Papers

A paper takes one result of the tree and writes it out in full for a reader who has not read the notes: a cover with title, date and lead, then the argument, every number traced to the code that computes it, in markdown that prints as a paged document.

The lead of each paper below says what it proves. The first editions on the shelf are deprecated and never edited.

An Unconditional Mertens Bound on the Half Interval

At every odd base `base >= 94939`, on the integers whose digits all lie in the lower half `{0, ..., (base-1)/2}`, the Mobius function cancels, `|M_F(x)| <= C A_F(x) exp(-c sqrt(log x))`, and the primes are counted, the von Mangoldt sum being `(base/(base+1)) A_F(x)` within the same error, both proved here with no hypothesis and computable constants; at a prime base `p` that set is `{k : p does not divide C(2k,k)}`, and a transfer operator whose cost per digit is `(2/pi) log base + O(1)` sets the base.

2026-10-02

A Moving Target for Erdos Problem 125

The sums of a base-3 number and a base-4 number, each written with the digits `0` and `1` only, have upper density at least `6/(11 liminf_k h_k(tau_k))`, where `h_k` is the cell energy of a Cantor convolution read at the scaling `tau_k = 4^(m_k)/3^k`; so the open half of Erdos problem 125 fails only if one orbit of an irrational rotation lands, at every late step, in a target of measure at most `190/M`, for every `M`.

2026-09-23

An Unconditional Mertens Bound at Large Base

The Mobius function summed over the integers that avoid one digit in base `q` is at most `C A_F(x) exp(-c sqrt(log x))` with no hypothesis at every base `q >= 584` by a proof, and at every base `q >= 115` on certificates computed base by base, with `C` and `c` effective; Dirichlet approximation cuts the frequencies into four regions, and only the minor arcs set the base.

2026-09-23

The First Base Below a Quarter

Base 21 missing the digit 0 is the least base at which a one-missing-digit set has the `l^1` exponent of its digit transform below the bar `1/4`, all 108 sets of the bases 3 to 20 certified above it; 34 is the least base at which every digit clears and every base from 34 clears at every digit, from 125 by a proof; under the generalized Riemann hypothesis each clearance is a power saving for the Mobius sum over the set.

2026-09-21

The Franel Threshold Is a Mean Square

On the Farey fractions whose denominators lie in a digit design, Franel's square-root threshold is equivalent to square-root cancellation, in mean square over the dilates of the design, of their Mertens sums, with explicit logarithms in the sandwich; the full set recovers Franel.

2026-09-21

The Stack Hears the Zeros

The novelty of the stack, read through a smooth window, has an error that is a sum of waves at the frequencies of the zeros of zeta; its decay exponent is a Riemann hypothesis meter, and a sharp window hears the primes instead.

2026-09-21

The Menger Sponge Is Not Minkowski Measurable

The Menger sponge is not Minkowski measurable, by the criterion of Kombrink, Pearse and Winter with the open unit cube as feasible set, which needs no pluriphase hypothesis, and an interval-arithmetic certificate that puts its periodic function at two phases at least 0.012296 apart.

2026-09-21

The Spin Spectrum Reads a Pair Census

The spin spectrum of a design is one fixed linear function of its pair census, so some designs are homometric and no spin reading one level of refinement can tell them apart.

2026-09-14