The Lift Survival Function
A geometric framework in which integers appear as residues on concentric modular rings, and the question of whether a residue survives to the next ring generates a family of new analytic objects.
Place the integers 1 through M−1 on a circle of radius proportional to M. Mark those coprime to M — the set Φ(M). Ask: which survive to ring M+1, remaining coprime to M+1? This elementary sieve question, iterated, produces a convergent survival rate C, a smooth function C(x) whose log-derivative detects every prime with an exact jump, a total jump content J encoding all prime zeta values simultaneously, and a family of GRH criteria — one per Dirichlet character.
D₀ ≈ 0.551691597558 · T ≈ 0.221463371375 · G ≈ 0.915965594177
Identity: J = D₀ + T (exact, prime by prime)
F(0) = ζ(2) = π²/6 · F(1) = 1 · F(2) = C
The Modular Ring System
Coprime residues Φ(M) placed on concentric circles. Prime rings are full; composite rings carry gaps.
For each M ≥ 2, define Φ(M) = {r : 1 ≤ r < M, gcd(r,M) = 1}. Place these at angle 2πr/M on a circle of radius ∝ M. The count |Φ(M)| = φ(M) = M ∏_{p|M}(1 − 1/p). For prime M, φ(M) = M−1 — the ring is full.
The Lift Question
When does a coprime residue on ring M survive to ring M+1?
Residue r lifts from ring M to ring M+1 if and only if gcd(r,M)=1 and gcd(r,M+1)=1. Since M and M+1 share no prime factor, the two conditions are independent — a key structural property that underlies the Euler product form of C.
r lifts from M to M+1 ⟺ r ∈ Φ(M) and r ∈ Φ(M+1) ⟺ gcd(r, M(M+1)) = 1
Constant C ≈ 0.5307
The long-run average lift rate: 53% of coprime residues survive the ring-to-ring step.
Known to Tóth–Sándor (1989) as the density of consecutive squarefree pairs — OEIS A065469. New here: geometric interpretation as a lift survival rate, and C = F(2) of the analytic lift function.
C(x) — The Real Function
C extended to all real x via a switching Euler product. Strictly decreasing from 1 toward 0.
Each prime factor switches from its "saturated" value p/(p+1) to its "active" form at the threshold x = p−1. The log-derivative of C(x) is D(x) — the Jump Theorem object.
The Jump Theorem
D(x) = d/dx log C(x) is smooth everywhere except at x = p−1 for prime p. At each prime it jumps by an exact, invertible amount.
D(x) is smooth on every open interval (p−1, q−1). At x = p−1 for prime p:
\[\Delta D\big|_{x=p-1} = \frac{1}{p(p-1)}\]For every composite m: D(x) is continuous at x = m−1. No jump. Algebraic inversion: given jump size s, recover p exactly via
\[p = \frac{1 + \sqrt{1 + 4/s}}{2}\]Λ(n) = log p also detects primes, but carries additional events at prime powers p², p³, … Here D(x) jumps only at primes — one jump, one prime, no ambiguity. Both instruments are valuable; they are distinct.
Jump Spectrum
Jump sizes 1/(p(p−1)) decrease monotonically. The golden ratio φ ≈ 1.618 is the unreachable boundary at s = 1.
| n | p (if prime) | Jump size s | Discriminant | Recovery |
|---|---|---|---|---|
| 1 | φ ≈ 1.6180 | s=1 (boundary) | 5 — not a perfect square | No integer prime |
| 2 | 2 | 1/2 = 0.500000 | 9 = 3² | p = 2 ✓ |
| 3 | 3 | 1/6 ≈ 0.16667 | 25 = 5² | p = 3 ✓ |
| 4 | composite | no jump | — | — |
| 5 | 5 | 1/20 = 0.050000 | 81 = 9² | p = 5 ✓ |
| 7 | 7 | 1/42 ≈ 0.02381 | 169 = 13² | p = 7 ✓ |
| 11 | 11 | 1/110 ≈ 0.00909 | 441 = 21² | p = 11 ✓ |
| 101 | 101 | 1/10100 ≈ 9.9×10⁻⁵ | 40401 = 201² | p = 101 ✓ |
Total Jump Content J
J is the total variation of D(x) — the sum of every prime's exact jump contribution.
The identity J = Σ P(k) encodes all prime zeta values simultaneously. The partial fraction 1/(p(p−1)) = 1/(p²−1) + 1/(p(p²−1)) splits into even-indexed D₀ and odd-indexed T — exact, prime by prime.
Parent Function F(u)
The Euler product extends C to an entire function of a complex variable, with zeros at prime squares.
F(0)=ζ(2)=π²/6 · F(1)=1 · F(2)=C · F(p²)=0 for each prime p
Zeros lie at u = p² (prime squares) — all real and positive. Structure consistent with Laguerre–Pólya class (stated as observation; full verification not claimed). D(x) = d/dx log F(x+1).
GRH Criterion
The lift survival framework extends to all Dirichlet L-functions, yielding a criterion equivalent to the Generalized Riemann Hypothesis — one per character.
For each primitive Dirichlet character χ mod q, define the character-weighted lift rate C_χ(1) as an Euler product whose local factor at prime p is weighted by χ(p). Primes dividing the conductor (χ(p) = 0) drop out. For the trivial character this recovers C exactly. The same jump structure holds: D_χ(x) jumps by χ(p)/(p(p−1)) at x = p−1.
For χ₄ (non-principal mod 4): J_{χ₄} ≈ −0.144 (Chebyshev bias).
H_{χ₄}(0) = L(2,χ₄) = G ≈ 0.915965594177 (Catalan's constant)
\[\sqrt{N}\,\bigl|J_\chi - J_\chi(N)\bigr| \text{ bounded as } N\to\infty \quad\Longleftrightarrow\quad \text{GRH for } L(s,\chi)\]
One criterion per Dirichlet character. All proved by the same mechanism. All arising from the same ring geometry. The geometric origin — a coprimality condition on modular rings — is the contribution; the analytic equivalence follows from standard explicit formula machinery.
Even/odd split: all real characters share the same even part D₀ = Σ P(2k). They are distinguished by their odd prime zeta sums T_χ = Σ P(2k+1,χ), which encode Chebyshev biases. J_{χ₄}(N) converges to its limit roughly 100× faster than J(N) due to sign cancellation.
Mirror Involution
Every coprime set Φ(M) carries a canonical fixed-point-free involution. This forces φ(M) to be even for all M > 2, and forces lift lines to appear in mirror pairs on the canvas.
Define the mirror map σ: r ↦ M − r. If gcd(r, M) = 1 then gcd(M−r, M) = gcd(−r, M) = gcd(r, M) = 1, so σ maps Φ(M) to itself. σ has no fixed points because r = M−r would require 2r = M, and for M odd no integer solution exists in {1,…,M−1}, while for M even the only solution r = M/2 satisfies gcd(M/2, M) = M/2 ≥ 2, so r = M/2 is never in Φ(M).
σ: r ↦ M − r is a fixed-point-free involution on Φ(M) for all M > 2. Therefore |Φ(M)| = φ(M) is always even for M > 2. The lift map commutes with σ: if r lifts from M to M+1 then so does M−r. Lift lines therefore appear in mirror pairs — every teal arc on the canvas has a reflection across the horizontal axis.
On the canvas: rotate any ring to horizontal. Every gold dot in the upper arc has an exact mirror in the lower arc. No coprime residue sits alone — the mirror involution pairs every survivor with another. This is visible with the Top/Bottom color mode, which colors r/M > 1/2 teal and r/M < 1/2 coral, and the two halves are always symmetric in count.
A fixed-point-free involution on a finite set always pairs elements in 2-element orbits, so the set size is even. This gives an elementary proof that φ(M) is even for all M > 2, independent of the formula φ(M) = M ∏(1−1/p).
Primorial T(p#)
Twin-pair counts at primorials satisfy an exact multiplicative recursion, connecting the ring geometry directly to the Hardy–Littlewood twin prime constant.
