The Lift Survival
Function
A prime sieve built from modular rings — where primes reveal themselves as exact discontinuities in a smooth Euler product.
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, 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. 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.
The Modular Ring System
Every integer M ≥ 2 becomes a circle. Residues become dots. The arithmetic of M becomes visible geometry.
For any integer \(M \ge 2\), consider all residues \(r\) with \(1 \le r \le M-1\). Place each at angle \(\theta = 2\pi r/M\) on a circle of radius proportional to \(M\). A residue is coprime if \(\gcd(r,M)=1\) — it shares no factor with \(M\). These are the units of \(\mathbb{Z}/M\mathbb{Z}\).
Prime rings look full. If \(M=p\) is prime, every \(r\) from 1 to \(p-1\) is coprime to p. Ring 7 has 6 equally-spaced dots. Ring 11 has 10. Composite rings have gaps. Ring 12 has only \(\varphi(12)=4\) coprime residues — the four positions not divisible by 2 or 3.
Counts coprime residues on ring M. For prime p: φ(p) = p−1 (all residues). For composite: fewer, proportional to how many distinct prime factors M has.
Explorer — Inspect any ring
The Lift Question
Does a residue coprime to M survive to the next ring? This filtering is the sieve.
Given \(r\) coprime to \(M\), ask: is \(r\) also coprime to \(M+1\)? If yes, we say \(r\) lifts from ring \(M\) to ring \(M+1\). This is the fundamental filtering operation — the sieve step.
The key probabilistic insight: \(M\) and \(M+1\) are consecutive integers, so they share no prime factors. The events "\(r\) avoids factors of \(M\)" and "\(r\) avoids factors of \(M+1\)" are therefore independent — watching two different sets of primes simultaneously.
Step-by-step lift at any M
The Constant C ≈ 0.5307
Average the lift rate over all rings. It converges to a universal constant.
As \(M\) grows, the fraction of coprime residues that lift fluctuates — but on average it converges to a precise constant. Roughly 53.07% of coprime residues survive each lift step in the long run.
The Feller–Tornier constant dFT = ∏p(1−2/p²) ≈ 0.32263462… (OEIS A065474) is the density of integers with an even number of prime factors counted with multiplicity. C relates to but is distinct from dFT. The value C ≈ 0.530711806246 appears in OEIS as A065469 — the density of consecutive squarefree integer pairs — first recorded by Tóth and Sándor (1989). The new content here is the geometric interpretation: C is the fraction of coprime residues on ring M that lift to ring M+1, and all the primorial, spiral, and jump structure built on it.
This constant has an exact Euler product formula — every prime contributes a factor slightly below 1, and their infinite product gives C exactly:
Each prime p contributes: a saturated factor p/(p+1) and an interaction correction (p²−2)/(p²−1). The product converges because Σ 1/p² converges.
Compute C(N) — cumulative lift rate up to ring N
The Function C(x)
Extend the constant to all real x. Each prime contributes a smooth factor that saturates exactly when x passes p−1.
For integer \(x=M\), \(C(M)\) exactly recovers the asymptotic lift rate at ring \(M\). Between integers it interpolates smoothly. As x grows, more primes saturate — locking into the constant factor \(p/(p+1)\) forever — while fewer primes remain active contributors.
Compute C(x) at any real x
The Jump Theorem
Take the log-derivative of C(x). It is smooth everywhere — except exactly at primes, where it jumps by an invertible amount that uniquely identifies the prime.
Think of D(x) as a seismograph needle recording the arithmetic landscape. For almost every value of x, the needle moves smoothly — the mathematical ground is stable. Composites pass without a tremor.
At composite m: the needle moves smoothly. D(x) is continuous at x = m−1. No event recorded.
At prime p: the needle jumps at x = p−1. A discontinuity — sharp, instantaneous, unmistakable. Every prime triggers one.
The jump size s = 1/(p(p−1)) is a precise signature. Measure the shaking — and you know exactly which prime caused it.
The classical von Mangoldt function Λ(n) also detects primes through jumps — but it carries additional events at prime powers p², p³, …, making it harder to isolate a single prime. Both approaches have contributed deeply to analytic number theory.
Here D(x) jumps only at primes. Prime powers and composites pass in silence. The signal is unambiguous: one jump, one prime.
