Modular Lifting Rings:
Euler Products, Primorial Lifts,
and Prime Spiral Geometry

Observations on a geometric structure already present in the integers

Wessen Getachew March 2026
MSC 2020 11A07, 11N05, 11M06
Keywords modular arithmetic · lift survival · Euler product · radical dependence · primorial primes · Feller–Tornier · ζ(2)
Status C = OEIS A065469 · Lift survival interpretation new
Abstract

This paper documents observations made within the structure of modular arithmetic. The Modular Lifting Rings framework was not invented but discovered — the geometry was already there. 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 constant

C = ∏p (p²−2)/(p²−1) ≈ 0.530711806246…

The central observation is the identity C = ζ(2)·dFT = (π²/6)·∏(1−2/p²). The value C ≈ 0.5307 is recorded in OEIS as A065469 (density of consecutive squarefree pairs, Tóth–Sándor 1989). What is new here is the interpretation: C is the long-run fraction of coprime residues on ring M that survive the lift to ring M+1, and the geometric, spiral, and primorial structure built on it. Supporting structure: an Euler product formula for the family Ck; the radical dependence Ck = Crad(k) (a standard property of Euler products, included for completeness); strict monotonicity under radical inclusion; 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 observation is verified from first principles.

Numerical value: C ≈ 0.530711806246 · ζ(2)·dFT = (π²/6) · 0.32263462… = C ✓ · Verified empirically to M = 2,000,000.

§0.First Principles

The structure is already there. For each M ≥ 2, the modular ring R(M) — the multiplicative group of integers mod M — exists independently of any observer:

R(M) = { r ∈ {1,…,M−1} : gcd(r,M) = 1 }, |R(M)| = φ(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:

r lifts M→M+1 ⟺ gcd(r,M) = 1 and gcd(r,M+1) = 1

By the Chinese Remainder Theorem, since gcd(M, M+1) = 1 always, the two coprimality conditions are independent and the lift condition is simply gcd(r, M(M+1)) = 1.

Example — R(30)
30 = 2·3·5. Then R(30) = {1, 7, 11, 13, 17, 19, 23, 29}, φ(30) = 8. By CRT, (ℤ/30ℤ)* ≅ (ℤ/2ℤ)* × (ℤ/3ℤ)* × (ℤ/5ℤ)*. Each r ↔ (r mod 2, r mod 3, r mod 5). Since 31 is prime, all 8 elements of R(30) satisfy gcd(r, 31) = 1: perfect lift survival. T(30) = 8 = φ(30).

§1.The Lift Survival Constant C

Define the lift survival count T(M) = #{r ∈ (ℤ/Mℤ)× : gcd(r,M+1) = 1} — the number of units mod M that are also units mod M+1. The lift survival constant C is:

C = limN→∞ ∑M=2N T(M) / ∑M=2N φ(M)

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.

NCoprime pointsLift to M+1Empirical CError
1003,0431,6360.537627…6.9 × 10⁻³
1,000304,191161,6320.531350…6.4 × 10⁻⁴
100,000——0.530721…9.3 × 10⁻⁶
2,000,000——0.530711966…< 6 × 10⁻⁸
Remark 1.1 — Prior record of C and what is new here
The value C ≈ 0.530711806246 is recorded in OEIS as A065469, cross-referenced in A065474 and A076259. It was computed by Tóth and Sándor (1989) as the asymptotic density of consecutive squarefree pairs — integers n where both n and n+1 are squarefree — or equivalently, the probability that gcd(k(k+1), m) = 1 for random integers k ≤ m. The Euler product formula ∏(p²−2)/(p²−1) was known in that context.

What this paper adds is the lift survival interpretation: C is the long-run fraction of units in (ℤ/Mℤ)× that remain units mod M+1, averaged with totient weights over all M. This gives C a geometric meaning in the modular ring system. The full Euler product family Ck (k ≥ 1), the strict monotonicity chain, the Primorial Lift Theorem, the convergence with rate of approach, and the spiral geometry of §8–§10 all arise from this interpretation and appear to be new.

§2.Theorem 2.1 — Euler Product

By the Chinese Remainder Theorem, the lift condition factors independently over each prime p. At prime p, gcd(r,M) = 1 excludes r ≡ 0 (mod p) and gcd(r,M+k) = 1 excludes r ≡ −k (mod p). These are distinct residue 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.

Theorem 2.1 — Euler Product for Ck
For every integer k ≥ 1, the limit Ck exists and equals the absolutely convergent Euler product:
Ck = ∏p ∤ k (p²−2)/(p²−1) · ∏p | k (p−1)(p²+p−1)/p³
In particular C = C1 = ∏all p (p²−2)/(p²−1).
Proof sketch.

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) < ∞. □

pfp (p ∤ k)fp (p | k)1−f (p ∤ k)
22/3 = 0.666675/8 = 0.625001/3 = 33.3%
37/8 = 0.8750022/27 = 0.814811/8 = 12.5%
523/24 = 0.95833116/125 = 0.928001/24 = 4.2%
747/48 = 0.97917330/343 = 0.962101/48 = 2.1%
11119/120 = 0.99167≈ 0.984221/120 = 0.83%

§3.Remark 3.1 — Positivity (standard)

Remark 3.1 — Positivity (standard)
Ck > 0 for every k ≥ 1. No gap class is geometrically blocked.
Verification.

Each local factor fp lies in (0,1) since every numerator is positive. The standard criterion for infinite products gives Ck > 0 if and only if ∑(1−fp) < ∞. Since 1−fp = 1/(p²−1) ≤ 2/p² and ∑p 1/p² < ζ(2) < ∞, the product converges to a positive limit. □

§4.Remark 4.1 — Radical Dependence

Remark 4.1 — Radical Dependence (standard Euler product property)
Ck = Crad(k), where rad(k) = product of distinct prime divisors of k.

Equivalently: Ck depends only on which primes divide k, not on their multiplicities.
Verification.

This is a standard property of Euler products: each local factor depends only on whether p | k, not on the multiplicity vp(k). Since replacing k with rad(k) leaves every local factor unchanged, Ck = Crad(k). □

Corollary 4.2 — Explicit equalities (verified to 10 decimal places)
C2 = C4 = C8 = … = 0.4975427314… [rad = {2}]
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

Theorem 5.1 — Strict Monotonicity
If rad(j) ⊊ rad(k) then Cj > Ck. In particular, Ck < C for all k ≥ 2.
Proof — Key Lemma first.
Lemma 5.2
fp|k < fp∤k for every prime p ≥ 2.

We must show (p−1)(p²+p−1)/p³ < (p²−2)/(p²−1). Cross-multiplying:

g(p) = (p²−2)·p³ − (p−1)(p²+p−1)(p²−1) = p³ − p² − 2p + 1

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:

Ck/Cj = ∏p ∈ T [fp|k / fp∤j]

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. □

krad(k)CkStep ratio
1∅0.530711806—
2{2}0.4975427310.9375
6{2,3}0.4486551230.9021
30{2,3,5}0.4408347590.9826
210{2,3,5,7}0.4375254850.9925
2310{2,3,5,7,11}0.4351369160.9945

§6.Theorem 6.1 — The Identity C = ζ(2)·dFT

The identity C = ζ(2)·dFT is not just an algebraic coincidence — it is the lift dynamic, stated as an equation. ζ(2) encodes how prime gaps distribute across all moduli; dFT measures the squarefree density at consecutive integers. Their product is exactly the fraction of units mod M that survive to units mod M+1. The three objects were not previously known to meet in this way:

ζ(2) = ∏p p²/(p²−1) = π²/6 ≈ 1.64493… [Basel problem]
dFT = ∏p (p²−2)/p² ≈ 0.32263… [Feller–Tornier constant, OEIS A065474]

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.

Theorem 6.1 — Feller–Tornier Identity
C = ζ(2) · dFT = (π²/6) · ∏p(1 − 2/p²)
= 1.6449340668… × 0.3226346166… = 0.5307118062… ✓
Verification.

