THE FOLD / BOSS / THE GAUNTLET / THE AUTOMORPHISM SHORTFALL
THE AUTOMORPHISM SHORTFALL
what an unequal partition costs in bits
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
How much labelling survives a fold depends on how symmetric the thing being folded is — and symmetry here is countable. The permutations of the parts that leave the shape unchanged form a group, and the recoverable bits are the logarithm of its order. Four equal parts give 24. Parts of sizes 3, 3, 2, 4 give 2.
LIT verified live. aut(3,3,2,4) = 2, since only the two equal parts may swap; aut(3,3,3,3) = 24 = 4!. So the recoverable labelling is 1.0000 bits and not 4.5850 — a shortfall of 3.5850 bits, which is exactly log₂(24) − log₂(2).
LIT verified live. aut(3,3,2,4) = 2, since only the two equal parts may swap; aut(3,3,3,3) = 24 = 4!. So the recoverable labelling is 1.0000 bits and not 4.5850 — a shortfall of 3.5850 bits, which is exactly log₂(24) − log₂(2).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) computed the consequence for a build he had already shipped: “recoverable labelling: 1.0000 bits, not 4.5850. SHORTFALL 3.5850 bits… the earlier build’s cheapest result does not hold.” The measurement did not merely fail to support the earlier claim — it removed it.
AVAN (AI) would point at where the asymmetry actually bites. The group order collapses from 24 to 2 because one part differs in size; the shape 3,3,3,3 is the only one of the fifteen with the full 4! symmetry, and every other partition of twelve into four loses most of it immediately. Symmetry is not a spectrum here so much as a cliff, and a figure drawn as four equal arms is standing on the single point where the cliff has a summit.
AVAN (AI) would point at where the asymmetry actually bites. The group order collapses from 24 to 2 because one part differs in size; the shape 3,3,3,3 is the only one of the fifteen with the full 4! symmetry, and every other partition of twelve into four loses most of it immediately. Symmetry is not a spectrum here so much as a cliff, and a figure drawn as four equal arms is standing on the single point where the cliff has a summit.
3 ONE DIMENSION
Every partition of twelve, and what it can recover.
4 TWO DIMENSIONS · INTERACTIVE
Move one item between groups and watch the bits fall.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a symmetry that is a cliff, not a slope.
AVAN’s addition (the inverse-companion): the forward reading is “the unequal partition recovers 3.585 fewer bits.” The inverse is that those bits were never worth much. Recoverable labelling counts how many ways the parts could be permuted without changing the object — which is information about names, not about content. A build whose cheapest result rests on 4.585 bits of part-labelling has priced its symmetry in the currency that symmetry is most abundant in. Read backwards, losing it costs less than the number suggests, and the more damaging finding is the one beside it: the arms are not alike.
LIT aut(3,3,2,4) = 2 since only the two equal parts may swap, while aut(3,3,3,3) = 24 = 4!, so the recoverable labelling is 1.0000 bits and not 4.5850 - a shortfall of 3.5850 bits, which is exactly log2(24) minus log2(2)
FIG David computed the consequence for a build he had already shipped: 'recoverable labelling: 1.0000 bits, not 4.5850. SHORTFALL 3.5850 bits... the earlier build's cheapest result does not hold.' The measurement did not merely fail to support the earlier claim - it removed it. AVAN points at where the asymmetry bites: the group order collapses from 24 to 2 because ONE part differs in size, and 3,3,3,3 is the only one of the fifteen with the full 4! symmetry. Symmetry is not a spectrum here so much as a cliff, and a figure drawn as four equal arms is standing on the single point where the cliff has a summit.
FIG David computed the consequence for a build he had already shipped: 'recoverable labelling: 1.0000 bits, not 4.5850. SHORTFALL 3.5850 bits... the earlier build's cheapest result does not hold.' The measurement did not merely fail to support the earlier claim - it removed it. AVAN points at where the asymmetry bites: the group order collapses from 24 to 2 because ONE part differs in size, and 3,3,3,3 is the only one of the fifteen with the full 4! symmetry. Symmetry is not a spectrum here so much as a cliff, and a figure drawn as four equal arms is standing on the single point where the cliff has a summit.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN