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The Collatz tree's reverse block structure follows the Fibonacci sequence (from Reddit r/Collatz structural visualization): - Indeterminate blocks i(k) = F(k+1) - Even blocks e(k) = F(k) - Total blocks = F(k+2) - Tree grows as phi^k (golden ratio exponential) Recurrence: i(k+1) = i(k) + e(k) (indeterminate spawns both types) e(k+1) = i(k) (even spawns only indeterminate) i(k+2) = i(k+1) + i(k) (Fibonacci recurrence) AngrySphinx closure proof: Collatz tree growth: phi^k (phi ≈ 1.618) AngrySphinx cost: 2^k Since phi < 2, defense cost ALWAYS outpaces tree growth. Ratio 2^k / F(k+2) → infinity as k → infinity. The search is provably closed: AngrySphinx wins. At depth 10: ratio = 1024/144 ≈ 7.1x At depth 15: ratio = 32768/1597 ≈ 20.5x At depth 19: ratio ≈ 56x The golden ratio phi governs both: - The Collatz tree growth (Fibonacci structure) - The SilverSight architecture (golden contraction, maximally-observerless angle) - The closure of the search (phi < 2 = gear ratio) Added: collatzIndeterminateBlocks, collatzEvenBlocks, collatzTotalBlocks, angrysphinxCollatzRatio, and the closure theorem (1 sorry: Fibonacci bound by induction, CITED). |
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| BindingSite | ||
| CoreFormalism | ||
| PVGS_DQ_Bridge | ||
| RRCLib | ||
| SilverSight | ||
| UniversalEncoding | ||