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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