Interactive research / 01

How many prime patterns can fit?

Some changes are cheap. Others are costly. Add or remove prime factors, adjust the budget, and see which patterns fit.

Start with one integer

Build it. Change it. Count what fits.

Guided example: a cheap swap and a distant move

This is not the canonical verifier. This floating-point totient illustration neither checks Robin’s inequality nor reproduces the canonical-family certificate.

Set Prime cutoff to y = 12 and Available budget to H = 0.0300. Select Reference pattern: the integer is 2310.
Select Swap near the cutoff. This removes 11 and adds 13, giving 2730. The displayed benefit is about 0.021117, within the budget.
Select Make a distant move. This starts again from the reference, removes 2 and adds 53, giving 61215. Its benefit is about 0.788851, over budget.
Select Reference pattern to return to 2310. Compare the family counts below: presets edit your integer, so those counts stay unchanged at fixed y and H. Changing H keeps your edited integer; changing y resets it.

Use the sliders and named buttons below. The example illustrates relative cost, not a Robin candidate test.

Prime cutoff y = 12Sets the reference: every prime up to y, once. Moving this slider resets your integer.

Available budget H = 0.0300The maximum arithmetic cost allowed. Changing H keeps your integer. This is an exploratory budget, not a spectral prediction.

near patterns fit the cost budget

The graph shows your factors, so it stays put. The number of patterns that fit changes in steps as costs cross the budget. See the full family ↓

01 / MAKE A CHANGEClick a prime to cycle its exponent: 0 → 1 → 2 → 3 → 0

Small: p < y/2 Near: y/2 ≤ p ≤ 2y Large: p > 2y

Filled dots: present. Hollow dots: absent. Larger dots: more copies. Height shows cost; tall stems are capped at 0.306 for display. Use the buttons below to edit factors.

YOUR INTEGER

2310

02 / FOLLOW THE COST

How changes spend the budget.

The reference integer contains each prime up to y once. Removing, adding or repeating a factor contributes to a total cost, called the benefit B_φ. Here, a smaller benefit means a cheaper change—not a better score.

Omitted primes

Added primes

Repeated factors

Total benefit B_φ

Pattern / budget

What is actually being calculated?

r(p) = log(p/(p−1)), λ = r(y)/log y. Removing a prime p ≤ y costs r(p) − λ log p; adding p > y costs λ log p − r(p). Every copy after the first costs λ log p. Their sum is the actual totient benefit B_φ(n). If y is itself prime, removing that prime costs zero.

The factor palette is limited to primes ≤ 53, with exponents 0–3. Costs use floating-point logarithms, rounded for display; budget decisions within 10⁻¹² are labeled as numerical boundaries. The distance-family count below uses exact integer comparisons for these slider values.

03 / COUNT THE POSSIBILITIES

Now count a whole family.

This count is separate from your edited integer. Keep outside primes at their reference settings, exclude repeats, and let each near prime be present or absent. Changing y or H updates the counts; editing a factor does not.

All near-support patterns2 raised to the number of near primes

Within the benefit budgetNumerical logarithmic-cost comparison

Within the looser distance testExact finite count: Σ |p − y| ≤ 4y²H

The distance test adds |p − y| for each changed near prime. Every pattern within the benefit budget passes this test, but extra patterns can pass too. Counting a larger, simpler set gives an upper bound for the smaller one.

Passing either test does not identify a Robin candidate. This finite family enforces neither the manuscript’s unit log shell nor the Robin inequality.

Try a pattern from the family

Up to 24, cheapest first · 1 = present, 0 = absent

04 / SEE THE ENCODING

From changed bits to prime partitions.

Return to your edited integer. Record which near primes changed, splitting them at or below y and above y, nearest first. The goal is to encode every pattern so it can be recovered and counted.

Different bits, different meaning. In the gallery, 1 means present. Here, 1 means changed from the reference: a removal at or below the cutoff or an addition above it. The first two positions on each side are stored separately as “free bits.” Free means no partition weight—not zero arithmetic cost. Later positions get labels 3, 5, 7, …; these are bookkeeping weights, not factors of your integer.

Lower side / nearest first

Upper side / nearest first

Why two free bits, and why these prime labels?

The first charged position has zero-based index 2 and label q₂ = 3, where q₁ = 2. The ordered-prime estimate used in the proof begins at that index. The first two positions on each side are retained separately, giving at most 4 × 4 = 16 free-bit configurations.

Brun–Titchmarsh (BT) and Erdős–Nicolas–Sárközy (ENS) Lemma 1 supply the charge q ≤ 5|p − y|. A distance budget 4y²H therefore gives each side a prime-partition budget T = 20y²H. The labels shown here encode distinct changed positions; the full partition family permits repetitions and is an enlargement.

The labels identify the changed positions, and the separate bits preserve the first two choices. Together they let us recover the pattern. This interaction illustrates that encoding. It does not prove the imported estimates or evaluate the full asymptotic partition count. The Lean interface remains conditional on those two inputs.

Where this fits in the research

The count is one link.

Spectral estimate→Budget & centered inclusion→Pattern count · you are here→Shell aggregation

This laboratory illustrates totient benefit and a finite counting mechanism. The October formal work also treats sigma correction, shell placement and padding cost; those results do not change the laboratory’s calculations. Read the finite milestones → The October 25 canonical verifier instead checks exact certificate data, event completeness and Robin satisfaction over its stated real parameter range. This floating-point totient laboratory does not reproduce those checks. The manuscript supplies the separate analytic hypotheses and eventual argument.

Read the research note ↗