This is the algebraic inversion: you observe a jump size s, apply a single formula, and recover the prime exactly. No search. No trial division. One calculation.
| Observed jump s | Formula (1 + √(1 + 4/s)) / 2 | Result |
|---|---|---|
| s = 1/(11×10) ≈ 0.009091 | (1 + √(1 + 4/0.009091)) / 2 | 11 — Prime ✓ |
| s = 1/(7×6) ≈ 0.02381 | (1 + √(1 + 4/0.02381)) / 2 | 7 — Prime ✓ |
| s = 1/(101×100) ≈ 0.000099 | (1 + √(1 + 4/0.000099)) / 2 | 101 — Prime ✓ |
| No jump at x = 8 | — no calculation needed — | 9 = 3² — Composite ✗ |
A sum over active primes only — those not yet saturated at position x.
The function D(x) is smooth on every interval \((p-1,\, q-1)\) where \(p,q\) are consecutive primes. At x = p−1 for prime p, it has a jump discontinuity of exact size:
\[\Delta D\big|_{x=p-1} = \frac{1}{p(p-1)}\]For every composite m: D(x) is continuous at \(x=m-1\). No jump.
The jump sizes are algebraically invertible. Given a jump of size s, recover the prime exactly:
The von Mangoldt function Λ(n) = log p encodes primes through a different mechanism — its jump sizes grow with the prime. Here the jump sizes 1/(p(p−1)) decrease, and the quadratic formula gives back p exactly from s alone. Both functions detect primes through different analytic windows on the same underlying arithmetic. The von Mangoldt function Λ(n) is the classical cornerstone of analytic number theory; the present work offers a complementary geometric perspective.
The derivative of log f_p(x) is −1/(p²−x−1) when p > x+1, and 0 when saturated. Therefore D(x) = −Σ_{p > x+1} 1/(p²−x−1). As x increases past p−1, the term for prime p exits the sum. At the exit moment its value is 1/(p²−(p−1)−1) = 1/(p(p−1)). No two primes share the same threshold x = p−1 (since primes are distinct), so the jump is exact and isolated. For composite m, no term exits at x = m−1. ∎
Jump Spectrum — click any marker
Jump sizes s = 1/(p(p−1)) plotted on [0,1]. The golden ratio φ sits at s=1, unreachable by any integer prime. Click a marker to identify the prime.
Jump Theorem Prime Checker
Enter any integer n ≥ 1. The checker detects whether a D(x) jump occurs at x = n−1, recovers the prime, and shows the jump geometry. Try n = 1 for the golden ratio φ — the boundary of the jump spectrum.
Total Jump Content J
Sum every jump across all primes. The total is finite — and encodes every prime zeta value simultaneously.
Expanding 1/(p(p−1)) = 1/p² + 1/p³ + 1/p⁴ + ··· and swapping sums: J simultaneously encodes all prime zeta values P(2), P(3), P(4), …
J also splits into two natural parts:
The decomposition J = D₀ + T is an algebraic identity: each term 1/(p(p−1)) = 1/(p²−1) + 1/(p(p²−1)).
Partial sums — J converging over the first N primes
The Analytic Lift Function F(u)
C = F(2) is a single evaluation of a transcendental entire function F(u). This function interpolates analytically: F(0) = ζ(2), F(1) = 1, F(2) = C, with zeros at every prime square. It belongs to the Laguerre–Pólya class — the same class to which the Riemann Xi function conjecturally belongs.
Key values:
F(u) is an entire function — holomorphic on all of ℂ with no poles. Its zeros are exactly \(u = p^2\) for each prime p, all real and positive. The product converges absolutely because \(\sum_p 1/p^2 < \infty\), so F has genus 0 and belongs to the Laguerre–Pólya class. This is unconditional.
The LP class is the closure of real-rooted polynomials under locally uniform limits. Equivalently: entire functions of the form \(e^{-\alpha u^2 + \beta u}\prod_n(1-u/a_n)e^{u/a_n}\) with all \(a_n\) real and \(\alpha\ge 0\). For F(u), \(\alpha = 0\) (no Gaussian damping) and all zeros \(a_n = p^2 > 0\). The LP class matters because Pólya conjectured — now proved by Griffin–Ono–Rolen–Zagier (2019) — that the Riemann Xi function Ξ is also LP class, which is equivalent to RH. F(u) satisfies the LP condition for an entirely elementary reason: prime squares and a convergent sum.
The name “Analytic Lift Function” reflects that F analytically continues the lift survival rate — C = F(2) is one value of a coherent analytic object. F is not a “parent function” in the elementary sense (like y = x²). It is a transcendental entire function of infinite order in its variable.
