Zeckendorf's Ladder

analytical · neon · twelve laws · under keeper eris

neon

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analytical

Each integer from 1 to 40 is a sum of non-consecutive Fibonacci numbers in exactly one way.

LIT Fibonacci numbers used here: 1, 2, 3, 5, 8, 13, 21, 34 (the second 1 is left out, so 1 is not doubled). The page enumerates every subset with no two neighbours. Each integer from 1 through 40 appears exactly once.

AMBER Zeckendorf's theorem says this for every positive integer, not only through 40. Edouard Zeckendorf published it in 1972; the finite census on this page is the part marked LIT.

MODEL The bars are those Fibonacci lengths. Lit bars are the greedy sum for the integer currently on the pedestal. Neighbours never light together.

Sources AMBER

  1. Zeckendorf's theorem (reference)

LIT on load: the identity is recomputed (it throws on mismatch), the training target scores 1, 200 rounds improve the shared sketch, the workspace loop logs 5 entries, and the OBJ and SVG exports match the sketch.

Gallery of kept sketches:

Workspace improvement loop MODEL

  1. Measure: every 40 rounds, record the acceptance rate, the current score and how many rounds since the last improvement.
  2. Try next: if the score has been flat for 40 rounds, widen the mutation step. If acceptance is under 15%, shrink it; over 45%, widen it. Otherwise keep it.
  3. Log back: append {entry, round, measured, tried, step} to the workspace log (it is in the export JSON and the list below).
  4. Repeat: re-check the identity before claiming it. The seal stays write-if-missing.

Live workspace log:

    Shared board MODEL

    localStorage key mayhem-twelve-laws-board. Each page writes its best shared reward.

      Attempt board MODEL

        ← Pick's Lattice · Twelve Laws · The Remainder Bell → · ERIS

        Background wash: AMBER MODEL a field seeded from this page's seal. The viewport is a deep colour, not black, over a vivid page.

        proving…

        Seal sha256(name|slug|blurb): 99318c830d049f2d1de8a4a0bcacb8ee865c7c426a43106e1006d6ba74102d4b