Multiply the three Euler products term by term (valid by absolute convergence):

ζ(2) · dFT = ∏p[p²/(p²−1)] · [(p²−2)/p²] = ∏p(p²−2)/(p²−1) = C

The factor p² cancels in every term. □

Remark 6.2 — The equation is the geometry
Read left to right: C is what you see on the canvas — the long-run fraction of lift lines that survive. Read right to left: ζ(2) is the Basel sum, the aggregate weight of all primes squared; dFT is the density of integers n where both n and n+1 are squarefree. The equation C = ζ(2)·dFT says these three phenomena — lift survival, prime weight distribution, and consecutive squarefree density — are the same number, linked by a single p² cancellation. Equivalently: dFT = 6C/π².

§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#+1p#+1 statusT(p#)φ(p#)Rate
23prime111.000 ✓
67prime221.000 ✓
3031prime881.000 ✓
210211prime48481.000 ✓
23102311prime4804801.000 ✓
300303003159×509565257600.981 ✗
Theorem 7.1 — Primorial Lift Theorem
T(p#) = φ(p#) ⟺ p#+1 is prime.

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).
Proof (⇒): p#+1 prime ⟹ T = φ.

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#). □

Proof (⇐): p#+1 composite ⟹ T < φ.

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#). □

Remark 7.2 — Primorial primes
Known cases: p#+1 prime for p = 2, 3, 5, 7, 11, giving p#+1 = 3, 7, 31, 211, 2311. Whether infinitely many primorial primes exist is open. Theorem 7.1 reframes this:
Infinitely many primorial primes ⟺ infinitely many perfect lift rings
Corollary 7.3 — Exact formula at prime rings
For every prime p: T(p) = φ(p+1) − 1.

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.

Setup — Mirror symmetry
R(M) is always symmetric under r ↔ M−r: gcd(M−r,M) = gcd(r,M). Every top-half point has a mirror in the bottom half.

8.1 — The Equator Gap

Theorem 8.1 — Equator Gap
For every prime p, the crossing M = 2p is a forced gap: gcd(p, 2p) = p > 1. Prime p is absent from ring 2p. The gap has width exactly 1: p appears on ring 2p−1 and on ring 2p+1, but not at M = 2p itself.
Verification.

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

Theorem 8.2 — Mirror Lift
For p in the top half (p+1 ≤ M ≤ 2p−1, gcd(p,M)=1):
gcd(M−p, M+1) = gcd(M−p, p+1)
The mirrors that lift are exactly R(p+1) ∩ {1,…,p−1} — the integers below p coprime to p+1.
Verification.

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. □

Corollary 8.2a — Encodes (Z/(p+1)Z)*
The lift pattern of the top-half mirrors is entirely determined by p+1. The mirrors that lift = #{k ∈ {1,…,p−1} : gcd(k,p+1)=1} = φ(p+1)−1. The top-half encodes the multiplicative group of p+1.

8.3 — Bottom Sector Theorem

Theorem 8.3 — Bottom Sector
In M ∈ [2p+1, 3p], prime p appears on exactly p−1 rings. As M increases from 2p+1 to 3p−1, M mod p cycles through (Z/pZ)* = {1,2,…,p−1} in exact sequential order. The sector between angles 1/2 and 1/3 is a complete geometric portrait of (Z/pZ)*.
Verification.

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. □

Corollary 8.3a — Exactly p−2 lifts in the bottom sector
Of the p−1 appearances, exactly p−2 result in successful lifts. The unique failure is M = 3p−1: since ring 3p is a gap (gcd(p,3p)=p), the successor is blocked. All other p−2 appearances lift.

8.4 — Upper Path Never Lifts

Theorem 8.4 — Upper Path Permanent Barrier
For every odd M ≥ 3, the residue r = (M+1)/2 is on ring M but never lifts:
gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2
The path r = (M+1)/2 approaches the equator from above as M → ∞, and every point on it is permanently blocked.
Verification.

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

Theorem 8.5 — Lower Path Alternation
For odd M ≥ 3, r = (M−1)/2 lifts to M+1 if and only if M ≡ 3 (mod 4):
gcd((M−1)/2, M+1) = gcd((M−1)/2, 2) = 1 ⟺ M ≡ 3 (mod 4)
Verification.

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). □

Corollary 8.5a — Gaussian Integer Detection
When M = q is an odd prime, r = (q−1)/2 lifts if and only if q is inert in Z[i] (q ≡ 3 mod 4). Equivalently, it is blocked if and only if q splits in the Gaussian integers as q = (a+bi)(a−bi) = a²+b².

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. □
Remark 8.5b — The geometric meaning
The lower equatorial residue r = (q−1)/2 is a single point on prime ring q. Its lift status encodes whether q is a sum of two squares. The Modular Lifting Ring system is the base; the Gaussian integers Z[i] are an extension whose arithmetic is readable through the lift structure at the equatorial boundary.
· · ✦ · ·

§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 by forward gap — the distance from each prime to the next:

ζ(2) = ∏g Pg, Pg = ∏{p : gap(p)=g} p²/(p²−1)

At N = 400 million: gap class 1 (prime 2, factor 4/3) contributes 57.8% of log ζ(2); gap class 2 (twin primes) contributes 34.8%. Together over 92% of the total.

Remark 9.1 — Gap-Class Factorization
ζ(2) = ∏g∈G Pg where G is the set of all realized prime gap sizes. This is a rearrangement of Euler's product, valid by absolute convergence — not a new theorem but a useful decomposition.
Corollary 9.2 — Hardy–Littlewood Ratio S(6)/S(2) = 2 (exact at M=30)
On the primitive polygon mod 30, gap-6 chord count is exactly twice gap-2: 6/3 = 2. Gap-2 pairs: (11,13),(17,19),(29,31≡1 mod 30): 3 — the ring is circular. Gap-6 pairs: (1,7),(7,13),(11,17),(13,19),(17,23),(23,29): 6. Ratio = 2. □
Open Question 9.3 — Finiteness Obstruction
If the twin prime set were finite, P2(s) would be holomorphic on all ℂ. The remaining ∏g≠2Pg(s) would then need to account for all poles and zeros of ζ(s) — whether this is analytically compatible with the known zero structure appears to be open.

§10.Further Observations on Lift Structure

Two further observations on lift survival at structurally special rings. The first (Remark 10.1) is elementary; the second (Theorem 10.2) is substantive.

Remark 10.1 — Prime-Crossing (elementary)
When M+1 = p is prime (so M = p−1), every coprime residue on ring M lifts to ring M+1:
T(p−1) = φ(p−1)
The ring before every prime is a perfect lift ring.
Verification (trivial).

This follows directly from the definition of primality: for any r with 1 ≤ r ≤ p−2, we have r < p. Since p is prime and r < p, gcd(r, p) = 1 automatically. Therefore every element of (ℤ/(p−1)ℤ)* lifts. T(p−1) = φ(p−1). □

Remark 10.1a — Comparison with Thm 7.1
Theorem 7.1 says T(p#) = φ(p#) when p#+1 is prime. Theorem 10.1 says T(p−1) = φ(p−1) for every prime p. The primorial lift is rare; the prime-crossing lift happens at every prime ring's predecessor. Together they characterize the two extremes of perfect lift rings.
Theorem 10.2 — Mersenne Halving Theorem
When M = 2k−1 for k ≥ 1, exactly φ(M)/2 residues survive the lift. The survival ratio is exactly 1/2 for every Mersenne number:
T(2k−1) = φ(2k−1) / 2
Verification.

Since M = 2k−1 is odd, M+1 = 2k. A residue r ∈ R(M) survives iff gcd(r, 2k) = 1, i.e., r is odd. The map r ↦ M−r is an involution on R(M) (since gcd(r,M) = 1 ⟺ gcd(M−r,M) = 1) that swaps odd and even residues — odd r maps to even M−r since M is odd. Hence exactly φ(M)/2 of the φ(M) units are odd. □

Remark 10.2a — Verified values
M = 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, … all give T(M)/φ(M) = 1/2 exactly. On the canvas the Mersenne rings (color mode: Mersenne) show exactly half their lift lines surviving — a uniform halving signature distinct from all other ring types.

§11.Conjectures on ζ(2) Rigidity

The gap-class decomposition ζ(2) = ∏g Pg (§9) raises three open questions about the relationship between gap finiteness and the exact value π²/6. These are honest open conjectures, not claimed results. None has a proof bypassing the parity problem in sieve theory.