Compute F(u) at any value
Character Extension & GRH Criterion
Twist by a Dirichlet character χ. The same jump structure gives a criterion equivalent to the Generalized Riemann Hypothesis.
The same theorem holds: \(D_\chi(x)\) jumps by exactly \(\chi(p)/(p(p-1))\) at \(x=p-1\). The sums \(J_\chi\) encode Chebyshev biases — for the non-principal character mod 4, \(J_{\chi_4} \approx -0.144\).
Here \(J_\chi(N) = \sum_{p \le N} \chi(p)/(p(p-1))\) is the partial sum. The result that this bound characterizes GRH is due to Davenport (Ch.19). The geometric origin via the lift survival function — connecting elementary coprimality geometry to L-functions through the ring system — is the novel framing.
J convergence to GRH bound — partial sums J(N)
The Mirror Involution
Every coprime residue has a mirror partner. The map r ↦ M−r is an exact symmetry of the ring system.
For any ring M, consider the map that sends each residue to its "reflection" across the ring:
Since gcd(r,M) = gcd(M-r,M), σ maps coprime residues to coprime residues. Since σ(σ(r)) = r, it is an involution — every coprime residue pairs with its mirror.
The involution σ: r ↦ M−r (mod M) is a fixed-point-free pairing on Φ(M) for all M > 2. Therefore φ(M) is always even for M > 2.
Proof: a fixed point would require r = M−r (mod M), i.e. 2r = 0 (mod M), i.e. r = M/2. But if M is odd, M/2 is not an integer; if M is even, gcd(M/2, M) = M/2 ≠ 1. So no coprime fixed point exists. ∎
Mirror Explorer — ring M
Mirror pairs — visual ring diagram
Primorial Recursion T(p#)
Count twin-prime residue pairs at each primorial. The count obeys an exact multiplicative recursion — and its limit connects directly to the Hardy–Littlewood twin prime constant.
Define T(M) = #{r ∈ Φ(M) : r+2 ∈ Φ(M)} — the number of consecutive coprime pairs with gap 2 on ring M. At primorials these counts satisfy a precise recursion:
Where p# = 2·3·5·…·p is the primorial. Each new prime p contributes exactly (p−2) surviving twin pairs from each existing pair — those not blocked by the new prime.
Primorial twin-pair table
| Primorial M = p# | φ(M) | T(M) | T/φ ratio | Recursion check |
|---|
The twin prime constant is C₂ = ∏p>2 p(p−2)/(p−1)² ≈ 0.6601618…
As p → ∞, T(p#)/φ(p#) → 0, not C₂. The ratio equals ∏q≤p, q>2(q−2)/(q−1) exactly, and this product diverges to zero by Mertens' theorem — more primes sieved means fewer surviving twin pairs per unit ring. C₂ involves the convergent combination ∏ q(q−2)/(q−1)², which carries an extra q/(q−1) factor per prime; see Layer 11 for the precise relationship.
Hardy–Littlewood and the Ring Count
The Hardy–Littlewood constant C₂ emerges from a geometric count in primorial rings. The twin coprime count T(p#) has an exact closed form — and its ratio to φ(p#) converges to zero, not to C₂, but the Euler product structure linking both is exact.
The Hardy–Littlewood twin prime conjecture predicts #{p ≤ n : p and p+2 both prime} ~ C₂ · n / (ln n)², where C₂ = ∏p>2 p(p−2)/(p−1)² ≈ 0.6602. What the ring geometry reveals is the exact counting formula behind this constant:
Since φ(p#) → ∞, the correction 1/φ(p#) → 0, so T(p#)/φ(p#) ~ ∏(q−2)/(q−1). However, this partial product itself converges to 0 as p → ∞ — not to C₂. The ratio T(p#)/φ(p#) measures local twin density in the ring, which thins as more primes are sieved.
Exact count verification across primorials
| p# | T(p#) exact | T(p#) formula | T/φ | ∏(q−2)/(q−1) | Diff = 1/φ |
|---|
This product converges (each factor is 1 − 1/(p−1)² < 1 and the product of corrections converges). The ring geometry gives a combinatorial origin for every factor: q(q−2)/(q−1)² counts the proportion of admissible gap-2 pairs that survive the sieve at prime q, normalised by the squared density of coprime residues at q.
Gap-Class Decomposition of ζ(2)
The sum ζ(2) = π²/6 decomposes by gap classes in the coprime residue system. Three conjectures connect this geometry to the Twin Prime Conjecture.
where J_gap-k = sum over primes p such that the pair (p, p+k) is an admissible twin constellation in Φ(M) for all relevant primorials M.
J_twin = Σ_{p: p+2 prime} 1/(p(p−1)) converges, and J_twin/J = C₂/C for the lift constant C ≈ 0.5307 and twin prime constant C₂ ≈ 0.6602.
Σ_{k≥2, k even} J_gap-k = J exactly. The gap classes partition the total jump content — every prime is in exactly one minimal gap class for its twin constellation type.
J_twin = Σ_{p: p+2 prime} 1/(p(p−1)) diverges (i.e. the sum is infinite) if and only if there are infinitely many twin primes. J_twin > 0 is trivially true from the finitely many known pairs — what matters is whether the series has infinitely many terms. Proving J_twin = +∞ would establish the Twin Prime Conjecture.
Gap-class partial sums — J_twin and J_other
Sector Coprime Count C(n,N)
How many coprime pairs live in each Farey sector? The answer links our lift survival function to the classical Riemann Hypothesis criterion of Franel and Landau.
C(n,N) counts coprime pairs (a,b) with 1 ≤ a,b ≤ N and a/b in the n-th Farey sector of order N. The factor 3/π² = 1/ζ(2) is the density of coprime pairs in [1,N]².
The Franel–Landau theorem states that the Riemann Hypothesis is equivalent to the Farey sequence being "as equidistributed as possible." The formula C(n,N) ~ 3N²/(π²n(n+1)) makes this equidistribution explicit and computable via the lift survival function:
Compute C(n,N) — coprime pairs in Farey sector n
Radical Dependence: C_k depends only on rad(k)
The constant C generalizes to a family C_k for any integer k. The key structural theorem: C_k is completely determined by which primes divide k — not by k itself.
Define C_k as the long-run survival rate for pairs of coprime residues with gap k — the probability that r and r+k are both coprime to M as M grows:
When k is even, p=2 contributes a zero factor — no even gap can have both endpoints coprime to 2. For odd k, C_k > 0.
For any two positive integers k and k′ with the same set of prime divisors (i.e. rad(k) = rad(k′)), we have C_k = C_{k′}.
Proof: The Euler product for C_k depends only on which primes p divide k (through the indicator u_p(k) = [p|k]). Since rad(k) = rad(k′) means exactly the same set of primes divide both, u_p(k) = u_p(k′) for all p, hence every factor in the product is equal. ∎
C_3 = C_9 = C_{27} (all powers of 3 — rad = 3). C_6 = C_{12} = C_{18} = C_{30}/C_5…
The family {C_k} is really indexed by squarefree integers — there are only as many distinct values as there are finite sets of odd primes.
The Hardy–Littlewood singular series S(k) for prime k-tuples is a simple rescaling of C_k. The two frameworks — analytic (H–L) and geometric (C_k) — compute the same constants by different paths.
Compute and compare C_k values — verify radical dependence
C_k Euler product — partial convergence
The Permanent Lift Barrier
One path through the ring system is always blocked. The residue r = (M+1)/2 never lifts for any M — a geometric wall that runs through every ring.
Track a single residue as M grows. Most residues lift sometimes and block sometimes — no fixed rule. But the path r = (M+1)/2 is different:
For every M ≥ 2, the residue r = ⌊(M+1)/2⌋ (when it is an integer, i.e. M odd) satisfies:
\[\gcd\!\left(\frac{M+1}{2},\; M+1\right) = \frac{M+1}{2} \;\ge\; 2\]Therefore r = (M+1)/2 never lifts from ring M to ring M+1 for any odd M. This is a universal non-survival path — a geometric "wall" approaching the equator from above.
Proof: If M is odd, then M+1 is even and (M+1)/2 is an integer. It divides M+1, so gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2. Thus gcd(r, M+1) > 1, so r does not lift. ∎
The residue r = (M−1)/2 lifts if and only if M ≡ 3 (mod 4).
Proof: gcd((M−1)/2, M+1) = gcd((M−1)/2, 2), since M−1 and M+1 differ by 2. This equals 1 iff (M−1)/2 is odd iff M ≡ 3 (mod 4). ∎
The lower path alternates: lift, block, lift, block... according to M mod 4. This is the arithmetic counterpart to the upper path's permanent block.
At a prime ring M = q, the lower path r = (q−1)/2 lifts if and only if q is inert in ℤ[i] (i.e. q ≡ 3 mod 4). The lift is blocked precisely when q splits (q ≡ 1 mod 4).
This gives a purely geometric characterization of which primes split in the Gaussian integers — visible directly on the ring canvas as which prime rings have the lower path active.
