u/Background-Major4104

https://wessengetachew.github.io/Strip2/

I've added a "Preview the math" button to Strip Arithmetic II. Change the grid size, slope, width, or offset, click it, and you get a sheet with the exact numbers behind whatever is on screen. It's all computed live in your browser.

The images attached are those sheets. Each one is the math for that configuration:

Coprime density: on a 10,000 × 10,000 grid, 60.7950% of cells are coprime, against 6/π² ≈ 60.7927% in the limit.

Strips: in the window −1 < 2r − M < 1, the measured density is 0.50005 against a predicted 0.50000.

https://wessengetachew.github.io/Strip2/

Strip Arithmetic II

Part 2 of my post on the Goldbach lattice

https://wessengetachew.github.io/Partition/page2.html

That post showed one identity: gcd(r, M) = gcd(r, M-r), and the two sum-strips it produces on a G×G lattice — fully coprime exactly when a Goldbach split works.

Strip Arithmetic II generalizes it. Instead of one fixed slope and a single gap g, the strip now has a slope a/b and a width w, and the same residue count ν_p that decided primality on the diagonal now produces two different things depending on how you read it:

· On a block of w consecutive residues, it gives min(w, p) — the local factor behind the block-coprime density C(n) from Page 5.

· On the Goldbach anti-diagonal, it gives ν_p ∈ {1, 2}, which assembles into the Hardy–Littlewood singular series:

𝔄(N) = ∏_p (1 - ν_p/p) / (1 - 1/p)²

Same residue count, two slopes, two classical densities — that's the whole page.

The 10-minute video is just me exploring it live: switching between Grid, Ring, and Cayley, then spending most of the time dragging the offset and width sliders to show how smoothly the strip reshapes and which partitions fall in or out as you go.

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Try it here: https://wessengetachew.github.io/Partition/

Previous work and SOME non video submission

https://wessengetachew.github.io/smith/

This work is a side quest of Page 5 where we introduced the Diagonal gaps between block-coprime residues as n grows on the C(n) framework.

Here

The core identity is deceptively simple:

\gcd(r, M) = \gcd(r, M-r)

On an G \times G lattice (points (r, M) with 1 \leq r, M \leq G ), fixing a gap g defines two sum strips:

· c : r + M = g

· d : r + M = E - g , where E = 2(G+1)

and two difference strips (which are transposes of each other).

The key observations:

· Strip c is fully coprime iff g is prime.

Because along r + M = g , we have \gcd(r, M) = \gcd(r, g) , so the strip samples one full period of \gcd(\cdot, g) . It is full exactly when \varphi(g) = g-1 , i.e., g is prime.

· Strip d is fully coprime whenever s = E - g is prime, but it only sees a window of g-1 consecutive residues of the longer period s . This window can accidentally miss all multiples of a prime factor of s when s is composite – but only if g \leq \sqrt{E} .

For g > \sqrt{E} , that “leak” closes: the window is long enough to force a factor.

Thus, above the threshold,

\text{both sum strips fully lit} \quad \Longleftrightarrow \quad E = g + s \text{ is a Goldbach partition}.

The tool lets you explore this live. You can:

· Slide g (and G ) to sweep partitions in real time.

· Switch between Grid, Ring (radial), and Cayley (Möbius) projections.

· Toggle multi‑pair overlay to see all Goldbach partitions (or all partitions) at once – each pair gets its own color.

I’ve attached a sample export: G = 135 gives E = 272 , and the batch shows all 7 Goldbach pairs:

3+269,\ 31+241,\ 43+229,\ 61+211,\ 73+199,\ 79+193,\ 109+163 .

The “s ” refers to how s = E - g changes as you move the gap slider – the tool literally animates through every possible split of the even number upto 4096 visually, and you can watch the two strips fill or empty as g and s gain or lose primality.

I’d love feedback – especially on the mathematical framing, the interface, or any edge cases I might have missed. The underlying code uses a sieve to precompute coprimality, so the whole lattice updates in milliseconds even at G = 4096 .

Let me know know!

https://wessengetachew.github.io/smith/

A fascinating corner of mathematics where pure theory meets computational experimentation.

Pick a random whole number. Now ask: how likely is it — along with the next n numbers right after it — to all share no common factor with a fixed modulus M? That probability has a name, C(n), and it falls off in clean, predictable steps, one jump at every prime.

C(n) = ζ(2) · ∏ₚ (1 − min(n+1, p)/p²)

It starts at total certainty. C(0) = 1, exactly — every factor in that Euler product cancels its own inverse against ζ(2), so nothing is left to compute. Strip the ζ(2) normalization out and look at the raw density D(n) = C(n)/ζ(2) instead, and n=0 isn't quite so trivial: D(0) = 1/ζ(2) = ∏ₚ(1 − 1/p²) ≈ 0.607927101854 — no cancellation left, it's just the reciprocal of ζ(2).

