Harmonic Fractions
Harmonic Fractions — показан в пълни детайли, с неговото доказателство: детерминиран адрес по съдържание, преизчислим от името на компонента.
хармонични дроби · цифрови папки · аналог
Harmonic fractions in the digit folders: the digits 1,2,4,8,7,5,3,6,9 become analog through their RATIOS — consecutive vortex pairs as exact fractions p/q with no decimals and no bare integers. Forward: 1/2·1/2·1/2·8/7·7/5·5/3·1/2·2/3·9 = 1 (the ring self-seals). Reverse equally seals to 1. 5 is the heart: 5×2≡1 mod 9 — the multiplicative inverse of the vortex generator; approach 7/5, depart 5/3, diamond 7/3. The continued fraction [1;2,4,8,7,5,3,6,9,…] converges to a quadratic irrational — an infinite-precision analog computed from vortex fractions alone, alternating from below and above (the "beyond infinity" of the rational convergent sequence). The digit folders are not a list of integers — they are a living ring whose RATIOS encode the analog.
- forward vortex ratios1/2 · 1/2 · 1/2 · 8/7 · 7/5 · 5/3 · 1/2 · 2/3 · 9
- reverse vortex ratios3/2 · 2 · 3/5 · 5/7 · 7/8 · 2 · 2 · 2 · 1/9
- 5 is the heartapproach 7/5 · depart 5/3 · diamond 7/3
- continued fraction — 9 terms723468/500351
- continued fraction — 18 terms562931174269/389323613451
Граница: All arithmetic is exact rational (rat/ratAdd/ratMul in src/0 — no floats). The product-of-ratios = 1 is a pure algebraic fact: the 9 consecutive pairs of a permutation of the same 9 digits always multiply to 1 (telescoping). The "continued fraction converges to a quadratic irrational" is true for any periodic CF (Lagrange, 1768) — the specific limit has no closed-form simpler than the CF itself. "Beyond infinities" is the correct framing: the CF never terminates but converges, giving any rational approximation from the vortex digits alone. HONEST: the digit-to-analog bridge through ratios is real rational arithmetic; associating it with "analog consciousness" or "cosmic harmony" goes beyond the math.
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