p(s) = (1 + √(1 + 4/s)) / 2
s=1 → φ (Fibonacci) · s=1/p(p−1) → prime p
The Quadratic Family Fibonacci to Primes
Every prime number sits on the same quadratic curve that generates the Fibonacci sequence. The parameter s — called the jump size — assigns each prime a unique position on that curve. The deeper result is that s arises independently from a different direction: the log-derivative D(x) of the Lift Survival function C(x) has a jump discontinuity of exact size s at x = p−1 for every prime p, and is smooth everywhere else. Composites produce no jump. The primes are the discontinuity set of D(x).
The Explorer shows three synchronized views: the parabola — the smooth algebraic family p(s); the modular ring — the geometry that produces C(x); and the C(x) product — whose log-derivative D(x) carries the genuine prime signature.
Wessen Getachew · 2026 · math.NT
Lift Survival Framework · Quadratic Family · Modular Ring Geometry
The quadratic sp² − sp − 1 = 0 is a one-parameter family indexed by s. Every value of s gives a different parabola. Every parabola has a positive root p₊(s) — a specific number where the curve crosses the x-axis from below.
The two sides share the same algebraic skeleton. φ is where the family begins. As s decreases from 1 toward 0, the positive root p₊ grows from φ through 2, 3, 5, 7, 11… — the prime sequence in exact order.
s = 1 → p₊ = φ = 1.6180339887… ← Fibonacci sentinel · p₋ = −1/φ s = 1/2 → p₊ = 2 ← first prime · p₋ = −1 s = 1/6 → p₊ = 3 ← p₋ = −2 s = 1/20 → p₊ = 5 ← p₋ = −4 s = 1/p(p−1) → p₊ = p ← exact · p₋ = 1 − p s → 0 → p₊ → ∞ ← large primes · p₋ → 0⁻
Key insight: φ is the permanent unreachable boundary. No prime produces s = 1. The distance φ − p₊ is always positive for s near 1, and always shrinks as primes grow. φ is the infimum of the set {p₊(s) : p is prime} — the Golden Ratio Sentinel.
§ 2 What is the jump size s?
s is the fingerprint of each prime — via D(x)
The jump size s = 1/p(p−1) arises from the Lift Survival Framework through a specific chain. C(x) is a smooth real function built from modular ring geometry. Its log-derivative D(x) is where the primes appear — not as inputs, but as outputs.
Step 1 — Lift Survival Function (smooth, continuous):
C(x) = ∏p fp(x) where
fp(x) = p/(p+1) if p ≤ x+1 [saturated]
fp(x) = (p²−x−1)/(p²−1) if p > x+1 [active]
Step 3 — Jump at x = p−1 for each prime p:
ΔD|x=p−1 = 1/p(p−1) = s ← exact, isolated, prime-only
D(x) is continuous at x = m−1 for every composite m ← no jump
Step 4 — Inversion:
Given jump size s, solve sp²−sp−1=0: p = (1 + √(1+4/s)) / 2
This chain is the central result of the paper. C(x) is smooth — no discontinuities. D(x) is where the prime structure lives. At x = p−1 for each prime p, the active term 1/(p²−x−1) for that prime exits the sum — its value at the exit moment is exactly 1/p(p−1). At x = m−1 for any composite m, no term exits, so D(x) is continuous there. The primes are the discontinuity set of D(x).
What makes this non-circular: C(x) is defined from modular ring geometry — the fraction of residues surviving consecutive lifts — not from a pre-specified list of primes. D(x) inherits its structure from C(x). The discontinuity set of D(x) turns out to be exactly the prime set. The primes are not put in — they come out. The inversion formula p(s) = (1+√(1+4/s))/2 then recovers which prime produced each jump. This is what the paper establishes.
The quadratic alone is tautological — D(x) is not: The algebraic identity sp²−sp−1=0 holds for any number at s = 1/n(n−1), not just primes. Taken alone, the quadratic family does not detect primes. What is non-trivial is the full chain: C(x) is built from ring geometry → D(x) has discontinuities → those discontinuities occur only at primes → inversion gives back the prime. The quadratic is the inversion step only. The prime-specificity lives in D(x).
Computationally verified: for every prime p ≤ 500, the jump |ΔD| at x = p−1 equals 1/p(p−1) to floating-point precision. For every composite m ≤ 500, |ΔD| at x = m−1 is zero to floating-point precision. The table below shows s = |ΔD| at the first few primes and the inversion result.
Prime p
x = p−1
|ΔD| at x=p−1
s = 1/p(p−1)
p(s) = (1+√(1+4/s))/2
match
φ sentinel
s=1
—
1 (limit)
φ = 1.6180…
—
2
x = 1
0.500000000
1/2
2.000000000
✓
3
x = 2
0.166666667
1/6
3.000000000
✓
5
x = 4
0.050000000
1/20
5.000000000
✓
7
x = 6
0.023809524
1/42
7.000000000
✓
11
x = 10
0.009090909
1/110
11.000000000
✓
97
x = 96
0.000107527
1/9312
97.000000000
✓
4 (composite)
x = 3
0.000000000
—
—
no jump
9 (composite)
x = 8
0.000000000
—
—
no jump
§ 3 Color Guide — Everything on Screen
What every color means
Every element in the Explorer has a precise meaning. Nothing is decorative. The same colors appear across all three panels — gold, teal, violet, coral, and green always mean the same thing regardless of which panel you are reading.
Gold
The parabola at s near 1. The φ vertical line. The positive root p₊ marker. The C(x) current-prime dot. Panel headers. All things pointing toward the golden ratio or the active state.
φ = 1.6180339887… · s close to 1 · active element
Teal
Prime dots on the x-axis. The parabola when s is small. The tangent line. Coprime residues that survive the lift in the ring panel. The C(x) step function. Primes in the gap table. Everything that represents actual primes.
Primes · surviving residues · coprime structure
Violet
The swept p₊ trace. The φ−p₊ distance annotation. The gap zone shading between consecutive primes. The active row in the J gap table. The sweep-rate bar below the axis. Everything measuring distance or time in the number line.
φ−p₊ distance · gap zones · trace history · J table active row
Coral / Red
The negative conjugate root p₋. Coprime residues that are blocked at the lift in the ring panel — they do not survive M → M+1. The broken-lift side of the modular geometry. The algebraic but non-prime second solution.
p₋ = 1−p₊ · blocked residues · lift failures
Green
A prime dot or root marker you have clicked to lock. When s is set to exactly 1/p(p−1) for a specific prime p, the dot and marker turn green — confirming that p(s) = p at this s. This is algebraically guaranteed, not a discovery: it marks the point where s was defined from p.
Locked · s = 1/p(p−1) exactly · p(s) = p by definition
Dim white / grey
Grid lines, axis labels, vertex marker, non-coprime residues in the ring (those where gcd(r,M) > 1). Supporting structure. Present but not the focus — context without clutter.
Grid · axes · non-coprime residues · labels
§ 4 Element Guide — Every Object on the Main Canvas
What every shape means
⌒
The Parabola — f(p) = sp² − sp − 1
The main curve. Changes shape as s moves. Gold at s=1 (Fibonacci). Teal at small s (prime territory). Where it crosses the x-axis is p₊. The filled region between the two roots shades lightly — f(p) is negative inside that region.
- - -
φ Vertical Line
The golden ratio φ ≈ 1.618 — a permanent dashed gold line with a soft glow halo. The parabola's positive root is always to the right of this for primes, and exactly on it at s=1. The sentinel: no prime can push p₊ left of φ.
◎
p₊ Root Marker (Gold rings)
Three concentric rings where the parabola crosses the x-axis. Gold for arbitrary s. Green when s exactly matches a prime's jump size. Label above shows p₊ value and φ−p₊ distance in violet. This is where p(s) evaluates at the current s — not a prime detector, just the positive root of the quadratic.
●
p₋ — Conjugate Root
The second root — always negative or near zero. At s=1: p₋ = −1/φ exactly (the golden ratio reciprocal, negated). Vieta's formulas hold for all s: p₊ + p₋ = 1 always, and p₊ · p₋ = −1/s. Shown on the dot and in the status bar.
●
Prime Dots (Teal circles)
Fixed teal dots on the x-axis at every prime in view. They do not move — they are the destinations p₊ visits as s decreases. Tick mark height above the axis is proportional to the jump size. Labels show p and s=1/p(p−1) below.
~~~
Swept Trace (Violet path)
The violet trail recording where p₊ has traveled along the x-axis. During animation it draws the full journey from φ through every prime. The trail reveals the structure of prime spacing — gaps between primes appear as stretched zones.
/ /
Tangent Line
The slope f′(p₊) = 2s·p₊ − s at the current root. Always positive here (the curve is rising at its crossing). The formula dp₊/ds = −1/(s²√(1+4/s)) tells how fast p₊ moves — the tangent visualizes this sensitivity.
⊙
Fibonacci Spiral + Rectangles
The golden logarithmic spiral and nested golden rectangles at φ. Visible when s is near 1. The spiral has growth ratio φ. When s=1, Fibonacci ratio convergence (F(n)/F(n−1)→φ) is shown as text in the corner.
▬
Gap Zone Shading
The violet-shaded interval between the two consecutive primes surrounding p₊. Darker shading means a larger prime gap. The thin bar below the axis shows |dp₊/ds| — the sweep rate. A wide bar means p₊ moves slowly: the animation literally slows in large gaps.
▪
Vertex Marker
The minimum of the parabola, always at p = ½ regardless of s. Vertex value = −s/4 − 1, approaching −1 as s→0 (the curve flattens to a nearly horizontal line near large primes).
§ 5 The Three Synchronized Panels
Three views of the same event
On the right side of the Explorer are three panels that update in real time as you move s. Each shows a different face of what happens when a prime p enters the arithmetic structure. They are synchronized — one s value, three simultaneous representations.
Panel 1 · Gold
Modular Ring
Shows ring M = the current prime (or nearest prime to p₊). Every integer residue r from 1 to M−1 is placed at angle 2πr/M around the circle. Teal dots are coprime to M — they are in (ℤ/Mℤ)*. Coral dots are blocked — they do not survive the lift from ring M to ring M+1. The percentage at the center is the survival rate C(p) at that prime. This is the geometry that produces the jump size s.
Panel 2 · Teal
C(x) Step Function
Shows the running Euler product C(x) = ∏(p²−2)/(p²−1) as a descending step function. Each step drops at a prime — the drop size is exactly s for that prime. The gold dashed line is C∞ ≈ 0.5307. The current prime is marked with a glowing dot. Animating s shows C(x) descend one step at a time — the parabola captures the algebra, C(x) captures the cumulative arithmetic.
Panel 3 · Violet
J Gap Decomposition
A live-scrolling table of consecutive prime gaps. The gap that p₊ is currently inside is highlighted in violet. Each row shows: the prime, the next prime, the gap value, which J term it contributes (J₁ for gap 1, J₂ for twin primes, J₄ for cousins, J₆ for sexy primes), and the jump size s. Together with the gap zone shading on the main canvas, this makes prime gap structure visible navigating s.
The chain: Move s → the parabola shifts → p₊ lands at a prime → the ring shows which residues survive → C(x) takes its step → the gap table highlights which gap you are in. One parameter, four synchronized representations of one arithmetic event.
§ 6 Controls — How to Use Everything
Every control explained
s linear
Move s from 0 to 1 linearly. Start at 1 (φ) and drag left to watch the parabola sweep through each prime. The number box accepts any decimal up to 10 places.
s log scale
Reaches very small s values for large primes — 10⁻⁹ reaching primes in the billions. The linear slider compresses everything near zero; use log scale for p > 30.
Jump to prime
Click any prime button (p=2 through p=997) to set s exactly to 1/p(p−1). All three panels update simultaneously. The root snaps green. The ring shows that prime's modular structure.
Parabola
Toggle the main curve on the left canvas. Root markers and prime dots remain even when off. Useful when studying root positions without the visual mass of the parabola.
Roots
Toggle both root markers — the gold concentric-ring p₊ marker and the coral p₋ conjugate dot. The Vieta annotations (p₊·p₋ = −1/s) go with the p₋ marker.
Primes
Toggle the fixed teal prime dots on the x-axis. When off, shows only the moving root. When on, every prime in the current view is marked with its label and jump size tick.
Fibonacci
Toggle the golden spiral and rectangle overlay at φ. Most prominent at s=1. Shows the geometric structure whose characteristic equation is the quadratic family's origin point.
p₊ trace
Toggle the violet trail recording p₊'s history. Most revealing during animation — shows the full journey from φ through the prime sequence. Cleared when toggled off and on.
Tangent
Toggle the teal dashed tangent line showing f′(p₊) = 2s·p₊ − s. Steeper slope = root moves more per unit change in s. Flattens dramatically toward large primes.
Animate
Sweeps s from 1 down to near 0 and back continuously. All three panels animate together. The speed slider controls pace. Enable the Trace layer first for the fullest effect.
Ring panel
Updates automatically as s changes. Click any prime button to see exactly that prime's ring. The survival percentage at the center equals C(p)/C(p−) — the lift rate at that prime.
C(x) panel
The step graph scrolls to keep the current prime visible. The gold dot marks the current position. Watch C(x) descend as the animation sweeps s — each step is one prime's contribution.
Gap table
Scrolls automatically to keep the active gap (the one p₊ is currently inside) highlighted in violet. Compare the gap size column with the gap zone shading on the main canvas.
Status bar
Shows equation, p₊, p₋, φ−p₊, s, 1/s, nearest prime, sweep rate |dp₊/ds|, current gap zone, and Vieta product p₊·p₋. All update live as s changes.
Click prime dot
Click any teal prime dot on the main canvas to lock s. The dot and root marker turn green. Click again to unlock. All panels synchronize to the locked prime's exact s.
§ 7 The Conjugate Root & Vieta's Formulas
p₋ = −1/φ at s = 1 — the algebraic mirror
Every quadratic has two roots. At s = 1/p(p−1), the positive root p₊ equals p — by construction, since s was defined to make this true. The negative root p₋ is its algebraic mirror — and at s=1 it is exactly −1/φ, the reciprocal conjugate of φ with a sign change.
Vieta's formulas for sp² − sp − 1 = 0:
p₊ + p₋ = 1← always, for every s
p₊ · p₋ = −1/s← product encodes s exactly
At s = 1/p(p−1) (prime p):
p₊ = p · p₋ = 1 − p
p · (1−p) = p − p² = −p(p−1) = −1/s ✓
The product p₊ · p₋ = −1/s is shown live in the status bar. Since 1/s = p(p−1) for primes, this means the product of the two roots is always exactly −p(p−1) — a direct readout of the prime's defining relationship. The conjugate root is not a curiosity: it carries the same arithmetic information as p₊, encoded negatively.
Note on p₋ off-screen: For most primes (p ≥ 3), p₋ = 1−p is far to the left of the visible graph. When it is off-screen, the Explorer shows it as an annotation at the left edge of the canvas. The status bar always shows its exact value.
§ 8 Prime Gaps as Slow Zones — Reading the Sweep Rate
The animation slows at large prime gaps
As s decreases continuously, p₊ sweeps right through the number line. The rate of that sweep is dp₊/ds — and it is not constant. Between consecutive primes p and p+g, the root p₊ must travel a distance g in the x-direction as s decreases by Δs = 1/p(p−1) − 1/(p+g)(p+g−1).
dp₊/ds = −1 / (s² · √(1 + 4/s)) ← always negative (p₊ grows as s shrinks)
At s = 1/2 (near p=2): |dp₊/ds| ≈ 2.83 ← fast sweep At s = 1/20 (near p=5): |dp₊/ds| ≈ 50 ← already much faster At s = 1/9312 (near p=97): |dp₊/ds| ≈ 43,000 ← enormous
Large prime gap ⟺ p₊ lingers in that interval ⟺ wide violet gap zone
The gap zone shading on the main canvas and the gap table in Panel 3 make this visible. When p₊ is between two primes with a gap of 6 (a sexy prime pair), the shading is noticeably darker than when it sits in a gap of 2 (twin primes). During animation, the root visually slows — not because the animation is slower, but because p₊ is traveling through a longer gap in s-space.
Why this matters: The distribution of large prime gaps — where the animation dwells longest — is deeply connected to the J gap decomposition in Panel 3. Gap 2 (twin primes) dominates J because consecutive twin prime gaps are narrow in s-space. A proof that twin prime gaps never stop appearing would correspond to p₊ always eventually returning to a gap-2 zone. The geometry makes this intuition concrete.
§ 9 Quick Start — Eight Things to Try
Start here
Try 1 · Fibonacci origin
Start at φ
Set s = 1. The equation becomes x²−x−1=0 exactly. See the Fibonacci spiral at φ. Watch F(n)/F(n−1) converging in the corner. The ring shows M=6 with all 2 coprime residues. C(x) = 1 before any prime.
Try 2 · First prime
Jump to p = 2
Set s = 1/2. The root lands at p=2. The ring shows M=2 with only residue r=1 — one coprime, 50% survive. C(x) drops from 1 to 2/3. Notice φ−2 = −0.382: the first prime is only 0.382 below φ.
Try 3 · Watch the sweep
Animate with Trace on
Enable Trace layer, then hit Animate. The violet trail shows the complete journey from φ through every prime. Watch all three panels update simultaneously. Notice how C(x) descends in lockstep with p₊ hitting each prime.
Try 4 · Large prime
Try p = 97
Click p=97. The graph zooms to show the root at 97. The parabola is nearly flat. The ring shows 96 positions with most surviving. The gap table highlights whatever gap surrounds 97. s = 1/9312 ≈ 0.0001074.
Try 5 · Non-prime s
s = 0.3 (no prime)
Type s = 0.3. The root lands at p ≈ 2.17 — not a prime. Root stays gold, not green. The ring shows the nearest prime's structure. The gap table highlights the gap containing 2.17.
Try 6 · Conjugate root
Read p₋ = −1/φ
Set s=1. Look at the coral dot on the left of the canvas at p₋ = −1/φ ≈ −0.618. Check the status bar: p₊·p₋ = −1 = −1/s exactly. Then set s=1/6 (p=3) — p₋ becomes −2 and p₊·p₋ = −6 = −1/s.
Try 7 · Prime gap
Watch a gap of 6
Jump to p=23. Then slowly drag s between 23 and 29 (gap 6, sexy prime pair). Watch the violet gap shading darken. The gap table row highlights. The sweep-rate bar widens. Compare with the gap between p=29 and p=31 (gap 2).
Try 8 · Ring geometry
Study the ring at p=5
Jump to p=5. Panel 1 shows ring M=5: residues {1,2,3,4} all coprime, s=1/20. Check which survive lift to M+1=6: only those coprime to 6. The center shows the survival rate, which is exactly C(5)/C(4) = (25−2)/(25−1).
§ 10 Why Does This Work?
The algebra in plain language
Start with any prime p. Define s = 1/p(p−1). Then plug s into the quadratic sp² − sp − 1 = 0 and check whether p satisfies it:
The algebraic identity sp²−sp−1=0 holds at p when s = 1/p(p−1) — this follows from substitution and is tautologically true for any number, not just primes. That alone is not the result. The result is the full chain:
1. Ring geometry → C(x) C(x) = ∏p fp(x) (smooth, built from coprime counts) 2. C(x) → D(x) D(x) = −d/dx log C(x) = Σp > x+1 1/(p²−x−1) 3. D(x) → prime signature D has jump +1/p(p−1) at x=p−1 for each prime p
D is smooth at x=m−1 for every composite m 4. Jump → prime measure jump size s → apply p(s) = (1+√(1+4/s))/2
Steps 1–3 are the substance. Step 4 is the inversion formula — elementary algebra. At s = 1 the inversion gives φ, connecting the Fibonacci characteristic equation to the same family. The quadratic is not the discovery. The discovery is that D(x), defined from geometry, has its discontinuities exactly at the primes. The quadratic then gives a clean closed form for which prime each discontinuity encodes.
The sentinel property: φ ≈ 1.618 is less than 2 — smaller than every prime. So p₊ at any actual prime is always greater than φ. φ is the infimum of {p₊(s) : p prime} — permanently unreachable, approached from above as s→1, but never achieved by any prime. That is why it is called the Golden Ratio Sentinel.
The Farey sequence is a linear projection of all modular GCD-1 rings up to order N. Each denominator b defines a modular ring Φ(b) projected onto [0,1].
Farey Sector Count — Remark 12.1
C(n,N) ≈ 3N² / (π²·n·(n+1))
Sector Sn = (1/(n+1), 1/n]
from Mertens (1874)
Franel–Landau (1924): RH ⟺ Σ|FN(k) − k/|FN|| = O(N½+ε). The sector deviations are what the zeros of ζ(s) govern.
Connection to Lift Survival
The ring at radius M on the concentric view IS the modular ring in Panel 1 of the Explorer.
Coprime residues at angle 2πr/M → fraction r/M on Farey circle.
Survival rate C(M) = fraction of teal dots (gcd(r,M)=1 AND gcd(r,M+1)=1).
Jump size s = 1/p(p−1) = fraction that do NOT survive at prime p.
Gaussian connection (Cor 8.5a):
lower barrier lifts iff q≡3 mod 4
iff q inert in ℤ[i] (n=4 lattice)
Ford Circles
center(p/q) = (p/q, 1/2q²)
radius(p/q) = 1/2q²
Two Ford circles are tangent iff their fractions are Farey neighbors: |ad−bc|=1. The packing encodes the entire Farey adjacency structure. PSL(2,ℤ) acts by Möbius transformations.
The density 6/π² is the probability that two random integers are coprime — the same constant as the squarefree density, connecting back to C = ζ(2)·dFT.