The first real step is C(1), and there's nothing mysterious about it either: it's ζ(2) times the Feller–Tornier product, ∏ₚ(1 − 2/p²) ≈ 0.530711820472. Two known constants multiplied together, one known answer. D(1) — same product, no ζ(2) factor — is the Feller–Tornier constant itself, ≈ 0.322634098939. Sanity check: C(1)/D(1) = ζ(2) ≈ 1.644934, exactly like C(0)/D(0). The ζ(2) factor cancels cleanly at every step; it just isn't always trivial to see why.

OEIS ties it together nicely: A013661 (ζ(2)) × A065474 (∏(1−2/p²)) = A065469 (C(1)).

Everything after that first step — n=2, 3, 4, and beyond, where each new prime saturates the product one at a time — is where this actually gets interesting, and it's the part I've been digging into.

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An original, self-consistent framework for studying prime distribution through the lens of a single density function, developed from first principles, visualized interactively at every stage, and extended from the real line to the complex plane with the analytic structure carefully tracked. 15 pages may add more of living mathematics.

https://wessengetachew.github.io/FT/

P1 — C(n) = ζ(2) · ∏_p (1 − min(n+1, p) / p²)

P2 — C(n) = Σ_{d≥1} w(d) · C(n,d) where w(d) = density weight of diagonal strip |M−r|=d

P3 — d/dt log D(t)|_{t=p−1} jumps by exactly 1/p(p−1) where D(t) = ∏_p (1 − min(t+1,p)/p²)

P4 — C(n,d) = ∏_{p|d}(1−1/p) · ∏_{p∤d, p≤n}(1−1/p) — stratification by modular distance d

P5 — d(r, 2r+1) = p_min(r) − 2 where p_min(r) is the smallest prime factor of r

P6 — C_sD(n) = ζ(s) · ∏_p (1 − min(n+1,p) / p^s)

P7 — A(H, p#) = ∏_{q≤p} (q − ν_H(q)) exact integer identity

P8 — C(n) = ζ(2) · ∏_p (1 − min(n+1,p)/p²) — full precision calculator

P9 — Rail(G,n) = (π²/6) · ∏_{p≤G} (1 − min(n+1,p)/p²)

P10 — P_{>n+1}(s) = P(s) − Σ_{p≤n+1} p^{−s} via Möbius inversion of log ζ

P11 — C(t) = ζ(2) · ∏_p (1 − min(t+1,p)/p²), t ≥ 0 — continuous real curve

P12 — F(ξ) = Σ_p 1/p(p−1) · e^{i(p−1)ξ} — weighted, not unit weights

P13 — ord_{s=1} C(t,s) = π(t+1) + t

P14 — D_χ(x) = −Σ_{p>x+1, χ(p)≠0} χ(p)/(p²−x−1) and J_χ = Σ_p χ(p)/p(p−1)

P15 — surface: (Re(s), Im(s)) ↦ log|C(t,s)|

Each positive integer n is paired with an infinite, non‑repeating, transcendental constant C(n) that encodes the block‑coprime density of length n+1

https://wessengetachew.github.io/FT/?n=0

C(n) – block‑coprime density (normalised so C(0)=1)

· C(0) = 1.000000

Pairs (r, M) with gcd(r, M)=1 only. Raw density D(0)=6/π²≈0.6079, multiplied by ζ(2)=π²/6 gives 1.

· C(1) ≈ 0.530712

Also require gcd(r, M+1)=1 (block length 2). Local factor at p = 1−2/p². C(1) = ζ(2)·∏_p (1−2/p²).

(Note: the classical Feller–Tornier constant is (1+∏(1−2/p²))/2 ≈ 0.6613, not C(1).)

· C(2) ≈ 0.412828

Block length 3: gcd(r, M), gcd(r, M+1), gcd(r, M+2) all =1. Uses min(3,p)/p².

· C(3) ≈ 0.373657

Block length 4: condition extends to j=0,1,2,3. Uses min(4,p)/p².

Formula: C(n) = \frac{\pi^2}{6} \prod_{p} \left(1 - \frac{\min(n+1,p)}{p^2}\right)

As n→∞, C(n) ∼ ζ(2)·e⁻ᵞ / ln(n+1) → 0 (Mertens).

https://wessengetachew.github.io/FT/?n=1

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1. The Ring Geometry

https://wessengetachew.github.io/MODZ/

For each modulus M ≥ 1, define the coprime residue set

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

Each element r is placed on a unit circle at angle

θ(r, M) = 2π · r/M

In the concentric arrangement, ring M sits at radius proportional to M (scaled to fit the canvas). The result: nested circles, each carrying φ(M) dots. As M grows, the dot density per ring trends toward the average coprime density 6/π² ≈ 60.79%.