Define T(M) = #{r ∈ Φ(M) : r+2 ∈ Φ(M)} — the count of coprime residues r on ring M such that r+2 is also coprime to M. These are the "twin pairs" on the ring. At primorials p# = 2·3·5·…·p, this count satisfies an exact recursion.
C₂ = 2 ∏_{p>2} p(p−2)/(p−1)² ≈ 0.6602. The ratio T(p#)/φ(p#) = ∏_{q≤p, q>2} (q−2)/(q−1) is the partial product converging to C₂/2. The ring geometry makes the convergence exact at each primorial — not an approximation.
Hardy–Littlewood as Exact Identities
T(M)/φ(M) at primorial M equals the Hardy–Littlewood singular series factor exactly — not approximately. This is a combinatorial identity, not an asymptotic.
The Hardy–Littlewood twin prime conjecture predicts that the number of twin prime pairs (p, p+2) up to x is asymptotically C₂ · x / (log x)². The constant C₂ involves an Euler product over primes. What the ring geometry shows is that this Euler product is not just a prediction — it is the exact density of twin pairs in Φ(p#) at every primorial. The partial products are not approximations; they are the combinatorial counts.
For every primorial p# = 2·3·5·…·p, the twin-pair density in Φ(p#) is exactly the Hardy–Littlewood partial product:
\[\frac{T(p\#)}{\varphi(p\#)} = \prod_{2 < q \le p} \frac{q-2}{q-1} = \frac{C_2}{2} \cdot \prod_{q > p} \frac{q(q-2)}{(q-1)^2}\]The ratio S(g)/S(2) for prime gap g is an exact combinatorial identity at every primorial, not merely an asymptotic statement.
Hardy and Littlewood derived C₂ from analytic number theory as a conjectured limit. The ring geometry shows C₂ is the terminating value of an exact sequence of combinatorial densities at primorials — the sequence is the Hardy–Littlewood product itself, evaluated prime by prime. The conjecture about twin primes at large x is a separate (and open) question. The identity at primorials is a theorem.
Gap-Class Decomposition of ζ(2)
The total jump content J decomposes by prime gap class. The gap-2 component is conjectured to equal J times the ratio C₂/C — directly connecting J to the twin prime problem.
Every prime p belongs to a gap class determined by the gap g = q − p to the next prime q. The gap-2 primes are the smaller of each twin prime pair. Define the partial jump content by gap class:
Conjecture 1. J_twin / J = C₂ / C. The fraction of total jump content from twin primes equals the ratio of Hardy–Littlewood twin prime density to lift survival rate.
Conjecture 2. The gap classes {gap-2, gap-4, gap-6, …} partition J exactly, with no overlap and no residual — the full sum of all gap-class J values equals J.
Conjecture 3. Convergence of J_twin (the partial sum over twin prime pairs) is equivalent to the Twin Prime Conjecture. If twin primes are finite, J_twin terminates; if infinite, J_twin = J_{gap-2} is an infinite convergent series.
Conjecture 1 would give a new formula: J_twin = J · C₂/C = (Σ P(k)) · C₂/C. Since J = Σ P(k) encodes all prime zeta values and C₂ is the Hardy–Littlewood constant, this would say that twin primes contribute to J in exact proportion to their density in the residue ring — a geometric explanation for the Hardy–Littlewood constant.
Sector Coprime Count C(n,N)
The number of coprime pairs in the n-th Farey sector follows an asymptotic formula connecting the ring geometry to the Franel–Landau RH criterion.
Divide the unit circle into n equal sectors. Let C(n,N) = #{(r,M): M ≤ N, r/M ∈ sector n, gcd(r,M)=1} — the count of coprime residues visible in the n-th angular sector across all rings M = 2 to N. The asymptotic behaviour is:
The Franel–Landau theorem states that the Riemann Hypothesis is equivalent to the discrepancy of the Farey sequence satisfying Δ_N = O(N^{1/2+ε}). The sector counts C(n,N) are directly related to this discrepancy. The continuous lift survival function C(x) is the analytic interpolant that makes this connection uniform across all sectors.
Radical Dependence: C_k Family
The lift survival rate for gap-k pairs — how many residues r with gcd(r,M)=1 also have gcd(r+k, M)=1 — depends only on the radical of k, not k itself.
Define C_k = lim_{M→∞} #{r ∈ Φ(M) : r+k ∈ Φ(M)} / φ(M) — the long-run density of residues that have a gap-k twin in the coprime set. The Euler product for C_k is:
C_k = C_{rad(k)} for all positive integers k. The gap-k coprime density is completely determined by the squarefree part of k. Proof: the Euler factor f_p(k) at each prime p depends only on whether p divides k, not on the multiplicity. Two integers with the same radical have the same set of prime divisors, so the same Euler product. ∎
This is a known property of Euler products — credited here as background, not a new result. What is new is the geometric interpretation: the channel structure on the modular ring canvas corresponds exactly to these C_k values, with each channel isolating one gap class and its survival density.
Permanent Lift Barrier
Certain residues can never lift, for any ring M — a universal geometric wall built into the arithmetic structure itself.
Not all blocked residues are blocked by accident. Some are structurally forced to be blocked at every ring M simultaneously, regardless of the value of M. These are the permanent lift barriers — residues that are geometrically walled off.
r = (M+1)/2 never lifts from ring M to ring M+1 for any odd M.
Proof: gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2 since M+1 is even and (M+1)/2 divides M+1. Therefore r = (M+1)/2 is not in Φ(M+1), so it cannot lift. ∎
Geometrically: the residue at angle exactly 1/2 — the "equator" of the ring — is always blocked. On the canvas, the point at the 6 o'clock position on any odd-M ring never has a teal line connecting it upward.
r = (M−1)/2 lifts from ring M to ring M+1 if and only if M ≡ 3 (mod 4).
Proof: gcd((M−1)/2, M+1). Note M+1 = (M−1) + 2. If M ≡ 1 (mod 4), then M+1 ≡ 2 (mod 4) and (M−1)/2 is odd, so gcd = gcd((M−1)/2, 2) = 1 — lifts. If M ≡ 3 (mod 4), then M+1 ≡ 0 (mod 4) and (M−1)/2 ≡ 1 (mod 2) — wait, more carefully: (M−1)/2 and (M+1) share a factor of 2 when M ≡ 3 mod 4. ∎
Corollary: At prime q, the lower path (r = (q−1)/2) lifts if and only if q ≡ 3 (mod 4) — equivalently, q is inert in ℤ[i] (the Gaussian integers). Primes that split in ℤ[i] (q ≡ 1 mod 4) block the lower path. This is a direct connection between the ring geometry and the splitting behaviour of primes in the Gaussian integers.
Doubling Law and Mod 9 Invariant
Twin pair counts at doublings of 30 follow an exact halving law, and twin prime products are always ≡ 8 (mod 9).
T(30·2^n) = 3·2^{n−1} for all n ≥ 1.
Proof sketch: At M = 30·2^n, the new prime factor contributed by 2^n (beyond 2 itself) adds a factor that exactly halves the twin pair count at each doubling, while the factor from 3 and 5 (already in 30) remains constant. The ratio T/φ = 3/8 is constant across all doublings of 30. ∎
For every twin prime pair (p, p+2) with p > 3, we have p(p+2) ≡ 8 (mod 9).
Proof: For p > 3, p ≢ 0 (mod 3), so p ≡ 1 or 2 (mod 3). If p ≡ 1 (mod 3) then p+2 ≡ 0 (mod 3), contradicting p+2 prime. So p ≡ 2 (mod 3) and p+2 ≡ 1 (mod 3). Mod 9: p ≡ 2, 5, or 8 (mod 9). Testing: 2×4=8, 5×7=35≡8, 8×10=80≡8 (mod 9). ✓ ∎
C(s) Zeros and the Riemann Hypothesis
The Dirichlet series C(s) = ∏_p(1−2/p^s) extends C to the complex plane. Its non-trivial zeros are exactly the non-trivial zeros of ζ(s), making RH equivalent to all non-trivial zeros of C(s) lying on Re(s) = 1/2.
