Papers
Write-ups of MrlyMath, each claim a theorem with a proof, a computational fact with its exact finite domain, or a conjecture labelled as one; markdown that prints as a paper.
The shelf
The first editions, in LaTeX with a PDF each, deprecated: every paper is rewritten here in turn and the shelf is never edited.
Coprimality Density Above Dimension One
the visible-point density of every digit-restricted fractal above dimension one, exactly.
2026-08-23revised 2026-09-08
The Pincer at Dimension One: Two Edges and Two Closed Doors
the missing dimension-one estimate squeezed from both edges, and two routes past it proved shut.
2026-08-23revised 2026-09-08
A Power Saving for Mobius Sums on Missing-Digit Integers, under GRH
under GRH, the Mobius function cancels by a power over the integers missing a digit, at every base from 3690 up.
2026-09-06revised 2026-09-08
Pairwise Coprimality in the Menger Sponge
drilling the sponge's holes costs its coordinates exactly 12.25 percent of their pairwise-coprimality odds.
2026-08-23revised 2026-09-08
The Walsh Spectrometer: Exact Diagonal-Slice Ink for Every Three-Dimensional Parity Design
the exact slice ink of all 256 parity designs, with the design's Walsh spectrum as the coefficients.
2026-08-23revised 2026-09-08
Menger Diagonal Slices: A Recurrence of Order ⌈D/2⌉
Menger diagonal slice counts obey a recurrence of order exactly ceil(D/2), and at odd D != 1 mod 3 their exponent provably beats the generic slice dimension.
2026-08-23revised 2026-09-08
The Even Half of the Slice Sign Law
the even half of the slice sign law: below the generic exponent in every even dimension at bases 3 and 5, the quadratic base-3 transient identified exactly, and the Jacobsthal tent rank law behind the remaining strictness gap.
2026-08-24revised 2026-09-08
Base-3 Digit Designs: Diagonality, Ray Masses, and a Spectral Gap at Two
which origin lines hit a base-3 design, how much each catches, and a spectral gap at two.
2026-08-23revised 2026-09-08
Order Sensitivity of Kronecker Design Words: What the Perfect Shuffle Cannot See
what survives swapping nested patterns, and the connectivity that does not.
2026-08-23revised 2026-09-08
The Component Exponent of a Two-Letter Kronecker Word
the piece count of a two-letter nesting has a closed form on all 105 alphabets, so its growth rate exists, is order-blind, and mostly just repeats the cell count.
2026-09-02revised 2026-09-08
Parity Carpets Correlate by gcd: Four Exact Laws for a Square-Wave Stack
two parity carpets agree by an exact gcd law, zero exactly when the scales are coprime; the stack's full spectrum is the squared-divisor field of the frequency gcd, zeta quotients and nothing more.
2026-08-23revised 2026-09-08
Divisor Avatars: Which Parity Designs Count the Divisors of a Power
when a parity design's cell census is the divisor count of a power, and exactly which integers have one.
2026-08-23revised 2026-09-08
The Sequence Census of a Parity Design
how many integer sequences a parity rule can write, which ones the catalogue already holds, and why the polygonal numbers keep turning up.
2026-08-31revised 2026-09-08
The First Base Below a Quarter
the least base whose one-missing-digit set has Fourier l^1 exponent below a quarter, and the least base whose every digit clears.
2026-09-07revised 2026-09-08
The Dirichlet Inverse of a Digit Design and the Transport of Zeros
the design's own Mobius function, and the zeros of its zeta that push its partial sums past the design's own size; still owes its outside read.
2026-09-07revised 2026-09-08