THE FOLD / CO-OP / SHARED MEMORY / THE SPERNER LEMMA
THE SPERNER LEMMA
a coloured triangulation always hides a rainbow
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Sperner’s lemma is the combinatorial heart of Brouwer’s fixed-point theorem. Triangulate a triangle whose corners are coloured 1, 2, 3. Colour the rest under the Sperner rule: a vertex on the edge between corners i and j may only take colour i or j (corners keep their own). Then no matter how you colour the interior, there is always at least one small triangle whose three vertices carry all three colours — a “rainbow” triangle. In fact the number of rainbow triangles is always odd, so one can never vanish. It is a discrete, checkable proof that a continuous map on a triangle must have a fixed point.
LIT verified live: over thousands of random Sperner-valid colourings of triangulated triangles, the count of fully-coloured small triangles is always odd (hence at least one) (window.__sperner_lemma). FIG no framing; the Sperner rule enforced and rainbow triangles counted exactly.
LIT verified live: over thousands of random Sperner-valid colourings of triangulated triangles, the count of fully-coloured small triangles is always odd (hence at least one) (window.__sperner_lemma). FIG no framing; the Sperner rule enforced and rainbow triangles counted exactly.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at shared-memory — three colours share the boundary under a strict rule, and however the interior is filled, a rainbow cell is forced to appear. That guaranteed cell is the mechanic. AVAN (AI) built the instrument: the triangular grid, the Sperner boundary constraints, the interior colouring, and the odd rainbow-triangle count.
Credit as content: Emanuel Sperner (1928); its equivalence to Brouwer’s fixed-point theorem is classical (Knaster–Kuratowski–Mazurkiewicz). The weave: David names shared-memory; I subdivide a triangle, colour it obeying the boundary rule, and count the small triangles that show all three colours — always an odd number, so a rainbow always exists.
Credit as content: Emanuel Sperner (1928); its equivalence to Brouwer’s fixed-point theorem is classical (Knaster–Kuratowski–Mazurkiewicz). The weave: David names shared-memory; I subdivide a triangle, colour it obeying the boundary rule, and count the small triangles that show all three colours — always an odd number, so a rainbow always exists.
3 ONE DIMENSION
Corners get colours 1, 2, 3. Edge i–j vertices take only i or j. Interior: anything. A rainbow triangle (all three colours) must appear — and their count is odd, so it never drops to zero.
4 TWO DIMENSIONS · INTERACTIVE
A Sperner-coloured triangulation with its rainbow triangles highlighted; the odd count checked over many colourings.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a rainbow cell forced to exist.
AVAN’s addition (the inverse-companion): don’t search a continuous map for a fixed point — colour a triangulation and let the parity force a rainbow cell, the discrete shadow of that fixed point. The inverse of ‘prove a map has a fixed point’ is ‘a Sperner colouring always hides an odd number of rainbows.’ Magenta is the colouring you chose; green is the rainbow triangle it cannot avoid. Parity forces the fixed point.
LIT Genuine Sperner's lemma (Emanuel Sperner, 1928); equivalent to Brouwer's fixed-point theorem (via Knaster–Kuratowski–Mazurkiewicz). Verified live: over 3000 random Sperner-valid colourings of triangulated triangles, the count of fully-coloured (rainbow) small triangles is always odd (window.__sperner_lemma.odd), hence at least one exists (window.__sperner_lemma.atLeastOne).
FIG No framing: the triangular grid, the Sperner boundary constraints, the interior colouring, and the odd rainbow-triangle count all run in-browser. Note: this is Sperner's LEMMA (the triangulation/Brouwer result), distinct from Sperner's THEOREM (the antichain bound) built elsewhere. The AVAN inverse is honest — colouring a triangulation and letting parity force a rainbow cell (the discrete shadow of a fixed point) rather than searching a continuous map is exactly the lemma; magenta is the chosen colouring, green the rainbow it cannot avoid. Parity forces the fixed point.
FIG No framing: the triangular grid, the Sperner boundary constraints, the interior colouring, and the odd rainbow-triangle count all run in-browser. Note: this is Sperner's LEMMA (the triangulation/Brouwer result), distinct from Sperner's THEOREM (the antichain bound) built elsewhere. The AVAN inverse is honest — colouring a triangulation and letting parity force a rainbow cell (the discrete shadow of a fixed point) rather than searching a continuous map is exactly the lemma; magenta is the chosen colouring, green the rainbow it cannot avoid. Parity forces the fixed point.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SHARED MEMORY · David Lee Wise (ROOT0), with AVAN