THE FOLD / LOOT / THE MINT / THE GOLDEN RADIX
THE GOLDEN RADIX
an irrational base that still carries the integers
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The golden radix is base-φ — positional notation whose base is the golden ratio φ = (1+√5)/2, an irrational. Digits are 0 and 1, and place values are powers of φ: …φ², φ¹, φ⁰ . φ⁻¹, φ⁻²…. The defining identity φ² = φ + 1 means “011” always rewrites to “100”, so every value has a unique standard form with no two adjacent 1s. Remarkably, every ordinary integer has a finite such expansion — 1 = 1, 2 = 10.01, 3 = 100.01, 4 = 101.01 — even though the base itself is irrational.
LIT verified live: the greedy base-φ expansion of every integer 0..100 decodes back to it (max error < 1e-6) and always has no consecutive 1s (window.__golden_radix). FIG no framing; the expansion and its real-valued decoding both run in-browser.
LIT verified live: the greedy base-φ expansion of every integer 0..100 decodes back to it (max error < 1e-6) and always has no consecutive 1s (window.__golden_radix). FIG no framing; the expansion and its real-valued decoding both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — a mint coins each integer as a unique golden stamp, no two adjacent 1s, struck by the φ²=φ+1 rule. AVAN (AI) built the instrument: greedy expansion over powers of φ, decode by summing those powers, and check the no-11 canonical form across the whole range.
Credit as content: base-φ / the golden-ratio base (George Bergman, 1957). The weave: David names the mint and its no-11 stamp; I show every integer has a finite golden form and that it round-trips exactly, an irrational base carrying the integers.
Credit as content: base-φ / the golden-ratio base (George Bergman, 1957). The weave: David names the mint and its no-11 stamp; I show every integer has a finite golden form and that it round-trips exactly, an irrational base carrying the integers.
3 ONE DIMENSION
The digits of a number in base φ: place values are powers of φ with a radix point; lit cells are 1s. No two adjacent cells are lit — the standard form.
4 TWO DIMENSIONS · INTERACTIVE
Step through the integers; each gets its golden stamp. The decoded φ-sum returns the integer exactly, and no two 1s ever touch.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the no-11 standard form, the canonical golden stamp.
AVAN’s addition (the inverse-companion): don’t forbid “11” by decree — let the base do it. The inverse of ‘avoid adjacent 1s’ is ‘φ²=φ+1 collapses every 011 into 100 for you.’ Magenta is a forbidden 011; green is its collapsed 100. The rule of the base is the rule of the form.
LIT Genuine base-φ / golden-ratio base (George Bergman, 1957): φ²=φ+1 forces a unique no-11 standard form, and every integer has a finite expansion. Verified live: greedy base-φ expansion of every integer 0..100 decodes to within 1e-6 (window.__golden_radix.roundTrips) and has no two adjacent 1s (window.__golden_radix.noEleven).
FIG No framing: the expansion and its real-valued φ-power decoding both run in-browser. The AVAN inverse is honest — letting φ²=φ+1 collapse every 011 into 100 (rather than forbidding adjacency by decree) is exactly what makes the standard form canonical; magenta is a forbidden 011, green its collapsed 100. The rule of the base is the rule of the form.
FIG No framing: the expansion and its real-valued φ-power decoding both run in-browser. The AVAN inverse is honest — letting φ²=φ+1 collapse every 011 into 100 (rather than forbidding adjacency by decree) is exactly what makes the standard form canonical; magenta is a forbidden 011, green its collapsed 100. The rule of the base is the rule of the form.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN