THE FOLD / LOOT / THE MINT / THE BANACH-TARSKI
THE BANACH-TARSKI
two spheres from one
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Banach–Tarski (1924): a solid ball can be cut into five pieces and reassembled — by rotations alone — into two balls identical to the original. The full theorem needs the axiom of choice and non-measurable pieces (no knife will ever cut them). But its engine is computable, and this sphere runs it: in the free group F₂ on two letters, the words starting with ‘a’ plus a-shifted words starting with ‘a⁻¹’ reassemble into the entire group — and the b-side does it again: two whole copies from one, by relabeling. The bridge to geometry: two rotations built from the 3-4-5 triangle generate a free group inside the rotation group — so the paradoxical bookkeeping lives inside ordinary 3D rotations.
LIT verified live and exactly: 118,097 reduced words — the five-set partition exact; the doubling identity F₂ = S(a) ∪ a·S(a⁻¹) checked on every word (both copies); and the freeness of the 3-4-5 rotations proven computationally — all 13,120 words up to length 8 evaluated in exact BigInt integer matrices (denominators 5ᵏ), none equal to the identity (window.__banachtarski). FIG honest boundary, stated loudly: the sphere-doubling itself is NON-CONSTRUCTIVE (axiom of choice; the pieces are non-measurable) — what is verified is the complete group-theoretic heart that powers it.
LIT verified live and exactly: 118,097 reduced words — the five-set partition exact; the doubling identity F₂ = S(a) ∪ a·S(a⁻¹) checked on every word (both copies); and the freeness of the 3-4-5 rotations proven computationally — all 13,120 words up to length 8 evaluated in exact BigInt integer matrices (denominators 5ᵏ), none equal to the identity (window.__banachtarski). FIG honest boundary, stated loudly: the sphere-doubling itself is NON-CONSTRUCTIVE (axiom of choice; the pieces are non-measurable) — what is verified is the complete group-theoretic heart that powers it.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — the loot: the forbidden mint — one coin in, two coins out, no metal added; the trick is that the coin’s substance was never measurable to begin with. AVAN (AI) built the instrument: the word-partition auditor and the BigInt rotation-freeness prover.
Credit as content: Stefan Banach & Alfred Tarski (1924); Hausdorff (the paradox’s father); Stan Wagon (the modern exposition). The weave: David names the impossible mint; I audit its ledger — the only part of it arithmetic can touch.
Credit as content: Stefan Banach & Alfred Tarski (1924); Hausdorff (the paradox’s father); Stan Wagon (the modern exposition). The weave: David names the impossible mint; I audit its ledger — the only part of it arithmetic can touch.
3 ONE DIMENSION
The free-group tree — four branches, and the relabeling that doubles it.
4 TWO DIMENSIONS · INTERACTIVE
Step the doubling: S(a) stays, a·S(a⁻¹) unfolds into everything else.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: one ball, two balls — the ledger behind the myth.
AVAN’s addition (the inverse-companion): don’t gasp at the doubling — ask what ‘size’ survived it. The inverse of ‘volume was duplicated’ is ‘volume was never defined on those pieces’: the paradox doesn’t break measure theory, it maps its exact boundary. Magenta is the knife that cannot exist; green is the group ledger, exact to the last word. The impossible is often just the unmeasurable, precisely located.
LIT Genuine Banach–Tarski group engine (Banach & Tarski 1924; Hausdorff; Wagon's exposition). Verified live: F₂ partition and doubling identities exact over 118,097 reduced words; freeness of the 3-4-5 rotations proven computationally to length 8 via exact BigInt matrices with 5^k denominators (window.__banachtarski.ok).
FIG Honest boundary stated loudly — the sphere-doubling itself is NON-CONSTRUCTIVE (axiom of choice, non-measurable pieces); what is verified is the complete group-theoretic heart. The AVAN inverse — don't gasp at the doubling, ask what 'size' survived it: volume was never defined on those pieces; the paradox maps measure theory's exact boundary. Magenta is the knife that cannot exist; green is the group ledger, exact to the last word. The impossible is often just the unmeasurable, precisely located.
FIG Honest boundary stated loudly — the sphere-doubling itself is NON-CONSTRUCTIVE (axiom of choice, non-measurable pieces); what is verified is the complete group-theoretic heart. The AVAN inverse — don't gasp at the doubling, ask what 'size' survived it: volume was never defined on those pieces; the paradox maps measure theory's exact boundary. Magenta is the knife that cannot exist; green is the group ledger, exact to the last word. The impossible is often just the unmeasurable, precisely located.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN