THE FOLD / LOOT / THE JACKPOT / THE HIPPARCHUS
THE HIPPARCHUS
a count buried in Plutarch for two thousand years
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
103,049 sat unexplained in Plutarch’s Table Talk for nearly two thousand years: Hipparchus, says Plutarch, computed that from ten simple assertions the Stoics could form 103,049 compound statements. Historians shrugged — until 1994, when graduate student David Hough noticed that 103,049 is exactly the tenth little Schröder number: the number of ways to bracket ten items (equivalently, plane trees with ten leaves where every internal node has at least two children). Hipparchus, in the second century BCE, was counting what we now call Schröder–Hipparchus numbers — the oldest serious enumeration result known. Plutarch’s companion figure ‘310,952 on the negative side’ is two off from (s₁₀+s₁₁)/2 = 310,954, likely an ancient copying slip (Habsieger, Kazarian & Lando 1998).
LIT verified live: the recurrence (n+1)sₖ₊₁ = 3(2n−1)sₖ − (n−2)sₖ₋₁ and an independent plane-tree dynamic program agree at every n ≤ 11, both delivering s₁₀ = 103,049 and (s₁₀+s₁₁)/2 = 310,954 (window.__hipparchus). FIG honest boundary: the historical identification is Hough’s (via Stanley); the 310,952 discrepancy is reported as the cited scholarship has it. Companion sphere: the-schroder covers the LARGE Schröder path numbers — this is their halved sibling, met in an ancient book.
LIT verified live: the recurrence (n+1)sₖ₊₁ = 3(2n−1)sₖ − (n−2)sₖ₋₁ and an independent plane-tree dynamic program agree at every n ≤ 11, both delivering s₁₀ = 103,049 and (s₁₀+s₁₁)/2 = 310,954 (window.__hipparchus). FIG honest boundary: the historical identification is Hough’s (via Stanley); the 310,952 discrepancy is reported as the cited scholarship has it. Companion sphere: the-schroder covers the LARGE Schröder path numbers — this is their halved sibling, met in an ancient book.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-jackpot — the loot: a number mined out of a 2000-year-old text and found to be exactly right. AVAN (AI) built the instrument: the recurrence, the tree-counting DP, and the two-route agreement audit.
Credit as content: Hipparchus (2nd c. BCE); Plutarch; Ernst Schröder (1870); David Hough & Richard Stanley (1994/1997); Habsieger–Kazarian–Lando (1998). The weave: David names the jackpot; I confirm both routes land on 103,049.
Credit as content: Hipparchus (2nd c. BCE); Plutarch; Ernst Schröder (1870); David Hough & Richard Stanley (1994/1997); Habsieger–Kazarian–Lando (1998). The weave: David names the jackpot; I confirm both routes land on 103,049.
3 ONE DIMENSION
The sequence 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049 — climbing to Plutarch's number.
4 TWO DIMENSIONS · INTERACTIVE
Step n; recurrence and tree-DP race to the same value every time.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the bracketings of a small phrase, fanned out.
AVAN’s addition (the inverse-companion): don’t just verify the ancient number — notice what its correctness implies. The inverse of ‘Plutarch preserved a curiosity’ is ‘Hipparchus possessed a counting method we lost for two millennia’. Magenta is the two-off copyist’s slip at 310,952; green is 103,049 landing exactly. The oldest correct answer in combinatorics, waiting in a dinner-party anecdote.
LIT Genuine Schröder–Hipparchus identification (Hipparchus 2nd c. BCE via Plutarch; Ernst Schröder 1870; David Hough & Richard Stanley 1994/1997; Habsieger–Kazarian–Lando 1998). Verified live: recurrence (n+1)s_{n+1}=3(2n−1)s_n−(n−2)s_{n−1} and an independent plane-tree DP agree for all n≤11; s(10)=103,049; (s10+s11)/2=310,954 (window.__hipparchus.ok).
FIG Honest boundary — the historical identification is Hough's via Stanley; the 310,952-vs-310,954 discrepancy is reported as the cited scholarship has it. The AVAN inverse — don't just verify the ancient number, notice what its correctness implies: the inverse of 'Plutarch preserved a curiosity' is 'Hipparchus possessed a counting method we lost for two millennia'. Magenta is the copyist's slip; green is 103,049 landing exactly. The oldest correct answer in combinatorics, waiting in a dinner-party anecdote.
FIG Honest boundary — the historical identification is Hough's via Stanley; the 310,952-vs-310,954 discrepancy is reported as the cited scholarship has it. The AVAN inverse — don't just verify the ancient number, notice what its correctness implies: the inverse of 'Plutarch preserved a curiosity' is 'Hipparchus possessed a counting method we lost for two millennia'. Magenta is the copyist's slip; green is 103,049 landing exactly. The oldest correct answer in combinatorics, waiting in a dinner-party anecdote.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN