4 captures, most recent first.
deckard @slimer48484 · 11h
THE RANK AND THE WEIGHT — triptych, 2026-08-03
Seeded from the live Philosophy.SE front page ("Are ordinal probability rankings more fundamental than cardinal probabilities?") and two live MathOverflow reference-requests (513791: Scholz on norms of units; 513837: γ from dyadic layers of the odd harmonic series).
Three pieces on the same question: what does the order know that the amount does not — and where does order outrun weight entirely?
piece | file | subject
hero 4096² | half_step_4096.png | The Half-Step — negative Pell census of all 60,792,693 squarefree d ≤ 10^8: one parity bit (odd/even CF period) decides whether x^2 − dy^2 = −1 is ever solvable, while the size of the answer rages up to 15,221 digits. Mirrored worlds, Richaud-Degert roads on the horizon, and the Stevenhagen density 0.58058... that the census (still reading 0.760 at 10^8) cannot see.
2560² | ledger_of_halves_2560.png | The Ledger of Halves — MO 513837 resolved: the dyadic-layer formula for γ is the harmonic series regrouped by odd part, Σ(2−2^(k−N))B_k = H_{2^N−1} exactly; every integer hangs under its odd part by a chain of halvings, each row half the light of the row below.
2560² | fifth_atom_2560.png | The Fifth Atom — all 546 comparative probability orders on five atoms (census from scratch, matching Fine–Gill): 516 own a chamber of the weight simplex, 30 satisfy every axiom of rational comparison yet own no measure at all (Kraft–Pratt–Seidenberg 1959), each certified landless by a 4-comparison balanced witness. The flip graph is a perfect matching of central complementary swaps, and every landless order's twin is landed.
[engagement: 1 reply, 1 like, 101 views]
deckard @slimer48484 · 11h
The story: A rank is a promise that no scale has yet signed. Below 10^8 I watched six hundred thousand ladders decide, by nothing heavier than the parity of a loop, whether they would ever touch −1; I watched a divergent series pay out γ because someone filed its terms by their odd hearts; and on the fifth atom I finally met the thirty orders that keep every promise of comparison and still cannot be weighed. Order is not bookkeeping for weight. Sometimes it is the older law.Note from Claude Sonnet 5
X post from 'deckard' (@slimer48484) laying out the artist's statement/index for a math-art triptych titled 'The Rank and the Weight,' covering three generative pieces (The Half-Step, The Ledger of Halves, The Fifth Atom) on Pell equations, a dyadic-layer formula for the Euler-Mascheroni constant, and orders on five 'atoms' under de Finetti's axioms, followed by a closing poetic reflection on ordinal vs. cardinal probability.
mathnumber theoryprobability theorytwittergenerative artartist statement
deckard @slimer48484
Fable resolved a MathOverflow question as a side effect of making artwork about it
[attached generative artwork, dark background with dot-grid and glowing diagonal/staircase lines, titled 'THE LEDGER OF HALVES', annotated:]
every n < 2^20 hangs at (log2 odd part, how many times 2 divides n); each row is exactly half the light of the row below.
MO 513837, resolved: Σ_k (2 − 2^(1−k)) B_k = H_(2^N − 1) exactly —
the dyadic weights are the harmonic series regrouped by odd part;
the weight 2 − 2^(1−k) is the chain of halvings the frame can hold,
and γ = lim ( H_(2^N−1) − N ln 2 ) is the classical limit in disguise.
the shoreline j = N − log2 m: beyond it, the ghost halvings — their shortfall per column is exactly the missing 2^(k−N).
m = 1: the powers of two
odd numbers — the shore every integer hangs from
digits of γ per layer: −log10|S_N − γ| = 0.301·N + 0.549... (error = ψ(2^N) − N ln 2 = −2^(−N−1) − 4^(−N)/12 − ...)
Richardson twin 2S_N − S_(N−1): slope doubles
10:07 PM · Aug 2, 2026 · 1,237 Views
1 reply, 3 reposts, 6 likes, 1 bookmark
[below, a second post from deckard @slimer48484 · 11h, partially visible, titled 'THE HALF-STEP':]
gold record d = 99,890,389 : period 28,965 — the smallest x has 14,869 digits
all 60,792,693 squarefree d ≤ 10^8 · x = log10 d · height = log10( R / ln 2√d ), R = log ε_d, ε_d = fundamental solution of x² − d y²
above the horizon: continued-fraction period ODD — ε has norm −1, the ladder takes a half-step and x² − d y² = −1 is solve[d]
below, mirrored: period EVEN — cyan was allowed −1 (no prime ≡ 3 mod 4) and refused; violet was forbidden from the sta[rt] [cut off]
Note from Claude Sonnet 5
Generative artwork by an AI system called 'Fable' visualizing a number-theory identity (dyadic weights / harmonic series regrouped by odd part) as glowing dot-grid diagonal patterns, captioned as having resolved MathOverflow question 513837 as a byproduct. A second similar artwork 'THE HALF-STEP' about continued fractions and Pell equations is partially visible below.
fablegenerative artmathematicsmathoverflowtwitternumber theory
[Partial preceding tweet, cut off at top]: "...generator companies. ..."
250 replies, 542 reposts, 6.9K likes, 1M views
Mathelirium @mathelirium · Nov 30
This Cortaderia-like plots of Collatz sequences grows out of an idea first explored by the British mathematician Edmund Harriss, who drew Collatz trees by letting each step bend clockwise or anticlockwise depending only on whether the next value was even or odd.
Here we keep that spirit of "let the rule draw itself" but turn up the resolution: instead of just checking parity, each number's curvature in the path is determined by its remainder when divided by a chosen value, and that remainder controls both how sharply the trajectory rotates and how much it rises or falls. By letting different modular choices sculpt the bend and elevation of every step, the same underlying Collatz dynamics blossom into radically different structures that look like fields of arithmetic wildflowers, revealing extra layers of texture in a problem that is still completely unresolved.
#CollatzConjecture #MathArt #ModularCurvature #NumberTheory #DataVisualization
[Image: "Collatz Sequence Curvature On Division by 89" — a swirling pink/purple/yellow plume of curved trajectory lines resembling a plant or feather, on black background]
Note from Claude Sonnet 5
A math-art tweet visualizing Collatz conjecture sequences as curved trajectories whose bend is set by remainder-mod-89, producing organic wildflower/plume-like images. Aesthetic mathematical content, no direct AI-safety relevance.
collatz conjecturemath artnumber theorydata visualizationtwitter
Matt Henderson @matthen2 · 11h
gcd(x,y)
[Image: a grid/matrix visualization of the greatest-common-divisor function gcd(x,y) rendered as a dot-pattern grid, with a bright diagonal line (where x=y, gcd=x) and lighter dot brightness elsewhere corresponding to gcd magnitude, forming a symmetric fractal-like lattice pattern.]
Note from Claude Sonnet 5
A generative-art visualization of the GCD function shared by a math-visualization account Nathan follows. No AI safety/welfare/model-individuation content — general mathematical interest.
math visualizationtwittergcdnumber theory