Global Rotation

A global rotation angle α is applied to every point:

θ_displayed(r, M) = 2πr/M + α + (M−1)·δ

where δ is the per-ring rotation increment (ring rot slider). Default: α = π/2 (90°, entered as 1/4 × 360°). Labels optionally stay fixed at their unrotated positions regardless of α.

2. The Lift Condition

A residue r on ring M lifts to ring M+1 when

gcd(r, M+1) = 1

Every coprime residue satisfies gcd(r,M)=1 by definition. The lift condition adds the requirement for the next modulus. Lift lines: green when it lifts, red when blocked.

Chain-n Survival

Require r to lift through n consecutive rings:

gcd(r, M+j) = 1 for j = 1, 2, …, n

The chain slider restricts visible lift lines to residues satisfying all n conditions simultaneously. As n increases, fewer points qualify and the canvas thins.

3. Live Counters

Three quantities update in the status bar on every render:

φ / total

Σ φ(M) / Σ (M−1)

→ 6/π² ≈ 0.6079

lift / φ = C(N)

Σ T(M) / Σ φ(M)

→ C ≈ 0.530712

M range

M_min – M_max

ring count, point count

where T(M) = |{r ∈ R(M) : gcd(r, M+1) = 1}| is the count of lifting residues on ring M.

The Lift Survival Constant C

C = lim_{N→∞} Σ_{M=2}^{N} T(M) / Σ_{M=2}^{N} φ(M) = ∏_p (p²−2)/(p²−1) = ζ(2) · ∏_p(1−2/p²) = ζ(2) · d_FT ≈ 0.530711806246…

where d_FT = ∏_p(1−2/p²) ≈ 0.3226 is the Feller–Tornier constant (OEIS A065469). The status bar shows the empirical C(N) for the current M range, converging toward 0.530712 as M_max grows.

Coprime Density

Σ_{M=2}^{N} φ(M) / Σ_{M=2}^{N} (M−1) → 6/π² ≈ 0.607927

This is the density of coprime pairs among all integer pairs — the fraction of the full grid occupied by points on the canvas.

4. Color Modes

16 color modes control how every point is colored. Applied per-point at render time based on (r, M, θ).

5. Display Overlays

Prime Spiral

For a fixed prime p, the residue r=p appears on every ring M where gcd(p,M)=1 — all M not divisible by p. Its angular position θ=2πp/M sweeps as M grows, tracing a spiral. Three geometric features emerge:

Equator gap

At M=2p: gcd(p,2p)=p≠1. The spiral always skips ring 2p. The gap is visible as a break in the colored path.

Upper path r=(M+1)/2 — always red

gcd((M+1)/2, M+1) = (M+1)/2 ≥ 2. This residue never lifts to M+1. Always shown blocked.Lower path r=(M−1)/2 — alternating

gcd((M−1)/2, M+1) = gcd((M−1)/2, 2) = 1 iff M≡3(mod 4). At a prime q this is the condition for q to be inert in ℤ[i] — the primes not expressible as a sum of two squares.

Lift Lines

Green segment from (M,r) to (M+1,r) when gcd(r,M+1)=1. Red when blocked. Opacity and line width are adjustable. The chain slider restricts to n-consecutive-lift survivors.

N-gon Polygons

Connect the φ(M) coprime points on ring M in angular order — you get the coprime polygon, a geometric representation of (ℤ/Mℤ)×. Three modes:

Mode Vertices Example M=6

Coprime only φ(M) vertices at coprime r Triangle: r=1,5 (+ closure)

Full M-gon All M points Hexagon: all r=0…5

Both Both overlaid Triangle inside hexagon

Gap Chords

For a chosen gap value k, connect residues r and r+k on the same ring when both are coprime. k=2 shows twin-prime pairs geometrically; k=6 shows sexy pairs.

Non-Coprime Points

Points where gcd(r,M)>1 — the zero divisors of ℤ/Mℤ. Colored by their gcd value (hue = gcd×47 mod 360). Hoverable when inspect is on.

6. The Inspect System

With Inspect ON, clicking any point opens a panel showing:

Field Value / Formula

r / M Residue and modulus

r/M decimal Fractional position on circle

θ angle 2πr/M in degrees

Farey sector n ⌊M/r⌋ — sector containing r/M

Half r/M > ½ (top) or r/M ≤ ½ (bottom)

Lift to M+1 gcd(r, M+1) = 1 ✓ or ✗

gcd(r,M) Should be 1 for coprime points

gcd(r,M+1) 1 = lifts, >1 = blocked

φ(M) Number of coprime residues on this ring

M prime Whether the modulus is prime

Mirror M−r gcd(M−r, M+1) shown

Appearances How many rings r appears on in [M_min, M_max]

Connect-same-r: when a point is inspected, gold dashed arrows connect all rings where r appears as a coprime residue, with arrowheads at midpoints and M= labels.