Interactive companion · Erdős problem #1100

The layer, the blockers, and φ

A visual companion to “An Unconditional Golden-Ratio Lower Bound For Coprime Adjacent Divisors of Squarefree Integers” (M. M. Ross, 2026). Everything on this page is computed live in your browser.

§5 · The object of study

A cardinality layer, drawn live

Draw k independent Unif[0,1] weights and form all (kj) subset sums of size j. Each sum is a tick below. Consecutive sums whose index sets are disjoint are marked in gold: for squarefree integers these are exactly the coprime adjacent divisor pairs within the layer. Theorem 5.1 says their expected number is at least c_β (k−jj).

k (weights)

j (layer)

Full layer — every subset sum, sorted. Gold ticks are disjoint adjacencies.

Magnifier position (a 1% window of the layer)

Magnified window — individual gaps become visible; gold arcs bridge disjoint consecutive pairs.

layer size

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disjoint adjacencies N

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C(k−j, j)

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ratio N / C(k−j, j)

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disjoint adjacency other subset sum

§5 · The two obstructions

External and internal blockers

Fix a disjoint pair S, T whose sums land within a window of width θ of each other. The pair is adjacent unless another size-j set R lands in the gap. The proof treats the two kinds of intruder with different geometry — and the difference is one power of θ.

External: R uses a free coordinate

A coordinate outside S∪T survives the conditioning, so convolution smoothing bounds the conditional density and the hit probability scales with the window’s length.

cost per blocker ∝ θ · m^−1/2

Internal: R ⊆ S∪T, no free coordinate

The swap decomposition writes Δ = U + V with U, V independent. Blocking pins both into (0,θ] — a codimension-two event.

cost per blocker ∝ θ² · (r(j−r))^−1/2

With one localized condition the C(2j, j) ≈ 4^βk internal candidates would overwhelm the gap probability by an exponential factor. The second factor of θ turns their total into 2^{−(H₂(β)−2β)k} of it — exponentially negligible for every β ∈ (0, ½).

§6 · Where the golden ratio enters

The saddle layer

The exponential size of the disjoint-pair count at layer j = βk is (k−jj) ≈ e^kF(β)/√k. The curve below is F. Its unique maximum solves 5β² − 5β + 1 = 0, and there the value collapses algebraically — not numerically — to log φ, because β(1−β) = (1−2β)² = 1/5 at the critical point.

One layer suffices: the proof never sums over j, so no uniformity of constants near the saddle is needed.

Theorem 5.1 · Live verification

Monte Carlo, in the browser

The theorem is falsifiable by experiment: if the ratio E[N] / (k−jj) decayed with k, the constant c_β could not exist. Run the trials: the ratio holds near 0.72 across every layer, with no drift.

Idle — a full run takes about half a minute.

k	j	β	layer size	trials	E[N] est.	± se	C(k−j,j)	ratio

Ratio vs k. Constancy is the theorem; decay would be its refutation.

This page verifies the probabilistic heart of the proof (Theorem 5.1) and illustrates its mechanisms. It does not — and cannot — check the prime-block transfer (Theorem 3.1), which is exact for every k and rests only on the prime number theorem in fixed proportional intervals.. Single file, no dependencies, no network. Built with Claude Fable 5.