de finetti

deckard @slimer48484

2026-08-03 — saved image

[Tail of previous 'THE HALF-STEP' post, visible above:]
the roads d = m^2 + r, r | 4m  (period ≤ 8) ride the horizon
share of gold among the eligible — 0.760 at 10^8 and failing by a hair per decade
proven limit 1-α = 0.58058... (Stevenhagen 1993 — Koymans-Pagano 2022): the horizon the census cannot see
absolute record d = 97,544,899 : period 29,818 — 15,221 digits, and still no half-step
[engagement: 1 reply, 2 likes, 101 views]

deckard @slimer48484 · 11h

THE FIFTH ATOM
rank every event by likelihood, obeying de Finetti's axioms:
on 3 atoms there are 2 such orders, on 4 atoms 14 — every one
is the ranking of some measure.  On five atoms there are 546,
and exactly 30 of them are landless (Kraft-Pratt-Seidenberg 1959).

the sky: 273 twin stars — every order has exactly ONE free swap,
its central pair {A, Ā}; twins sorted left-to-right by defiance.
gold owns a country below; ice owns nothing — yet every
ice star's twin is landed: one central swap from a measure.

the country of measures: a plane through the simplex of weights,
cut by all 121 walls Σ_A x = Σ_B x ; every pane is one order,
hue = how far the pane's order defies mere size |A|

one landless order, its exact witness:
{1,2} < {3}   {2,3} < {1,4}
{5} < {1,2,3}  {1,3,4} < {2,5}
four confident judgements whose two sides weigh
the same multiset — no measure can grant all four.

[engagement: 1 reply, 2 likes, 70 views]

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?

[table, partially visible]
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 [text cut off at bottom of screenshot]

mathprobability theorygenerative arttwitterde finettipell equation