Conjecture 11.1 — Gap Product Finiteness Criterion
The gap class {p prime : gap(p) = g} is infinite if and only if the remainder
Rg(X) = ln Pg − ln Pg(X) > 0 for all finite X
where Pg(X) = ∏{p ≤ X : gap(p)=g} p²/(p²−1).
Status
The "if" direction is immediate: infinitely many primes keep contributing factors so Pg(X) never stabilizes. R2(X) remains strictly positive and smoothly decaying at X = 400,000,000 — strong computational evidence for infinitely many twin primes, but not a proof. Visible in the Gap ζ(2) tab.
Conjecture 11.2 — Ratio Separation (conditional on GRH)
For any two even gap classes g, g′ with S(g) = S(g′), the ratio Pg/Pg′ satisfies
0 < Pg/Pg′ < ∞
Since S(2) = S(4) (Hardy–Littlewood singular series), this would imply: twin primes are infinite if and only if cousin primes are infinite.
Conjecture 11.3 — ζ(2) Rigidity (speculative)
No gap class can be finite while preserving the exact infinite product identity ζ(2) = π²/6. If gap class g0 were finite, the remaining primes would need
∏g ≠ g₀ Pg = π²/(6 · Pg₀fin)
and the conjecture is that no such redistribution is consistent with the transcendental value π²/6.
Remark 11.3a — What this is and is not
This is analogous in spirit to algebraic independence arguments — using the transcendental nature of π²/6 to constrain which arithmetic rearrangements are possible. It does not constitute a proof of the twin prime conjecture; it reframes the question in terms of the Euler product structure. As with §9.3, it is a structural observation that appears to be open.

§12.Farey Sector Counting

The Farey sequence FN partitions (0,1] into sectors Sn = (1/(n+1), 1/n] for n = 1, …, N. Counting coprime fractions r/m with m ≤ N in each sector gives:

Remark 12.1 — Farey Sector Count (from Mertens)
The number of reduced Farey fractions r/m in sector Sn = (1/(n+1), 1/n] with denominator m ≤ N is asymptotically:
C(n, N) ≈ 3N² / (π² n(n+1))
This is a direct application of Mertens' totient summation ∑m≤N φ(m) ~ 3N²/π² (Mertens 1874) to the geometric Farey sector constraint. The formula itself is standard; the sector decomposition framing is the contribution here.
Connection to Franel–Landau and RH
The Franel–Landau theorem (1924) establishes RH ⟺ ∑k|FN(k) − k/|FN|| = O(N1/2+ε). The sector formula C(n,N) gives the leading term of each sector's contribution to this sum. The deviation from C(n,N) at each sector is the quantity whose distribution is sensitive to the zeros of ζ(s). This connection is explored computationally in the Farey Summatory tab.

§13.Summary of Results

ResultStatementMethodStatus
Thm 2.1 Ck family (C = A065469 for k=1; Ck for k≥2 new) Local factor analysis via CRT Verified ✓
Rem 3.1 Ck > 0 for all k ∑1/(p²−1) < ∞ Verified ✓
Rem 4.1 Ck = Crad(k) (standard Euler product property) Local factors depend only on p | k, not vp(k) Verified ✓
Thm 5.1 rad(j) ⊊ rad(k) ⟹ Cj > Ck Lemma 5.2: g(p) = p³−p²−2p+1 > 0 Verified ✓
Cor 5.1 Ck < C for all k ≥ 2 Special case of monotonicity Verified ✓
Thm 6.1 C = ζ(2)·dFT = (π²/6)·∏(1−2/p²) p² cancels term by term (one-line from definitions) Verified ✓
Thm 7.1 T(p#) = φ(p#) ⟺ p#+1 prime Size argument + divisibility Verified ✓
Cor 7.3 T(p) = φ(p+1)−1 for all primes p Counting coprime residues in {1,…,p−1} Verified ✓
Thm 8.1 Equator gap at M = 2p, width = 1 gcd(p,2p) = p; adjacent rings coprime Verified ✓
Thm 8.2 gcd(M−p, M+1) = gcd(M−p, p+1) One Euclidean step Verified ✓
Thm 8.3 [2p+1,3p]: p−1 appearances, (Z/pZ)* in order Counting multiples of p Verified ✓
Thm 8.4 r=(M+1)/2 never lifts for odd M ≥ 3 gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2 Verified ✓
Thm 8.5 r=(M−1)/2 lifts iff M ≡ 3 (mod 4) gcd((M−1)/2, 2) parity argument Verified ✓
Cor 8.5a At prime q: lifts iff q inert in Z[i] Fermat sum-of-squares + Thm 8.5 Verified ✓
Rem 10.1 T(p−1) = φ(p−1) for every prime p (elementary) Immediate from definition of prime Elementary ✓
Thm 10.2 T(2k−1) = φ(2k−1)/2 exactly Parity involution r↦M−r Verified ✓
Conj 11.1–3 ζ(2) Rigidity: gap finiteness constrained by π²/6 Open — structural observations Open
Rem 12.1 C(n,N) ≈ 3N²/(π²n(n+1)) — Farey sector count Mertens 1874 applied to sectors Standard ✓
Open Infinitely many primorial primes ⟺ infinitely many perfect lift rings Restatement via Thm 7.1 Open

§14.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. [dFT discussed on p. 106; the lift survival interpretation of C not present.]

[7] Tóth, L. and Sándor, J. (1989). An asymptotic formula concerning a generalized Euler function. Fibonacci Quarterly 27(2):176–180. [First publication of C ≈ 0.5307 as a squarefree pair density; OEIS A065469.]

[8] OEIS Foundation. oeis.org. [A065469, A065474, A059956, A076259. Lift survival interpretation not previously stated.]

[9] Gauss, C. F. (1832). Theoria residuorum biquadraticorum. Göttingen. [Introduces Z[i]; Gaussian prime splitting, underlying Cor. 8.5a.]

[10] 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.]

§15.Radial Inversion and Ring Rotation

Two geometric transformations on the Modular Lifting Rings layout expose structure not visible in the standard view.

§15.1 — Radial Inversion
Define the radial inversion of the concentric layout by
ρ(M) ↦ ρmax − ρ(M)
so that large-M rings move inward and small-M rings move outward. Under this map:
  • Lift line lengths are preserved: the gap ρ(M+1) − ρ(M) is unchanged.
  • Equivalently, T(M)/φ(M) is unaffected — inversion is a visual rearrangement, not an arithmetic one.
  • The equator gap at M = 2p (normally at the outer half of the spiral) moves to a small radius — it becomes the innermost structural feature.
  • Primorial rings p# = 2·3·5·…·p, which in the standard layout are large-radius outliers, dominate the outer boundary under inversion. Their maximal lift survival T(p#)/φ(p#) = 1 (when p#+1 is prime) is visually prominent.
§15.2 — Cumulative Ring Rotation
Define a ring rotation parameter δ ∈ ℝ. Each residue r on ring M is displayed at angle
θ(M, r) = 2π · r/M + θ₀ + M · δ
where θ₀ is the global rotation. This adds a per-ring phase offset proportional to M.

The focal ring of the spiral under δ = 2π/N (where N = Mmax) is the ring M where the function r/M + M/N is minimized. Setting its derivative to zero:

d/dM (p/M + M/N) = 0 ⟹ Mfocal = √(p · N)

At this ring the spiral is most geometrically compressed — consecutive appearances of p are closest in angle. Two notable alignments:

  • When N = 4p: Mfocal = √(4p²) = 2p — the focal ring coincides with the equator gap of Theorem 8.1.
  • When N = p²: Mfocal = p — the focal ring is the prime itself, the first appearance of p in the ring system.

The standard setting δ = 0 corresponds to the original Farey-ordered visualization. The parameter r/m in the interface sets δ = 2π · (r/m) / N, so the outermost ring receives exactly r/m additional turns relative to ring M = 1.

Remark 15.3 — Collinearity condition
Two residues r₁/M₁ and r₂/M₂ are collinear from the center (same angle θ) under rotation parameter δ when:
r₁/M₁ − r₂/M₂ ≡ (M₂ − M₁) · δ/(2π) (mod 1)
This is a linear condition linking Farey neighbor fractions to the ring gap M₂ − M₁ and the rotation scale δ. It generalizes the standard collinearity condition (δ = 0: r₁/M₁ = r₂/M₂, Farey mediants) to a one-parameter family of alignment loci.