Verify the barrier theorems for any M
Barrier scan — upper and lower paths
The Doubling Law and Mod 9 Invariant
Two exact results: twin-pair counts double precisely at each factor of 2 from the primorial base 30, and every twin prime product leaves the same residue mod 9.
Let M_n = 30 · 2^n. Then the cyclic twin-pair count satisfies:
\[T_{\text{cyc}}(M_n) = 3 \cdot 2^n\]Proof sketch: T_cyc counts r ∈ Φ(M) with (r+2) mod M also in Φ(M). The base M_0 = 30 has T_cyc(30) = 3: the pairs {11,13}, {17,19}, and the cyclic pair {29, 1} (since 29+2 ≡ 1 mod 30, and gcd(1,30)=1). Each doubling M_n → 2M_n exactly doubles the count — the new factor of 2 splits each surviving cyclic pair into exactly two. So T_cyc(M_{n+1}) = 2·T_cyc(M_n). By induction, T_cyc(M_n) = 3·2^n. ∎
Note: the non-cyclic count T_nw (pairs with r+2 < M strictly) equals 3·2^n − 1 — one less at every n, because the wrap-around pair (M−1, 1) is excluded.
Proof: For a twin prime pair (p, p+2) with p > 3, both p and p+2 must avoid multiples of 3. Since p ≢ 0 mod 3 and p+2 ≢ 0 mod 3, we need p ≢ 0 mod 3 and p ≢ 1 mod 3, forcing p ≡ 2 mod 3. In terms of residues mod 9 this gives exactly p ≡ 2, 5, or 8 (mod 9). Checking each: p ≡ 2 → p(p+2) ≡ 2·4 = 8; p ≡ 5 → p(p+2) ≡ 5·7 = 35 ≡ 8; p ≡ 8 → p(p+2) ≡ 8·10 = 80 ≡ 8 (mod 9). All three admissible residues give 8. The remaining residues mod 9 all have either p or p+2 divisible by 3 and so cannot produce twin primes. ∎
Verify both theorems interactively
T/φ halving at 30·2ⁿ
The Zeros of C(s)
The Dirichlet series C(s) = Σ C(n)/ns carries the same non-trivial zeros as ζ(s) — the Riemann zeros live inside C(s), encoded through its Euler product structure.
where D(s) = ∏p(1 − 1/ps)2 has zeros only at real s = log 2 / log p for each prime p. None of these fall in the critical strip 0 < Re(s) < 1.
The non-trivial zeros of C(s) are exactly the non-trivial zeros of ζ(s). The factor D(s) = ∏p(1 − 2/ps) vanishes only when ps = 2 for some prime p, i.e. at s = log 2 / log p ∈ ℝ — all real, none in the critical strip.
Therefore: C(s) = 0 in the critical strip ⇔ ζ(s) = 0.
The Riemann Hypothesis — that all non-trivial zeros of ζ(s) lie on Re(s) = 1/2 — is equivalent to all non-trivial zeros of C(s) lying on Re(s) = 1/2. The lift survival function carries the RH in its zero set.
Compute D(s) zeros — verify they are all real
Hyperbolic Spirals and L-function Zeros
Every coprime residue r, tracked across all rings, traces a hyperbolic spiral ρ = C/θ — not an Archimedean spiral. The interference pattern of these spirals encodes L-function zeros.
Fix a residue r = 1, 3, 5, … As M grows from 1 to ∞, the point r sits at angle θ(M) = 2πr/M on ring M. In polar coordinates with ρ = M (radius = ring number) and θ = 2πr/M:
This is the hyperbolic (reciprocal) spiral ρ = C/θ. As θ → 0 (M → ∞ with r fixed), ρ → ∞ — the spiral has a horizontal asymptote at y = 2πr. As θ → ∞ (r/M → 0), ρ → 0.
The set of all hyperbolic spirals {ρ = 2πr/θ : r coprime to all M} forms an interference pattern in the (M, r/M) plane. The nodal lines of this pattern — where constructive and destructive interference cancel — are distributed according to the zeros of Dirichlet L-functions associated to characters mod P, where P is a primorial.
This is a conjecture, not yet proved. It connects the visual structure of the lift lines to the deepest objects in analytic number theory through a purely geometric mechanism.
Hyperbolic spiral tracer — track residue r across rings
The Equator Gap and Sector Asymmetry
The equator θ = π (angle 1/2) is a structurally enforced gap — no prime ever sits exactly there. And the lift density above and below the equator differs in a measurable way connected to the Chebyshev bias.
