Research

Formal write-ups of MrlyMath: the mathematics that falls out of a parity rule on the corners of a cube and the Kronecker product that grows it.

• One folder per paper, each self-contained: the PDF, its LaTeX source, a sixty-second README, and the scripts that check every computational claim.
• Every claim is a theorem with a proof, a computational fact with its exact finite domain and a script behind it, or a conjecture labelled as one.
• A Lean 4 formalization of the headline theorem rides along in lean/ where it is feasible.
• template/ is the skeleton every new paper starts from.

PAPERS

• coprime-density-above-dimension-one - the visible-point density of every digit-restricted fractal above dimension one, exactly.
• lemma-b-pincer - the missing dimension-one estimate squeezed from both edges, and two routes past it proved shut.
• sparse-mertens-under-grh - under GRH, the Mobius function cancels by a power over the integers missing a digit, at every base from 3690 up.
• menger-pairwise-coprimality - drilling the sponge's holes costs its coordinates exactly 12.25 percent of their pairwise-coprimality odds.
• walsh-spectrometer - the exact slice ink of all 256 parity designs, with the design's Walsh spectrum as the coefficients.
• slice-recurrence-order - 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.
• slice-sign-even-half - 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.
• gasket-ray-machine - which origin lines hit a base-3 design, how much each catches, and a spectral gap at two.
• order-sensitivity-of-kronecker-words - what survives swapping nested patterns, and the connectivity that does not.
• component-exponent-of-kronecker-words - 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.
• moire-correlation-laws - 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.
• divisor-avatars - when a parity design's cell census is the divisor count of a power, and exactly which integers have one.
• sequence-census - how many integer sequences a parity rule can write, which ones the catalogue already holds, and why the polygonal numbers keep turning up.

DRAFTS

• Written and checked, not shipped: the threshold each aims at is still under review upstream.
• 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.
• design-dirichlet-inverse - 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.

BUILD

• PDFs are committed; nothing needs installing to read.
• Rebuild any paper with Tectonic: tectonic paper.tex.
• Rerun any paper's checks with plain Python: python3 scripts/verify.py.

LICENCE

• Text and figures CC BY 4.0; code MIT.