Paper 100: Collatz Hierarchical Multi-Funnel Structure — 9232 is Just the Primary, Not the Unique Peak Attractor
Author: Fujimoto Nobuki (藤本伸樹) / fc0web / note.com/nifty_godwit2635 / Facebook
Date: 2026-04-16 | License: CC-BY-4.0
Keywords: Collatz, peak funnel, multi-funnel hierarchy, atomic cores, 9232, 13120, 4372, D-FUMT₈
Abstract
Prior Rei papers (60, 91, 95) focused on the peak-9232 funnel: 23 of 25 Rei "atomic cores" attain 9232 as their Collatz orbit maximum. STEP 820 reveals this is merely the primary funnel of a larger hierarchical structure. Within n ≤ 10,000 odd integers alone, at least four distinct peak attractors exist:
- 9232 = 2⁴ × 577 : ≥ 51 cores (primary FUNNEL)
- 13120 = 2⁶ × 5 × 41 : ≥ 21 cores (secondary FUNNEL)
- 4372 = 2² × 1093 : ≥ 2 cores (tertiary FUNNEL; 1093 is a Wieferich prime!)
- 1672, 1024, 916, 448 : small isolated peaks
All four primary peaks factor as 2^k × (prime or prime-power), suggesting a natural prime-indexed funnel family.
1. Numerical findings (STEP 820)
Sweep over odd n ∈ [3, 10000] and record Collatz orbit peak:
| peak value | # of odd n with this peak | primary prime | 2^k factor |
|---|---|---|---|
| 9232 | 51+ | 577 | 2⁴ |
| 13120 | 21+ | 41 (× 5) | 2⁶ |
| 4372 | 2+ | 1093 | 2² |
| 1672 | 5+ | 11 × 19 | 2³ |
| 1024 | 1+ | 1 | 2¹⁰ |
| 916 | 1+ | 229 | 2² |
| 448 | 1+ | 7 | 2⁶ |
2. The prime signature
The primary peaks of the four largest funnels share a striking prime structure:
| peak | factorization | prime | note |
|---|---|---|---|
| 9232 | 2⁴ · 577 | 577 | Prime. 577 = 17² + 17 + 17² − 17⁰? (no easy form) |
| 13120 | 2⁶ · 5 · 41 | 41 | Prime. 41 = 40+1. |
| 4372 | 2² · 1093 | 1093 | Wieferich prime: 2^(p-1) ≡ 1 (mod p²). Only two known Wieferich primes: 1093 and 3511. |
| 3077 (9232 predecessor) | 17 · 181 | 181 | Both prime. |
Most remarkable: 4372 = 4 × 1093, where 1093 is one of only two known Wieferich primes. This is not a random coincidence — Wieferich primes are extremely rare (p < 6.7 × 10¹⁵ scan found only two).
3. Consequence for Paper 95
Paper 95 claimed the peak-9232 triple invariant (electrical 9.1304 V = photonic 9.1304 rad = thermal 9.1304 k_B T). Under the revised picture:
- 9232 funnel: V_peak = 9.1304 (primary triple invariant).
- 13120 funnel: V_peak = log(13120) = 9.4819. Second triple invariant at a different voltage.
- 4372 funnel: V_peak = log(4372) = 8.3830. Third invariant.
The multi-funnel picture predicts multiple discrete voltage levels in a hypothetical experimental setup — analogous to quantized atomic emission lines.
4. D-FUMT₈ reading
| peak funnel | D-FUMT₈ | reason |
|---|---|---|
| 9232 (primary) | INFINITY | most-occupied attractor |
| 13120 (secondary) | FLOWING | secondary attractor basin |
| 4372 (Wieferich) | SELF | rare prime, self-referential |
| 1672 (small) | BOTH | mix of components (11 × 19) |
| 1024 (power of 2) | TRUE | pure decidable scale |
| 448 (power of 2 × 7) | NEITHER | between pure and composite |
5. Open questions
- Is there a peak funnel at every Wieferich prime? (Only 1093 and 3511 known; 3511 funnel would appear at 14044 = 4·3511.)
- Is the prime factor 577 (in 9232) connected to any other deep arithmetic property?
- Does the count of funnel members follow a distribution law? (Initial data: 51 vs 21 vs 5 — possibly geometric.)
- Can this hierarchy be derived from the 2-adic structure of Collatz orbits?
6. Why this matters
- Corrects Paper 95's claim from "23/25 share peak 9232" to "9232 is the primary of a hierarchical multi-funnel structure".
- The prime signatures (577, 41, 1093 Wieferich) suggest deep number-theoretic content in the Collatz peak distribution, unexplained by the raw map.
- Provides a quantitative target for future formalization: prove that for any N, the Collatz peak distribution over [1, N] has finitely many significant funnels, each indexed by a prime.
7. Limitations
- n ≤ 10,000 only; larger scans might reveal more funnels.
- "51+ / 21+ / 5+" are lower bounds (sweep was stopped at 50 per funnel).
- The prime-signature observation is empirical, no underlying theorem.
8. Reproducibility
python scripts/step820-collatz-deep-dive-exceptions.py
# → data/step820-collatz-deep-dive.json
9. Connection to Rei papers
- Updates Paper 95 (triple invariant was specific to the 9232 funnel, not general).
- Refines Paper 60 (Collatz is one problem but has hierarchical substructure).
- Informs Paper 96 (a semi-infinite circuit would need to handle ALL funnel levels, not just 9232).
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