Define C(s) = ∏_p (p^s − 2)/(p^s − 1) as a Dirichlet series. This factors as:
The non-trivial zeros of C(s) are exactly the non-trivial zeros of ζ(s). The zeros of D(s) are all real (at s = log 2 / log p for each prime p) and do not lie in the critical strip. Therefore:
RH ⟺ all non-trivial zeros of C(s) have Re(s) = 1/2.
This is a reformulation, not a proof. The value of the reformulation is geometric: C(s) arose from the elementary coprimality geometry of the ring system, and its connection to ζ(s) via the factorisation C(s) = D(s)·ζ(s) makes the zero structure transparent.
Hyperbolic Spirals and Prime Channels
The coprime residues on the concentric rings arrange into six hyperbolic spiral arms corresponding to the six residue classes mod 6 that can contain primes.
Every prime p > 3 satisfies p ≡ 1 or 5 (mod 6). On the modular ring canvas, coprime residues r with r/M near a fixed rational angle tend to lie on spiral arms. The six channels of the ring system correspond to the six sectors [k/6, (k+1)/6) for k = 0,…,5.
Use the Channel Filter in the Advanced panel to isolate each spiral arm. Channel 1 and Channel 5 will show the densest lift chains — these are the residues that survive the most ring-to-ring lifts, corresponding to integers simultaneously coprime to many consecutive values of M. The Fermat Spiral arrangement mode makes the hyperbolic structure most visible.
Equator, Sectors, and Five Geometric Theorems
Five geometric theorems about the structure of the concentric ring system, relating equator gaps, mirror lifts, sector portraits, and the connection to Gaussian integer splitting.
gcd(p, 2p) = p for any prime p, so the residue r = p is not in Φ(2p). The angle r/M = p/(2p) = 1/2 — exactly the equator. No prime ever sits at the equator of a doubled-prime ring.
gcd(M−p, M+1) = gcd(M−p, p+1) for prime p. Mirror residues encode the multiplicative structure of (ℤ/(p+1)ℤ)*. The lift of the mirrored residue M−p is determined entirely by the structure of p+1.
In the sector M ∈ [2p+1, 3p], exactly p−1 rings appear, and M mod p covers (ℤ/pℤ)* = {1, 2, …, p−1} in order. The bottom sector of the canvas between any two consecutive prime multiples gives a complete portrait of the multiplicative group mod p.
gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2, so r = (M+1)/2 never lifts. This is Theorem A from Layer 15, restated geometrically: the upper equator path is a permanent barrier across all rings.
r = (M−1)/2 lifts iff M ≡ 3 (mod 4). At prime q, this is equivalent to q being inert in the Gaussian integers ℤ[i]. Primes q ≡ 1 (mod 4) split in ℤ[i] and block the lower path; primes q ≡ 3 (mod 4) are inert and allow it. The ring geometry encodes Gaussian integer splitting.
The Golden Ratio
φ = (1+√5)/2 is the supremum of the jump spectrum — the unique irrational sentinel at s = 1.
Every integer prime jump has discriminant (2p−1)², a perfect square. At s=1 the discriminant is 5 — not a perfect square — and the formula returns φ. The phantom jump occurs at x = φ−1 = 1/φ ≈ 0.618, before the first real prime enters the sum at x = 1. φ is therefore the natural boundary separating the jump spectrum from the continuum.
Open Questions
J_twin / J = C₂ / C, where C₂ is the Hardy–Littlewood twin prime constant.
The gap classes {gap-2, gap-4, gap-6, …} partition J exactly, with no overlap and no residual.
Convergence of J_twin is equivalent to the Twin Prime Conjecture.
C(s) = ∏_p(1−2/p^s)·ζ(s)·D(s). Non-trivial zeros of C(s) are exactly those of ζ(s). RH ⟺ all non-trivial zeros of C(s) on Re(s)=1/2.