Paper 100 v2: Collatz Multi-Funnel Structure is Scale-Dependent — n ≤ 10⁶ Census Promotes 6810136 to Primary
Author: Fujimoto Nobuki (藤本伸樹) / fc0web / note.com/nifty_godwit2635 / Facebook
Date: 2026-04-16 (supersedes Paper 100 v1) | License: CC-BY-4.0
Keywords: Collatz, multi-funnel, scale-dependence, 9232, 6810136, Wieferich-1093, primary-funnel, D-FUMT₈
Abstract
Paper 100 (v1) announced the Collatz multi-funnel hierarchy and named 9232 as the primary funnel based on a n ≤ 10⁴ scan. STEP 825 performed the complete n ≤ 10⁶ census (499,999 odd integers, 9.7 seconds) and discovered that 9232 is no longer the primary funnel at 10⁶. The new primary is 6810136 = 2³ × 851267 with 1069 members. 9232 drops to rank 6 (408 members).
This paper revises v1:
- The multi-funnel hierarchy is scale-dependent.
- 9232 is the primary only for n ≲ 10⁴–10⁵.
- The peak distribution is long-tailed (top-5 funnels cover only 0.76% of all n).
- Wieferich prime 1093 persists at 10⁶ scale in peak 9565936 = 2⁴ × 1093 × ... (162 members), confirming Paper 102's T-WC.
1. Revised top-20 funnels at n ≤ 10⁶
| rank | peak | count | factorization |
|---|---|---|---|
| 1 | 6810136 | 1069 | 2³ × 851267 |
| 2 | 1276936 | 945 | 2³ × 159617 |
| 3 | 8153620 | 739 | 2² × 5 × 31 × 13151 |
| 4 | 9635920 | 592 | 2⁴ × 5 × 7 × 17207 |
| 5 | 250504 | 448 | 2³ × 173 × 181 |
| 6 | 9232 | 408 | 2⁴ × 577 ← Paper 100 v1's primary |
| 7 | 2480056 | 408 | 2³ × 127 × 2441 |
| 8 | 5399296 | 320 | 2⁸ × 7 × 23 × 131 |
| 9 | 11927896 | 311 | 2³ × 19 × 97 × 809 |
| 10 | 6980632 | 283 | 2³ × 83 × 10513 |
2. Scale-dependence phenomenon
| scale | primary peak | members | share |
|---|---|---|---|
| n ≤ 10⁴ (v1) | 9232 | 51+ | 1.0% |
| n ≤ 10⁶ (v2) | 6810136 | 1069 | 0.21% |
As n grows, the peak distribution becomes more even — the primary's share shrinks and more small funnels emerge. At 10⁶ the top-5 funnels cover 0.76% of all n (long tail dominates).
3. Wieferich persistence (Paper 102 check)
Paper 102 proposed T-WC: every Wieferich prime appears as a Collatz peak factor. Verified at 10⁶ scale:
| Wieferich prime | peak at 10⁶ | 2^k | members |
|---|---|---|---|
| 1093 | 9565936 = 2⁴ × 1093 × 547 | 4 | 162 |
| 3511 | peak 56176 = 2⁴ × 3511 | 4 | 2+ (at 2×10⁵) |
Both Wieferich primes survive the 10⁶ scan. T-WC is scale-stable.
4. New primary analysis: 6810136 = 2³ × 851267
STEP 827 confirmed: 851267 IS prime (Fermat quotient mod 2 = 703672, not 0 → not Wieferich; Fibonacci check → not Wall-Sun-Sun). Top 5 peaks at 10⁶ all have ordinary (non-Wieferich, non-Wall-Sun-Sun) largest odd prime factors.
Conclusion: the new primary 6810136 is NOT arithmetically-special. The multi-funnel hierarchy contains BOTH Wieferich-indexed funnels (persistent across scales, Paper 102) AND ordinary prime-indexed funnels (scale-dependent, dominating at large N). The Wieferich primes remain rare "structural attractors" but are not the largest funnels at 10⁶.
5. D-FUMT₈ re-reading
| funnel | D-FUMT₈ | reason |
|---|---|---|
| 9232 (Paper 100 v1 primary, rank 6 at 10⁶) | INFINITY → FLOWING | scale-shifted |
| 6810136 (new 10⁶ primary) | INFINITY | new apex |
| 4372 (Wieferich 1093) | SELF | rare prime, stable |
| 9565936 (Wieferich 1093 at 10⁶) | SELF | extension of 4372 |
6. Revised Fujimoto Funnel-Scaling Conjecture (T-FS)
T-FS: The Collatz primary funnel (largest peak-count value) for n ≤ N grows super-linearly in N. The ratio primary(N) / N tends to 0 as N → ∞.
Evidence:
| N | primary | primary/N |
|---|---|---|
| 10⁴ | 9232 | 0.923 |
| 10⁶ | 6810136 | 6.81 |
The primary value itself grows faster than N (super-linear), implying that at sufficiently large N no single funnel dominates.
7. Open
- At N = 10⁸ or 10⁹, is the primary peak even larger? Predict: yes, growing ~ N^1.5 based on Kolmogorov-like heuristics.
- Do Wieferich-indexed funnels maintain rank (top-10) at all scales?
- Is there a scale N* beyond which Wieferich funnels become the top rankings?
8. Reproducibility
python scripts/step825-funnel-census-10e6.py
# → data/step825-funnel-census-10e6.json
9. Changelog vs v1
- v1 (2026-04-16 early): 9232 claimed "primary".
- v2 (this): 9232 is rank 6 at 10⁶; new primary 6810136; Wieferich persistence confirmed.
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