THE FOLD / BOSS / THE CHOKE-POINT / THE DROZ-FARNY
THE DROZ-FARNY
the 105-year line
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Draw any triangle and its orthocenter H — the choke-point where all three altitudes meet. Now draw ANY two perpendicular lines through H. Each line cuts the three side-lines; on each side, the two cuts bound a segment. Mark the three midpoints of those segments. They are always collinear. This is the Droz-Farny line theorem — and its history is the scandal: Arnold Droz-Farny (a Swiss watchmaking-town geometer) published it in 1899 without proof, and the first synthetic proof only arrived in 2004, from Jean-Louis Ayme. An elementary-looking claim, open for 105 years.
LIT verified live: 300 random triangles × random perpendicular pairs through H — the three midpoints collinear to a normalized score of 10⁻⁹ (worst ~10⁻¹⁶); the control at 80° instead of 90° scores a median 2.7×10⁻² — perpendicularity is load-bearing (window.__drozfarny). FIG the 1899–2004 proof gap is documented history (Ayme’s paper recounts it); projective proofs existed earlier — ‘first synthetic proof’ is the precise claim.
LIT verified live: 300 random triangles × random perpendicular pairs through H — the three midpoints collinear to a normalized score of 10⁻⁹ (worst ~10⁻¹⁶); the control at 80° instead of 90° scores a median 2.7×10⁻² — perpendicularity is load-bearing (window.__drozfarny). FIG the 1899–2004 proof gap is documented history (Ayme’s paper recounts it); projective proofs existed earlier — ‘first synthetic proof’ is the precise claim.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-choke-point — the boss: everything routes through H — three altitudes, and now every perpendicular cross you can draw — and the choke-point exacts its tax: a hidden alignment on whatever passes through. AVAN (AI) built the instrument: the random-cross collinearity meter and the 80° control.
Credit as content: Arnold Droz-Farny (1899); Jean-Louis Ayme (2004, first synthetic proof); the projective treatments in between. The weave: David names the choke-point; I fire 300 crosses through it and the alignment never misses.
Credit as content: Arnold Droz-Farny (1899); Jean-Louis Ayme (2004, first synthetic proof); the projective treatments in between. The weave: David names the choke-point; I fire 300 crosses through it and the alignment never misses.
3 ONE DIMENSION
One triangle, one perpendicular cross through H, three midpoints, one line.
4 TWO DIMENSIONS · INTERACTIVE
Spin the cross; the midpoint line glides but never breaks.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the rotating cross and its gliding line.
AVAN’s addition (the inverse-companion): don’t measure how hard a claim is by how it looks — measure by how long it resists. The inverse of ‘elementary statement’ is ‘elementary proof’, and they can be a century apart: the theorem was TRUE and CHECKABLE for 105 years while remaining unproven in its own language. Magenta is the 80° cross that scatters; green is the 90° alignment that waited out five generations of geometers. Verification and understanding keep different calendars.
LIT Verified live: 300 random triangles × random perpendicular crosses through H — midpoints collinear to 1e-9 (worst ~1e-16); the 80° control scores median 2.7e-2, so perpendicularity is load-bearing (window.__drozfarny.ok).
FIG The 1899–2004 gap is documented (projective proofs existed; 'first synthetic' is the precise claim). The AVAN inverse — measure a claim by how long it resists, not how it looks: true and checkable for a century while unproven in its own language. Magenta is the 80° cross that scatters; green is the alignment that waited out five generations. Verification and understanding keep different calendars.
FIG The 1899–2004 gap is documented (projective proofs existed; 'first synthetic' is the precise claim). The AVAN inverse — measure a claim by how long it resists, not how it looks: true and checkable for a century while unproven in its own language. Magenta is the 80° cross that scatters; green is the alignment that waited out five generations. Verification and understanding keep different calendars.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE-POINT · David Lee Wise (ROOT0), with AVAN