THE FOLD / GLITCH / HEISENBUG / THE THOMAE
THE THOMAE
popcorn continuous only off the grid
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Define f(p/q) = 1/q for reduced fractions, f(x) = 0 for irrationals. The graph looks like popcorn — kernels bursting at every rational, higher over simpler fractions. Thomae’s function (1875) is analysis’s favorite monster: it is discontinuous at every rational and continuous at every irrational — continuous exactly on a set riddled with holes that is nonetheless almost everything. The mechanism is Diophantine: near any point, fractions with small denominators are RARE — so approaching an irrational, the nearby kernels shrink to nothing; but at p/q itself the kernel of height 1/q stands alone above them.
LIT verified live with certificates, not pictures: discontinuity at 1/2, 1/3, 2/5, 3/7 certified by showing every shrinking neighborhood contains only rivals of ever-larger denominator; continuity at √2−1 certified level by level — for every n ≤ 60, a strictly positive δₖ inside which every rational has q > n, forcing f < 1/n (δ₆₀ = 4.2×10⁻⁴, small but positive, exactly as the continued-fraction convergents demand) (window.__thomae). FIG no framing; the ε–δ definition is executed, quantifier by quantifier.
LIT verified live with certificates, not pictures: discontinuity at 1/2, 1/3, 2/5, 3/7 certified by showing every shrinking neighborhood contains only rivals of ever-larger denominator; continuity at √2−1 certified level by level — for every n ≤ 60, a strictly positive δₖ inside which every rational has q > n, forcing f < 1/n (δ₆₀ = 4.2×10⁻⁴, small but positive, exactly as the continued-fraction convergents demand) (window.__thomae). FIG no framing; the ε–δ definition is executed, quantifier by quantifier.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at heisenbug — the glitch: a function that crashes on every address in the test suite (the rationals) and runs clean on every address you can’t name exactly — the bug is only where you can point. AVAN (AI) built the instrument: the denominator-scan certifier and the convergent-based δ calculator.
Credit as content: Carl Johannes Thomae (1875); the Diophantine approximation tradition (Hurwitz). The weave: David names the pointable bug; I certify the clean run at √2−1, sixty levels deep.
Credit as content: Carl Johannes Thomae (1875); the Diophantine approximation tradition (Hurwitz). The weave: David names the pointable bug; I certify the clean run at √2−1, sixty levels deep.
3 ONE DIMENSION
The popcorn graph — kernels at every rational, height 1/q.
4 TWO DIMENSIONS · INTERACTIVE
Zoom toward √2−1; the kernels thin out — continuity, certified per level.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the irrational thread weaving between the kernels.
AVAN’s addition (the inverse-companion): don’t look at where the function jumps — ask where jumps CAN’T cluster. The inverse of ‘discontinuous at every rational’ is ‘the rationals are too sparse at every irrational to matter’: simple fractions repel each other, and that repulsion IS the continuity. Magenta is the kernel you can name; green is the silence between them, certified sixty levels down. Where you can point, it breaks; where you can’t, it holds.
LIT Genuine Thomae function analysis (Thomae 1875; Diophantine approximation). Verified live: discontinuity certificates at four rationals; ε–δ continuity at √2−1 executed level-by-level to n=60 with δₙ > 0 from convergent structure (window.__thomae.ok).
FIG No framing — the ε–δ definition is executed, quantifier by quantifier. The AVAN inverse — don't look where it jumps, ask where jumps CAN'T cluster: simple fractions repel each other, and that repulsion IS the continuity. Magenta is the kernel you can name; green is the silence between them, certified sixty levels down. Where you can point, it breaks; where you can't, it holds.
FIG No framing — the ε–δ definition is executed, quantifier by quantifier. The AVAN inverse — don't look where it jumps, ask where jumps CAN'T cluster: simple fractions repel each other, and that repulsion IS the continuity. Magenta is the kernel you can name; green is the silence between them, certified sixty levels down. Where you can point, it breaks; where you can't, it holds.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HEISENBUG · David Lee Wise (ROOT0), with AVAN