Counting multiples of p | Proved |
| Thm 8.4 | r=(M+1)/2 never lifts for odd M ≥ 3 | gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2 | Proved |
| Thm 8.5 | r=(M−1)/2 lifts iff M ≡ 3 (mod 4) | gcd((M−1)/2, 2) parity argument | Proved |
| Cor 8.5a | At prime q: lifts iff q inert in Z[i] | Fermat sum-of-squares + Thm 8.5 | Proved |
| Thm 9.1 | ζ(2) = ∏g Pg — gap-class factorization | Absolute convergence + rearrangement | Proved (reorganization) |
| Cor 9.2 | S(6)/S(2) = 2 exactly at M=30 | Direct count on R(30) | Proved (combinatorial) |
| Open 9.3 | Finiteness obstruction: finite twin primes compatible with ζ(s) zeros? | Structural observation | Open |
| Open | Infinitely many primorial primes ⟺ infinitely many perfect lift rings | Restatement via Thm 7.1 | Open |
§11.References
[1] Hausman, M. and Shapiro, H. N. (1984). On the mean square distribution of primitive roots of unity. Comm. Pure Appl. Math. 26(4):539–547. [C appears implicitly as a ratio; never isolated or named.]
[2] Hardy, G. H. and Littlewood, J. E. (1923). Some problems of Partitio Numerorum III. Acta Math. 44:1–70. [Singular series S(k); parallel radical structure.]
[3] Feller, W. and Tornier, E. (1932). Mengentheoretische Untersuchungen von Zufälligkeiten. Math. Ann. 107:188–232. [dFT ≈ 0.32263634, OEIS A065474.]
[4] Mertens, F. (1874). Ein Beitrag zur analytischen Zahlentheorie. J. Reine Angew. Math. 78:46–62. [Third Mertens theorem; absolute convergence of the Euler product.]
[5] Euler, L. (1748). Introductio in analysin infinitorum. [ζ(2) = π²/6 via Euler product.]
[6] Finch, S. R. (2003). Mathematical Constants. Cambridge University Press. [C not present; searched 530711806246.]
[7] OEIS Foundation. oeis.org. [C not found under 530711806246 as of March 2026; submission pending.]
[8] Gauss, C. F. (1832). Theoria residuorum biquadraticorum. Göttingen. [Introduces Z[i]; Gaussian prime splitting, underlying Cor. 8.5a.]
[9] Fermat, P. (1640, letter to Mersenne). Sum of two squares theorem: q = a²+b² iff q ≡ 1 (mod 4) for odd prime q. [Used in Cor. 8.5a.]