Research

The mathematics of the mrly tree: a parity rule on the corners of a cube, substituted into itself by the Kronecker product, and everything that falls out of those two moves. One subject, one tree, and

Discoveries

The Apollonian gasket

The circle face of the stack: the Descartes reflection as exact integer arithmetic on `(k, k x, k y)`, the strip packing's line-tangent circles proved to be exactly the Ford circles so the Farey stack is the packing's shadow with brightness `floor(Q sqrt(2/k))`, the curvature census, root quadruple excluded, whose exponent by ratio lands at `1.305` against the rigorous residual dimension, the eight residues mod 24, and why no design of base at most 100 carries that dimension.

The automata

An elementary cellular automaton is a three-dimensional design read as a rule and Life a nine-dimensional one: what the identity buys, and what it cannot.

What base 3 hides

What base 3 hides: an Eisenstein L-value where base 2 hid pi, and the collapse theorem that turns any multiplicatively dependent family of bases into one design in base `r^lcm(e_i)`.

Beneath a design

The object beneath a design as three slots, accept, place and glue: the memory dial priced by a transfer matrix with `kappa` zero on exactly the non-empty product rules over `19563` live classes, the Collatz carry landed on that dial as a four-state transducer of unbounded radius whose zero-carry sets are the golden and supergolden rules at `3n + 1` and `7n + 1`, and the radix dial that keeps every count, carries the gasket, the dragons and the Koch curve as codes with a twist vector, and loses the fill law the moment a digit turns; the Mobius meter of every width-three rule to `2^30` against its own mass, the memory zeta as a matrix ladder with one pole comb per eigenvalue, and the design census up to similarity and up to affine conjugacy.

A design is a Boolean function

Designs are Boolean functions up to cube symmetry; the strongest theorem in the tree.

Two bases

What a second multiplicatively independent base does to a design: Cobham and Cobham-Semenov at their sources, every proper dim 1 design proved base-locked by density alone, the semilinear designs of dim at least 2 pinned between a proved count condition and the block designs with the diagonal as the standing counterexample, the global planar budget Refuted twice and replaced by one budget per axis for product designs, the base-2 gasket against the base-3 gasket counted exactly to `3^24`, and a three-base object on the line whose budget is negative, with five members and no sixth below a height of `38170` decimal digits.

Complexity

Boolean complexity of the catalog, and the Laplacian spectra of the fractals it builds.

Structure against noise

Self-similar designs raced against matched random cell sets on components and boundary.

The coprimality spine

The coprimality spine: exact base-local factors on every design, the census behind them, the window at dimension one with the band automaton's critical out-degree and its algebraic block rate, and primes on a design, where the least base below the quarter threshold is 21 at one missing digit and 32 at two, the one-digit family closes at 34 and the two-digit family closes at 21 against the third.

The core

What a design is, the headline counts, and the three genera.

Crop

The circle on a design: the corner disc count, its log-periodic main term, and the crossing shell read as a rooted tree whose transfer operator is derived from the geometry.

Cuts

The six-gasket theorem: a diagonal cut through one parity solid is Sierpinski all the way down.

Complex dimensions

Complex dimensions: every design in the lattice class, the carpet proved not Minkowski measurable with its explicit profile, and the arithmetic pole at `s_0 + 2 pi i/log 3` certified genuine.

The stack is an RH-observable

The stack's moire is a Farey resonance diagram; each scale `n` adds exactly `phi(n)` bright nodes, the stack is an address rather than a construction (any depth evaluates in closed form, which buys rendering and provably nothing toward RH), and restricting the Farey sequence to a digit design empties a fixed sixth of the line at base 3 `{0,1}` when both coordinates are restricted, while restricting the denominator alone leaves the Franel-Landau shape reading the same.

The hexagon moire

The moire of the stacked diagonal slices: the exact cut-ink laws, the doubling sign law, the quarter-line bands, and the ghost star, whose cell-frame decay is a closed form at every band width, `-1/4` at the arm and `-1/8` in the limit, every layer-pair constant an exact rational, and no frame-free coefficient.

Information

A render as data: the rearranged SVD reads a design's code out of noise and peels a magic word, while the fractal codebook loses to deflate.

Integers

The census the other way round: which integers the whole registry writes, which it never writes, and which it writes thousands of times.

Magic words

A different design at every scale: the word grammar, what collapses by block reduction, and what only the letter order can see.

Method

How the results here are produced and checked, worked through on the odd-side fill polynomial.

The Mobius meter across digit designs

The Mobius meter across digit designs: exact transfer between scaled columns, the 47-column cancellation census, a power saving on the dense columns under GRH, the pair route with its major-arc lemma and its `l^1` threshold `1/4`, and square-root cancellation as the open exponent.

Pi out of the stack

Pi recovered from the density of coprime points in the stacked set, two disjoint ways.

REFS

Every named reference, sequence id, theorem and attribution on this tree's root pages, resolved to a canonical URL. Ids that appear only inside a lab study are resolved in that study's own pages.