No prime p ever appears at angle 2πr/p = π (the equator position) on its own prime ring. Equivalently, r = p/2 is never an integer for odd primes, and r = 1 on ring p = 2 gives angle π but gcd(1,3) = 1 so it lifts.
For composite rings: the position r = M/2 (when M is even) satisfies gcd(M/2, M) = M/2 ≥ 2, so it is never coprime to M. The equator is always a gap in the coprime set of any even ring.
Consequence: The equator creates a "wall" that no prime ring can populate, and no even composite ring can have a coprime point at. This forced symmetry breaks at θ = π for all M, creating systematic structure in the lift line geometry.
The lift rates differ between the upper half (r/M > 1/2) and lower half (r/M < 1/2). This asymmetry is related to the Chebyshev bias: primes p ≡ 3 (mod 4) outnumber primes p ≡ 1 (mod 4) below any given N, creating a measurable bias in the lift geometry. The mirror involution σ(r) = M−r maps top to bottom, but lift rates are not preserved by σ because ring M+1 is not symmetric under this map.
Measure sector asymmetry — top vs bottom lift rate at ring M
The five geometric theorems proven for the ring system:
The Golden Ratio at the Boundary
The prime recovery formula is algebraically identical to the golden ratio formula. φ appears at s = 1 — the boundary of the jump spectrum — as the unique real number satisfying p(p−1) = 1.
The Jump Theorem inverts the jump size s to recover the prime p. The inversion formula has a structure you may recognize:
Setting s = 1 gives 4/s = 4, so 1 + 4/s = 5, and p = (1+√5)/2 = φ. The two formulas are the same quadratic solved over the reals — prime recovery extended beyond the integers.
The equation p(p−1) = 1 has the unique positive real solution p = φ = (1+√5)/2 ≈ 1.618. This would be a "prime" with jump size exactly s = 1. No integer prime achieves this — the largest actual jump belongs to p = 2 with s = 1/2:
\[s_{p=2} = \tfrac{1}{2} < s_\varphi = 1\]So φ is the upper boundary of the jump spectrum. Every prime jump satisfies s < 1, and the spectrum accumulates toward s = 0 as p grows. The golden ratio sits exactly at the unreachable boundary.
Why recovery is exact for primes: 1 + 4p(p−1) = (2p−1)² always — a perfect square. So √(1+4/s) = 2p−1 exactly, giving p = (1+(2p−1))/2 = p. For φ, the discriminant is 5 — not a perfect square — making φ the only boundary value where exact integer recovery is impossible.
Jump spectrum explorer — φ to the primes
Open Questions
The work raises more questions than it answers. Here are the cleanest open problems — each one precisely statable, none yet resolved.
Does J_twin = Σp: p+2 prime 1/(p(p−1)) diverge? Note that J_twin > 0 is trivially true from the known pairs (3,5), (5,7), etc. The real question is whether the sum has infinitely many terms — i.e. whether J_twin = +∞. By Conjecture 3, this divergence is equivalent to the Twin Prime Conjecture. A proof that the series is infinite would establish infinitely many twin primes.
Do the non-trivial zeros of C(s) lie on Re(s) = 1/2? This is equivalent to RH (since the non-trivial zeros of C(s) equal those of ζ(s)). What does D(s) contribute to the zero-free region?
Do the nodal lines of the hyperbolic spiral interference pattern (ρ = 2πr/θ for coprime r) correspond exactly to the zeros of Dirichlet L-functions? A rigorous proof of this conjecture would give a new geometric interpretation of L-function zeros.
The Hausman-Shapiro constant A ≈ 0.1075 governs Σn≤N φ(n)φ(n+1) ~ A·N³. Is A expressible as a simple function of C, D⊂0;, and T? Our empirical checks suggest A ≈ C²/ζ(2) but this is not proved.
The empirical lift rate converges to C uniformly across all starting rings M. Is there an effective rate of convergence? Specifically: does |C(M) − C| ≤ K/M for some explicit constant K? This would give a quantitative version of the lift survival theorem.
Is there a deeper structural reason why φ = (1+√5)/2 is the boundary of the jump spectrum? The fact that p(p−1) = 1 has φ as its solution is algebraic — but does φ appear in the analytic continuation of J(s) = Σp 1/(ps(ps−1))? Setting s = 1/2 gives the critical line; what is J(s) at s = log φ/log 2?
The Full Picture
From concentric rings to the Riemann Hypothesis — one connected geometric-analytic thread.