Sequences

Integer sequences fall out of the fractal work: cell counts, coprimality counts, Euler characteristics, design counts. This page is the ledger they are cited from. Every entry below meets the same

Slices

The diagonal slice of the solid cube: the `6n` census, centered-hexagonal vertices, and the splitting-prime rule.

Spectra

The tile grammar of the diagonal slice at every odd base: the two-tile claim, the closed forms, and the mod-4 split.

Spin

A design turned about its centre: the ripple identity, the complete spin spectrum, the sponge's opaque diagonal, and a Gaussian Farey.

The spirograph

The roulette a design draws: the coincidence law that counts its distinct curves, the node count `2ab C(k,2) + k a(b-1)` inside the window `min(1, A)` with the plane cut into `nodes + 2` regions, the exact crossing staircase above the loop threshold whose thresholds are algebraic numbers obeying a reciprocity, the alignment reaches where four of the carpet's curves run through one point, and the walls, whose cover no closed form tried survives.

The algebra of the stack

The algebra of the stack: stacking is Dirichlet convolution and the group knows nothing new, the exact spun stack is a Gaussian Farey whose Fourier `L^2` discrepancy obeys Franel's identity one field up and is equivalent to the Riemann hypothesis for `zeta(s) L(s, chi_-4)`, its hexagonal twin puts `L(2, chi_-3)` in a proved node constant, the layers are a dilation system whose symbol is `zeta(1 + s)` so no reweighting moves the line, and the lean is the operation outside the Dirichlet group that still lands a closed form.

The walk dimension

The walk-dimension census: same mass, different music.

Weighted designs

Weighted designs: what weights move on the mass side, what they never move on the length side, and the pressure that closes both.

The zeta function of a digit design

The design's own zeta function: an explicit zero-free half plane and the census it closes, a residue comb whose teeth are Rouche-certified one by one, a second family that is no critical line and obeys no counting, symmetry, contraction or gain law, the ordinate shadow as a constant-free Newton step from each zeta zero, the multiplicativity wall with the three products that stand where the Euler product does not, and the identity zeta_F N_F = 1 whose Mertens function runs the wrong way.

The mathematics of the mrly tree: a parity rule on the corners of a cube, substituted into itself by the Kronecker product, and everything that falls out of those two moves. One subject, one tree, and this page is its law; the index of the notes is built from their front matter at mrly.net/research.

THE LAW

  • Type folders, not topic folders: a topic is one slug that repeats across notes/, claims/, papers/, lab/ and ../figures/, never a box of its own.
  • notes/<slug>.md is a note: timeless, present tense, no dates and no story, opening on a front matter block of title, lead, figure and slug; the index of the notes is built from that block and nothing is registered anywhere.
  • claims/<slug>.md is a claims file: one dated line per claim, - YYYY-MM-DD [Tag] the claim. Witness: its generator, append only; the built page at /research/discoveries/ reads every file and filters by tag, topic and date.
  • Every claim carries exactly one tag: Proved, Verified, Conjecture, Refuted; a claim is never deleted because someone else published it first, a dated cross-reference line is added, and only a counterexample moves a claim.
  • Every printed number names its generator: a crate function in ../crates, a study in lab/rs/ or lab/py/, a sequence row in sequences.md, or an OEIS entry; a number with no generator keeps its claim at Conjecture.
  • A theorem from the literature is cited at its source, resolved to one URL in REFS.md; a claim of this tree never rests on a citation alone.
  • sequences.md is one generated ledger, written by cargo run -p mrlylab --bin ledger: a sequence is a row only when a claim cites it or the OEIS holds it, its id the canonical JSON name, its anchor the id's hash; never a page per sequence, and the enumerated millions are data, not pages.
  • papers/<slug>.md is a paper: markdown, a cover in front matter (title, lead, date, figure), print CSS for the paged document, no LaTeX; the shelf of first editions is deprecated and never edited.
  • ../figures/ is one crate: a figure is src/bin/<name>.rs, cargo discovers it, scripts/figures.sh presses it to ../files/figures/<name>-dark.png and <name>-light.png, and a reel to .gif; a figure is drawn by code, never by hand, and a page names a figure by its name alone.
  • lab/rs/<study>/ is a Rust crate in the root workspace and lab/py/<study>/ a Python study run with uv; a study README says what it computes, how to run it and which claim lines it witnesses, and a study is deleted the day a crate function or a demo computes its numbers.
  • The demos in ../site/demos run the same crates through wasm; a note links the demo that shows it, and every demo is linked from some page.
  • Math is written in backticks so it renders everywhere; lines never wrap; no comment in code beyond a section delimiter, no em-dash anywhere.
  • Author line MrlyProd on every route: Carlo directs, Claude writes and computes, and /method/ says so.
  • One check: bun run check in ../site passes on this tree or fails on a bad line, and nothing else gates it.

LICENCE

Text and figures CC BY 4.0; code MIT. The sequences themselves belong to the OEIS and its contributors.